Research Tools in Natural Resource and Environmental Economics 9789814289221

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recognized experts in environmental economics, this book is the first of its kind and a valuable reference and textual source for graduate students and active researchers. It draws together the pedagogical discussion of the key tools used to conduct theoretical and empirical research in natural resource and environmental economics. With contributions by prominent international researchers like Robert Ayres, Charles Perrings, and Anastasios Xepapadeas, the book will be useful for researchers who wish to learn new techniques or change their area of research emphasis within natural resource and environmental economics or those who wish to familiarize themselves with these tools.

Batabyal Nijkamp

ISBN-13 978-981-4289-22-1 ISBN-10 981-4289-22-1

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A collection of scholarly accounts and articles written by

Research Tools in Natural Resource and Environmental Economics

Research Tools in Natural Resource and Environmental Economics

Research Tools in Natural Resource and Environmental Economics

Amitrajeet A Batabyal Peter Nijkamp editors

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Research Tools in Natural Resource and Environmental Economics

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Amitrajeet A Batabyal Rochester Institute of Technology, USA

Peter Nijkamp VU University Amsterdam, The Netherlands

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RESEARCH TOOLS IN NATURAL RESOURCE AND ENVIRONMENTAL ECONOMICS Copyright © 2011 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

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ISBN-13 978-981-4289-22-1 ISBN-10 981-4289-22-1

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In memory of Balarka Amarnath Batabyal (1966–2010)

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Contents Acknowledgments

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Contributors

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Part I. Introduction

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1. Introduction to Research Tools in Natural Resource and Environmental Economics

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Amitrajeet A. Batabyal and Peter Nijkamp

Part II. Theoretical Tools

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2. Dynamic Analysis

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Ray G. Huffaker 3. Stochastic Analysis: Tools for Environmental and Resource Economics Modeling

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Anastasios Xepapadeas 4. Game Theory: Static and Dynamic Games

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Hassan Benchekroun and Ngo Van Long

Part III. Empirical and Experimental Tools 5. Introduction to Economic Valuation Methods

141 143

Sabah Abdullah, Anil Markandya and Paulo ALD Nunes 6. The Hedonic Method: Value of Statistical Life, Wage Compensation, and Property Value Compensation Philip E. Graves vii

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7. Environmental Valuation: The Sum of Specific Damages Approach

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Philip E. Graves 8. Econometric Estimation of Non-linear Continuous-Time Models of Intertemporally Optimizing Agents

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Kieran P. Donaghy and Clifford R. Wymer 9. Computable General Equilibrium Models for the Analysis of Economy–Environment Interactions

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Ian Sue Wing 10. Experimental Methods and Environmental and Natural Resource Policy

307

Todd L. Cherry and Michael McKee

Part IV. Interdisciplinary Tools 11. Modeling the Economics of Ecosystem Services at Watershed Scale: A Spatial Model of Land Use Externalities and the Regulating Functions of Wetlands

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Silvio Simonit and Charles Perrings 12. Geographical Information Systems Models and Spatial Data Analysis

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Robert P. Haining and Jane Law 13. Materials Balance Models

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Robert U. Ayres and Gara Villalba M´endez 14. Industrial Ecology: Reflections of an Environmental Economist

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Finn R. Førsund Index

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Acknowledgments This book would not have been possible without the cooperation of several researchers scattered throughout the world. As such, we would like to thank the contributors of the individual chapters of this book for their enthusiastic participation in this time-consuming project. The enthusiasm of the contributors is amply on display in the high-quality chapters they have written on the various research tools that are the subject of this book. Batabyal would like to acknowledge the support he received from his wife Swapna B. Batabyal and from his daughter Sanjana S. Batabyal during the long gestation period of this book. In addition, he would also like to acknowledge the financial support he received from the Gosnell endowment at the Rochester Institute of Technology. Nijkamp thanks the Department of Spatial Economics at the VU University, Amsterdam for providing an excellent support framework during the preparation of this book. Amitrajeet A. Batabyal Rochester, New York Peter Nijkamp Amsterdam, The Netherlands July 2010

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Contributors Sabah Abdullah Department of Economics, University of Bath, Bath, UK. E-mail: [email protected] Robert U. Ayres Economics and Political Science Area, INSEAD, Fontainebleau, France. E-mail: [email protected] Amitrajeet A. Batabyal Department of Economics, Rochester Institute of Technology, Rochester, New York, USA. E-mail: [email protected] Hassan Benchekroun Department of Economics, CIREQ, McGill University, Montreal, Canada. E-mail: [email protected] Todd L. Cherry Department of Economics, Appalachian State University, Boone, North Carolina, USA. E-mail: [email protected] Kieran P. Donaghy Department of City and Regional Planning, Cornell University, Ithaca, New York, USA. E-mail: [email protected] Finn R. Førsund Department of Economics, University of Oslo, Oslo, Norway. E-mail: fi[email protected] Philip E. Graves Department of Economics, University of Colorado, Boulder, Colorado, USA. E-mail: [email protected] Robert P. Haining Department of Geography, University of Cambridge, Cambridge, UK. E-mail: [email protected] xi

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Contributors

Ray G. Huffaker Department of Food and Resource Economics, University of Florida, Gainesville, Florida, USA. E-mail: rhuffaker@ufl.edu Jane Law School of Planning, University of Waterloo, Waterloo, Ontario, Canada. E-mail: [email protected] Ngo Van Long Department of Economics, CIREQ, McGill University, Montreal, Canada. E-mail: [email protected] Anil Markandya Department of Economics, University of Bath, Bath, UK. E-mail: [email protected] Michael McKee Department of Economics, Appalachian State University, Boone, North Carolina, USA. E-mail: [email protected] Gara Villalba Mendez Department of Chemical Engineering, Universitat Aut` onoma de Barcelona, Barcelona, Spain. E-mail: [email protected] Peter Nijkamp Department of Spatial Economics, Free University, Amsterdam, The Netherlands. E-mail: [email protected] Paulo A.L.D. Nunes Department of Economics, University of Venice, Venice, Italy. E-mail: [email protected] Charles Perrings ecoSERVICES Group, Arizona State University, Tempe, Arizona, USA. E-mail: [email protected] Silvio Simonit ecoSERVICES Group, Arizona State University, Tempe, Arizona, USA. E-mail: [email protected] Ian Sue Wing Department of Geography and Environment, Boston University, Boston, Massachusetts, USA. E-mail: [email protected]

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Clifford R. Wymer Department of City and Regional Planning, Cornell University, Ithaca, New York, USA. E-mail: [email protected] Anastasios Xepapadeas Department of International and European Economic Studies, Athens University of Economics and Business, Athens, Greece. E-mail: xepapad@ aueb.gr

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PART I

Introduction

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Chapter 1

Introduction to Research Tools in Natural Resource and Environmental Economics Amitrajeet A. Batabyal∗ and Peter Nijkamp† ∗

Rochester Institute of Technology [email protected] † Free University, The Netherlands [email protected]

1.1.

Preliminaries

Economists typically think of natural resources as being either exhaustible1 or renewable. Exhaustible natural resources include most minerals such as coal and iron ore and renewable natural resources commonly include fisheries, forests, and rangelands. Because all natural resources are ultimately renewable, the key criterion that determines whether a natural resource is exhaustible or renewable is the rate at which it regenerates relative to typical human life spans. From a contemporary standpoint, the environment is a more general concept than is the concept of a natural resource. Even so, until fairly recently, most environmental problems were basically pollution problems of

1 Exhaustible natural resources are also sometimes referred to as depletable or as nonrenewable natural resources.

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one sort or another2 and hence, from an economic standpoint, the analysis of environmental problems was very closely linked to the study of externalities and, in particular, to the study of external diseconomies. However, today, environmental concerns loom large in our quotidian lives and hence the field of environmental economics transcends pollution problems and encompasses research questions in areas as diverse as benefit–cost analysis, economic valuation, and game theoretic analysis. Alongside the above mentioned developments, there have been significant developments pertaining to natural resources and the environment in related disciplines such as ecology, geography, mathematics, and the physical sciences. Consider, for instance, research developments in ecology and in natural resource and environmental economics. As noted by Batabyal (2008), since the publication of Gordon’s (1954) seminal paper, research by the biologist Hardin (1968), the economist Daly (1968), and the mathematician Clark (1973; 1976), has increasingly led to the view that what economists call renewable natural resources and what ecologists more generally call ecological systems are really jointly determined ecological–economic systems whose evolution over time is dependent on intertemporal and stochastic forces that are partly ecological and partly economic in nature. This view has gained currency in the last three decades and hence, in contemporary times, natural resource management is really all about the management of ecological–economic systems. Similarly, developments in natural resource and environmental economics and in engineering and the physical sciences have led to the development of the so-called materials balance and industrial ecology models.3 Finally, developments in natural resource and environmental economics and in geography have led to the development of models that are based on geographical information systems (GISs) and to models that involve spatial data analysis.4 Given these many and varied research developments, two questions that now arise naturally are the following. First, what are the most common research tools that economists and other scholars have used to frame and 2 Readers wishing to learn more about the history of the environmental movement in the western world should consult Carson (1962), Dales (1968), Ehrlich (1968), and Meadows et al. (1972). 3 The reader should consult Kneese et al. (1971) and Graedel and Allenby (2002) for more on these sorts of models. 4 See Goodchild and Haining (2004) and Haining (1990; 2003) for additional details on these types of models.

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analyze the natural resource and environment related questions that have interested them? Second, rather than going to myriad sources in the extant literature, how can one familiarize oneself and thereby learn these most common research tools by perusing a single source? The answers to these two questions comprise the basic subject matter of this book. As such, we now move to a more detailed and systematic discussion of these two questions. Following this introductory chapter which comprises Part I of this book, there are 13 chapters and each of these chapters — written by an expert or by a team of experts — discusses a particular tool that is widely employed by researchers working in the field of natural resource and environmental economics. For ease of comprehension, we have divided these 13 chapters into three parts. Part II of this book focuses on theoretical tools and this part consists of three chapters that provide a detailed discussion of dynamic analysis (Chap. 2), stochastic analysis (Chap. 3), and game theoretic analysis (Chap. 4), respectively. Part III is concerned with empirical and experimental tools and this part consists of six chapters. Chapter 5 discusses the tools that are typically used for the purpose of economic valuation. Chapter 6 focuses on the so-called hedonic approach. Chapter 7 provides a concise commentary on the relatively novel “sum of specific damages” approach. Chapter 8 is concerned with the intricacies of econometric estimation and simulation. Chapter 9 discusses the uses of computable general equilibrium models. Finally, Chap. 10 is dedicated to a discourse on experimental economic models. The four chapters that comprise Part IV of this book are devoted to a discussion of what we call interdisciplinary research tools. Specifically, Chap. 11 focuses on the modeling of what we have previously called ecological–economic systems. Chapter 12 discusses models involving the use of GISs and the analysis of spatial data. Chapter 13 talks about what have now come to be known as materials balance models. Finally, Chap. 14 provides a discourse on industrial ecology models. Note that the chapters in Part IV of this book delineate research tools that are not tools in the sense of the tools described in Parts II and III of this book. In other words, these tools in Part IV do not constitute mathematically distinct topics. Instead, these tools utilize quite a few of the previously described mathematical tools to model natural resource and environmental phenomena in distinct ways. For instance, an ecological– economic model of the sort described in Chap. 11 may well utilize some of

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the tools described previously in Chap. 2 on dynamic analysis. However, what is distinct about and hence makes this utilization noteworthy is that an underlying natural resource — such as a wetland — is modeled as a jointly determined ecological–economic system whose behavior over time is determined by both ecological and economic forces. No book can reasonably be expected to cover every conceivable research tool and, in this regard, our book is no exception. Our primary objective in this book is to provide a thorough perspective on the most commonly used research tools in natural resource and environmental economics today. Having said this, the following five points deserve some mention. First, there are no chapters in this book on either benefit–cost analysis or on multicriteria decision analysis (MCDA). Benefit–cost analysis is now a wellestablished methodology in virtually all fields of economics and not just in natural resource and environmental economics. As a result, this methodology is well explained in a number of existing books such as Boardman et al. (2001) and hence does not require any new coverage. MCDA is certainly a useful research tool but, in our opinion, relative to the tools that are discussed in this book, this is a less central tool. Second, although the topics of environmental sustainability and poverty have rightly attracted a considerable amount of attention in contemporary times, these topics can be well studied by using one or more of the research tools that are covered in detail in this book. Third, readers interested in exploring research questions about seemingly broader issues such as cultural heritage, urban landscapes, or biological diversity ought to be able to undertake such exploration by focusing, inter alia, on the research tools described in Parts III and IV of this book. Fourth, issues concerning the distribution of resources per se do not require specific research tools for analysis and the substantive issues here can be studied meaningfully with a combination of the research tools delineated in this book. Finally, a thorough understanding of stochastic analysis — discussed in Chap. 3 — in particular and the research tools that are the subject of the other chapters in this book will certainly help a researcher comprehend not only climate change modeling issues but also contemporary public policy debates concerning the problem of climate change. With the five points of this and the preceding paragraph in mind, we now proceed to discuss the intellectual contributions of the individual chapters in this book.

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1.2. 1.2.1.

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Theoretical Tools Dynamic analysis

Theoretically inclined natural resource and environmental economists have used the tools of static optimization for a long time. However, given the fact that natural resource use and management related questions are fundamentally intertemporal in nature, researchers have used dynamic analysis to study the use and the management of exhaustible natural resources at least since Gray (1914) and Hotelling (1931). Chapter 2 by Ray G. Huffaker begins by noting that most introductory and advanced textbooks purporting to cover dynamic analysis such as Kamien and Schwartz (1991), Leonard and Long (1992), and Caputo (2005) provide good coverage of the trinity of the calculus of variations, optimal control theory, and dynamic programming. In particular, optimal control theory is typically presented in these books as a generalization of the calculus of variations. Also, dynamic programming as formulated by Bellman (1957) is typically presented to point out and then exploit the fact that in discrete time, many dynamic economic problems have a recursive structure to them. What Chap. 2 does effectively is to shed valuable light on two less well covered but nonetheless very salient topics in dynamic analysis. The first topic concerns the asymptotic stability properties of the equilibria of dynamic economic models. This topic is particularly significant because even though many textbooks on dynamic analysis do provide coverage of situations in which the local stability properties of the solutions to a, for instance, optimal control model can be discerned by linearizing the system dynamics in the vicinity of the equilibria, there is typically little discussion of what analytic steps need to be taken when these linearization methods fail. The second topic pertains to the stability properties of the solution to a dynamic economic model when, all else being equal, a single parameter of the underlying model is varied. To motivate this topic, Huffaker provides a nice discussion of a non-linear fisheries model described in Clark’s (1976) justly famous book titled Mathematical Bioeconomics. Specifically, he demonstrates that bifurcation analysis can be used to describe the ways in which the dynamics of a particular economic system change in response to small perturbations in a parameter of interest.

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Stochastic analysis

Researchers now understand that although dynamic analysis of natural resource and environmental phenomena is useful, more often than not, this analysis must be informed by a recognition of the basic “challenge posed by the complex time behavior and in-built uncertainty of . . . ecologicaleconomic systems” (Perrings et al., 1995, p. 302). In addition, risk and uncertainty5 are fundamental notions in the field of natural resource and environmental economics. To see this clearly, consider, for instance, the growth of renewable natural resource stocks or the evolution in the stocks of one or more pollutants. The growth and the evolution are typically uncertain. Similarly, in order to study the problem of global warming or climate change, it is imperative that we first comprehend, as best as we are able, the uncertainties surrounding the accumulation of greenhouse gases. The discussion in the preceding paragraph tells us that comprehending and analyzing risk and uncertainty is of great significance in the field of natural resource and environmental economics. Therefore, Chap. 3 by Anastasios Xepapadeas presents a comprehensive account of the modeling issues that typically arise when a researcher seeks to construct and analyze stochastic models of one or more natural resource and/or environmental phenomena. This chapter builds on the previous chapter on dynamic analysis in the sense that Xepapadeas’s main focus is to show how probabilistic tools and methods can be used to analyze dynamic problems since, as he rightly contends, intertemporal analysis is the most interesting and germane way of studying natural resource and environmental management issues. To this end, Chap. 3 begins by first presenting some general results from probability theory and then moves on to a discussion of stochastic calculus, stochastic differential equations, and Itˆ o’s lemma. Next, this chapter presents a fine discussion of stochastic optimal control and this section of the chapter pays particular attention to the so-called Hamilton– Jacobi–Bellman equation which is a central concept in stochastic dynamic optimization. References to specific natural resource and environmental problems enliven this discussion as does the pedagogical emphasis on the 5 An

environment is characterized by the presence of risk when whatever is random in this environment can be described by a cumulative probability distribution function. In contrast, an environment is characterized by the presence of uncertainty when whatever is random cannot be delineated by any known cumulative distribution function. See Knight (1921) for a more detailed discussion of this basic distinction.

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relatively straightforward linear-quadratic and the constant relative risk aversion (CRRA) problems. Xepapadeas concludes this chapter with a lucid discussion of irreversibilities, option values, optimal stopping rules, and nonexpected utility models. The discussion of this last topic nicely points out that it is now possible to account for many of the objections to the maximization of expected utility that have been raised in the extant literature by studying, inter alia, robust control models inspired by the prior work of Hansen and Sargent (2001; 2007) in which agents confront model ambiguity and, in particular, concerns about model misspecification. 1.2.3.

Game theoretic analysis

Game theory is now firmly established as a standard theoretical research tool for scholars who are interested in studying the interactions between two or more economic agents either at a point in time or over time when these interactions are strategic in nature. Very recently, Patrone et al. (2008) and Sumaila et al. (2008) have noted that researchers have now begun to analyze natural resource and environmental issues as diverse as international fisheries management, the management of traditional grazing rights in semiarid African pastoralist systems, and the usefulness of coalitions in the design of international environmental agreements, in game theoretic terms. Because of the extensive contemporary use of game theory to model and analyze all manner of research questions in natural resource and environmental economics, it is now essential for theoretically minded scholars to have a good understanding of the ways in which game theory not only can be but also has been used to model problems in natural resource and environmental economics. This understanding is provided in Chap. 4 by Hassan Benchekroun and Ngo Van Long who, broadly speaking, focus their competent discussion of game theoretic analysis into a part on static games and a second part on dynamic games. The discussion of static games focuses on two areas of application that are of considerable interest in contemporary natural resource and environmental economics. The first such application area pertains to the interactions between players with opposing objectives in the context of pollution emissions and abatement games and the related question of the design of international environmental agreements (IEAs) — where both emissions and abatement play key roles — with certain desirable properties. The authors show that from a modeling perspective, it is often the case that

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the objective function of a pollution abatement game is an increasing affine transformation of the objective function in the pollution emissions game. Further, the discussion of IEAs demonstrates that the sustainability of an IEA from a game theoretic perspective depends very much on what stability criterion one uses to gauge sustainability. The second application area is about the use of network theory to ascertain the extent to which polluting oligopolists will cooperate in research and development (R&D) concerning pollution abatement technologies. Benchekroun and Long point out that when the strategic interaction between two polluting firms (countries) is such that the difference between the firm (country) specific pollution tax rates is small, R&D collaboration between these firms (countries) occurs. In contrast, when this difference exceeds a certain threshold, there is no intercountry R&D collaboration although such collaboration does occur between the various polluting firms within a particular country. The discussion of dynamic games in Chap. 4 first provides a clear commentary on the distinctions between open loop and feedback Nash equilibria. Next, this discussion demonstrates the ways in which dynamic games can and have been used in the extant literature to study the exploitation of both exhaustible and renewable natural resources and the interactions between polluting firms. The authors conclude this discussion by rightly paying careful attention to a frequently misunderstood concept, namely, the nature of Stackelberg equilibria in dynamic games. They point out that there are several notions of dynamic Stackelberg leadership and hence each of these notions need to be distinguished clearly. However, even with these distinctions, we learn that in general, it is not possible to identify a feedback Stackelberg equilibrium which is “global” in the sense that the follower’s move is an optimal reaction to the set of possible decision rules employed by the leader. 1.3. 1.3.1.

Empirical and Experimental Tools Economic valuation

Economic valuation has been used for a long time to place a value on all manner of goods and services. When these goods and services are traded in markets, the task of valuation is relatively straightforward. However, in the realm of natural resources and the environment, some goods — such as the output of a mine — are clearly market goods and hence it is not difficult to

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assign a value to these kinds of goods. However, many environmental goods and particularly services are not traded in any market. Although this fact clearly does not mean that such goods and services have no value, it does mean that ascertaining the value of such goods and services is an involved undertaking. In his influential paper “Conservation Reconsidered,” Krutilla (1967) suggested that we pay careful attention to the irreversibilities associated with the development of natural environments, that the value of what he called “natural scenic wonders” may well be rising over time and that there might be an existence value associated with certain natural phenomena. These and other observations by scholars — see Hanemann (1992; 1994) and Portney (1994) — have led natural resource and environmental economists to take the difficult task of economic valuation seriously. As a result, the scholarly journals in natural resource and environmental economics are now filled with economic valuation exercises of one sort or another. This state of affairs makes it necessary for scholars with an interest in valuation to comprehend the tools that are typically used to conduct the underlying valuation exercises. Chapter 5 by Sabah Abdullah, Anil Markandya, and Paulo Nunes provides a lucid account of economic valuation as it is carried out today. Focusing first on the common valuation techniques, the authors distinguish between use and non-use values, they provide a typology of the different kinds of market valuations, and then, after pointing out the differences between revealed and stated preference approaches, they proceed to discuss the methods that are routinely used to undertake non-market valuation. It is worth noting that the authors of Chap. 5 provide a detailed, step-by-step approach that can be followed by researchers interested in conducting their own valuation exercises. In addition, this chapter also takes a prospective look at the subject of economic valuation and points to challenges for future studies on valuation. Perusing this discussion, we learn that if the subject of economic valuation is to be conceptually and empirically rigorous and complete then researchers will have to adequately address three challenges. First, it will be necessary to come up with ways to increase the response rates to surveys and address the various biases that the extant literature has identified in order to increase both the validity and the reliability of economic valuation exercises. Second, for valuation exercises to be truly useful, researchers will need to come up with marginal or incremental values and not total values. Finally, with regard to benefit transfers, scholars will need to credibly determine the extent to which values, either

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from a site or of a service, can be transferred to another site or to another type of service. 1.3.2.

The hedonic method

The hedonic method is a revealed preference method of valuation. Specifically, this method of valuation employs surrogate markets to place a value on, for instance, environmental quality. The housing market is the most commonly used surrogate in the hedonic “pricing” of environmental values. In the context of the housing market, the hedonic method relies on information provided by consumers in making their housing choice decisions. We all know that economic agents derive utility from living in nice homes and locations and, in this regard, when the demand for land and housing increases, the price of housing also increases. In addition to determining the value of environmental amenities, the hedonic method can also be used to estimate the premium placed by agents on “nice” jobs. Applications of the hedonic method date back to the 1960s and Ridker (1967) and Ridker and Henning (1967) were the first to empirically show that air pollution negatively influences property values. Since the publication of the two contributions mentioned in the preceding paragraph, the hedonic method has been used widely to measure the value of a statistical life (VSL) and to study wage and property value compensation and therefore, unsurprisingly, this is where Philip E. Graves picks up his discussion of the hedonic method in Chap. 6. Focusing first on the VSL, Graves notes that although some studies have estimated VSLs from non-labor markets, most studies of VSLs focus on labor markets and in this latter group of studies the basic objective is to analyze the implicit wage compensation for the varying risk of death. The author rightly notes that studies in the extant literature exhibit incredibly large differences in VSLs and hence researchers need to be very careful about measurement errors and omitted variable and selectivity biases if their VSL studies are to be both credible and useful for the conduct of public policy. Graves next proceeds to discuss wage and property value compensation. With regard to wage compensation studies, the author rightly notes that although such studies can be very useful, a researcher must always be mindful of the fact that environmental values generated using the hedonic method can either overstate or understate true values. For instance, if environmental quality differences are misperceived or if economic agents are unaware of the ways in which environmental quality affects them then the

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true benefits of cleaning up the environment will be understated by the hedonic method. In contrast, the value of the environment can also be overstated by the use of the hedonic method. Graves illustrates this point by using an example from property value compensation. Chapter 6 concludes with the following salient message about the hedonic method. We learn that until fairly recently, researchers typically believed that one could ascertain the value of, for instance, clean air either by examining the variation in property values in land markets or by examining the variation in wages in labor markets. In other words, the wage compensation and the property value compensation approaches were viewed as substitute approaches. However, this view is incorrect because economic agents generally do not adopt a two stage procedure for choosing a location in which they first select a location among alternate labor markets and then select a location within the selected labor market based on housing prices and variations in pollution. 1.3.3.

The sum of specific damages approach

The sum of specific damages (SSD) approach to environmental valuation is a rather straightforward but insufficiently understood research tool that is based on the simple idea that the benefits of cleaning up pollution — or some other bad — are essentially equivalent to reductions in damages. So, in a simple “clean up pollution” context, this approach would simply add up the damage reductions and then put a monetary value on, as the name suggests, the sum of these individual damages. Chapter 7 by Philip E. Graves nicely explains that even though this SSD approach is intuitively straightforward, there are a number of complications that arise when one attempts to operationalize the approach. For instance, because there are many different types of damages associated with pollution such as morbidity, mortality, materials damage, and crop damage, determining what the physical damage reductions will be if environmental quality is improved necessarily requires the expertise of individuals from many different disciplines such as biology, economics, and epidemiology. In addition, there is the issue of “perception.” In other words, the SSD approach assumes that people do not perceive damages as being related to pollution. Instead, the maintained assumption in this approach is that people behave as if these negative physical effects simply occur, with more or less negative outcomes occurring when, for instance, pollution levels are

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either high or low. If this were not the case and people were aware of the relationship between damages and pollution, they would engage in what the extant literature calls “averting behavior.” If this were to happen, then a researcher interested in applying the SSD approach in a given situation must recognize that people are using scarce resources to avoid a damage that otherwise would have occurred. If this last fact involving costly mitigating expenditures is not recognized and a researcher counts only the damages that continue to occur then (s)he will end up understating the true benefits of cleaning up pollution. As Graves rightly points out in Chap. 7, this is the sense in which a reliance on the SSD approach to value the benefits of environmental policies might result in the production of too little environmental quality. 1.3.4.

Econometric estimation and simulation

The prior work of many natural resource and environmental economists involves the analysis of well articulated models — see, for instance, Dasgupta and Heal (1979) and Pindyck (1984) — in which economic agents are optimizing objective functions subject to one or more constraints over time. Therefore, it is important to comprehend the ways in which econometric tools can be used to estimate non-linear, continuous time models of intertemporally optimizing economic agents. This is the rationale adopted by Kieran P. Donaghy and Clifford R. Wymer in their able discussion of econometric estimation techniques in Chap. 8. A key feature of this chapter is that it shows how extant software programs such as Wysea can be used to empirically implement the specific methods discussed by the authors. Given their objective, Donaghy and Wymer begin the proceedings by first demonstrating the way in which one is to delineate a non-linear, continuous time economic system and then they provide a review of the econometric techniques that they use to estimate such models. This part of the chapter includes a discussion of both linear and non-linear models and the authors pay considerable attention to boundary conditions and to the intertemporal and the stability properties of the models under consideration. The authors conclude their hitherto largely conceptual discussion with an interesting application. This application involves the use of Nigerian data to first specify and then estimate a particular version of a prominent model of the exploitation and the production of an exhaustible resource originally

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due to Pindyck (1978). Even though the authors have to deal with a small sample size, this application is revealing in large part because the authors do a nice job of pointing out the issues that arise in the estimation of such a model. We conclude our discussion of this chapter by highlighting two of these issues. The first issue concerns the time horizon of the model being estimated. The authors note that although the theoretically specified model has an infinite horizon, in practice, it often makes good sense to choose a long but finite horizon and then observe the extent to which the obtained estimates change when one extends the initially picked long but finite time horizon. The second issue relates to identification. We are told that the use of unobserved variables makes the issue of parameter identification an important one. In particular, we learn that a researcher interested in estimating models of the sort discussed in this chapter must worry about what the authors call “formal underidentification” in which some parameters are unidentified for any sample and “underidentification” which arises in the context of a particular sample. 1.3.5.

Computable general equilibrium models

Computable — also known as applied — general equilibrium (CGE) models fall into a category of economic models that use actual economic data to estimate how an economy might react to one or more policy changes. CGE models are descended from the now classic input–output models first studied by Leontief (1936; 1937). However, beginning with the early work of Johansen (1960), most contemporary CGE models are significantly more elaborate than Leontief’s models and, as well, they place much more weightage on the role of prices in an economy. Even though CGE models are now quite popular among natural resource and environmental economists, many in the field either do not understand the workings of such models or they view such models with suspicion because CGE models are supposedly “black boxes.” The criticism here is twofold. First, because the more realistic models are also complex, it is often difficult to identify the precise nexuses between the model outputs and the model inputs, the algebraic structure of the model, and the method of solution. Second, it has also been claimed that the complex internal workings of CGE models permit a researcher to pay insufficient attention to assumptions that may, in fact, end up driving some of the obtained results.

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Given this unsavory state of affairs, what is needed now is a clear and well thought out chapter that explains all the intricacies of CGE models to scholars interested in this research tool in a simple and intuitive but rigorous manner. This is exactly what the comprehensive Chap. 9 by Ian Sue Wing does. The author begins the proceedings by first exploiting the circular flow of economic activity to algebraically derive the equilibrium conditions that are at the core of every CGE model. He then shows how a CGE model of an economy can be constructed by utilizing the conditions emanating from consumer utility and firm profit maximization to generate a system of non-linear equations that represent this economy. Next, Sue Wing uses a so called social accounting matrix (SAM) to numerically calibrate the aforementioned algebraic model structure. Finally, in the main section of this chapter, Sue Wing demonstrates how what he calls “economy–environment” interactions can be analyzed formally and rigorously and, in addition, how a variety of research questions in natural resource and environmental economics can be efficaciously modeled using the CGE framework. Although the focus of Chap. 9, in terms of applications, is specifically on questions in natural resource and environmental economics, this chapter does highlight the fact that CGE models have wide applicability in the realm of public policy. This savory state of affairs arises because of two key reasons. First, there frequently is consistency between the results obtained with CGE models and the results obtained with other kinds of analyses. Second, CGE models provide a consistent framework to examine the linkages and the tradeoffs among many different policy measures. 1.3.6.

Experimental economic models

Experimental economics, a relatively new field in economics, is all about the application of experimental methods to study economic questions. In this field of inquiry, experiments are used to test the validity of economic theories and to analyze the utility of novel market mechanisms. These tasks are typically undertaken by using cash motivated subjects — often undergraduate students — to create real-world scenarios that will help us better comprehend, for instance, the actual working of markets. Although experimental economics has dramatically gained both acceptance and popularity, it is only very recently that natural resource and environmental economists have begun to comprehend the true scope of

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experimental methods in shedding light on research questions concerning the use and the management of natural resources and the environment. As such, the interface between experimental economics and natural resource and environmental economics remains a much talked about but insufficiently studied area. Given this state of affairs, Chap. 10 by Todd L. Cherry and Michael McKee provides a lucid account of experimental methods and their relevance for the conduct of natural resource and environmental policy. This account has two basic objectives. First, the authors pay attention to the principles of experimental design and then provide an outline of the most commonly used methods in experimental economics. Second, the authors point to specific areas within natural resource and environmental economics that are “especially fertile grounds” for the use of experiments as an effective research tool. This chapter challenges the much discussed recent claim — see List and Levitt (2007) — that in order for experiments to be truly useful they ought to be conducted in the field and not in a laboratory. As the authors point out, most of the points that have been raised to advance the “in the field” claim can be adequately addressed by incorporating the rules of experimental design enunciated by Smith (1982). Cherry and McKee acknowledge that while a laboratory setting may indeed be subject to “selfselection effects,” they claim that such effects arise in field settings as well. With regard to the uses of experimental methods in natural resource and environmental economics, this chapter rightly points out that the scope and the complexity of the present regulatory system for environmental quality is, almost certainly, going to be a “growth industry.” This realization has led to great interest among natural resource and environmental economists in exploring the behavioral aspects of compliance with extant regulations. Specifically, the authors note that because there is little or no field data, the experimental method frequently enables a researcher to directly examine the behavioral responses to specific regulatory policies. This chapter concludes with two plausible points. First, it notes that one of the most fruitful uses of experimental methods in natural resource and environmental economics is likely to be in the area of the valuation of environmental amenities and damages. Second, it tells us that experimental methods can be extremely useful in ascertaining the unintended consequences of specific regulatory schemes such as the tradable emission permits scheme.

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Interdisciplinary Tools Ecological–economic models

As a result of the increased interactions between natural resource and environmental economists on the one hand and ecologists on the other, a key common perspective has now emerged. Specifically, natural resource and environmental economists now no longer treat the environmental resource base “as an indefinitely large and adaptable capital stock” (Dasgupta, 1996, p. 390; emphasis in original). Similarly, ecologists now understand that it would be a big mistake to “regard the human presence as an inessential component of the ecological landscape” (Dasgupta, 1996, p. 390). Instead and as noted in Sec. 1.1 of this chapter, resources such as fisheries, forests, and rangelands are now viewed as jointly determined ecological–economic systems whose evolution is determined by forces that are partly ecological and partly economic in nature. This welcome development has led to the flourishing of the new field of ecological economics. Increasingly, natural resource and environmental economists, ecological economists, and ecologists have begun to focus research attention on the hitherto neglected but nonetheless salient topic of valuing the many services provided by ecological–economic systems. The modeling and the analysis of these “ecosystem” services provided by wetlands is the subject of the interesting Chap. 11 by Silvio Simonit and Charles Perrings. This chapter first points out that wetlands play an important role in nutrient absorption and this absorptive function has a role to play in sewage treatment, freshwater provision, and the production of reeds. The chapter then goes on to note and then analyze the fact that the “main value” of nutrient absorption by wetlands is the contribution that this service makes to the regulation of fish supplies. The research tool that this chapter showcases is the explicitly interdisciplinary model of the relationship between land use change, wetland area, water quality, and fish stock biomass. Concentrating on the Yala wetland/swamp on the Kenyan side of Lake Victoria, the authors note that it is possible to identify the contribution of each hectare of land in the catchment to the nutrient load entering the wetland and to then use this information to ascertain the merits of alternate policy proposals involving the potential conversion of the Yala for agricultural purposes. The central message of Chap. 11 to researchers is that with careful modeling, it is possible to uncover the value of what the authors call the “regulating ecosystem services” by comprehending the contribution they

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make to the production of things that humans care about directly, namely, the Millennium Assessment’s (2005) so-called provisioning and cultural services. Interpreted differently, the value of these regulating services arises from the value of the provisioning and the cultural services that they protect. Finally, because the sustainability of these aforementioned services depends on the capacity of an ecological–economic system such as a wetland to deliver those services over a range of environmental conditions, researchers must understand that the value of the regulating services will change with the value of the protected service and the variability of the underlying environmental conditions.

1.4.2.

GIS and spatial data analysis

In the process of discussing key research issues at the interface of the environment and regional science in the golden anniversary issue of the journal Papers in Regional Science, Batabyal and Nijkamp (2004) noted the existence of barriers to the integration of spatial data in environmental analyses and discussed ways in which these barriers might be dealt with. In the time since the publication of this paper, many of these barriers have now been lifted and, in addition, many new developments in GISs and spatial data analysis have occurred. Therefore, Goodchild and Haining (2004) point out that although GIS and spatial data analysis started out as two more-or-less separate areas of research, they have grown closer over time. Given these developments, the use of GIS and spatial data analysis to address research questions in natural resource and environmental economics has also grown over time. This means that researchers interested in this interdisciplinary “marriage” between geography and natural resource and environmental economics must increasingly be familiar with the tools that scholars have been using to shed light on questions concerning natural resources and the environment that also have a distinct spatial dimension to them. This is where Chap. 12 by Robert Haining and Jane Law comes in. Specifically, this chapter provides the reader with a thorough grounding in the models that arise when GISs and the tools and methods of spatial data analysis are used jointly. The authors begin their chapter by first reviewing the main types of spatial data that are encountered by researchers and the ways in which these data map into statistical theory. We learn very quickly about the necessity of making assumptions about the manner in which the data have been generated. Without making such assumptions, the authors tell us that

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it is simply impossible to progress beyond descriptive summaries of the spatial data. Next, this chapter discusses both exploratory and confirmatory spatial data methods and their relationship to GIS models. We learn that exploratory spatial data analysis provides what we might think of as a starting point for the statistical analysis and that this statistical analysis may or may not lead eventually to the fitting of a model for carrying our confirmatory spatial data analysis. Because of the contemporary popularity of Bayesian inference, the authors spend some time familiarizing readers with some of the more significant issues in this kind of inference. Particular attention is paid to the issues of missing data, measurement errors, data incompatibility, and the integration of individual and aggregate data. Three noteworthy conclusions arise from the detailed discussion of GISs and spatial data analysis by the authors. First, models must play a key role not only in the construction of GIS databases but also in their analysis. Second, a GIS is in essence the technological support for any research in which places and the spatial relationships between places and the characteristics associated with these places matter in order to comprehend observed outcomes. Finally, the introduction of spatial relationships into models of the sort discussed in this chapter can not only yield new insights about economic outcomes but they can also lead to the questioning of the conventional wisdom about these outcomes. 1.4.3.

Materials balance models

Following the discussion in Ayres et al. (1970) and Ayres (1999), the notion of a material or mass balance refers to the application of the conservation of mass to the analysis of physical systems. In particular, by explicitly accounting for the materials entering and departing a physical system, flows of mass can be established which otherwise either might not have been established or might have been difficult to measure without using this particular technique. The precise conservation law that is used in the analysis of a particular system depends, ultimately, on the problem being studied but all such laws revolve essentially around the idea that matter can neither appear nor disappear spontaneously. Since the 1960s, materials balance models have been used frequently to study questions at the interface of economics and engineering. Even so, because these sorts of models are not a standard part of the educational curriculum in most doctoral programs in natural resource and

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environmental economics, the models themselves remain inadequately understood by many researchers who would, in principle, like to know more about the interdisciplinary underpinnings of such models. To this end, Chap. 13 by Robert U. Ayres and Gara Villalba Mendez provides a useful interdisciplinary perspective on materials balance models. This chapter begins by describing the prominent first and the second laws of thermodynamics. The authors explain that a salient consequence of the second law — sometimes also know as the entropy law — is that the economically valuable products from any process tend to be more ordered and have lower entropy than the corresponding raw material inputs. At the same time, the waste residuals from economic processes tend to be more disordered and have higher entropy than the inputs to the process. Interpreted differently and following the work of Georgescu-Roegen (1971), we learn that economic processes tend to convert low entropy raw materials into high entropy wastes. Practically speaking, the reason why this is significant is that high entropy waste residuals have no positive market value to anyone but, at the same time, they also do not disappear by themselves. Therefore, they are commonly disposed off in suboptimal ways. This chapter then moves on to explain the differences between useful and useless wastes, the so-called “life cycle” view of materials or mass flows, and the utility of the idea of mass balance in the conduct of life cycle analyses (LCAs). The authors conclude their discussion of materials balance models with the following salient point. Some of the biggest contemporary environmental problems — such as climate change — are the result of materials extraction, conversion processes, utilization, and waste generation. Hence, it is very important to quantitatively measure the material flows and the waste mass flows in both absolute terms and in terms of annual changes and changes between different elements. Material flow and material flow analysis (MFA) is a very useful way of conducting these important exercises. 1.4.4.

Industrial ecology models

The roots of industrial ecology lie in systems analysis and the field has sometimes been referred to as a “higher level systems approach” for framing and studying the interactions between industrial and natural systems. More specifically and consistent with the discussion in Graedel and Allenby (2002), industrial ecology is the study of the physical, chemical, and biological interactions and interrelationships both within and between industrial

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and natural (ecological) systems. Some researchers have pointed out that a key objective of industrial ecology ought to be the identification and the implementation of policies that will lead to industrial systems being more like natural (ecological) systems. In Chap. 14, Finn R. Førsund provides an environmental economist’s perspective on recent developments in industrial ecology. His specific goal in this chapter is to explain to the reader the ways in which standard economic tools might be used to accomplish the stated objectives of industrial ecology. To this end, the author begins his discussion by first stating the goals of industrial ecology and then he lays out what he calls “the basic approach of environmental economics.” He then proceeds to document the problems with the way in which multioutput processes are modeled within natural resource and environmental economics. We are told that even though the standard formulation for modeling multioutput processes in environmental economics is logical when we are in the realm of goods such as bicycles or cars, this same formulation is problematic because it cannot capture what the author calls the “unavoidability”of residuals generation. Interpreted differently, whatever formulation is used by a researcher must not violate the materials balance principle discussed in Chap. 13. Given this viewpoint, the author proceeds to discuss a better approach to multioutput modeling that is consistent with the above mentioned materials balance principle. This is the “factorially determined multioutput production” approach. The use of this approach permits a researcher to explicitly account for the need to include residuals as a part of the underlying production process. In addition, we learn that when end-of-pipe purification occurs, it is better to represent this “purification division” as a separate activity where the residuals from the production activity are inputs that are processed using resources. This leads to a new set of residuals that must be accounted for in a way that is consistent with the materials balance principle. The upshot of the analysis presented in Chap. 14 is threefold. First, industrial ecology models that are devoid of economics are of little use because, operating by themselves, industrial ecology models have no way of selecting between residuals reduction and tradeoffs between environmental qualities and the production of human-made goods. Second, environmental damages from the discharge of residuals can be internalized effectively be levying an apposite Pigovian tax on each of the involved residuals. Finally, for global pollutants like greenhouse gases, from a practical standpoint, it may be easier to put in place a “cap-and-trade” policy.

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This notwithstanding, the author conjectures that the difference between a “cap-and-trade” policy and a pollution tax may well be very small. 1.5.

Conclusions

There is no gainsaying the fact that although many different research tools have been used by scholars seeking to shed light on the gamut of research questions in natural resource and environmental economics, it is difficult to find a thorough and compelling treatment of the most commonly used such tools in a single volume. As suggested in Sec. 1.1 of this introductory chapter, this less than ideal state of affairs is regrettable. In addition, this disagreeable state of affairs also imposes needless entry costs on scholars seeking to conduct research in the vibrant field of natural resource and environmental economics. Given the above delineated situation, our objective in this book is to provide thorough and compelling accounts of the most commonly used research tools in natural resource and environmental economics that are written by experts. These experts have immense credibility because of two salient reasons. First, they are active researchers themselves. Second, they are the practitioners of the very tools that they have written about in the different chapters of this book. In this introductory chapter, we have attempted to provide a holistic and coherent context within which one may view the emergence of the various research tools that are discussed in this book. The salience and the utility of these research tools are now firmly established within the field of natural resource and environmental economics. Therefore, in the coming years, one may look forward to many interesting and policy relevant developments in this field that involve the use of one or more of this book’s research tools. Acknowledgments The first author acknowledges financial support from the Gosnell endowment at RIT. The usual disclaimer applies.

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Portney, PR (1994). The contingent valuation debate: Why economists should care. Journal of Economic Perspectives, 8, 3–17. Ridker, RG (1967). Economic Costs of Air Pollution. New York: Praeger, New York. Ridker, RG and JA Henning (1967). The determinants of residential property values with special reference to air pollution. Review of Economics and Statistics, 49, 246–257. Smith, VL (1982). Microeconomic systems as an experimental science. American Economic Review, 72, 923–955. Sumaila, UR, A Dinar and J Albiac (2008). Game theoretic applications to environmental and natural resource problems. Environment and Development Economics, 14, 1–5.

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PART II

Theoretical Tools

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Chapter 2

Dynamic Analysis Ray G. Huffaker University of Florida rhuffaker@ufl.edu

2.1.

Introduction

Several textbooks introduce economic dynamics. In general, dynamic analysis is motivated by the need to model stock-adjustment flows governing, for example, macroeconomic phenomena such as inflation (Shone, 2002), or the optimal management of natural resources and the environment (Clark, 1990). Three basic approaches to dynamic optimization are covered: the calculus of variations, optimal control theory, and dynamic programming. The approaches are shown to produce the same solution, and their relative utility in analytical economics is compared. Optimal control theory is presented as a generalization of the calculus of variations, and thus capable of generating more economic insight. Dynamic programming is presented as an approach primarily designed to exploit the recursive nature of problems in discrete time, with limited application in continuous time due to difficulties of solving the Hamilton–Jacobi–Bellman partial differential equation (Chiang, 1992; Intrilligator, 1971; Kamien and Swartz, 1991; Leonard and Long, 1992). This chapter draws attention to two questions essential to the analysis of dynamic economic models that receive relatively less attention in introductory textbooks. First, are the equilibria of a dynamic model asymptotically stable or unstable? Introductory textbooks generally pose non-linear optimal control formulations generating solutions whose local stability properties can be established by linearizing system dynamics in the

29

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vicinity of equilibria. The associated eigenvalues are calculated, and their signs are linked with various types of stability properties. The discussions stop short of covering the problems that arise, along with the opportunities for more in-depth analysis, when familiar linearization methods fail. This occurs increasingly as dynamic economic models become more complex in application. Other techniques from dynamic systems theory, relatively unfamiliar to economists, must be substituted to investigate stability properties. Second, do stability properties abruptly shift when a parameter in the model is varied cateris paribus? When familiar linearization methods fail, stability properties are not robust, and may shift in systematic ways. Consider, for example, the zero-cost non-linear fisheries model (Clark, 1990). The basic model is an optimal control formulation, whose necessary conditions for optimality are given by a pair of non-linear differential equations. Linear stability analysis in the neighborhood of the single equilibrium demonstrates it to be a saddle-point (Clark, 1990, Fig. 4.3). A slightly more complex version of the model generates multiple equilibria. The phase diagram (Clark, 1990, Fig. 6.10) shows that the configuration and stability characteristics of equilibria abruptly change when a revenue parameter exceeds critical values at which linear stability analysis fails. Characterizing the optimal harvest policy becomes much more complicated. The study of how system dynamics systematically change in response to small perturbations in underlying parameters is the realm of bifurcation analysis. This chapter investigates advanced topics in stability and bifurcation analysis at an introductory level — building them from the “ground up.”1 Hopefully, the reader becomes better able to analyze increasingly complex dynamic economic models to explain a broader array of real-world behavior. The chapter begins with a brief discussion of the stability of linear firstorder differential equations designed to show why eigenvalues impart information on the stability of solutions. The stability techniques are applied to analyze the stability of the Solow growth model. The chapter then discusses the stability of two-equation systems of differential equations to lay the foundation for linear stability and bifurcation analysis of nonlinear dynamic economic systems. These analytical techniques are applied to analyze formally the stability properties of the more complex zero-cost non-linear fishery model depicted in Clark (1990, Fig. 6.10). 1 See Tu (1994) for presentation of advanced stability and bifurcation topics in economic modeling in an intermediate- to advanced-level textbook.

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2.2.

2.2.1.

31

The Role of Eigenvalues: Stability of First-Order Linear Differential Equations Stability requirements in a nutshell

The first-order linear homogeneous differential equation (dy)/(dt) = by(t) has solution y(t) = y(0)ebt , where b is a fixed parameter, and y(0) is the initial value of y(t). The asymptotic behavior, or “stability”, of y(t) is controlled by b (the “eigenvalue”) because it is the value exponentiated in the solution. When b = 0, solution dynamics of y(t) are “shut down” since y(t) remains at its initial condition y(0) through time, i.e., limt→∞ [y(t) = y(0)e(0)t ] = y(0). When b > 0, solution dynamics are unstable since y(t) explodes as t → ∞, i.e., limt→∞ [y(t) = y(0)e(b>0)t ] = ∞. Finally, when b < 0, solution dynamics are stable since y(t) dampens toward the equilibrium, y e = 0, i.e., limt→∞ [y(t) = y(0)e(b 0 y eq unstable

(1)

For example, the phase diagram for b < 0 appears in Fig. 2.1(a). For y(t) < y eq , (dy)/(dt) > 0, which means that y(t) evolves toward equilibrium as

dy dt

•

k

y(t)

0 (a)

k max

k*

k(t)

(b)

Fig. 2.1: Phase diagrams for first-order differential equations: (a) stability (b < 0), and (b) the Solow Growth Model.

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t → ∞. For y(t) > y eq , (dy)/(dt) < 0, which means that y(t) decreases toward equilibrium as t → ∞. In sum, the solution is stable. The stability properties of the Solow growth model

2.2.2.

The techniques for first-order linear differential equations can be used to analyze the stability properties of the non-linear Solow growth model: k˙ = sf (k(t)) − (n + g + δ)k(t). The net change in capital held per unit of effective labor, k(t), is given by the difference between the flow of investment in new capital (first term) and the flow of various ways in which capital is consumed (second term). Model parameters are: s (saving rate), g (growth rate of technology), n (growth rate of labor), δ (physical depreciation rate), and k0 (initial condition). The production of k(t) is given by f (k(t)). The production function is characterized by positive but diminishing marginal returns, (df )/(dk) > 0 and (d2 f )/(dk 2 ) < 0, and the Inada df df = ∞ and limk→∞ dk = 0, which guarantee an interior conditions, limk→0 dk equilibrium. The phase diagram solution for the Solow growth model is shown in Fig. 2.1(b). The graph has a maximum at k max such that n+g+δ df max (k . )= dk s Equilibria occur where the graph cuts the horizontal axis, i.e., k e = 0, k ∗ . The origin is an unstable equilibrium because the associated eigenvalue is positive: df ∂ k˙ eq (k = 0) = s (0) − (n + g + δ) > 0. ∂k dk Alternatively, the interior equilibrium, k ∗ , is a stable equilibrium because the associate eigenvalues is negative, df ∂ k˙ eq (k = k ∗ ) = s (k ∗ ) − (n + g + δ) < 0, ∂k dk ˙

since ∂∂kk (k max ) = 0, k ∗ > k max , and thus f  (k ∗ ) < f  (k max ) by the assumption of diminishing marginal returns.

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2.3.

2.3.1.

33

The Role of Eigenvalues: Stability of Systems of Linear Differential Equations Stability requirements in a nutshell

Consider a two-equation system of linear homogeneous differential equations    dx    ˙ = a11 x(t) + a12 y(t) a11 a12 dt = x x˙ x(t) ⇒ = (2) y˙ y(t) dy a21 a22 = y ˙ = a x(t) + a y(t) 21 22 dt   A coefficient matrix

with solution       a12 a12 x(t) λ1 t λ2 t + k2 (x0 , y0 )e = k1 (x0 , y0 )e (λ1 − a11 ) y(t) (λ2 − a11 )     ev1

(3)

ev2

where x0 and y0 are initial levels of x and y (“initial conditions”); k1 and k2 are constants tying the solution to the initial levels; λ1 and λ2 are the eigenvalues of coefficient matrix A in Eq. (2), and ev1 and ev2 are the eigenvectors associated with λ1 and λ2 , respectively. Since there are an infinite number of initial conditions, Eq. (3) gives an infinite number of solutions. These solutions can be depicted in a two-dimensional phase diagram that plots x(t) against y(t) from various initial points x0 and y0 as time t advances. System (2) has a single equilibrium for which x˙ = y˙ = 0 at the origin: (xe , y e ) = (0, 0). The eigenvalues λ1 and λ2 , determine system stability because they are exponentiated in solution (3). The eigenvalues solve the “characteristic equation”

Tr(A) ± Tr(A)2 − |A| 2 C(λ) = λ − Tr(A)λ + |A| = 0 ⇒ λ1,2 = 2 (4) where Tr(A) = a11 + a22 (the trace of A), and |A| = a11 a22 − a21 a12 (the determinant of A). The eigenvectors ev1 and ev2 solve →

(A − λi I)evi = 0

(5a)

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→

where 0 is a two-dimensional row vector of zeros, and i = 1, 2. The solution to the eigenvector problem is composed of two linearly dependent equations. Consequently, there are infinitely many solutions    a12  − (a − λ ) c 11 i evi =   c

(5b)

where c is an arbitrary constant set equal to (a11 − λi ) in Eq. (3). Solution trajectories given by Eq. (3) run parallel to eigenvectors as they asymptotically approach, or depart from, equilibrium. There are 11 possible stability cases depending on whether the eigenvalues generated by Eq. (4) are real, complex, or nonzero (Fig. 2.2). Some cases that arise in the economic application below are illustrated in phase diagrams in Fig. 2.3. When eigenvalues are real and negative, the equilibrium is a “stable node” that attracts solution trajectories from all initial values (Fig. 2.3(a)).

λ+ , =

Tr ( A)2 − 4 A Tr ( A) ± 2 2

Real

Complex

Tr ( A) 2 − 4 A ≥ 0

Tr ( A) 2 − 4 A < 0

λ+, =

Repeating

Distinct

α

Tr ( A) 2 − 4 A = 0

Tr ( A) 2 − 4 A > 0

λ+ , λ < 0

(1)

λ+ , λ > 0

(2)

λ+ > 0, λ < 0

(3)

λ = 0, λ < 0

(9 )

Single degenerate

λ = 0, λ > 0

(10 )

Single degenerate

Fig. 2.2:

2 Tr(A) ABS[Tr(A) − 4 A ] i ± 2 2

λ+ = λ− = Tr ( A) / 2

λ >0

(4) (5 )

λ =0

(11)

λ 0, and R (H > H m ) < 0, where R = (∂R)/(∂H). The fish population growth rate is a logistic function such that: F  (X msy ) = 0, F  (X < X msy ) > 0, and F  (X > X msy ) < 0, where F  = (∂F )/(∂X) and X msy is the population level generating the maximum sustained yield (MSY). The current-valued Hamiltonian is H(t) = R(H(t)) + λ(t)[F (X(t)) − H(t)]

(12)

where λ(t) is the marginal present value of the fish stock at t. The necessary conditions for optimization are ∂H = R (H) − λ = 0 → λ(t) = R (H) ∂H ∂H λ˙ = −λF  (X) → F  (X) = δ − λ˙ − δλ = − ∂X λ ∂H = X˙ = F (X) − H ∂λ

(13a) (13b) (13c)

where time subscripts are dropped to simplify notation. The optimality principle (13a) equates the marginal present value of capital with the marginal revenue of harvest along the optimal path. The adjoint condition (13b) is an arbitrage condition equating the rate of return to investing capital (left-hand side) to the rate of return to consuming it (right-hand side). Condition (13c) imposes the evolutionary dynamics of the fish population on the optimal solution. Conditions (13(a–c)) reduce to a two-dimensional system of non-linear differential equations2 : R (H) H˙ =  [δ − F  (X)] R (H) X˙ = F (X) − H

(14)

For purposes of analysis, the following functional forms are substituted  into system (14): R(H) = (2H m − H)bH and F (X) = rX 1 − X K , where H m is the harvest rate maximizing the flow of revenue, b is a positive revenue parameter, r is the intrinsic growth rate, and K is the carrying 2 System

(14) is obtained by taking the time derivative of λ in Eq. (13a), substituting it ˙ along with λ into Eq. (13b), and solving for H.

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capacity of the fish population. The fish stock MSY, F (X), is X msy = K/2, with associated harvest rate H msy = F (X msy ) = (rK)/(4). The optimality system becomes   2r H˙ = (H − H m ) δ − r + X K (15)   X X˙ = rX 1 − −H K Optimality system (15) has at most three equilibria:  m K − K 2 − 4KH r e m e H = H , X− = 2  m K + K 2 − 4KH r e m e H = H , X+ = 2 X gr =

K(r − δ) , 2r

H gr =

K(r2 − δ 2 ) 4r

(16a)

(16b)

(16c)

e e In Eqs. (16a) and (16b), H e = H m sets H˙ = 0, and X− and X+ set X˙ = 0. m msy , the equilibrium fish stocks coalesce at the population When H = H e e = X− = (K)/(2). The equilibria vanish level producing MSY, X msy : X+ m msy . In Eq. (16c), X gr is the “golden-rule” equientirely when H > H librium — equating the marginal productivity of the fish stock with the discount rate — that sets H˙ = 0, and H gr sets X˙ = 0.

2.5.1.

Linear stability analysis

Following Eq. (7), the linearized version of optimality system (15) expanded about equilibrium (H e , X e ) is  2r  2rX e (H e − H m ) δ−r+  K   K       dH˙ (H e ,X e ) ˙ dH e e     dH dX (H ,X )    H˙  H(t)  = (17a)      X(t) X˙ e   2X   −1 r 1−    K  ˙    dX dH



(H e ,X e )



˙ dX dX

J(H e ,X e )

(H e ,X e )



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e The linear expansion particular to (H m , X− ) is



r δ − K  =  X˙

  H˙

 K2

−1





4KH m − r 

   H(t)  4KH m  X(t)

0 r K

 K2 −

r

(18)



e ) J(H m ,X−

e ) is a lower triangular matrix whose eigenvalues are the diagonal J(H m , X− elements:

λ1 = δ −

r λ2 = K

r K

 K2 −

4KH m r



4KH m K2 − r

 < 0 for H m < H gr    = 0 for H m = H gr    > 0 for H m > H gr

 >0    =0    H

(19b)

msy

The signs of the eigenvalues depend on the level of H m (the fixed harvest rate parameter maximizing the flow of revenue) relative to the harvest rate sustaining the golden-rule fish stock, H gr , and the harvest rate sustaining the MSY fish stock, H msy . Linear stability analysis demonstrates the equie ) to be a saddle point when H m < H gr , and an unstable librium (H m , X− node or focus when H gr < H m < H msy . The equilibrium is nonhyperbolic for H m = H gr and H m = H msy . Linear stability analysis fails, and stability properties must be analyzed along a center manifold. e ) is The linear expansion particular to (H m , X+ 

r δ + K  = ˙  X

  H˙



 K2 −1

4KH m − r r − K  e) J(H m ,X+



0  K2 −

   H(t)   4KH m  X(t) r



(20)

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e J(H m , X+ ) is a lower triangular matrix whose eigenvalues are the diagonal elements:  < 0 for H m > H gr  m  r 4KH  K2 − (21a) λ1 = δ + = 0 for H m = H gr  K r   m gr > 0 for H < H  > 0 for H m > H msy  m  r 4KH  λ2 = − K2 − (21b) = 0 for H m = H msy  K r   m msy < 0 for H < H e Linear stability analysis demonstrates the equilibrium (H m , X+ ) to be a m gr gr saddle point when H < H and a stable node or focus for H < H m < H msy . The equilibrium is nonhyperbolic for H m = H gr and H m = H msy . The linear expansion particular to (H gr , X gr ) is        2r K(r2 − δ 2 ) m ˙ H − H  H(t) 0 K 4r = (22)  X(t) X˙ −1 δ   J(H gr ,X gr )

The eigenvalues solve the characteristic Eq. (4):   2r K(r2 − δ 2 ) 2 m −H =0 C(λ) = λ − δλ + K 4r

(23)

When H m = H gr =

K(r2 − δ 2 ) , 4r

C(λ) becomes: λ2 − δλ = 0. Consequently, one of the eigenvalues must be zero, so that the equilibrium (H gr , X gr ) is nonhyperbolic at H m = H gr . The signs of the eigenvalues when H m = H gr can be inferred using the rule that the product of the roots of an even-powered polynomial equals the constant term, which, in the case of C(λ), is the determinant of the linearization matrix:   ! ! 2r K(r2 − δ 2 ) − Hm (23a) λ1 λ2 = !J(H gr , X gr )! = K 4r The determinant is positive, |J(H gr , X gr )| > 0, when H m < H gr . Consequently, the eigenvalues can both be positive or negative, or may

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Hm

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H m < H gr

Summary of linear stability results. H m = H gr

Stability Saddle point e (H m , X− )

Nonhyperbolic

Stability Saddle point e (H m , X+ )

Nonhyperbolic

Stability Unstable Nonhyperbolic (H gr , X gr ) node/focus

H gr < H m < H msy Unstable node/ focus Stable node/ focus Saddle point

H m = H msy

H m > H msy

Nonhyperbolic

Vanishes

Nonhyperbolic

Vanishes

Saddle point

Saddle point

be a complex conjugate pair (whose product is always positive). Negative eigenvalues can be ruled out by Descares’ Rule of Signs. For λ < 0, the sign sequence of the terms of C(λ) is: (+, +, +). Thus, there are zero negative roots. Alternatively, for λ > 0, the sign sequence of the terms of C(λ) is: (+, −, +). Thus, there are at most two or zero positive roots. In the latter case, the roots are complex with positive real parts. In sum, the equilibrium (H gr , X gr ) is either an unstable node or focus when H m < H gr . The determinant is negative, |J(H gr , X gr )| < 0, when H m > H gr . This implies that the eigenvalues are real and of opposite sign. Thus, the equilibrium (H gr , X gr ) is a saddle point. Table 2.1 summarizes the linear stability results for all equilibria of optimality system (15). 2.5.2.

Center manifold analysis

The stability properties of the non-linear fisheries model abruptly shift when H m (the harvest rate maximizing revenue flow) reaches two critical levels: H gr (the golden-rule harvest rate), and H m (the MSY harvest rate). Linear stability analysis fails, and stability properties must be studied along a central solution manifold confining system dynamics. The first task is to compute the center manifold in the neighborhood of a non-hyperbolic equilibrium of interest. This chapter investigates the abrupt change in dynamics in the neighborhood of (H e , X e ) = (H m , X gr ) when H m reaches critical value: H m = H gr . 2.5.2.1. Deriving the normal form Computation begins by converting linearized optimality system (17a) to normal form (8). The equilibrium, (H m , X gr ), and critical value,

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H m = H gr , are shifted to the origin by introducing the following new variables: h(t) = H(t) − H m , x(t) = X(t) − X gr , and hm = H m − H gr . Substituting the new variables into Eq. (17a) leads to   2r m       (h − h ) h˙  K 0 0 h  = +   2 x −1 δ (24) rx x˙   − K A h˙m = 0 (see Appendix A for derivation). Normal form (8) lastly requires that matrix A be diagonalized. This is accomplished by introducing a second set of new variables:      h δ 0 u = ⇒ h = δu, x = uv x 1 1 v   P

where the columns of P are eigenvectors of A. Substituting the new variables into system (24) gives   2r (u + v)(δu − hm )      δK      u˙   m 0 0 u f (u,v,h )  + =  2r r m 2 (25) 0 δ v v˙  − (u + v)(δu − h (u + v) ) −     δK K   D g(u,v,hm )

h˙m = 0 (see Appendix B for derivation). Equation (25) fully meets the requirements of normal form (8) since D has the eigenvalues of A along the diagonal; and non-linear functions f (u, v, hm ) and g(u, v, hm ), and their first derivatives with respect to (u, v, hm ), vanish when evaluated at equilibrium (ue = v e = hm = 0). 2.5.2.2. Computing the center manifold equation Coordinates on the center manifold (Eq. (9)) must satisfy the center manifold equation, given by3 v = qc (u, hm ) 3 Variable

(26)

u serves as the independent variable since the zero eigenvalue λ1 is found in the u˙ equation.

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and the associated dynamics (remembering that h˙m = 0 in Eq. (25)) ∂qc ∂qc ˙m ∂qc u˙ + u˙ h = ∂u ∂hm ∂u

(27)

" # ∂qc N qc (u, hm ) = u˙ − v˙ = 0 ∂u

(28)

v˙ = or equivalently

Substituting v˙ and u˙ from Eq. (25) into Eq. (28) gives   ∂qc 2r m m N [qc (u, h )] = (u + v)(δu − h ) ∂u K   

u˙

 2r r (u + v)(δu − hm ) − (u + v)2 = 0 − δv − δK K  

(29)

v˙

Equation (29) is a partial differential equation that is solved for the center manifold equation, v = qc (u, hm ). An approximation theorem allows qc (u, hm ) to be computed to any desired degree of accuracy by solving Eq. (29) to the same degree of accuracy (Wiggins, 2000, Theorem 18.1.4). The following power series expansion is generally posed for qc (u, hm ): v = qc (u, hm ) = a1 u2 + a2 uhm + a3 (hm )2 + · · ·

(30)

where a1 , a2 , and a3 are unknown constants. Substituting Eq. (30) into Eq. (29), and retaining only terms of the same order as Eq. (30) gives     2r 2r N [qc (u, hm )] = δa1 + u2 + δa2 − uhm + δa3 (hm )2 = 0 (31) K δK (see Appendix C for the full substitution generating Eq. (31)). Constants a1 , a2 , and a3 are solved for by equating coefficients 2r 2r = 0 ⇒ a1 = − K δK 2r 2r = 0 ⇒ a2 = 2 term: δa2 − δK δ K

u2 term: δa1 + uhm

⇒ a3 = 0

(hm )2 term: δa3 = 0 The resulting center manifold equation is v = qc (u, hm ) = −

2r 2 2r u + 2 uhm δK δ K

(32)

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u

47

u

•

u =0

Unstable node/focus

•

u =0

Unstable node/focus

•

hm , u = 0

•

hm , u = 0 H m = H gr

bifurcation point

•

•

u =0

u =0

Convergent eigenvector of saddle points gr gr

Convergent eigenvector of saddle pointsp

(H

,X

)

( H m , X −e )

(a)

(b)

Fig. 2.4: Abrupt changes in system dynamics at the golden-rule equilibrium. Notes: (a) bifurcation diagram of Eq. (33b) and (b) relationship to original system (15).

2.5.2.3. Dynamics reduced to the center manifold The dynamics of Eq. (25) reduced to the center manifold are obtained by substituting the center manifold equation (32) into u˙ in Eq. (25):     2r 2 2r 2r    u + 2 uhm  (δu − hm ) u˙ = u + − δK  δK δ K    

(33a)

v=qc (u,hm )

  1 m 2r u u− h u˙ = K δ

(33b)

where Eq. (33b) results from retaining only terms of the same order as Eq. (30). 2.5.2.4. Bifurcation analysis Equation (33b) is the normal form for a “transcritical” bifurcation. A bifurcation diagram graphed in (hm , u) space shows the following (Fig. 2.4(a)): • There are two curves of equilibria obtained by setting Eq. (33b) equal to zero: ue = 0 (the horizontal axis) and ue = 1δ hm (the positively sloped line). These curves pass through the critical “bifurcation” point at the origin (ue , hm ) = (0, 0).

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• The stability of equilibria resting on ue = 0 is determined by the eigenvalue
m 2r m < 0 for h > 0 du˙ e (u = 0) = − h (34a) du δK > 0 for hm < 0 Equilibria ue = 0 are unstable when the parameter hm is negative (the dashed curve), but become stable (the solid curve) after hm passes through the bifurcation point, (ue , hm ) = (0, 0). • The stability of equilibria resting on ue = 1δ hm is determined by the eigenvalue  $  1 du˙ 2r m > 0 for hm > 0 h (34b) u e = hm = < 0 for hm < 0 du δ δK Equilibria ue = 1δ hm are stable when the parameter hm is negative (the dashed curve), but become unstable (the dashed curve) after hm passes through the bifurcation point, (ue , hm ) = (0, 0). In sum, the system displays the defining characteristic of a transcritical bifurcation; namely, an exchange of stability when the two curves of equilibria coalesce at the bifurcation point: (ue , hm ) = (0, 0). The implications of the bifurcation diagram for the dynamic behavior of the original system (15) are summarized in Fig. 2.4(b). In terms of the original variables, the bifurcation point at the origin is H e = H m = H gr . Negative values of hm correspond to H m < H gr , and positive values correspond to H m > H gr . The equilibria along the horizontal axis of the bifurcation diagram are the golden-rule equilibria, (H gr , X gr ) (defined in Eq. (16c)), in the original system. The equilibria along the upward sloping curve, e ) (defined in Eq. (16a)), in the original system. ue = 1δ hm , are (H m , X− m gr For H < H , the golden-rule equilibria are unstable nodes/foci, e ) are saddle points (Table 2.1). This is demonstrated in the and (H m , X− phase diagram for this case (Fig. 2.5(a)). The straight dashed lines are zero-change curves H˙ = 0, and the bell-shaped dash curve is the zeroe e ) and (H m , X+ ), change curve X˙ = 0. The outer equilibria, (H m , X− are saddle points. The inner equilibrium is the golden-rule equilibrium, (H gr , X gr ), which for the underlying parameters, is an unstable focus. A trajectory emanating from the unstable focus at (H gr , X gr ) provides the downward-sloping convergent path connecting it with the saddle point e ). For this reason, the bifurcation diagram considers the equilibria (H m , X− e ) as stable in the neighborhood of (H gr , X gr ). (H m , X− For H m > H gr , the golden-rule equilibria are saddle points, and m e ) are unstable nodes/foci (Table 2.1). The bifurcation treats the (H , X−

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Fig. 2.5: Phase diagrams of original system (15). Notes: (a) H m < H gr , (b) H m > H gr and (c) H m > H msy . The straight dashed lines are zero-change curves H˙ = 0, and the bell-shaped dash curve is the zero-change curve X˙ = 0.

saddle points at (H gr , X gr ) as stable for the same reasons explained above. As demonstrated in the phase diagram for this case (Fig. 2.5(b)), a trajece ) provides the tory emanating from the interior unstable focus at (H m , X− downward-sloping convergent path connecting it with the outer saddle point (H gr , X gr ). Consequently, the bifurcation diagram views saddle-point equilibria (H gr , X gr ) as locally stable. 2.5.2.5. Implications of bifurcation theory for selecting optimal policy Bifurcations have important implications for characterizing the optimal policy from an optimal control formulation. In the non-linear fishery model analyzed above, bifurcations occur because revenues are a bell-shaped function of the harvest rate, i.e., marginal revenue can be zero at a harvest rate equal to H m . In a more basic version, marginal revenue is decreasing in harvest, but never negative (Clark, 1990, Fig. 4.3). The basic version is characterized by a single saddle-point golden-rule equilibrium that is always hyperbolic. The optimal infinite-time policy is to select an initial harvest

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rate resting on the trajectory asymptotically approaching the golden-rule equilibrium directly above the initial fish population. This steady-state configuration and optimal policy only hold in the more complex model when the harvest rate maximizing the flow of revenue exceeds the MSY rate: H m > H msy (Fig. 2.5(c)). In short, the potential for negative marginal revenue is not binding on the solution. As demonstrated by bifurcation diagram (Fig. 2.4(a)), the stability of the golden-rule equilibrium shifts abruptly when the harvest rate maximizing the flow of revenue falls below the golden-rule level: H m < H gr . The golden rule equilibrium is an unstable node/focus that is no longer optimal or approachable (Fig. 2.5(a)). There are two outer saddle-point equilibria that offer potentially competing optimal policies. The lower e ) can be approached by the convergent population saddle point (H m , X− trajectory emanating from the lower left corner of the diagram, or the convergent trajectory emanating downward from the golden-rule equilibe ) can be approached by rium. The higher population saddle point (H m , X+ the horizontal convergent trajectory from both directions. Problems arise when an initial fish population rests under two trajectories converging to different equilibria, as happens for population levels e and X gr . Is the optimal policy to set the initial harvest rate, between X− H0 , between the golden-rule rate, H gr , and the rate maximizing the flow of revenue, H m , to follow the convergent trajectory emanating from the golden-rule equilibrium? Or, is the optimal policy to set the initial harvest on the horizontal trajectory converging to the upper population saddle point? In this case, extra marginal analysis must be applied to determine which convergent trajectory constitutes the optimal policy (Davidson and Harris, 1987). Appendix A. Derivation of Eq. (24) Substituting new variables h(t), x(t) and hm into H˙ (Eq. (15)) gives    2r    (x + X gr ) h˙ = (h + H gr ) − (hm + H gr ) δ − r +      K   ˙ H

H(t)

=

Hm

X(t)

2rx (h − hm ) K

where the fact was used that δ − r +

2r gr KX

= 0 (by Eq. (16c)).

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Substituting the new variables into X˙ (Eq. (15)) gives  x + xgr       X(t)   gr    x˙ = r x + X  1 −  + H gr )  K  − (h      

˙ X

X(t)

= δx −

H(t)

rx2 −h K

+ , gr2 where the fact was used that rX gr 1 − XK − H gr = 0 (by Eq. (16c)). To move toward the normal form underlying central manifold analysis, the ˙ x) transformed system (h, ˙ is linearized about the new equilibrium x = h = m h = 0, and the non-linear terms (including any term with perturbation parameter hm ) are placed into an added vector:   h˙



2rx  K =  x˙ −1

 !  ! 2r 2r m m !   (h − h ) (h − h )!  K h K  ! + !    2 x 2rx rx ! δ− − ! e e m K x =h =h =0 K

Evaluating the linearization matrix at equilibrium gives Eq. (24).

Appendix B. Transforming System (24) to Diagonalizing Variables Substituting the diagonalizing variables into system (24) gives   2r   (u + v)(δu − hm ) u   P = AP + K  r v v˙ − (u + v)2 K       2r (u + v)(δu − hm )    u˙ 1/δ 0  K 0 0 u  ⇒ + =   r v˙ −1/δ 1 0 δ v − (u + v)2     K P −1 AP P −1   u˙

Completing matrix multiplication gives system (25).

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Appendix C. The Full Substitution Generating Eq. (31) N [qc (u, hm )]









 2r  2 m  = (2a1 u + a2 hm )  u + a1 u2 + a2 uhm + a3 (hm c )  (δu − h )  K    v=qc (u,hm )

∂qc ∂u

# − δ a1 u2 + a2 uhm + a3 (hm )2   "

 +

v=qc (u,hm )



2r  2 m m 2 m u + a1 u + a2 uh + a3 (h )  (δu − h )   δK v=qc (u,hm )



2   r  + u + a1 u2 + a2 uhm + a3 (hm )2  = 0 K   v=qc (u,hm )

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References Chiang, A (1992). Elements of Dynamic Optimization. New York: McGraw-Hill, Inc. Clark, C (1990). Mathematical Bioeconomics: The Optimal Management Renewable Resources, 2nd Ed. New York: John Wiley & Sons, Inc. Davidson, R and R Harris (1987). Non-convexities in continuous-time investment theory. Review of Economic Studies, 48, 235–253. Glendinning, P (1995). Stability, Instability, and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. New York: Cambridge University Press. Intrilligator, M (1971). Mathematical Optimization and Economic Theory. Englewood Cliffs, N.J.: Prentice-Hall, Inc. Kamien, M and N Schwartz (1991). Dynamic Optimization: The Calculus of Variations and Optimal Control in Economics and Management. New York: North Holland. Leonard, D and N V Long (1992). Optimal Control Theory and Static Optimization in Economics. New York: Cambridge University Press. Shone, R (2002). Economic Dynamics, 2nd Ed. New York: Cambridge University Press. Solow, R (1956). A contribution to the theory of economic growth. Quarterly Journal of Economics, 70 (February), 65–94. Tu, P (1994). Dynamical Systems: An Introduction with Applications in Economics and Biology, 2nd Ed. New York: Springer-Verlag. Wiggins, S (2000). Introduction to Applied Nonlinear Dynamic Systems and Chaos 2d. New York: Springer.

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Chapter 3

Stochastic Analysis: Tools for Environmental and Resource Economics Modeling Anastasios Xepapadeas Athens University of Economics and Business [email protected]

3.1.

Introduction

Risk and uncertainty are fundamental concepts for the understanding of basic characteristics and properties permeating the field of environmental and resource economics. The impact of risk and uncertainty can be identified across basic features of environmental economics and resource management, such as the evolution of resource stocks or stocks of pollutants, market or technological conditions.1 For example, uncertainty associated with the growth of renewable resources or the existing stocks of exhaustible resources may have serious repercussions on resource depletion or resource extinction and the design of policies related to harvesting and the management of ecosystems and biodiversity. The understanding of uncertainties associated with the accumulation of greenhouse gases and the mechanisms of global warming and its socioeconomic impacts is crucial for analyzing climate

1 There

is vast literature related to risk and uncertainty in environmental and resource economics. We refrain from citing this literature here. A very useful survey of the field and relevant references can be found in M¨ aler and Fisher (2005).

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change and designing appropriate policies. Uncertainties associated with future market and technological conditions are central in characterizing option values of environmental resources and current exploitation policies. The purpose of this chapter is to present some of the basic tools for introducing and analyzing risk and uncertainty in environmental economics and resource management. The main focus will be tools and methods which can be used to study dynamic problems since dynamic analysis is probably the most interesting and relevant framework for analyzing environmental and resource management issues. The rest of the chapter is organized as follows. Section 2 presents some general results from probability theory, Sec. 3 introduces methods of stochastic calculus and presents the fundamental concept of stochastic differential equations and the celebrated Itˆ o’s lemma. Section 4 discusses dynamic optimization in a stochastic framework and presents the Hamilton–Jacobi–Bellman (HJB) equation which is central in optimal stochastic control. Section 5 discusses irreversibilities, option values, and optimal stopping rules, and introduces the concept of free boundary in decision making under uncertainty. Section 6 introduces the concepts of non-expected utility in decision making and discusses approaches of modeling uncertainty through multiple priors and robust control methods. The last section concludes the chapter.

3.2. 3.2.1.

Results from Probability Theory Probability space and random variables

We start with the definition of some fundamental concepts.2 An experiment E is the operation through which several possible outcome or results can be produced. A probability space is a triplet (Ω, F , P ) which can be associated with an experiment. Ω is an arbitrary set of points ω ∈ Ω. Each point ω is called a sample point or outcome of the experiment, for example head (H) or tail (T ) in a single toss of a coin where Ω = {H, T }, or a set of objects with detectable characteristics and/or properties. The set Ω is called the sample space, while a subset of Ω is called an event. Thus, an event is a collection of some of the possible outcomes of an experiment. The set of all events is the power set of Ω which contains 2Ω events. For example, if we 2 For details see, for example, Priestley (1981), Malliaris and Brock (1982), and Chang (2004).

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toss two coins Ω = {HH , TT , HT , TH } and its power set contains 24 = 16 events. F is a class of subsets (or events) of Ω, i.e., F ⊂ 2Ω which is called a σ-algebra and which has the following properties: 1. The sample space Ω belongs to F , or Ω ∈ F (equivalently ∅ ∈ F). 2. If A ∈ F, then the complement of A also belongs to F , or Ac = {ω ∈ Ω:ω∈ / A} ∈ F. 3. For any sequence Ai ∈ F, i = 1, 2, . . . the union of all Ai belongs to F , or ∪∞ i=1 Ai ∈ F. The pair (Ω, F ) is called a measurable space. Let P a set function P : F → R. This function is called a probability measure if it satisfies 1. P (∅) = 0 and P (Ω) = 1. (∅ is usually called the “impossible event”) 2. 0 ≤ P (A) ≤ 1 for all A ∈ F. 3. If Ai ∈ F and the Ai ’s are mutually disjoint, i.e., Ai ∩ Aj = ∅ if i = j,
∞ then P (∪∞ i=1 Ai ) = i=1 P (Ai ). The real number P (A) which is assigned to the event A is interpreted as P (A) = “the probability that event A occurs” The classical definition of probability, as opposed to the axiomatic approach developed above, can be presented as follows. Suppose that an experiment has a finite number of N possible outcomes ω1 , . . . , ωN . We say that event A occurs if the outcome is any of ω1 , . . . , ωM , thus A = ω1 ∪ ω2 . . . ∪ ωM . If all outcomes ω1 , . . . , ωN are equally likely then P (A) = M/N . The classical definition follows from the properties 1–3 above of the probability measure. Let (Ω, F ), (Ω , F  ) denote two measurable spaces. A mapping X : Ω → Ω is (F , F  ) measurable, if for every meaningful event ω  ∈ Ω there exists a meaningful event ω ∈ Ω, or for each A ∈ F  , X −1 (A ) = {ω : X(ω) ∈ A } ∈ F. If the image space Ω is the real line R then the measurable real function X : Ω → R is called a random variable. That is, a random variable is a measurable function from the sample space Ω to the real line R, such that for each event ω ∈ Ω there corresponds a unique real number X(ω). If X : Ω → Rm and it is measurable then it is called a random vector, X(ω) = (X1 (ω), . . . , Xm (ω)). Let X be a random variable, its distribution function which is denoted by F (X) is defined as F (x) = P ({ω ∈ Ω : X(ω) ≤ x}) = P [X ≤ x]

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All distribution functions satisfy: 1. 0 ≤ F (x) ≤ 1. 2. limx→−∞ F (x) = 0, limx→∞ F (x) = 1. 3. F (x) is a non-decreasing function. That is, for any h ≥ 0, F (x + h) ≥ F (x). Distribution functions could be discrete, continuous, or mixed. We focus on the purely continuous case. In this case, F (x) is assumed continuous and differentiable for almost all x. X is called a continuous random variable. The probability density function of a continuous random variable X is the derivative f (x) of the distribution function F (x), or  x dF (x) f (u)du, f (x) = F (x) = dx −∞ with the properties: 1. f (x) ≥ 0. ∞ 2. −∞ f (x)dx = 1. 3. For any a, b (a ≤ b), P [a < X ≤ b] =

b a

f (x)dx .

Let X be a continuous random variable X with probability density function f (x). The mean µ and the variance σ of f (x) are defined as:  ∞ µ= xf (x)dx −∞

σ2 =



∞

−∞

3.2.2.

(x − µ)2 dx

Stochastic processes

A stochastic (or random) process {X(t) : t ∈ T } is a family of random variables indexed by the t, where t belongs to a given index set T . Let the index set T denote time. If T is such that t = 0, 1, 2, . . . then {X(t)} is said to be a discrete time (or discrete parameter) stochastic process. If T = [0, ∞) or T = [0, 1], then {X(t)} is said to be a continuous time (or continuous parameter) stochastic process. Since X(t) is a random variable, each time the experiment is performed a different value of X will in general be observed. An observed record of a stochastic process is a collection of observed outcomes at different points in time which is called a realization,

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sample path, or trajectory of the stochastic process. The collection of all possible records is called the ensemble of the stochastic process. A stochastic process can be regarded as a function defined on T × Ω, or X : T × Ω → R. For a given t ∈ T, Xt (·) : Ω → R is a random variable. For a given ω ∈ Ω, X(·, ω) : Ω → R is a function from T to R. This function is the sample path. The range of X is the state space, while the value X(t, ω) is the state at time t, and X is the state variable. In environmental and resource economics, a stochastic process usually describes the stock (or biomass) of a natural resource or the pollutants generated through production or consumption. Thus, a realization will be an observed record of resource biomass, or deposits, or stocks of pollutants at different points in time. These observations are the state variables. For each t, X(t) will have a probability distribution and, when X(t) is a continuous random variable, a probability density function ft (x). The mean and the variance of X(t) will be given by 

∞

mean{X(t)} = E{X(t)} = −∞

xf t (x)dx = µ(t) 2

variance{X(t)} = E[{X(t) − µ(t)} ] =



∞

−∞

(x − µ)2 dx = σ 2 (t)

A stochastic process is called stationary up to order 2 (or simply stationary) if: 1. It has the same mean µ at all points in time, or E[X(t)] = µ. 2. It has the same variance σ 2 at all points in time, or var[X(t)] = σ 2 . 3. The covariance between the values of X at any two points of time t and s, defined as cov{X(t)X(s)} = E[X(t)X(s)] − µ2 , depends only on the difference t − s. For a stationary process the autocovariance function R(τ ) is defined as R(τ ) = E[{X(t)X(t + τ )}] − µ2 while ρ(τ ) = R(τ )/R(0) is the autocorrelation function of X(t). A stochastic process {X(t)} is called Gaussian or normal, if for any n and a subset {t1 , t2 , . . . , tn } ∈ T , the joint probability distribution of {X(t1 ), X(t2 ), . . . , X(tn )} is multivariate normal and is completely determined by its mean value function E[X(t)], and covariance function cov{X(t)X(s)}.

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A discrete parameter stochastic process {Xt }, t = 0, ±1, ±2, . . ., is called a purely random process or white noise if it is a sequence of uncorrelated random variables. That is, cov{Xt , Xs } = 0 for all t = s. A white noise process will be denoted by {εt }. A continuous parameter white noise {ε(t)} is a process such that for any subset of times t1 , t2 , . . . , tn , the sequence ε(t1 ), ε(t2 ), . . . , ε(tn ) forms a set of uncorrected random variables. Assuming E[ε(t)] = 0 and var{ε(t)} = σε2 , the autocovariance and autocorrelation function for the white noise are defined as  2  σε τ = 0 1 τ =0 , ρε (τ ) = Rε (τ ) = 0 τ = 0 0 τ = 0 We present below stochastic process with specific characteristics which have been extensively used in the analysis of environmental and resource economics in stochastic environments. 3.2.2.1. The Wiener process A stochastic process W (t), t ∈ [0, ∞) is a Wiener process or a Brownian motion process, if it satisfies the following properties: 1. W (0) = 1 with probability 1 (w.p.1), i.e., the process starts at zero. 2. For any two points of time t and s (0 ≤ s < t), the increment W (t) − W (s) is not influenced by the increment W (s) − W (0). Thus, the increment {∆Wt } = {W (t) − W (t − 1)} for an integer t is identically and independently distributed (i.i.d). 3. The increments of the process between any two points of time t and s (0 ≤ s < t) are distributed normally, or W (t) − W (s) ∼ N (µ(t − s),

σ 2 (t − s))

if we standardize the process, that is µ = 0, σ 2 = 1, then W (t) − W (s) ∼ N (0, t − s). 3.2.2.2. The Markov process A stochastic process satisfies the Markov property, if the probable future state at any time t > s is independent of the past behavior of the state variable at time t < s, and depends only on the current state s. A discrete parameter stochastic process {Xt : t = 0, 1, 2, . . .} defined on a countable state space B is said to be a Markov chain if it satisfies the

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Markov property, that for all x ∈ B and t = 0, 1, 2, . . . P [Xt+1 = x | X0 , X1 , . . . , Xt ] = P [Xt+1 = x | Xt ] The Markov property says that the probability that the random variable X will be at state x at time t + 1 conditioned on the past behavior of the random variables X0 , X1 , . . . , Xt , i.e., conditioned on the information provided by the realization of the stochastic process, is equal to the probability that the random variable X will be at state x at time t + 1 conditioned only on current (or present) information provided by Xt . The Markov property implies that what matters for determining the future state of the system is only its current state. It does not matter how the stochastic process reached the current state. The probability that Xt+1 is at state x, given that Xt is at state z (Xt = z), is called the one-step transition probability: t,t+1 = P [Xt+1 = x | Xt = z] Pzx

For a continuous stochastic process, let {X(t) : t ∈ T = [0, T¯ ] ⊂ [0, ∞)}, and denote Ft = σ(Xs : s ∈ [0, t]). Ft is a family of increasing σ-algebras in the sense that Fs ⊂ Ft , s < t. The stochastic process {X(t) : t ∈ T = [0, T¯ ] ⊂ [0, ∞)} defined on a probability space (Ω, F , P ) is Markov, if for any s, t such that 0 ≤ s ≤ t ≤ T¯ , it satisfies w.p.1 one of the following: 1. P [X(t) ∈ B | Fs ] = P [X(t) ∈ B | X(s)] 2. P [A | Fs ] = P [A | X(s)] for A ∈ F 3.2.2.3. The Poisson process A stochastic process X(t), t ∈ [0, ∞) is a Poisson process with parameter λ if it satisfies the following properties: 1. X(0) = 1 w.p.1, i.e., the process starts at zero. 2. For 0 < t1 < t2 < · · · < tn , the increments {∆Xti } = {X(ti ) − X(ti−1 )}, i = 1, . . . , n, t0 = 0 are independent. 3. The increments of the process between any two points of time t and s (0 ≤ s < t) have a Poisson distribution, or P [W (t) − W (s) = k] =

[λ(t − s)]k [−λ(t−s)] e , k!

k ∈ N = {0, 1, 2, . . .}

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3.3. 3.3.1.

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Anastasios Xepapadeas

Stochastic Calculus Stochastic differential equations

In environmental and resource economics, dynamic analysis implies modeling the system under study as a dynamical system. In the deterministic case and when one state variable is studied, e.g., the biomass of a renewable resource X or the stock of a pollutant S, this can be obtained with the help of a deterministic differential equation. The evolution of environmental state variables can be modeled using deterministic differential equations for example as   X dX ˙ ≡ X = rX 1 − (1) , X(0) = X0 , or dt K   X bX 2 ˙ X = rX 1 − (2) − 2 K a + X2 S˙ = Z − mS + f (S),

S(0) = S0

(3)

where X is biomass, r is intrinsic rate of growth, K is carrying capacity, (bX 2 )/(a2 + X 2 ) is a non-linear predation term, S is stock, for example, of greenhouse gases (GHGs), Z is emissions of GHGs, m is environments absorbing capacity of GHGs, and f (S) a non-linear feedback term. These deterministic differential equations can be written in a general form as dX = µ(t, X), dt

or dX = µ(t, X)dt

This deterministic differential equation is extended to a stochastic differential equation,3 as dX (t) = µ(X(t), t)dt + σ(X(t), t)dW (t), or dX t = µ(t, Xt )dt + σ(Xt , t)dW t ,

X(0) = X0 given

X0 given

(4) (5)

where Wt is a Wiener process. In (4) the stochastic process X(t, ω) and the Wiener process W (t, ω) are defined for ω ∈ Ω and t ∈ T where (Ω, F , P ) a probability space. The stochastic differential Eq. (4) is called an Itˆ o 4 stochastic differential equation. 3 For

details of stochastic calculus see, for example Malliaris and Brock (1982), Oksendal (2003), and Chang (2004). 4 If we are dealing with a deterministic dynamical system, the corresponding concept in a stochastic environment is a system of stochastic differential equations.

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The Itˆo stochastic differential Eq. (4) can be interpreted heuristically as showing that in a small time interval ∆t, the change in Xt is an amount which is normally distributed with expectation µ(Xt , t)∆t and variance σ 2 (Xt , t)∆t. The change in Xt is independent of the past behavior of the process, since as we showed above the increments of a Wiener process are independent and normally distributed. The function µ is referred to as the drift coefficient, while σ is called the diffusion coefficient. The stochastic process Xt is called a diffusion process. The stochastic differential Eq. (4) can be associated with the stochastic integral equation  Xt − X0 =

0



t

µ(t, Xt )dt +

t

σ(t, Xt )dW t

0

(6)

t The integral 0 σ(t, Xt )dW t is an Itˆ o integral. A widely used example of the Itˆ o stochastic differential equation is the geometric Brownian motion defined as dX t = µXt dt + σXt dW t ,

X0 given

(7)

In terms of the environmental models presented in (1), (3), the corresponding Itˆ o stochastic differential equation would be   Xt (8) dX t = rX t 1 − dt + σ(Xt )dW t , X0 given K dS t = (Zt − mS t + f (St ))dt + σ(St , Zt )dW t ,

S0 given

(9)

Model (8) indicates that the expected rate of change of biomass is 2 t rX t (1 − X K ) and the variance of this change is σ (Xt ), while (9) indicates that the expected rate of change of the stock of GHGs is (Z − mS + f (S)) and the variance of this change is σ 2 (Xt , Zt ). Note that in the GHGs case the variance of this stochastic process is modeled to depend both on the stock and emissions of GHGs. 3.3.2.

Itˆ o’s lemma

Consider the integral equation  X(t) = X0 +



t

µ(s)ds + 0

0

t

σ(s)dW s

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which, under appropriate assumptions about the functions µ and σ, defines a stochastic process X(t) with continuous sample paths. It is said that the process X(t) possesses a stochastic differential on t ∈ [0, T ], defined as dX (t) = µ(t)dt + σ(t)dW (t) Itˆ o’s lemma is the basic stochastic calculus rule for computing stochastic differentials of composite stochastic (or random) functions. Itˆ o’s lemma: Let Y (t) = u(t, X(t)). Then the process Y (t) also has a differential on t ∈ [0, T ] given by:   1 ∂ 2 u(t, X) 2 ∂u(t, X) ∂u(t, X) + µ(t) + σ (t) dt dY (t) = ∂t ∂X 2 ∂X 2 ∂u(t, X) σ(t)dW (t) (10) ∂X If the function u does not explicitly depend on t, i.e., Y (t) = u(X(t)), then ∂u(t,X) = 0 in (10). If the stochastic differential of X(t) is given by ∂t +

dX = µ(X)dt + σ(X)dW (t) 5

and Y = u(X), Itˆ o’s lemma implies that   1 dY = u (X)µ(X) + u (X)σ 2 (X) dt + u (X)σ(X)dW (t) 2

(11)

Itˆo’s lemma can be obtained by direct computation using a second-order Taylor expansion of the function Y = u(X). For the case leading to (11) we have, using the second-order Taylor expansion 1 dY = u (X)dX + u (X)(dX )2 2 = u (X)[µ(X)dt + σ(X)dW (t)] 1 + u (X)[µ(X)dt + σ(X)dW (t)]2 2 Differential (11) is obtained by using the “multiplication table”: × dW dt 5t

is omitted to simplify notation.

dW dt 0

dt 0 0

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3.3.3.

65

Existence and uniqueness of solutions, dependence on parameters, and initial conditions

A solution X(t) of the stochastic differential Eq. (4) is a stochastic process X(t), which has stochastic differential (4) and which satisfies for all t ∈ [0, T ] the integral Eq. (6). The following result can be stated. Suppose that: 1. The functions µ(t, x) and σ(t, x) are measurable with respect to x ∈ R and t ∈ [0, T ]. 2. For x, y ∈ R and t ∈ [0, T ] there exists a constant K such that (a) |µ(t, x) − µ(t, y)| + |σ(t, x) − σ(t, y)| ≤ K|x − y| (b) |µ(t, x)|2 + |σ(t, x)|2 ≤ K(1 + |x|2 ) 3. X0 is given. Then, there is a solution X(t) of the stochastic differential Eq. (4) defined on [0, T ] which is continuous w.p.1, such that sup E[X 2 (t)] < ∞

[0,T ]

Furthermore this solution is pathwise unique, i.e., if X and Y are two solutions P [sup |X(t) − Y (t)| = 0] = 1 [0,T ]

Assume that the stochastic differential Eq. (4) depends on a parameter α, or dX (α, t) = µ(α, t, X(t))dt + σ(α, t, X(t))dW t

(12)

and that the functions µ and σ satisfy the conditions for existence and uniqueness of a solution, then under certain assumptions about the structure of the functions µ and σ with respect to α, the solution of the stochastic differential Eq. (4) depends on the parameter α and the initial condition ∂µ ∂σ , ∂α are continuous and bounded then X0 . Furthermore, if the derivatives ∂α ∂X(t,α) the derivative Y (t) = ∂α exists where X(t, α) is the solution of (12).

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The geometric Brownian motion

We clarify some of the above concepts using the geometric Brownian motion (7). A closed form solution of (7) is    σ2 X(t) = X0 exp µ − t exp[σW (t)] 2 The mean and the variance of the geometric Brownian motion are given by E[X(t)] = X0 eµt 2

var[X(t)] = X02 e2µt (eσ t − 1) 3.4.

Optimal Stochastic Control

Optimal stochastic control6 studies problems where the evolution of the stochastic process X(t) is influenced by another stochastic processes u(t) called a control process, which belongs to an appropriately defined control space U . For the Itˆ o stochastic differential Eq. (4), the control process can be introduced into the drift and the variance functions as dX (t) = µ(X(t), u(t), t)dt + σ(X(t), u(t), t)dW t ,

X(0) = X0 given (13)

In this case (13) is called a controlled stochastic differential equation, while the process X(t) which satisfies (13) is said to be a controlled Markov diffusion process. Given a finite time interval t ≤ s ≤ T the optimal stochastic control problem is to choose the control process u(t) to maximize (or minimize), subject to the dynamics imposed by (13), a criterion or objective defined as

 T L(X(s), u(s), s)ds + ψ(T, XT ) (14) J = Et t

where L and ψ are continuous functions satisfying appropriate assumption so that J is well defined. L is referred to as a running benefit (or cost) function and ψ is referred to as a terminal benefit (or cost) function. For 6 For

a detailed analysis of optimal stochastic control see, for example Kushner (1971), Fleming and Rishel (1975), Fleming and Soner (1993), Malliaris and Brock (1982), and Chang (2004).

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an infinite horizon problem with discounting the criterion to be maximized (or minimized) is defined as   ∞ ∞ −ρs e L(X(s), u(s), s)ds , ρ > 0 (15) J = E0 0

subject to (13). The solution to the optimal stochastic control problem is obtained by using Bellman’s principle of optimality to obtain the dynamic programming (DP) equation or the HJB equation. 3.4.1.

The HJB equation

Assume that in the finite time interval t ≤ s ≤ T the controller observes the states X(t) of the controlled process, and the initial data X0 ≡ x are known and given. In DP the supremum (or infimum), or maximum (or minimum) of the quantity J in (14) to be maximized is regarded as a function V (x, t) of the initial data (x, t). For any initial data V is defined as V (X, t) = sup J(t, X; u) u∈UC

where UC is the class of controls admitted. V is said to be the value function. Bellman’s principle of optimality states that

 t+h L(X(s), u(s), s)ds + V (X(t + h), t + h) V (X, t) = sup Et u∈UC

t

The principle of optimality leads to the DP or the HJB equation, which for the one-dimensional problem (13)–(14) can be written as7   ∂V ∂ 2 V ∂V (X, t) = H t, X, , (16) ∂t ∂X ∂X 2    ∂V ∂ 2 V ∂V , µ(X, u, t) H t, X, = max L(X, u, t) + u∈U ∂X ∂X 2 ∂X  1 ∂2V 2 + σ (X, u, t) (17) 2 ∂X 2 V (T, X) = ψ(T, XT )

(18)

The HJB equation is a non-linear partial differential equation of second order (17), with boundary condition (18). It can be shown (Kushner, 1971) 7 For

a proof see, for example, Fleming and Soner (1993).

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that if a value function which is twice continuously differentiable in x (or class C 2 ) exists then a solution to problem (13)–(14) is a solution to the HJB equation. Formally, the HJB equation is connected to the optimal stochastic control problem through the verification theorem (Fleming and Soner, 1993), which can be informally stated as follows. Let W (t, X) be continuously differentiable in t and twice continuously differentiable in X (or of class C 1,2 ). Let W (t, X) be a solution to (17)–(18). Then • W (t, X) ≥ J(t, X) for any admissible control u ∈ U and initial data (t, X). • If a u∗ (s) exists such that (19) below is satisfied  ∂V ∗ µ(X ∗ (s), u, s) u (s) ∈ arg max L(X ∗ (s), u, s) + ∂X  1 ∂2V 2 ∗ + σ (X (s), u, s) (19) 2 ∂X 2 then W (t, X) = V (t, X) = J(t, X; u∗ ) and u∗ (s) is optimal. X ∗ (s) is the solution of (13) corresponding to u∗ (·) with X ∗ (t) = X. Condition (19) suggests that the optimal feedback control should satisfy  ∂V µ(X, u, s) u∗ (s, X) ∈ arg max L(X, u, s) + ∂X  1 ∂2V 2 + σ (X, u, s) (20) 2 ∂X 2 3.4.1.1. Infinite time horizon Infinite time horizon problems are very often encountered in environmental and resource economics applications. In these problems it is assumed that the problem is time autonomous, which means that the drift and variance in the dynamics (13), and the running function in (14) do not explicitly depend on time. The optimal stochastic control problem is written as  ∞  e−ρt L(X(t), u(t))ds , ρ > 0 (21) max J ∞ = E0 u

0

s.t. dX (t) = µ(X(t), u(t))dt + σ(X(t), u(t))dW t

(22)

X(0) given

(23)

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The HJB equation for this problem is ρV (X) = H(X, V  (X), V  (X)) 



H(X, V , V ) =

(24) (25)

 1 max L(X, u) + V  µ(X, u) + V  σ 2 (X, u) u∈U 2

(26)

lim E[e−ρt V (Xt )] = 0

(27)



t→∞

with (27) being a transversality condition at infinity. The optimal feedback policy satisfies   1  2 ∗  u (X) ∈ arg max L(X, u) + V µ(X, u) + V σ (X, u) (28) 2 3.4.1.2. Environmental and resource economics modeling The set up presented above is very useful for modeling stochastic dynamic problems in environmental and resource management.8 The stochastic differential Eqs. (8), (9) can be regarded as a controlled stochastic process if we assume that a certain policy can be introduced as a control. Thus, if emissions Zt is regarded as a control variable in (9), and the net flow of benefits from emissions are defined as U (Zt , Xt ) = B(Zt ) − D(Xt ), where B(Zt ) is an increasing concave benefit function and D(Xt ) an increasing convex damage function, then the optimal stochastic emission control problem can be written, assuming in order to simplify the absence of non-linear feedbacks (f (St ) ≡ 0) and σ(St ) = σSt , as  ∞ max E0 e−ρt [B(Zt ) − D(Xt )]dt (29) Zt

0

subject to dS t = (Zt − mS t )dt + σSt dW t ,

S0 given

The HJB equation is   1 ρV (S) = max B(Z) − D(S) + V  (Zt − mS t ) + V  σ 2 S 2 Z 2 8 For

(30)

stochastic modeling in dynamic environmental and resource economic models see also, for example, Plourde and Yeung (1989), Clark (1990), Conrad (1999), and Olson and Roy (2000).

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B  (Z) + V  (S) = 0,

Z>0

(31)

The feedback emission rule implies that Z = h(V  (S)), which after substitution into (30), results into the HJB equation 1 ρV (S) = B(h(V  (S))) − D(S) + V  (S)(h(V  (S)) − mS t ) + V  σ 2 S 2 2 (32) which is a non-linear ordinary differential equation (ODE) of second order which should be solved with the transversality condition at infinity (27). If harvesting u(t) is introduced into (8), then under the assumption that harvesting affects both the drift and the variance of the biomass rate of change, the controlled stochastic differential equation can be written as     Xt (33) − ut dt + σ(Xt , ut )dW t , X0 given dX t = rX t 1 − K If the flow of profits is defined as: π(ut , Xt ) = R(ut ) − c(Xt )ut , where R(ut ) is a concave revenue function and c(Xt ) is a differentiable convex non-increasing unit cost harvesting function of the biomass stock, then the harvesting problem can be formulated as an infinite horizon optimal stochastic control problem:  ∞ e−ρt π(ut , Xt )dt , subject to (33) max E0 ut

0

The HJB equation for this problem is       X 1  2  ρV (X) = max R(u) − c(X)u + V rX 1 − − u + V σ (X, u) u K 2 (34) where the optimal feedback harvesting rule satisfies R (u) − c(X) − V  + V  σ(X, u)

∂σ = 0, ∂u

for u > 0

(35)

Assume, to simplify, that R (u) = u, and σ(X, u) = σX, then (35) becomes u = c(X) + V  (X). After substituting into (34), the HJB equation becomes ρV (X) = R(c(X) + V  (X)) − c(X)[c(X) + V  (X)]a + V  (X)     X 1  rX 1 − − c(X) − V (X) + V  (X)σ 2 X 2 K 2

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This is again a non-linear ODE of second order which should be solved with the transversality condition at infinity (27). 3.4.1.3. The multidimensional problem We generalize the previous results by assuming that the range of the processes Xt is a subset of the n-dimensional Euclidean space and that the control space is m-dimensional. The controlled stochastic differential equation system for an autonomous finite time horizon problem is dXs = µ(X(s), u(s))ds + σ(X(s), u(s))dWs ,

X(0) given

(36)

and the objective is  max J = E u



T

L(X(s), u(s))ds + ψ(X(T ))

(37)

t

where Ws is a normalized vector Wiener process with E[(Wt − Ws )(Wt − Ws ) ] = I(t − s) where I is the identity matrix. Define the matrix [Sij (X, u)] = S(X, u) = σ  (X, u)σ(X, u), i, j = 1, . . . , n. The operator n ∂ 1 ∂2 L = µi (X, u) + Sij (X, u) ∂Xi 2 i,j=1 ∂Xi ∂Xj i=1 u

n

(38)

is known as the differential generator of the controlled vector processes Xs (Kushner, 1971). The system of (36)–(38) has a well-defined solution for sufficiently smooth µ, σ, and u, so that the control u determines the process Xs . The HJB equation for this general problem is −

∂V (X,t) = max{L(X, u) + Lu V (X,t)} u ∂t

(39)

where Lu V =

n i=1

µi (X, u)

n ∂V 1 ∂2V + Sij (X, u) ∂Xi 2 i,j=1 ∂Xi ∂Xj

(40)

For the infinite time horizon problem, the HJB equation becomes ρV (X) = max{L(X, u) + Lu V (X,t)} u

(41)

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Linear-quadratic and CRRA models

A procedure for solving a stochastic optimal control problem would be to use the optimality condition (20) or (28) to derive the feedback optimal control as a function of the state variable, the value function, and its first and second derivatives, i.e., to derive u∗ = u∗ (X, V, V  , V  ), and then substitute this function into the HJB equation. Thus, the HJB equation turns out to be a non-linear partial differential equation (PDE), or a system of PDEs, with boundary conditions for the finite time case, or a non-linear ODE, or a system of ODEs, with transversality conditions at infinity. Obtaining closed form solutions of these non-linear equations is not possible in most of the cases. The solution is, however, facilitated if the functional form of the value function is known. Knowledge of the functional form of the value function is possible if the criterion of the problem and the controlled diffusion process characterizing the dynamics have a specific structure. Two of the most common structures which have been extensively used in economics are presented here. The first is the linear-quadratic model (LQM) where the criterion is quadratic and the dynamics linear in the state and the controls, and the CRRA model, where the criterion is a function exhibiting constant relative risk aversion (CRRA)9 and the dynamics are linear in the state and the controls. 3.4.2.1. A linear-quadratic problem The solution of an LQM problem is presented using example (29). Let the benefit function be B(Z) = α0 − 12 α2 Z 2 , and the damage function be D(S) = 12 γS 2 , (α0 , α2 , γ) > 0. In this problem, Z is to be interpreted as ¯ so that negative values derivations from a base benchmark emission level Z, ¯ Then, the HJB for Z are interpreted as reductions from benchmark level Z. equation will be   1 1 2 1  2 2 2  ρV (S) = max α0 − α2 Z − γS + V (Zt − mSt ) + V σ S Z 2 2 2 We try a quadratic function form for the value function or V (S) = A0 + AS 2 . Then, the optimal feedback emission rule should satisfy 2A α2 Z = V  (S), or Z = S α2 9 CRRA

functions belong to the class of hyperbolic absolute risk aversion models (HARA). HARA models, when combined with linear dynamics, result in value functions with generally known functional forms.

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which is a linear feedback rule. Substituting this rule into the HJB equation, we obtain   2A 1 S − mS + Aσ 2 S 2 , or ρ(A0 + AS 2 ) = α0 − (AS )2 − γS 2 + 2AS 2 α2    4 1 2 2 0 = α0 − ρA0 + − 1 A + (σ − 2m − ρ)A − γ S 2 α2 2 (42) Since (42) should be satisfied for all values of S, the unknown parameters (A0 , A) of the value function should satisfy α0 − ρA0 = 0, 

or A0 =

α0 ρ

 4 1 − 1 A2 + (σ 2 − 2m − ρ)A − γ = 0 α2 2

(43) (44)

Let the negative real root of (44) be AN , then the optimal emission rule in N a feedback form will be Z ∗ = 2A α2 S indicating the optimal reductions from ¯ Substituting the optimal emission rule the benchmark emission level Z. into the stochastic differential equation for the emission stock we obtain the geometric Brownian motion   2AN − m St dt + σSt dW t , S0 given dS t = α2 with solution  St = S0 exp

2AN σ2 −m− α2 2

  t exp[σWt ]

3.4.2.2. A CRRA problem One of the most well-known examples which use a CRRA utility function in the criterion is Merton’s model of wealth allocation between consumption and investment in a riskless and a risky asset in the absence of transaction costs (e.g., Kamien and Schwartz, 1991, Chap. 21). Following standard notation let W : total wealth, w: fraction of wealth invested in a risky asset, r: return on the riskless asset, µ expected return on the risky asset, σ 2 : variance per unit time of return on risky asset, c: consumption, U (c) = cb /b, 0 < b < 1 a CRRA utility function with 1 − b the degree of relative

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risk aversion, and zt a Wienner process. The change in wealth is described by the following controlled stochastic differential equation dW t = [r(1 − wt )Wt + µwt Wt − ct ]dt + wt Wt σdz t ,

W0 = W 0

(45)

The objective is to determine time paths {ct , wt } for the controls c and w, in order to solve  b  ∞ ct −ρt e max dt (46) c,w b 0 subject to (45)

(47)

The HJB equation for this problem is 

cbt + V  (W )[r(1 − wt )Wt + µwt Wt − ct ] c,w b  1 2  + (wt Wt σ) V (W ) 2

ρV (W ) = max

(48)

The optimal consumption policy c and investment policy w should satisfy 1

c = [V  (W )] b−1 ,

w=

V  (W )(r − µ) σ 2 WV  (W )

(49)

We try the functional form V (W ) = AW b for the value function. Substitution of this functional form into (48) and (49) results, after some simplifications, in

Ab =

(r−µ)2 2σ2 (1−b) b)b−1

ρ − sb − (1 −

(50)

Thus, the unknown parameter A of the value function is determined by (50). Substituting the value function into (49), we obtain the optimal feedback consumption and investment policies as 1

c∗t = Wt (Ab) b−1 ,

w∗ =

µ−r (1 − b)σ 2

The optimal policy rule indicates that consumption is a constant proportion of current wealth, and that the fraction of wealth invested in the risky asset is constant over the time horizon.

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3.5.

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Irreversibility and Option Value

The interaction between uncertainty and irreversibility is an issue that has received considerable attention in the literature of finance and investment and subsequently in the environmental and resource economics literature.10 One fundamental proposition, established by Arrow and Fisher (1974) in the area of environmental economics, is that an option values exists associated with refraining from an irreversible decision now, when next period benefits or losses due to the decision are uncertain, even if the decision maker is risk neutral. Closely associated with the above concepts is the concept of timing of the irreversible decision, and the question of whether the decision maker should postpone action until more information is acquired in the future. Recent approaches to the solution of this type of problem focus on the derivation of a free boundary derived by the solution of the associated HJB equation. The underlying intuition behind the free boundary concept in decision making under uncertainty can be described in the following way. If the decision about undertaking an irreversible action depends on the value of a parameter that evolves stochastically in time, then there will be a critical value of this parameter such that it will be optimal to undertake the irreversible action when the observed value of the parameter is on the one side of the critical value, and not to undertake the irreversible action when the observed parameter value is on the other side of the critical value. The curve that determines this critical value for any point in time is the free or exercise boundary. Thus, the basic property of the boundary is that it divides a certain strategy space into two regions. Depending on the region of the space in which a stochastic variable is realized, the decision maker decides whether or not to undertake the irreversible action. In this section, we present this approach by deriving the free boundary for the problem of irreversible development of an environmental resource under uncertainty (Xepapadeas, 1998). We consider an environmental asset of fixed size L which can be developed into a new use. In the existing use, the asset has an environmental or intrinsic value B(L). The asset could be, for example, a scenic land that can potentially undergo tourist development. In the undeveloped stage, the land provides benefits, for example, as an ecosystem or in terms of sight-seeing or hiking. 10 See Pindyck (1991, 2002), Dixit and Pindyck (1994), and Scheinkman and Zariphopoulov (2001).

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Assume that there is a potential developer of the site, and at each point in time he/she develops h(t) ≥ 0. Thus, total cumulative development at time t is defined as  t hi (s)ds, D(t) ≤ L (51) D(t) = 0

Since hi (t) ≥ 0 development is irreversible. After the development, the environmental value of the undeveloped asset is defined as B(L − D(t))

with BD < 0,

BDD < 0

Thus, the development of the site reduces its intrinsic value at an increasing rate. However, the development generates a net flow of services, say tourist services, according to an increasing and strictly concave function: f (D(t)),

f  (D(t)) > 0,

f  (D(t)) < 0

Uncertainty is introduced into the model by assuming that the market price of the developed resource evolves stochastically according to a geometric Brownian motion: dP (t) = aP (t)dt + σP (t)dz (t),

P (0) = P0

(52)

where {z(t)} is a Wiener process, and a and σ are constants. Thus, the developer’s net revenues at each instant of time are defined as P (t)f (D(t)) The cost of developing one unit of the resource is assumed to consist of two parts. One part, c, is fixed, while the other part depends on the amount of the resource already developed up to this time, and is given by φ(D(t)), φDi > 0, φDD > 0. Thus, the cost associated with the existing development can be regarded as some kind of stock effect in the cost function. More development reduces, for example, the available supply of the land if we are examining tourist land development, and drives land prices up. It also increases operational costs through the increased demand for skilled personnel. These costs are assumed to be convex. Consider the decision to develop the site by ∆D = D0+ − Do from the existing development level Do . Then, D0+ = Do + ∆D. The cost of this change in the development is then defined as c(D0+ − Do ) + φ(D0+ )

(53)

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Given this model, we examine the optimal development strategy for the developer which takes the form of a free boundary, P = P (D), relating price and development. When observed price Pob < P (D), no development is undertaken, while when Pob > P (D) enough investment is undertaken to restore equality on the boundary. Given a discount rate ρ, the developer seeks the non-decreasing process D(t), which will maximize the present values of net revenues less the cost of development. The value function associated with this problem can be written as  ∞ e−ρt P (t)f (D(t))dt (54) V (D, P ) = max E0 D

0

subject to (51) and (52) By the concavity of f (D), the value function is concave in D. At each instant of time the developer has two choices: to preserve the site or to develop. The time interval, when no development is taking place and the previously acquired development is used to generate net revenues, can be defined as the continuation interval. A stopping time is defined as time τ at which development is undertaken. Let D∗ (τ ) be the optimal development process at time τ . If τ is a stopping time, then  τ e−ρu P (u)f (D(u))du + e−ρτ V (D∗ (τ ), P (τ )) V (D, P ) = max E0 D

0

(55) Assume that in the time interval [0, θ], the developer undertakes no new development, but keeps it constant at the level Do . By the principle of DP, the value function should be no less than the continuation payoff in the interval [0, θ], plus the expected value after θ, or  V (D, P ) ≥ E

θ

e−ρu P (u)f (D)du + e−ρθ V (D(θ), P (θ))

(56)

0

with equality if Do is the optimal policy in [0, θ]. Applying the Itˆ o lemma to the value function on the right-hand side of (56), dividing by θ and taking limits as θ → 0, we obtain ρV ≥

1 2 2 σ P VPP + aPV P + Pf (D) 2

with equality if D(t) = Do in the interval [0, θ].

(57)

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Consider now the decision to develop instantaneously by ∆D = D0+ − Do . Then from the definition of the optimal stopping time, we have V (D, P ) ≥ E[V (D0+ , P ) − c(D0+ − Do ) − φ(D0+ )]

(58)

Since the value function is concave in D, the optimal development can be obtained by maximizing the right-hand side of (58). Assuming that constraint (51) is not binding, we obtain the necessary and sufficient conditions for the optimal choice of development as VD (D, P ) − c −

∂φ(D) ≤ 0, ∂D

with equality if ∆D > 0

(59)

Solution of (59) for the interior solutions determines equilibrium value of development. Thus when no development is optimal, (57) is satisfied as equality, while when development is optimal, (59) is satisfied as equality. Combining (57) and (59), the HJB equation can be written as11     1 ∂φ(D) min ρV − σ 2 P 2 VPP + aPV P + Pf (D) , − VD − c − =0 2 ∂D (60) We examine the optimal policy, i.e., the structure of the free boundary. The optimal free boundary will divide the (P, D) space into two regions: the no development region, say region I, and the development region, say region II. In region I the first term of the HJB Eq. (60) is zero, since ∆Di = 0, and the second term of the HJB equation is positive by (59), thus 1 ρV − σ 2 P 2 VPP + aPV P + Pf (D) = 0 2

(61)

The general solution of (61) can be obtained as V (D, P ) = A1 (D)P β1 + P where δ = ρ − a and β1 = 12 − σa2 + root of the fundamental quadratic Q= 11 See

f (D) δ

( σa2 − 12 )2 +

2ρ σ2

(62) > 1 is the positive

1 2 σ β(β − 1) + aβ − ρ = 0 2

Dixit and Pindyck (1994) for the procedure for deriving the HJB equation.

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In region II, the second term of (60) is satisfied as zero and ∆D > 0 or VD (D, P ) − c −

∂φ(D) =0 ∂D

(63)

Using (62) and (63), the unknown functions A1 (D) and P = P (D) can be determined. To do this the “value matching” and the “smooth pasting” conditions are used.12 The value-matching condition means that on the boundary separating the two regions the two value functions should be equal. Solving (62) for P , we can obtain the yet unspecified function for the boundary P = P (D). Then we have, combining (62) and (63) and substituting for P ∂A1 (D) β1 f  (D) P +P ∂D δ ∂φ(D) = c+ , P = P (D) ∂D

VD (D, P ) =

(64)

The smooth-pasting condition means that the derivatives of the value functions with respect to P on the boundary are equal or ∂A1 (D) β1 −1 f  (D) P =0 + ∂D δ with P = P (D)

VDP (D, P ) = β1

(65)

Combining (64) and (65), we can solve for the unknown functions P (D) 1 (D) to obtain and ∂A∂D   ∂φ(D) δ c + ∂D β1 (66) P (D) = β1 − 1 f  (D)  β1 −1  β ∂A1 (D) f  (D) 1 β1 − 1 =− (67) ∂D β1 δ c + ∂φ(D) ∂D

Relationship (66) is the equation of the free boundary for the developer. A more clear interpretation can be given to the equation of the boundary by writing (66) as   β1 P (D)f  (D) ∂φ(D) = c+ δ β1 − 1 ∂D 12 For a presentation of these conditions see, for example Pindyck (1991) and Dixit and Pindyck (1994).

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The left-hand side of this expression is the expected value of the revenue flow from the marginal development unit, discounted at the rate δ = ρ − a, where a is the expected rate of price change and ρ is the discount rate for is the cost of future revenues. On the right-hand side the term c + ∂φ(D) ∂D the marginal development unit. Thus, marginal development is justified if the present value of the incremental revenue of the development exceeds its incremental costs by the factor (β1 )/(β1 − 1). Since β1 > 1, this term exceeds unity and reflects the cost of waiting. Define the expected present value of a marginal development unit as:
E[PV ] = ( Pf δ(D) ), and define the incremental cost of this development as ∆C = c + ∂φ(D) ∂D , then the benefit–cost ratio for this investment is defined ] as BC = E[PV ∆C . As indicated by the equation of the boundary, the optimal investment rule requires β1 >1 BC = β1 − 1 The fact that the benefit–cost ratio for the marginal project exceeds one, as compared to the traditional rule of BC = 1, reflects the option values of keeping the status quo development level. Thus, more benefits are required from the marginal development unit in relation to its cost when the option values of waiting is taken into account, relative to the case when such an option values is ignored. By Eq. (64) the incremental development is justified if the discounted
value of the incremental development marginal value product, Pf δ(D) , covers i (D) development costs, c + ∂φ∂D , plus the opportunity cost of the option to i

1 (D) 1 (D) wait, ∂A∂D P β1 . By (67), the marginal option values ∂A∂D is negative; i.e., it represents a cost. The evolution of the development policy can be described as follows. In region I, (61) is satisfied as strict inequality and no development is undertaken. For any given D, random price fluctuations move the point (D, P ). If the point crosses the boundary and enters region II, then development is immediately undertaken so that the point shifts back on the boundary. Thus, optimal development proceeds gradually. In the terminology of Dixit and Pindyck (1994), this is a barrier control policy.

3.6.

Non-expected Utility Models

Optimal stochastic control methods in economics are used as a tool for the study of the fundamental issue of analyzing decision making and

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of predicting and explaining behavior. The basic assumption is that an individual facing a decision problem under uncertainty should maximize expected utility subject to probabilities beliefs, which are given by a single probability measure defined over the state space. This measure is the Bayesian prior which, when new information arrives, should be updated to a posterior according to Bayes’s law. Expected utility maximization subject to a stochastic differential equation, as modeled in optimal stochastic control, expresses this assumption. This implies that the basic framework of analyzing decision making under risk and uncertainty is the subjective expected utility maximization (SEU) of Savage (1954) where the probability measure is known by the decision maker. There is however evidence, which is associated with the Allais and Ellsberg paradoxes, suggesting that decision makers do not behave in a way which is consistent with SEU. These ideas can be associated with the concept of uncertainty discussed by Knight (1921). Knight made the distinction between risky events, for which a true probability measure can be specified, and a worse type of ignorance where a unique probability measure is not available, which he called uncertainty. Gilboa and Schmeidler (1989), motivated by Elsberg paradox, formulated, in an atemporal setting, a set of appropriate axioms and incorporated the idea of uncertainty or ambiguity aversion into decision making, thus formally introducing a way of modeling decision making under uncertainty which is not based on expected utility maximization. Dynamic non-expected utility maximization models in which agents are adverse to model ambiguity have been constructed by Epstein and Wang (1994) and Chen and Epstein (2002).13 In the recent literature we can distinguish two main, although interrelated, approaches for dealing with ambiguity: the multi priors and the robust control approaches. 3.6.1.

Uncertainty aversion and multiple priors

Let the sample space Ω, and consider an individual observing some realization ωt ∈ Ω. The basic idea underlying the multiple priors approach is that beliefs about the evolution of the process {ωt } cannot be represented by a probability measure. Instead, beliefs conditional on ωt are too vague to be represented by such a single probability measure and are represented by a set of probability measures (Epstein and Wang, 1994). Thus for each 13 For

a recent review of non-expected utility maximization models in environmental and resource economics, see Shaw and Woodward (2008).

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ω ∈ Ω, we consider P(ω) as a set of probability measures about the next period’s state. Formally P is a correspondence P : Ω → M(Ω) assumed to be continuous, compact valued and convex valued and M(Ω) is the space of all Borel probability measures. The individual ranks uncertain prospects or acts which are denoted by α. Let u be a von Neumann–Morgenstern utility function. The utility of any act α in an atemporal model is defined as  (68) U (α) = min u(α)dQ Q∈P

In continuous time framework, recursive multiple priors utility, in a finite time setting, is defined as   T

Vt = min EQ Q∈P

e−β(s−t) u(α)ds

(69)

t

where the subjective set of priors P on a space (Ω, F ), is uniformly absolutely continuous with respect to Q ∈ P.14 These definitions of utility in the context of multiple priors corresponds to an intuitive idea of the “worst case”. Utility is associated with the utility corresponding to the least favorable prior (LFP). With utility defined in this way, decision making by using the maxmin rule follows naturally, since maximizing utility in the multiple priors case implies maximizing the utility which corresponds to the least favorable prior. Analyzing decision making according to this maxmin rule constitutes a departure from expected utility maximization. The individual’s set of priors can be further specified for the purposes of the analysis by the so called k-ignorance approach. In this case, the individual considers the reference probability measure P and another measure Q ∈ M(Ω). The discrepancy between the two measures is defined by the relative entropy    +∞ 1 e−δt EQ ε2t dt (70) R(Q//P ) = 2 0 where ε is a measurable function associated with the distortion of the probability measure P to the probability measure Q. According to the k-ignorance 14 Uniformly absolutely continuous means that for every ε > 0 there is a δ > 0 such that EF and Q(E) < δ implies that P (E) < ε ∀P ∈ P.

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approach the individual incorporates into her/his decision-making problem the instantaneous relative entropy constraint Q(τ ) = {Q : EQ [ 12 ε2t ] ≤ τ , for all t}, which means that probability measures differing from the reference measure P by at least as much as τ should be taken into account. If Q is a probability measure associated with the least favorable outcome, then k-ignorance embodies worst case scenario decision making.15 3.6.2.

Robust control methods

Another way to embody decision makers’ ambiguity and in particular their concerns about model misspecification, is to use robust dynamic control, which is also a minmax approach, that is a non-expected utility maximization approach, which has been introduced in economics by Hansen and Sargent (see, for example, Hansen and Sargent, 2001). In this case the decision maker suspects that his/her model is misspecified, in the sense that there is a group of approximate models that are also considered as possibly true given a set of finite data. These approximate models are obtained by disturbing a benchmark model, and the admissible disturbances reflect the set of possible probability measures that the decision maker is willing to consider, or otherwise how ambiguous the decision maker is about the initial estimated model. The objective of this approach is to choose, using a minmax criterion formulated in terms of a differential game where one agent is “nature” that “chooses” the LFP, a rule that will work well under a range of different model specifications. The robust control method which can be regarded as an approach for deriving optimal dynamic policy rules under model uncertainty is presented below using the wealth allocation model of Sec. 3.4.2.2.16 Following Hansen and Sargent, model (45) is regarded as a benchmark model. If the decision maker was sure about this benchmark model then there would be no concerns about robustness to model misspecification. Otherwise, concerns for robustness to model misspecification can be 15 Another way to specify the set of priors is the so-called e-contamination approach (Epstein and Wang, 1994), where the set of priors is a convex combination of the probability measure P and the measure Q. Thus

P
= {(1 − )P + Q : Q ∈ M(Ω),  ∈ [0, 1]} 16 The

application follows Vardas and Xepapadeas (2007). For an application to a water management problem which requires, however, a more complex framework of analysis which is beyond the purpose of this chapter, see Roseta-Palma and Xepapadeas (2004).

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reflected by a family of stochastic perturbations, so that the probabilities implied by (45) are distorted. The measure P is replaced by another probability measure Q. The perturbed model is constructed by replacing zt in (45) with  zt = zˆt +

t

h(s)ds

(71)

0

where {ˆ zt : t ≥ 0} is a Brownian motion and {h(t) : t ≥ 0} is a measurable drift distortion. Therefore using Eq. (71), the corresponding Eq. (45) for wealth dynamics becomes dWt = [wt (µ − r + σh)Wt + (rWt − ct )]dt + W σwd zˆt

(72)

The discrepancy between the distribution P and Q is measured as the relative entropy:  2  ∞ h −ρu eQ E R(Q) = du (73) 2 0 Under model misspecification, a multiplier robust control problem can be associated with the problem of maximizing the present value of lifetime expected utility, or  ∞ e−ρt U (ct )dt max E0 wt ,ct

0

In this case, the multiplier robust control problem becomes    ∞ h2 −ρt e U (c) + θ J(θ) = sup inf EQ dt 2 w,c h 0

(74)

subject to (72). In criterion (74) θ is the so-called robustness parameter which is the Lagrangian multiplier at the optimum, associated with the entropy constraint Q(τ ) = {Q ∈ Q : R(Q P ) ≤ τ } and takes values greater than or equal to zero. A value of θ = ∞ indicates that we are absolutely sure about the measure P , with no preference for robustness or concerns about model misspecification, or equivalently no ambiguity aversion. This case can be regarded as the risk aversion case and the problem is reduced to the standard Merton problem with the objective function given by (46). Lower values for θ indicate preference for robustness under model misspecification, or uncertainty (ambiguity) aversion,

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where a value of θ = 0 indicates that we have no knowledge about the measure P . Using the results of Fleming and Souganidis (1989) regarding the existence of a recursive solution to the multiplier problem, the problem (74) can be transformed into a stochastic infinite horizon two-player game between the investor and Nature. Nature plays the role of a “mean agent” and chooses a reduction h in the mean return of assets to reduce the investor’s lifetime utility. The Bellman–Isaacs conditions for this game imply that the value function V (W, θ) satisfies the following equation  h2 + [w(µ − r + σh)W + rW − c]VW ρV (W, θ) = max min U (c) + θ w,c h 2  1 2 2 2 (75) + W σ w VW W 2 The solution of the above problem is characterized by the following equations U  (c) = VW w∗ σVW W θ (r − µ) 1 VW   w∗ = 2 σ W VW W 1 − VW2 θVW W h=−

(76) (77)

U  (c) VW =  VW W U (c)(∂c/∂W ) where w∗ denotes the fraction of wealth invested in the risky asset when there are concerns about model misspecification and the decision maker tries to find robust decision rules. If we compare w and w∗ given by (49) and (77) respectively, we have — because of the concavity of the value function with respect to wealth — that 2 w1 VW =1− >1 ∗ w1 θVWW

(78)

Thus independently of the utility function and the value of the robustness parameter, concerns for model misspecification decrease the fraction of wealth invested in the risky asset relative to the standard Merton case. Moreover as, θ → ∞, the robust portfolio weight tends to Merton’s optimal

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weight and the utility maximizer acts as if he/she knows the initial benchmark model with certainty. For the LQM model presented in Sec. 3.4.2.1, a robust control approach implies that the corresponding Issacs–Bellman condition (75) can be written as  1 1 h2 + VS (Z − mS + σhS ) ρV (S, θ) = max min α0 − α2 Z 2 − γS 2 + θ Z h 2 2 2  1 + S 2 σ 2 VSS 2 with optimality conditions Z=

VS (S, θ) , α2

h=

VS (S, θ)σS θ

The optimal policy could be specified by an appropriate choice of a functional form for the value function. 3.7.

Summary

This chapter presented tools for analyzing risk and uncertainty in dynamic models of environmental economics and resource management. Starting with the fundamental concepts of probability spaces, random variables, and stochastic processes, the chapter moves to stochastic differential equations and stochastic calculus. Optimal stochastic control, which is the core of dynamic models of environmental economics and resource management follows, with some examples related to the way to solve the HJB equation for environmental management and investment problems. The optimal stochastic control methodology is further extended to cover issues of option values and optimal timing in the management of natural resources. The last part introduces non-expected utility models, which is a departure from the expected utility paradigm, and can be used to analyze uncertainty in the Knightian sense and study cases where model or scientific uncertainty prevails.

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Chapter 4

Game Theory: Static and Dynamic Games Hassan Benchekroun∗ and Ngo Van Long† ∗

4.1.

McGill University, Canada [email protected] † [email protected]

Introduction

The concepts and techniques of static and dynamic games have been important tools in the analysis of many strategic issues in natural resource and environmental economics. While formal analysis of static games began with von Neuman (1928), dynamic games was introduced to economics by Roos (1925; 1927). The main difference between static games and dynamic games is that in a dynamic game, economic agents (players of the game) operate in an environment that changes over time. The environment in a dynamic game is represented by one or several state variables (such as a stock of pollution, a biomass) and a description of how actions taken by players affect the evolution of these variables. In analyzing any problem of strategic interactions, generally one is well advised to begin with the simplest model. This often means that one should, as a first step, abstract from dynamic considerations. Static game theory is rich enough to shed lights on many issues of interest. On the other hand, many problems in resource and environmental economics are intrinsically temporal problems, and eventually the temporal dimension must be taken into account. This is why dynamic game models are often discussed in scientific journals in this field. In Sec. 4.2, we illustrate the usefulness of static games in environmental economics, focusing on two issues, namely (i) international environmental 89

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agreements, and (ii) networks and R&D cooperation in an international oligopoly. In Sec. 4.3, we turn to dynamic games. We briefly explain various equilibrium concepts in dynamic games, and provide a number of illustrations in natural resources and environmental economics. 4.2.

Static Games

Static game theory has proven to be an essential tool to model environmental and resource problems where a few players interact strategically. There are excellent comprehensive reviews of applications of game theory in environmental economics and management. We refer the reader to the influential books of Cararro and Sinisalco (1997), Batabyal (2000), Finus (2001), and the survey in Wagner (2001). We focus here on two applications of game theoretical tools related to the formation and sustainability of cooperation in pollution games. In the first part we review emission games and abatement games, and the comprehensive analytical treatment by Rubio and Ulph (2006) of the influential model of international environmental agreements initiated by Barrett (1994). In the second part we consider an application of network formation theory to the case of R&D cooperation between polluting oligopolists, as proposed by Benchekroun and Claude (2007). 4.2.1.

Self-sustaining international environmental agreements: An analytical treatment

We begin with two non-cooperative games: an emission game and an abatement game. Then, we turn to the question of how cooperation can be achieved and an analysis of stable coalitions. 4.2.1.1. An emissions game Consider a world consisting of N countries i = 1, . . . , N . A strategy for country i is a non-positive level of emissions qi ≥ 0. The payoff of country i is given by b 1 πi (qi , Qi ) ≡ aq i − qi2 − (qi + Q−i )2 2 2 where Q−i ≡ Σk=i qk and a and b are two positive parameters. The term aq i − 2b qi2 can be interpreted as the gross benefit from consumption and the

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term 12 (qi + Q−i )2 can be interpreted as the environmental damages each country suffers from the total emissions Q. Note that since the marginal damage from emissions is normalized to one, a large value of b represents a small marginal benefit or a large marginal damage cost. When countries choose their emissions simultaneously, country i considers Q−i as given and chooses qi to maximize πi (qi , Qi ). The unique Nash equilibrium is a qi = q ∗ = b+N and the equilibrium payoff is π∗ =

1 (−N 2 + 2N + b) 2 a 2 (N + b)2

Remark: It can easily be checked that the level of emissions that maximizes world welfare is a a qc = < q∗ = b + N2 b+N and that welfare under cooperation is higher than Nash equilibrium welfare πc = πc − π∗ = =

1 a2 1 (−N 2 + 2N + b) 2 > π∗ = a 2 2N +b 2 (N + b)2 1 (−N 2 + 2N + b) 2 1 a2 − a 2 2N +b 2 (N + b)2 1 N 2 (N − 1)2 a2 2 (N 2 + b)(N + b)2

Note that the denominator of the above expression is a strictly increasing function of the parameter b and thus the gains from cooperation is a decreasing function of b. It is when b is close to zero that the gains from cooperation are most substantial. The above game, analyzed in Rubio and Ulph (2006), is a game of pollution emissions. This is in contrast with the abatement emissions’ game analyzed in Barrett (1994). 4.2.1.2. An abatement game In an abatement game, the strategy of country i is its level of abatement qi and the payoff function of country i is given by
 b Q2 πi (qi , Q−i ) = aQ − − cq 2i N 2

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where Q ≡ Σqi and where a, b, and c are positive parameters. The term 2 b(aQ − Q2 )/N represents the benefit of country i from a world abatement effort Q, whereas the term cq 2i represents the cost to country i of exerting an abatement effort qi . Thus, the marginal benefit of the first unit of abatement is (ab)/N . This game has a unique Nash equilibrium given by q∗ =

a N (1 + γ)

where γ ≡ c/b and the Nash equilibrium payoff is π∗ = b

(N − 2γ + 2N γ) a2 2 (γ + 1) 2N 2

There exists a correspondence between the emissions game and an abatement game (see Appendix 1 in Rubio and Ulph (2006) or Diamantoudi and Sartsetakis (2006)). Consider N identical countries and assume that each country hosts a polluting industry. In the absence of abatement activities, each country generates a level of pollution e¯ and suffers from the global level of emissions. Countries can abate pollution. Abatement by country i is denoted as qi and corresponds to e¯ − ei . Total abatements are given by e − ei ) = N e¯ − E where E = Σi ei . Abatement is a public good Q = Σi (¯ shared by all countries and its benefit is given by
 b (N e¯ − E)2 Bi (E) = a(N e¯ − E) − N 2 The cost of the abatement level e¯ − ei is Ci (ei ) =

c (¯ e − ei )2 2

Thus, each country’s payoff is given by b 2 c b(N e¯ − a) E− E + πi (¯ eei − e2i + e) πi = c¯ 2 N 2N Barrett argues that global emissions may not be smaller than N e¯ − a since the marginal abatement is zero at a. In fact, the benefit from abatement reaches its maximum level when all emissions N e¯ are abated. Given the functional form of abatement, the marginal abatement is zero when total abatement is a. These observations imply that when a units are abated all pollution is abated, i.e., a = N e¯ or e¯ = a/N .

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Thus, each country’s payoff becomes πi =

a ca c b 2 ei − e2i − E + πi N 2 2N N

which can be rewritten as 
a b B cB a B N cB 2 1 2 πi = ei − ei − E + πi N bB 2bB 2 N where B is used to refer to the parameters in Barrett (1994). A change of notation gives 
a bB b 2 1 2 πi = ae i − ei − E + πi N 2 2 N where a=

cB a B = γB aB , bB

b=

N cB = γB N bB

Thus, the objective function of an abatement game is an increasing affine transformation of the objective function considered in an emissions game. 4.2.1.3. Sustaining cooperation in a non-cooperative world Since non-cooperative equilibria are in general inefficient, there is room for improvement over a non-cooperative equilibrium outcome. It is then natural for the members of a subgroup of the players to consider coordination of their respective strategies to improve on their non-cooperative equilibrium payoff. Cooperation between two or more countries is called an International Environmental Agreement (IEA). In the absence of a supranational authority, any agreement to improve on the Nash equilibrium outcome must be self-sustaining. An IEA game is a metagame where an emissions game or an abatement game is preceded by an initial stage where countries decide whether to join a coalition or not. An IEA consisting of M ≤ N members chooses a vector of M strategies, one for each coalition member, that maximizes the sum of the payoffs of the coalition members. The case where M = N is called the grand coalition. There are several criteria of stability that have been used in the theory of the formation of coalitions. The predominant criterion of stability used in the IEA literature is that of internal and external stability (Carraro and

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Siniscalco, 1993; Barrett, 1994; Na and Shin, 1998 among many others). This criterion is borrowed from the theory of cartel formation and the seminal paper of d’Aspremont et al. (1983). When a player considers the gain from defection, she/he assumes that all remaining countries in the IEA would continue to cooperate and choose their respective strategies to maximize their joint welfare. Let π s (M ) and π ns (M ) denote the equilibrium payoffs of the signatory and non-signatory countries when the number of signatories is M ≤ N . A given IEA with M members is said to be internally stable if no inside member wants to leave the IEA, i.e., π s (M ) ≥ π ns (M − 1) and externally stable if no outside member wants to join the IEA, i.e., π ns (M ) ≥ π s (M + 1) An IEA is stable if and only if it is both internally and externally stable. Once an IEA forms, two scenarios are possible: (i) the IEA members are assumed to act simultaneously with the other non-signatories, or (ii) the IEA members are assumed to have the ability to move first and announce their emissions policies before the non-signatories choose their respective emissions policies. The predominant strand of literature assumes the latter scenario, i.e., the IEA members play a leadership role in the emissions game which follows the membership game. We report here the analysis of the case where signatories move first, as in Rubio and Ulph (2006). This game is solved using backward induction. We first determine the reaction function of the non-signatories and determine the optimal strategy of the signatories. Suppose the first M countries adhere to the IEA (and the remaining N − M countries are non-signatories). The problem of a non-signatory country k > M is
 b 2 1 2 max aq k − qk − (qk + Q−k ) qk ≥0 2 2 The reaction function of country k is thus   a − Q−k qk = max ,0 b+1 The signatories choose their emissions to maximize the sum of their payoffs, taking into account the reaction function of the non-signatories:  M
 b 2 1 2 max aq i − qi − (qi + Q−i ) q1 ,...,qM ≥0 2 2 i=1

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 qk = max

a − Q−k ,0 b+1

 for k = N − M + 1, . . . , N

Since countries are identical, we symmetry assumption:
b max M aq s − qs2 − qs ≥0 2

can rewrite this problem using the 1 (Mq s + (N − M )qns )2 2



subject to  qns = max

a − Mq s ,0 b+N −M



where qs and qns denote respectively the emissions of a signatory and a non-signatory country. Rubio and Ulph (2006) show that there are three possibilities. Let g(b, M ) ≡ b2 − (N − M )(M − 2)b + (N − M )2 and h(b, M ) ≡ b2 + (N + M 2 − 2M )b − (N − M )M (i) We have interior solutions if and only if g(b, M ) > 0 and h(b, M ) > 0. The interior solutions are given by qs =

ag(b, M ) bω

qns =

ah(b, M ) bω

and

where ω ≡ ((b + N − M )2 + bM 2 ) The equilibrium payoffs are given by a2 π (M ) = 2b s


bN 2 1− 2 ω

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π ns (M ) =

a2 2b


(b + 1)N 2 (b + N − M )2 1− ω2

(ii) If g(b, M ) ≤ 0, then we have a signatory corner solution, i.e., qs = 0 and qns =

a b+N −M

with equilibrium payoffs π s (M ) = −

a2 (N − M )2 2(b + N − M )2

and π ns (M ) =

a2 (b − (N − M )(N − M − 2))2 2(b + N − M )2

(iii) If h(b, M ) ≤ 0, then we have a non-signatory corner solution, i.e., qns = 0 and qs =

a M

with equilibrium payoffs π s (M ) = −

a2 (b + M (M − 2)) 2M 2

and π ns (M ) = −

a2 2

The functions g and h can never have non-positive values and thus qs = qns = 0 cannot yield a Stackelberg equilibrium. This is because the marginal benefit at zero pollution level is equal to a > 0, whereas the marginal damage of pollution at zero is nil. When M = 1 or M = N we have an interior solution. Which solution occurs depends on b, N , and M . The creation of a coalition or a change in the size of a coalition can result in a change from an interior to a corner solution or vice versa. Therefore, before tackling the issue of coalition formation it is important to clarify how the signs of the functions g and h depend on the model parameters. Rubio and Ulph fix N

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b14 12 10 8 6 4 2 0 0

1

2

Fig. 4.1:

b

3

4

5

6

7

8

9

M

10

Contour plot of g(b, M ) when N = 10.

2.0

1.5

1.0

0.5

0.0 0

1

2

3

Fig. 4.2:

4

5

6

7

8

9

M

10

Contour plot of h(b, M ).

and study the sign of g and h as a function of b and M . The analysis can be summed up in the figures below. In Fig. 4.1, we have the level curve g(b, M ) = 0 and the interior of the curve represents all (b, M ) such that non-signatories have zero emissions. In Fig. 4.2, we have the level curve h(b, M ) = 0 and the interior of the curve represents all (b, M ) such that signatories have zero emissions.

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4.2.2.

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4.2.2.1. The stale coalitions We now turn to the analysis of stable coalitions within the emissions game. Main proposition: There exists b1 (N ), b2 (N ) such that: (i) If b < b1 (N − 1), the unique stable IEA of the Stackelberg model with non-negative emissions is the grand coalition (Proposition 3 in Rubio and Ulph, 2006). (ii) If b ∈ [b2 (N − 2), b2 (4) = N − 4], there exists an upper bound given by the smallest integer no less than n3 that belongs to a self-enforcing IEA. This upper bound decreases with b (Proposition 4 in Rubio and Ulph, 2006). (iii) If N > 5 and b > N − 4, the maximum level of cooperation that can be achieved by a self-enforcing IEA is three. (Proposition 5 in Rubio and Ulph, 2006). (iv) If b is large enough, the equilibrium is interior and the largest size of a stable coalition is 2. From this analysis one can conjecture the following. Conjecture: The size of the largest stable IEA is a decreasing function of b. Thus it is possible to get the grand coalition as a stable IEA, and this occurs when b is small enough. The possibility of a stable grand coalition is due to the leadership advantage of the IEA. Emissions are strategic substitutes (i.e., best responses are downward sloping). This in itself gives an incentive for a leader to increase its quantities relative to the case where it moves simultaneously with the non-signatories. The instability in the scenario where countries move simultaneously is due to the reaction of the outsider who increases its emissions following the creation of the IEA of (N − 1) members. When the IEA is a leader, it decreases its overall level of emissions by a smaller amount than if it were not a leader (in which case it possibly increases its emissions) and therefore the outsider’s increase in emissions is smaller than under the simultaneous-move game (where an outsider possibly decreases its emissions). This moderate reaction of the outsider is the reason why a grand coalition can be stable under a leadership model. The externality of pollution induces the IEA to reduce its emissions with respect to the Nash equilibrium. In fact, we can show that under the Nash

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equilibrium the IEA may well end up increasing its emissions, resulting in a decrease of the non-signatories’ production of emissions (possibly to a zero level). This is more likely to happen when b is small which explains the sustainability of the grand coalitions for small values of b. While in a Nash equilibrium the payoff of non-signatories is always larger than that of signatories, in a Stackelberg equilibrium this is no longer necessarily true. It is interesting to note that in the emissions game, the range of parameters (b, M ) under which an IEA is sustainable corresponds to the range where the gains from cooperation are the largest. 4.2.2.2. Some related studies McGinty (2007) extends the analysis of an IEA in an abatement game ` a la Barrett (1994) to the case where countries are asymmetric. Countries have specific marginal abatement costs and benefits. McGinty shows that when the covariance between the benefit shares and the slopes of the marginal abatement cost is negative the non-cooperative level and stable coalition abatement is greater than a symmetric model predicts. The full cooperation level of abatement is unambiguously higher when nations differ. Batabyal (2000) departs from the main stream of the literature and considers the problem of the formation of an IEA when the IEA does not have perfect information about polluting firms. A supranational authority deals with polluting firms through their governments. It is shown that the supranational authority may still induce the optimal level of emissions through appropriate transfer schemes.1 While most of the literature on self-enforcing environmental agreements uses the concept of internal and external stability as a criterion of sustainability, Diamantoudi and Sartsetakis (2002) and Osmani and Tol (2009) have considered the concept of farsighted stable coalitions. Recall that in the definition of internal and external stability, when a country contemplates a unilateral deviation (joining or leaving the coalition) it assumes that the other players do not change their membership status. This concept of stability has been described by de Zeeuw (2008) as myopic stability 1 The

implementation of cooperation through transfers has also been investigated in the case where signatories and non-signatories move simultaneously (see Carraro and Siniscalco, 1993; Petrakis and Xepapadeas, 1996) for the case of asymmetric countries. Hoel and Schneider (1997) show that the possibility of receiving a transfer reduces the incentive to commit to cooperation.

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since a player does not foresee any change that might occur in the coalition following his/her defection. It could be argued that this behavior is rather simplistic. A more sophisticated player would anticipate that by defecting, other players’ incentives to defect might be affected. In their analysis of an IEA, Diamantoudi and Sartzetakis (2002) and Osmani and Tol (2009) recognize this possibility. (See also Zeeuw, 2008 for the case of a dynamic IEA.) The notion of farsighted stable IEAs is borrowed from coalition formation theory (see e.g., Xue, 1998). In their pioneering work on farsighted stable IEAs, Diamantoudi and Sartsetakis (2002) identify conditions under which there always exists a unique set of farsighted stable IEAs. They show that farsighted stable IEAs can be significantly larger than the stable IEAs under the internal and external stability criterion. They also consider the possibility of groups of countries jointly leaving or entering the agreement and remarkably fully characterize the coalitionally farsighted stable IEAs.2 Osmani and Tol (2009) deal with the case of asymmetric countries and study the stability of multiple farsighted stable coalitions. They consider the welfare functions of 16 world regions taken from the Climate Framework for Uncertainty, Negotiation and Distribution (FUND) model. They find all farsighted stable coalitions. They show that a single farsighted stable coalition can improve welfare and abatement by 27% and 97% respectively, relative to the internally and externally stable coalition. In the case where multiple coalitions can form, they show that 11 of the 16 regions can cooperate. The resulting cooperation increases welfare by 100% and abatement by 400% compared to the non-cooperative equilibrium outcome. Despite this positive note, these gains in welfare and increase in abatements are still a fraction of the full potential of cooperation, i.e., welfare and abatement under the grand coalition. Finus and Rundshagen (2009) consider the impact of institutional arrangements and in particular examine the role played by the open membership assumption. They show that it is easier to sustain cooperation under exclusive than under open membership and point to the higher degree of consensus necessary to form a coalition.

2 Another pioneering work on the farsighted stability concept is Eyckmans (2003) which investigates the coalitional stability of the Kyoto Protocol in a stylized dynamic integrated assessment model that is similar to the RICE model in Nordhaus and Yang (1996).

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Kolstad (2007) includes uncertainty in environmental costs and benefits, and learning about these costs and benefits. It is shown that systematic uncertainty decreases the size of an IEA.3 Learning can either increase or decrease the size of an IEA. 4.2.3.

Networks and R&D cooperation among polluting oligopolists

This section borrows from the literature on network formation under strategic behavior in general (see, e.g., Goyal, 1993; Jackson and Wolinsky, 1996; Bala and Goyal, 2000) and is related in particular to the recent literature on collaborative network formation (Kranton and Minhart, 2001; Goyal and Joshi, 2003). It also belongs to the literature on unilateral adoption of environmental and resource regulations (e.g., Hoel, 1991; Golombek and Hoel, 2004; Benchekroun, 2003). It determines the impact of unilateral regulations on R&D collaboration to abate emissions. In the example on IEAs that we considered above, when a coalition forms the characteristics (e.g., the payoff functions) of the players do not change, only each player’s strategy choice changes. It is possible that by forming a coalition the payoff functions of players are affected. For example, cooperation over abatement may well result in sharing of technology of abatement and ultimately result in improved abatement technologies. In this case, the cost function of abatement of each polluter can be affected by the formation of a coalition. Network formation theory offers adequate modeling tools to capture the main elements of this type of situations. Benchekroun and Claude (2007) apply the network formation framework in the seminal paper of Goyal and Joshi (2003) to the problem of R&D agreements in abatement technologies. Although they treat the case of firms, the model can be reinterpreted as a model of cooperation of countries to form an R&D agreement. They examine how the difference in the taxes on pollution emissions across countries affects the structure of stable networks of collaboration to abate pollution. They consider two countries. Firms in each country are charged a tax on their pollution emissions and the tax rate on pollution emissions may differ across countries. Consider a two-stage game in which firms first sign bilateral R&D agreements in order to achieve cost savings and then compete (` a la Cournot) on the output market. Emissions are taxed and R&D agreements between firms 3 See

also Ulph (2002; 2004).

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achieve cost savings through a decrease in the emission/output ratio and the resulting decrease in the tax the firm pays per unit of output produced. Benchekroun and Claude (2007) show that, when the difference between the country-specific tax rates on pollution is small enough, the complete network is pairwise stable: international R&D collaboration occurs. When the difference between tax rates is beyond a certain threshold, the pairwise stable graph is such that within each country all firms collaborate with one another; however, there is no cross-country R&D collaboration. This has an important policy implication. Suppose a subgroup of countries agree to unilaterally adopt more stringent environmental policies such as higher taxes on pollution emissions than in the countries that are not part of the agreement. The positive impact on the environment of such an agreement may be diminished (and outweighed) by the negative impact on the severance of international collaboration links to abate pollution emissions. 4.2.3.1. The model The model extends Goyal and Joshi (2003) to allow for two regions. Consider an industry composed of a set N = {1, 2, . . . , n} of identical firms, with n ≥ 3. Firms compete ` a la Cournot in the same product market and produce a single homogeneous good. They are located in two different countries, A and B. The set N is partitioned in two subsets NA and NB of sizes nA and nB . Firms have complete information about market structure and competitors’ technology. The inverse demand function is given by p = α−Q where α > 0, Q ≡ Σni=1 qi where qi is the quantity produced by firm i. In the absence of an environmental policy, firms use an identical technology characterized by a fixed emission/output ratio s0 > 0. When governments introduce emission taxes, firms have an incentive to reduce their emission/output ratio prior to competing in the market. For simplicity, assume that individual producers have no technology available for abating pollution. However, firms can make a more effective use of their technology and reduce their pollution emissions through collaborative R&D and information exchange. There are two stages. In the first stage, firms have an opportunity to form pairwise collaborative links with other firms and acquire a cleaner technology (i.e., an emission/output ratio smaller than s0 ). In the second stage, all firms compete in the output market. The bilateral relationships that may occur in the first stage are described in graph-theoretic terms. The binary variable gi,j ∈ {0, 1} indicates whether the pair of firms (i, j) ∈ N conducts joint R&D activities. The value of gi,j is 1 when firm i and firm j establish a collaborative link (i.e., conduct

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joint R&D activities) and 0 when they don’t. A network g = {(gi,j )i,j∈N } provides an inventory of all bilateral ties that have been established within the industry. We let G denote the set of all networks. Furthermore, we let g + gi,j (respectively, g − gi,j ) denote the network obtained from g by addition (deletion) of the link between the pair (i, j). Let ηi (g) denote the number of links established by firm i. A network g ∈ N is said to be symmetric if every firm has the same number of links. Examples of symmetric networks include the empty network, g e , in which ηi (g e ) = 0, ∀i ∈ N and the complete network, g c , in which ηi (g c ) = n − 1, ∀i ∈ N . Let g A+B denote the network where any two firms that belong to the same country are linked but no cross-country link is formed: g A+B = {gi,j = 0, ∀(i, j) such that i ∈ NA , j ∈ NB

and gi,j = 1, otherwise}

Interfirm cooperation affects the emission technology of the firms. For an arbitrary network g ∈ G, let firm i’s emission/output ratio be denoted by εi (g). A firm’s emission/output ratio depends exclusively on the number of bilateral links it has with other firms: εi (ηi (g)) = s0 − s1 ηi (g),

∀i ∈ N

with s0 , s1 > 0 and such that s0 − s1 (n − 1) ≥ 0. Note that εi (ηi (g e )) = εi (0) = s0 and that εi (ηi (g)) > εi (ηi (g) + 1) ≥ 0, for all i ∈ N and all network g ∈ G. Since lower emission level entails reduced tax payments, each network g ∈ G generates a different cost configuration for the industry. Let τi denote the tax per unit of emissions that firm i pays. When producing a unit of output, firm i incurs a constant marginal cost c ≥ 0 and must pay the tax amount τi εi (ηi (g)) on the pollution emitted to produce that unit of output. Hence, firm i’s effective marginal cost is cei (g) ≡ c + τi εi (ηi (g)) and the cost configuration of the industry is defined as ce (g) ≡ {cei (g)}i∈N . Let πi (g) denote firm i’s (equilibrium) profit at the second stage. We say that a network g ∈ G is pairwise stable if the following conditions are satisfied for gi,j = 1,

πi (g) > πi (g − gi,j ) and πj (g) > πj (g − gi,j )

for gi,j = 0,

if πi (g + gi,j ) > πi (g),

then πj (g + gi,j ) ≤ πj (g)

In other words, a network structure g ∈ G is said to be stable if every pair of linked firms has a strict incentive to maintain its link and no pair

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of unlinked firms has a strict incentive to set a new link. This definition, borrowed from Goyal and Joshi (2003), adapts the definition of pairwise stability in Jackson and Wolinsky (1996) to capture the idea that the cost of forming a link is small but positive. 4.2.3.2. Market equilibrium In stage 2, given an arbitrary network g ∈ G, firms compete in the output market. It can be shown that the equilibrium quantity produced by firm i is given by4    1 (α − c) + qiN (g) = τj εj (ηj (g)) − nτi εi (ηi (g)), ∀i ∈ N (n + 1) j=i

and its equilibrium profit is πi (qiN (g)) = (qiN (g))2 ,

∀i ∈ N.

Each firm’s equilibrium profit is increasing in the number of its links and decreasing in the number of links of its competitors. The formation of a link generates externalities. When two firms agree to collaborate, they both benefit from a symmetric reduction in their emission/output ratio. Each signatory firm thus has a lower effective marginal cost than non-signatory firms. The marginal effect of the formation of a link between a pair of firms (r, k) ∈ N on firm i’s equilibrium quantity (i = r, k) is given by ∆qi (g + gr,k ) =

1 [τk ∆εk (ηk (g + gr,k )) + τr ∆εr (ηr (g + gr,k ))] < 0 (n + 1)

Since ∆πi (g + gr,k ) = (qi (g) + qi (g + gr,k ))∆qi (g + gr,k ) < 0 such a link between firms r and k affects negatively any other firm i in the industry. The effect of a link between firms i and k on firm i’s equilibrium quantity is equal to ∆qi (g + gi,k ) = 4 Given

1 [τk ∆εk (ηk (g + gi,k )) − nτi ∆εi (ηi (g + gi,k ))] (n + 1)

α, c, and (εk )k∈N the set of parameters (α, c, nA , nB , (τk )k∈N and (εk (.))k∈N ) such that qiN (g) ≥ 0 for all i ∈ N is nonempty. Indeed for any nA , nB , (τk )k∈N and (εk (.))k∈N we have qiN (g) ≥ 0 for all i ∈ N and all g ∈ G provided α − c is large enough.

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Remark 1: The stability conditions of a graph g can be stated in terms of changes in the equilibrium quantities produced by each firm. A link between firms i and k is formed if and only if ∆qi (g + gi,k ) > 0 and ∆qk (g + gi,k ) > 0. 4.2.3.3. Stable networks From the expression of ∆qi (g + gi,k ), it can be verified that the marginal effect of an additional agreement between firm i and firm k on firm i’s equilibrium quantity is given by ∆qi (g + gi,k ) =

s1 (nτi − τk ) (n + 1)

and, by symmetry, we have ∆qi (g − gi,k ) = −∆qi (g + gi,k ). Assume that τi = τA for all i ∈ NA and τj = τB for all i ∈ NB . In the case where τA = τB , it is straightforward to deduce from Proposition 3.1 in Goyal and Joshi (2003) that the unique stable network is the complete network g c . The question now is whether the stability of the complete network is robust to tax-induced cost differentials. Benchekroun and Claude (2007) show the following proposition. Proposition: Let τi = τA for all i ∈ NA and τj = τB for all j ∈ NB then: (i) for (τA , τB ) such that

1 n c


0, the equilibrium total pollution emissions (i.e., from firms in both countries) may not be a decreasing function of τA ; it can jump upward at τA = nτB :   εk (ηk (g c ))qkN (g c ) − εk (ηk (g A+B ))qkN (g A+B ) < 0 when τA = nτB k∈N

k∈N

When τA is increased from nτB− to nτB firms break the cross countries collaborative links, which increases their emissions/output ratio. Consider

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the case where pollution is transboundary and suppose for example that both countries have a lax environmental regulation and are using an identical tax on pollution emissions τl that is positive but small5 (e.g., close to zero) and that Country A wishes to reduce the total level of pollution emissions (e.g., due to political pressure from environmental lobby groups). If Country B refuses to change its tax on pollution emissions, then Country A can consider unilaterally raising its tax on pollution emissions. However, such a unilateral increase of τA may have the exact opposite effect than the one intended: it may result in an increase in the level of pollution emissions (if τA is set beyond nτl ). This is illustrated in Fig. 4.3 below for the case where nA = nB = 2. 4.2.4.

Notes on some static games with private information

The applications of game theory to the environmental and resource economic problems are numerous. We have limited this section to applications of coalition formation theory in the area of environmental economics.

Fig. 4.3: 5 More

Total equilibrium emission levels.

precisely, τl ∈ {τl : qi (g A+B ) ≥ 0 for all i ∈ NA when τB = τl and τA = nτl }.

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Several important applications of game theory have been omitted. In particular, we would like to mention the review in Lewis (1996) of possible approaches to protect the environment when benefits and costs of pollution are private information. It is shown that due to private information at the (constrained) optimum, marginal abatement cost and benefit are not equalized. Duggan and Roberts (2002) consider a problem of efficient allocation of pollution emissions under asymmetric information where firms are perfectly informed but the regulator ignores the relevant characteristics of the firms. They construct a mechanism that endogenously results in the efficient level of total pollution as well as an efficient allocation of the emissions, taking into account that firms can strategically manipulate the price paid for pollution emissions. Montero (2008) considers the case where polluters are also incompletely informed. Montero constructs a simple mechanism, a uniform-price sealed-bid auction of an endogenous quantity of permits of pollution with a fraction of auction revenues returned to firms, that implements the first-best allocation of pollution rights. These are important and promising contributions of implementation theory to environmental and resource economics.

4.3.

Dynamic Games

Natural resource and environmental problems usually involve interactions that take place over time, and the physical environment in which agents interact typically changes over time. For this reason, dynamic games have been used extensively in the analysis of resources and environmental problems.6 Dynamic games are also called state-space games. In a state-space game, the environment is represented by a vector of state variables, which directly or indirectly affect the payoffs of agents. Agents influence the evolution of the state variables by using their control variables. A dynamic game can be formulated in discrete time or in continuous time. In the former case, there are T periods, where T may be finite or infinite. In the latter case, T denotes the length of the time horizon, which can be finite or infinite. A dynamic game normally displays the following properties. First, the players may receive payoffs in every period (or at every point of time). Second, the overall payoff for a player is the sum (or integral) 6 For a recent comprehensive survey of dynamic games in the economics of pollution see Jørgensen et al. (2010).

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of his/her discounted payoffs over the time horizon, possibly plus a terminal payment called scrap value which depends on the terminal stocks. When agents optimize at time t, they take into account the stream of present and future payoffs, but the payoffs they have received prior to t are no longer relevant. Third, the payoff that a player receives in a period may depend both on the actions taken in that period and on the state of the system in that period, as represented by the state variables. Fourth, the state of the system changes over time, and the rate of change of the state variables may depend on the actions of the players, as represented by their control variables. Fifth, the rate of change of each state variable is described by a difference equation or a differential equation. In what follows, the term differential games include both dynamic games in continuous time (where differential equations are involved), and dynamic games in discrete time where the evolution of each state variable is described by a difference equation. Let us give a fairly formal description of a differential game in continuous time. (The readers are referred to Dockner et al. (2000) for a more precise formulation.) Time is represented by t. The game starts at time zero and ends at time T .7 There are n state variables, denoted by xi where i = 1, 2, . . . , n. The vector of state variables is x = (x1 , x2 , . . . , xn ) ∈ X ⊆ Rn . The set S ≡ X ∪ [0, T ] is called the state-date space. An element (x, t) is called a (state, date) pair. The number of player is an integer N . Player j has a vector of m control variables, denoted by cj . Assume that cj (t) ∈ Cj ⊆ Rm . We call Cj player j’s control space. We define the control space by C ≡ Πj Cj . The evolution of the system is described by a system of s differential equations x˙ i (t) = Fi (x(t), c1 (t), c2 (t), . . . , cN (t), t),

i = 1, 2, . . . , n

where xi (0) = xi0 is given. In vector notation x(t) ˙ = F(x(t), c1 (t), c2 (t), . . . , cN (t), t) Player j’s instantaneous payoff at time t is uj (t) = Uj (x(t), c1 (t), c2 (t), . . . , cN (t), t) In what follows, we will suppress the time argument t when there is no risk of confusion. The overall payoff of player j is T e−rj t Uj (x, c1 , c2 , . . . , cN , t)dt + e−rj T Sj (xT , T ) 0

7 See

also Jørgensen and Zaccour (2007).

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where Sj (xT , T ) is his/her scrap value function and rj is a non-negative constant, called the discount rate of player j. Each player j seeks to maximize his overall payoff. In order to do this, he/she must have some ideas about what other players are doing. A Nash equilibrium is a strategy profile such that each player’s strategy maximizes his/her own overall payoff given what is predicted for the other players. Such prediction depends on what strategy space each player is restricted to. The following section discusses the main types of strategies. 4.3.1.

Open-loop versus feedback Nash equilibrium

For our present purposes, we consider only two types of strategies: path strategies and Markovian decision-rule strategies (or feedback strategies). A path strategy (or open-loop strategy) pj is a function that determines player j’s actions at each time t as a function of t only. It is as if each player makes a commitment right at beginning of the game never to deviate from his/her planned time path of actions (his/her control variables). Let Pj be the set of open-loop strategies that are available to player j. Let P ≡ Πj Pj . Once all players have chosen their open-loop strategies, the evolution of the state variables is described by x˙ i (t) = Fi (x(t), p1 (t), p2 (t), . . . , pN (t), t),

i = 1, 2, . . . , n, xi (0) = xi0

or, in vector notation x(t) ˙ = F(x(t), p(t), t),

x(0) = x0

where p(t) ≡ (p1 (t), p2 (t), . . . , pN (t)) We assume this equation has a unique solution x∗ (t). The overall payoff for player j is then Wj (x0 , p) =

0

T

e−rj t Uj (x∗ (t), p(t), t)dt + e−rj T Sj (x∗T , T )

ˆ= We define an open-loop Nash equilibrium (OLNE) as a strategy profile p ˆ 2, . . . , p ˆ N ) ∈ P such that no player can make himself/herself better (ˆ p1 , p off by choosing a different open-loop strategy, i.e., ˆj, p ˆ −j ) Wj (x0 , p) ≥ Wj (x0 , p

for all j

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To find an OLNE, one uses the maximum principle to derive the necessary conditions of each player’s optimal control problem, taking as given the time path of the vector of control variables of other players. Then, one ˆ such that all the necessary conditions for all players finds a fixed point p are satisfied. Next, one verifies that the sufficient conditions are satisfied at that fixed point. One of the main advantages of the concept of OLNE is that such an equilibrium is relatively easy to compute. Open-loop Nash equilibria are also attractive because they are time consistent. To see this, suppose the game is played and everyone has followed his/her Nash equilibrium strategy. Suppose at some time t1 > 0, when the state vector takes the value x(t1 ) as anticipated, player j asks himself/herself whether he/she can make himself/herself better off by switching to a different strategy. Clearly, the answer is no, because his/her original choice of strategy obeys Bellman’s principle of optimality.8 Thus, we conclude that open-loop Nash equilibria are time consistent. On the other hand, if by mistake some players have deviated from their planned course of action, so that the stock size x(t1 ) is different from what was anticipated at time zero, then at t1 players will in general find that they would be better off by switching to another strategy. We conclude that open-loop Nash equilibria are not robust to trembling hand deviations (Selten, 1975). One may say that open-loop Nash equilibria are not subgame perfect (even though the concept of a subgame is problematic in continuous time). For this reason, we now turn to the concept of Markov-perfect Nash equilibrium (MPNE). We define a Markovian decision rule strategy (or simply Markovian strategy in short) as a function that determines at each (state, date) pair what action to take. Let φj be player j’s Markovian strategy, then cj (t) = φj (x(t), t) Let Qj be the set of Markovian strategies that are available to player j. Let Q ≡ Πj Qj . Once all players have chosen their Markovian strategies, the evolution of the state variables is described by x˙ i (t) = Fi (x(t), φ1 (x(t), t), . . . , φN (x(t), t), t),

i = 1, 2, . . . , n, xi (0) = xi0

8 See e.g., Leonard and Long, 1992, Chap. 5 for a brief introduction to the principle of optimality.

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x(t) ˙ = F(x(t), φ(t), t),

x(0) = x0

where φ(t) ≡ (φ1 (t), φ2 (t), . . . , φN (t)) We assume this differential equation has a unique solution for any initial condition (xt1 , t1 ). We define the performance index for player i at the (state, date) pair (x, t) by T e−rj (τ −t) Uj (x(τ ), φ(x(τ ), τ ), τ ))dτ + e−rj (T −t) Sj (xT , T ) Jj (x, t, φ) = t

We define a MPNE (also called a feedback Nash equilibrium) as a strategy profile φˆ = (φˆ1 , φˆ2 , . . . , φˆN ) ∈ Q such that, at any (state, date) pair (x, t) ∈ X ∪ [0, T ], no player can make himself better off by choosing a different strategy, i.e., ˆ ≥ Jj (x, t, φj , φˆ−j ) Jj (x, t, φ)

for all j

It is important to stress the requirement that this inequality hold for all possible (state, date) pair (x, t) ∈ X ∪ [0, T ], not just for the initial pair at time zero, (x0 , 0). As Reinganum and Stokey (1985) point out, a decisionrule Nash equilibrium for a given (x0 , 0) is not necessarily Markov-perfect. To be Markov-perfect, a Nash equilibrium in decision rules must satisfy the additional property that the continuation of the given decision rules constitutes a Nash equilibrium when viewed from any future (date, state) pair. Dockner et al. (2000, Example 4.2) give an example of a Nash equilibrium in decision rules that fails to be Markov-perfect. To find an MPNE, the usual method is to make use of the Hamilton– Jacobi–Bellman (HJB) equations that the value function of each player must satisfy. The HJB equation for player j is  ∂Vj (x, t) = max Uj (x, cj , φˆ−j (x), t) rVj (x, t) − cj ∂t  ∂Vj (x, t) F(x, cj , φˆ−j (x), t) + ∂x with the terminal condition Vj (x, T ) = Sj (x, T )

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lim e−rt Vj (x(t), t) = 0

t→∞

It is worth noting that OLNE and MPNE can be thought of as based on two alternative assumptions about the ability of players to precommit. In an OLNE, players commit to a whole time path of actions. In an MPNE, players cannot precommit at all. Reinganum and Stokey (1985) argue that in some cases, players may be able to commit to actions in the near futures (e.g., by forward contracts), but not to actions in the distant future. They develop a simple model where a game begins at time 0 and ends at a fixed time T , and there are k periods of equal lengths δ, where kδ = T . At the beginning of each period, agents can commit to a path of action during that period. The special case where k = 1 corresponds to the open-loop formulation, and OLNE is then the appropriate equilibrium concept. At the other extreme, where δ → 0, the appropriate equilibrium concept is MPNE. The choice of equilibrium concepts is to some extent dependent on tractability. The relative ease of finding an OLNE is one of its attractive features. For some examples of OLNE in the economics of natural resources, see Gaudet and Long (1994; 2003), and Benchekroun et al. (2009; 2010). 4.3.2.

Dynamic games in the exploitation of natural resources

4.3.2.1. Renewable resources In the early stage of development of dynamic games, most authors focused on open-loop Nash equilibria, partly because they are easier to compute. To illustrate, let us consider a variant of a model by Clark and Munro (1975) who study a game among n fishermen who have access to a common fish stock X. The price of fish is a constant p > 0. Fisherman i’s effort is Ei (t) ≤ E. The opportunity cost of effort is w > 0 per unit. His harvest is hi (t) = qEi (t)X(t), where q > 0 is the catchability coefficient. The evolution in the stock obeys the differential equation X˙ = G(X) −

n  i=1

qE i X ≡ G(X) −

n 

qE i X

i=1

where G(X) is a strictly concave function that describes the natural growth of the biomass in the absence of exploitation. Assume that G(0) = 0 and G (0) > 0.

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πi = pqE i X − wE i Assume the utility of profit is U (πi ) =

πi1−α 1−α

where α ≥ 0. Fisherman i takes as given the time path of effort of all other agents and chooses his time path of effort Ei (t) ∈ [0, E] to maximize ∞ Vi = e−δt πi (t)dt 0

To find the OLNE, optimal control theory is used. The Hamiltonian for agent i is   n  1 (pqE i X − wE i )1−α + ψ G(X) − qE j X  H= 1−α j=1 where ψ is the costate variable. Assuming an interior solution, the first-order conditions are (pqE i X − wE i )−α (pqX − w) = ψqX ψ˙ = δψ − ψG (X) + ψq(Ei + (n − 1)Ej ) − pqE i (pqE i X − wE i )−α X˙ = G(X) −

n 

qE j X

j=1

The transversality condition is lim e−δt ψ(t)X(t) = 0

t→∞

In a symmetric OLNE, Ei (t) = E(t) for all i. Focusing on the steady state, ˙ Then, we get nqE = G(X)/X and we set X˙ = 0 = ψ.

 pqXqE 0 = ψ δ − G (X) + nqE − pqX − w OL that satisfies the The symmetric OLNE results in a steady state X∞ following externality-distorted modified golden rule: 
 OL w ) 1 G(X∞  OL − (n − 1) δ = G (X∞ ) + n X pqX OL ∞ −w

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The externality in this model is that each fisherman does not take into account the fact that his effort today will raise other fishermen’s costs tomorrow via its effect on tomorrow’s stock. If the fishermen cooperate and coordinate their efforts to maximize the sum of their payoffs, the resulting (optimal) steady-state stock, denoted by so , satisfies the modified golden rule X∞
  so G(X∞ w ) so )+ δ = G (X∞ X pqX OL ∞ −w OL so Clearly, X∞ < X∞ , indicating overexploitation. It can be verified that the steady-state profit in the OLNE is too low relative to the social optimum. A recent empirical study by McWhinnie (2009) has supported the theoretical predictions of Clark and Munro (1975). While the fishery model by Clark and Munro (1975) offers a nice framework for illustrating the strategic elements associated with the concept of OLNE, it does not seem possible to find a clear analytical characterization of feedback Nash equilibria for that model. To illustrate the use of feedback Nash equilibrium in a simple fishery model, we turn to Levhari and Mirman (1980). Two countries, i and j, have common access to a stock of fish, denoted by yt . Let their harvests be qti and qtj . Assume

yt+1 = (yt − qti − qtj )α ,

0 0. It follows that the coefficient of the term ln y must be zero, i.e., Ai =

1 1 − βα

(2)

Substituting for Ai , using Eq. (2), into Eq. (1) yields player i’s feedback strategy ci = (1 − αβ)(1 − γ j )y ≡ γ i y This yields γ i as a function of γ j γ i = (1 − βα)(1 − γ j )

for 0 ≤ γ j ≤ 1

In a symmetric feedback equilibrium, γ i = γ j = γ, where γ=

1 − βα 2 − βα

Under this feedback equilibrium, the stock evolves as follows
α βα α yt+1 = (yt − 2γyt ) = ytα 2 − βα Thus the steady-state biomass in the feedback Nash equilibrium is FB = y∞




αβ 2 − αβ

α/(1−α)

which is smaller than the steady-state stock under cooperation. This confirms that the tragedy of the commons can occur even if the harvesting function depends only on the effort and is independent of the stock.9 For further extensions of this fish-war model, see Fischer and Mirman (1986), and Kolouvatianos and Mirman (2007). While Levhari and Mirman (1980) formulated the fish-war game in discrete time, a number of authors explored feedback Nash equilibrium in a fishery in continuous time. See, for example Clemhout and Wan (1985; 1994), Martin-Herran and Rincon-Zapareto (2005), and Gaudet and Lohoues (2008). 9 This

result is in sharp contrast with the open-loop result found by Chiarella et al. (1984).

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The above fishery models are based on the assumption that individual fishermen have no impact on the market price. This assumption is relaxed in a number of papers, including Dockner et al. (1989), Jørgensen and Yeung (1996), Benchekroun (2003; 2008), and Fujiwara (2009; 2010). Benhabib and Radner (1992) consider a fish-war model where players can resort to trigger strategies. They show that under certain conditions, trigger strategies can support a socially efficient outcome. Long and Sorger (2006) show how the tragedy of the commons is affected when agents can deposit their savings in offshore bank accounts. Long and Wang (2009) investigated the effect of status-consciousness on the tragedy of the commons. 4.3.2.2. Non-renewable resources Salant (1976) applies dynamic games to the world oil market. He uses the concept of OLNE. While his model admits an oligopoly structure, he focuses on the special case of Nash–Cournot competition between a cartel and a fringe of small firms. The cartel versus fringe competition can alternatively be formulated as a leader-follower model, see for example Ulph and Folie (1980). Lewis and Schmalensee (1980) and Loury (1986) characterize OLNE involving non-identical oligopolists. Gaudet and Long (1994) show how a non-renewable resource duopoly reacts to a redistribution of stocks. For further generalization of results for open-loop oligopoly in nonrenewable resources, see Benchekroun et al. (2009; 2010) and Benchekroun and Witagen (2010). Long et al. (1999) compare open-loop and feedback equilibria in a class of resource extraction games. For some results on feedback oligopoly in non-renewable resources, see Benchekroun and Long (2006). They show that when firms use feedback strategies, windfall gains can be a curse. In their models, all firms eventually exhaust their stocks. In an alternative feedback oligopoly model, Salo and Tavohnen (2001) assume that extraction costs rise as the stock dwindle, and they become prohibitively expensive so that eventually firms abandon their deposit without physically exhausting them. In other words, there is economic exhaustion, but not physical exhaustion. The economic exhaustion formulation turns out to be a very useful hypothesis. This formulation has been used to study feedback equilibrium in a game of trade between resource-poor economies and a resource-rich country, see e.g., Rubio and Escriche (2001), Liski and Tavohnen (2004), and Chou and Long (2009).

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Resource-extracting firms need not be always noncooperative or always cooperative. Benchekroun et al. (2006) study the case where firms cooperate as a cartel, but anticipate that at some future time the cartel may break up and the firms will become Cournot rivals. Benchekroun and Gaudet (2003) show how deviation from a Nash equilibrium (because of production perturbation) may affect the profitability of an exhaustible resource oligopoly. 4.3.3.

Pollution games

Following Long (1992) and Ploeg and de Zeeuw (1992), let us consider a world consisting of two countries. Let Qi (t) be country i’s output at date t. Assume that emissions are proportional to output, Ei (t) = Qi (t). Let P (t) denote the stock of pollution. Assume P˙ (t) = E1 (t) + E2 (t) − δP (t) where δ > 0 is the decay rate. The pollution damage suffered by country i at time t is (cP 2 )/2. The net utility of country i is 1 c Ui (t) = AE i (t) − (Qi (t))2 − (P (t))2 2 2 and its social welfare is

Wi =

0

∞

e−ρt Ui (t)dt

where ρ > 0 is the rate of discount. Let us find the OLNE of this model. Since countries use path strategies in the open-loop formulation, let us suppose that country i believes that country j’s emission strategy is Ej (t) = gjOL (t). Then, it seeks to solve the following optimal control problem

 ∞ 1 c max e−ρt AE i (t) − (Ei (t))2 − (P (t))2 dt 2 2 Ei (·) 0 subject to P˙ (t) = Ei (t) + gjOL (t) − δP (t),

P (0) = P0

Applying the maximum principle, we obtain the necessary conditions −(E˙ i − ρ(Ei − Ai )) = −ci P − δ(Ei − Ai ) P˙ = Ei + gjOL − δP,

P (0) = P0

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and the transversality condition is lim e−ρt (Ei (t) − Ai )P (t) = 0

t→∞

Since the two countries are identical, we obtain the following system of two differential equations E˙ = cP + (ρ + δ)(E − A) P˙ = 2E − δP,

P (0) = P0

with the transversality condition lim e−ρt (E(t) − A)P (t) = 0

t→∞

OL OL , E∞ ) where There is a unique steady-state pair (P∞ OL P∞ =

2A(δ + ρ) 2c + δ(δ + ρ)

OL E∞ =

OL δP∞ Aδ(δ + ρ) = 2c + δ(δ + ρ) 2

Comparing with the case where the two countries cooperate and maximize the sum of their welfare, we see that the steady-state stock of pollution OL is too high. P∞ What happens if countries use feedback strategies? Suppose country i believes that country j employs a feedback emission strategy, Ej (t) = gjFB (P (t)), so that its rate of emissions at t is conditioned on the currently observed level P (t). Then, country i maximizes

 ∞ 1 c max e−ρt AE i (t) − (Ei (t))2 − (P (t))2 dt 2 2 Ei (·) 0 subject to P˙ (t) = Ei (t) + gjFB (S(t)) − δP (t),

P (0) = P0

Realizing that gjFB (P ) is a function of the pollution stock, country i knows that it can indirectly manipulate country j’s emissions at t by influencing the evolution of P . This strategic consideration was absent in the openloop case.

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To find the feedback Nash equilibria of this game, we make use of the HJB equations. The HJB equation for country i is 

1 2 c 2  ρVi (P ) = max AE i − Ei − P + Vi (P )(Ei + Ej (P ) − δP ) Ei 2 2 where Ej (P ) is country j’s feedback strategy, and Vi (P ) is country i ’s value function. The transversality condition is lim e−ρt Vi (P (t)) = 0

t→∞

(3)

The first-order condition with respect to Ei is Ei = A + Vi (P ). This equation gives Ei = Ei (P ), i.e., country i’s emissions depend only on P . Appealing to symmetry, we get the HJB equation ρV (P ) =

c 1 2 [A + 4AV  + 3(V  )2 ] − δPV  − P 2 2 2

This equation and the transversality condition (3), identify the set of possible Markov-perfect Nash equilibria. Let us conjecture that the value function is quadratic V (P ) = −

ωP 2 − πP − µ 2

(4)

Then, V  (P ) = −ωP − π and hence the feedback strategy is linear E(P ) = A − π − ωP

(5)

It is plausible to expect that ω > 0, i.e., a higher stock will make countries choose lower emissions, and π > 0, i.e., if P = 0, the marginal effect on welfare of an exogenous increase in P is negative. Making use of (4) and (5), the HJB equation gives a quadratic equation of the form λ0 + λ1 P + λ2 P 2 = 0 where λ0 , λ1 , and λ2 are expressions involving the parameters δ, ρ, and c and the coefficients ω, π, and µ. Since this equation must hold for all P , it follows that λi = 0 for i = 0, 1, 2. Using these three conditions, we can solve for ω, π, and µ. First, we obtain     ρ 2 ρ 1 δ+ − δ+ + ω= + 3c 3 2 2

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(To ensure convergence to a steady state, the positive root ω > 0 is selected.) Then π= µ=

2Aω δ + ρ + 3ω

(A − π) (3ω − δ − ρ) ≡ µm 2ρ

The linear feedback strategy is E=

A(δ + ρ + ω) − ωP δ + ρ + 2ω

It follows that 2A(δ + ρ + ω) − (2ω + δ)P P˙ = δ + ρ + 2ω For P to converge to a steady state, it is necessary that 2ω + δ > 0. This inequality is satisfied if and only if the positive root for ω is selected. The steady-state pollution stock under the MPNE with linear feedback strategies is FB = P∞

2A(δ + ρ + ω) (δ + ρ + 2ω)(2ω + δ)

OL is lower than the Clearly the OLNE steady-state pollution stock P∞ FB MPNE steady state P∞ . This result is dependent on the fact that we have focused on a quadratic functional form for the value function Vi (P ). Dockner and Long (1993) show that there are other value functions that satisfy the HJB equation. These value functions result in non-linear emission strategies. In fact, there is a continuum of non-linear strategies, and some of them outperform the OLNE in the sense that both countries would be better off under such strategies. When there are multiple equilibria, it is not clear which one is likely to prevail.10 Myerson (2009, p. 1111) points out that the existence of such games with multiple equilibria is a pervasive fact of life that needs be appreciated and understood, not ignored by economists.11 Multiplicy of equilibria is also found in a discrete-time 10 Multiplicity of equilibria occurs in nature as well. As Gould (1995, p. 28) noted, places with apparently identical vegetation, moisture, and temperature might harbor shells of maximally different form. 11 For a treatise on how to select a plausible Nash equilibrium, see Harsanyi and Selten (1988).

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transboundary pollution model by Dockner et al. (1996), who show that there exists equilibria where the time path of the pollution stock display chaotic behavior. This result is consistent with the tragedies of the commons described in Dutta and Sundaram (1993). A number of authors have linked pollution with the extraction of fossil fuel resources, and studied dynamic games between an oil cartel and a resource-importing economy that suffers from the pollution. See Wirl (1994; 1995); Wirl and Dockner (1995); Tavohnen (1995; 1996), and Rubio and Escriche (2001). Jørgensen and Zaccour (2001) show how a downstream country can induce an upstream country to reduce its polluting activities. They find conditions for time-consistent side payments. The effects of population growth and technology changes on emissions strategies are studied by Dutta and Radner (2006) in a model of global warming. For games of joint implementation of environmental projects, see Breton et al. (2006).

4.3.4.

Stackelberg equilibrium in dynamic games

Let us turn to situations where players are not symmetrically placed. Consider a two-player game, and assume that player 1 can make a commitment on what strategy he/she will use before player 2 can choose his/her strategy. We call player 1 the Stackelberg leader, and player 2 the follower. The resulting equilibrium is called a Stackelberg equilibrium. In dynamic games, players make their moves in each period. There are several concepts of dynamic Stackelberg leadership which must be distinguished. 4.3.4.1. Open-loop Stackelberg equilibrium and time consistency issues When player 1 can announce at the start of the dynamic game the whole sequence of his/her moves and is able to commit to it, we say he/she is the open-loop Stackelberg leader. An early model of open-loop Stackelberg leadership in natural resource economics is Kemp and Long (1980). They assume there are three players: player 1 is a strategic oil-importing country (called Home), player 2 is the collection of perfectly competitive oilextracting firm located in the second country (called Foreign), and player 3 are residents of the rest of the world (called ROW). All consumers have the same demand curve, and their demand for oil is zero if the price of oil is greater than or equal to α (we call α the choke price). Assume zero

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extraction cost. Then, Hotelling’s rule informs us that producer’s price must increase over time at a rate equal to the rate of interest. Assume player 1 acts as an open-loop Stackelberg leader. It announces right at the beginning of the game a time path of per unit tariff rate. Its optimal tariff rate must also increase over time at the rate of interest (so that in Home consumer’s price rises at the rate of interest, i.e., Hotelling’s rule also applies to consumers). It follows that in the Stackelberg equilibrium, at some finite time T the tax-inclusive price for consumers in Home reaches α, while in ROW the price is still below α. From T onwards, oil is consumed only in ROW, until the resource stocks are exhausted. The open-loop Stackelberg equilibrium that we describe above is well defined. However, it is based on the assumption that player 1 is committed to the sequence of tariff rates that it initially announces. This assumption does not seem plausible: at time T , player 1 will have an incentive to reduce the tariff to allow imports into Home. In this sense, the initially announced tariff path is time inconsistent.12 Karp (1984) explores the issue of time inconsistency in open-loop Stackelberg leadership in more detail, and proposes that a restriction be imposed on the Stackelberg leader’s behavior so that the strategy he/she announces at the beginning is time consistent. Karp’s method has been applied to a number of interesting problems, see Batabyal (1996a,b). For futher discussion of the time-consistency issue in the context of international trade in exhaustible resources, see Karp and Newbery (1991, 1993). 4.3.4.2. Stagewise Stackelberg equilibrium Under what conditions would the strategy of a Stackelberg leader be time consistent? Clearly, time consistency is ensured in the special case where the followers are not intertemporal optimizers while the leader is, as in Crabb´e and Long (1993). Alternatively, one can consider a two-player dynamic game where even though both the players are intertemporal optimizers, the leader only leads in each period. Both players know this, and they solve their problems backwards, beginning with the last period, T −1. The equilibrium strategies and equilibrium payoffs for period T − 1 clearly depend on the state variable at the beginning of that period, say yT −1 . Knowing this equilibrium, both players know what to do in period T − 2, and so on. This 12 The time-inconsistency issue arises in many other contexts, see e.g., Kydland and Prescott (1977) and Cohen and Michel (1988).

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scenario has been referred to as stagewise Stackelberg equilibrium.13 The following models in the fishery literature illustrate the concept of stagewise Stackelberg equilibrium. Recall the fish-war model of Levhari and Mirman (1980) in the preceding section. Now, let us assume that in each period, country 1 is the first mover: it announces how much it will catch in that period before country 2 makes its move. To find the stagewise Stackelberg equilibrium of this game, let us begin with the simplest case, where there is only one period to go, i.e., only one decision to make: how much to catch this period, given that next period (the terminal period) the game ends and each player k receives an exogenously fixed share sk of the final stock. Country 2 takes the leader’s catch q 1 as given, and chooses q 2 to maximize ln q 2 + β ln s2 [y − q 1 − q 2 ]α This yields country 2’s reaction function q2 =

y − q1 1 = φF 1 (y, q ) 1 + αβ

where the superscript F indicates that this is the follower’s strategy, and the subscript 1 indicates that there is only one period to go. Country 1, knowing this reaction function, chooses q 1 to maximize

α y − q1 1 1 1 ln q + β ln s y − q − 1 + αβ This yields country 1’s decision rule g1L (·): q1 =

y = g1L (y) 1 + αβ

where the superscript L indicates that this is the leader’s strategy, and the subscript 1 indicates that there is only one period to go. Substituting this decision rule into the follower’s reaction function, we obtain our prediction of the follower’s catch L q 2 = φF 1 (y, g1 (y)) =

13 See

αβy ≡ ψ1F (y) (1 + αβ)2

Mehlmann (1988) and Basar et al. (1985). Some authors use the term feedback Stackelberg equilibrium for stagewise Stackelberg equilibrium, e.g., Basar and Olsder (1995).

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Note that ψ1F depends only on y while the reaction function φF 1 depends 1 F on both q and y. We call ψ1 the follower’s (one-period-to-go) equilibrium feedback strategy (as he/she predicts correctly the leader’s strategy). Then, the terminal stock to be shared is y − q1 − q2 =

α2 β 2 y (1 + αβ)2

Armed with this information, we know that for the game where there are two periods to go, the follower will seek to maximize ln q 2 + (αβ)(1 + αβ) ln(y − q 1 − q 2 ) + constant To find the follower’s reaction function and the leader’s decision rule, a similar procedure applies. Repeating this process, we can see that in the limit (i.e., letting the time horizon tend to infinity), the stationary equilibrium feedback strategies of the leader and the follower are, respectively q 1 = (1 − αβ)y

and q 2 = αβ(1 − αβ)y

This indicates that the leader has a higher payoff than the follower. The after-harvest stock is y − q 1 − q 2 = α2 β 2 y We can see that the stock will approach a steady-state level S y∞ = [α2 β 2 ] 1−α α

S It is easy to see that the steady-state stock y∞ under the stagewise StackelF under the feedback berg leadership is smaller than the steady-state stock y∞ Nash equilibrium. Let us turn to a continuous-time model of stagewise Stackelberg leadership in a fishery, based on Benchekroun and Long (2002b). Two countries have access to a common fish stock denoted by St . In each period, the fish travel along country 1’s coastline before reaching country 2. Country 1 is therefore a natural stagewise Stackelberg leader. If it chooses effort level ω1t , its harvest is ω1t St and its profit is (ω1t St )σ where 0 < σ < 1. Country 2, having observed ω1t , chooses its effort level ω2t . We assume that its profit function is (ω2t St )σ (1 − ω1t ), because country 1’s prior harvest makes it more costly for country 2 to catch fish. Assume that the transition

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equation is S˙ t = AS t −

2 

ω1t St ≡ F (St , ω1t , ω2t )

i=1

where we assume A > 0 and r > σA. Suppose country 2 believes that country 1’s strategy is ω1t = g1 (St ). Its HJB equation is rV 2 (S) = max{(ω2 S)σ (1 − g1 (S)) + V2 (S)F (S, g1 (S), ω2 )} ω2

Suppose that g1 (S) = ω1 , a constant. Try the following value function for country 2: V2 (S) = BS σ It can be verified that if ω1 > 1, then B = 0 = ω2 and, if 0 ≤ ω1 ≤ 1, then ω2 =

r − σA + σω1 ≡ ω2 (ω1 ) 1−σ

and
B=

1−σ r − σA + σω1

1−σ (1 − ω1 )

We now turn to country 1. Its HJB equation is rV 1 (S) = max{(ω1 S)σ + V1 (S)F (S, ω1 , ω2 (ω1 ))} ω1

Again, try the functional form V1 (S) = B1 S σ . Then, provided that r−σA < 1 − σ, country 1’s HJB function is satisfied with
B1 = (1 − σ)

1 − σ + σ2 r − σA

1−σ

and ω1 =

r − σA ω1 1−σ

Notice that Eq. (6) may be interpreted as country 2’s reaction function, which displays strategic complementarity, namely, the reaction curve is upward sloping. In equilibrium, the follower exercises greater effort than the leader. It is interesting to note that had country 1 not known that ω2 depends on ω1 , the result would have been a na¨ıve Nash–Cournot equilibrium, and country 1’s effort would be greater, at a level ω 1 defined by ω 1 =

r − σA r − σA > 1 − 2σ 1−σ

(provided 1 − 2σ > 0). We may, therefore, conclude that the stagewise Stackelberg leader exercises restraint because it knows the follower’s effort is an increasing function of that of the leader. Benchekroun and Long (2002b) consider the possibility of an exogenous deviation from this equilibrium. For example, suppose the government of country 1, responding to pressures from conservationist groups, agrees to reduce its effort ω1 over a specified number of years. How would country 2 react? The authors find that the answer depends on the length of country 1’s period of committed reduction in harvesting. If the period of commitment is short, country 2’s optimal reaction is to increase its effort ω2 . If it is long but finite, country 2 will initially reduce its harvesting effort, but will eventually increase it when the end of the commitment period is near. Finally, if the period of commitment is infinite, country will reduce its effort permanently. 4.3.4.3. Global feedback Stackelberg equilibrium Under stagewise leadership, the leader perceives that in each period, the follower’s move is a reaction to his/her own move. This is not the same thing as saying that the follower’s decision rule is a reaction to the leader’s decision rule. A truly global feedback Stackelberg equilibrium would require the leader’s knowledge of how the follower’s choice of decision rule responds

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to each possible decision rule of the leader. This is a very difficult problem to formulate in general terms. Fortunately, for a small class of problems, a global feedback Stackelberg solution can be found. As an example, a government may want to design a tax rule that corrects for stock externalities. Benchekroun and Long (1998) show that in the case of a pollution stock contributed by emissions of oligopolists, there exists an optimal tax rule where the rate of tax per unit of emissions is made dependent on the state variable. This tax rule achieves the socially optimal outcome, which corresponds to the first-best control and command scenario.14 Because the first best is achieved, there is no reason why the government would change the tax rule at a later date. In other words, the global feedback Stackelberg equilibrium found by Benchekroun and Long (1998) is time consistent. Benchekroun and Chaudhuri (2011) show that the use of a tax on emissions can have an impact on the market structure: e.g., the grand coalition or the cartel made up of all the firms in the industry can become stable and social welfare under a tax on emissions and cartel formation can be smaller than under a business-as-usual scenario (i.e., the tax is zero). Similarly, in a model of international trade involving bilateral monopoly in an exhaustible resource market where the importing country acts as the global Stackelberg leader, Fujiwara and Long (2010) find a time-consistent optimal tariff that is conditioned on the state variable (the stock of resource that remains). 4.3.5.

Empirical models of dynamic games in resource and environmental economics

The purpose of empirical models is to quantify the likely impacts of proposed policies on specific regions for a specific period of time. Realworld data are used to calibrate parameters of demand and cost functions. The equilibrium paths of the model are then solved by numerical methods. Kaitala et al. (1991; 1992a,b; 1995) model transboundary pollution games involving a number of countries, including Finland and Russia. Bernard et al. (2008) set up a dynamic game model of trading in pollution permits where the two dominant players are Russia and China. The passive players are other Annex B countries.15 Russia has a large stock of pollution permits to sell, and China is likely to earn a lot of permits through the clean development mechanism established under the Kyoto 14 Benchekroun

and Long (2002a) show the multiplicity of optimal tax rules. the Kyoto protocol, Annex B countries consist of OECD countries plus Russia and central and eastern European countries. 15 Under

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protocol. Russia has a stock of permits x1 (t) that are banked at time t. The stock evolves according to the following law of motion x1 (t + 1) = x1 (t) − u1 (t) + q1 (t) + h(t) where u1 (t) is the amount of permits it sells in the world market, q1 (t) is its level of emissions abatement, which earns new tradable permits, and h(t) is its exogenous flow of permits (called hot air).16 The stock x2 (t) of the second dominant player (China) follows a similar law of motion, except that China does not have a hot air flow h(t). The equilibrium price of a permit p(t) must equate the demand by the rest of the world to the combined supply of permits by Russia and China: DR (p(t)) = u1 (t) + u2 (t)

(7)

where the function DR (.) is derived from the competitive equilibrium conditions. Inverting the market clearing condition (7) one gets −1 (u1 (t) + u2 (t)) ≡ P (u1 (t) + u2 (t)) p(t) = DR

Both China and Russia behave like duopolists in an exhaustible resource market where the resource stocks are x1 and x2 . The authors assume that neither China nor Russia cares about the damage cost of global warming. Player i maximizes T −1 

βit [P (u1 (t) + u2 (t))u1 (t) − ci (qi (t))] + βiT πi xi (T )

t=0

where βi is the discount factor, ci (.) is the abatement cost function, and πi is the scrap value per unit of the final stock of permits. The search for an equilibrium is formulated as a non-linear complementary problem, for which efficient algorithms have been developed (Ferris and Munson, 2000). The authors numerically compute the OLNE. (It would be much more difficult to compute feedback Nash equilibria.) For calibration, they use simulation results of a computable general equilibrium model, namely GEMINI-E3 (Bernard and Vielle, 1998; 2000; 2003), and a partial equilibrium model of the world energy system, namely POLES (Criqui, 1996). GEMINI-E3 provides the data to estimate demand for pollution permits 16 After

the fall of the Soviet Union, Russia experienced a sharp decline in output, which gives rise to emission credits. These are called hot air because they are available at no cost.

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by Annex B countries, and marginal abatement costs for Russia. POLES is used to estimate the marginal abatement costs for China. The authors compare the Nash equilibrium with the outcome of an alternative scenario where only Russia behaves strategically. It is found that duopolistic competition between Russia and China lowers the permit price significantly. 4.3.6.

Dynamic games with imperfect information

Some authors have introduced asymmetric information in dynamic games. In Gaudet et al. (1995), a government designs a royalty rule to maximize social welfare when it faces potential extractive firms that have private information about their costs. This dynamic adverse selection problem is a game between the government and the resource-extracting firm. Long and Sorger (2009) studied a dynamic moral hazard problem. They formulated a game between a principal who owns a resource stock and an agent who is offered a contract to exploit it. The action of the agent is not observed by the principal, but he/she can imperfectly infer it from the evolution of the stock which itself is subject to a random disturbance represented by a Brownian process. The principal offers the agent a contract which contains a rule that prescribes how the amount he/she will pay the agent at each point of time depends on the observed stock level at that time. The authors characterize the optimal rule that maximizes the principal’s payoffs subject to the participation constraint of the agent. Karp and Livernois (1994) considered a pollution tax adjustment rule that would achieve a long-run pollution target when the regulator does not have full information about abatement costs. The authors show that when firms are identical and play open-loop strategies, in the steady state the per unit tax is lower than the tax under full information. If firms are not identical, then under the OLNE the steady-state allocation of emissions among firms is not efficient. If firms use feedback strategies, there is a continuum of equilibria even in the case of identical firms. With heterogeneous firms, under certain conditions there exist candidate Markov-perfect equilibria that allocate emissions efficiently at the steady state. However, under the proposed tax adjustment rule, efficiency is unlikely outside the steady state. Another source of asymmetric information is that the regulatory agency cannot observe the rate of discharge of each polluter. The authority would then have to rely on a tax scheme whereby a firm’s pollution tax liability depends only on aggregate emission of the whole industry. Is there a tax

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scheme that would induce firms to achieve the socially optimal rate of abatement? This problem is addressed in a static setting by Segerson (1988), who proposed that firms be charged a unit tax that depends on the overall level of pollution, the so-called ambient tax. Xepapadeas (1992) extends the analysis to a dynamic and stochastic framework, but under the assumption that the taxing authority aims only at making firms achieve the optimal steady-state pollution stock. It is unclear if there exists a tax scheme that would induce firms to achieve the efficient time path of pollution both at the steady state and along the approach path. Karp (2005) studied the implications for tax revenue when the regulator adjusts the ambient tax rate to the deviation of the actual pollution level from the socially efficient one. He focuses on open-loop equilibrium and flow pollution. List and Mason (2001) considered an asymmetric version of the transboundary pollution model of Dockner and Long (1993). In their model, there are two regions that have different parameter values for the regional damage function and production function. They show that the outcome of a game between the two regions may be superior to the outcome under an imperfect central planner who forces the two regions to have the same emission rate. Similar results are obtained by Mason and List (1999) for a problem where the central planner lacks information about the synergistic effects of two types of pollutants. Are price instruments better than quantity instruments when the regulator does not have full information? Weitzman (1974) has considered this issue in a static context. Moledina et al. (2003) looked at this issue in a context where firms behave strategically. The authors employ in the first instance a two-period model, where firms try to manipulate the regulator’s beliefs about their abatement costs. In this model, the regulator does not know that firms behave strategically. It is shown that in the case of emission tax, the firms will overabate in period one in order to induce the regulator to set a lower tax in period two. It should be noted that Moledina et al. (2003) are not dealing with a mechanism design problem.

4.4.

Conclusion

Both static and dynamic games have been successfully employed to shed light on many resource and environmental issues involving strategic interactions among a number of players. The insights generated by game-theoretic models can potentially be used to help design mechanisms for improving

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economic efficiency. In particular, empirical models are useful tools for policy making. Conversely, issues in resource and environmental economics have provided opportunities for researchers to sharpen their tools and develop new concepts and techniques for dealing with emerging issues. References Bala, V and S Goyal (2000). A non-cooperative model of network formation. Econometrica, 68, 1181–1231. Barrett, S (1994). Self-enforcing international environmental agreements. Oxford Economic Papers, 46, 878–894. Basar, T, A Haurie and G Ricci (1985). On the dominance of capitalist leadership in a feedback Stackelberg solution of a differential game model of capitalism. Journal of Economic Dynamics and Control, 9, 101–125. Batabyal, AA (1996a). Consistency and optimality in a dynamic game of pollution control I: Competition. Environmental and Resource Economics, 8, 205–220. Batabyal, AA (1996b). Consistency and optimality in a dynamic game of pollution control II: Monopoly. Environmental and Resource Economics, 8, 315–330. Batabyal, AA (2000). On the design of international environmental agreements for identical and heterogeneous developing countries. Oxford Economic Papers, 52, 560–583. Benhabib, J and R Radner (1992). The joint exploitation of a productive asset: A game theoretic approach. Economic Theory, 2, 155–190. Benchekroun, H (2003). Unilateral production restrictions in a dynamic duopoly. Journal of Economic Theory, 111(2), 214–239. Benchekroun, H (2008). Comparative dynamics in a productive asset oligopoly. Journal of Economic Theory, 138, 237–261. Benchekroun, H and R Chaudhuri (2011). Environmental policy and stable collusion: The case of a polluting oligopoly. Journal of Economic Dynamics and Control, 35(4), 479–490. Benchekroun, H and D Claude (2007). Tax differentials and the segmentation of networks of cooperation in oligopoly. The B.E. Journal of Theoretical Economics, 7(1), (Topics), Article 26. Benchekroun, H and G Gaudet (2003). On the profitability of production perturbation in a dynamic natural resource oligopoly. Journal of Economic Dynamics and Control, 27(7), 1237–1252. Benchekroun, H, G Gaudet and NV Long (2006). Temporary natural resource cartels, Journal of Environmental Economics and Management, 52(3), 663–674. Benchekroun, H, A Halsema and C Withagen (2009). On nonrenewable resource oligopolies: The asymmetric case. Journal of Economic Dynamics and Control, 33, 1867–1879.

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Empirical and Experimental Tools

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Chapter 5

Introduction to Economic Valuation Methods Sabah Abdullah University of Bath, UK [email protected] Anil Markandya University of Bath, UK and Basque Centre for Climate Change, Spain [email protected] Paulo A.L.D. Nunes University of Venice, Italy and Fondazione Eni Enrico Mattei (FEEM), Italy [email protected]

5.1.

Introduction

Economic valuation has been widely used in different sectors, for example in: health, transport, and the environment. The use of valuation methods has increased, owing to the number of interest groups, corporations, governments and researchers demanding economic values for environmental goods. The World Bank over three fiscal years (2000 to 2003) conducted an average number of 6 to 9 projects per year in environmental valuation (Silva and Pagiola, 2003). Also, there have been various incidents at the global level that have compelled and accelerated the valuation of environmental goods and services. One example was the much publicized incident in 1989, involving the Exxon/Valdez oil tanker that struck a reef in Prince 143

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William Sound, Alaska, spilling at least 11 million gallons of crude oil and killing many birds and mammals.1 Establishing the usefulness to the local area of cleaning up the spill required a comprehensive process to examine the economic value of environmental goods, such as birds or biodiversity, which was not available. The economic valuation of environmental assets in general is among the most pressing and challenging issues confronting environmental economics. One may wonder for what reason such monetary assessments of environmental resources are undertaken. Four main reasons can be identified: performing cost–benefit analysis, environmental accounting, assessing natural resource damage, and carrying out proper pricing.2 This chapter has three main objectives: (1) to present common economic valuation techniques, (2) to examine guidelines and recommended frameworks in carrying out economic valuation approaches, and (3) to provide practical applications and future outlook of some valuation approaches. The next sections discuss: the varied economic valuation techniques (Sec. 5.2), steps in carrying out a valuation study (Sec. 5.3), challenges in the application of economic valuation approaches in the case of biodiversity and ecosystems as evidenced by the call from the European Commission as the preparatory work for the global study, The Economics of Biodiversity and Ecosystems (TEEB) (Sec. 5.4). Section 5 concludes.

5.2.

Economic Valuation Approaches

A market is defined as a place where goods and services are transacted using monetary instruments and monetary information is available to users. However, there are other goods and services, such those in the environmental sector, which lack a market, hence these are known as non-market goods (or services). Such goods may not be traded in a day-to-day market, because they do not have a price against them. However, placing an economic price on these goods or services, which reflects a value, is important. By valuing these goods and services, users can acknowledge their presence and their influence on their livelihoods. In this section, we introduce 1 This incident was followed up by the media as well as the US Federal government and researchers. The information was found at the US Environmental Protection Agency (EPA) website. 2 For further reading of these objectives, see Nunes et al. (2004).

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the concept of use and non-use values which is relevant in welfare estimation for users of environmental goods and services. 5.2.1.

Introduction

Valuation aims to confer accurate economic values on non-market goods and services, but in order to place an economic value on a non-marketable, say an environmental good or service, the various components that make up its total economic value (TEV) need to be identified. The TEV of environmental goods consists of use value and non-use value. The use value (UV) is a value related to the present or future use of a particular habitat by individuals. It can be subdivided into direct use values and indirect use values. Direct use values are derived from the actual use of a resource either in a consumptive way or a non-consumptive way (e.g., timber in forests, recreation, fishing); indirect use values refer to the benefits derived from ecosystem functions (e.g., watershed protection or carbon sequestration by forests). The non-use values (NUVs) are associated with the benefits derived simply from the knowledge that a natural resource — such as a species or habitat — is maintained. By definition, such a value is not associated with the use of the resource or the tangible benefits deriving from its use. It can be subdivided into two parts that overlap according to its definition. First, there are existence values, which are not connected to the real or potential use of the good, but reflect a value that is inherent in the fact that it will continue to exist independently from any possible present or future use of individuals. Second, bequest values are associated with the benefits of the individuals derived from the awareness that future generations may benefit from the use of the resource. These can be altruistic values, when the resource in question should in principle be available to other individuals in the current generation. A separate category is made up by option values, which are values attributed by individuals given the knowledge that a resource will be available for future use. Thus, it can be considered like an assurance that a resource will be able to supply benefits in the future. The quasi-option values, which is sometimes classified as a NUV, represents the value derived from the preservation of the future potential use of the resource, given some expectation of an increase in knowledge. The quasi-option value is important when the decisions on consumption are characterized by a high reversibility. Other values worthy of mention, but not discussed further in

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this study include: functional, anthropocentric, biocentric, assigned, and held values (Lockwood, 1998) and psychological (Nunes et al., 2004). An example of TEV can be exemplified by considering a forest which is part of an ecosystem. A UV might be a planned recreational visit to the forest either now or later. The NUV may consist of unplanned or nonusable sources of welfare related to the forest, where one is willingness to pay (WTP)/willingness to accept (WTA) for the existence of the forest or to leave it for others.3 On one hand, an individual might feel empowered to protect this forest (stewardship) and on the other, may have the urge to let the forest be available to the present generation (altruism) and/or allow it to be available for future generations (bequest). The use of valuation methods varies according to the different forms of value, i.e., whether UV or NUV is estimated. Valuation approaches estimate the nature of the goods and services according to the benefits being considered. Therefore, we refer to market and non-market values and against this background, where we divide the valuation approaches according to market data and non-market methods of revealed and stated preference. For market approach the production function and damage cost methods are used for welfare estimation, whereas the non-market includes: stated preference (SP) and revealed preference (RP) methods. SP is direct and is hypothetical in nature whereas the RP is an indirect approach that uses market data based on established and actual recorded behavior. Taking the forest example mentioned earlier in this section, Table 5.1 illustrates the classification of economic values derived from a forest and the various market and non-market approaches used to value such an ecosystem. In Secs. 5.2.2. and 5.2.3, the market and non-market approaches are further addressed, respectively.

5.2.2.

Types of market valuations

The market-based methods attract little attention from environmental economists, perhaps because they are considered straightforward and do not pose interesting methodological challenges. They are, however, of considerable importance and a good part of the value of ecosystems is 3 Carson

et al. (2001) attributed the differences between WTP and WTA to property rights, whereas Hanemann (1991) argued that the gap between WTP and WTA estimates for public goods is determined by income and substitution effects.

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Table 5.1:

Classification of economic values provided by forests.

Value Components

Use value (UV)

Examples of Benefits 8 Direct use value > > > > > > (DUV) > > > > > > > > > > > < > > > > > > > > > > > > > > > > Indirect use value > : (IUV)

8 Bequest value > > > > (BV) > > > > >
(NUV) > > > > > > Existence value > > : (EV)

a Market

147

Most Suitable Valuation Technique

Recreational benefits and tourism e.g., hiking, hunting, camping and wildlife viewing Forestry resources with commercial value e.g., timber harvesting

Revealed preference methods e.g., travel cost methods

Forestry ecosystem and ecological functioning e.g., erosion control, water purification, carbon sequestration

Production function, damage cost methods

Legacy benefits e.g., preservation of forestry for future generation

Stated preference methods e.g., contingent valuation, choice experiment

Existence benefits e.g., knowledge guarantees that some forests resources are not extinct

Stated preference methods e.g., contingent valuation, choice experiment

Aggregate price analysisa

price valuation technique.

in fact represented by commercial and financial gains and losses. Some of the market valuation approaches include: averting behavior, replacement/restoration, production factor method, and dose–response. Averting behavior (AB) or preventive expenditure The preventative expenditure technique measures the expenditure incurred in order to avert damage to the natural environment, human infrastructure or to human health. The technique can be used to measure the impacts of biodiversity loss on both marketed and non-marketed goods and services, with the exception of NUV. In terms of costing biodiversity loss impacts, preventative expenditure should be seen as a minimum estimate of impact costs since it does not measure the consumer surplus (i.e., the additional

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amount above actual expenditure that consumers would be willing to pay in order to protect a particular good or service from the impacts of biodiversity loss). In the context of climate change, preventative expenditure, if undertaken, would in reality be an adaptation cost, since it is an expenditure aimed at reducing the impacts of climate change (which may also include the loss of biodiversity). As such, great care needs to be taken if using the technique in the context of a cost–benefit analysis of adaptation options. Replacement/restoration The replacement/restoration cost technique can be used to measure the costs incurred in restoring or replacing productive assets or restoring the natural environment or human health as a result of the impacts of environmental degradation. As with preventative expenditure, restoration costs is a relatively simple technique to use and has the added advantage over preventative expenditure of being an objective valuation of an impact — i.e., the impact has occurred, or at least is known. Use of the replacement costs method relies on replacement or restoration measures being available and the costs of those measures being known. As such, the method is unlikely to be appropriate for costing the impacts on irreplaceable assets such as biodiversity loss. Another shortcoming with the technique is that actual replacement or restoration costs do not necessarily bear any relationship to willingness of individuals’ to pay to replace or restore something. For example, in the context of climate change, the potential health impacts of an increase in the air temperature, and the associated additional health service costs incurred to restore the health of someone made ill by a tropical disease, may be less than that person’s WTP to avoid getting the disease in the first place. Production factor method This estimates the economic value of an environmental commodity through an “impact-pathway” approach, in which a change in the environmental attribute is linked to impacts on “endpoints” that are relevant for human wellbeing. For example the benefits of tree planting via reduced erosion are measured first by the link between soil cover and erosion rates and then by the link between erosion rates and agricultural productivity. Such methods can be very useful to value many services provided by ecosystems, including forestry (timber and nontimber), agriculture (value of diversity in crops and use of genetic material) and marine systems (losses from overfishing, species invasion).

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Dose–response (DR) This technique involves a change in environmental quality affecting the output of goods (or services). The cause is the source (dose) impacting on the environment (the response). For instance, significant amounts of sulfur dioxide (dose) in the air leads to acid rain, which pours into streams and rivers (the response) causing acidification. However, it is difficult to distinguish the various causes that affect the receptors, hence there needs to be a strong association between the dose and its impact. The DR function can be used to estimate use values either directly or in association with SP and RP approaches. 5.2.3.

Types of non-market valuation methods

Market prices and costs can provide estimates of the increase in the value of commercial activities, such as timber extraction, fishing etc., the value of revenues from tourism activities related to visits to natural areas and the value of contracts signed by firms and governmental agencies, also known as bioprospecting contracts. However, some environmental goods and services do not affect markets and market data are not available to value them. In such cases, methods have been developed to derive consumers’ preferences namely: RP and SP methods. A number of studies have used SP methods when market information has been unavailable, in converse situations, where such data exists, RP is applied. Moreover, the SP is frequently used to elicit the NUV, as these values do not have recorded behavior, unlike the UV that can be better measured by RP. Table 5.2, shows the differences between RP and SP approaches. The SP approaches include: (a) Contingent valuation (CV) CV is currently the most used technique for the valuation of environmental goods where individuals state their WTP/WTA for a good or service. One important reason for this is because only SP methods like CV can elicit the monetary valuation of the NUVs, which typically leave no “behavioral market trace.” Furthermore, CV allows environmental changes to be valued even if they have not yet occurred (i.e., ex ante valuation). It allows the specification of hypothetical policy scenarios or states of nature that lie outside the current or past institutional arrangements or levels of provision.

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Sabah Abdullah, Anil Markandya, and Paulo ALD Nunes Differences between revealed preferences (RP) and stated preference (SP).

Revealed Preference (RP)

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Stated Preference (SP)

(1) Portrays the world as it is i.e., the current market equilibrium

(1) Describe hypothetical or virtual decisions context (flexibility)

(2) Consist of inherent relationship between attributes (technological constraints are fixed)

(2) Control relationships between attributes (permits utility functions with technologies)

(3) Only existing alternatives as observables

(3) Include existing, and/or proposed and/or generic choice alternatives

(4) Represent market and personal limitations on decision maker

(4) Does not represent changes in market and personal limitations effectively

(5) High reliability and face validity

(5) Appears reliable when respondents understand, commit-to and respond to tasks

(6) Yield one observation per respondent

(6) Yield multiple observations per respondent

Source: Adapted from Louviere et al. (2000).

Finally, CV allows one to enrich the information base by submitting the process of value formation to public discussion. Against this is the criticism that the values are hypothetical (payments are not actually made or cash paid out) and that the method is subject to many biases. (b) Conjoint choice or Choice experiment (CE) Conjoint choice is also a commonly used SP method, and the relative merits of this against CV are much discussed in the literature. This method elicits information on values by asking individuals to choose between alternatives; conjoint ranking, where individuals rank alternatives in order of preference and conjoint rating, which indicates their strength of preference on a cardinal scale. A number of studies, ranging from environment to health and the transport sectors, have used the SP approach and/or a combination of both CV and CE. The differences and similarities between these two techniques should not dictate the superiority of one technique over another or support one for another; rather these techniques complement each other. Although, for some SP studies there are reasons for choosing, say, CV over CE and are determined by a combination of factors, such as the researcher’s interest,

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funding (or sponsorship) issues, the sampling framework, and what part of the TEV of the good needs to be valued. According to Bateman et al. (2002), choosing which of the two SP approaches to use depends on the kind of value needed (i.e., total or relative), information availability (CV has greater literature), welfare and/or welfare-consistent estimates, cognitive processing, and sampling means (number of responses per individual). Freeman (2003) was “cautiously optimistic” about the SP method and reported that others are attracted to SP because of the “relatively easy and inexpensive way to get usable values for environmental resources.” In a similar vein, Whittington (2002) concluded that SP is vital to a developing country’s policy application, but it is far from being a high-quality option at a low cost. The RP methods include: (a) Hedonic pricing (HP) This estimates the economic value of an environmental commodity, say, clean air or an attractive view, by studying the relation between such attributes and house prices (Palmquist, 1991). Hedonic price estimation has been applied to elicit environmental/ecosystem values associated with recreation, landscape values, and genetic and species diversity. Hedonic techniques are particularly employed in valuing visual amenity, quality of soil assets, and exposure to air pollution. (b) Travel cost method (TCM) This estimates the economic value of recreational sites by looking at the generalized travel costs of visiting these sites (Bockstael et al., 1991). The valuation is then based on deriving a demand curve for the site in question, through the use of various economic and statistical models. Where the individual makes a choice involving more than one site, the discrete choice models have used the random utility theory framework to value not only visits to different sites but also the attributes of sites, such as water quality. The travel cost technique has been widely applied, especially in North America, where Parsons4 has assembled a list of over 120 such studies. There are three dimensions in TCM: the quality of the good to be valued, the number of visits and duration, and the substitutes to other sites (Kolstad, 2000). 4 http://www.aere.org/resources/parsons.pdf,

[22 November 2007].

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Finally, benefit transfer (BT) consists of exporting previous benefit estimates (either from SP or RP) from one site to another, at one point in time, with regards to the researcher’s area of interest. In BT estimates, there are three possible forms of transfers: transfer of an average of WTP estimates from one primary study, transfer of the WTP function, and transfer of WTP estimates by aggregating other WTP estimates employing metaanalyses (Bateman et al., 2002). According to Rosenberger and Loomis (2001), BT involves the use of economic values from one specific area, with a known resource and policy conditions, to another site in similar circumstances. Generally, the first site is known as the “study site” and the second as the “policy site.” Sites differ in characteristics and one has to be cautious when applying these from one site to another. On the one hand, this method reduces the cost of starting a completely new valuation study, whereas on the other hand, the compilation of a comprehensive database often proves costly. In a paper to compare the stated and revealed preference estimates from 83 studies conducted from 1966 to 1994, Carson et al. (2001) found that CVM estimates were lower than RP estimates i.e., CVM estimates were around 30% lower than multisite travel cost estimates. All in all, valuation studies conducted in both developed and developing countries have used a combination of RP and SP. Researchers have combined both SP and RP, where the application of SP and RP approaches is that some strength exists in each method and, when combined, they provide a useful toolbox in valuing environmental goods and services. Despite these differences, the use of both SP and RP methods in valuing environmental goods has increased in recent years, for instance in fisheries (Whitehead, 2006), water (Urama and Hodge, 2006; Hanley and AlvarezFarizo, 2003), transportation (Espino et al., 2006; Memon et al., 2005; Dissanayake and Morikawa, 2002; Polydoropoulou and Ben-Akiva, 2001), recreation (Earnhart, 2004; Park et al., 2002), forestry (Adamowicz et al., 2004), animal husbandry (Scarpa et al., 2001), and cultural artifacts (Boxall et al., 2003).

5.3.

Conducting Non-Market Valuation Studies

This section discusses the recommended steps in conducting non-market valuation studies, since for the market methods the existence of information and data is available unlike the non-market approaches.

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There are various methodologies that have been elaborated by nonmarket valuation practitioners in particular for CV (see Bateman et al., 2002; Champ et al., 2003, Markandya et al., 2002; Alberini and Kahn, 2006; Mitchell and Carson, 1989; Whittington, 2002; Arrow et al., 1993). Of all, the most venerated are the guidelines as recommended by the National Oceanic and Atmospheric Administration (NOAA) panel (see Arrow et al., 1993). Most of these frameworks require one to follow specific steps: define the study objectives, design the questionnaire and survey, create a database and analyze data, estimate the WTP/WTA values, and validate the results. 5.3.1.

First step

The study needs to name the goods (or services or policies) to be valued, the respondents to be interviewed, and the unit of measurement. Mitchell and Carson (1989) state that it was unclear from valuation literature whether interviewees should answer for themselves (individually) or at household levels. According to Quiggin (1998), WTP from households is less than aggregated WTP from individuals when it involves altruism motives; nevertheless, the household is viable as a measure of analysis in three possible ways: aggregating individual values, treating household as a unit, and applying referendum method. 5.3.2.

Second step

The questionnaire design is divided into several major parts, with the common format being a three-part questionnaire. The procedure consists of a “warm up” section, followed by a WTP scenario and finally the socioeconomic and demographic questions. The WTP questions involve hypothetical scenarios, where respondents are requested to value non-market goods by providing an amount against the good (or service) being considered. For instance, the questionnaire design in the CV method requires a careful and clear research methodology and design. The guidelines provided in most of the literature on CV questionnaire design emphasize that hypothetical scenarios need to be realistic, feasible, understandable, and simple for respondents to relate to. In some cases, the use of visual aids — photographs, maps, etc. — may provide more relevant information on the good in question, although the use of photographs is known to influence the WTP amount (Arrow et al., 1993), hence the importance of pretesting such items.

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Moreover, the SP study can be supplemented by employing additional tools, to enhance the reliability of their research: For example, opinion surveys, an awareness survey, a local impact study, and an attitudinal questionnaire. Such attitudinal questions help to construct CV validity. Most of the studies included some socio-economic and demographic information, like head of household, age, race, education levels, employment, urbanization, marriage status, with/out children, home owner (or renter), and membership of an environmental organization. The opt-out or status quo choice, as suggested by the NOAA, can be implemented in SP studies. Moreover, a follow-up question can be used to obtain further information, with regard to “no” responses to WTP questions, as recommended by the NOAA guidelines. Additionally, debriefing respondents and reminding them of the budget constraints is recommended by the NOAA panel (Arrow et al., 1993). The use of focus group discussions (FGDs) is advocated during questionnaire design to help the researcher understand respondents’ preferences and also questionnaire characteristics (Bateman et al., 2002). The pretesting of questionnaire is one of the effective ways of testing reliability (Mitchell and Carson, 1989; Arrow et al., 1993). Another consideration of interest in valuation studies is that of whether to offer incentives to the respondents. On one side, incentives, such as those in monetary forms encourage respondents to answer the question but on the other side, these incentives can bias estimates. The effects of these incentives depend on the type of incentives; hence, it is advisable to pretest the questionnaire with the incentive on offer beforehand. An important feature in estimating WTP involves elicitation formats for SP studies, such formats include: open-ended, closed-ended, dichotomous choice (double-bounded and multiple bounded questions), bidding game, and take-it-or-leave-it questions. The use of these formats is known to influence WTP values and some questionnaire formats are “cognitive burden” i.e., to say are a tiresome activity for respondents. For example, respondents may find open-ended questions more strenuous to answer than closed-ended questions. Another item of interest is the type of the payment vehicle, where this refers to the form of payment for a good or service in a valuation exercise. Similar to the question format type, the type of payment vehicle is reported to influence WTP values. There are various types of payment vehicle such as tax, price increase in a bill, and fee for the good or service. To determine the right payment vehicle, it needs to be credible, relevant, acceptable, and coercive (Bateman et al., 2002).

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5.3.3.

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Third step

This involves the survey design and conducting the interviews by various elicitation modes such as mail, personal interviews, telephone, and most recently the internet. There are varied costs and response rates involved in administering these methods and the most popular methods of data collection are mail surveys and face-to-face interviews (Carson et al., 2001). Generally, mail surveys are inexpensive to administer compared to other forms of elicitation; however, the response rate is about 25–50% (Bateman et al., 2002). According to Mitchell and Carson (1989) one shortcoming of this data collection method relates to non-response bias, where respondents are unwilling to answer questions. Non-response bias can be for a variety of reasons, in particular, illiteracy and self-selection are evident in mail surveys, wherein the latter case respondents who are interested in the good (or service) respond more than others (Bateman et al., 2002). SP practitioners may consider combining and conducting several different modes (telephone, mail, or personal interviews) to reduce the biases, namely, selfselection bias and non-response bias. For instance, the telephone may be used followed by a mail survey or alternatively mail and personal interview can be employed. 5.3.4.

Fourth step

This involves data collection, database creation, and analyzing information gathered during the survey. The biases that occur in this stage are associated with selection bias, sample non-response, and item non-response prior to analyzing of the data collected. According to Whitehead (2006), these biases can be resolved by constructing weights to reflect the population weights for the sample nonresponse; a follow-up in case of selection bias; and data imputation for the item nonresponse. After the biases are adjusted, the data is cleaned, coded, organized, and entered in a computer program ready for analysis. 5.3.5.

Fifth step

This involves the estimation of the WTP or WTA values according to the specified models which may vary on the elicitation format types such as open-ended question, bidding, and others. Additionally, in this phase the estimation of annual individual and population WTP/WTA estimates is important in calculating the total benefits. Finally, a good practice calls for

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validity and reliability in SP studies. Carson et al. (2001) defined validity as conformity between what one hopes to measure and what is measured in real life, and reliability is demonstrated when results are upheld in repeated cases, in the case of CV using the same method. For validation, the estimates can be tested both internally and externally. The difference between internal and external validation is that the former is conducted within one valuation method, unlike the latter, which compares one valuation method with another, i.e., SP against RP. There are three types of validation for valuation methods, namely content, convergent, and criterion.5 Content validation evaluates whether the questions referred to the appropriate good or service in a clear and understandable manner, whereas convergent validity is a comparison process of crossreferencing whether the estimates from one SP study reports similar results to another SP study or otherwise reports dissimilar results, which vary in an expected form. The criterion validity likens the CV studies with the HP method, using the actual market information or predicted market information.

5.4.

Application of Economic Valuation Approaches

In this section, we review and discuss the application of the wide range of economic valuation methods addressing the assessment of biodiversity benefits and ecosystems services based on the preparatory work for the global study, which is named The Economics of Ecosystems & Biodiversity (TEEB), Phase I. The remarks found in this report were contributed by worldwide scientific and practitioners in valuation studies (see Markandya et al., 2008 for more details). Following the classification of the valuation methods described in Sec. 5.2.1, we shall classify the wide range of contributions that report the use of market data and non-market methods of RP and SP. 5.4.1.

Market biodiversity values and market valuation methods

The main market-based value studies submitted to the TEEB report are shown in Table 5.3. A further set of recent European market-based studies 5 See

Mitchell and Carson (1989) for more discussion.

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Introduction to Economic Valuation Methods Table 5.3:

Submissions on the market value of biodiversity and biological services.

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Submission By SM Gautier

Service

Region

Sanderson and Prendergast K¨ alberer

Forest ecosystems Commercial harvest losses Commercial use of wild plants

Canada United States/Canada England

Beech Treesa

Germany

Worm et al.

Marine ecosystema

Global

Conservation Int. William Marthy RSPB UK

Bird keeping

Indonesia

United Kingdom

Lilian Spijkerman

Employment, tourism, health, economy-wide Forest land services

Williams et al.

All ecosystem services

Scotland

BirdLife/RSPB

Values of “natural services” Forest services

Global

Willis et al. LIFE Priolo Project CLIBIO

Maryanne GriegGran (IIED) Hoppichler et al. K¨ alberer

Bernstein

157

Brazil

GB

Value of the Priolo (Azorean Bullfinch) Climate related losses in forest services Hydrological functions of ecosystems

Azores

Alpine ecosystem servicesb Value of nature overall

Austria

Forest values as a carbon store

Tropical

Europe

Various

Netherlands

Focus Mountain Beetle Pest/invasive species Valuation in terms of livelihoods supported. Measures replacement cost Measures restoration value Biodiversity losses traded against economic benefits. Quantifies socio-economic benefits Opportunity cost of agricultural development Based on Costanza et al.’s methodology.

Carbon and health benefits Values in terms of tourism and associated benefits. Ecosystem function loss valued using market data Losses due to soil erosion

Tourism, replacement costs Tourism, water management, nature itself Relative to opportunity cost (Continued)

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Table 5.3: Submission By

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Brotherton English nature

Brotherton

Graham

Service

(Continued) Region

Blanket peat as a carbon store Salt marshes Hedgerows

England

Nutrient sinks in the Danube floodplains Forest services

Various

United Kingdom

England

Woodward and Wui Willis

Wetlands

United Kingdom

Forest services

United Kingdom

Farber

Coastal wetlands

United States

Ten Kate and Laird Webber et al. Van den Hove and Moreau

Pharmaceutical services Geodiversity

Global

Deep-sea life

Global

Costanza et al.

17 ecosystem services across 16 Biomes Values of landscapes through auctions

Global

Daan Wensing

United Kingdom

Netherlands

Focus Other services also noted Flood protection Fruits harvested Reed beds for thatch reed Based on benefit transfer Health benefits of air pollution absorbed Flood defence functions Replacement and mitigation costs of water supply Values loss in terms of damages from storms Net value of genetic material Tourism value Total benefits of food production, oil and gas, and nutrient recycling Total values assuming all is lost. Ooijpolder nature area landscapes were “sold”

a Indicates

valuations are in terms of physical replacement units or changes in productivity. b Also includes some non-market valuations.

of biodiversity value and the value of ecological services relevant to the report is presented in Box 5.1. A number of points should be noted in this regard as discussed below. First, estimates of market values are partial and generally not comparable. Sometimes the figures are given as net income, sometimes as gross income. Some report gains in terms of employment or increased local

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Box 5.1: European studies of ecosystem service value based on market data. 1. In the coastal waters of several European countries there is a notable loss of crayfish populations (Austropotamobius pallipes, A. astacus, A. torrentium) because of pollution, habitat loss, overfishing, and the introduction of alien species (mainly North American). Services lost include food (domestic varieties fetch twice the price of nonnative varieties), regulating services (trophic effects on prey and predators), recreation, and cultural services. Although estimates of the loss are not made, estimate of costs of restoration are. They indicate modest costs (around 225,000 per stream over 5 years in France) that would be well below the value of recovered crayfish populations. 2. In Germany and Romania, the Danube has lost a number of ecosystem services due to dam construction. These include wetlands that have been permanently flooded, a reduction in biodiversity due to lost fauna (including populations of a number of fish species), and poorer water quality. Estimates of the annual values of these services in market terms are around $16 million for the fisheries, $131 million for the increased cost of water treatment to obtain drinking water, and $16 million from the tourism services the wetland could provide. 3. In Greece, Lake Karla, which was a wetland site, has gradually been transformed as a result of human intervention that dates back to 1936 and is now a drained agricultural land. Partial restoration is now under way at a cost of around 152 million. Estimates of the benefits, however, are not available, although the main services that will be restored have been identified — commercial fisheries and farming. Farmers are finding that current land use is unsustainable. 4. Overfishing in the North Sea is a major threat to its biodiversity and ecosystem health and stocks of a number of fish are now under stress. Estimates of the benefits of recovery plans that would increase populations are about 600 million a year. This excludes benefits from fish processing and recreational fisheries. 5. Peat bogs in the United Kingdom are being lost as a result of intensive livestock farming. The result is a loss of carbon sequestration services, potable water services, and habitat for species. The (Continued)

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Box 5.1: (Continued)

6.

7.

8.

9.

additional costs of water treatment as a result of the loss are estimated at 1.8 to 3.6 million a year. Other benefits, including carbon sequestration or tourism have not been estimated. Plantation of non-native monocultural forests (eucalyptus and pine) in Portugal has resulted in a loss of biodiversity and an increased risk of fire as these species are more fire prone compared to the oaks they substituted. Other services lost include soil protection, water provisioning, game, non-timber forest products, all of which are less well provided in monocultural forests of this type. The cost due to fires alone in 2001 was 137 million, although we do not know how much of this was due to the expansion of monocultural afforestation. The annual value of forest ecosystem services is estimated at 1.33 billion. Eutrophication of coastal marine ecosystems in Sweden is well known and studied. Services lost as a result of eutrophication include provisioning for commercial fishery species and reduced efficiency in regulating services such as cycling and depositing nutrients. In addition, there is a loss of cultural services, notably recreation. The value of the loss of regulating services is estimated at 6– 52 million a year for the Stockholm archipelago (for a 1 m improvement in summer secci depth (depth to which water can be seen with the naked eye)) while the value of provisioning services is estimated at 6– 8 million a year for the Kattegat and Skagerrak fishery areas. This is based on a reduction in the output of plaice juveniles as a result of eutrophication. Finally, the loss of cultural services is based on the costs of removing algae, which is estimated at 7000 for the Swedish West Coast. The Osprey is a highly valued bird that experienced a sharp decline in the United Kingdom in the 19th century. The restoration that has taken place since the 1950s has generated benefits of 4.8 million to the local economies in Scotland in the areas where the birds nest. Clam fishing in the lagoon of Venice, Italy is a highly profitable activity but it is currently carried out using a technology (vibrating rake) that damages the resilience of the ecosystem. If the present system continues, yields will decline rapidly. A shift to a manual (Continued)

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Box 5.1: (Continued) collecting system would result in a lower income for the fisherman immediately but would decline more slowly over time. Depending on the discount rate, a fisherman may adopt the manual system. Alternatively, the calculations show how much compensation we would have to give the fisherman to adopt the less damaging system. Source: Kettunen and ten Brink (2006).

economic activity, which are not necessarily economic benefits or at least not in full (that depends on what alternative employment opportunities exist and what alternative economic activities are possible). With the current state-of-the-art, however, it is possible to carry out proper valuations based on economy-wide impacts of biodiversity loss or conservation and some studies have done that. Second, in some cases estimates are based on the costs of restoring lost services. While useful, such estimates could be higher or lower than the market value that is lost. Indeed, one of the purposes of valuing loss of biodiversity is to see if replacement or restoration is justified. Using the latter as a measure of value does not allow answering that question. Third, underlying the market-based approach are scientific studies linking the estimated impacts on biodiversity to certain causes (e.g., the effect of pests on forests, of forests on air pollution, etc.). We should recognize that there are still considerable uncertainties regarding these links, which should be reflected in the reported benefits. Fourth, the majority of studies refer to marginal changes in local areas. At the same, time, there are a few that value the broad scale of services provided globally. The numbers from these latter studies are extremely large. Finally, the purpose of many of the studies was to show that the services provided by nature are significant and either merit protection (where biodiversity is threatened) or merit expansion (where there is potential for that). Estimates of the “opportunity cost” of land — i.e., what it would be worth if it were not conserved — are often much lower than the value of the biodiversity services provided if it were conserved.

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Sabah Abdullah, Anil Markandya, and Paulo ALD Nunes

Revealed preference valuation methods and non-market biodiversity values

There is a large body of empirical studies on the values attached to different ecosystem services. Nijkamp et al. (2008) list some 75 biodiversity studies that had been carried out in Europe from 1981 to 1997, and doubtless there have been many more since. The EVRI6 database of environmental valuation studies using non-market methods had about 460 records for European studies in June 2008, with a heavy dominance of UK research (37% of the total). Just under half of the total methods were based on revealed preference (47%) and just over half were stated preference. Topics covered included: biodiversity loss, wildlife, national parks and nature reserves, water courses (non-fishing use), recreational fisheries, landscape, endangered species, wetlands, and woodlands. According to the TEEB report, a small number of contributions from the use of RP methods were received. Hence, we can distinguish two types of biodiversity values. The first type are values elicited using a travel cost model and are mainly associated to the recreational values provided by the conservation of areas rich in biodiversity, including natural parks, see Moons et al. (2000) (Table 5.3). The second type are biodiversity values elicited by the use of hedonic price models, see Garrod and Willis (1994) and Ruijgrok (2004a) (Table 5.4). In this context, the biodiversity benefits are estimated by assessing the impact of neighborhood characteristics, including nature/biodiversity ones, on the housing price. For example, Willis and Garrod (2001) studied the value of a waterside location using property sale prices for Greater London and the Midlands over a five-year period (1985– 1989). The estimations rendered for properties located on the waterside a premium of £2689 for Greater London and £2238 for the Midlands. 5.4.3.

SP valuation methods and non-market biodiversity values

The most populated method in the report is the SP which is in accordance to the EVRI database of environmental valuation studies — where more than half of the 2003 records refer to non-use or passive use values estimates. These values are only possible to elicit using SP valuation studies — typically with the use of contingent valuation and stated choice surveys. 6 www.evri.ec.gc.ca/EVRI,

[23 November 2007].

Belgium

Forest valuation

Valuation of the recreational, hatching, and regulating services of nature and landscape as a result of a nature restoration project Valuation of Flemish public parks Ecosystem services such as biomass production, water provision, recreational services, residential amenity, prevention of soil erosion, waste treatment, carbon fixation, biological control of production, natural risk regulation, conservation of biodiversity

T Cerulus (study authors: Moons et al.) T Cerulus (Bogaert et al.)

Spain

Research Tools in Natural Resource. . .

Introduction to Economic Valuation Methods (Continued)

Recreational and aesthetic value of the parks RP and SP VANE project designs ad hoc methodologies and guidelines for the spatial representation of economic values. Analysis regarding aggregation and commensurability of results, treatment of intertemporal valuations and loss of ecosystem services as a result of land-use changes

9in x 6in

Belgium

Italy

Case 1: carbon-storage benefits of conserving forests in the Cardamom Mountains in Cambodia. Case 2: costs of hydropower developments due to increases in sedimentation in Costa Rica. Valuation methods are not specified Valuation of biodiversity in Cansiglio forest and St. Erasmo area in terms of biodiversity zoning and spatial stratification. SP method Three categories of values considered: the recreational value, the non-use value, and the indirect use value. RP, SP, and value transfer methods are used. The study includes a cost–benefit analysis, a cost-effectiveness analysis, and a financial analysis. Valuation methods like contingent hierarchy and benefit transfer are used.

Focus

9:44

T Cerulus (Liekens et al.) D Azqueta

Belgium

Total economic value of biodiversity

AL Notte

Worldwide

Various aspects of biodiversity values at local and global scales

M Grieg-Gran

Region

Service

Submissions on the non-market value of biodiversity and biological services.

Submission By

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163

Ecosystem services

Eftec

Defra

Ecosystem services provided by natural environment, including life-support services, economic activity support, health and quality of human life, etc. Ecosystem services

England and Wales

United Kingdom

United Kingdom

United Kingdom

Changes in biodiversity in the UK countryside

Research Tools in Natural Resource. . .

(Continued)

Development of an introductory guide to valuing ecosystem services with an emphasis on the value of changes in the services provided by the natural environment. Development of guidance to value habitats within the context of flood and coastal erosion risk management projects and strategies, involving a full-scale benefits transfer process.

9in x 6in

RSPB

Indonesia

Brazil

Worldwide

Cost-effective and ecologically effective compensation payments for species protection. Valuing ecosystem goods and services provided by protected areas at various scales. Analysis of the opportunity costs for maintaining forest cover against pressure of agriculture conversion. Focus on the link between the conservation of biodiversity and the livelihoods of rural people living close by the protected areas. Recommending the use of contingent valuation method for the valuation of biodiversity programs and the choice experiment method for biodiversity attributes. Benefit transfer method is not advocated. Reversing wildlife declines and restoring degraded landscapes/ecosystems will deliver significant benefits for society and the economy.

Focus

164

Ecosystem functions and services

Ecosystem services of protected natural areas Atlantic forest land, biodiversity “hotspot areas”

L Spijkerman

Germany

Region

(Continued)

9:44

L Spijkerman (Study authors: Chomitz et al.) L Spijkerman (Study authors: Pattanayak and Wenland) Christie et al.

Species protection

Service

F Watzold

Submission By

Table 5.4:

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Sabah Abdullah, Anil Markandya, and Paulo ALD Nunes

Welfare benefits enhancement due to nature conservation Wildlife protection related various welfare benefits

RSPB

RSPB

United Kingdom

Worldwide

Worldwide

Worldwide

Worldwide

Introduction to Economic Valuation Methods (Continued)

The wild living resources include the commercial fish species, the game animals, the marine mammals and birds, and hardwoods. Insurance value of biodiversity again the uncertain provision of ecosystem services. Natural resource modeling. Conceptual ecological–economic model on biodiversity and insurance. Examples of coffee plantation and rainforest. External benefits of agro-ecosystem and agro-biodiversity by employing a conceptual ecological–economic model. Nature conservation can help to enhance human health, contribute to economic development, etc. Nature conservation improves the quality of people’s life in terms of sustaining and enhancing human health, offering education opportunities, and contributing to sustainable communities and economic activities. Review the social benefits provided by biodiversity.

A literature review of the economic, social, and ecological value of ecosystem services in the context of global change and ecosystem degradation. Two case studies in Indonesia and Uganda. Costanza methodology

Focus

Research Tools in Natural Resource. . .

Biodiversity

Agro-biodiversity

Baumg¨ artner and Quaas

RSPB

Biodiversity

Quaas and Baumg¨ artner

Worldwide

United Kingdom

Scotland

Worldwide

Region

9in x 6in

Baumg¨ artner

Murray and Simcox

Scotland’s ecosystem services and natural capital Commercialized wild living resources in the United Kingdom Biodiversity

Ecosystem goods and services

Service

(Continued)

9:44

Williams et al.

Submission By

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165

Biodiversity conservation

Economic impact of the natural environment in Wales Economic impacts of England’s natural environment

BirdLife International

National Trust

Impacts of nature conservation on local economies in the United Kingdom Impact of RSPB nature reserves on local economies in the United Kingdom

Rayment and Dickie and RSPB

Shiel et al.

Marine environment in the United Kingdom

GHK and Wilson

United Kingdom

United Kingdom

United Kingdom

England

Covering tourism and direct employment.

(Continued)

The study identifies the activities responsible for building and maintaining the stock of natural capital and activities that benefit from the quality of the natural environment. Investigating the potential benefits for economic activities in the United Kingdom of a system of marine spatial planning. 12 case studies regarding nature conservation benefits to the rural economies.

9in x 6in

Wales

Presenting 26 case studies of the contribution of wildlife to wellbeing in EU, covering the following countries: Spain, United Kingdom, England, Wales, Poland, Romania, Austria, Portugal, Scotland, France, Germany, Mediterranean, Turkey, Denmark, Belgium, Slovakia, and Czech Republic. 9 case studies regarding the role of nature conservation in improving livelihoods and fighting poverty in South Africa, Kenya, Uganda, Brazil, Burkina Faso, Ecuador and Peru, Jordan, Indonesia, and Cambodia. Contribution of the Welsh environment to the economic growth and quality of life in Wales.

Focus

166

Worldwide

EU

Region

(Continued)

9:44

GHK and GFA RACE

Biodiversity

Service

BirdLife International

Submission By

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Sabah Abdullah, Anil Markandya, and Paulo ALD Nunes

The social and environmental benefits of forestry in Great Britain. Services provided by marine biodiversity

Valuing the environment in small islands

Willis et al.

Van Beukering et al. United Kingdom overseas territories in the Caribbean

United Kingdom

Great Britain

United Kingdom

England and Scotland

United Kingdom

Region

Research Tools in Natural Resource. . .

Introduction to Economic Valuation Methods (Continued)

Economic sector: wildlife tourism. Involves the following bird species: white-tailed eagles, ospreys, red kites, bee-eaters, choughs, peregrines, capercaillies, Montagu’s harriers, hen harriers, and seabirds. Concerning two most important habitats for supporting livelihoods and activities in the area: woodlands and hedgerows and wetlands. A review of studies on valuing biodiversity across all types of woodland in the United Kingdom. SP methods are used to estimate the WTP for various aspects of biodiversity value. Empirical estimates of marginal benefits of various social and environmental benefits and total value across forests and woodlands in Great Britain. The report provides estimates of the annual economic value of a range of goods and services in the United Kingdom, including provision of food and raw materials, recreation, nutrient cycling, gas and climate regulation, disturbance prevention, cognitive values and NUV. It includes two case studies: the North Sea and Skomer Island. The study provides guidance on how the value of the environment on small islands can be estimated and incorporated into planning and development decisions (EEWOC project). Economic valuation studies and monetary damage estimates are included.

Focus

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Hanley et al.

The commercial uses of wild plants in England and Scotland Value of biodiversity in United Kingdom forests

Economic impact of spectacular bird species in the United Kingdom

Service

(Continued)

9:44

Sanderson and Prendergast

Submission By

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167

Socio-economic valuation of nature, water, soil, and landscape

Ruijgrok et al.

Wild geese in Scotland

Economic benefits of biodiversity

Scotland

Conservation value of forests

Asociaci´ on Guyra Paraguay and WWF MacMillan et al.

The Netherlands

EU

Research Tools in Natural Resource. . .

(Continued)

Economic costs and benefits of wild geese in Scotland. RP method is used to estimate total annual WTP for alternative goose management policies. CLIBIO project aiming at assessing the economic impacts of climate change on biodiversity and human wellbeing. Its first year’s report presents data for the total economic value of ecosystem goods and services produced by European forests. Providing an overview of physical interventions, ecosystem functions, changes in physical conditions and their socio-economic effects, as well as a valuation methodology to include economic ecosystem valuation in SCBA.

168

Priolo, Italy

9in x 6in

Carraro et al.

Paraguay

Economic benefits and impact of the LIFE Priolo project

Dickie

Total economic value derived from the ecosystem functions in Tuz G¨ ol¨ u, especially protected area in Turkey. RP method is performed, considering a range of ecosystem services like water quality, protection against landslides and floods, carbon sequestration, leisure and tourism, educational and scientific services, in addition to ecotourism. Ecosystem services are particularly important in helping to offset market costs. Significant services from the SPA relating to water resources, flood/landslide protection, carbon storage, conservation, educational, resilience, and scientific services. Definition of the high conservation value (HCV) forests method in Paraguay.

Focus

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Priolo, Italy

Economic benefits and impact of a conservation project

LIFE Priolo project

Turkey

Region

Ecosystem functions

Service

(Continued)

Waliczky

Submission By

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Sabah Abdullah, Anil Markandya, and Paulo ALD Nunes

Socio-economic valuation of nature, water, soil, and landscape Various aspects of the natural environment in the Netherlands

Ruijgrok et al.

Specific ecosystem services provided by biodiversity

Species conservation

Diversity of biodiversity

Valuing particular species

Austrian Alps

Martin-Lopez et al.

Christie et al.

Christie et al.

Hoppichler et al. Austria

Worldwide

United Kingdom

Spain

Spain

Research Tools in Natural Resource. . .

Introduction to Economic Valuation Methods (Continued)

Providing more than 400 values designed to calculate the welfare effects of changes in biodiversity and ecosystem functions. Studies cover the value of cultural heritage (RP and SP), the existence value of natural river and canal banks (SP), the value of the effects of acidification measures on nature (SP), the benefits of water quality improvement (SCBA), various, economic evaluation of dike management, and the application of the Dutch national guideline for monetarizing ecosystem values. Estimating the influence of individuals’ environmental behavior and knowledge on their WTP for sustaining specific ecosystem services provided by biodiversity, with a case study in the Donana National and Natural Park, Spain (RP). Estimating the WTP for biodiversity conservation of 15 selected species in the Donana National Park (RP). Focusing on biodiversity conservation and enhancement on farmland in English with two case studies: Cambridgeshire and Northumberland. List of literatures containing values of various species in United Kingdom, Scotland, EU, and Finland, looking at different aspects of the species, including their own estimations (SP). Valuation studies cover main issues, including tourism, the regulating functions of forest, a number of national parks, and the exploration of Alpine water resources (RP and SP).

Focus

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Martin-Lopez et al.

The Netherlands

The Netherlands

Region

(Continued)

9:44

Witteveen and Bos

Service

Submission By

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169

Brotherton (Study authors: Fisher et al.) Brotherton (Study authors: Christie et al.) United Kingdom

Great Britain

United Kingdom

Value of biodiversity enhancement in Cambridgeshire and Northumberland

Provisioning services and regulating services provided by various English ecosystems Blanket peat in the English uplands

English Nature

Worldwide

Jamaica

Biological goods and services

Bernstein

Research Tools in Natural Resource. . .

(Continued)

(1) Estimating the WTP for a number of policies that would avoid or reduce biodiversity decline (SP). (2) Assessing the value of four attributes of biodiversity: familiar species of wildlife, rare unfamiliar species of wildlife, species interactions within a habitat, and ecosystem services (SP).

Estimating the important role of blanket peat in agricultural, grouse management, recreation, culture, water catchments, and carbon storage. Increasing biodiversity is important to insure a supply of ecosystem services.

170

The Netherlands

Demonstrating the fundamental role of measuring biodiversity value in introducing incentives for resource conservation. Concerning a number of ecosystem services, including tourism and recreation, water management, real estate auctions, production and products, regulating services and other intrinsic values of nature. Regulating services provided by tropical forests; economic contribution by world’s ecosystems; and agriculture-related services providing sustainable livelihoods for people. Including the value of salt marshes in flood defence, tourism benefits from visits to the country, and commercial harvesting of fruits and reedbeds.

Focus

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Insurance value of biodiversity

Economic benefits derived from nature

K¨ alberer

Worldwide

Region

(Continued)

9:44

Brotherton (Study author: Crowle)

The economic value of biodiversity

Service

Pearce

Submission By

Table 5.4:

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Sabah Abdullah, Anil Markandya, and Paulo ALD Nunes

Case study of the Norfolk Broads (SP).

United Kingdom

(Continued)

Case study of the Cley Marshes in Norfolk (SP).

United Kingdom

Total benefits of avoiding flood damage in intertidal habitats and wetlands Benefits of avoiding further damage to habitats from coastal flooding

WTP for wetland flood control function (metaanalysis).

Estimates the mean WTP to avoid the final losses of flooding. (SP). Concerning wetland services: flood defence, storm defence, and recreational functions (metaanalysis).

United Kingdom

United Kingdom United Kingdom

Soil erosion caused by dredging of stream channels, sediments washing onto roads, etc. (NS).

Wetland service

The value of individual wetland services

Cost of flooding

United Kingdom

Benefits transfer

The results provide WTP for four different forest management standards for achieving different levels of biodiversity (SP). Valuing in terms of forgone net pollution costs or net benefits of having trees.

Focus

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Brotherton (Study authors: Clarkson and Deyes) Brotherton (Study author: RPA) Brotherton (Study authors: Woodward and Wui) Brotherton (Study authors: Brouwer et al.) Brotherton (Study authors: Klein and Bateman) Brotherton (Study authors: Bateman et al.)

Great Britain

Mortality and morbidity benefits of air pollution absorption by woodland Value of the Danube floodplains as a nutrient sink Cost of soil erosion in the United Kingdom United Kingdom

United Kingdom

Region

Biodiversity benefits of woodlands

Service

(Continued)

9:44

Brotherton (Study authors: Grarrod and Wills) Brotherton (Study authors: Powe and Willis) Brotherton (Study author: Gren)

Submission By

Table 5.4:

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Introduction to Economic Valuation Methods 171

United Kingdom United Kingdom

Value of a waterside location on housing prices

Value of knowledge about geodiversity

Costanza et al.

Van den Hove and Moreau

17 ecosystem services across 16 biomes Biodiversity related primary production

(Continued)

Research Tools in Natural Resource. . .

North America

Worldwide

United Kingdom Worldwide

Four geological locations are studied: the whole site at Wren’s Nest NNR, the Seven Sisters cavern within the NNR, the Jurassic Coast WHS, and the Isle of Wight (NS, but likely RP). Shooting benefits and cost of biodiversity management and conservation. Concerning 13 deep-sea habitats and ecosystems and 9 related goods and services, e.g., food production, deep-sea oil and gas wells, nutrient cycling from the oceans, etc. Provide an estimate of the current global economic value of ecosystem services. Estimating the impact of increase in biodiversity on the value of ecosystem services in warmer climates.

Two locations are studied: Greater London and the Midlands (RP).

Different geographical settings are considered (SP).

172

Recreational values from hunting Deep-sea habitats and ecosystems

England

Value of the wooded landscape

9in x 6in

Danby

United Kingdom

Recreational value of RSPB reserves

Brotherton (Study authors: Rayment and Dickie) Brotherton (Study author: Garrod) Brotherton (Study authors: Garrod and Willis) Webber et al.

Estimating the incremental effect of wetlands using the impact-pathway approach. The study is based on data from the most extensive valuation study of woodland recreation, including 6 woodland sites across England (RP + market price). RP

Focus

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United Kingdom

United States

Region

Wind storm protection of coastal wetlands Recreational function of woodlands in the United Kingdom

Service

(Continued)

Brotherton (Study author: Farber) Brotherton (Study author: Scarpa)

Submission By

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Sabah Abdullah, Anil Markandya, and Paulo ALD Nunes

Valuing the conservation of ecosystems and natural capital Dynamics and values of ecosystem services Landscape, wildlife, and recreational amenities of woodlands

Balmford et al.

England

Source: Markandya et al. (2008). (RP): Revealed preferences methods; (SP) Stated preferences methods. (NS): Valuation method has not been specified.

Brotherton (Study author: Bishop)

Worldwide

Worldwide

Worldwide

Region

Two woodland sites are studied: Derwent Walk and Whippendell Wood in England (SP).

Concerning global spatial distribution of marketed and non-marketed economic value. Estimation of the rates of change in the extent of six biomes to estimate the net loss in value from ecosystem conversion. GUMBO model

Focus

9in x 6in

Boumand et al.

Ecosystem services

Service

Sutton and Costanza

Submission By

(Continued)

9:44

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Sabah Abdullah, Anil Markandya, and Paulo ALD Nunes

Christie et al. (2004) note that there have been a lot of studies (mostly in the United States using SP techniques) looking at valuing particular species — see Nunes and van den Bergh (2001). Valuations per species range from $5 to $126 per household per year and for multiple species from $18 to $194 per household per year. In the United Kingdom, MacMillan et al. (2001) looked at wild goose conservation in Scotland and White et al. (1997; 2001), looked at four mammals — otters, water voles, red squirrels, and brown hare. MacMillan et al. (2001) looked at the reintroduction of species (beaver and wolf) in the native forests of Scotland. They also note that there have been a range of studies on habitats — using either recreation/tourist value approaches or valuation using SP methods (e.g., CV). Work includes those of Garrod and Willis (1994), and Hanley and Craig (1991) on upland heaths in Scotland, and Macmillan and Duff on restoring pinewood forests in Scotland. Willis et al. (2003) extend this work to examine public values for biodiversity across a range of UK woodland types. Other studies have assessed public WTP to prevent a decline in biodiversity. For example, MacMillan et al. (1996) measured public WTP to prevent biodiversity loss associated with acid rain; while Pouta et al. (2000) estimated the value of increasing biodiversity protection in Finland through implementing the Natura 2000 programme. White et al. (1997; 2001) examined the influence of species characteristics on WTP. They concluded that charismatic and flagship species such as the otter attract significantly higher WTP values than less charismatic species such as the brown hare. For the remaining non-market biodiversity values that are estimated with the use of SP valuation methods, a number of points should be noted as discussed below. First, topics covered included wildlife, national parks and nature reserves, water courses (non-fishing use), recreational fisheries, landscape, endangered species, wetlands, and woodlands. So, the question is: are we valuing environmental resources (including biological resources) in the name of biodiversity (the diversity of biological resources)? Second, there is a heavy dominance of studies from the United Kingdom or parts of the United Kingdom (England, Wales, Scotland), with particular emphasis on significant countryside habitats, forested areas, or charismatic and flagship species. Another emerging question is: how reliably can we use these estimates for policy, especially when we use the value transfer methods to other sites and countries, which are not characterized by the same flag species?

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Gaps in knowledge, and suggestions for future research

The TEEB report highlighted the following points with respect to gaps in knowledge found among various valuation studies. The value of indigenous knowledge in the conservation of biodiversity and biodiversity value of marine resources — especially deep sea resources — is an underresearched topic. Primary valuation studies are still lacking for ecosystem services provided by deserts, tundra, ice/rock, and cropland. There are few studies on the value of changes in the delivery of goods and services arising from conversion of natural habitat. In particular few suitable studies were found for 10 of the natural biomes, including rangelands, temperate forests, rivers and lakes, and most marine systems. Most valuation topics covered included wildlife, national parks and nature reserves, water courses (non-fishing use), recreational fisheries, landscape, endangered species, wetlands, and woodlands. There is poor distinction between the valuation of biological resources for their biodiversity and for their contributions as biological resources in themselves. More work is needed to provide the keys for an accurate interpretation of the difference between the two. Also, there is a heavy dominance of UK studies as noted earlier, with particular emphasis on significant countryside habitats, forested areas, or charismatic and flagship species. It is unclear how reliable these estimates may be for policy, especially when we transfer the value to other sites and countries, which are not characterized by the same flagship species. There are still major gaps in the science and economics of the valuation of ecosystem services, including the definition of ecosystem boundaries, the need to improve environmental understanding and the effective valuation of genetic material. Others have noted the gap in studies that consider marginal effects in a proper way (rather than assuming they are equal to the average impact). There is a need to properly integrate our knowledge of ecosystem dynamics into an economic assessment of land-use options. Moreover, most of the economic evidence available concentrates on a single use of a single good of a given ecosystem, while most ecosystems provide multiple services that are interrelated in complex ways. Also, any economic review of the costs of biodiversity must factor in the climate change. We contribute to significant climate change impacts on biodiversity, which in turn is responsible for important feedbacks on human welfare. There are gaps in the public’s understanding and awareness of biodiversity.

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By improving information and awareness, we may achieve significant gains in conservation at a modest cost. There is a fundamental uncertainty regarding the minimum level of ecosystem structure needed to provide a continual flow of services. Related to this is the importance of recognizing the discontinuities in the biophysical relationships that govern ecosystem service provision and that are critical to a valuation of their loss. More accurate information is needed on the populations who stand to gain or lose from changes in biodiversity and ecological services. An example is given about the benefits of woodlands, where it is difficult to identify the populations for which the different categories of social and environmental benefits are to be aggregated. Present valuation systems based on individualistic and reductionist approaches are inadequate and what is needed is a different paradigm. One suggestion is to focus on valuing the limitations imposed on the development of human societies by ecological constraints, such as the “steady-state economy” model of Hermann Daly. Another is to adapt a valuation system based on “fairness” and collective values. Methods are needed to identify priority areas for biodiversity conservation that minimize conflict with agricultural productivity. In fact, a similar point has been made in recently published TEEB report (TEEB, 2008). In fact, it states that notes that protecting the largest area within an ecosystem does not equate to the greatest protection for its biodiversity. Related to the above, there is a need to know more about how to identify the extent to which the goals of safeguarding biodiversity and securing ecosystem services are consistent, and at what point there is a trade-off between the two. Most of the valuations address the issue of biological resources rather than biodiversity per se. We are all diminished by a world that is increasingly homogenized, but there is as yet no credible estimation of the cost of diversity loss. This is arguably an important research question. Only a few agricultural studies demonstrate true farm system costs for the loss of genetic diversity in land races and domestic breeds. Costs associated with the loss of naturally occurring diversity are uncertain. With regard to suggestions for future research one can obviously note the need to fill the gaps identified in the list. The following were pointed out by report contributors as the future directions: (1) Closing the gap between ecological economics and other areas of economics using experimental approaches and models of behavioral economics.

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(2) The greater use of market-based approaches including auctions as a means of eliciting values and determining the demands for conservation. Such an approach was applied in the Netherlands (e.g., Triple E) and is reported in the market research. (3) Researching the emerging issues of how the depletion of biological resources may heighten global insecurity and conflict. Most of the times, the distributional aspect regarding the allocation of costs and benefits reveals to be crucial in accepting or rejecting a given conservation practises. Hardly any research is available on this. (4) Future research in relation to ecosystem services should consider a range of interdependent ecological functions, uses and economic benefits at a given site; or track changes in site values across different states of ecological disturbance. Future research should investigate the necessary conditions for incentive measures and their relative merits in relation to different locations, ecosystems, and types of ecosystem stress. (5) The present analysis is static (concerning the global spatial distribution of marketed and non-marketed economic value). It needs to be extended to a dynamic analysis in order to provide more useful information on the trends in the value of ecosystem services and sustainability. (6) Most economic valuation exercises miss a common platform of biodiversity analysis. According to the ecosystem approach, one needs to provide a detailed catalog of ecosystem services and address the value of ecosystems and ecosystem services, as the basis for understanding the value of the biodiversity that underpins those services. (7) Ecosystem management options should be evaluated by coupling service change assessments with valuations of these changes. From the evidence of the gaps and future direction, a critical analysis of the submitted evidence suggests that we are witnessing a progressive loss of biodiversity. This is the cause of significant welfare damages. Second, one can also conclude that the economic valuation of changes of biodiversity, and its impacts on the provision of ecosystem goods and services, requires, inter alia, that a clear biodiversity change scenario is formulated, that changes are within certain boundaries, and that the perspective on biodiversity value is made explicit. So far, relatively few valuation studies have met these requirements. In fact, most studies lack a uniform, clear perspective on biodiversity as a distinct, unequivocal concept. Against this background, the Millennium Ecosystem Assessment is now recognized as a key reference for assessing the economics of biodiversity

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loss. However, to the present date, we have insufficient knowledge about, for example, how the functioning of ecosystems relates to the production of ecosystem goods and services and what is the underlying role of biological diversity within this complex relationship, so that for this reason alone it is very difficult, if not factually impossible, to assess the total economic value of biodiversity. Even if one admits that we could place a value on a set of goods and services represented by all ecosystems, and remembering that at present scientists still do not have sufficient knowledge to map and calculate the full range of ecosystem goods and services (across all the different types of world ecosystems), we would still be unable to answer the question “what is the value of biodiversity?” To answer this question, one would also have to include: (a) the role of genetic variation within species across populations and its impact on the provision of ecosystem goods and services, (b) the role of the variety of interrelationships in which species exist in different ecosystems on the provision of ecosystem goods and services, and (c) the role of functions among ecosystems on the overall level of provision of ecosystem goods and services. Therefore, and without any doubt, a full monetary assessment will be impossible or subject to much debate. An important reason for the latter is that global level values are difficult to compare due to an equal international income distribution. All in all, the available economic valuation estimates should be considered at best as a lower bound to an unknown value of biodiversity, and always contingent upon the available scientific information as well as the global socio-economic context.

5.5.

Challenges for Valuation Studies

There are challenges facing valuation studies both conceptually and practically. Based on the TEEB submissions and this group’s work, we would argue that the following issues need to be addressed in valuation work: gaps in the studies, the application of valuation studies using surveys, incremental value, addressing multiple services, and the adding up and benefit transfer. First, after all the work on valuation, there still remain areas where credible and reliable economic values are not available. As noted these include many species, marine ecosystems, cultural and spiritual values, and dynamic dimensions of all values. They also include damages to ecosystems

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from air and water pollution (Ecolas, 2007). In all cases further work on valuation is worthwhile, using innovative combinations of market and nonmarket techniques. For damages from air and water pollution, Ecolas has already made a number of recommendations for such studies. In other cases, a similar list should be drawn up to fill the gaps that have been referred to in this paper. But it is only realistic to assume that it will be some time before a comprehensive set of estimates is available. In the interim, we need to rely more on cost effectiveness tools, and to focus the development of these tools around the indicators for the biodiversity strategy for the EU, as reflected in the set of indicators drawn up by the European Environmental Agency (EEA). Not all those indicators are amenable to such an analysis but many are. It should be a matter of urgency to develop these tools and make them available to decision makers responsible for allocating funds and introducing and implementing conservation policies. Second, the other challenges facing application of valuation studies in particular to SP surveys include: low response rates, biases and effects, restricted budget amounts, weak questionnaire design, and lack of validity and reliability. Of these challenges, the biases and/or effects tend to influence the WTP/WTA values. Typically, these biases distort the WTP/WTA estimates. Some of these biases include: starting point bias, strategic bias, hypothetical bias, as well as the question order, and temporal embedding effects. For CV studies, there are four main sources of biases: (1) use of a scenario that has strong incentives for respondents to misrepresent their true WTP; (2) use of scenarios that have strong incentives to prompt respondents to depend on the scenario for WTP; (3) misidentification and wrongly describing parts of the scenario, and (4) incorrect sampling design (or execution) and improper ways of aggregating benefits (Mitchell and Carson, 1989). The guidelines provided by the NOAA panel with respect to CV were drawn up under consideration of situations found in the United States and developed countries. It is worth noting that developing countries differ from developed countries in their social-economic and political structures, making the NOAA recommendations relatively difficult and costly to implement in the former, as against the latter. Despite these challenges facing valuation researchers, the future application of these methods in both developed and developing countries seem plausible as long as the gaps and challenges discussed earlier are acknowledged and addressed appropriately. Third, the valuation of goods or service to be useful there has to be an incremental value. There is little advantage in knowing the total value of an ecosystem unless there is a threat to eliminate it or a policy to reconstruct it

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in its entirety, which is rarely the case. Yet, many valuation studies provide estimates of the total costs of whole systems and there is even one of the value of the whole world’s ecosystems (Costanza et al., 1997). Carrying out an incremental analysis (which may entail estimating significant nonmarginal changes in ecosystems), however, is not as easy as it might sound. If one is using revealed preference methods, a link has to be established between a change in the environmental attribute and the demand for a visit to a site or the value of a property. If one is carrying out an SP analysis, the respondent has to understand the nature of the incremental change, which is more difficult than asking for the value of access to a site or use of a particular recreational facility. Incremental analysis under SP is perhaps a little easier with CE rather than CV. Fourth, valuation studies need to address multiple services and the “adding up” problem. Many ecosystem services that individuals receive are multidimensional and there is an adding up problem. The value attached to one forest area for recreational or other use is not independent of whether another forest nearby is conserved or not. The implication is that studies need to be undertaken allowing for substitution effects, which makes them more specific to a particular application and less capable of being transferred to other applications. Finally, the issue of benefit transfer leaves a debating question: to what extent can these values be transferred from one site to another and from one type of service to another? Economists have devoted a great deal of effort to see how far such transfers are possible, given that full valuation studies are expensive to conduct. The most comprehensive way to carry out transfers is to use a “metaanalysis,” which takes all existing studies and estimates a relationship which gives changes in the benefit values as a function of site characteristics, attributes and size of the population affected, type of statistical method used, etc. in the sample of existing studies. This is then transferred to the policy site in a procedure referred to as value transfer, which can provide a single value for the policy site or a “value function,” which gives a range of values depending on the characteristics of the object of valuation. Metaanalyses are available for urban pollution, recreational benefits, recreational fishing, water quality, wetlands, visibility improvement, price elasticity of demand and travel cost, valuation of life, and valuation of morbidity (Nijkamp et al., 2008).7 These provide the best 7 Nijkamp et al. (2008) also refer to other methods of making a transfer such as rough set analysis, fuzzy set analysis, and content analysis. Since there is no assessment of

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method of transfer, although one should note that many values are very location specific and the transfer can never be perfect. Although use is made of metaanalyses and value transfers in estimating costs we do not have an idea of the extent of such use. Transfers of estimates are unlikely to enter databases such as EVRI, because they do not involve original estimation. It would be helpful to know how many policyrelated studies have indeed used proper metaanalyses to derive environmental values in a policy-making context. Recent discussions on the subject seem to suggest that while value transfer is easier for some ecosystem services it is less easy for others. It should be possible, for example, to derive estimates of some categories of recreational benefits (including recreational fishing), improvements in water quality, carbon sequestration, and perhaps visibility. It is more difficult to carry out credible benefit transfer, except in a very localized way (e.g., estimates for one landscape or land-use pattern to another that is close by) for other categories of value. As noted under the “adding up problem” different combinations of benefits cannot be valued by adding up the individual benefits of ecosystem services. Notwithstanding such difficulties, the international community urgently seeks estimates of the foregone benefits from biodiversity at the European Union (EU) and global level. The current initiative on the economics of biodiversity loss explicitly states that it will seek to estimate the “economic significance of the global loss of biodiversity.”8 Given that there are thousands of ecosystems and sites of importance within the EU, let alone the whole world, it is impossible to conduct individual studies to obtain the relevant information in a timely way.9 Hence, some kind of benefit transfer will be essential if the goal of obtaining national, regional, and global estimates of the damages from biodiversity loss in the absence of any action is to be obtained. The same applies, a fortiori, to estimating the reductions in such damages when some actions to protect the ecosystems are implemented. The answer to this problem consists of working in parallel at two levels. The first is to develop rules of thumb for acceptable estimates of the overall

their application to biodiversity and ecosystems one cannot comment on how useful they might be. 8 Background Paper for the Working Group, 1st Meeting on the Review of the Economics of Biodiversity Loss, European Commission, Brussels, 21st November 2007. 9 Taking the Nature 2000 sites in Europe and dividing them into the 27 habitats, we have over 75,000 ecosystems whose services need to be valued.

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costs of biodiversity loss and the second is to improve the application of benefit transfer for specific evaluations of ecosystems service benefits. To tackle the problem first, rules of thumb will be needed, based on rough values, for all the ecosystems under threat. The cost of policy inaction (COPI) project that is just being delivered to the European Commission has this as its objective (COPI, 2008). It cannot be expected that this project will arrive at a real scientific assessment of the costs of inaction but it may be able to provide credible orders of magnitude, based significantly on the market-based losses of services from the expected degradation of ecosystems. Coverage of non-market costs will be much less complete. Whatever estimates are obtained should be subjected to as much scrutiny from the scientific community as possible and revised in the light of responses to provide some headline figures that are broadly correct. The exercise will necessarily ignore many of the guidelines for benefit transfer. In the light of that, it will need some way of deriving approximations in the right ballpark and further work in this area will almost certainly be required. To tackle the second approach we need to establish a clear set of guidelines about which kinds of benefit transfer are possible. Such guidelines need to stipulate not only the kind of ecosystem services but also the areas and countries where the transfer can be carried out given the available set of valuation studies. Second, further research should be carried out on how “packages” of ecosystem services may be valued without undertaking whole new studies. It may be possible to develop approximations for adding up benefits that can be individually transferred but where there is an adding up problem. Third, where transfer is not possible toolkits should be developed that can be used to carry out location-specific studies. Given the large database of existing studies, these can help simplify and demystify the process of valuation so that it can be conducted more routinely and more cheaply. Finally, an inventory of all major ecosystems should be drawn up and the loss of services expected under different scenarios should be prepared. Some of this is underway for some ecosystems but not for all the important ones.

5.6.

Conclusion

This chapter introduced the various valuation approaches for both market and non-market values and recommended important steps in conducting valuation exercises using the SP methods. Also, from the evidence

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submitted to TEEB, we can conclude that while methods have been developed and used widely for some environmental values (especially market values as well as recreation and amenity), there are still several gaps to cover. Notable gaps in biodiversity and ecosystem application are among valuation of loss of species other than headline species; marine ecosystems; cultural and spiritual values; and the dynamic aspect of all ecosystems and values — changes over time are at the core of all ecosystems. Also, the future outlook seems promising particularly for SP studies in developing countries, where with the lower costs in administering these studies using face-to-face interviews. In summary, it is likely that governments or multilateral organizations will continue to carry out more market and nonmarket approaches, to evaluate welfare impacts related to the conservation of biodiversity and the provision of ecosystem goods and services. This signals, in turn, a cultural change in the field since we are no longer working within the academic and research grounds, we are now providing crucial information to support the definition of policy programs in the environmental sector, and therefore play a significant role along with all the policy practitioners.

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White, PCL, AC Bennett and EJV Hayes (2001). The use of willingness-to-pay approaches in mammal conservation. Mammal Review, 31(2), 151–167. Willis, KG and GD Garrod (2001). Estimating the benefits of alleviating low flow in the River Darent. In Economic Valuation of Water Resources: Policy and Practice, D Moran and P McMahon (eds.). Chartered Institution of Water and Environmental Management (CIWEM), London. Willis KG, G Garrod and R Scarpa (2003). Social and Environmental Benefits of Forestry Phase 2: The Social and Environmental Benefits of Forests in Great Britain. Report to the Forestry Commission, Edinburgh.

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Chapter 6

The Hedonic Method: Value of Statistical Life, Wage Compensation, and Property Value Compensation Philip E. Graves University of Colorado [email protected] In this chapter, we examine the hedonic approach to inferring the benefits of an environmental or resource policy. Initial emphasis will be given to value of a statistical life (VSL) turning later to the use of hedonics for more general valuations of environmental quality. For this method to work well in any context, the nature of perceptions is of crucial importance, as discussed further below. For accurate valuations of a statistical life, households must have very good, ideally perfect, perceptions of risks of alternative courses of action (e.g., probabilities of workplace death by occupation or probability of death with and without automobile seat belt or smoke detector use). Similarly, accurate valuations of an environmental improvement require that households have perfect perceptions of both (1) where it is clean and dirty and (2) what various levels of environmental quality mean to our health and welfare. Under such strong assumptions, one would expect people to ponder how to avoid risks of death, on the one hand, or pollution damages, on the other. Indeed, as long as the marginal costs of avoiding damages are less than the marginal benefits of avoiding damages, we would expect people to continue to avoid damages until marginal costs and benefits are equated, the insight that underlies the hedonic approach to valuation.

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The technical details of the hedonic valuation methodology, whether applied to VSL or more general environmental values, have been the subject of many articles and reviews (see Bateman et al., 2001; Blomquist et al., 1988; Day, 2001; McConnell, 2006; Palmquist, 1991; 2004; Roback, 1982; Rosen, 1975; 1979; Sheppard, 1999; Viscusi and Aldy, 2003; Taylor, 2008, among many contributions). As a consequence, and due to the desire for brevity in this chapter, discussion is limited to the intuition given at the beginning of each section below. Suppressing the technical apparatus, which will be familiar to most readers, allows greater focus on important issues of appropriate interpretations of the hedonic results, interpretations that are often ignored, if indeed they are even widely known. 6.1.

The Value of a Statistical Life (VSL)

While there are several hedonic analyses of VSL that employ non-labor markets (e.g., seat belt use, automobile markets, and smoke detectors) the bulk of studies in this area obtain their insights from labor markets, examining the implicit wage compensation for varying risk of death. In these studies the dependent variable is wages (or ln wages1 ) of individual workers which is regressed upon a vector of individual personal characteristics (e.g., age, education, race, sex, experience) and job characteristics (e.g., occupation, industry, union). The risk of death, although quite controversially measured, is then included. In the linear specification, for example, letting risk of death be measured as annual deaths per 10,000 workers, a coefficient on that variable of $.25/h would imply that averting one statistical death is worth $5 million dollars (i.e., $.25 times 2000 h equals $500/year compensation for a 1/10,000 increase or decrease in the probability of death). The preceding example should not be interpreted as if all workers were identical in their attitudes toward risk, and as if producers had no control over on-the-job risks. If this were the case, arriving at an accurate measure of VSL would be much more straightforward than it is in practice. Holding all other causes of wage variation constant statistically, the analyst could estimate the representative individual’s indifference curve between risk and wage compensation. Such an indifference curve would likely be non-linear 1 Some

studies use a Box–Cox transformation of the dependent variable for greater specification flexibility. Typically estimated relations lie between the log and linear specifications.

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with constant increments to risk requiring ever-larger amounts of compensation for that risk. But this is not what is being estimated; rather, the estimated equations are reduced form expressions reflecting the interaction of many individual buyers and sellers of labor, hence represents a marginal price locus of the individual equilibria. It is sometimes the case that a second-stage analysis is undertaken to identify the actual individual demand curves for risk and how those might vary with price, income, age, and so on. Most typically, the data requirements of the second-stage preclude that estimation, and it is, in any event, the marginal price that is of primary relevance in a benefit–cost analysis of risk-reducing policies for relatively small changes in risk. There have been a large number of studies attempting to estimate the value individuals place on these small changes in the probability of death (Viscusi and Aldy, 2003, for example, reviewed 60 studies, while Mrozek and Taylor (2002) provided a metaanalysis of over 40 studies). Viscusi and Aldy argue that perusal of the VSL literature leads to a VSL range of $4 million to $9 million, with an average value of $7 million, while Mrozek and Taylor feel that a more appropriate range would be $1.5–2.5 million. Moreover, the individual studies underlying such surveys exhibit much greater variance from very small to very large VSLs variance that would seem to be larger than expected even allowing for sampling from an underlying perhaps log-normal distribution of individual values.2 What accounts for such implausibly large differences? Real-world studies suffer from a number of limitations that give rise to widely varying estimates of wage increases necessary to compensate for an increment/decrement in risk of death. A list of these limitations would include: (1) Measurement error in key variables. The single most critical variable in VSL research is the measure of risk of death on-the-job. In an ideal world, one would wish to employ subjective measures of risk as perceived by individuals. This information is seldom available, particularly for the large samples necessary to obtain precision in what must certainly be a small determinant of overall compensation. In studies 2 One

would additionally expect the individual values to vary according to income, as well as age and cohort. As to income Viscusi and Aldy find an income elasticity of around 0.5 or 0.6 for VSL this accounts for the controversially different VSLs in the developed versus the developing world. Aldy and Viscusi (2008) also find important age and cohort effects.

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employing objective probabilities, some have used overall risk of death, rather than the lower on-the-job risk of death, which would lead to strong downward bias in resulting VSLs. (2) Omitted variable bias. Of particular concern is the probability of nondeath injury; this variable would likely have a high positive correlation with probability of death and its omission in a great many studies results in overstatement of VSLs. Additionally, there are many lessobvious omitted variables that are likely to be correlated with wages, and small biases can lead to large changes in VSL.3 (3) Selectivity/Publication bias. Many of the studies employ subgroups of the general working population for which sorting on the basis of risk preferences might be expected to occur. Jobs known to be very risky, for example, might attract those less averse to risk and conversely. Moreover, quite unusual results, either very low or very high VSLs in the present context, are not as likely to be published, which makes the high observed variance in VSLs perhaps even more disturbing. (4) Dread/Unfamiliarity variation. Since the way in which individuals would be likely to die in different jobs would vary, some causes of death would be expected to require greater compensation than other causes of death.4 This would suggest that policy-makers might wish to assign higher values for VSL for policies that might lower the probability of particularly unpleasant deaths, though the extant literature is insufficiently developed to allow this at present.

3 For example, Viscusi and Aldy observe that most studies of the US labor market find that union affiliation is positively correlated with a greater wage-risk trade-off while international evidence is much more mixed. They provide several arguments as to why the risk premium might be higher for union members (marginal versus average worker preference, the quasi-public good nature of workplace safety, and better safety information for the unionized). An alternative explanation, concentrated union membership in undesirable rustbelt locations where wages must be higher to compensate, can account for (1) the apparent higher risk premium in union jobs in the United States, (2) the failure to find that gap in the international setting (e.g., Canada, England), and (3) the anomalous finding in several papers that non-union workers, concentrated in sunbelt locations, appear to have negative compensating differentials for risk (See Graves, 2009 for further details). In addition to locational correlations, job traits themselves (e.g., harsh environments, intense seasonal fishing, dirt, or smell) must be controlled in ways that are unlikely in practice. 4 VSLs would appear to be lower for more common or pleasant risks (e.g., driving versus flying, seat belt/helmet use, smoking, or rock climbing) while some ways of dying would likely yield very large VSLs (e.g., radiation or pesticide poisoning) despite extremely low relative probabilities of death.

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Despite the concern expressed here and in many discussions of VSL, public policy-makers are in an inescapable bind; any decision, including the decision to do nothing, is going to have physical effects on the probability of death. Thinking seriously about the valuation of those effects is likely to lead to better decisions than not assigning them a value at all, implicitly leaving that to decision makers. While not yet perfect, current ranges of VSL found in the literature (about $2–9 million with a mean of $4–6 million in 2009 dollars, with a point guestimate of $5 million, although most government agencies are currently using about $7 million) offer some guidance in choosing among a wide range of potential projects. Some projects will be easily rejected or accepted with any reasonable number for VSL, while others will be sufficiently marginal that perhaps a sensitivity analysis to choice of VSL would be informative for decision makers. 6.2.

The Hedonic Valuation of Environmental Quality

Two ways that people can avoid pollution damages are by locating in cleaner towns and/or by locating in cleaner parts of a given town. We discuss the appropriate use of this method in detail in this section, since it is commonly misused and that misuse generally leads to downward-biased estimates of environmental values. The fundamental notion underlying all hedonic methods is merely that people like to make themselves as well off as possible, exactly the assumptions that we make about their behavior in ordinary markets. Other things equal, we would all prefer to live in a cleaner town or live in a cleaner part of a given town. The idea with hedonic methods is to examine how much households are willing-to-pay in land and/or labor markets to live in cleaner locations, since they will in general have to pay, as we shall see. The main ideas are really quite simple, but to gain a clear understanding of this method we shall first consider wage and rent compensation separately (as is often done, although this is in general incorrect as will become apparent). We then describe in considerable intuitive detail an integrated model that was first formally presented by Roback (1982) and later implemented empirically by Blomquist et al. (1988). We turn first to the labor-market approach since that is identical to the approach employed to obtain VSL. The only difference is that instead an interest in wage compensation for risk of death, the focus turns to wage compensation for undesirable environmental amenity levels which vary across labor markets.

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Hedonic Methods: Wage Compensation

Some labor market regions are more polluted than others, and households will have to be compensated for the pollution they experience to be willing to work in dirtier cities. That is, if City A (one of two otherwise identical cities) has higher pollution levels than City B, residents would move from A to B reducing the labor supply in A (raising wages) while increasing the labor supply in B (lowering wages). The movements would continue to occur until the wage differential just compensated people for the higher average pollution in City A. One extremely desirable feature of this approach is that it gives us exactly what we want, the marginal willingness-to-pay in dollar terms, which can then be compared to the marginal costs of policies yielding that amount of cleanness. The process for the wage compensation approach is virtually identical to that employed in the efforts to obtain VSL discussed in the preceding section: (1) First, obtain as much data as possible on individual wages as well as the determinants of wages for people at various locations (education, experience, age, occupation, region, union, etc.) and add measures of environmental quality levels in those locations to the independent variables. (2) Next, perform a regression analysis that statistically relates the wage (as the dependent variable) to those determinants. As was the case for VSL (and will be seen below to be the case for property values as well), there is little theoretical guidance on functional form degree of linearity, interactions among variables, and so on. This raises the possibility that researchers inadvertently, and advocates deliberately, might distort environmental values by their choices along a number of dimensions, discussed in greater detail below. (3) The coefficients on the environmental quality variable will indicate how much impact a given change in environmental quality has on wages, holding constant other wage determinants. In this way, as with VSL, the trade-offs between environmental goods and other goods that people also value can be directly measured. Since higher levels of environmental quality are a desirable trait of a labor market area, we would expect that wages would be lower in the high-environmental quality locations since the supply of labor would be greater to such areas.

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As was the case for VSL labor market studies, environmental values generated in this way can either overstate or understate true values. Omitted other goods that might be positively correlated with environmental quality, as for example the presence of an ocean or a symphony orchestra, would tend to overstate the value of environmental quality. Alternatively, if environmental quality differences across labor market regions are not perceived or if people are unaware of how environmental quality affects them, the true benefits of cleaning up will be understated by this method one would not expect households to accept wage cuts for unperceivable benefits. However, a large number of wage studies (see Bockstael and McConnell, 2007 for a nice review in the present context) indicate that households are willing to give up wages to live in cleaner locations. 6.4.

Hedonic Methods: Property Value or Rent Compensation

The property value or rent compensation method of hedonic valuation translates the logic that underlies the labor market studies to the housing market.5 How much a house will sell or rent for is clearly related to the nature of the traits that the house possesses. Some of those traits are structural, such as whether it is constructed from stone or wood, square footage, number of bathrooms, size of lot, presence of pool or tennis court, type of heat, and so on. Other traits relate to location such as neighborhood variables (school quality, freedom from crime, access to various destinations, and so on). These latter traits are location-fixed public goods whose prices end up being bundled together into the price of the house along with its structural traits.6 Environmental quality, viewed from this perspective, is just another location-fixed trait that is desirable from a household’s perspective. 5 Most studies are actually conducted with property values, but there will be similar relationships between either rents or property values and various amenities for either measure (and some studies merge rents and property values, with an assumed discount rate this is somewhat problematic in that areas of expected growth have higher than average property value to rent ratios, while areas of expected decline have lower than average property value to rent ratios; since areas of expected growth would, of course, also be higher amenity locations the process of using a single discount rate introduces measurement error into the dependent variable). We use both terms interchangeably, since most of the present discussion is qualitative rather than quantitative in nature. 6 Note that the location specificity essentially transforms such public goods into private goods, bought in a housing bundle.

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Assuming perceptions are perfect and that we have a competitive housing market, the value of clean air must be paid for in higher prices for houses in areas having higher air quality. If we can determine how much people are willing to pay for an otherwise identical home in a clean location versus a dirty location, we will again have a measure of exactly what we want, the marginal dollar willingness-to-pay for environmental quality, which can then be compared to the dollar marginal cost of environmental quality. The process is quite similar to the wage hedonic approach: (1) First, obtain as much information as possible about the traits: structural, neighborhood, and environmental quality of all houses (in what is hopefully a large sample), along with their property values and/or contract rent. In an ideal world, the property value (the dependent variable) would be the actual sales price, but sometimes information is used from multiple-listing books, scaled up or down by the going ratio of list price to exchange price. (2) Next, regress the dependent variable, property value, against its structural and neighborhood determinants. Note that this examination involves many possible functional forms and that non-linearities, synergisms, etc. may be important.7 As with the wage hedonic approach, there is little theoretical guidance on the nature of the functional relationship between property values and their determinants which enables researchers accidentally, and advocates intentionally to publish very different conclusions, even from identical raw data.8 (3) The coefficients on the environmental quality variables reveal how much impact a given change in environmental quality has on property values for average households.9 That is, the trade-off between environmental quality and other goods can be directly measured, and since higher 7 One might generally expect marginal damages to be an increasing function of the level of pollution and this can be readily tested for by putting in quadratic terms, using logarithms, etc. Interaction terms can explore, for example, whether the damages from, say, sulfur oxides depend synergistically on the amount of fine particulate present. See Graves et al. (1988) for a review of the robustness of property value hedonic estimates to a host of alternative treatments. 8 See Krumm and Graves (1982) for an application to air quality of a methodology devised by Zellner and Siow (1980) to eliminate biases when theoretical guidance on functional form is limited. 9 As with the wage hedonic method, a second stage in this analysis can reveal how individual demands for location-fixed amenities vary by price, income, family size, and other

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environmental quality is a desired trait, we expect to observe higher house prices or rents in cleaner areas, other things equal. As with wage studies, property value studies suffer from problems stemming from either data limitations or from the assumption of perfect information. If some other amenities are positively correlated with the environmental measure and those other amenities are omitted from the equation the value of the environment will be overstated. For example, suppose that the less polluted parts of a city are also more desirable for several other reasons (less crime, better schools, less graffiti, better streets, better lighting, more parks, etc.) and these other traits are omitted from the equation. By not including the other goods that are correlated with environmental quality, the impact of environmental quality will seem to be larger than it is, since the effects of the other non-included variables will be at least partially attributed to environmental quality.10 For a variety of reasons (e.g., that the cleaner parts of a city tend to be occupied by richer people) one would expect other spatially varying traits (e.g., school quality) to be positively correlated with environmental quality. With constantly improving data collection, this problem should become less important over time. Suppose, however, that people don’t fully perceive the impact of pollution on their health and well-being or how the pollution levels vary across locations or both. This is quite plausible, since even the experts have widely varying opinions about the amount of physical damage, particularly health damage stemming from pollution (see the sum-of-specific damages approach of Chap. 7 which follows the present chapter). Moreover, since many pollutants are odorless, colorless, and tasteless in ambient concentrations commonly encountered, it might be difficult for the average person to even know whether a particular house is in a high-pollution or low-pollution location. If buyers don’t properly perceive all of the damages from pollution or if they cannot tell which locations are dirtier, the benefits estimated by this approach will be understated. As with the case of wage compensation, people will not pay for something without tangible benefits. individual-specific variables. In other words, information can be obtained on the underlying demands for environmental quality that underlie the locus of observed equilibrium market matches between the many buyers and sellers. 10 The environmental quality coefficient will appear to be larger by the effect on property values associated with the omitted variable times the positive correlation of that variable with environmental quality.

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What is the net effect of these potential biases, one suggesting overvaluation, one suggesting undervaluation? Nobody knows with great certainty, although we shall take this up in greater detail in closing. Many studies, however, show strong positive relationships between property values and environmental quality; indeed, many more studies analyze environmental values via property value compensation than analyze them via wage compensation (e.g., Nelson, 2004 provides a metaanalysis of more than 20 studies of airport noise). The property value approach is particularly useful for valuing spatially concentrated environmental damages, for example the impact of toxic waste dumps on surrounding land values. As we shall see, however, sorting out the likely direction of bias is more complicated than it seems to this point.

6.5.

Wage and Property Value Differentials are not Alternatives

Until fairly recently,11 the preceding hedonic approaches to valuing environmental improvements were viewed as alternative approaches. That is, it was thought that one could find out what clean air was worth either by examining property value variation in land markets or by examining wage variation in labor markets. The approaches were viewed as alternative ways of measuring the same environmental preferences. Indeed, if the values happened to be similar under the two methods, greater confidence was placed in either as a measure. It turns out that this is incorrect under plausible assumptions about people’s behavior when evaluating locations. Indeed, for this view to be valid, it must be the case that people follow a two-stage procedure in picking a location. First, only looking at wages (and average pollution levels), they pick a location among alternative labor markets; only then, having settled on a labor market, do they select a location within that labor market based on housing price and pollution variation within that area. This would clearly be irrational since one would do much better in general by looking at the combination of wages, rents, and amenities available in all locations prior to selecting their location. 11 For

virtually any new theoretical insights (e.g., Roback, 1982 and operationalized by, Blomquist et al., 1988), it takes about two decades for those insights to be incorporated into decisions by politicians and practitioners.

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Another way to think about this is that, between two otherwise identical locations, the one that is more polluted will be less attractive — so, people will move from the more-polluted to the less-polluted location until they are equally well off in both locations. But, as they move into the less-polluted location they both increase the supply of labor (driving down wages) and increase the demand for land (driving up rents). Hence, the true value of the less-polluted locations is the sum of what is being paid for reduced pollution in both the labor and land markets.12

6.6.

A Graphical Exposition of the MultiMarket Hedonic Method

Imagine, initially, that the entire world were a flat, featureless plain, where all locations are literally identical. There is no variation in closeness to ocean, scenic views, and so on. There would be no reason to pay more for one location than for another. Moreover, imagine again initially (we will be dropping these restrictive assumptions shortly) that all households have the same preferences and all firms have the same profit functions. In particular, there is no variation in desires for lot size or income by households and no variation in the land or labor intensity of production processes. This case is depicted in Fig. 6.1. The upward-sloping curve, labeled V, represents rent and wage combinations that would be equally attractive to households i.e., if rents are higher in one location than another residents at that location would have to be compensated by a higher wage. If wages were not higher residents would be worse off in the high-rent city and would leave, driving down rents. With 12 There are many Rand-McNally Places Rated almanacs that rank cities according to quality of life. The approach taken is to focus on some number of traits (parks, school quality, crime, and so on) assigning a number from 1 to 5, with higher numbers being better. The numbers are added up and the city with the highest number is said to be best. There are many problems with this approach (notably, it weighs all traits equally, when people would presumably care much more about some traits than others). Also, and interestingly, under this approach, high rents and property values are usually taken to be a bad thing and result in a lower ranking (e.g., in one such study, Newark, NJ ranked much higher than Santa Barbara, CA in large part because of the higher cost-ofliving in Santa Barbara but of course it was not a higher cost-of-living but rather a higher benefit-of-living, a benefit that we just have to pay for!). The economic approach merely argues that the more we are willing to pay for the traits associated with a location, the better that location must be. For a well-conducted economic empirical study of quality of life, see Blomquist et al. (1988).

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Equilibrium rent and wage, when there is no amenity variation.

no variation in amenities, higher rents would have to be compensated for by higher wages.13 Similarly, from the firm’s perspective, since there is no variation in amenities that affect productivity over space (no deep-water ports or differential access to mine mouths), if rents were higher in a location, wages would have to be lower. If wages were not lower in that location, firms would be less profitable there and would leave, lowering labor demand and causing wages to fall. In such an incredibly boring world, we would observe identical rent levels and wage rates in all locations. If a location offered any wage/rent combination not at the intersection of the V and C curves in Fig. 6.1, it must be the case that either firms or households are better off or worse off at that location, a non-sustainable situation. If, for example, a location offered R0 rents but a wage rate that was greater than W0 , households would be better off at that location, while firms would be worse off at that location. Households would enter (increasing the supply of labor) and firms would exit (decreasing the demand for labor) and wages would fall. Firms leaving would cause rents to fall and households entering would cause rents to rise, as the wages fall. Whether rents rise or fall during the movement to equilibrium depends on whether households move in faster than firms move out. But, in a true equilibrium, satisfaction 13 The V curves are the locus of all wage-rent combinations that give exactly the same level of satisfaction. Normally, goods (public or private) are presumed to be what enters household utility, and we assume that more goods are preferred to less and that more leisure is preferred to less. The V curves are indirect utility functions since they are written in terms of prices rather than goods. Higher prices for goods (rents) are bad and higher prices for labor (wages) are good. Similarly, the C curve is the locus of wage-rent combinations that give exactly the same level of unit costs, which corresponds to profits for a good sold on national markets with low shipping costs.

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must be the same everywhere or people will move and profits must be the same everywhere or firms will move. If there is no variation in amenities that affect households or firms, rents and wages will become equal in all locations. Now let us begin dropping these unrealistic assumptions, first by introducing variation in an amenity that households care about, but which does not affect firm profitability at all. Perhaps, a scenic view comes into existence at one location, or one location becomes sunnier or lower humidity (or in our context, air quality improves, in some way not related to the costs of firms, perhaps via a light consistent breeze) vis-`a-vis other locations. At the ubiquitous initial wages and rents, the nicer location offers a higher level of satisfaction, so households would be expected to migrate to it. The influx of households will raise the demand for land (raising rents) and increase the supply of labor (lowering wages). Indeed, households would continue to enter until the wage decreases and rent increases rendered the nicer location no nicer than elsewhere; in other words, they would enter until the lower wages and higher rents exactly compensated for the nicer amenities.14 This case is depicted in Fig. 6.2, where the V1 curve shows the wage and rent combinations that would make households just as well off in the nice location as in other less-nice locations.15 That is, with the higher level of amenity, a
, in the nice place, households will move into that place until the increase in rents (R1 − R0 ) and decrease in wages (W1 − W0 ) just exactly compensate households for the niceness of the location. It should now be clear that treating rent hedonic compensation and wage hedonic compensation estimates as alternatives is generally incorrect to accurately measure the value of the amenity; the wage and rent compensation must be added together. Further clarifying, in the light of Fig. 6.2, a rent hedonic equation would indicate that the value of the amenity was only (R1 − R0 ), when if all 14 Two things should be noted. First, as people enter they might also lower levels of endogenous disamenities (e.g., air pollution or congestion in the Los Angeles case) along with raising rents and lowering wages. As long as all amenity variables are included in the analysis, this is not a problem, since the net niceness of the city will still be captured by rents and wages. Second, it might not take too many people actually moving to result in full compensation for the city’s niceness. This is akin to the fact that only a few drivers need to move from slow lanes to fast lanes on the freeway at rush hour to make all lanes equally fast. 15 The set of V curves, for various amenity levels, are called level sets because they all give the same level of satisfaction in equilibrium.

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Wage and rent compensation for a desirable household amenity.

compensation occurred in the housing market the true value of the amenity would be the much larger entire vertical distance from R1 to the original V0 curve. Similarly, a wage hedonic would suggest that the value of the amenity was only (W1 − W0 ) when the true value is the much larger horizontal distance from the V0 curve to the V1 curve. One might be tempted to say that the nice city has a higher cost-of-living, but, while this is commonly said, it is misleading. It actually has a higher benefit-of-living and competitive bidding in land and labor markets force households to pay for the benefits.16 Firms can have variation in important amenities as well as households. If a location is more productive for some reason, profit-seeking firms would want to move into that location to make greater profits. In fact, rational firms would move to the more productive location as long as they can achieve higher profits by doing so. But, as they move in they increase the demand for labor driving up wages and they directly drive up industrial rents (and indirectly drive up residential rents by hiring more employees). Eventually, these higher labor and land costs will offset the dollar value of the productivity-enhancing amenity. The situation is shown in Fig. 6.3, which is very similar to the previous figure except that now the amenity affects only the firm, while in Fig. 6.2 it was only the household that was affected by the amenity. As firms move into the location that is desirable for them, perhaps because of a deepwater port or nearness to a raw material, they will drive up both rents and 16 If some location trait is a disamenity to households, and is neutral to firms, the preceding discussion merely reverses all signs. Households must be compensated for the disamenity via some mix of higher wages or lower rents, with the V curve shifting to the right rather than to the left.

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Wage and rent compensation for a desirable firm amenity.

wages. They will continue to move into the productive location until its productivity is completely offset by the higher prices that must be paid for land and labor. Note that even though rents have gone up for households, they are no worse off than before. This is because they are being compensated for the higher rents by the higher wages that they receive in equilibrium. In this case, since the location is not nicer from the household’s perspective, the higher rents do represent a higher cost-of-living, but one that is completely offset by higher wages. Households real satisfaction is unaffected by the presence of the firm amenity. While not of great importance for purposes of this book, the value of the amenity to the firms could also be calculated as the sum of what they are paying for it in the land and labor markets.17 The real world is not so cut-and-dried as the two preceding extreme cases would suggest. Sometimes an amenity from a household’s perspective will be a disamenity from the firm’s perspective and vice versa. Or, an amenity for a household might also be an amenity for a firm (e.g., a deep-water port that offers the firm transportation savings while providing recreational benefits for households). We shall consider two cases of particular relevance to hedonic valuation as applied to policy decisions in environmental economics. Suppose that city leaders of a location pass a law that forces firms to clean up pollution. This renders that location less desirable to firms due to the higher production costs associated with the pollution controls. But the 17 As with the household, if a location specific trait is a disamenity to firms they would require compensation in the form of some mix of lower rents or lower wages. Firms would exit the location until that compensation made the undesirable area as profitable as other locations.

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law will result in a cleaner environment which will make the location more desirable to households. What will happen in this case? Since firms will be leaving one would expect both rents and wages to fall, but since households will enter the more attractive area, one would expect rents to rise and wages to fall. Wages will clearly fall (since both effects work in the same direction), but the effect on rents is in general ambiguous. We do not know which effect dominates, without further information about how desirable the clean air is to people or how undesirable the cost effect is to firms. The city might get larger (if firm cost impacts of the law are negligible and households greatly value the environment) or it might get smaller (if cost impacts of the law are large and environmental benefits are small). This case is depicted in Fig. 6.4, where the effects are drawn as offsetting from the perspective of city size, hence rents. For this specific case, it turns out that the full benefits of cleaning up the air are captured in wages, with no changes in property values or rents. This would, of course, be a fluke in the sense that there would generally be rent effects, positive or negative.18 What is quite clear in this case is that a hedonic property analysis will greatly understate the value of the cleaner air. When the amenity bundle at a location is good for both households and firms, both would want to move in. Suppose, for example, that a nation-wide law is passed that subsidizes firms to clean up in areas where there is nonattainment of air pollution standards, with no subsidy in areas meeting current air pollution standards. This situation would cause rents

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W0

C0 Wage

When the household amenity is a firm disamenity.

shares of compensation for an amenity are not bounded by zero or one when both households and firms are affected, so it would be theoretically possible, though unlikely in practice, that a single market hedonic approach could overstate the benefits of cleanup.

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to unambiguously rise in nonattainment areas (both households and firms move in, increasing directly and indirectly the demand for land), while the wage effect is ambiguous firms moving in demanding labor would tend to drive wages up, but households moving in supplying labor would tend to drive wages down. Again, the net effect would depend on whether the amenity is more important to firms than to households (that is, the net effect depends on which curve shifts up the most in the graphical setting); however, the value of the amenity will appear largely in land markets, but again only as a fluke would there be no labor market effects. In this case, the choice of a hedonic wage analysis will greatly understate the value of the cleaner air. It has been assumed to this point that all households and all firms are all homogeneous in the preceding discussion. This is of course not the case. Land-intensive firms would not be expected to be found in locations where land is expensive (which is why corn is not seen growing in downtown New York). Similarly, those households who have unusually large preferences for land, perhaps those with large families or pronounced gardening desires, would not locate where land is very expensive, perhaps locating in suburbs or exurbia rather than in more central areas. If a firm’s labor demands are unusually large, it would avoid locations with unusually high wages. If a household does not supply labor (e.g., the retired19 ), they would want to locate where amenities are mostly paid for in wages rather than rents. This would also be the case for those who have very high demand for services. Conversely, those households that supply lowskilled labor to service industries are likely to be priced out of very desirable and high rent locations (e.g., Malibu, CA, Aspen, CO, or Key West, FL) and will have to be compensated in higher wages to locate there or commute in to work i.e., the low-skilled may actually have higher wages in desirable locations. As the preceding discussion makes clear and as even casual reference to the real world verifies there is a very rich tapestry of locational choices when the full implications of the role of firm and household amenities are considered. This is even more the case when endogenous amenities are considered, amenities such as the amount of similar people present in a community (e.g., the ethnic neighborhoods of large cities that often make them much more attractive to particular types of people than would otherwise be the case).

19 For

greater detail on this argument, see Graves and Waldman (1991).

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Turning specifically to environmental concerns of bias in the use of the hedonic method to value environmental projects affecting households there are several issues to ponder. In early property value and wage compensation studies data limitations or carelessness often resulted in the omission of many variables, other than pollution, that affect household welfare. This would not be a problem if the omitted variables were uncorrelated with the pollution variables of interest (their omission would just add white noise, reducing the precision with which property values or wages were explained, but not biasing the environmental coefficients). But, in many cases one would expect other goods to be correlated with environmental quality that is, one might expect environmental quality to be higher in suburbs than in central cities. Suburbs, however, are also likely to have better schools and lower crime than central cities. If school quality or safety are omitted from the equation, the environmental variable will pick up their effects, with overstatement of the value we place on environmental quality.20 Similarly, in wage studies, climate amenities might well be correlated with pollution (i.e., the Rustbelt industrial areas are generally both more polluted and less desirable climatically than the Sunbelt hence wages in the Sunbelt will be lower for both reasons, and failing to include climate variables will result in overstatement of the value placed on environmental quality). In early studies, omitted variable bias was a major concern but the past two decades have seen far better data availability and data analysis.21 There are four reasons, however, why it seems likely that hedonic methods would understate the value of environmental quality improvements. The first, and most obviously damaging, is that the benefits of environmental quality must be fully perceived by households for them to be willing to pay more for cleaner locations. Even the world’s foremost health experts have spirited debates about the role various pollutants play in human disease and death. It seems very implausible that ordinary people would be able to accurately perceive such things; moreover, since 20 The bias on the coefficient of environmental quality variable would be equal to the effect of the omitted variable on property value (or wage) times the correlation of that variable with environmental quality in the data. If the correlation is small the bias would be small, even with important omitted variables; similarly, if the correlation is high (closer to one) then the full effect of any omitted variable would be attributed to environmental quality. 21 GIS modeling enables the merging of many disparate datasets, while use of something called fixed effects modeling (where dummy variables, taking on the value 1 or 0 if an observation is or is not in a specific location, implicitly holds constant a host of unmeasured variables).

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many pollutants are odorless, colorless, and tasteless in normal ambient concentrations, it is possible that ordinary people would be unable to distinguish the clean places from others. Why do hedonic studies show such large environmental effects then? It is certainly the case that people will perceive localized smells, bad visibility, and other impacts of pollution that are inevitably revealed by our five senses. Yet, it is precisely such perceived damages that are ignored in the sum of specific damages (SSDs) approach discussed in the chapter which follows this. The environmental valuation method described in Chap. 7 assumes that damages (typically health damages) are unperceived and just occur to people at greater rates in dirtier locations. Despite assumptions about perceptions that should suggest adding hedonic values to SSD values, it is commonly viewed as a good corroborative finding when estimates from hedonic studies show damages of the same or nearly the same magnitude as those from SSD studies. Given the nature of the assumptions about preferences the two approaches clearly cannot be viewed as alternatives. A much stronger case can be made for adding together the damages estimated from an SSD study to those of an hedonic study to get the true damages, those both perceived and unperceived. Such a procedure might result in some double counting, since an area that is unhealthy might also smell bad, but it is likely that the two methods pick up largely unrelated damage categories, those perceivable and those that are not perceivable by households. This point is quite important in practical environmental situations, whether in regulatory rulings or in court testimony. The benefits of environmental cleanup are estimated either from SSD types of approaches or hedonic types of approaches, but the estimates are never added together which would in many cases double the estimated benefits of cleanup. The second reason to expect the hedonic method to understate true benefits of environmental improvements is that, with notable exceptions (e.g., Blomquist et al., 1988), the studies do not add the benefits measured in property values to those measured via wage compensation. As is clear from Fig. 6.2, considering either market separately will generally undervalue the amenity. This observation, emphasized here, is not merely a theoretical possibility, as an innovative recent empirical approach, applied to San Francisco and Sacramento data, by Kuminoff (2008) reveals. In a nested analysis comparing results from a traditional property value approach, he finds that his new “dual-market” framework increases estimates for the average per/household marginal willingness-to-pay by as much as 110%.

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The third reason for expecting the hedonic method to understate true benefits is that the hedonic method, even properly conducted, only captures use benefits of the environmental resources of concern, since the amenities are bundled with housing and jobs. Non-use benefits might well be of greater magnitude in particular environmental settings and policies allocating the environmental resource should, on efficiency grounds, encourage highest value usage even if that results in nonuse of the environmental resources. Illustrating, is the California Coastal Commission properly allocating scarce ocean locations? It is clear that in the absence of this regulatory authority virtually the entire coast of California would be lined with high-rise condos, looking much more like Miami than at present. But, the scenic Pacific Coast Highway has value to all who drive it, and to a large extent that value has been perceived as being of greater importance than the (admittedly very large) benefits households would receive if the coast were opened to unrestricted development. The final reason why hedonic methods might be expected to understate the benefits of environmental cleanup stems from the supplies of clean locations relative to the demands for clean locations. The hedonic method results, at least in principle, in zero spatial consumer surplus. That is, if one location is nicer than another location, households will continue to move to the nicer location, until it is no longer nicer, until identical locations have identical prices. There will be no consumer surplus over space, and indeed this is one of the reasons the hedonic method is desirable in that the full benefits that are perceived are measured. But, the fact that people are very different means that understatement of environmental benefits (damage reduction) can occur if there are more locations with the amenity than there are people strongly desiring the amenity. Suppose, for example, that there are very few households containing really unhealthy individuals, individuals with weakened cardiopulmonary systems who would be highly damaged by pollution. Such households might be willing to pay a great deal for a very clean location, but they might only have to pay a much smaller amount, if the number of somewhat clean locations is large relative to the number of these households. They will get, in other words, consumer’s surplus over space. Inferring the value of cleaning up the environment from the average person in this case would ignore the high marginal benefits received by these households.22 When one 22 As an illustration of the possible/occasional importance of this point, suppose that a large city, based on an hedonic analysis suggesting that its mass transit system has low

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considers the very large number of traits that can matter to a heterogeneous population with very diverse preferences, it becomes clear that a great deal of consumer surplus can remain in the hedonic equilibrium. In the case of incrementable environmental goods, the unobserved consumer surplus corresponds to a higher marginal value that might if observed justify a policy intervention to increase levels of the public good.

6.7.

Can Single-Market Hedonic Studies Ever Produce Accurate Amenity Measures?

In certain specific cases, cases that are useful to examine, single market studies can provide accurate measures of the value of an environmental amenity. Consider this case (quoting Taylor, 2008, p. 2 of an excellent, although lengthy, review of the hedonic method): Imagine the following hypothetical scenario in which there are two identical lakes, each with 100 identical homes surrounding them. All homes are lakefront, and all the characteristics of the homes themselves, the land, and the neighborhoods are identical across the properties. At the current equilibrium price of $200,000 per house, all 200 homes on either lake are equally preferred. Now, let’s imagine that water clarity at one lake, Lake A for example, is improved. We assume that the improved water clarity is preferred by all households. Now, if any home on Lake A were offered at the original equilibrium price of $200,000, consumers would uniformly prefer this house to any house on Lake B. In other words, at the current prices, there would be excess demand for the houses located on Lake A, and as such, the price of these houses must rise to bring the market into equilibrium. The price differential that results from the change in water clarity at Lake A is the implicit price consumers are benefits, is considering dropping that costly system. Handicapped individuals might be receiving extremely large benefits from that transit system, but since they are small in number their values of access to the transit system are not picked up in the hedonic analysis. Similarly, for a city contemplating the addition of a mass transit system, the high demands of the handicapped might not be properly considered. Illustrating further, it is only at rush hour when those desiring to travel at high rates of speed are particularly harmed and have high marginal benefits for an additional lane relative to typical drivers. During normal traffic flow periods, when everyone can travel at the speed they wish, the consumer surplus received by the speeders has no bearing on the marginal benefits of adding a lane. The marginal benefit of an incremental lane is zero when everyone can travel at any speed they wish. The hedonic method is like rush hour in that it assumes there are fewer desirable fast lanes than there are people wanting fast lanes, with movement among lanes occurring to make all lanes equally (un)desirable.

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willing to pay for that incremental increase in water clarity. This willingness to pay for water clarity is indirectly revealed to us through the market prices of the homes. For instance, if in the new equilibrium, houses on Lake A sell for $210,000, while houses at Lake B sell for $200,000, the implicit price associated with the increased water clarity is $10,000.

What is somewhat remarkable about this example is that it is very narrow in ways that reduce the applicability of the single market approach. First, there are 100 identical homes around the lakes, which implies a fixity in the housing capital stock that is unusual. In terms of the discussion of previous sections, this also fixes city size; in other words the spatial system is closed versus open, not allowing net in-migration from one lake to the other or to the clean lake from any other locations. More important, at least in the short run, is the implicit assumption (not explicitly discussed) that the location of employment of the home owners does not matter. This example can only correctly measure the water quality improvement if and only if all property owners around the two lakes fall into one of three categories: (1) employed at some distant location vis-`a-vis the two lakes, (2) employed at a point equidistant between the lakes, or (3) retirees, out of the job market entirely. It turns out that the houses in this valuation exercise were primarily summer homes of people living distantly, whether retired or not. Hence, a reasonably strong argument can be made that the property value approach can properly value the improvement in environmental quality in this case. More generally, however, cleaning up one lake implies that people would want to move from the other lake to the clean one (and from other locations entirely) until utility were reequalized, and this would involve more houses around the now cleaner lake with more employment in the clean lake area i.e., people move in, simultaneously increasing the demand for housing (more of which would be built to equalize profits over space) and increasing the supply of labor, lowering wages. It would be, then, the combination of the higher prices of houses and the lower wages that would measure the cleaner lake’s value. 6.8.

Conclusions: Hedonic Analysis as Practiced is Biased Against the Environment

Each of the damage estimation methodologies discussed in this and the following chapter separately understates damages as typically conducted.

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The approaches of this chapter require perfect perceptions of environmental benefits (or risk in the VSL case) along with perfect knowledge of how environmental quality varies over space (or risk over jobs in the VSL case). The SSD approach assumes zero damage perception for its accuracy and, moreover, tends to omit minor health effects (e.g., watery eyes) and emphasizes acute damages rather than the more difficult to study chronic damages.23 In addition, it is the case that expert legal testimony and typical regulatory practice still typically employs either a property value study or a wage study, despite our having known for more than two decades that compensation for environmental amenities and disamenities will generally occur in both the land and labor markets as discussed in detail in this chapter. The extent to which damages appear in land versus labor markets would generally vary according to many things,24 but considering either market separately is likely to greatly underestimate the damages from pollution. A very important reason for the popularity of the hedonic method is that the benefits of an environmental project that improves the environment come in convenient dollar-denominated units that make it much easier to compare those benefits to the costs of the project. But, convenience per se is of little consequence when it comes along with bias, bias that suggests that the environment is being undervalued.

23 In principle, all physical effects associated with an environmental improvement should be included in the SSD approach, as discussed more fully in the following chapter. This would include all forms of morbidity benefits along with mortality benefits as well as materials damage, crop damage, damage to ornamental plants, esthetics, and so on. Note that, from the perspective of the present chapter, the more inclusive the damage categories in the SSD approach the more likely that perceived damages would be included, increasing the probability of double counting damages. As a practical matter, it is rare to extend the SSD approach beyond a small range of clearly agreed upon health effects (e.g., asthma attacks due to ozone). 24 If an environmental pollutant were highly concentrated (e.g., a hazardous waste dump) one would expect a greater percentage of its damage to appear in property values, while the damages from more regionally ubiquitous pollutants might be expected to appear primarily in wage rates. The existence of firm amenities and disamenities complicates the generation of firm conclusions, though the conclusion remains that using only one of the two markets in which environmental quality is valued generally results in understatement of environmental values.

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References Aldy, JE and WK Viscusi (2008). Adjusting the value of a statistical life for age and cohort effects. The Review of Economics and Statistics, 90(3), 573–581. Bateman, IJ, B Day and IR Lake (2001). The effect of road traffic noise on residential property values: A literature review and hedonic pricing study. UK: Scottish Executive Development Department: Edinburgh. Blomquist, GC, G Berger and J Hoehn (1988). New estimates of the quality of life in urban areas. American Economic Review, 78, 89–107. Bockstael, NE and KE McConnell (2007). Hedonic wage analysis. In Environmental and Resource Valuation with Revealed Preferences: A Theoretical Guide to Empirical Models. The Economics of Non-Market Goods and Resources, Vol. 7, pp. 151–187. Netherlands: Springer. Day, B (2001). The theory of hedonic markets: Obtaining welfare measures for changes in environmental quality using hedonic market data. London: Economics for the Environment Consultancy. Graves, PE (2009). A note on the union effect in VSL studies. Manuscript. Graves, PE and D Waldman (1991). Multimarket amenity compensation and the behavior of the elderly. American Economic Review, 81(5), 1374–1381. Graves, PE, JC Murdoch and MA Thayer (1988). The robustness of hedonic price estimation: Urban air quality. Land Economics, 64, 220–233. Krumm, R and P Graves (1982). Morbidity and pollution. Journal of Environmental Economics and Management, 9(4), 311–327. Kuminoff, NV (2008). Recovering preferences from a dual-market locational equilibrium. Virginia Tech Working Paper 2008-13. Mrozek, RR and LO Taylor (2003). What determines the value of life? A metaanalysis. Journal of Policy Analysis and Management, 21(2), 253–270. Nelson, JP (2004). Meta-analysis of airport noise and hedonic property values: Problems and prospects. Journal of Transport, Economics and Policy, 38, 1–28. Palmquist, RB (1991). Hedonic methods. In Measuring the Demand for Environmental Quality, JB Braden and CD Kolstad (eds.), 77–120. Amsterdam: North Holland. Palmquist, RB (2004). Property value models. In Handbook of Environmental Economics: Valuing Environmental Changes, Vol. 2, K-G Maler and JR Vincent (eds.). Amsterdam: North Holland. Polinsky, AM and DL Rubinfeld (1977). Property values and the benefits of environmental improvements: Theory and measurement. In Public Economics and the Quality of Life, L Wingo and A Evans (eds.), pp. 75–104. Baltimore: Johns Hopkins. Polinsky, AM and S Shavell (1975). Air pollution and property value debate. Review of Economics and Statistics, 57, 100–105. Polinsky, AM and S Shavell (1976). Amenities and property values in a model of an urban area. Journal of Public Economics, 5, 119–129.

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Polinsky, AM, DL Rubinfeld and S Shavell (1974). Economic benefits of air quality improvements as estimated from market data. In Air Quality and Automotive Emissions, NAS/NAE, Vol. 4, Chap. 4, Sections 3–5. Washington, DC: National Academy of Science/National Academy of Engineers. Portney, PR (1981). Housing prices, health effects and valuing reductions in risk of death. Journal of Environmental Economics and Management, 8 (March), 72–78. Ridker, RG and JA Henning (1967). The determinants of property values with special reference to air pollution. Review of Economics and Statistics, 49, 246–257. Roback, J (1982). Wages, rents, and the quality of life. Journal of Political Economy, 90, 1257–1278. Rosen, S (1974). Hedonic prices and implicit markets: Product differentiation in pure competition. Journal of Political Economy, 82, 34–55. Rosen, S (1979). Wage-based indexes of the quality of life. In Current Issues in Urban Economics, P Mieszkowski and M Straszheim (eds.), pp. 75–104. Baltimore: Johns Hopkins University Press. Sheppard, S (1999). Hedonic analysis of housing markets. In Handbook of Urban and Regional Economics, Vol. 3: Applied Urban Economics. Amsterdam: North Holland. Smith, VK (1983). The role of site and job characteristics in hedonic wage models. Journal of Urban Economics, 13, 296–321. Smith, VK and JC Huang (1995). Can markets value air quality? A metaanalysis of hedonic property value models. Journal of Political Economy, 103, 209–227. Taylor, LO (2008). Theoretical foundations and empirical developments in hedonic modeling. In Hedonic Methods in Housing Markets, A Baranzini, J Ramirez, C Schraerer and P Thalmann (eds.), pp. 15–38. Netherlands: Springer. Viscusi, WK (1993). The value of risks to life and health. Journal of Economic Literature, 31, 1912–1946. Viscusi, WK and JE Aldy (2003). The value of a statistical life: A critical review of market estimates throughout the world. Journal of Risk and Uncertainty, 27(1), 5–76. Zellner, A and A Siow (1980). Posterior odds ratios for selected regression hypotheses. In Bayesian Statistics: Proceedings of the First International Meeting Held in Valencia, JM Bernardo, MH Degroot, DV Lindley and AFM Smith (eds.), pp. 585–603. Valencia, Spain: University of Valencia Press.

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Chapter 7

Environmental Valuation: The Sum of Specific Damages Approach Philip E. Graves University of Colorado [email protected] The sum of specific damages (SSDs) approach to valuing environmental improvements recognizes explicitly that the benefits of cleanup are equivalent to reductions in damages. This approach is sometimes referred to as the health effects model, since the damages usually considered are related to morbidity and/or mortality and because epidemiological evidence on dose–response figures prominently in the physical effects of concern. Although this approach is remarkably simple and intuitive, we shall see that it is fraught with two types of uncertainty and is, moreover, likely to provide downward-biased damage estimates. To a large degree, the claim of downward bias stems from the implicit assumptions, discussed further below, regarding the nature of perceptions under this approach vis-` a-vis the hedonic approach of Chap. 6. The idea under the SSD approach is to first gauge how much an environmental policy will reduce physical damages, ∆Di , of a wide variety.1 There are hundreds of studies relating various levels and types of pollution 1 Atmospheric modelers for the case of air or hydrologists for the case of water would first be required to model how much environmental quality is predicted to change in various locations as a result in the changed pattern of residuals resulting from the environmental policy under consideration. That is, there would have to be some impact on residuals (possibly just moving them to a location where they do less damage) for there to be any impact on environmental quality, and what that impact will have on environmental quality at various locations will need to be modeled. The damages reduced will then depend on how many damage receptors are present in locations with improved environmental quality.

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(e.g., particulates, sulfur dioxide, ozone, or lead) to physical damages taking many forms, such as asthma, cancer, cardiovascular disease, chronic bronchitis, hospital admissions, lead neurotoxicity and blood pressure effects, mortality, respiratory infections, and work loss (see the reference list here for a representative sample and Ostro (1994) for a lengthy list). A dollar value, $Vi , is then placed on each category of damage, with for example a prevented life lost being valued at perhaps $5 to $7 million, and the prevention of an asthma attack much less.2 The marginal benefits to be compared to the marginal costs of the policy will, then, be the sum of all of the reductions in physical effects times their respective values: Marginal benefits = Σ(∆Di )$Vi

(1)

The reduction in physical damages is usually further decomposed into (for greater detail in an interesting developing world context see Ostro, 1994) ∆Di = bi ∗ POPi ∗ ∆EQ

(2)

∆Di = change in population risk for health effect i, bi = slope of the dose–response function for health effect i, POPi = population at risk for health effect i, and ∆EQ = change in environmental quality, measured as pollution reduction. Illustrating, suppose that an environmental policy is enacted that is expected to lower pollution fine particulate levels by 5 micrograms per cubic meter in some populous region.3 Assume that this change in air quality will save one life per million people and will eliminate 100 cases of chronic bronchitis per million people. If there are 8 million people in the region affected by the policy, then 8 lives will be saved and 800 cases of chronic bronchitis will be eliminated. Further, assume that a saved life is worth $3–7 million, with a best point estimate guess of $5 million (see Chap. 6 for where

2 Ideally, for at least rough consistency with modern economic welfare theory, the values should be willingness-to-pay for marginal reductions in damages, but in the literature other valuation methods have been used (e.g., medical treatment costs or the value of lost productive days and years, which as subcomponents must be lower than MWTP). 3 A microgram is one-millionth of a gram. To get a sense of relative magnitude, a gram of sugar is roughly one-fourth of a teaspoon, with one-millionth of that amount floating typically in Brownian motion for very small particles in one cubic meter of air.

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the details of value of statistical life (VSL) calculation) and an eliminated case of chronic bronchitis is worth $50,000 (perhaps based on contingent valuation or some other stated preference mechanism, discussed in Chap. 5). Then, the policy would have benefits of 8 × $5,000,000 + 800 × $50,000 = $80,000,000. If these are the only benefits of the policy and it can be put in force for $80 million or less, it would be efficient to adopt the policy since it would have marginal benefits greater than or equal to marginal costs. The preceding example can be used to illustrate all three major problems with the SSD approach. First, the physical effects due to the policy, ∆Di , are highly uncertain; although we supposed that 8 people would not die and 800 would not acquire chronic bronchitis if the policy is put into effect, such estimates are very uncertain. In testimony prior to the implementation of the environmental policy, some experts may argue that the damages prevented will be large, while others will argue that the damages prevented may be very small. In part, this stems from advocacy positions; an expert working for the American Lung Association is more likely to predict more bronchitis cases prevented by the policy than an expert working for the National Association of Manufacturers. The final determination of damages will likely depend on some mix of the credibility/credentials of the experts and the quality of the analyses they present. Where do experts of either stripe get their information? There are three primary approaches (toxicological, clinical, and epidemiological) with epidemiological studies tending to carry the most weight. Clinical studies are used to address research questions that can be well examined in laboratory settings. In a human clinical study, scientists investigate the effects of individual air or water pollutant doses by measuring a variety of health effects (e.g., lung function, heart rate variability, blood component analysis). Clinical studies are themselves usually initiated in response to prior biological studies, either in vitro or in vivo in animal surrogates for humans. The latter provide information about the way pollutants generate their molecular effects, and such animal and in vitro studies are particularly important when human data is unavailable or when such data cannot be ethically obtained. Epidemiological studies, while less rigidly controlled, offer more natural settings through the statistical analysis of data from human populations or by field studies. In some cases, researchers follow fairly large groups of individuals and use detailed questionnaires to relate the incidence of various disease endpoints to pollutant levels. Field studies involve fewer individual observations and employ repeated assessments of health effects of pollution

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exposure. The smaller numbers of subjects involved in field studies allow researchers to extend the information obtained in large-scale epidemiological studies by including measurements of clinical health endpoints. Various epidemiology studies have implicated particulate matter in premature death among elderly individuals with cardiopulmonary disease and to increased use of medications, doctor visits, and hospital visits for individuals with pulmonary disease such as asthma. Toxicology studies attempt to identify and study the specific properties and constituents of various pollutants that are responsible for causing adverse health effects. Toxicologists test the molecular, cellular, and systemic effects of pollutants in experimental settings using cell and tissue cultures, animals, and computer models. As already indicated, findings of dose–response effects from a toxicology study might prompt the initiation of either or both clinical trials and epidemiological investigations. Knowledge is gained from the various approaches, but there remains great uncertainty at the policy level about how physical effects relate to pollution exposures. This is particularly so for chronic pollution effects, such as perhaps a long-latency cancer, vis-` a-vis the more immediate acute effects.4 When certain physical effects are difficult, for various reasons, to tie to pollution, they will tend to be ignored in the SSD approach, leading to understatement of damages. Death or cancer at least have clear definitions, but certain forms of pain, dermatitis, neurological effects, various endocrine disruptions, and the like are difficult even to quantify, let alone relate to pollution, hence are likely to be ignored in practice. Returning to the example of how Eq. (1) might be used (or misused), the second source of uncertainty is on what values to place on the physical effects that are predicted to occur. Is the VSL $5 million? Or, is it one-tenth or ten times that? Could the value of a chronic bronchitis case be an order of magnitude greater or smaller than the $50,000 used in the illustration?5 One 4 Acute effects are easier to study, clinically or epidemiologically because study participants can be tracked with reasonable accuracy over short time periods. Diseases that take many years of exposure to emerge are more difficult to study because study participants move to different locations. This is a particular problem in that those with the weakest immune systems are the most likely to move to less polluted places, places where they remain more likely to die or exhibit other morbidity effects at higher rates than normal. Hence, any health impact endpoint associated with dirty air will be understated because the least healthy individuals will have moved from dirty to clean places. 5 Ackerman and Heinzerling (2004) place particular emphasis on the notion that the prices used in benefit–cost analysis are so meaningless that the approach is fatally flawed and that refinements of the type discussed in this book are unlikely to make a meaningful

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might argue that values such as these are at least plausible, and one could make a fairly strong case for the argument that there is greater uncertainty regarding the physical effects estimated by the epidemiologist than there is regarding the values placed on them by the environmental economist. Neither of the uncertainties discussed to this point would seem to point to any obvious downward bias in damage estimates. There are two important reasons to suspect that such a downward bias exists, however. First, the physical effects should be all of the physical effects that will occur as a result of the policy, not just (a portion of) the health effects. If a policy cleans up the air or water, it will have physical benefits of a wide variety, not just mortality and morbidity benefits. There will generally be ecosystem improvements, agricultural crop yield benefits, material damage reductions (e.g., house painting with less frequency), benefits for pets, as well as esthetic effects (e.g., smells, visibility). Since we get all of those effects as a result of the policy they all should be counted, yet in practice they never are.6 The situation is as depicted in Fig. 7.1. In the figure, health benefits are depicted as comprising a portion of total benefits, and if decisions are based only on (typically a subset of) health effects, EToo little will be produced relative to the optimum at E∗ . The welfare loss from an environmental quality that is too low is depicted by the shaded area in the graph. There is another theoretical and practical problem, mentioned briefly in Chap. 6, with the SSD approach that strengthens the claim that too little environmental quality will be produced if this approach is used to estimate the benefits of environmental policies. For this method to work well as a measure of pollution damages, people have to be unaware that pollution has any impact on the damages. That is, the impact of pollution on, say,

difference to that conclusion. However, benefit–cost analysis, at a minimum, aids in the ranking of the vast number of potential policies that could be pursued. Additionally, it offers the potential to identify really bad or really good policies, in terms of their efficiency. For a detailed book review of Ackerman and Heinzerling, see Graves (2005) (also at http://spot.colorado.edu/%7Egravesp/BookReviewPriceless.PDF). 6 One of the reasons many whole categories of damages are ignored in benefit–cost analysis is that we have (irrationally) broken our environmental quality standards into primary standards (relating to human health) and secondary standards (relating to human welfare considerations other than health). This is irrational because an environmental improvement gives us benefits that come in many forms and it is inappropriate to compare some arbitrary subset of benefits to the costs of a policy rather than all of the benefits. The distinction between primary and secondary standards should be eliminated.

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MB, MC

MC Welfare loss

MBTotal MBHealth E Too little Fig. 7.1:

E*

Environmental quality

Welfare loss from failure to include all marginal benefits in SSD.

MB

EQ0

EQ1

V D

A

0

Fig. 7.2:

H0

H1

Healthy days

SSD assumes zero perceptions of the cause of damages that occur.

health has to either be unknown to households or they must be unable to determine where it is clean and dirty. The environmental source of the damages has to be unperceived. Illustrating (Fig. 7.2), it is implicitly presumed that the physical effects just happen to people; household members get sick or die or do not but they do not know why. The SSD method implicitly assumes zero perception of the cause of the number of healthy days we experience. In Fig. 7.2, the physical benefits of an environmental improvement, e.g., going from EQ0 to EQ1 , is that households get a larger number of healthy days, H0 going to H1 , after the environmental improvement. The marginal benefits of the environmental policy yielding the improved health outcomes in the SSD approach are seen as area A, the area between H0 and H1 , and below V . The value assigned to a healthy day should, of course, in principle expected to be declining in healthy days as reflected by the demand curve, D, but for marginal changes V may provide a good approximation to marginal willingness-to-pay.

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Environmental Valuation: The Sum of Specific Damages Approach

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However, if households have any idea as to what determines healthy days one would expect them to engage in mitigating behavior (typically referred to as averting behavior in the literature). In particular, if we suspect that air or water pollution is damaging us, we would engage in (costly) behavioral changes to avoid those damages. We might not, for example, exercise outside on high pollution days, we might install dust filters or air conditioning in part to avoid air pollution, we might move to a less-preferred but cleaner location, and so on. In the case of water, we might buy distilled water, or install water filters, as a means of avoiding damages from polluted surface reservoirs or aquifers. In all of these cases, we are expending our scarce resources to avoid a damage that otherwise would have happened. To merely count the damage that continues to occur, ignoring such costly mitigating expenditures, understates true damages, hence understates the benefits of cleanup. This high perception case is depicted in Fig. 7.3 where healthy days are produced, in much the same way that we produce home-cooked meals. In this view of the nature of perceptions, changes in environmental quality resulting from an environmental policy shift the marginal provision cost of healthy days, allowing any quantity to be produced at lower cost. There is no clear relationship between the benefits of the environmental policy calculated under the different assumptions about perceptions depicted in Figs. 7.2 and 7.3. In Fig. 7.3, the benefits of the environmental policy as seen to be the sum of areas A and B. That is, we benefit by the lower cost of acquiring the initial level of healthy days, H0 , plus we obtain the net benefits of the larger number of healthy days that are optimal when they become less expensive as a consequence of the policy. These areas could be large or small relative to area A in Fig. 7.2 depending on the slopes and locations of the marginal provision cost curves.

MB, MC

MCEQ0 MC EQ1

MB0 = MC0 MB1 = MC1

B A D (MB) H0

Fig. 7.3:

H1

Healthy days

The production of healthy days, when damage cause is perceived.

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However, it is unlikely that many people think about producing healthy days in the same way that they produce home-cooked meals, combining various inputs according to an optimal equimarginal principle to arrive at an efficient outcome. Rather, such high levels of perception are only likely for readily observable dimensions of environmental quality (smell, visibility, and the like). In Chap. 6, the hedonic method was examined, and was found to be most relevant for precisely these types of observable dimensions of environmental quality. This was why it was argued there that a case could be made for adding together the perceived benefits (from a generally multimarket hedonic analysis) and the unperceived benefits (from an SSD analysis). As indicated in Chap. 14, while the addition might result in some double counting, that prospect is likely to be offset by the omission of many damage categories which is typical of the SSD approach. References Ackerman, F and L Heinzerling (2004). Priceless: On Knowing the Price of Everything and the Value of Nothing. New York: New Press. Chestnut, LG and RD Rowe (1987). Ambient particulate matter and ozone benefit analysis for Denver. Prepared for U.S. Environmental Protection Agency, Region 8. September. Chestnut, LG, BD Ostro and RD Rowe (1987). Santa Clara criteria air pollutant benefit analysis. Prepared for the U.S. Environmental Protection Agency, Region 8, San Francisco, CA. Graves, PE (2005). Book Review: Priceless: On Knowing the Price of Everything and the Value of Nothing, by F Ackerman and L Heinzerling. New York: The New Press 2004, in The Journal of Economic Literature, 43(1), 188–190. Hall, JV, AM Winer and M Kleinman (1992). Valuing the health benefits of clean air. Science, 255(5046), 812–817. Krupnick, A and R Kopp (1989). Appendix: The health and agricultural benefits of reductions in ambient ozone in the United States, in Catching our Breath: Next Steps for Reducing Urban Ozone. Office of Technology Assessment, U.S. Congress, Washington, D.C. Krupnick, AJ and PR Portney (1991). Controlling urban air pollution: A benefit– cost assessment. Science, 252(5055), 522–528. Ostro, B. (1994). Estimating the health effects of air pollutants: A method with an application to Jakarta. Policy Research Working Paper 1301, The World Bank, Policy Research Department, Public Economics Division, 1818 H Street NW, Washington, D.C. 20433. Portney, PR and J Mullahy (1986). Urban air quality and acute respiratory illness. Journal of Urban Economics, 20, 21–38.

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Chapter 8

Econometric Estimation of Non-linear Continuous-Time Models of Intertemporally Optimizing Agents Kieran P. Donaghy∗ and Clifford R. Wymer† Cornell University [email protected] † [email protected]

∗

8.1.

Introduction

Many models used in the study of natural resource and environmental economics assume that agents are engaged in some type of intertemporally optimizing behavior. (See, e.g., Pindyck, 1978; Dasgupta and Heal, 1979; Conrad and Clark, 1987; and other chapters of this volume.) Arguably, in testing hypotheses of the behavior underlying the use of natural resources or the natural environment, or for analysis of strategic behavior in a planning or policy context, the models employed will be based on the assumption of intertemporal optimization, so the parameters of these models should be estimated by imposing the full set of implied constraints that characterize the models’ solutions. Many models of intertemporally optimizing behavior are also formulated in continuous time and, to the extent they are based on non-trivial economic theory about consumption, production, and trade, they are usually formulated in terms of non-linear functional forms.

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This chapter sets out and illustrates an approach to estimating nonlinear continuous-time models that are predicated on the assumption of underlying intertemporally optimizing behavior. In so doing, it draws extensively on material already in the literature — our primary purpose not being to present novel research results. Results presented in this chapter have been obtained with well-documented software programs — Wysea (various dates), developed by the second author — that implement the methods here discussed and which have been available to researchers for at least a decade. Although the estimated model discussed here is derived within the constraints of a strict, non-robust, Hamiltonian approach, the estimator allows the use of much more general models. In particular, robust optimization in which the derivatives of the control variables enter the objective function gives a second-order system of the form also discussed below. The proportional, integral, and derivative (PID) approach (see, for instance, Phillips, 1954) helps to offset some of the problems that arise with nonrobust optimization and provides a model which is more stable and more realistic. The knife-edge properties of the more constrained model are not realistic and such models are often rejected by the data. A similar but modified approach consistent with the assumption of intertemporal optimization leads to an adaptive model via the Riccati equations; it is not necessary to estimate the Riccati equations explicitly but their coefficients may be derived from the full-information estimates of the restricted model. Thus, the hypotheses underlying the model in this chapter may be generalized within the same framework and provide more realistic models (see, for example, Balta et al., 2007; Saltari et al., 2009). In addition to this introductory section, this chapter has five other parts. In the second section the general notation for describing non-linear continuous-time economic systems is presented and the third provides a review of the econometric theory of the estimation of continuous-time models — linear, non-linear, and with boundary point conditions. The fourth section contains a brief outline of the analysis of the dynamic properties and stability of such models, and the next gives the estimation of Pindyck’s (1978) model of the optimal exploration and production of non-renewable resources from annual data on Nigerian oil exploration and extraction. An appendix to the chapter provides a discussion of the estimation of a more general (mixed second-order) continuous-time model which is likely to be relevant in this field and of which Pindyck’s model is a special case.

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8.2.

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System Representation1

For the purposes of this chapter and for simplicity, but without loss of generality, it is assumed that the economic system of interest can be represented by a recursive system of mixed second-order mixed stock-flow non-linear differential equations: D2 y1 (t) = ϕ1 {Dy1 (t), y2 (t), y1 (t), z(t), θ} + υ1 (t) Dy2 (t) = ϕ2 {Dy1 (t), y2 (t), y1 (t), z(t), θ} + υ2 (t)

(1)

where the yi (t) are vectors of mi endogenous variables, with the column vector y(t) = {y1 (t) y2 (t)}, z(t) a vector of exogenous variables and θ a vector of p parameters. D is the differential operator d/dt and ϕ is a vector of continuous and differentiable functions. If appropriate, z(t) may include first-order derivatives of other exogenous variables. The ν(t) are vectors of white noise disturbances so that their integral is a homogeneous random process with uncorrelated increments. A more precise definition from a stochastic point of view is given below and a rigorous definition of the disturbances is given by Bergstrom (1983). The variables y(t) and z(t) may be stocks or flows; this distinction becomes important for estimation. In this model, the y1 (t) are called second-order variables in that this is the highest order in which they appear in the model; similarly, y2 (t) are called first-order variables. Some equations may be identities, in which case the corresponding ν(t) are zero. The model may also include zero-order equations, i.e., equations defining zero-order variables, which need not be recursive, although in many models they are likely to have a causal interpretation. For the purposes of this chapter and again for simplicity only, it is assumed below that any zero-order equations have been eliminated from the model. Defining additional variables such that Dyi (t) = yj (t), (1) can be written as a general first-order model of the form Dy ∗ (t) = ϕ∗ {y ∗ (t), z(t), θ} + υ ∗ (t)

(2)

where y ∗ (t) = {Dy1 (t) y2 (t) y1 (t)} is the (column) vector of m∗ = 2m1 +m2 endogenous variables, and υ ∗ (t) = {υ1 (t) υ2 (t) 0}. System (1) can be extended immediately to mixed rth-order mixed stock-flow systems and 1 This section and the following two draw extensively on material in Wymer (1993b, 1997).

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these can be reduced in the same way by the definition of additional variables to the first-order system (2). Importantly for the type of research discussed later in this chapter, this framework can be extended to allow boundary-point constraints to be imposed so that, for example, some dynamic equations can define forward-looking variables. A simple example is a model that includes the present value of a future income stream. Let the (discounted) income stream be given by f (y(t), t) so that its present value is given by yp (t) =
Th t f (y(s), s)ds where Th is the required horizon and yp (Th ) = 0. Although yp (t) is not observed, it is defined by the dynamic equation Dyp (t) = −f (y(t), t) with the end- or boundary point condition yp (Th ) = 0 and hence may be included as one of the endogenous variables y(t) in the model. Although it is not observed, this relationship is fully taken into account in the solution of the model. Hamiltonian systems, rational expectations processes, some forms of game theory models, and models including other forms of boundary constraints can be modeled in a similar way. 8.3.

Estimation

To provide an outline of the properties of the estimators of these models, the general non-linear model (1) can be defined more precisely as the recursive system of first-order stochastic equations dy(t) = ϕ{y(t), z(t), θ}dt + ζ(dt)

(3)

where y(t), z(t), ϕ, and θ are defined as in (1). This is the same as (1) when m1 = 0 but can be extended to the system (2) or higher-order ζ(dt) is a vector of white noise disturbances so that the integral
systems. t ζ(ds) is a homogeneous random process with uncorrelated increments; t−δ thus E[ζ(dt)] = 0, E[ζ(dt)ζ  (dt)] = |dt|Ω where Ω is a positive semidefinite matrix of order m, and E[ζ(∆1 )ζ  (∆2 )] = O for any disjoint sets ∆1 and ∆2 . Some equations may be identities with the corresponding elements of ζ(dt) and Ω zero; in this case the submatrix of Ω corresponding to stochastic equations will be positive definite. Let the rank be mR ≤ m. Higher- and mixed-order systems may be defined by introducing additional variables and identities as above. The system may also contain zero-order equations but for simplicity only that will be ignored here. The variables y(t) and z(t) may be stocks or flows; this distinction becomes important for estimation. For simplicity in exposition, the formulae below are derived for the strictly

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first-order system and ignore the complications that arise in mixed-order models but those formulae can be generalized as indicated below. The white noise assumption may be generalized. Following Bergstrom (1983), Gaussian
t estimators may be obtained under the assumption that the integral t−δ ζ(ds) of the white noise innovation process is a Gaussian process although the innovations ζ(dt) themselves are not assumed to have that property. Thus, the innovations may be a mixture of Brownian motion and Poisson processes which provides a plausible representation of economic behavior in that the innovations may come as discrete jumps at random intervals. Although the parameters of the non-linear model may be estimated directly using either an approximate discrete or exact discrete estimator, the costs are high and properties of the exact estimator are not well known and have to be inferred heuristically. On the other hand, the estimators of linear models are well developed and their asymptotic properties well known, so that the non-linear model can be approximated by a Taylorseries expansion about some appropriate point, such as the sample mean or some path, such as the steady state, to give a linearized model that then can be estimated subject to all of the restrictions inherent in the non-linear model and the linearization. This provides estimates of the parameters θ of the non-linear model and those estimates can be used for hypothesis testing, analysis, and forecasting. The approximation error inherent in the linearization means, however, that even if the white noise innovations or their integrals are Gaussian, this may not be true of the disturbances in the linearized model. These issues are discussed in Wymer (1993b). 8.3.1.

Linear estimators

The linear, or linearized, model corresponding to (3) can be written as dy(t) = A(θ)y(t)dt + B(θ)z(t)dt + ζ(dt)

(4)

where the elements of the matrices A and B are functions of θ, and in the linearized case, the point or path around which the system has been linearized. This has the solution  t δA(θ) y(t − δ)dt + e(t−s)A(θ) B(θ)z(s)ds y(t) = e 

t−δ t

+ t−δ

e(t−s)A(θ) ζ(ds)

(5)

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Assuming the continuous variables are observed every δ time units (so that δ is the length of the observation interval in terms of the basic time unit of the model), and let xτ be the discrete observation of the continuous variable at time t, so that xτ = x(τ δ). The exact discrete model is then  τδ e(τ δ−s)A(θ) B(θ)z(s)ds yτ = eδA(θ) yτ −1 + 

(τ −1)δ

τδ

+

e(τ δ−s)A(θ)ζ(ds)

(6)

(τ −1)δ

If the z(t) are analytic functions of time the integral of the term involving exogenous variables can be evaluated exactly, otherwise they need to be approximated, generally by a third-order process. The error in this approximation is of O(δ 4 ) as δ tends to zero and under suitable regularity and smoothness conditions leads to an asymptotic bias in the parameter estimates obtained by a maximum-likelihood procedure of O(δ 3 ). Given the properties of the disturbances ζ(dt) above, the
τ δ as (τdefined δ−s)A(θ) ζ(ds), are disturbances of the discrete model (6), ξt = (τ −1)δ e such that  δ
esA(θ) ΩesA (θ) ds, E[ξt ] = 0, Ξ = E[ξt ξt ] = (7) 0 E[ξt ξs ] = O

for all t, s, t = s.

It must be emphasized that even if Ω is a diagonal matrix this will not be true of the variance matrix of errors of the exact discrete model E[ξt ξt ] so the ξt are not independent but, given the properties of the innovation process ζ(dt), they will be serially uncorrelated. For this reason, and because of the cross-equation restrictions on the coefficients of the system, it is necessary to use a simultaneous-equation estimator. Observations generated by the continuous model (4) will satisfy the exact discrete model (6) irrespective of the observation interval δ, so the sampling properties of the parameters θ may be derived from the sampling properties of (6). A full-information maximum-likelihood (FIML) estimator of (6) allows all of the restrictions inherent in the underlying continuous model, and in any linearization of that model, to be imposed, and thus provides consistent and asymptotically efficient estimates. Moreover, for higher-order models in particular, the Gaussian estimator of continuous models is superefficient in that the estimator of parameters

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and their asymptotic standard errors converges at a rate of O(1/T ) as T → ∞ where T is the sample size, whereas FIML estimators of stan√ dard discrete models converge at a rate of O(1/ T ) as T → ∞. Moreover, as mentioned above, these estimators may be obtained under the assumption that the integral of the white noise innovation process is a Gaussian process although the innovations themselves are not assumed to have that property. Exact estimators of mixed-order mixed stock-flow models also can be derived but are much more complex because flow variables, and derivatives of other variables, cannot be observed
at a point but only as an integral t over the observation period, i.e., ytf = t−δ y f (s)ds and if for some “stock” variable that is observed at a point in time, dyis (t) = yj (t)dt, observations on yj (t) are given by yjt = yit − yit−δ (See Wymer, 1997 for illustrations.) The serial correlation in the observations of continuous variables arising from the moving-average process introduced by the measurement of flow variables must be taken into account during estimation in order to prevent asymptotic bias in the estimators. Exact estimators of mixedorder, mixed stock-flow models have been derived and used by Bergstrom et al. (1992) but these are extremely complex. To avoid the complications that arise from moving-average processes, an approximation to this moving average that is independent of the parameters of the system may be derived for rth-order models as in Wymer (1972). This approximation, which has fixed coefficients, is of the same order as the moving average in the exact model. This approximate moving average can be inverted and truncated after a few terms and used to transform all of the series in the sample to eliminate the serial correlation inherent in the observations, at least to an approximation. This transformation provides a system similar to the exact discrete model but in which the disturbances can be treated as serially uncorrelated. If the matrix Ω were unrestricted and the model were linear in y(t) the log-likelihood function of the differential equation system would be T 1   −1 mT ln(2π) − ln det Ξ − (ξ Ξ ξt ) 2 2 2 t=1 t T

ln(θ, Ω) = −

where Ξ is defined as above; hence ln(θ, Ω) = −

T T mT ln(2π) − ln det Ξ − trΞ−1 V 2 2 2

(8)

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 with V being the residual covariance matrix V = T1 Tt=1 ξt ξt . The Gaussian estimator may then be obtained by maximizing ln(θ, Ω) with respect to Ω and θ and as Ω is unrestricted this is equivalent to minimizing ln det V with respect to θ. It is assumed throughout that the parameters are identified. If not, and sometimes in the derivation of (1) or (4) from some underlying theory information is lost which leads to a pair of parameters which always appear in the same way, the model will need to be modified. Although there is potentially a problem of aliasing in these models, this is usually precluded by the restrictions on the elements of A, which will often be non-linear and cross-equation. In comparing ordinary discrete models (i.e., models that are not stochastically equivalent to a continuous process) it should be noted that even if Ω is diagonal this will not be true of error covariance matrix Ξ of the exact discrete model (6) except for extreme cases. Thus, a simultaneous estimator is required. Second, even if A has relatively simple constraints, this will not hold for the coefficients of the lagged endogenous variables in (6). In any case, the elements of A are often highly non-linear functions of the parameters, and imply cross-equation restrictions on the parameters so a full-information estimator is needed. With higher-order systems, the introduction of additional variables to reduce an rth-order system to first order has to be taken into account in (7) and (8). An rth-order differential system will lead to a discretevariable VARMAX model of the same order. In such models, Ω is of rank mR only (augmented with O matrices) while A and eA(θ) are of rank and order m∗ . Assuming a linear model defined as in (2), for example, the solution for y3 (t) in (5) may be inverted to give y1 (t − 1), providing the corresponding submatrix of eA is nonsingular which will almost certainly hold, and y1 (t − 1) = Dx1 (t − 1) eliminated from the right-hand side (RHS) of (5). As Bergstrom (1983) has shown, if x(t) = {x1 (t) x2 (t)}, successive substitutions allow (5) to be written (ignoring the exogenous variables) as the second-order VARMA process (see Appendix A) x(t) = F1 x(t − 1) + F2 x(t − 2) + ξ(t)

(9)


t
t−1 where ξ(t) = t−1 P0 (t − s)ζ(ds) + t−2 P1 (t − 1 − s)ζ(ds) and Pi and Fi are matrix functions of submatrices of eA(θ) . Hence, ξ(t) is now a first-order moving-average process with ξ(t) and ξ(t − 1) being serially correlated with

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a correlation matrix that can be derived from above but E[ξ(t)ξ  (s)] = O for all t, s, |t − s| > 1.

8.3.2.

Non-linear estimators

Although most of the theoretical development of continuous-time estimators has been within the context of linear models, in practice most of the models being estimated have been linearizations of some underlying nonlinear system. This approach is justified in that the estimated parameters are those of the non-linear system and the linear model is estimated subject to all of the restrictions inherent in the theoretical structure and the linearization. This does, however, introduce an approximation into the models being estimated and may cause serial correlation in the disturbances of the linearized model. See Wymer (1995) for further discussion. Even when the disturbances of the non-linear model are Gaussian, this need not be true of the disturbances of the linearized model. In addition, linearization may be true of the disturbances of the linearized model. In addition, linearization may prevent some parameters from being identified, although they are identified in the non-linear model, so that estimates of those parameters must be obtained in some other way and these estimates usually will be inconsistent. Even where parameters are formally identified in the linear model, however, the way in which they enter the linearized model may mean that they are poorly determined and the nonlinear model may provide more robust, and more precise, estimates of these parameters. For these reasons, FIML estimators analogous to the estimators of linear models have been developed to estimate the parameters of non-linear systems directly as discussed by Wymer (1993b, 1995). Both a non-linear exact discrete estimator and approximations to it are available. While some asymptotic properties of these estimators are known, others have to be inferred heuristically from the properties of the linear estimators. Although the exact discrete estimator for linear models is derived using the analytical solution to the differential-equation system, for non-linear models the system must be solved by numerical integration, but the principle is the same in both cases. In the pure case, for instance, where all variables are observed at a point in time, the estimator will be consistent and efficient and is analogous to the exact discrete estimator for linear models that provides estimates that are superefficient. If this estimator was applied to a linear

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model, the estimates would be the same as with the exact linear discrete estimator. This pure exact non-linear discrete estimator requires the exogenous variables (or forcing functions) to be analytic functions of time so that the integration procedure is exact. Under those conditions, the exact linear and non-linear discrete estimators are the same for linear models but this will not be true for more general exogenous variables. In most econometric models, the exogenous variables are defined only by a series of discrete observations. As in the linear estimators, the continuous path of exogenous variables may be approximated using a polynomial fitted to nearby observations. Although this introduces an asymptotic bias into the estimator, this bias is known to be small in the linear case and a similar result can be expected in non-linear systems. The derivation of the exact estimator of the non-linear model (3) is analogous to that of the linear estimator. Let ξ(t) be the vector of errors in the trajectories of the system such that ξ(t) = y(t) − yˆ(t) where y(t) is the solution to (4) given initial conditions y(t − δ) and yˆ(t) satisfies Dˆ y (t) = ϕ{ˆ y (t), z(t), θ} given the same initial conditions. Thus, ξ(t) which will be the integral of a function of a (non-linear) matrix function J[ϕ{y(t), θ}] of the vector on the RHS of (3) with where J is the Jacobian ∂ϕ{y(t),θ} ∂y(t) respect to y(t), and let Ξt = E[ξt ξt ] =



1

esJ[ϕ{y(t+s−1),θ}]ΩesJ




[ϕ{y(t+s−1),θ}]

ds

(10)

0

Thus, again the errors ξ(t) are interdependent but serially uncorrelated. In the “pure” case, the endogenous variables y(t) are assumed to be observable at a point in time, such as stocks, prices, interest, and exchange rates, and the exogenous variables to be given analytic functions of time. For a given set of initial values of the parameters θ and a set of initial values for the variables at the point t − 1, the solution trajectories of (3) can be found by a numerical integration procedure. Let the vector of solution trajectories at t given the observed y(t − 1) as initial values and the given set of θ be yˆ(t; θ). The residuals corresponding to ξ(t) can then be calculated and used to form the likelihood function that can then be maximized to give FIML estimates of the parameters. The difficulty which arises is that Ξt in (10) is a function of y(t) and ˜ is the block diagonal matrix with (block) so may vary over time. If Ξ  elements Ξ1 , · · · , ΞT and ξ˜ = [ξ1 , · · · , ξT ] the corresponding column vector

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of residuals, the log-likelihood function may be written  T mT ˜ ˜ −1 ξ) ˜−1 ln(2π) − ln det Ξ (ξ˜ Ξ 2 2 2 t=1

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T

ln(θ, Ω) = −

(11)

˜ is positive definite there exists a lower triangular matrix L with posiAs Ξ ˜ = LL . Thus, as in Bergstrom (1985), tive diagonal elements such that Ξ the Gaussian log-likelihood becomes 1 2 mT ln(2π) − (ε + 2 ln ii ) ln(θ, Ω) = − 2 2 t=1 i mT

(12)

where ii is the ith diagonal element of L and ε = [ε1 , · · ·, εmT ] is a vector ˜ The factorwhose elements may be calculated recursively from Lε = ξ.  ˜ into LL is very fast as is the recursive calculation to find ε. ization of Ξ Equation (11) can be maximized with respect to θ and Ω. With higher-order systems, again the introduction of additional variables defined as an identity must be taken into account in a way similar to the linear model but via the Jacobian. Asymptotic standard errors of the parameters may be calculated either algebraically for the linear (or linearized) model or numerically for the nonlinear model. As usual with full-information models, likelihood-ratio and Wald tests may be used to test the joint hypotheses underlying the model and it’s consistency with the data generated by the economic system. A comparison of the linear and non-linear estimators shows that, although many parameter estimates are not significantly different, some estimates are quite different; this occurs particularly with parameters that appear to be poorly identified in the linearized model. Although the costs can be high in terms of computing time, especially if the model is defined in the implicit form rather than the recursive form (3), the estimates suggest that there are substantial benefits in using the non-linear estimator in that the asymptotic standard errors are very small relative to those of the estimators of the linearized model and to the approximate discrete estimator of the non-linear model. The mean-square residuals are also smaller. It is considered that these results are due partly to the use of an exact rather than approximate estimator, which would be consistent with the estimators of linear models, but also because the specified relationships, and especially identities, are estimated directly without introducing errors of linearization and, perhaps more importantly, eliminating any serial correlation arising from linearization.

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8.3.3.

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Estimating models with boundary conditions

The exact non-linear estimator has been extended to allow the estimation of non-linear two-point boundary problems (or even multipoint), which would cover Hamiltonian systems, rational expectations, differential games, and other problems with forward-looking variables. The numerical solution of such models, some of which are highly non-linear, with given parameters is now well developed and is being used routinely. Such a procedure has been embedded within the non-linear estimator discussed above. The question of whether the parameters are identified remains, especially in the case of data generated by an optimally controlled differential-equation system. Assume that the basic time unit in the system is chosen such that δ = 1, so that the sample consists of a set of observations y(t) = yt , t = 1, · · ·, T . An essential part of the non-linear estimation procedure above is the integration of the system (3) to find the solution yˆ(t) that satisfies Dˆ y (t) = ϕ{ˆ y (t), z(t), θ} given initial conditions y(t) = yt for each observation. The vector of errors ξ(t) = y(t) − yˆ(t) in the estimated trajectories of the system then can be used to derive the error covariance matrix of the system and to calculate Gaussian or quasi-maximum-likelihood estimates of the parameters. Instead of solving the system over each observation interval given initial conditions only, the solution can be obtained subject to more general boundary conditions defined at (ta , tb ); let these points be (t − 1, t + Th ) where Th is fixed and often will be large. In some cases, such as the oil extraction model below, the system will be autonomous but otherwise exogenous variables must be defined over the interval (0, T + Th ) perhaps as analytical functions of time. Assume that the first-order system (3) consists of ma endogenous variables ya (t) for which initial values are assumed to be given and mb endogenous variables yb (t) defined by the endpoint conditions. Thus, the first-order system becomes Dya (t) = ϕa {ya (t), yb (t), z(t), θ} + υa (t) Dyb (t) = ϕb {ya (t), yb (t), z(t), θ} + υb (t)

(13)

which can be solved numerically for the vector [ya (t) yb (t)] for each t = 1, · · ·, T given ya (t−1) and an appropriate set of boundary-point conditions defined at some point t + Th . All this presumes that the boundary point problem is well defined. For simplicity, it is assumed that any zero-order equations, such as those that arise in optimal control problems, have been eliminated but the estimation and solution procedures being used do not

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require this. The errors in the trajectories can be calculated and used to give a quasi-maximum-likelihood estimator of the parameters of the system. It must be emphasized that the properties of the errors, particularly their serial correlation properties, and the properties of the estimator have not been investigated at this stage. It also must be emphasized that to obtain consistent estimates it is necessary to reinitialize the solution procedure for each observation and that the horizon Th should be constant; if appropriate, estimates with different values of Th may be used to test whether Th is sufficiently large. While at present the asymptotic properties of the estimator are unknown, they may be inferred from full-information Gaussian estimators of non-boundary point models. Estimators of those models are consistent and stochastically efficient, and for some models, generally of a separate class from those here, are superefficient as the estimator converges at rate √ 1/T rather than 1/ T . At least some models of the robust control class are in the superefficient class.

8.4.

Stability Analysis

A study of the properties of the system includes not only determining its stability for the estimated or given parameter values, but also the question of whether it is structurally stable over some relevant range of parameter values, such as the confidence intervals of the parameter estimates. Essentially, a model is structurally stable if small changes in parameters do not produce a qualitative change in its dynamic properties; points at which parameter changes do have a qualitative effect are bifurcation points. In very simple models with few parameters, knowledge of the model may be sufficient to indicate the relevant intervals of the parameter set where bifurcation points will exist. In larger models, sensitivity analysis can be used to determine the parameters that crucially affect the dynamics of the system and that are likely to be relevant. The existence of influential parameters may have policy implications. For example, if a model is structurally unstable for some parameter values, and if this can lead to some behavior of the system that is undesirable, it may be possible to introduce policies that will shift the relevant parameters away from critical values. In previous work, the asymptotic properties of the non-linear system (1) have been studied by deriving some transformation of the variables, such as deviations about the steady state, which allows the model (2) to be written

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as an autonomous system — i.e., as a system in which the time variable is not itself an argument. Thus, if (1) has a steady state such that y(t) = y ∗ eρt (assuming that the exogenous variables are analytical functions of time), the system may be solved for y ∗ and ρ as functions of the parameters θ. It then may be possible to define a set of variables — say x(t), perhaps as deviations of y(t) about its steady state — to produce an autonomous system. Differentiation of the equilibrium point y ∗ (θ) with respect to the parameters θ allows an analysis of the steady state itself, and dynamical behavior in the neighborhood of the steady state may be investigated using the eigenvalues and eigenvectors of a linearization about the steady state. The structural stability of the model, i.e., the effect on the dynamic properties of changes in the parameter values, may be studied using sensitivity analysis by differentiating the eigenvalues (and eigenvectors if required) with respect to the parameters (Wymer, 1993a). These techniques can be extended directly to a study of more general dynamical behavior in non-linear systems through the analysis of attractors via the computation and analysis of Lyapunov characteristic exponents. (See Wymer, 1997 for a theoretical and methodological discussion and Piras et al., 2007 and Wymer, 2009 for empirical applications.) In an analysis of the dynamics of these models generally, a distinction must be made between dissipative systems which always have an attractor or repeller, such as fixed points, tori, or strange attractors, and conservative or Hamiltonian systems which do not but have an infinity of closed orbits so that any initial point will always lie on one of these orbits. A Hamiltonian model will usually be known by construction, but the two types of systems can be distinguished using the generalized divergence of the functions ϕ{y(t), θ} of (2) or (3). Let a set of initial conditions be contained in a vanishingly small hyperellipsoid V in n-dimensional space. This volume will change as a function of t as y(t) changes, so that dV = dt

 V

   n ∂ϕi ··· dy1 , · · ·, dyn ∂yi i=1

(14)

The summation term is the generalized divergence or Lie derivative of ϕ; dissipative systems are characterized by contracting volumes, i.e., dV /dt < 0, while in conservative or Hamiltonian systems V is constant. If a dissipative system is structurally stable and converges on a strange attractor the system will be chaotic, i.e., its long-run trajectory will have aperiodic fluctuations. All conservative or Hamiltonian systems are structurally unstable and do not possess attractors but chaotic behavior can

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arise in the neighborhood of the Hamiltonian orbit when the system is perturbed; this is local turbulence and hence of a different nature from that which arises from strange attractors.

8.5.

Example: Estimation of a Model from the Economics of Exhaustible Resources

Pindyck’s (1978) model of exploration and production of non-renewable resources provides a small concise example of a non-linear continuous-time model of intertemporally optimizing behavior, which may be estimated with empirical data using the approach outlined in the third section of this chapter.2 The model is based on the assumption that it is possible to augment proven reserves of a non-renewable resource through exploration and that the volume of discoveries grows as a function of drilling activity and secondary or tertiary recovery of existing fields. Total reserves to be exploited in the future are given by cumulative discoveries less extraction. The state equations of the model are X˙ = rws e−uX

(15)

R˙ = X˙ − q

(16)

where X denotes cumulative discoveries, R proven reserves, w exploratory activity measured by wells drilled, and q is the extraction rate. The intertemporal objective functional to be maximized  tf e−δt {(a − bq)q − (c/R)q − (m + nw)}dt t0

gives the net present value of exploratory and extractive activity. In the integrand, the first term is the value of production, i.e., product of price and quantity, where a − bq represents a price-dependent demand function. The second term implies that the cost of extraction is decreasing in proven reserves, while the third term is the cost of exploration. The Hamiltonian for this optimization problem is H = e−δt {(a − bq)q − (c/R)q − (m + nw)} + µ1 rws e−uX + µ2 (rws e−uX − q) 2 The

model used (initially) in this chapter is Conrad and Clark’s (1987) specification of Pindyck’s model and employment time-series data for Nigeria (1980–2008) published by the Energy Information Administration of the United States Department of Energy.

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The first-order necessary conditions for optimization include the two state equations, the following costate equations, and optimality conditions, and (the implied) initial and transversality conditions: ∂H/∂X = −µ˙ 1 = −(µ1 + µ2 )urws e−uX

(17)

∂H/∂R = −µ˙ 2 = e−δt cqR−2

(18)

∂H/∂q = e

−δt

(a − 2bq − c/R) − µ2 = 0

(19)

∂H/∂w = −e−δt n + s(µ1 + µ2 )rws−1 e−uX = 0

(20)

The transversality conditions are limt→∞ µi = 0. The first-order conditions (15)–(20) may be estimated by a continuous FIML estimator under the assumption that those equations hold with an additive disturbance term, whose integral is a Gaussian innovation process. The sample used in this chapter was drawn from the Nigerian oil economy with annual observations from 1980 to 2005. Initial estimates of this model suggested that the errors did not satisfy the assumptions and were likely to be heteroskedastic. This could not be tested reliably owing to the small sample size. On these grounds, however, the model was rewritten by defining X, R, and w as logarithms but without changing the objective function nor the first-order conditions. The stochastic error process is then of a logarithmic form. Defining X  = ln X, R = ln R, w = ln w, and letting µi = µi eδt , i = 1, 2, the model becomes



X

X X˙  = resw e−ue /eX = re(sw −ue




X







R˙  = eX −R X˙  − qe−R = re(sw −R −ue X





µ˙ 1 = ur(µ1 + µ2 )esw −ue

−X
) )




− qe−R

+ δµ1




µ˙ 2 = −cqe−2R + δµ2


q = (a − µ2 − ce−R )/2b w =

(15a)


1 {ln[sr(µ1 + µ2 )/n] − ueX } 1−s

(16a) (17a) (18a) (19a) (20a)

Although the model above is deterministic, for any econometric study it is assumed that the state equations do not hold without error and that there is imperfect control. The imperfection can arise from stochastic errors in the state equation or from errors in the objective function or optimizing

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process. In particular, control of such a system must be considered imperfect with the model representing an abstraction derived from economic theory. The costate variables µi are unobservable and (16) is an identity so the rank of the error variance matrix is three in this model. The model may be estimated by a Gaussian FIML estimator subject to the transversality conditions limt→∞ µi = 0 by taking the observed values of the state and control variables X  (t), R (t), q(t), w (t) for each observation t, and from initial values of the parameters and the µ (t), the boundary point (BP) problem can be solved by numerical integration within a minimization procedure to give the values of µ (t) such that the transversality conditions hold at some point t + Th ; Th is some horizon usually relative to t but it could be absolute. The system is then integrated forward from t to t + 1 to give the residuals x(t + 1) − x ˆ(t + 1) for the observed variables and hence the residual covariance matrix. A Newton minimization procedure or equivalent may then be used to optimize the log-likelihood function relative to the parameters of the model.3 In this case, the underlying stochastic innovations matrix Ω as in (3), relevant to the disturbances ζi (dt) of the stochastic dynamic equations, is of order one only, but the error covariance matrix of the whole system, made up of this augmented by the errors of the zero-order equations, is three. If the system as a whole is well defined, the submatrix of the Jacobian corresponding to the zero-order variables in the zero-order equations will be nonsingular (in this case of rank two) and thus can be used to eliminate (or concentrate) the corresponding elements of the submatrix of the Jacobian of the first-order equations (differentiated with respect to the zeroorder variables). This gives a Jacobian for the first-order equations of order four. The submatrix of this (concentrated) Jacobian corresponding to the first-order variables determined in non-stochastic equations (i.e., first-order identities and costate variables) will again be nonsingular if the system is well determined, and so can be used to concentrate the Jacobian to a matrix of order one (in this case). It is this matrix that is used in (7) to obtain Ξ. Ξ is then augmented by variance and covariance terms corresponding to

3 During the iterative procedures in estimating this model the argument of the logarithm in (20a), the specification of w
, could become negative. To avoid this, the argument was squared and the logarithm multiplied by 1/2. This is valid, providing the argument at the end of each iteration is positive, as it was here. Problems of this nature are not uncommon in estimating (or even solving or simulating) non-linear models. Although there are often ways of getting around such problems, there is no general solution.

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the stochastic zero-order equations and the log-likelihood evaluated as in (11) and (12). This is discussed in Appendix A. Several issues arise with this procedure, some computational and some of a deeper nature. Horizon: It is presumed these models do not have a closed form solution or at least do not have one that can be calculated readily. Although the horizon in the theory is infinite, in general a finite horizon can be chosen that is a very good approximation. Re-estimating with a longer horizon and comparing the estimates will indicate whether the chosen horizon is sufficiently large. In the model estimated for this chapter the horizon was specified as 50 years. Control and other zero-order equations: In the model here the control equations are sufficiently simple that they can be solved algebraically for the control variables. If the hypotheses of the model are correct, in a continuous system it can be assumed that observations are on or close to the optimal trajectory at all times. Thus, in simple cases it would be possible to eliminate the costate variables from the system and solve a model with observed variables only. In this case, for example, a model y˙ = f (y) where y = [XR q w] could be derived by using Eqs. (19a) and (20a) to eliminate µ1 and µ2 . In general, however, the zero-order equations will be highly non-linear and must be solved numerically. The elimination of the costate variables is implicit but the solution for all variables in the system takes this into account. This does complicate the estimation procedure since these zero-order non-linear equations must be solved for every point in the integration procedure but usually this is not a serious difficulty. The models being estimated are usually, but not necessarily, linear in y(t), ˙ or close to it so the dynamics are essentially recursive. Identification: In addition to the usual issue of identification of parameters in any model, the use of unobservable (or at least unobserved) variables in this model makes this question even more important. In these models, where the equations are derived as the first-order conditions of an intertemporal optimizing process, some or many of the parameters in the costate equations will occur elsewhere. Hence, they might be identified independently of the costate equations. More importantly, however, the costate equations are not merely dropped from the model for estimation but are at the core of the BP solution; any change in a parameter in a costate equation will affect the whole system and hence the residual covariance matrix for all observed variables. As this is a full-information procedure, identification of

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the parameters and the power of the estimator will be enhanced by these equations while conversely the inclusion of these equations requires a fullinformation estimator. Two forms of identification can arise, “formal” underidentification in which some parameters are unidentified for any sample, and underidentification which arises for a particular sample. It is possible that theory provides a specification where two or more parameters only occur in the model in the same functional form; a simple example is where two parameters α and β occur only as α + β. In such cases, the function may be identified but not the components. The latter case arises where parameters are identified in a formal sense but a lack of variation in a variable in the sample being used may mean that some parameter cannot be estimated. A weaker form of this causes very poor estimates with a parameter having very large standard deviation. Two practical issues that arise with estimation of these models revolve around initialization of the iterative procedures that are required. Estimation involves up to four nested iterative procedures: the Newton procedure for maximization of the Gaussian likelihood with respect to the parameters, including explicit or implicit parameters which make up the error innovation matrix Ω, the procedure for solving the BP problem for the initial values of the (usually) costate variables, the integration procedure, and an iterative procedure for solving non-linear equations, particularly any zero-order equations. These models are usually recursive in the first-order derivatives of the state and costate variables, but zero-order equations, such as those specifying the first-order conditions for the control variables, are often non-linear.4 Initialization: A likely problem with estimation of Hamiltonian systems are that costs or similar parameters in the objective function are not known nor are the values of the costate variables at a given point in the economy, in particular for these purposes the initial values of the costate variables at the beginning of the sample. Thus, estimation of these models involves two sets of initialization: specifying a reasonable set of initial values of the parameters and finding a corresponding suitable set of initial values of the costate variables. Given a set of initial values of the parameters, a search can be 4 The

basic model (1) can be generalized so that the functions ϕ1 , ϕ2 may include y¨1 (t) and y˙ 2 (t). It is assumed in this chapter that this is not the case but the discussion above is fully consistent with (2) having the form A(θ)y˙ ∗ (t) = ϕ∗ {y ∗ (t), z(t), θ} + ζ ∗ (dt) where A is lower triangular. For simplicity, A is assumed in this chapter to be diagonal.

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made for the initial values of the costate variables. Solving the BP problem to find the values of the costate variables at the first observation, that is µ(0) given observed values x(0) of all other variables, the value µ(1) on that trajectory can be taken as the initial value for the costate variables for solving the BP problem for observation 1. If this is successful, initial values can be found for the costate variables for each observation for the initial parameter set. Often, however, the initial parameter set does not permit full initialization of the costate variables and it becomes necessary to search over some parameter space to fully initialize the costate variables. Limited experience suggests that once a full set is found these will allow the parameters to be varied according to a Newton procedure, for instance, with the initial values of costate variables being updated as the parameters change. Initial values of the costate variables seem to be quite robust once the procedure has been initialized. Means of residuals: An assumption in deriving estimators for these models is that the disturbance or stochastic innovation process has expected value zero. This implies that the mean of residuals should also be zero. In single equation linear models this is enforced by using deviations about means or something equivalent for estimation. This can be extended to linear simultaneous equations provided it is consistent with the constraints, perhaps nonlinear and cross-equation, on the parameters; constraints on the parameters arise from the theory underlying the specification of the model. It is possible even in a linear model that these constraints are such that for a particular sample not all means of residuals are zero. Imposing the constraint that the mean of residuals of a non-linear model, i.e., a model non-linear in variables, is generally not possible and hence the assumption becomes another joint hypothesis which must be tested. Table 8.1: Estimation of oil model for Nigeria, 1980–2005. a. Estimates of parameters

a b c n s r u δ

Point Estimate

Asymptotic Standard Error

4.454 2.684 6.034 0.020 0.472 0.379 −0.0019 0.0036

0.121 0.093 0.535 0.002 0.058 0.127 0.009 0.005

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b. Data and residuals

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Variable Observations

X
µ
1 µ
2 R
q w


Residuals

Mean

Standard Deviation

Mean

Standard Deviation

3.298

0.325

−0.0038

0.075

2.967 0.694 2.383

0.232 0.138 0.447

0.0011 −0.0057 −0.0537

0.112 0.123 0.663

Initial values of variables X
(0) µ
1 (0) µ
2 (0) R
(0) q(0) w
(0)

2.846 0.018 0.410 2.815 0.523 2.797

In a linear model, the likelihood ratio of the FIML estimates relative to the likelihood of the unrestricted reduced form may be used to test whether the model is consistent with the data. As shown above, an rthorder continuous linear model is equivalent to a highly constrained rthorder VARMAX process so that forms the basis for testing in the linear case. There is no clear corresponding test of consistency with the data of the joint hypotheses that comprise a non-linear model as there is no equivalent in non-linear systems to an unrestricted reduced form of a linear model. An approximation to testing an second-order non-linear model would be to use a linear unconstrained VARMAX model (autoregressive of order two, moving average of order one) as this would be equivalent to a firstorder power series expansion of the non-linear functions; second- or even higher-order expansions should provide a better approximation, provided the sample size is sufficiently large. For example, for a non-linear first-order differential equation model, this would entail calculating the log-likelihood of the discrete model   C3 yt−1 + yt−2 C4 yt−2 + ut + Gut−1 yt = C1 yt−1 + C2 yt−2 + yt−1

plus corresponding terms for any exogenous variables including time or a  . The matrices constant term and perhaps cross-product terms in yt−1 yt−2 C and G are unrestricted. This will have a very large number of coefficients

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relative to the parameters of the non-linear model. This does not take the BP problem into account; the impact of that will be something equivalent to increasing the order of the VARMAX process by one. The likelihood ratio, given by 2 (LU − LR ), has a Chi-square distribution with the number of degrees of freedom equal to the difference between the number of parameters in the unrestricted and constrained models. The difficulty with using this test in non-linear cases is that if the models are not nested there is no measure of the number of overidentifying restrictions. Hence, the ratio must be considered indicative only. For the Nigerian oil model above, the maximum log-likelihood is 137.4; and the log-likelihood of the unrestricted first-order linear VAR model is 175.6 and the second-order linear VAR process 191.5. As the number of degrees of freedom is relatively small, this comparison suggests that the hypothesis that the non-linear model is consistent with the data must be rejected at the 5% level. This is a weak test as only the first-order terms in the power series expansion have been used; any higher-order expansion would make the rejection even more definite. It must be noted, however, that the sample size is small and there are aspects of the model which are clearly at odds with the time series. Also, the purpose of this chapter is to estimate and test a given theoretical model and no attempt has been made to modify the theory, to introduce robust control, or additional factors such as the institutional structure. The relatively high errors of the (control) equation for wells drilled w suggests that either the state equation determining the change in cumulative discoveries or the objective function is misspecified at least with respect to wells drilled. As wells drilled essentially represents high-risk investment the way in which w enters the objective function is likely to be more complex than in the model above. (The probability of success in drilling a marginal well is likely to decrease over time.) The model needs to be extended to provide a more accurate representation of the Nigerian oil industry. The last two observations of cumulative discoveries jumped from about 41 to 52 billion barrels with a corresponding increase in proven reserves. The model is too simple to take account of that. Exploratory activity, measured by number of wells drilled, fell from around 160 per annum at the beginning of the sample to around 60 to 80 for 12 years before increasing to around 180, or even much more, in the next 12 years. Again, the current model does not capture that fluctuation. Cumulative discoveries (X) and proven reserves (R) are in billion barrels, extraction rate (q) is in billion barrels per annum, and exploration

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activity (w) in 10 wells drilled per annum. All variables except q are in natural logarithms.

8.6.

Potential Improvements in Model Specification

The estimation of this model, and in previous work such as that in the macro-economic field of Saltari et al. (2009), suggests that the constraints of a full Hamiltonian model are too rigid or too stringent to satisfactorily explain economic behavior. The Saltari model also showed that the introduction of differential costs in the objective function was not sufficient to overcome this rejection of the model. One way to address this intrinsic difficulty may be to incorporate integral action in the specification of the model. In the 1950s, Phillips advocated the use of policies based on “proportional, integral, and derivative (PID) action” to stabilize macroeconomies manifesting cyclical or unstable behavior (Phillips, 1954). More recently, Salmon, among others, has advocated the incorporation of integral action in the specification of macroeconomic (and other) models as a means of generating specifications with endogenously determined stabilizing adjustment dynamics.5 (See, e.g., Marcellino and Salmon, 2002.). If these models are to be used for policy analysis or simulation, or to investigate the strategic implications of some aspect of the economy or market, the model should have a firm theoretical basis and it is essential that the joint hypotheses embedded in the model are validated, i.e., that they are consistent with the data. Moreover, an understanding of the nonlinear dynamics of economic models requires knowledge of the parameter values. It appears that the Hamiltonian specification of existing models, not only the one above, is too simplistic to explain observed dynamic behavior. Observed behavior is more robust than the models. Introducing integral action to the specification of intertemporal optimization behavior is but one way to render more robust the decision behavior modeled. As modelers have come to appreciate that intertemporally optimizing agents do not have — and know that they do not have — complete knowledge of the economy or decision environments in 5 Integral

action can be introduced to a representative agent model by introducing to the agent’s objective functional an expression that involves a derivative of a control variable. Solution of the model must then incorporate an integral of the derivative expression. Marcellino and Salmon (2002) show that in such cases the model must converge to a stable solution (if otherwise well specified and estimated with data from a stable regime).

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Research Tools in Natural Resource. . .

which they operate, other ways of characterizing decision behavior have been suggested in a burgeoning literature on “robust control” theory. (See, e.g., Hansen and Sargent, 2007, in general, and in natural resource and environmental economics, see, e.g., Xepapadeas and Roseta-Palma, 2003, and Roseta-Palma and Xepapadeas, 2004.) Non-linear continuous-time models of intertemporally optimizing agents embodying characterizations of robust control can now be estimated and analyzed by the approaches set out above with available software. (See Wymer, various dates.) Appendix A Let a mixed second-order linear model, without exogenous variables, be d[Dy1 (t)] = A11 (θ)Dy1 (t)dt + A12 (θ)y2 (t)dt + A13 (θ)y1 (t)dt + ζ1 (dt) (A.1) dy2 (t) = A21 (θ)Dy1 (t)dt + A22 (θ)y2 (t)dt + A23 (θ)y1 (t)dt + ζ2 (dt) (A.2) and define dy1 (t) = y3 (t)dt

(A.3)

as a temporary variable so the model may be written as the first-order system dy ∗ (t) = A(θ)y ∗ (t)dt + ζ ∗ (dt)

(A.4)

where 

 y3 (t) y ∗ (t) = y2 (t) , y1 (t)



A11 A =  A21 I

A12 A22 0

 A13 A23  , 0

  ζ1 (t) and ζ ∗ (t) = ζ2 (t) 0

Note the reordering of variables as in y ∗ (t). The solution as in (5) is  t ∗ A(θ) ∗ y (t − 1)dt + e(t−s)A(θ) ζ ∗ (ds) (A.5) y (t) = e t−1

As shown in Bergstrom (1983), once the intermediate variable y3 (t) is eliminated, this solution gives a second-order VAR, and first-order MA process in y1 (t) and y2 (t). Let K(t − s)ij be the (i, j) sub-matrix of e(t−s)A(θ) .

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As (eA )31 is almost certain to be nonsingular, the equations for y3 (t) may be inverted to give ∗ A −1 A ∗ A −1 A ∗ y1∗ (t − 1) = (eA )−1 31 y3 (t) − (e )31 (e )32 y2 (t − 1) − (e )31 (e )33 y3 (t − 1)  t A −1 e(t−s)A(θ) ζ1 (ds) (A.6) − (e )31 t−1

and y1∗ (t − 1) eliminated from the equations y1∗ (t) and y2∗ (t). The equations for y1 (t) may be lagged so ∗ y1∗ (t − 1) = (eA )11 (eA )−1 31 y3 (t − 1) A ∗ + [(eA )12 − (eA )11 (eA )−1 31 (e )32 ]y2 (t − 2)

A ∗ + [(eA )13 − (eA )11 (eA )−1 31 (e )33 ]y3 (t − 2)  t−1 + [e(t−s)A(θ) ]11 ζ1∗ (ds) − (eA )11 (eA )−1 31 t−2



t−1

×

[e t−2

(t−s)A(θ)

]11 ζ1∗ (ds)



t−1

+

[e

(t−s)A(θ)

t−2



]12 ζ2∗ (ds)

(A.7) and y1∗ (t − 1) eliminated from the model to give the second-order autoregressive, first-order moving average process as in (9) y(t) = F1 y(t − 1) + F2 y(t − 2) + ξ(t)

(A.8)

where



t

ξ(t) = t−1

 y1 (t) y(t) = , y2 (t)  t−1 ¯ P0 (t − s)ζ(ds) + P1 P1 (t − 1 − s)ζ(ds) t−2

P¯i and Fi are matrix functions of submatrices of K = eA(θ) , and Pi (s) is a matrix function of submatrices of K = esA(θ) , where   −1 K33 + K31 K11 K31 K32 F1 = , −1 K23 + K21 K11 K21 K22   −1 −1 K31 K13 − K31 K11 K31 K33 K31 K12 − K31 K11 K31 K32 F2 = , −1 −1 K21 K13 − K21 K11 K31 K33 K21 K12 − K21 K11 K31 K32

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 K32 , K22

K31 K21

 K31 P¯1 = K21 so

−1 −K31 K11 K31 −1 −K21 K11 K31

 Ξ0 (t) =

 P1 =

0

1



K11 K31

P0 (s)ΩP0 (s)ds  Ξ1 (t) =

0

1

K12 K32

 + 0

1

 and

P¯1 P1 (s)ΩP1 (s)P¯1 ds

and

P¯1 P1 (s)ΩP0 (s)ds

The derivation of the error covariance matrices of the non-linear model is, in essence, the same although the system must, in general, be solved numerically. Let the more general mixed second-order model, following from (A.2) but written as a non-linear form of (A.3), be dDy1 (t) = ϕ1 {Dy1 (t), y(t), z(t), θ}dt + ζ1 (dt) dy2 (t) = ϕ2 {Dy1 (t), y(t), z(t), θ}dt + ζ2 (dt) 0 = ϕ3 {Dy1 (t), y(t), z(t), θ} + ζ3 (dt)

(A.9)

0 = ϕ4 {Dy1 (t), y(t), z(t), θ} where y(t) is the vector of the (levels) of endogenous variables in the system with elements yi (t), i = 1, . . . 4 and the disturbances as defined in (A.2) are the column vector ζ(t) = [ζi (dt)] with E{ζ(dt)ζ  (dt)} = Ω. Zero-order variables and equations have been introduced where the y3 (t) are determined by stochastic equations with error innovations ζ3 (dt) and the y4 (t) are determined by identities. Some of the first-order equations defining y2 (t) may be identities so that the corresponding ζ2 (dt) will be zero for all t. For simplicity in the derivation below only, any zero-order variables y4 (t) in the identities will be eliminated by substitution. Introducing an intermediary variable Dy3∗ (t) = y1∗ (t) and with some renumbering of variables the model may be rewritten as the first-order system dy1∗ (t) = ϕ∗1 {y ∗ (t), z(t), θ}dt + ζ1∗ (dt) dy2∗ (t) = ϕ∗2 {y ∗ (t), z(t), θ}dt + ζ2∗ (dt) dy3∗ (t) = y1∗ (t)dt 0=

ϕ∗4 {y ∗ (t), z(t), θ} + ζ4∗ (dt) 0 = ϕ∗5 {y ∗ (t), z(t), θ}

(A.10)

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where   ∗   Dy1 (t) y1 (t) y ∗ (t)  y (t)    2   2     ∗  =  y1 (t) , y ∗ (t) =  y (t)   3     ∗   y4 (t)  y3 (t)  y5∗ (t)

y4 (t)

 ζ1 (dt)   ζ2 (dt)    ζ ∗ (t) =   0    ζ4 (dt) 0 

This system (A.9), or equivalently but more accurately (A.10), may be solved by numerical integration. The y(t) in the discussion and formulae below are taken from the trajectory of the numerical integration. The difficulties that arise in the non-linear case relative to the linear model are in the derivation of the integral defining Ξt in (A.10), especially where the system is of order two or more. The Jacobian J ∗ of the vector of functions ϕ∗ differentiated  with respect ∂ϕ∗ ∗ i {y(t),θ} ; in this model to the y ∗ (t) has (matrix) elements Jij (t) = ∂yj∗ (t) ∗ ∗ J31 = I, J3j = 0 for j = 2, . . . , 5. The Jacobian of the zero-order iden∗ will be tities with respect to the corresponding zero-order variables, J55 nonsingular if the system is well defined and so may be used to concentrate the Jacobian. Thus, the matrix elements of the Jacobian will become ∗ ∗ ∗ ∗−1 ∗ J˜ij = Jij − Ji5 J55 J5j for i, j = 1, . . . , 4; hence, in this model the matrix ∗ elements J3j are unchanged. For simplicity, and without loss of generality, it will be assumed that the model contains no zero-order identities and so this concentration of the Jacobian is unnecessary; in the following no ˜ distinction is made between J and J. The Jacobian of the zero-order stochastic equations with respect to ∗ will also almost certainly be nonsingular so that can be used y4 (t), J44 ∗ ∗ = Jij − to further concentrate the Jacobian; the elements become J¯ij ∗ ∗−1 ∗ Ji4 J44 J4j for i, j = 1, . . . , 3. Since this corresponds to a stochastic equation, the corresponding disturbance terms now become ζ¯i (dt) = ζi (dt) − ∗ ∗−1 ∗ ∗ J44 ζ4 (dt) for i = 1, 2. The matrix elements J3j are again unchanged Ji4 and the disturbance term ζ3 (dt) is identically zero. As in the linear case, ¯ ¯ defining K(t)ij = (eJ {y(t),θ} )ij as the (i, j) matrix element of eJ{y(t),θ} , the integrals of the disturbance process may be written  t ∗ ∗ ∗−1 ξi (t) = Ki1 (t − s){ζ1 (ds) − J14 J44 ζ4 (ds)} t−1



t

+ t−1

∗ ∗−1 Ki2 (t − s){ζ2 (ds) − J24 J44 ζ4 (ds)}

for i = 1, 3.

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The relevant integrals ξ3∗ (t) corresponding to y1 (t) and ξ2∗ (t) corresponding to y2 (t) are derived as above by eliminating the intermediate variable y3∗ (t) from the solution. Again, as K31 is almost certain to be nonsingular, the equations for y3∗ (t) may be inverted, at least implicitly, −1 (t − 1)× to give y ∗ (t − 1) which will contain the stochastic term −K31 2
t 3 ∗−1 ∗ ∗ K (t − s){ζ (ds) − J J ζ (ds)}. Note ζ (t) = ζ (t) so the 3i i 3 3 4 i4 44 i=1 t−1 expression can be phrased in terms of the original innovation matrix Ω. The terms corresponding to y3 (t − 1) in the Jacobian may then be substituted out from the equations y1∗ (t) and y2∗ (t) to give error covariance matrices of the same form as above but with a different definition of K. Thus  

  t ∗ ∗−1 ζ1 (ds) − J14 J44 ζ3 (ds) ξ1 (t) P0 (t − s) = ∗ ∗−1 ξ2 (t) ζ2 (ds) − J24 J44 ζ3 (ds) t−1    t−1 ∗ ∗−1 ζ1 (ds) − J14 J44 ζ3 (ds) + , P1 (t − 1 − s) ∗ ∗−1 ζ2 (ds) − J24 J44 ζ3 (ds) t−2 where the Pi (t) are matrix functions6 of sub-matrices of K. Crucially, unlike the linear case, the matrices J and K in these expressions are functions of t and must be evaluated at t or (t − s) for 0 < s ≤ 1 as appropriate. Separating out the terms in ζ3 (ds), the disturbances in the zero-order equations, gives

  t   t−1  ζ1 (ds) ζ1 (ds) ξ1 (t) P0 (t − s) P1 (t − 1 − s) = + ξ2 (t) ζ2 (ds) ζ2 (ds) t−1 t−2    t ∗ ∗−1 J44 ζ3 (ds) J14 P0 (t − s) − ∗ ∗−1 J24 J44 ζ3 (ds) t−1    t−1 ∗ ∗−1 J14 J44 ζ3 (ds) P1 (t − 1 − s) − ∗ ∗−1 J24 J44 ζ3 (ds) t−2 where the minus signs are simply the result of the way the zero-order equation was defined in (A.9). The corresponding integral of the disturbances 6 Note that the J (and hence K) in these expressions are defined as the concentrated form of the Jacobian derived by eliminating the non-stochastic zero-order variables y5 .

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of the zero-order equations is  t ∗−1 ∗ ξ3 (t) = − J44 J41 {K11 ζ1 (ds) + K12 ζ2 (ds)} t−1



t

− t−1



t

− t−1

∗−1 ∗ J44 J42 {K21 ζ1 (ds) + K22 ζ2 (ds)} ∗−1 ∗ J44 J43 {K31 ζ1 (ds)



t

+ K32 ζ2 (ds)} − t−1

∗−1 J44 ζ3 (ds)

Augmenting P0 , P1 to allow for the zero-order equations gives        t  t−1 ξ1 (t) ζ1 (ds) ζ1 (ds) ξ2 (t) = P0 (t − s) ζ2 (ds) + P1 (t − 1 − s) ζ2 (ds) t−1 t−2 ξ3 (t) ζ3 (ds) ζ3 (ds) where P0 and P1 are now   K31     P0 (t−s) =  K21    3  −J ∗−1  J ∗ K 4i

44

−

K32

−

K22 i1

∗−1 −J44

i=1

3 

 2   i=1 2 

 ∗ K3i Ji4

   ∗−1 , J44     

 ∗ K2i Ji4

i=1 ∗−1 −J44

∗ J4i Ki2



∗−1  J44 

i=1

and P1 (t − 1 − s) 2

−1 K31 K11 K31 K31

6K31 K11 − 6 6 6 =6 6 −1 6K21 K11 − K21 K11 K31 K31 4

K31 K12 −

−1 K31 K11 K31 K32

−1 K21 K12 − K21 K11 K31 K32

0

0

2 X



3

∗−1 J44 7 7 i=1 7 7

2  7. X ∗−17 ∗ 7 − K2i Ji4 J44 5

−

∗ K3i Ji4

i=1

0

 Hence, from the definition of Ξr = E[ξt ξt+r ] as above and the properties of the stochastic innovations ζ(ds)  t  t−1  P0 (t − s)ΩP0 (t − s)ds + P1 (t − 1 − s)ΩP1 (t − 1 − s)ds, Ξ0 (t) = t−1



t−1

Ξ1 (t) = t−2

t−2

P1 (t − 1 − s)ΩP0 (t − s)ds

and Ξr (t) = 0 for |r| > 1.

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References Balta-Ozkan, N, KP Donaghy and CR Wymer (2007). Effects of trade on emissions in an enlarged European Union: Some comparative dynamics analyses with an empirically based endogenous-growth model. In: Globalization and Regional Economic Modeling, RJ Cooper, KP Donaghy and GJD Hewings (eds.), pp. 353–392. Heidelberg: Springer. Bergstrom, AR (1983). Gaussian estimation of structural parameters in higherorder continuous-time dynamic models. Econometrica, 51, 117–152. Bergstrom, AR, KB Nowman and CR Wymer (1992). Gaussian estimation of a second-order continuous-time macroeconometric model of the United Kingdom. Economic Modelling, 9, 313–351. Conrad, JM and CW Clark (1987). Natural Resource Economics: Notes and Problems. New York: Cambridge University Press. Dasgupta, PS and GM Heal (1979). Economic Theory and Exhaustible Resources. Cambridge: Cambridge University Press. Hansen, LP and TJ Sargent (2007). Robustness. Princeton: Princeton University Press. Isard, P (1995). Exchange Rate Economics. Cambridge: Cambridge University Press. Marcellino, M and M Salmon (2002). Robust decision theory and the Lucas critique. Macroeconomic Dynamics, 6, 167–185. Phillips, AW (1954). Stabilization policy in the closed economy. Economic Journal, 67, 290–323. Phillips, PCB (1991). Error correction and long-run equilibrium in continuous time. Econometrica, 59, 967–980. Pindyck, RS (1978). The optimal exploration and production of nonrenewable resources. Journal of Political Economy, 86, 841–861. Piras, G, KP Donaghy and G Arbia (2007). Nonlinear regional economic dynamics: Continuous-time specification, estimation and stability analysis. Journal of Geographical Systems, 9, 311–344. Saltari, E, G Travaglini and CR Wymer (2009). Investment, productivity and employment in the Italian economy. In The Economics of Imperfect Markets, G Calcagnini and E Saltari (eds.), pp. 1–28. Berlin: Springer-Verlag. Wymer, CR (1972). Econometric estimation of stochastic differential equation systems. Econometrica, 40, 565–577. Wymer, CR (1993a). Continuous time models in macro-economics: Specification and estimation. In: Continuous Time Econometrics: Theory and Applications, G Gandolfo (ed.), pp. 35–79. London: Chapman and Hall. Wymer, CR (1993b). Estimation of non-linear continuous time models from discrete data. In: Models, Methods, and Applications of Econometrics, P Phillips and V Hall (eds.), pp. 91–114. Oxford: Basil Blackwell. Wymer, CR (1995). Advances in the estimation and analysis of non-linear differential equation models in economics. In: Methods and Applications

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of Economic Dynamics, L Schoonbeek, E Sterken and SK Kuipers (eds.), pp. 41–76. Amsterdam: North-Holland. Wymer, CR (1997). Structural non-linear continuous-time models in econometrics. Macroeconomic Dynamics, 1, 518–548. Wymer, CR (2009). Aperiodic dynamics in the Bergstrom/Wymer model of the United Kingdom. Econometric Theory, 25, 1099–1111. Wymer, CR (various dates). WYSEA (Wymer Systems Estimation and Analysis) Programs and Manuals.

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Chapter 9

Computable General Equilibrium Models for the Analysis of Economy–Environment Interactions Ian Sue Wing Boston University [email protected]

9.1.

Introduction and Motivation

This chapter is an introduction to the subject of computable general equilibrium (CGE) modeling in environmental and resource economics. CGE models are a widely used tool for the quasi-empirical analysis of environmental externalities — and policies for mitigating them — which are large enough to influence prices across multiple markets in the economy. We provide a simple, rigorous, and practically oriented exposition that develops the framework of a CGE model from microeconomic fundamentals, outlines how the resulting algebraic structure may be numerically calibrated using the economic and environmental data and then solved for the equilibrium values of economic variables, and illustrates how CGE simulations may be used to analyze the economy-wide impacts of environmental externalities and associated mitigation policies. Walrasian general equilibrium prevails when supply and demand are equalized across all of the interconnected markets in the economy. A CGE model is an algebraic representation of the abstract Arrow–Debreu general equilibrium structure which is calibrated on economic data. The resulting numerical problem is solved for the supplies, demands, and prices which support equilibrium across a specified set of markets, which can range from a single subnational region to multiple groups of countries interacting within 255

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E1. Product markets

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Expenditure Goods and services

Goods and services A. Households

F.

C. Government

F.

B. Firms

E2. Intermediate goods mkts.

Taxes

Taxes

Profits/ factor income D. Factor markets H1. Externalities

H1. Externalities G. Environment

H2. Externalities Goods and factors

Fig. 9.1:

H2. Externalities Payments

/

Non-market effects

The circular flow of the economy, with environmental interactions.

the global economy. Notwithstanding this, every economy represented in these models typically has the same basic structure: a set of producers, consumers, and governments whose activities are linked by markets for commodities and factors as well as taxes, subsidies, and perhaps other distortions. The conceptual framework at the core of every CGE model is the familiar circular flow of the economy, which is illustrated in Fig. 9.1. The circular flow stems from the market interactions among three institutions: (A) households, which are assumed to own the factors of production, (B) firms, which rent factors from the households for use in the production of commodities subsequently purchased by households and other firms, and (C) the government, which provides government goods that are consumed by households and firms. These interinstitutional transfers of goods and factors are indicated by solid black arrows, each of which represents a distinct market. In equilibrium, the value of goods and services traded in each market is balanced by a compensating financial transfer, indicated by a dashed black arrow. Factor rentals generate a stream of

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income for households (D), which finances their expenditure on commodity purchases. The latter, in turn, generates revenue for firms (E1). Likewise, an individual firm’s purchase of intermediate inputs generates revenue for the other firms which produce those commodities (E2), and public provision is financed out of taxes on firms and households levied by the government (F). In general, most economies will also include a foreign sector, with which firms and households will exchange flows of goods and payments. For the sake of brevity we do not go into detail about open economy CGE modeling, as that constitutes an entire area of research all its own. The chapter’s principal motivation is to elaborate the interactions between the circular flow of the economy and the environment (G). Diminished ecological functioning or increases in the scarcity rent of nominally unpriced natural resources occurs as a result of firms’ and households’ emissions of residuals or consumption of environmental services as inputs (H). These effects constitute classic negative externalities, as environmental services which assimilate pollution or provide inputs to production tend neither to be traded within markets nor tabulated in the economy’s accounts, resulting in a divergence between the private and social costs of the goods and factors with whose quantity and quality they are linked. While earlier surveys by Conrad (2002), Bergman (2005), and B¨ ohringer and Loschel (2006) have sought to organize the broad and diverse literature on environmental CGE modeling along a variety of functional and topical lines, Fig. 9.1 suggests a simple structural classification based on the scope and character of the environment–economy interactions represented within the model of the relevant externality. Most CGE studies in the past decade either (i) quantify how changes in the circular flow due to endogenous processes, policy changes, or exogenous shocks affect the level of environmental quality,1 or (ii) shed light on the consequences for the circular flow of policies that seek to increase the level 1 This body of work includes investigations of consequences of trade liberalization — principally in developing countries, its environmental and economic costs, and the effectiveness of various policy options for managing them (Dessus and Bussolo, 1998; Abler et al., 1999; Jansen, 2001; Faehn and Holmoy, 2003; He, 2005; Li, 2005; Vennemo et al., 2008), examinations of the interactions among environmental externalities and economic growth, income distribution and labor markets, (Abler et al., 1998; Berck and Hoffmann, 2002; Nugent and Sarma, 2002; Coxhead and Jayasuriya, 2004; Taylor et al., 2009), and studies of resource management practices in the areas of water quality and allocation (Seung et al., 2000; Roe et al., 2005; Diao et al., 2005; 2008; Brouwer et al., 2008; Strzepek et al., 2008; van Heerden et al., 2008) as well as agricultural and forest land — with particular emphasis on the impacts of macroeconomic shocks deforestation, land

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of environmental quality by altering the allocations of polluting residuals or the consumption of environmental resources.2 As the solid gray arrows (H1) imply, these analyses typically specify a one-way linkage from the economy to the environment, treating the latter as a more or less exogenous boundary condition on economic activity. Qualitatively similar linkages in the reverse direction can be found in a less common but rapidly growing arena of application: (iii) assessments of the economic impacts of changes in environmental quality (including disasters of anthropogenic or natural origin), indicated by the dashed gray arrows (H2).3 Lastly, comparatively few analyses (iv) bring the environment degradation (Abdelgalil and Cohen, 2001; Bashaasha et al., 2001; Cattaneo, 2001; 2005; Ianchovichina et al., 2001; Wiig et al., 2001; Fraser and Waschik, 2005). 2 The bulk of the policy literature lies in this area, focusing on the general equilibrium economic costs of policies to abate pollution in various media. There are comparatively few papers on air (Morris, 1999; O’Ryan et al., 2003; 2005; Cao et al., 2009) and water (Xie and Saltzman, 2000) pollution, the vast majority focus on the problem of climate change, in particular the consequences of greenhouse gas (GHG) emission limits for energy markets and the implications for aggregate welfare. This topic has itself spawned several lines of research beyond standard single-economy analyses of the growth and welfare effects of abatement measures (e.g, Garbaccio et al., 1999; Kamat et al., 1999; Nwaobi, 2004; Fisher-Vanden and Ho, 2007; Telli et al., 2008), in particular examinations of the economic effects of environmentally motivated taxes, especially their interactions with preexisting market distortions (Parry et al., 1999; Bye, 2000; Bovenberg and Goulder, 2001; Babiker et al., 2003; Andre et al., 2005; Bovenberg et al., 2005; Conrad and Loschel, 2005), the environmental and economic consequences of long-run increases in R&D and factor productivity, energy-saving technological progress or the emergence of GHG abatement or alternative energy technologies with the potential for radical savings in compliance costs (Dellink et al., 2004; Grepperud and Rasmussen, 2004; Schafer and Jacoby, 2005; Jacoby et al., 2006; McFarland and Herzog, 2006; Allan et al., 2007; Berg, 2007; Otto et al., 2007; Fisher-Vanden and Sue Wing, 2008), the nexus between GHG abatement, the global trading system, and leakage of emissions from abating to nonabating developing countries (B¨ ohringer and Rutherford, 2002; 2004; Babiker, 2005; Babiker and Rutherford, 2005), the conditions for endogenous emergence of coalitions of abating regions in a game theoretic setting (Babiker, 2005; Carbone et al., 2009), the interactions between emission reduction measures such as tradable permit schemes and imperfections in energy markets (Klepper and Peterson, 2005; Hagem et al., 2006), and the distributional impacts of schemes to share the burden of GHG emission reductions, such as allowance allocations in tradable permit systems, across countries and regions (B¨ ohringer, 2002; B¨ ohringer et al., 2003; B¨ ohringer and Welsch, 2004; Kallbekken and Westskog, 2005; Klepper and Peterson, 2006; Nijkamp et al., 2005; B¨ ohringer et al., 2007; B¨ ohringer and Helm, 2008), firms and industries (Bovenberg et al., 2005; 2008) and income groups within regions (Rose and Oladosu, 2002; van Heerden et al., 2006; Oladosu and Rose, 2007; Ojha, 2009). 3 This literature utilizes the outputs of global climate models to simulate the impacts of global warming on various sectors of the economy as economic shocks (Winters et al., 1998; Dessus and O’Connor, 2003; Bosello et al., 2006; 2007; Aunan et al., 2007; Hubler

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fully within the ambit of the circular flow, simultaneously capturing H1 and H2 by representing both the impacts of economic activity on the environment and the simultaneous feedback effects on production, consumption, and welfare.4 The reason is the difficulty involved in representing physical processes in such a way that they can be integrated within CGE models’ abstract equilibrium structure.5 The chapter’s secondary motivation is pedagogic. Despite their popularity as a research tool, CGE models continue to be unfairly dismissed by some in the field as “black boxes,” whose complex internal workings obfuscate the linkages between their outputs and features of their input data, algebraic structure, or method of solution, and worse, allow questionable assumptions to be hidden within them that end up driving their results. The problem can largely be traced to a lack of familiarity with CGE models on the part of economic researchers, a situation which is exacerbated by the dearth of graduate methods courses on, and rigorous yet accessible articlelength introductions to, the subject. Although descriptions of CGE models’ underlying structure, calibration, and solution methods, and techniques of

et al., 2008; Vennemo et al., 2008; Boyd and Ibarrar´ an, 2009). Similar types of assessments are also used to quantify the impacts of large-scale natural hazards on regional economies (Rose and Liao, 2005). 4 These include quasi-benefit–cost analyses of pollution control policies when the environmental externalities in question affect consumers’ health (Williams, 2002; 2003; Matus et al., 2008; Mayeres and Van Regemorter, 2008), an assessment of the impacts of nonseparability between the environment and commodities or factors — especially labor, with the goal of developing new model calibration techniques (Carbone and Smith, 2008), and, in a particularly exciting development, the coupling of a CGE model of the economy with a general equilibrium simulation of an ecosystem (Finnoff and Tschirhart, 2008), where the latter models predator–prey interactions as arising out of the net energymaximizing behavior of a variety of representative organisms (Tschirhart, 2000; 2002; 2003). 5 Many studies have found it easier to employ a wholly separate physical impacts model in an iterative manner, using the precursors of environmental damage generated by a CGE model for its inputs, and then using its outputs to specify shocks to the CGE model’s simulated economy. But this solution often proves computationally challenging. For example, the MIT Integrated Global System Model (IGSM) is a system of “softlinked” models in which the MIT EPPA CGE simulation generates future emission trajectories which then drive large-scale global climate and terrestrial ecosystem simulations in an open-loop fashion, without feedback effects on the economy (Sokolov et al., 2005). The challenges involved in closing this loop are illustrated by B¨ ohringer et al.’s (2007) recasting of Nordhaus’s (1994) DICE model of global warming as an intertemporal CGE model, in which a highly stylized physical model of atmospheric GHG accumulation and climate change is approximated using the Jacobian of the global mean temperature response to the simulated path of emissions.

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application abound, they tend to be divided across different literatures, each of which focuses on its own aspect of the subject.6 The aim of this chapter is to remedy this lacuna by unifying these disparate elements to provide a systematic, practical, and intuitive explanation of the methods by which CGE models are constructed, calibrated, solved, and employed in the analysis of economy–environment interactions. By laying bare the simple yet elegant structural underpinnings common to the multitude of the studies cited above, we hope to make CGE models more accessible to the broader community of natural resource and environmental economists, and to encourage their use as a tool for research in the discipline. The rest of the chapter is organized as follows. Section 9.2 begins by exploiting the circular flow to algebraically derive the equilibrium conditions at the core of every CGE model. Section 9.3 demonstrates how a CGE model of the economy is constructed by imposing the axioms of consumer and producer optimization on this framework to generate a system of non-linear equations which represents an economy of arbitrary dimension. Section 9.4 outlines the techniques of using an economic database known as a social accounting matrix to numerically calibrate the foregoing algebraic structure. The meat of the chapter is in Sec. 9.5, which introduces economy– environment interactions and illustrates several techniques for modeling problems which arise in environmental and resource economics. Section 9.6 concludes with a summary and brief remarks on key directions for future research. 9.2.

The Algebra of General Equilibrium

For expositional clarity we consider a hypothetical closed economy made up of an unspecified number of firms which are grouped into distinct 6 Of the numerous articles cited above, the majority merely discuss only those attributes of their models that are pertinent to the application at hand, or present their model’s equations with little explanation to accompany them. Books and manuals devoted to modeling techniques (e.g., Ginsburgh and Keyzer, 1997; Lofgren et al., 2002) tend to be exhaustively detailed, and articles focused on the operations research aspects of modeling (e.g., Rutherford, 1995; Ferris and Pang, 1997) often involve a high level of mathematical abstraction, neither of which makes it easy for the uninitiated to quickly grasp the basics. Finally, while pedagogic papers (e.g., Devarajan et al., 1997; Rutherford, 1999; Paltsev, 2004) often provide a lucid introduction to the fundamentals, they tend to focus either on models’ structural descriptions or the details of the mathematical software packages used to build them, while glossing over CGE models’ theoretical basis or procedures for calibration. For an exception, see Kehoe (1998).

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industry sectors — each of which is assumed to produce a single commodity, and an unspecified number of households which are assumed to jointly own an endowment of distinct primary factors of production. When the economy is in equilibrium, the circular flow embodies the physical principle of material balance, whereby the quantity of a factor with which households are endowed, or of a commodity produced by firms, must be completely absorbed by firms or households (respectively) in the rest of the economy. It also embodies the accounting principle of budgetary balance. Firms’ expenditures on inputs must be balanced by the value of the revenue generated by the sale of the resulting product, households’ expenditures on goods must be balanced by their income, and each unit of expenditure has to purchase some amount of some type of commodity or factor. These principles are the reflection of the Karush–Kuhn–Tucker complementary slackness conditions for the optimal allocation of commodities and factors and the distribution of activities in the economy (Mathiesen, 1985a,b). Indeed, we build on Paltsev’s (2004) intuitive exposition to demonstrate that CGE models are at their core the mathematical expression of this fact. Before we proceed, it should be noted that the financial transfers in Fig. 9.1 may be deduced from the equilibrium price and quantity allocation of goods and services. For this reason it suffices to model the equilibrium of an economy at a particular instant of time in terms of barter trade in commodities and factors, without explicitly keeping track of money as a commodity. However, by the same token, it is necessary to denominate the relative values of the different commodities and factors in terms of some common unit of account. This is accomplished by expressing the flows of goods in units of a single commodity (the so-called numeraire good) whose price is taken to be a fixed reference point. CGE models embody this device by solving only for relative prices, a point about which more is discussed below. An additional feature of our exposition is that it holds government as an institution offstage while capturing the distortionary impacts of its activity on private markets. On the one hand, government’s role in the circular flow is often passive — to levy taxes and disburse the resulting revenues to firms and households as subsidies and transfers, subject to exogenous rules of budgetary balance. On the other, its impact on private markets is critical, as every tax (subsidy) distorts the supply–demand balance for the commodity in question and simultaneously generates a stream of revenue that increments (decrements) the income of consumers. The essence of this process may be captured by assuming that tax receipts (subsidy payments)

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are immediately transferred to (assessed upon) households in a lump-sum fashion, which allows these flows to be represented as income transfers from (to) the institution being taxed (subsidized). Our first step is to introduce some notation. Let the indices i = {1, . . . , N }, j = {1, . . . , N } and f = {1, . . . , F } denote the set of commodities, industry sectors, and primary factors, respectively. We assume that households’ use of commodities satisfies d = {1, . . . , D} types of demands, which for simplicity we lump into two broad categories, d = {C, O}: consumption, C, from which households derive utility, and all other final uses, O, which are assumed to be exogenous. With this infrastructure in hand we may specify the key variables in the economy which need to be kept track of: pi and wf , which are the equilibrium prices of good i and factor f ; yj and xi,j , the quantities of industry j’s output and its use of sector i’s output as an intermediate input; Vf and vf,j , households’ aggregate endowment of factor f and industry j’s use of that factor; gi,d , households’ use of commodity i to satisfy final demand d; and u and µ, the indexes of households’ utility and aggregate expenditure per unit of utility. We also define the output-normalized unit quantities of intermediate goods and primary factors, x
i,j = xi,j /yj and v
f,j = vf,j /yj , the utility-normalized unit quantity of consumption of the ith good,
gi,C = gi,C /u, and the unit expenditure-normalized prices of commodities and factors, p
i = pi /µ and w
f = wf /µ. Turning to taxes and subsidies, it is typical in CGE models for these instruments to be specified in an ad-valorem fashion, which determines the fractional change in the price of the commodity or factor to which it is being applied. Thus, a tax rate τ on the output of industry j drives a wedge between the producer price of output, pj , and the consumer price, (1 + τ )pj , in the process generating revenue for the government from the yj units of output in the amount of τ pj yj . Note that a subsidy which lowers the consumer price relative to the producer price may also be modeled in this way, by specifying τ < 0. Taxes and subsidies may appear in any number of markets, enumerated by the index h = {1, . . . , H}. For simplicity we consider only two of these in our exposition: the markets for the output of the industry sectors (indicated by the superscript Y ), and the markets for factor inputs to industries (indicated by the superscript F ), using τjY and τfF to indicate the distortions associated with the outputs of the various industries and households’ endowments of various factors.

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We now formally analyze the firms (B). Budgetary balance implies that the value of a unit of each commodity must equal the sum of the values of all the inputs used to produce it, which is simply the unit cost of the inputs of intermediate materials as well as the payments to the primary factors employed in its production. The guiding principle is the complete accounting of the sources of economic value, which simultaneously reflects constancy of returns to scale in production and perfectly competitive markets for firms’ inputs and outputs, and ensures that in equilibrium producers make zero profit. Recognizing that producers value their outputs on a net-of-tax basis and their inputs in on a gross-of-tax basis, we have pj y j =

N F       1 + τiY pi xi,j + 1 + τfF wf vf,j f =1

i=1

⇔ pj y j =

N F       1 + τiY pi x 1 + τfF wf v
f,j yj
i,j yj + i=1

(1)

f =1

A conceptual sleight-of-hand makes it possible to apply similar reasoning to the households (A). Consumption of commodities may be thought of as the production of a “utility good” whose value, given by the product of the utility level and the unit expenditure index or “utility price,” must equal the sum of the gross-of-tax values of the commodities consumed by the household (i.e., the inputs to utility production): µu =

N N       1 + τiY pi gi,C ⇔ µu = 1 + τiY pi g
i,C u i=1

(2)

i=1

We now turn to the product market (E). Under the assumption that commodities are not freely disposable, firms’ outputs are fully consumed by households, and households’ endowment of primary factors is in turn fully employed by firms. The implication is that, for a given commodity, the quantity produced must equal the sum of the quantities that are demanded by the other firms and households in the economy. Analogously, in the market for a given factor (D), the aggregate quantity demanded by the various firms must completely exhaust the aggregate supply with which the households are endowed. This is the familiar condition of market clearance, which implies that in each market in equilibrium the value of sellers’

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receipts must equal the value of purchaser’s outlays on a net-of-tax basis: yi =

N 

xi,j +

j=1

Vf =

N 

D 

gi,d ⇔ pi yi =

j=1

d=1

vf,j ⇔ wf Vf =

j=1

N 

N 

pi xi,j +

D 

pi gi,d

(3)

d=1

wf vf,j

(4)

j=1

In equilibrium, the flows of revenue in the markets for commodities and factors are fundamentally linked. The return to households’ endowments of primary factors, which is equal to the value of their factor rentals to producers, constitutes consumers’ primary income, which, along with recycled tax revenue, they exhaust on purchases of commodities. The implication is that the revenue gained by renting out primary factors, plus the value of tax receipts transferred to households by the government (shown in parentheses below), balances the gross expenditure on the satisfaction of demands, reflecting the principle of balanced-budget accounting known as income balance:   F F N N  D     wf Vf +  τfF wf Vf + τjY pj yj  = pi gi,d (5) I = f =1

f =1

j=1

i=1 d=1

This expression highlights the fundamental equivalence between aggregate Y disposable income (given by I D = I − N i=1 (1 + τy )pi gi,O ) and the value of utility defined by (2), which implies the following supply–demand balance condition for utility goods:   F F N    w
f µVf +  τfF w
f µVf + τjY p
j µyj  µu = I D ⇔ µu = f =1

−

N 

f =1

(1 + τiY )
pi µgi,O

j=1

(6)

i=1

The algebraic framework of a CGE model is derived from equilibrium conditions (2), (3), and (6). The key feature of each condition is that the residual equation which results from eliminating the common factor exhibits complementary slackness with respect to the variable that is common to both sides. The economic intuition behind this is straightforward (Paltsev, 2004). Looking first at the zero-profit conditions (1) and (2), firms earning

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negative profit will shut down with an output of zero, while those earning zero profit can continue to produce a positive quantity of output. Similarly, households whose spending on consumption of goods exceeds the value of utility generated by consumption will cease activity with utility level of zero, while households whose consumption expenditure exactly matches the value of utility thus generated can continue to consume with a positive level of utility. In both cases the common factor is the activity level of the firm or household, with unit profit in goods production complementary to the relevant producer’s level of output, and unit expenditure complementary to the level of utility: pj
0

µ
0 i=1

(7b) Turning to the market clearance conditions (3) and (4), any commodity or factor which is in excess supply will have a price of zero, while a good or factor which is neither in excess demand nor excess supply will have a positive price. Therefore, the balance between supply and demand for each of these inputs is complementary to the corresponding price level: yi >

N 

xi,j + gi,C + gi,O , pi = 0

or yi =

j=1

N 

xi,j + gi,C + gi,O , pi > 0

j=1

(7c) Vf >

N  j=1

vf,j , wf = 0

or Vf =

N 

vf,j , wf > 0

(7d)

j=1

Finally, market clearance in utility (6) exhibits complementary slackness with respect to the unit expenditure index. The intuition is that if the supply of utility goods exceeds the demand generated by consumption out of disposable income then unit expenditure on commodities will be zero, while

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if the supply just matches the demand then unit expenditure is positive: u > I D /µ, µ = 0,

or u = I D /µ, µ > 0

(7e)

We note that disposable income does not exhibit complementary slackness with respect to any of its constituent variables, and moreover is made redundant by (6), which relegates it to the simple role of an accounting identity. This is often made explicit by specifying an additional expression in which I D exhibits complementary slackness with respect to its own definition, and by designating the unit expenditure index as the numeraire price by fixing µ = 1. Doing so has the effect of dropping Eq. (7e) by recasting the utility index as the value of aggregate consumption, u = I D . The core of a CGE model is given by the system of Eqs. (7c)–(7e). However, in order to actually compute the general equilibrium of an economy it is necessary to elaborate this structure by specifying the primal quantity allocation of commodity and factor uses in terms of the dual commodity and factor prices. It is this task which we tackle next. 9.3.

From Equilibrium Conditions to a CGE Model: The CES Economy

The algebraic specification of a CGE model in terms of the dual results from the imposition of the axioms of producer and consumer maximization on the market-clearance, zero-profit, and income balance conditions derived above. In so doing, the key simplifying assumption is to model the economic decisions of the disparate households collectively, by treating them as a representative agent, and to model the economic decisions of the disparate competitive firms in each industry collectively, by treating sectors as representative producers. To provide a concrete illustration of how this may be done we employ the device of a “constant elasticity of substitution (CES) economy” in which the representative agent has CES preferences and each industry has CES production technology. On the demand side of the economy, the representative agent derives utility from the consumption of commodities according to a CES utility function with technical coefficients αi,C and elasticity of substitution ω. The agent’s problem is to maximize utility by allocating expenditure over commodities subject to the constraint of disposable income, taking goods prices parametrically. We recast this problem in terms of the dual, and model the agent as allocating the input quantities of consumption goods

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necessary to generate a unit of utility so as to minimize unit expenditure subject to the constraint of her consumption technology:   ω/(ω−1) 

N        (ω−1)/ω 1 + τiY pi gi,C ≤ I D αi,C gi,C max u =  gi,C    i=1

i

  ω/(ω−1) 

N  N        (ω−1)/ω 1 + τiY pi g
i,C 1 = αi,C g
i,C ⇔ min µ =  gi,C  e  i=1

i=1

The solution to this problem is the instantaneous unit demand for commodities and associated conditional final demands: gi,C = αω
i,C



 −ω ω µ , 1 + τiY pi

gi,C = αω i,C



 −ω ω µ u 1 + τiY pi

(8)

On the supply side, the representative firm in industry j generates output by combining the instantaneous uses of intermediate goods and factors according to a CES production technology with technical coefficients βi,j and γf,j , and elasticity of substitution σj . The producer’s problem is to maximize profit πj by allocating expenditure among i intermediate inputs and f factor inputs, taking the prices of inputs and output parametrically. Recasting this problem in terms of the dual, we treat the firm as allocating input quantities of intermediate goods and factors necessary to generate a unit of output so as to minimize unit production cost subject to the constraint of its production technology:  max

xi,j ,vf,j

 N F         Y F 1 + τi pi xi,j − 1 + τf wf vf,j  πj = pj yj −  f =1

i=1

yi =

N  i=1

⇔ min

x ei,j ,e vf,j

1=

N  i=1

(σ −1)/σj βi,j xi,jj



+

F 

σj /(σj −1)  (σ −1)/σj γf,j vf,jj

f =1

 N F         Y F 1 + τi pi x 1 + τf wf

i,j + vf,j  pj =  f =1

i=1

(σ −1)/σj βi,j x
i,jj

+

F  f =1

σj /(σj −1)  (σ −1)/σj γf,j v
f,jj

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The solution to this problem is the instantaneous unit demands for commodities and factors and associated conditional demands:  −σj σj  −σj σj σ  σ  pj , xi,j = βi,jj 1 + τiY pi pj y j x
i,j = βi,jj 1 + τiY pi v
f,j =

σ  γf,jj 1

 −σj σj + τjF wf pj ,

vf,j =

σ  γf,jj 1

(9a)   −σ σ j + τfF wf pj j y j (9b)

The crucial step is to substitute the demands (8) and (9) into the equilibrium conditions (7a)–(7e). Doing this yields the system of 2N + F + 3 non-linear inequalities, each paired with one of the 2N + F + 3 unknown variables, shown in Table 9.1. This is what is referred to as “a CGE model.” Letting Ξ(·) denote the pseudoexcess demand correspondence of the economy made up of the stacked vector of Eqs. (10a)–(10f), and using z = {p, w, y, µ, u, I D } to indicate the stacked vector of unknowns, the system may be expressed compactly as Ξ(z) ≥ 0,

z ≥ 0,

z
Ξ(z) = 0

(10)

or, using the shorthand notation “⊥” to indicate the complementarity between a variable and its matched equation, Ξ(z) ⊥ z. The mathematical problem defined by Eq. (10) is highly non-linear, with the result that it is not possible to obtain a closed-form solution for z. This is the reason for the “C” in CGE models: to find the general equilibrium of an economy with realistic utility and production functions, the system of equations that describes equilibrium must be transformed into a numerical problem that can be solved using optimization techniques. In the operations research literature, the square system of numerical inequalities (10) is known as a mixed complementarity problem or MCP (Ferris and Pang, 1997; Ferris and Kanzow, 2002), which is easily solved using algorithms that are now routinely embodied in modern, commercially available software systems for optimization (Dirkse and Ferris, 1995; Ferris and Munson, 2000; Ferris et al., 2000).7 All that remains to be demonstrated is how to generate a numerical problem by taking the foregoing algebraic framework to the data. 7 The interested reader is referred to Sue Wing (2009) for a sketch of the basic computational approach. In modern software implementations, the solution to the MCP is computed to numerical tolerances that are routinely six orders of magnitude smaller than the value of the aggregate income level.

i=1

N X

αω i,C

f =1

σj `` 1 γf,j

+

ID =

+

f =1

+

i=1

j=1

N X

τjY

pi gi,O p j yj ,

´ ´−σj σj wf p j yj ,

N X

τfF

,

I D ≥ 0,

µ ≥ 0,

wf ≥ 0,

pi ≥ 0,

u ≥ 0,

−

σ

βi,jj

σj `` 1 γf,j

f =1

+

3

−

j=1

N X

τjY

i=1

pj yj 5 = 0.

3

pi gi,O

3 ´ ´−σj σj wf p j yj 5 = 0

N X

τfF

wf Vf + τfF wf Vf

f =1

F X

/µ = 0.

j=1 i D

N X

F X

5=0

1/(1−ω) 3

`` ´ ´−σj σj p j yj 1 + τiY pi

1−ω αω i,C pi

11/(1−σj ) 3 ´ ´ 1−σj A 7 + τfF wf 5=0

´ ´−ω ω µ u − gi,O 5 = 0 1 + τiY pi

I D 4I D −

2

µ u−I

h

wf 4Vf −

2

``

j=1

N X

− αω i,C

pi 4yi −

2

i=1

N X

σj `` γf,j 1



f =1

u 4µ −

2

+

F X

i=1

(10f)

(10e)

(10d)

(10c)

(10b)

(10a)

Research Tools in Natural Resource. . .

τfF wf Vf

wf Vf −

f =1 F X

F X

Income Definition:

/µ,

j=1 D

N X

+

´ ´−σj σj p j yj pi

`` ´ ´−ω ω µ u + gi,O , 1 + τiY pi

σ `` βi,jj 1

+ αω i,C

j=1

u≥I

Vf ≥

yi ≥

N X

τiY

`` ´ ´1−ω 1 + τiY pi

1/(1−ω)

11/(1−σj ) ´ ´1−σj F A , + τf wf

yj ≥ 0,

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Market Clearance:

µ≤



+

σj `` γf,j 1

i,j

0 N X ´ ´1−σj σ `` 6 yj 4pj − @ βi,jj 1 + τiY pi

2

The equations of the CGE model.

9:44

F X

i=1

Zero Profit: 0 N X ´ ´1−σj σ `` β j 1 + τiY pi pj ≤ @

Table 9.1:

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Computable General Equilibrium Models 269

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270

Fig. 9.2:

9.4.

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A social accounting matrix.

Social Accounting Matrices and Numerical Calibration

The circular flow of an economy in instantaneous equilibrium can be completely characterized by four data matrices. The market for firms’ intermediate inputs (E2) is described by an N × N input–output matrix of industries’ uses of commodities, X, the factor market (D) is described by an F × N matrix of primary factor inputs to industries, V, the product market (E1) is described by an N × D matrix of commodity uses by final demand activities, G, and tax and subsidy distortions are summarized by the H × N matrix T. Arranging these matrices in the manner shown in Fig. 9.2 results in an accounting tableau known as a social accounting matrix, or SAM. A SAM is the cash-flow statement for an economy in equilibrium, which gives a snapshot of the interindustry and interactivity financial flows over an interval of time — typically one year. Each cell element is an input–output account that is denominated in the units of value of the period for which the flows in the economy are recorded, typically the value of the currency in the year in question. Each account is uniquely defined by its row and column, and records the payment from the account of a column to the account of a row. Thus, an account’s components of income of (e.g., the value of receipts from the sale of a commodity or rental of a factor) appear along its row, and the components of its expenditure (i.e., the values of households’ purchases or the production of a good) appear along its column (King, 1985; Reinert and Roland-Holst, 1997).

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The structure of the SAM reflects the principle of double-entry bookkeeping, which requires that for each account, total revenue — the row total — must equal total expenditure — the column total. This is apparent from Fig. 9.2, where the sum down any column of the left-hand matrices X, V, and T is equivalent to the expression for zero profit in production (1), the sum across any row in the uppermost matrices X and G is equivalent to the expression for goods market clearance (3), and the sum across any row in the middle array V is equivalent to the expressions for factor market clearance (4). Furthermore, once these conditions hold, the sums of the elements of G on one hand and V and T on the other should equal one another, reflecting the fact that in a closed economy GDP (the sum of the components of expenditure) equals value added (the sum of the components of income), which is equivalent to the income balance condition jointly specified by Eqs. (5) and (6). Formally

yj =

yi =

N 

xi,j +

i=1

f =1

N 

D 

xi,j +

j=1

Vf =

F 

N 

v f,j +

H 

th,j

(11a)

h=1

g i,d

(11b)

d=1

v f,j

(11c)

j=1 F  f =1

Vf +

H  h=1

Th =

N  D 

g i,d

(11d)

i=1 d=1

To numerically calibrate our example CES economy, it is necessary to establish equivalence between the systems of Eqs. (10) and (11). Dawkins et al. (2001) describe a variety of approaches to address this problem depending on what kind of information is available in addition to the SAM. For example, Kehoe (1998) describes a procedure when data exist on benchmark prices, however, more often than not such information is simply unavailable. In a typical data-constrained environment, the simplest method to fit Eq. (10) to the benchmark equilibrium in the SAM is to treat the price variables as indices with benchmark values of unity: pi = wf = µ = 1, and treat the activity and income variables as real values which are set equal to the entries in the SAM.

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To establish the latter equivalence, observe that the simplified tax structure in our CES economy partitions T into a vector of payments for indirect taxes by industry, tY,j = τjY pj yj , and a matrix of tax payments associated with each industry’s use of the f factors, tf,j = τfF wf vf,j . We may, therefore, write the zero-profit condition (11a) as y j − tY,j =

N 

xi,j +

i=1

F 

(v f,j + tf,j )

f =1

in which the left-hand side gives j’s net-of-tax value of output, and terms on the right-hand side are the gross-of-tax value of j’s uses of intermediate and factor inputs. At benchmark prices these imply that yj = y j − tY,j , (1 + τiY )xi,j = xi,j and (1 + τfF )vf,j = v f,j + tf,j , with benchmark tax rates on output and factors given by τ Yj = tY,j /yj and τ F f = T f /(V f + T f ). Moreover, inspection of G and V reveals that (1 + τiY )gi,d = gi,d , u = GC and Vf = V f + T f . The technical coefficients of the cost and expenditure equations are then easily found by substituting these conditions into the demand functions (8) and (9): αi,C = (g i,C /GC )1/ω

(12a)

βi,j = (xi,j )1/σj (y j − tY,j )−1/σj

(12b)

γf,j = (v f,j + tf,j )1/σj (y j − tY,j )−1/σj

(12c)

We draw attention to the fact that the calibrated values of the technical coefficients are not independent of the values of the elasticities of substitution. The fact that the SAM contains no information on the values of σ and ω makes the calibration problem (12) underdetermined, with more free parameters than there are model equations or observations of benchmark data. This is a problem which is magnified in CGE models which specify industries’ cost functions and consumers’ expenditure functions using hierarchical CES functions, each of which contains multiple elasticity parameters. For this reason, elasticity parameters are almost always exogenously specified by the analyst, but because relevant empirical estimates are frequently lacking, modelers all too often resort to selecting values based on a mix of judgment and assumptions. The ad hoc character of this procedure has come in for criticism by some empirical economists (e.g., Jorgenson, 1984; McKibbin and Wilcoxen, 1998; McKitrick, 1998) who advocate an econometric approach to CGE modeling in which the pseudoexcess demand correspondence is built up

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273

from statistically estimated cost and expenditure functions. However, the main drawback of this alternative is that the econometric estimations are data-intensive, requiring time-series of input–output matrices which are a challenge to construct and therefore rarely available. In response, sophisticated limited-information methods have been developed to calibrate the substitution elasticities and technical coefficients based on more widely available ancillary data in addition to the SAM (e.g., Arndt et al., 2002), but so far these techniques have not seen widespread adoption. In any event, specifying values for σ and ω makes it possible to calibrate the technical coefficients and substitute these two sets of parameters into the expressions in Table 9.1 to generate a system of numerical inequalities which constitutes the actual CGE model. We emphasize that to satisfy the resulting expressions with equality, all one has to do is simply set the price variables equal to unity and the quantity variables equal to the corresponding values in the SAM. This procedure, known as benchmark replication, permits the analyst to verify that the calibration is correct. The intuition is that since a balanced SAM represents the initial benchmark equilibrium of the economy, plugging the values in the SAM back into the calibrated numerical pseudoexcess demand correspondence should reproduce that equilibrium. All our calculations thus far have assumed a realistic setting in which the economy under consideration is initially ridden with tariffs and/or subsidies. But it is worth noting that in the absence of benchmark distortions σj σj ω (T = 0, τ Yj = τ F f = 0), Eq. (12) replaces the terms αi,C , βi,j and γf,j in Eq. (10) with coefficients given by the ratio of the relevant cells of the SAM and the corresponding column totals, i.e., the value shares of the inputs to consumption and production. The key implication is that the values selected for the substitution elasticities have no practical impact on the benchmark equilibrium, which makes intuitive sense because the SAM completely specifies the model’s initial equilibrium, which in turn is consistent with an infinite number of potential values for σ and ω. The corollary is that the substitution possibilities in the economy — i.e., the degree of adjustment of economic quantities in response to changes in prices, both within and between sectors — are fundamentally determined by the SAM.

9.5.

Modeling Applications: Integrating the Environment

The astute reader will perhaps find the analysis so far unsatisfying, because all that has been done is to demonstrate how a CGE model can be

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constructed to replicate the economic dataset used for its calibration, without mentioning the environment. This section redresses this imbalance by elucidating several techniques for incorporating various kinds of economy–environment interactions within CGE models. We begin with a note about data. As mentioned in Sec. 9.1, social accounting matrices compiled by statistical agencies generally do not record either the value of environmental resources or the environmental damage costs of the economy’s residuals. Although there is a literature on the construction of the so-called “environmentally extended SAMs” which seek to integrate flows of resources and residuals into the traditional economic input–output framework,8 the majority of CGE studies employ satellite accounts of pollution or the use of environmental factors. This is the approach that we use below. As a general matter, to use CGE model for any kind of research or policy analysis, the analyst must first capture the initial effect of a policy shock by perturbing one or more of the exogenous parameters of the economy, and then compute a counterfactual equilibrium based on the new parameters. To evaluate the impacts of the shock, the analyst compares the initial and counterfactual equilibrium vectors of prices and activity levels, and the level of utility, and uses the results to draw inferences about the shock’s effects in the real world, subject to the caveats of the accuracy and realism of the model’s assumptions. CGE models’ principal advantage is their ability to measure the ultimate impact of the shock on consumers’ aggregate well-being in a theoretically consistent way, by quantifying the changes in the consumption of the representative agent which result from the myriad supply–demand adjustments across the various markets in the economy. In particular, given an initial level of utility u implied by the SAM, a counterfactual level of utility u
, and unit expenditure set as the numeraire, the welfare change (u
/u − 1) measures the impact of the shock in terms of equivalent variation, expressed as a percentage of initial expenditure. But ironically, it is this very functionality which exposes the kernel of truth to the black box criticism articulated in Sec. 9.1. CGE models’ comprehensive representation of the economy, combined with their popularity as a tool for prospective policy analysis, has earned them a reputation as a sort of economic crystal ball. Table 9.1 emphasizes that the 8 See,

e.g., Alarc´ on et al. (2000), Xie (2000), Lenzen and Schaeffer (2004) and Mart´ınez de Anguita and Wagner (2010).

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non-linearity and dimensionality of the pseudoexcess demand correspondence often make it difficult to intuitively understand the net impact of a change in a single parameter, even in models that are structurally simple and represent only a modest number of sectors and/or households. A rigorous accounting of the myriad interactions that determine the character of the counterfactual equilibrium can therefore entail a substantial amount of sensitivity analysis and testing, with the result that ex-post evaluations of the veracity of model projections are rare. Moreover, the few such assessments point to a disturbingly poor forecasting record,9 which suggests that CGE models’ usefulness as a research tool owes less to their predictive accuracy than their ability to elucidate the mechanisms responsible for the transmission of price and quantity adjustments among markets. CGE models should therefore be properly regarded as computational laboratories within which to study the dynamics of the economy–environment interactions and the manner in which they ultimately give rise to welfare impacts. With this admonition, we outline how the environment may be integrated into the CES economy in Table 9.1. We consider two types of externalities, pollution and the depletion of nominally unpriced non-market environmental goods and services used as inputs to production. Our first step is to specify the link between economic activity and each externality. Pollution is easily represented, with the most common modeling device being exogenously specified emission coefficients, which, when multiplied by (for example) the outputs of dirty industries represented in the model, yield the sectoral and aggregate pollution loads associated with economic activity. Using e ⊂ j and φe to indicate the polluting subset of sectors and their associated emission coefficients implies sectoral emissions εe = φe ye and an overall pollution level E = e εe . The use of non-market environmental factors is more challenging to represent because the aggregate demand for such inputs — say, — must be specified as a function of endogenous variables within the model. In particular, if firms are able to substitute market inputs for non-market factors, then it becomes necessary to adjust the model’s the core algebraic framework and its calibration procedures. An example below provides a detailed illustration of the kinds of changes involved. The second step is to capture the initiating impulse of shocks. It is common to represent these through exogenous parameters whose values 9 See

e.g., Panagariya and Duttagupta (2001), Kehoe (2005) and Valenzuela et al. (2007).

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may be specified by the analyst. By far the most important of these is the endowment of primary non-reproducible factors, Vf , which are nominally determined by the row totals of the factor supply matrix in the SAM and defines the overall scale of economic activity, thereby playing the role of a boundary condition in the resulting equilibrium. Other parameters are policy variables which may be price-based — taxes and subsidies, as we have seen, or quantity-based — e.g., constraints on commodity and factor demand and/or supply. They can also take the form of technology parameters which represent improvements in productivity, changes in the rate and/or bias of technical progress, or even structural shifts in consumers’ preferences. The third step is to propagate the shock through the economy. In most cases the equilibrium in the SAM implies benchmark values for the types of shock parameters discussed above, so that a counterfactual scenario may be simulated by simply changing the values of these parameters and solving the CGE model for a new equilibrium. The impact of the various shocks on environmental quality may then be computed in an open-loop fashion. In the case of pollution the emission coefficients may be applied to the new industry activity levels to make an ex-post calculation of the pollution load, while the change in demand for non-market inputs will typically be computed as part of the counterfactual simulation. The final step in the analysis is to evaluate the welfare impact of the shock in question. The disutility of pollution and depletion of environmental factors may be represented through the specification of an environmental damage function, D, denominated over E or , which is exogenous to the general equilibrium problem. This device permits aggregate welfare to be computed as (u − D), and then compared across benchmark and counterfactual scenarios. Fundamental to this approach is the implicit treatment of environmental damage as separable from other economic variables, principally consumption. However, Espinosa and Smith (1995) and Carbone and Smith (2008) demonstrate that this assumption, which is ubiquitous in the literature is not innocuous — alternative assumptions of complementarity or substitutability between environmental quality and consumption or leisure can result in a given shock having dramatically different welfare impacts. In line with our focus on CGE models as a research tool we go on to provide step-by-step examples of model formulation, specification, and calibration in the four archetypical research areas surveyed in the

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introduction: (i) The consequences of liberalization and expansion of the economy for pollution and the demand for unpriced environmental resources (Sec. 9.5.1), (ii) The economic impacts of policies for environmental resource conservation (Sec. 9.5.2) and pollution control (Sec. 9.5.4–9.5.7), (iii) The impacts of changes in environmental conditions and availability of resources on the equilibrium of the economy (Sec. 9.5.3), and (iv) Full integration of an environmental amenity within the general equilibrium framework (Sec. 9.5.8). 9.5.1.

Environmental consequences of liberalization and growth

Our first example examines the consequences of economic liberalization and expansion for pollution and the consumptive use of environmental resources. Given the present closed-economy setting, it is only possible to simulate the effects of trade liberalization in the most heuristic of ways. In the case where the exogenous commodity demands gi,O represent net exports, the partial impacts of balance-of-payments shocks on goods market clearance and aggregate income can be modeled by applying augmentation factors AO i > 0 to the benchmark values of these demands in the SAM, i.e., gi,O = AO i g i,O . Exogenous reductions in import demand and increases in export supply can therefore be modeled by setting AO i ≷ 1 if gi,O ≷ 0. Liberalization of the domestic economy is easily simulated by reducing the tax rates τ Y and τ F from their benchmark calibrated values — or, where these values are negative, setting them to zero to model the elimination of subsidies. Expansion of the economy is typically captured through increases in the supplies of labor and capital (indicated by the subscripts L and K), implemented by augmenting these components of the endowment vector relative to the values given by the SAM. This is easily accomplished by introducing aggregate productivity growth parameters AVL , AVK > 1, which relax the boundary condition for the economy: Vf = AVf V f (f = L, K). Efficiency-improving technical progress which reduces the industries’ unit production costs can be modeled in the same fashion, by multiplying the sectoral cost functions by neutral productivity parameters AYj < 1.

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As mentioned earlier, the equilibrium in the SAM implies benchmark values of unity for the elements of AO , AV , and AY . Perturbing these parameters and solving the model yields counterfactual values of industry outputs from which the new level of pollution and associated environmental damage, D(E), may be computed. Modeling the impacts of shocks on the use of unpriced environmental resources is more complicated because the failure of the SAM to record the value of these inputs implies that their use by firms entails an unmeasured cost of production. Although the symmetry of the SAM indicates that the latter constitutes missing factor returns, these neither redound to the households nor exert income effects on consumption. Constant returns to scale (CRTS) in market inputs therefore suggests that industries’ production technology is subject to diminishing returns once non-market inputs are accounted for. Our exposition deals with the simplest case of a single non-market factor, whose effect on equilibrium may be captured by including an additional term in the sectoral cost functions. Then, assuming that the resource has a “virtual price”  and a production coefficient δj , the industry zero profit and commodity market clearance conditions become10   σ   1−σj j Y pj ≤ Aj βn,j 1 + τiY pi i

+ yi ≥





σ  γf,jj 1

f σ

AYj βi,jj

j

+ αω i,C

 1−σj σ + τfF wf + δj j 1−σj

1/(1−σj ) ⊥ yj

(13a)

⊥ pi

(13c)



 −σj σj pj y j 1 + τiY pi



 −ω ω 1 + τiY pi µ u + gi,O

Now, by Shepard’s Lemma, Eq. (13a) implies the additional marketclearance condition:  σ σ

≥ δj j −σj pj j yj ⊥ (13g) j

In a manner analogous to Eq. (10d), this expression casts the variable

in the role of an exogenously determined endowment of the environmental resource. But our purposes require a different sort of boundary condition: to 10 These

expressions highlight the need for care when modifying the equilibrium conditions of the model, as consistency among the conditions typically requires that a change in any one equation be matched by adjustments in one or more of the other model equations.

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investigate how the use of the resource adjusts endogenously to shocks, we need to transform from a fixed factor into one that is in perfectly elastic supply at a constant virtual price. This price level is a fundamental unknown: zero is incompatible with the computation of general equilibrium because it permits the supply of the environmental factor — and therefore the size of the economy — to grow arbitrarily large, while economic theory provides no guidance on what the appropriate positive level should be. A reasonable assumption is that the private opportunity cost of a nominally unpriced input will be one or more orders of magnitude smaller than the prices of market goods. We therefore assume an arbitrary target price, denoted by the exogenous parameter   1, which constitutes a price constraint with respect to which exhibits complementary slackness. The assumption is that a virtual price higher than  chokes off demand for the resource, so that either  > , = 0 or  = , > 0, expressed as the additional auxiliary equation: ≥

⊥

(13h)

The CGE model with non-market resources is made up of the three conditions (13a), (13c), (13g), and (13h) in conjunction with Eqs. (10b) and (10d)–(10f), which we relabel (13b) and (13d)–(13f). The resulting algebraic structure can be calibrated by applying the techniques in Sec. 9.4 to a standard SAM, setting =  = 0. The technical coefficient on the non-market input may be calibrated using the share of the non-market factor in the true (i.e., market + nonmarket) cost of industry j’s producσ σ tion, given by δj j /(1 + δj j ), whose value can be assumed or imputed from the results of valuation studies. The key numerical trick is to calibrate the implicit baseline quantity of resource use, , while approximating the benchmark equilibrium in the SAM. This is done by solving the calibrated model with  set close to zero (say 0.001). Counterfactual scenarios may then be simulated using the procedure outlined above, with ex-post evaluation focusing on the impact of shocks on relative to its baseline value. A damage function, D( ), which is sufficiently steeply sloped admits the possibility that economic expansion facilitated by increased use of non-market environmental factors is welfare-diminishing. 9.5.2.

Environmental resources: Scarcity and conservation

Our second topical example is resource scarcity and management, which the model in Eq. (13) is eminently suited to examine. Scarcity of the

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environmental factor is easily introduced by replacing the target price level in the auxiliary constraint (13h) with an upward-sloping supply curve, Ψ:  ≥ Ψ( )

⊥

(13h
)

The function Ψ may be specified is different ways to model a diverse array of economy–environment interactions, in particular the economic implications of critical thresholds in resource use. We may want to examine situations where exceeding such a threshold, indicated by the parameter  , causes the private opportunity cost of the environmental factor to rise steeply enough to exert a non-negligible drag on economic activity. A classic example is a fishery where rapid declines in fish stocks warrant increasing expenditures on search and harvesting effort. Parameterizing the effect of the threshold as Ψ = [1 + ψ1 ( /  )ψ2 ],

with 0 < ψ1  1, ψ2  1

leads to increases in  inducing substitution of capital, labor, and intermediate goods for fish stocks in fish production, and, ultimately, choking off demand for . The crucial piece of information provided by the model is how far in excess of the threshold this equilibrium lies. Policies to manage resources may be modeled in an analogous fashion. A Pigovian tax that narrows the gap between the private and social opportunity costs of the non-market resource can be introduced on the right-hand side of both Eqs. (13h) and (13h
) through the addition of a tax parameter, τ  , whose value is specified by the analyst. This change necessitates a corresponding adjustment to the definition of income to account for the effects of the revenue raised by the tax on the consumption of the representative agent:     1 + τiY pi gi,O + wf Vf − τfF wf Vf ID = f

+



i

τjY

f 

pj y j + τ

⊥ ID

(13f
)

j

The resource management CGE model is made up of (13a)–(13e), (13f
) and (13g), along with either (13h) or (13h
) inclusive of τ  . By solving for the counterfactual equilibria associated with a range of values of τ  and tracing out the resulting sectoral and aggregate reductions in the

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environmental factor, one can construct marginal abatement cost (MAC) curves which not only identify which industries respond most elastically to the tax but also indicate the direct cost of conservation.11 The same procedure also makes it possible to characterize the dual welfare impacts of bringing non-market environmental factors within the ambit of the price system. In addition to the intuitive beneficial effect of reduced environmental demand on D, there is the income effect of recycled tax revenue in Eq. (13f
), whose impact is ambiguous. Resource tax payments constitute a new stream of income which offsets the rising price of market goods due to higher costs of environmental inputs, and attenuate the reduction in u. Simultaneously, however, increases in the unit production costs and declines in the output of relatively resource-intensive industries are likely to reduce revenue from preexisting taxes on output, amplifying welfare losses. This framework also allows us to compute the optimal Pigovian tax, which should equal the marginal environmental damage of resource extraction, D
( ). Somewhat heuristically, it is convenient to think of the tax itself as exhibiting complementary slackness with respect to this condition, which can be exploited to transform τ  into an endogenous variable through the following auxiliary equation:12 τ  = D
( )

⊥ τ

(13i)

In principle, including this condition as an additional component of the pseudoexcess demand correspondence allows the optimal value of τ  to be computed along with the other economic variables. However, in practice introducing such a constraint can make a CGE model difficult to solve.13 11 The total direct cost of conservation is approximated by the area under the MAC curve up to the observed quantity of resource reduction. 12 Of course, the real problem we wish to solve is max (u−D), but the resulting first-order  condition u
() = D
() cannot be used in Eq. (13i) because no closed-form expression exists for the marginal utility of the resource. 13 The underlying problem is max
(u − D) subject to (13a)–(13h
), which is known as a τ mathematic program with equilibrium constraints, or MPEC. Optimal tax problems of this kind pose a challenge for numerical optimization algorithms because the correspondence between u and τ  may not be smooth or even locally monotonic in the presence of preexisting distortions τ Y and τ F . One important reason is discrete jumps in the values of the tax revenue components of income induced by a marginal change in τ  as firms’ and households’ substitution patterns determine the economy’s price and quantity allocations, which can seriously degrade the performance of gradient-based iterative schemes like Newton’s method. The development of methods for the specification and solution of these sorts of problems is an active area of computational economic research.

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A less elegant but potentially more effective alternative is to keep τ  exogenous, use brute-force computational power to repeatedly solve the model over a grid of values of this parameter, and finally select the value of the tax at which (u − D) is maximized. 9.5.3.

Climate impacts and environmental disasters

In contrast to the bidirectional interaction in the previous example, environmental changes are generally be modeled as exerting a one-way influence on the economy. These kinds of shocks can operate through several channels, including induced changes in consumer expenditure patterns (e.g., increased demand for space conditioning and residential energy use due to greater temperature extremes arising from climatic change), changes in the productivity of primary factors in various industries (e.g., increases or decreases in crop yields due to shifting temperature or precipitation regimes), and reductions in the aggregate endowments of capital and labor (e.g., disasterrelated property losses and morbidity or mortality). We capture these three types of influences within the model by introducing shock paramF V eters AC i , Af,j , and Af into the representative agent’s expenditure function, industry sectors’ cost functions, and the market-clearance condition for factors, respectively. The second and third channels are important for modeling large shocks such as extreme climate impacts, the incidence of which is likely to be concentrated in a small number of vulnerable activities. The CES economy’s treatment of inputs to production and consumption as fungible, combined with the assumption of perfect intersectoral factor mobility, suggests that a significant adverse shock to industrial productivity or factor supplies may generate only modest increases in the unit costs of exposed industries. By contrast, concerns about irreversible climate impacts center on specific inputs with limited substitution possibilities.14 A convenient way of implementing the latter within a CGE model is to introduce sector-specific factors which are subject to environmental shocks. By designating a subset of primary factors as intersectorally immobile through the index s ⊂ f , whose elements may be assigned to sectors, j
(s), we may perturb the endowments of specific factors in the same 14 For

example, sites of special cultural, historical, ecological, or aesthetic significance in the tourism sector, and specific combinations of climate and arable land in the production of various kinds of crops.

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way as in the model of economic growth: Vs = AVs V s , except with AVs < 1. Letting s
= j\s indicate intersectorally mobile factors, the unit cost and expenditure functions and the factor market-clearance condition become:  pj ≤



σ

j βn,j



 1−σj 1 + τiY pi

i

+



σj  AF f,j γf,j 1

+

τfF

 1−σj wf

1/(1−σj ) ⊥ yj

(14a)

⊥u

(14b)

⊥ pi

(14c)

f

 µ≤ yi ≥





ω AC i αi,C

i σ

βi,jj



1+

τiY

 1−ω pi

1/(1−ω)



 −σj σj pj y j 1 + τiY pi

j

  −ω ω ω Y + AC µ u + gi,O i αi,C 1 + τi pi     σj
−σj
σj
F F  pj
y j
 As,j γs,j
1 + τs ws  −σj σj Vf ≥  F σj   As
,j γs
,j 1 + τsF
ws
pj y j  

s ⊂ f, j
= j
(s) s
⊂ f

⊥ wf

(14d)

j

The environmental impact CGE model is made up of Eqs. (14a)–(14d), with remaining conditions (14e) and (14f) identical to (10e) and (10f), and can be calibrated in the usual way with the parameters AC , AF , and AV set to unity. In cases where the factor account in the SAM does not record the payments to sector-specific factors, it is customary for the latter to be imputed as a proportion of the benchmark capital remuneration in the relevant industries, often based on a combination of judgment and ancillary data. The productivity-reducing impacts of environmental shocks are F simulated as increases in the values of various elements of AC i and Af,j , and V reductions in Af , with the analyst controlling the sectoral specificity of the shock through the values of the parameters AVs . As this type of application typically does not consider damage to the environment, the welfare impact is easily calculated as the decline in u that results from the income effects of diminished factor supplies and the substitution effects of commodity price changes.

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Pollution control: Pigovian taxation

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We now turn to a model of the aggregate costs of reducing pollution, using the classic case of a Pigovian tax τ ε as a starting point. Under the assumption that emissions are linked to output, τ ε translates into a vector of industry-specific output tariffs φe τ ε , each element of which constitutes an additional markup on the gross-of-tax consumer price of dirty good e, and generates a stream of tax receipts to households in the total amount ε e φe τe ye . To incorporate the tax into the model in a way that minimizes notational clutter we introduce an additional variable, pj , to denote the Pigovian tax-inclusive consumer price of commodity j. For polluting goods, pe = (1 + τeY )pe + φe τ ε , while for the subset of clean goods which emit no pollution, indexed by e
= i\e, the consumer price is the same as in Eq. (10): pe
= (1 + τeY
)pe
. In the same way that each industry’s activity level exhibits complementary slackness with respect to its zero-profit condition, we must introduce an additional variable, yj , which indicates the aggregate demand for j’s output and is complementary to the definition of pj . Furthermore, in the same way that each commodity’s price exhibits complementary slackness with respect to its market-clearance condition, we must explicitly state that pi is complementary to the aggregate supply–demand balance for good i: yi = yi . Intuitively, we can think of ye as representing the activity level of a vacuous “toll-booth” sector whose sole purpose is to convert each unit of untaxed dirty good at a price pe into the same quantity of taxed dirty good at a price (1 + τeY )pe + φe τ ε . These adjustments imply the new pseudoexcess demand correspondence: 1/(1−σj )   σ   σ 1−σ   1−σj  γf,jj 1 + τfF wf ⊥ yj (15a) pj ≤  βi,jj pi j + i

 µ≤

f



1/(1−ω) αω 1−ω i,C p i

i

yi ≥ gi,O + ID =

 f

+

 j

wf Vf −

 j

−σj σj pj y j

σ

βi,jj pi  i

τjY pj yj +

pi gi,O +  e

+ αω i−ω µω u i,C p 

⊥u

(15b)

⊥ pi

(15c)

⊥ ID

(15f)

τfF wf Vf

f

φe τ ε ye

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with the remaining conditions (15d) and (15e) identical to Eqs. (10d) and (10e), and the two additional equations:  (1 + τeY )pe + φe τ ε e ⊂ j ⊥ yj pj ≤ (15g) (1 + τ Y
)p

e ⊂ j e e yi = yi

⊥ pi

(15h)

which together make up the Pigovian pollution control CGE model. As per the discussion in Sec. 9.5.1, τ ε may be varied over a range of values to generate equilibria which can be employed to compute MAC curves for pollution as well as the optimal tax on emissions. 9.5.5.

Pollution control: Quantitative emission targets

A simple modification transforms the previous model into a test-bed for the analysis of quantitative pollution control measures such as performance standards and tradable permit systems. We introduce the primal variable E to indicate the constrained aggregate pollution load associated with the dual Pigovian tax. This allows us interpret the tax as the Lagrange multi plier on a limit on emissions, e φe ye ≤ E, which is essentially a market clearance condition for pollution with respect to which τ ε exhibits complementary slackness:  φe ye ≤ E ⊥ τε (15i) e

This expression enables the analyst to specify the emission limit E as a policy parameter, which in turn transforms the tax into an endogenous variable, the implicit “quota-equivalent” price of pollution. Finally, notice that the performance target could as easily have been specified in terms of the pollution intensity of aggregate income by dividing the left-hand side of (15i) by I D . The variable τ ε can also be interpreted as the endogenous marketclearing price of emission allowances in an economy-wide cap-and-trade scheme. Interestingly, in the present representative-agent setting the two main methods for allocating allowances — auctioning by the government and grandfathering to firms — are identical in their algebraic expression and economic impacts. Grandfathering allowances is equivalent to defining a new factor of production, which, while notionally improving firms’ profitability, is actually owned by their shareholders, so that the

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returns accrue as income to the representative agent. Likewise, auctioning allowances generates additional government revenue which is then immediately recycled to the representative agent in a lump sum. This is the fundamental symmetry of general equilibrium: a given value of τ ε , or the constrained pollution level E consistent with this value, will generate identical general equilibrium allocations of emissions, production activity, prices, and welfare.

9.5.6.

Pollution control: Elastic factor supply and the double dividend

Up until this point in the chapter, taxes on primary factors have played a minor role in the determination of equilibrium. The reason is that all of the models thus far assume that factors of production are in perfectly inelastic supply, with the result that factor taxes have no income effects, and only serve to distort industries’ demands for different factors in accordance with the relative magnitudes of the elements of τ F .15 This is highly unrealistic, especially in the case of labor, a fact which is the key to the voluminous literature on the “double dividend” of environmental taxation. Payroll taxes distort labor supply decisions by inducing households to overallocate their endowment of time to the consumption of leisure. However, the revenue raised by taxing pollution may be used to partially replace payroll tax receipts, facilitating revenue-neutral reductions in payroll taxes which generate welfare improvements that offset the macroeconomic costs of abating pollution. In this setting, the double dividend refers to the dual benefits of reduced environmental damage and the efficiency gain from labor-market liberalization. The magnitude of the latter depends crucially on the tax elasticity of labor supply, which in a CGE model is determined by the coefficient on leisure and the substitutability of leisure for goods in consumers’ utility functions. To simulate the choice between leisure and supplying labor into the framework of the Pigovian CGE model, we assume that the representative agent possesses an exogenous endowment of time, T , of which an endogenous quantity  is consumed as leisure. This implicitly defines the economy’s aggregate labor endowment as an endogenous variable, T ≥ VL + . Leisure 15 Full

employment is known as a neoclassical macroclosure rule. The alternative Keynesian closure rule assumes perfectly elastic labor supply with the wage which fixed at its benchmark level. See, e.g., Robinson (2006).

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substitutes for the consumption of commodities in the generation of utility, which we model by specifying the representative agent as having a twotiered nested utility function whose lower tier is a CES subutility function denominated over goods consumption similar to Eq. (15b), and whose upper tier is a CES aggregation of consumption of commodities and leisure. The upper tier’s indexes of activity and unit expenditure are u and µ, as in (15), and we assume that leisure has a technical coefficient λ an elasticity of substitution ξ with the composite of consumed goods. The lower tier’s indexes of activity and unit expenditure are u  and µ . The upper and lower tiers of the corresponding dual expenditure function are complementary to the activity levels u and u , while the supply–demand balance for u  is complementary to µ , the demand for utility u is the same as Eq. (15e), and the supply–demand balance for leisure is complementary to its own activity level. The distorting effect of a payroll tax (τLF ) is the wedge that it drives between the consumer prices of labor and leisure: the representative agent demands leisure at the pretax wage while firms demand labor at the tax-inclusive wage. The implication is that the value of the representative agent’s endowment of leisure is wL , while his/her income from wages and recycled payroll tax receipts are wL (T − ) and τLF wL (T − ). The CGE model with a Pigovian pollution tax and labor-leisure choice is given by the following equations 1/(1−ω)   1−ω ω αi,C pi ⊥u  (16a) µ ≤ Vf ≥



i

 f = L  

γf,jj ((1 + τfF )wf )−σj pj j yj σ

σ

j

T ≥



j γL,j ((1 + τLF )wL )−σj pj j yj + 

σ

j

ID =



wf Vf −

f =L

σ

 i

pi gi,O +



τfF wf Vf +

f =L

+ wL T + τLF wL (T − ) +



   

⊥ wf

(16b)

τjY pj yj

j

φe τ ε ye

⊥ ID

(16c)

1−ξ µ ≤ (λξ wL + (1 − λ)ξ µ 1−ξ )1/(1−ξ)

⊥u

(16d)

−ξ µξ u u  ≥ (1 − λ)ξ µ

⊥µ 

(16e)

−ξ ξ  ≥ λξ wL µ u

⊥

(16f)

e

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along with Eqs. (15a), (15c), (15e), and (15g)–(15h), which we relabel (16a), (16c), (16e), and (16g)–(16h). The double dividend is modeled by reducing the labor tax burden to the point where the additional pollution tax revenue just offsets the loss in payroll tax revenue, so that the sum of pollution and labor tax receipts remains at the benchmark level, T L . As this condition is satisfied by appropriately decrementing the labor tax rate for a given value of τ ε , we model τLF as an endogenous variable which exhibits complementary slackness with respect to the revenue neutrality constraint, which is implemented as an auxiliary equation:  T L = τLF wL (T − ) + φe τ ε ye ⊥ τLF (16g) e

The critical uncalibrated parameters in this model are the benchmark time endowment, the coefficient on leisure and the goodsleisure substitution elasticity, whose values are traditionally calculated from ancillary data on the S , and the ratio of the represenuncompensated labor supply elasticity, ηL tative agent’s time endowment to his/her labor supply, R = T /(T − ). Multiplying the numerator and denominator of R by the wage yields the benchmark values of the time endowment and leisure demand, T = RV L and  = (R − 1)V L . Using Eqs. (16d) and (16f), the labor supply elasticity can be expressed as     1−ξ wL ∂(T − ) (1 − λ)ξ µ S ηL = =ξ 1−ξ T −  ∂wL T − λξ wL + (1 − λ)ξ µ 1−ξ in which the fraction in parentheses is R − 1, the ratio of the benchmark values of leisure and labor income, and the term in square braces is the share of commodities in the value of total (leisure plus nonleisure) consumption, which at benchmark prices is simply GC /(GC + wL ) = GC / (GC + (R − 1)V L ). The elasticity of goods-leisure substitution and the coefficient on leisure may then be calibrated as S ξ = ηL

GC + (R − 1)V L (R − 1)GC

 and λ =

(R − 1)V L GC + (R − 1)V L

1/ξ

We close our discussion of this topic with two observations. First, the time endowment ratio is a key uncertain parameter, so that given its importance for the value of the calibrated parameters — and, ultimately, the magnitude of the double dividend — it is frequently the object of sensitivity

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analysis.16 Second and more substantively, implicit in the foregoing analysis is the strong assumption of no involuntary unemployment, which in CGE models is typically the result of wage rigidities that drive a wedge between households’ time endowment and their labor supply. Crucially, there is no utility benefit associated with these non-leisure, non-work hours, which we denote T U . Perhaps, the simplest way of operationalizing involuntary unemployment is to specify the quantity of “dead time” as the dual of a minimum wage, w L , via the auxiliary complementarity condition:17 wL ≥ wL

⊥TU

(16h)

Integrating this with our previous formulation necessitates an adjustment of the market-clearance condition for hours (16b) to accommodate unemployed time:  σ σ j γL,j ((1 + τLF )wL )−σj pj j yj +  + T U ⊥ wf (16b
) T ≥ j

and also a change in the calibration formulas. The attractive feature of this approach is that ancillary information on the level of unemployment, U , may be used to approximate the unobserved quantity of unemU = U /(1 − U )V L . From there, ployed time in the benchmark as T a redefinition of the time-endowment ratio to include unemployed time, R = T /(T − T U − ), allows the representative agent’s endowment of time to remain the same, with a reduced initial leisure endowment:  = (R − U 1−U − 1)V L . Updated values for the goods-leisure elasticity and the coefficient on leisure may then be derived by following the calibration procedure outlined above. 9.5.7.

Pollution control: Technology policies

Another shortcoming of the canonical model of pollution control is its implicit assumption of a Leontief (fixed-coefficients) relationship between 16 Ballard et al. (1985, p. 135) set R = 1.75 “. . . to reflect that individuals typically work a forty-hour, out of a possible seventy-hour week.” Ballard (2000) shows that in CGE studies this parameter can range from as low as 1.5 to as high as 5, and develops a calibration technique which uses econometrically estimated values of compensated and uncompensated labor supply elasticities in place of R. 17 For an alternative specification, see Balistreri (2002).

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pollution and output. While this formulation correctly represents the stoichiometric relationship between carbon dioxide and fossil fuels in analyses of climate policy (e.g., Goulder et al., 1999), is not generally applicable to most pollutants, which are often capable of being at least partially abated through investments in pollution control. Availability of the control option means that the eth polluting industry faces the choice of either abating pollution and associated tax liabilities on residual emissions by reducing output — and revenue, or installing pollution control capital. Cost minimization implies that the industry will choose to undertake the latter up to the point where the marginal cost of installation equals the marginal revenue savings from the ability to expand output. Representing the outcome of this process requires us to specify the pollution control technology, which for the sake of simplicity we model as a CRTS Leontief transformation technology which combines inputs of abatement capital and the dirty good to produce “remediated,” i.e., pollutionfree, output. Abatement generated by tax-induced output reductions and by pollution control are assumed to be perfect substitutes. Therefore, the control activity is modeled as demanding inputs at their ad-valorem taxinclusive consumer prices while supplying output at the Pigovian taxinclusive price of the dirty good in question. Assuming a production coefficient χ on the use of capital for pollution control, this formulation F )wK + (1 − χ)(1 + τeY )pe . implies the zero-profit condition pˆe ≤ χ(1 + τK As in prior models, this condition must be specified as complementary to a primal activity level. We therefore introduce the variable qˆe to indicate the operating activity of sectors’ pollution controls, which in turn determines the input demands for capital and the dirty good, χqe and (1 − χ)qe . A few points about this formulation are worth noting. The most important is that the pollution control activity competes with the dummy tollbooth activity mentioned above, as they both demand the output of dirty sector e as an input while their outputs jointly satisfy aggregate demand for dirty good e. The effect of pollution control will be to shift industry e’s MAC curve downward over some portion of its domain, but the specifics depend on the capital intensity of the control technology (determined by χ) as well as ancillary information, without which it is not possible to make a priori predictions about the extent to which qˆe will displace yˆe . The latter is found by simulating the model, which is easiest to do in the case where pollution is uncontrolled in the benchmark equilibrium. This sidesteps the potentially significant complications that attend the calibration of pollution

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control technology as a subcomponent of the dirty sectors.18 But in either situation care must be exercised in accounting for emissions: here, the tollbooth sector should be thought of as the source of pollution, in the form of residual emissions released after each dirty industry abates up to the point where the marginal cost of doing so equals the Pigovian tax.19 We therefore specify the model with the pollution control activity as a counterfactual alternative, by replacing Eqs. (15c), (15d), and (15h) with  σ  σ qe + ye ≥ ge,O + βe,jj pe−σj pj j yj + αω e−ω µω u e ⊂ i  e,C p   j  σj −σj σj ⊥ pi ω βe
,j pe
pj yj + αω e−ω e
⊂ i ye
≥ ge
,O + 
µ u e
,C p  j

Vf ≥

γ σj ((1 + τ F )w )−σj pσj y f j j f f,j 

yj =

(17a)

 σ  σ j F γK,j ((1 + τK )wK )−σj pj j yj + χ qe e

f = K

⊥ wf (17b)

ye + (1 − χ) qe

e⊂j

ye


e
⊂ j

⊥ pj

(17c)

and including the remaining Eqs. (15a), (15b), (10e), (15f), and (15g) — which we relabel (17a), (17b), (17e), (17f), and (17g), along with the additional condition of zero profit in pollution control F )wK + (1 − χ)(1 + τeY )pe pe ≤ χ(1 + τK

⊥ qe

(17d)

The present structure opens up a wealth of possible applications. Especially relevant in the context of US environmental policy is the general equilibrium benefit–cost assessment of controlling pollution through technology mandates. This instrument is modeled by replacing Eq. (15i) with a constraint that specifies an upper bound, say ϑ, on the fraction of each dirty industry’s output which may be produced without control technology. The associated complementary variable is the shadow price on the standard, τeqb, which may be interpreted as an implicit subsidy on the (more 18 See,

e.g., the methodology developed by Hyman et al. (2003) to calibrate technologies for non-CO2 GHG emission control using marginal abatement cost curves derived from bottom-up engineering analyses. P 19 This is the reason for not specifying the left-hand side of Eq. (15i) as e φe ye , which in the present setting overstates emissions.

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expensive) production with pollution control. The associated rents represent a transfer of revenue from dirty production to the pollution control activity, and are revenue-neutral with respect to both the government and the income balance of households. We therefore drop the terms in τ ε from the tax-inclusive goods price and income definition equations (15g and 15f), and add the cost-reducing effect of the subsidy to the pollution control activity’s zero-profit condition (17d). The resulting technology mandate model is made up of Eqs. (17a)–(17c) along with the following equations 

wf Vf −





(17e)

pj ≤ (1 + τjY )pj

⊥ yj

(17f)

F pe ≤ χ(1 + τK )wK + (1 − χ)(1 + τeY )pe − τeqb

⊥ qe

(17d)

qe + ye ) qe ≥ ϑ(

⊥ τeqb

(17g)

f

i

pi gi,O +



⊥ ID

ID =

τfF wf Vf +

f

τjY pj yj

j

Lastly, it is also possible to extend the basic pollution control model to a encompass a range of discrete technology options, some of which may be operated in the benchmark equilibrium, some of which may be unprofitable and inactive — as in the present example — but may become active in response to changes in relative prices induced by policy or other kinds of shocks. While the latter are comparatively easy to model, and indeed are a mainstay of prospective technological analyses (e.g., Sue Wing, 2006), the former are more difficult to calibrate because of the need to reconcile the economic accounts in the SAM with often incommensurate engineering descriptions of technologies’ operating characteristics and performance (e.g., Sue Wing, 2008; B¨ohringer and Rutherford, 2008). 9.5.8.

A non-separable environmental amenity

Our final example illustrates the challenges which attend modeling of both the benefits as well as the costs of pollution control when the latter are nonseparable from consumption. Following Carbone and Smith (2008), we introduce an environmental amenity as a public good within the general equilibrium framework of (10). The amenity (in Carbone and Smith, air quality) has an activity level A , defined as the inverse of the aggregate pollution load (1/E), and a dual marginal willingness to pay, or virtual price, θ. To keep the exposition simple we assume that the amenity is a substitute for aggregate goods consumption, and enters the utility function

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in the same manner as leisure in the model of Eq. (16). In the resulting two-level nested CES utility function, we use ζ to denote the elasticity of substitution between goods and the amenity and ν to denote technical coefficient on the amenity. Finally, the value of the amenity to the representative agent is reflected as an additional term in the income definition equation, which points to the interpretation of I D as “virtual income”, i.e., market income plus the value of environmental quality. The non-separable amenity CGE model is made up of the following equations  µ ≤



1/(1−ω) αω i,C ((1

i

ID =

 f

+

wf Vf −



+



τiY

1−ω

⊥u 

)pi )

(1 + τiY )pi gi,O +

i

τjY pj yj + θA



(18a)

τfF wf Vf

f

⊥ ID

(18b)

j

1−ζ )1/(1−ζ) µ ≤ (ν ζ θ1−ζ + (1 − ν)ζ µ

⊥u

(18c)

−ζ ζ

⊥µ 

(18d)

A ≥ ν ζ θ−ζ µζ u  −1  A = φe ye

⊥θ

(18e)

⊥A

(18f)

u  ≥ (1 − ν) µ  ζ

µ u

e

in addition to Eqs. (10a) and (10c)–(10e), which we relabel (18a) and (18c)– (18e). In contrast to the double-dividend model, calibrating the present structure requires that the analyst make a number of ad hoc assumptions. Chief among these is the substitutability of the amenity for goods consumption, which is currently a wide open empirical question that renders the value of ζ a matter of judgment.20 The public good character of the amenity implies that its benchmark virtual price θ is the value of marginal willingness to pay for the amenity (e.g., as estimated by econometric studies), summed over the number of individuals or households in the economy, while the benchmark quantity of the amenity, A , is simply 20 It seems reasonable to assume 0 < ζ < 1, which ensures that goods and the amenity are both necessary inputs to consumption.

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the inverse of the ambient pollution load estimated for the benchmark year. The technical coefficient on the amenity may then be calibrated as ν = (θA /(GC + θA ))1/ζ . In this example, increasing taxes on the output of dirty industries e ⊂ i
from their benchmark levels (say to some τeY > τeY ) will by (18f) and the last term in (18b) generate an extra positive income effect that offsets the increases in dirty commodities’ consumer prices, potentially creating a net welfare gain. The payoff to this integrated approach is that it obviates an external environmental damage function. Indeed, just about any analog of D(E) can be used to construct Eq. (18f). Moreover, studies such as B¨ohringer et al. (2007) demonstrate the potential for replacing this expression in its entirety with the output of a detailed natural-science simulation run in tandem with the CGE model, in a scheme where the two models pass values of E and A back and forth to one another in an iterative fashion until their solutions converge. The environment in Figure 9.1 is represented by the natural-science simulation, which computes the physically derived quantity of the amenity based on inputs of the pollution load generated by the CGE model (H1). The quantity of the amenity thus computed is then used as an input to the next iteration of the CGE model (H2). Such a scheme is made necessary by the feedback of the income and substitution effects of changes in A on the demand for dirty goods, the activity levels of dirty industries, and, ultimately, the level of pollution.

9.6.

Summary, Caveats, and Future Research Directions

This chapter has provided a lucid, rigorous, and hands-on introduction to the fundamentals of computable general equilibrium models and their application in natural resource and environmental economics. The objective has been to demystify CGE models by developing a simple, transparent, and comprehensive framework within which to conceptualize their structural underpinnings, numerical parameterization, mechanisms of solution, and techniques of application. The circular flow of the economy was used as the starting point to develop the equilibrium conditions of a CGE model, and it was demonstrated how imposing the axioms of producer and consumer maximization on this conceptual edifice facilitate the complete algebraic specification of an economy of arbitrary sectoral dimension. There followed a description of the techniques of numerical calibration and an overview of

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the general procedures for the use of the resulting numerical model as an analytical tool. The meat of the chapter focused on techniques of application, illustrating a range of structural modifications that enable CGE models to project the implications of changes in the circular flow for environmental quality or resource use, as well as to analyze the economy-wide impacts of environmental change and resource scarcity, and to evaluate the incidence of policies for limiting pollution and conserving environmental resources. Throughout, the exposition has been kept deliberately simple for the sake of practicality: with a only minimal data gathering beyond the SAM presented in Sue Wing (2009, Fig. 14.6), it is a straightforward task to specify all of the models discussed herein a high-level language such as GAMS (Brooke et al., 1998) and solve the resulting numerical problems as an MCP. Moreover, different elements of the template models showcased above may be easily combined, facilitating the analysis of a range of problems arising in natural resource and environmental economics. But in spite of the broad swath of intellectual territory covered by this chapter, space constraints have precluded discussion of many important topics. The chapter’s closed-economy focus has paid scant attention to the specification and calibration of multiregion models which combine SAMs for individual economies with data on trade flows, and their application to transboundary environmental issues such as policy coordination, abatement coalition formation, and emission leakage. We have also given short shrift to the intra-regional distribution of the burden of environmental protection or resource conservation, which can be implemented by generalizing our simple CES economy to incorporate multiple households with different levels of income and endowment of traditional and environmental factors. As well, our models’ maintained assumption of frictionless competitive markets has precluded consideration of the important role of market imperfections — particularly imperfect competition — in influencing both externalities and the level and distribution of the economy-wide costs of internalizing them. The multiplicity of methods available for representing imperfect competition within CGE models (Conrad, 2002), coupled with the divergence of the results generated by different formulations (Roson, 2006), highlights the need for methodological reconciliation. An even less adequately served area of study is the analysis of natural resource depletion using forward-looking CGE models specified in the complementarity format outlined here (cf. Lau et al., 2002), into which the Hotelling model has not yet been incorporated. Calibrating such a model is especially challenging, so much so that prior

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analyses have relied either on recursive-dynamic simulations (e.g., Bailey and Clarke, 2000), or dynamic models with alternative specifications of the depletion process (Babiker et al., 2009). A final research frontier which lies well beyond the scope of this review is the application of the general equilibrium framework to environmental science, in particular the study of trophic interactions within ecosystems (Tschirhart, 2000), which raises the exciting prospect of the development of integrated economic–ecological CGE models (e.g., Tschirhart, 2003). Hopefully, the base of practical and theoretical knowledge developed here will lay the groundwork for the reader to go on to apply CGE models in these and other areas of environmental and resource economics.

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Chapter 10

Experimental Methods and Environmental and Natural Resource Policy Todd L. Cherry∗ and Michael McKee† Appalachian State University ∗ [email protected] † [email protected]

10.1.

Introduction

Environmental economists have found it increasingly useful to apply the tools of experimental economics to analyses of contemporary environmental and natural resource policy issues. Motivating these efforts are at least two considerations. First, it is often the case that data required for assessments of policy initiatives are unavailable and field trials are not economically feasible.1 In such instances, one may appeal to experimental economics. When properly designed and implemented, laboratory experiments provide a powerful, low-cost mechanism for the generation of data that can be useful for the evaluation of proposed policy interventions.2 A second motivation 1 This was not always the case. The 1960s was a “hey day” of social experimentation. Several programs were subject to field trials. Some examples are income transfer programs (as in the Gary, IN trials), elementary and secondary education delivery (performance contracting experiments by the Office of Equal Opportunity or OEO). More recently, ethical considerations, as well as cost, have limited the use of such trials. Laboratory experiments are often the only means remaining of obtaining real data from real human decision makers. 2 The use of laboratory experiments is the potential “educational” benefit that results from the use of experiments to illustrate the effects of policy initiatives B data from

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has been the investigation of the underlying behavioral issues that drive responses to stated preference surveys.3 Such responses are often essential to mandatory benefit cost analyses of proposed policies. Environmental economists have increasingly recognized the promise of experimental economics to provide a powerful scientific approach in generating policyrelated hypotheses and the data required to test these hypotheses. Environmental economists have also increasingly recognized the ability of experimental economics to provide an effective non-scientific method that demonstrates, in compelling ways, the economic efficacy of policy alternatives to policy/decision makers. This chapter has two primary purposes. First, we wish to provide the reader with a sketch of methods used in experimental economics with an emphasis on adherence to the precepts of experimental design. Such adherence is necessary to forestall many of the recent criticisms, as well as some of the persistent ones, of laboratory experiments.4 Our second purpose is to discuss areas in environmental economics that are especially fertile grounds for the use of experiments as a research tool. We see timely opportunities for the application of the experimental method in the area of valuating environmental amenities/damages (Sec. 3) and policy design issues related to enforcing compliance with environmental rules and regulations (Sec. 4). We discuss an area of past success and wrap up with conclusions (Sec. 5).

10.2.

Laboratory Experiments: Design Requirements

Any microeconomic system consists of a set of fundamental building blocks (Smith, 1982). There is an environment consisting of economic agents and “goods.” There is an institution consisting of a system of property rights, rules of exchange, and payoffs from actions, which can include new or changed laws and regulations. There are outcomes as generated by the experiments can be used to demonstrate the consequences of policy alternatives to decision makers in ways that theory cannot. Many policy makers are not trained in economic theory and empirical arguments often carry more weight than the theoretical predictions alone. Absent field data, the only alternative for empirical demonstrations are laboratory experiments. Experiments may be viewed as a means of improving the decision-making process of policy makers by providing information that would be otherwise unavailable or by furthering the evaluation of decision criteria such as benefit measurement. 3 The use of stated preferences via, for example, contingent valuation method or CVM, is not without controversy and many of the behavioral issues relating to truthful and accurate revelation have been addressed in the laboratory. 4 See Harrison and List (2004) for a discussion of the merits of “field experiments” versus laboratory experiments.

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behavior of agents, and the behavior is determined by the interactions between the environment and the institution. The essence of laboratory experiments in economics is the (usually simplified) creation of an institution, the explicit control of agents’ preferences, and observations of outcomes or agents’ behavior that arise from an introduced change in the institutional setting, introduction of a new good, or inclusion of new agents. A brief elaboration of these essential components of an economic experiment follows. The control of agents’ preferences is accomplished by the use of what is referred to as “induced preferences” by which preferences are “induced” by the experimenter through fixing the value or cost of a good for each individual agent. For example, given a “good” with no intrinsic value (a common requirement for experiments),5 the “value” of this good to a buyer is established by the experimenter’s commitment to redeem any units of the good acquired by the agent at a fixed dollar amount. The buyeragent’s income then becomes the fixed redemption price minus whatever amount he or she must pay to acquire a unit of the good. The “cost” of a unit of the good to a seller is established by the experimenter’s commitment to provide the seller with a unit of the good at a fixed dollar amount. The seller-agent’s income then becomes the price received from a buyer for a unit of the good minus the cost of acquiring the unit from the experimenter. By setting the redemption price (to buyers) and the acquisition price (to sellers), the experimenter controls agents’ preferences. The creation of an institution involves the explicit description of all contextual aspects that may “substantively” affect an agent’s behavior, such as property rights, rules governing exchange and payoffs (positive and negative), and information. As a simple example, consider the widely observed posted offer market as implemented in an experiment. The environment 5 Neutrality increases the experimenter’s control over subject preferences and avoids leading subjects to invoke different “mental scripts,” which may enable them to fill in (potentially) missing information in the instructions but which also may unpredictably influence their choices. It is sometimes claimed that the use of neutral instructions limits the ability to generalize from the experimental to the naturally occurring setting. In fact, however, it is not possible to generalize beyond the laboratory unless one uses neutral instructions, since the experimenter cannot control the values that subjects associate with loaded terms. This is not to say, however, that all experiments must be conducted with neutral terms. Indeed, there may be instances that beg for the use of non-neutral terms. For example, experiments which seek to observe cultural effects on behavior may prescribe the use of non-neutral terms precisely to enable the researcher to observe the effects of culture and/or social norms on behavior in settings that are close to the field setting. Examples include work investigating cultural attitudes toward gambling and altruism.

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consists of buyer-agents and seller-agents who, respectively, buy and sell “units” of a good under rules wherein: information available to each agent is limited to the agents’ own values/costs and the posted asks (offers) of the sellers. Payoffs to buyer-agents are straightforward: redemption value minus price paid for each unit purchased; payoffs to seller-agents are posted ask price less the acquisition cost for each unit sold. All factors relevant to the agents have been specified: property rights, rules of exchange, and payoffs from the decision. The theoretical prediction that is investigated concerns the extraction of the available surplus by the buyers and sellers. A key to the transferability of the experimental results to inform discussions of the naturally occurring world is adherence to a basic set of precepts and a set of procedural guidelines that have become a key part of the experimental economics methodology.6 Smith (1982) lists five precepts necessary for experimental validity: reward salience, payoff dominance, privacy, nonsatiation in reward medium, and parallelism.7 The reward to making decisions must be salient to the subject; the link between the decision taken and the reward must be clear to the subject. The payoffs to decisions in the experiment must be sufficiently high that they dominate other factors such as a desire to please the experimenter. Though simplified, the experimental setting must maintain parallelism with the naturally occurring setting being investigated. The payoffs must be made privately in order to reduce the potential for interpersonal comparisons that can potentially influence behavior. The (arguably) most critical precept for informing a policy debate is that the design of an institution applied in the experiment parallels the real-world conditions that it is intended to simulate. If policy makers are 6 Following Davis and Holt (1993), a few of the more important procedural conditions relevant for the creation of an institution include the following: the experiment should be administered in a uniform and consistent manner to allow replication; the experiment should not be excessively long or complicated, since subjects may become bored or confused; subjects must believe that the procedures described to them are the procedures actually followed; and instructions provided to subjects should be understandable, should avoid the use of examples that lead subjects to anchor on certain choices, and should be phrased in neutral rather than loaded terms. 7 The precepts are: saliency (the level of rewards received by subjects must be directly related to their decision); reward (payoff) dominance (rewards much be large enough to offset any subjective costs or benefits that subjects place on their participation in the experiment); and privacy (each subject must know only his or her own payoffs so that they do not receive any subjective value from payoffs of other subjects). The privacy condition may be relaxed in settings where the experimenter seeks to investigate the role of such motives as fairness or equity.

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to accept experimental findings as informative and meaningful in policy analyses, both the environment and the institution used in the laboratory must closely and obviously establish a relationship between conditions represented in the experiment and naturally occurring, field conditions in which the policy is intended to operate. The greater the experimenter’s desire to produce results that might influence policy, the greater the degree of parallelism is needed since policy makers are generally unaccustomed to high levels of abstraction. While we are discussing parallelism it is useful to consider an, often cited, procedural requirement for the use of neutral language in the experiment instructions and setting. The effect of language used in the experiment on agent behavior has been the subject of debate in experiments, especially those involving a policy-specific investigation. For policy exercises that involve routine individual decisions, such as tax reporting or purchase decisions, it may be useful to invoke field language since this will make the setting more realistic for the subjects. A few studies have explored the impact of language on behavior and results are mixed. Alm et al. (1992a–c) report no difference in behavior for student subjects in tax compliance experiments for alternative forms of tax language while Wartick et al. (1999) report differences in the behavior of adult subjects. With an appropriate environment and institution, the end result of an experiment involves observations of agent behavior. Typically, the researcher’s interest is focused on how agent behavior is affected — how it changes — for exogenously imposed changes in institutional parameters. For example, a rules-compliance experiment focuses on how agent behavior (compliance with the rule) changes with changes in such things as standards, the level of resources expended for rules enforcement, and/or the level of fines and penalties. Two sources for critiques of experimental methods that are germane to the issue of how experiments influence the policy debate are the common use of student subjects and the collateral issue of a subject’s, trained or untrained, abilities to perform complex tasks that may be an integral part of the experiment. The role of subject training has been addressed by several authors including Ben-David et al. (1998), Muller and Mestelman (1998), and Harrison et al. (1990). While it is costly to train subjects, these studies all point to the desirability of such training in settings requiring complex decision tasks. Ben-David et al. show that the experimenter may be able to utilize the data obtained in the simpler (training) sessions through careful construction.

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While it is common to criticize experiments for their usual reliance on student subjects, it is less common to articulate the rationale for dismissing results based on student decisions. It is true that students represent a “convenient” pool but it is also the case that these subjects meet other criteria for successful navigation through an economic institution or system. That is, university-level students have steep learning curves, follow directions, and can adapt to the interface used for the lab experiment.8 Unless one can argue that these traits are undesirable in evaluating behavior, it would seem that student subjects are ideal. However, it is useful to study differences in behavior across subject pools. In some cases, experimenters have intentionally included alternative types of subjects in the design. When experiments have been conducted with both student subjects and non-student subjects, the results have generally supported the finding that the pool does not materially affect the results. See Plott (1987), Smith et al. (1988), and Dyer et al. (1989) for arguments and empirical evidence supporting the achievement of parallelism with the use of student subjects. Suppose it is observed that some differences in agent behavior are identified across the alternative institutions. Additional experiments (replications) might be particularly productive in allaying concerns that the “artificial” setting of the laboratory per se may affect subject behavior.9 A second approach that might be used to explore the parallelism issue would focus on the perhaps more subtle potential effects on subject behavior associated with what might be viewed as intrinsic, or innocuous experimental conditions (a major source of concern to Smith and Mansfield, 1998). This approach, which closely follows practices used in the natural sciences, would test the extent to which behavior obtained with a given experimental design by one set of researchers in a given laboratory is replicated when the design is used by other sets of researchers in different laboratories. It is useful at this point to recall Plott’s description of the value of economic experiments: “While laboratory processes are simple in comparison to naturally occurring processes, they are real processes in the sense that real people participate for real and substantial profits and follow real

8 There is an argument worth bearing in mind. Individuals with experience in institutions that approximate the experiment setting may apply heuristics, such as representativeness (Tversky and Kahneman, 1974), that are inappropriate to the specific environment in the laboratory. 9 See, e.g., Smith and Mansfield (1998) who offer the conjecture that the experimental laboratory is intrinsically an artificial setting and that generalizations to the field are implausible.

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rules in doing so. It is precisely because they are real that they are interesting” (Plott, 1982; p. 1486). The bulk of what experimental methods can offer as a means for allowing the environmental economist to inform the policy debate derives from simplicity. There are those, however, who reject results obtained with such simplicity as providing meaningful insights for analyses of the efficacy of public policies. Recently, there has been an assertion that the lab is a less than reliable venue and that experiments need to be conducted in the “field.” While the idea of field experiments is not new (see footnote 1), the more recent literature focuses on the field venue for testing basic economic propositions. List and Levitt (2007) enumerate several factors that they assert limit the experimenter’s ability to generalize behavior beyond the laboratory. These include the presence of moral and ethical considerations, the context in which the decision is embedded, self-selection of individuals participating in the laboratory, and the stakes of the game. Most, if not all, of these issues are mitigated by incorporating the precepts of experimental design articulated by Smith (1982) and ensuring that payoffs are salient and that the financial rewards dominate such factors as the subjects’ desires to please (or punish) the experimenter. Context, stakes level, and moral considerations are minimized through the adherence to reward saliency and payoff dominance. While the laboratory may be subject to self-selection effects, this is also the case in field settings. Risk-averse individuals will self-select into safe jobs, for example. We close this section with what, in our view, is a most succinct and appropriate response to the critique that laboratory experiments are too artificial to credibly be used in the policy debate. Plott offered this argument, in reference to his experimental tests of a particular policy model: The experiments were designed to check the accuracy of that model. If the model advocated because of its generality failed to be reliable in the simple case of the experimental markets, the burden of proof would presumably rest on the advocate to explain why it did not work. . . If the model performs sufficiently badly in the experiment, the burden is on the model’s advocate to explain why the experiments were ‘special’ or ‘different’ from the complex case in which the model is supposed to work (Plott, 1987, p. 205).

10.3.

Laboratory Experiments Inform the Elicitation of Values for Non-priced Goods

In many instances, conducting benefit–cost analysis of environmental policy prescriptions requires that we obtain values of goods, such as environmental

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amenities, not traded in formal markets. While it may be possible to rely on indirect methods of valuation (e.g., travel cost and hedonic pricing), such methods cannot be applied to a large class of environmental goods of primary interest to environmental economists. In these cases, researchers often rely upon a survey method, the contingent valuation (CV) method, which involves the direct elicitation of values from individuals. The CV methodology involves creating a constructed market in the minds of respondents and asking them to report the price they would be willing to pay (willing to accept) for increases (decreases) in an amenity in this constructed market. The respondent must solve two (possibly interrelated) problems: value formation and value reporting. The value formation problem requires that the survey instrument induce the respondent to contemplate the good or setting and determine a maximum willingness to pay (or minimum willingness to accept) for the good. This cognitive task is complex since the individual is often faced with an unfamiliar setting and must factor in other demands on the budget. The value reporting problem requires that the respondent be induced to report a willingness to pay that corresponds to his/her “true” underlying demand for the good. The question as to the extent to which applications of the CV method produce what they intend to produce — unbiased measures of agents’ maximum willingness to pay (accept) for an increment (decrement) of an environmental good — has been and continues to be the subject of intense and often acrimonious debate. At the center of this debate is the issue of hypothetical bias in CV surveys, and the question remains: do people respond to a hypothetical survey as if the market was real and budget constraints were binding.10 The experimental method provides an effective approach for environmental economists to investigate hypothetical bias and other critical issues related to the valuation of non-priced environmental goods. Until fairly recently, efforts to design CV studies that might provide unbiased valuation estimates for non-priced environmental goods have focused on developing means to calibrate the values derived from a CV study. For example, Blackburn et al. (1994) and Fox et al. (1998) employ the experimental method to explore calibration by having subjects respond to hypothetical and then real valuation questions. A calibration function is subsequently estimated which relates differences in responses obtained in 10 Those interested in further discussion on this debate can consult Haneman (1994), Portney (1994), and Diamond and Hausman (1994).

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the two treatments to subject characteristics. Typically, efforts to calibrate CV values were confounded by a general finding that calibration weights estimated for one particular environmental good could not be generalized to other goods. For example, Fox et al. (1998; p. 2) note that “As our results show, calibration appears to be commodity-specific, and as such, (the calibration approach) must proceed on a case-by-case basis until enough evidence exists to reveal any systematic bias that can eventually lead to a general calibration function.”11 Without predictable, stable calibration functions, this ex post approach does not hold much promise for the conduct of ex ante policy analysis. A second approach, which is ex ante in nature, is to directly motivate subjects to provide unbiased responses to hypothetical valuation questions. Early attempts along these lines (Loomis et al., 1994; Neill, 1995) attempted to remove hypothetical bias by adding text in the CV script that asks subjects to consider budgetary substitutes in responding to the valuation question. While these attempts were ineffective, subsequent empirical experimental inquiries have revealed potentially rich lines of future research that could substantively affect the viability and credibility of the CV method, and therefore our ability to provide decision makers with more palatable (and hopefully less controversial) measures for environmental values. Cummings and Taylor (1999) and Bjornstad et al. (1997) suggest potential uses of experimental methods to improve the design of CV questionnaires, resulting in demonstrably robust, unbiased valuation responses. Both papers employ a starting point referendum design introduced by Cummings et al. (1997). They set the benchmark as the voting responses elicited in the referenda that involved actual subject payments and modify the hypothetical referenda in such a way that generates results indistinguishable from the real referenda (thus assuming the real referenda are incentive compatible). Cummings and Taylor (1999) use cheap talk to explain the issue of hypothetical bias to subjects, why it might occur, and how it affects responses in surveys. Using four different public goods, and several experimental designs, they find results in the hypothetical 11 Similar results are commonly reported in the marketing literature. For example, Louviere (1996, p. 170) observes that “I think it very unlikely that one simple and generalizable approach to “adjusting” (stated preference) numbers will be found, despite clear evidence supporting strong monotonic links between stated preferences and actual choices in real markets...differences in product awareness, learning, etc. necessarily imply that no one magic constant can exist across all product categories”.

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referenda that included the cheap talk script to be indistinguishable from responses in the real-payment referenda. Similarly, Bjornstad et al. (1997) use a “learning design” to allow subjects to learn first hand the effects of hypothetical surveys on responses; as compared to surveys involving actual payments. Again, they find that hypothetical surveys that were part of their learning design elicited voting responses that were not significantly different than responses to the same question asked of different samples in real-payment referenda. Taylor et al. (2001) conduct what is likely the first induced value test of the dichotomous choice referendum elicitation mechanism. Using a closed referendum setting they report that the values obtained from the votes of the participants were identical in the hypothetical and real settings over a wide range of value distributions. Vossler and McKee (2006) extend this work to investigate a range of elicitation mechanisms beyond the referendum to include various forms of payment card and multiple bounded decision settings. The referendum performs best overall and is least likely to be subject to any sort of hypothetical effects. Cherry et al. (2003) provide an alternative approach to investigating hypothetical bias by studying the role of markets in yielding more accurate and consistent preferences and values. The experiments build on previous work that shows rationality is a social phenomenon that results from market repetition and discipline (Arrow, 1987; Smith, 1991; Evans, 1997). This calls into question the ability of isolated individuals responding to a survey, real or hypothetical, to accurately state their preferences and values. They use a preference reversal framework, in which an individual states his or her preference and values for two lotteries with similar expected values but varying levels of risk. A well-documented result, the so-called preference reversal, is that individuals typically reveal inconsistent preferences and values, indicating a preference for the safe lottery while stating a greater willingness to pay for the risky lottery. However, research has shown an individual’s inconsistent preferences and values become consistent when he or she is subject to market-like discipline that arbitrages his or her inconsistency (Chu and Chu, 1990). Cherry et al. (2003) find when individuals resolve the inconsistency of their reversal, it is typically done so by lowering their willingness to pay, not by changing their preferences. This highlights the ability of markets, and social dynamics in general, to mitigate hypothetical bias. Cherry et al. also extends this line of research by designing experiments to test whether the rationality that is induced by market-like discipline can spill over to positively influence rationality in non-market settings.

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Their results do find evidence of a so-called rationality spillover, rationality gained in markets can be transferred to non-market settings. They find the transfer of rationality is robust when directed towards a contingent valuation context, and the increased rationality in the non-market setting seemingly results from a reduction in hypothetical bias (i.e., lower stated values, not change in preferences). Future research may continue to explore the extent and limits of rationality spillovers to improve non-market stated preferences and values.12 A second use of laboratory experiments in CV survey design uses induced value experiments as a means for investigating the causes of behavioral regularities that have been observed in field studies and the refinement of mechanisms that may be used in stated choice surveys. Bohara et al. (1998), Collins and Vossler (2009), and Cherry et al. (2004) provide examples of how the laboratory may be used in these ways. Bohara et al. (1998) experimentally investigates how changes in the amount and type of information provided in CV surveys may affect valuation responses. Their results suggest that the open-ended (OE) elicitation format generates valuation responses that are sensitive to information concerning the cost and group size while the dichotomous choice (DC) elicitation format does not. Since cost and group size information are not related to individual valuation, Bohara et al. suggest that the DC format is preferred since it yields more consistent results across information provided.13 A second example of how the lab may be used to investigate behavior in stated preference surveys is provided in Collins and Vossler (2009). They take the choice method elicitation mechanism (Louviere, 1996) to the laboratory to investigate the question of robustness in hypothetical question settings (surveys). In setting up their experiment to address possible differences in hypothetical versus real responses to a choice model survey instrument, it was revealed that the choice model instrument lacks a rule for provision of the good. Thus, the CM, as typically implemented in the field, cannot satisfy the requirement that the survey be seen as “consequential” by 12 Cherry and Shogren (2007) extended this work by showing that market induced rationality can transfer to different classes of decisions. Settle et al. (2008) reports an early test of harnessing market-like discipline by taking the laboratory to Yellowstone National Park to estimate preferences and values of environmental lotteries. 13 Brown et al. (1996) conducted OE and DC surveys and well as OE and DC realpayment surveys. An important difference between Brown et al. and Bohara et al. is that Brown et al. simply compare responses with no underlying principle to test. Bohara et al. offer a theoretical underpinning, and then test it in the lab.

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the respondent (Carson et al., 2000). Incentive compatibility in the survey response requires that the individuals treat the survey “as if” the results will affect policy decisions. A key to this perception is that the link between the responses and the policy action be clear to the survey respondents.14 Collins and Vossler, in addition to testing the real versus hypothetical nature of the choice method, also refined the choice method to implement a provision rule. Cherry et al. (2004) highlight the importance of using an induced-value experimental design to maintain control in the laboratory. Past experimental efforts used private goods to investigate valuation and preferences in the laboratory, including validating hypothetical surveys (Neill et al., 1994; Cherry et al., 2003), examining the WTP/WTA disparity (Kahneman et al., 1990; Shogren et al., 1994), and eliciting individual discount rates (e.g., Harrison et al., 1995). The use of private goods, which have a market price outside the laboratory, creates the potential of outside options to influence observed laboratory valuation behavior. They find that bidders, whether the setting is real or hypothetical, account for outside options when formulating bids in the laboratory. Subjects shaved their bids at the outside option, which suggests the valuation of private goods in the laboratory may be truncated at the market price found outside the laboratory. The result illustrates the need for researchers to consider potential hidden incentives in the laboratory. Most, if not all, of the experimental research that has focused on the hypothetical bias question compares subject responses to valuation questions obtained under conditions where stated WTP required actual payment of the WTP amount (a “real” survey) with those obtained under conditions where payment of the stated WTP was hypothetical (a hypothetical survey). The maintained assumption in these works is that the “real” survey is incentive compatible; that payment of stated WTP induces subjects to accurately and fully reveal their preferences, and that resulting value estimates are unbiased. This assumption then justifies the author’s attribution of differences observed between the two surveys as a “bias.” This assumption can be challenged on the basis of two related arguments. First, to the extent that subjects may have incentives to free-ride in “real” surveys, there are no compelling reasons to expect symmetrical free-riding behavior in the hypothetical survey. Thus, “overbidding” that is reported to occur 14 This

is a good time to point out that a major contribution of the experimental method is that the act of designing the experiment forces the researcher to confront the gaps in the theory or the policy instrument being evaluated.

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in hypothetical surveys may in fact be underbidding by subjects in the real surveys. Second, the referenda used in both the Cummings and Taylor (1999) and the Bjornstad et al. (1997) works cited above were not closed referenda. In other words, provision of the public goods used in these studies was not necessarily limited to the level that would be provided by the outcome of the specific referendum in which the particular subjects were participating. The good could be provided by sources external to the group of subjects participating in the referendum. Arguably, these conditions could invite forms of free-riding behavior, more so (perhaps) in the real surveys than in the hypothetical surveys. Taylor (1998) provides a discussion of this issue and an empirical inquiry as to the prevalence of free riding in the open referendum. Taylor et al. (2001), as described above, correct this design element with the result that the hypothetical “bias” disappears. Their results provide further support for the adherence to the theoretical underpinnings when implementing an elicitation mechanism and the need to impose the consequentiality property when conducting field surveys to elicit aggregate values for environmental amenities. Thus, the challenge for future research is to develop a mechanism for valuing public goods that can be shown to be demand revealing in real payment settings and which can be implemented in field-CVM conditions. Laboratory experiments using induced values will be crucial to the development of these instruments. Only when such an instrument is designed, one that is demonstrably demand revealing in a real-payment situation, may it be tested for incentive compatibility in hypothetical valuation situations. 10.4.

Regulatory Institutions and Compliance

Many environmental policies rely upon regulatory institutions that are intended to control the behavior of firms and individuals. Such institutions are typically characterized by regulations, sanctions for noncompliance with the regulations, and an enforcement mechanism that determines the likelihood that noncompliance on the part of any given agent will be detected. As examples, the Clean Water Act and the Clean Air Act establish standards for emissions into the air and the water, provide for specific penalties for noncompliance with these standards, and charges the U.S. Environmental Protection Agency (EPA) with the responsibility for enforcing the provisions of the Act (the probability of a non-complying entity being

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detected is, in turn, determined by the EPA’s allocation of resources to this activity). The scope of environmental regulations has continually expanded over the last two decades, and we have every reason to expect this expansion to continue in future too. Looking to the future, past experiences will surely motivate expectations for a future characterized by expanding and changing regulatory regimes for activities such as the transportation and disposal of toxic wastes, toxic materials, the disposal of scrap tires, food chains, ocean disposal of all classes of materials, ad infinitum. We feel that these expectations suggest regulatory compliance as an area of research that will become of increasing importance to decision makers and, therefore, one that will offer the environmental economist opportunities for contributing to the process of policy design. The substance of these contributions can be materially enhanced with the use of experimental methods if the challenges posed by a number of difficult parallelism problems somewhat unique to compliance issues can be effectively met. The scope and complexity of present regulatory system for environmental quality, and the strong likelihood that environmental regulations will be a “growth industry” over the next several decades, have led to the recent interest of environmental economists in the behavioral aspects of compliance, and the experimental method is serving as a central tool in this research.15 Murphy and Stranlund (2006) and Evans et al. (2008) provide examples of how experiments can contribute to the understanding of environmental regulation and compliance issues. Murphy and Stranlund (2008) design experiments to investigate the performance of voluntary discovery and disclosure policies. The general objective of these policies is to decrease the cost of enforcing environmental regulations by encouraging greater compliance with lower penalties for violations that are voluntarily discovered and reported to authorities. With little or no field data, the experimental method enables a direct investigation of the behavioral responses to these policies. In the baseline setting, subjects were responsible for making a costly decision about the level of care taken to reduce the likelihood of a violation occurring. There was an exogenous probability of an 15 We do find efforts by environmental economists to extrapolate experimental results from the tax compliance literature to environmental compliance; see Segerson and Tietenberg (1992) and Gabel and Sinclair-Desagne (1993). This dearth in experimental research related to regulatory compliance is not limited to environmental regulations. See Pagan (1998) for an innovative effort to explore compliance with immigration laws via experiments.

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audit, which would identify whether a violation occurred, and subjects only faced a penalty if a violation was revealed through an audit. The experimental design varied this basic setting by introducing the opportunity for subjects to voluntarily disclose their violations under different conditions. The design also varied the penalty of voluntarily disclosed violations and the cost of determining one’s compliance status. Murphy and Stranlund report that subjects responded strongly to the disclosure incentive, though the number of disclosures was negatively related to the level of the penalty. They also find that the reduction in penalties for self-disclosed violations led to less care taken to reduce the chance of a violation, which points to a tradeoff between greater violation disclosure and reduced environmental quality. Evans et al. (2008) investigate whether an agency facing a limited budget and required to conduct audits that discipline the firms targeted in the environmental regulation can design a selection mechanism for choosing firms to be audited. Using the voluntary reporting that is the basis of the toxic release inventory (TRI) program, they apply laboratory experiments to investigate whether the internal structure of the firm can be used to inform the selection process for audits. They find that reliance on rank order tournament compensation (commonly used as an incentive mechanism) will increase the likelihood of cheating on the reporting regulations. Thus, laboratory experiments can explore and make explicit the impact of such a reporting scheme on the incentives of division leaders under a tournament-based compensation mechanism. Emissions releases (for which excess are fined at a given rate) would be observed. One would then have to establish an institution in which the broader range of reporting described above is required, and in which audits of reports occur with some probability. Reported emissions releases under this institution are observed. A comparison of the level of cheating (underreporting) in both settings would provide a measure of the efficacy of the proposed policy, and would demonstrate such efficacy in a way that (we assert) would be particularly compelling to decision makers. Compliance with legal reporting requirements represents a formal institution but many environmental agreements are voluntary. These agreements may form to forestall externally imposed (presumably more costly) regulation. McEvoy (2009) uses a series of laboratory experiments to investigate the conditions under which such voluntary agreements are selfenforcing. The central idea is that avoiding future regulation via voluntary agreements is equivalent to voluntarily providing a public good. Most

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coalitions formed to increase contributions to a public good do not require full participation by all users of the public good and this creates incentives to free riding. Given the opportunity to opt out of a voluntary coalition, in theory, agents should try to be among the first to do so, forcing the remaining undecided agents to bear the cost of participating in the coalition. McEvoy reports a test of the predicted sequence of participation decisions in voluntary coalitions using real-time threshold public goods experiments. He finds that subjects’ behavior is more consistent with the theoretical predictions when the difference in payoffs between coalition members and free-riding nonmembers is relatively large. This suggests that such agreements will be more likely among more homogeneous players. Since compliance is a subset of a mechanism design problem, it is interesting to consider a different class of problems where the laboratory can be used to evaluate a proposed mechanism. Parkhurst and Shogren (2008) illustrate how the laboratory can testbed a policy instrument that does not yet exist, a primary benefit of the laboratory. They report results from a series of experiments that investigated whether a set of endogenously determined conservation subsidies can induce landowners to voluntarily comply with a socially optimal spatial pattern of conservation. Conservation goals often require contiguous reserves and corridors to support viable species populations and ecological processes, but creating contiguous protected areas typically requires voluntary cooperation of private landowners. To achieve the targeted spatial pattern of conservation, the subsidy instrument spatially links individual land units so that the subsidy paid to one conserved land unit depends on the conservation of the bordering land unit. In essence, monetary incentives — subsidies and penalties — are set to encourage specific patterns of land preservation. Parkhurst and Shogren’s laboratory investigation reveals evidence that spatially linked conservation subsidies can achieve a targeted conservation patterns at costs lower than the standard approach of compulsion and fixed-fee subsidies, though the success of the instrument depends on subjects gaining enough familiarity to overcome the complexity of the coordination problems. To clarify the experimental design challenge associated with the environmental compliance, consider the following critical aspects of an experimental design. First, the relationship between agent activities and outputs which are the subject of compliance must be clearly described. Second, the means for verifying, or “auditing,” elements of agent activities must closely parallel field conditions. Third, plausible mechanisms that affect expectations for audits must parallel those extant in the field. Finally, the

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incentives structure that drives the compliance decision must be reasonably transparent and must, again, parallel those that exist in the field. To provide perspective for these elements of design, consider the following discussion of experiments that have focused on related issues of tax compliance, a set of issues for which the above conditions are readily satisfied. Such experiments rely upon the following simplified conceptual framework. Suppose that an individual receives a fixed amount of income I, and must choose how much of I he or she wishes to declare to the tax authorities. Declared income D is taxed at the rate t. Unreported income is not taxed; however, the individual may be audited with probability p, at which point a fine f is imposed on each dollar of unpaid taxes. If underreporting is detected, the individual’s income IC equals I − tD − ft (I − D ), while, if underreporting is not detected, income IN is I − tD. The individual chooses D to maximize the expected utility EU(I) of the evasion gamble, or EU (I) = pU (IC ) + (1 − p)U (IN ), where utility U (I) is assumed to be a function only of income. This optimization generates the first-order condition pU  (IC )(f − 1)t − (1 − p)U  (IN )t = 0, where a prime denotes a partial derivative. From this model, the implications of increasing enforcement (higher probability and/or fines) are easily seen. In making the compliance decision, an individual faces a portfolio choice problem, with evasion as the risky, high return asset and disclosure as the riskless asset. By reducing the return to the risky act (noncompliance) the policy maker obtains higher compliance.16 Now, consider typical designs for experiments intended to explore various dimensions of tax compliance behavior within the context of the above conditions. The output of interest is income. In earlier research, income is simply “given” to subjects in the experiment on the premise that how the income is earned will not affect compliance.17 More recently, compliance experiments have incorporated earnings tasks to enhance 16 As stated above, this is a very simple model. Experiments that make use of more complex, realistic, institutions have been conducted by many researchers. As examples of experiments in which audit probabilities are endogenous, see Alm et al. (1993a,b); issues related to the uncertainty of enforcement are treated in Alm et al. (1992a); issues related to communication of enforcement efforts are reported by Alm et al. (2009). 17 We must recognize the potential for a parallelism problem in this instance, however. This problem is described as the found money problem which may arise in instances where subject decisions involving money given to them as a part of the experiment are different from decisions that they might make which involve average levels of their discretionary income.

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parallelism. This also emphasizes the opportunity cost of disclosure and brings such experimental settings closer to one appropriate for environmental regulation compliance investigations. Second, the methods for auditing declared income, and therefore declared taxes, are reasonably straightforward. In the experiment, a subject = s “real” income is known in the event of an audit. In the field, the tax authorities have access to an audit process designed to elicit information regarding an agent’s true tax liabilities. Third, in the field, there are millions of taxpayers, data from which are used to develop taxpayer profiles that are used to define target conditions for selecting who is to be audited. In the theoretical development above, the taxpayers are presumed to be aware of baseline probabilities of an audit and, at least in general terms, of limits to reporting beyond which the probability of an audit increases. This awareness of the chance of an audit is “paralleled” in the tax compliance experiment by simply telling subjects the probability that their report will be audited. More recently, there has been work examining the means by which taxpayers may learn of audit probabilities: formal announcements of the audit probability by the authority, communication among agents being audited, and reports of past audit activity by the enforcement agency (Alm et al., 2009). Finally, in the main, the incentives structure that drives the compliance decision is utility maximization. Imbedded in this argument are considerations related to such things as risk aversion. We acknowledge the potential for compliance incentives other than income. For example, the relevance of an agent = s perception of how taxes are to be used has been demonstrated in experiments reported by Alm et al. (1993a,b). While, as demonstrated above, parallelism between laboratory and field institutions can (arguably) be reasonably attained for tax compliance experiments, this setting obviates the problems that can attend efforts to adapt these designs to issues related to compliance with environmental regulations. Consider the following as examples of some of the more important issues. First, the production process by which inputs result in the output “emissions” is often very complex and, perhaps most importantly, data relevant for this process may be difficult or impossible to obtain. For experiments that meaningfully parallel the field, it will generally be necessary to include the process in which agents can directly affect emissions by changes in inputs and/or technologies. Even if they are somehow simplified, the cognitive tasks required by subjects may be excessive. Second, in the field, auditing processes exist for large, point-source emitters such as power plants — firms are required to maintain emissions measurement equipment,

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results from which must be reported to the environmental manager. For smaller point-source emitters, and for the wide range of emissions that come from non-point sources, meaningful auditing processes may be problematic (Segerson and Tietenberg, 1992). Until measurement mechanisms for these classes of emissions are put into place, the role of compliance experiments will be very limited. Next, in contrast to the tax compliance case where data from millions of agents are available for the purpose of developing signals to drive audits (and therefore the probability of an audit), the number of agents relevant for environmental compliance will be smaller — very much smaller. The recent work by Evans et al. (2008) has focused on the use of the structural characteristics of the firm, its internal organization, and incentive structure, to target the more likely offenders. Finally, while sanctions related to many environmental regulations will typically be applied at the level of the firm, a firm = s compliance with rules and regulations may be given parallel treatment in the laboratory as being analogous to individual compliance with tax laws. The plausibility of this treatment, in terms of parallelism, relies on the extent that we (reasonably) expect that sanctions would have implications for promotion and/or job security for the individual in the firm with responsibility for making compliance decisions. These considerations may then rationalize experiments in which the compliance decision is made operational at an individual level. The challenges to our ability to design meaningful “parallel” experiments for the exploration of environmental compliance policies just described are certainly formidable. Some progress has been made since Cummings et al. (2001) raised these questions. This is discussed below. The potential payoffs for meeting these challenges, in terms of advancements in our ability to offer policy insights, are substantial. A place to begin this process is suggested by lessons learned from tax compliance experiments. Such experiments make clear the critical importance of effective audits for high compliance rates. More specifically, for example, we know that: • more frequent audits encourage greater compliance, although the impact of a greater probability of detection is small and non-linear. This suggests that enforcement efforts be directed in a systematic manner rather than simply increasing the frequency of inspection; and • related to the above, conditional audits produce higher compliance rates than purely random audits.

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It is the use of systematic audits that motivates the work of Evans et al. (2008). Faced with a budget constraint, the environmental enforcement agency will wish to concentrate audit resources on firms likely to be noncompliant. Evans et al. (2008) show that compensation schemes based on rank order tournaments are more likely to lead to malfeasance if such behavior increases the likelihood of winning the tournament. This suggests that firms comprised of multiple divisions in which the division leaders compete for prizes (promotions and/or bonuses) are more likely to cheat on regulations that increase the division costs, reducing the profitability of the division, and the likelihood of the division leader winning the tournament. The closest regulatory structure to the voluntary disclosure of the income tax system would be the toxic release inventory (TRI) mandated under the Emergency Planning and Community Right to Know Act (1986) that requires that firms in certain standard industrial classification (SIC) categories provide annual statements of on- and off-site emissions (or waste generation) of a large number of toxic materials. The intent of the legislation would appear to be to provide the market (investors) with information that would allow them to form a conjecture as to the potential value of the firm. Firms with poor records of toxic releases would face downward pressure on the price of their securities and would have their access to capital compromised. In theory, it sounds like an appropriate use of the market to enforce environmental standards. Hamilton (1995) finds that stock prices do reflect announcements of releases. Firms clearly have an incentive to underreport releases. In the income tax system, for the detection of cheaters we rely on audits the effectiveness of which rely in turn on the existence of matching paperwork. The TRI program does not provide equivalent data that allows for a strict accounting of toxic materials. This is a materials-balance issue that can be seen in the following way. A firm receives as inputs a wide variety of materials that contain toxic materials. The question becomes: what happens to these toxic materials? At the end of the day, these toxic materials are: (i) embodied in the product produced by the firm; (ii) have been released to the environment or are contained in waste materials that are on hand or have been disposed of; and (iii) or remain in the firm = s inventory of input materials. Under the TRI program, we have no way of “auditing” the firm = s reports of (ii) because there are no provisions under which we can know (i) and (iii). One policy alternative is then immediately suggested which would result in the generation of “matching paperwork” akin to what we have with our tax system. Firms are required to account for the toxic content of

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materials that they receive (and for which paperwork exists) as inputs, and they must report the status or disposition of such materials as described in (i)–(iii). This information might or might not affect the likelihood of an audit; it would certainly affect the likelihood that an audit would detect unreported releases. Such a reporting system coupled with an effective audit regime would increase the likelihood that forms would report honestly and the financial markets could complete their role as mechanisms to punish malfeasant firms.18 Laboratory experiments could be used to explore and make explicit the impact of such a reporting scheme in the same manner as work on the incentives of division leaders under a tournament-based compensation mechanism (Evans et al., 2008) discussed above. One would begin with an experimental setting that parallels the present TRI setting: only releases are reported. Emissions releases (for which excess are fined at a given rate) would be observed. One would then have to establish an institution in which the broader range of reporting described above is required, and in which audits of reports occur with some probability. Reported emissions releases under this institution are observed. A comparison of the level of cheating (underreporting) in both settings would provide a measure of the efficacy of the proposed policy, and would demonstrate such efficacy in a way that (we assert) would be particularly compelling to decision makers.

10.5.

A “Success Story” for the Laboratory and Some Concluding Remarks

In this chapter, we have focused on the criteria necessary for economics laboratory experiments to inform the policy debate and in doing so we have emphasized the necessity of following design precepts that ensure the validity of the experiment for its intended purpose. We have addressed many of the critics of laboratory experiments and argued that many of the criticisms are defused by adherence to the precepts of good experimental design. We have described in some detail areas of environmental economics 18 We thank Robert Ayres who suggested this in private conversation regarding the TRI program. Ayres noted that it is not as effective as it might be due to the lack of a mass balance approach, which would require firms to report their intake and use of toxic materials. Anything unaccounted for would be deemed to be a release. Taxpayers are placed in exactly this setting under a self-reporting tax scheme as we have in the United States.

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and policy that are especially suitable for the application of experimental methods. These are valuation of environmental amenities/damages and compliance with environmental rules and regulations. With the move to the use of markets to serve as regulatory institutions, came the introduction of tradable emission permits. Since there were no existing field examples, this presented an opportunity to use the laboratory to testbed the trading institutions and, most importantly, identify potential unintended consequences. In principle, there is clearly nothing new about the idea of using market-based mechanisms to the ends of environmental management. Indeed, debate during the 1960s concerning the possible design for such mechanisms, and social implications that might be associated with their use, played an important role in the evolution of the subdiscipline of environmental economics.19 The character of this debate has changed markedly over the last three decades, however, in terms of both breadth and depth. The contemporary reliance on market-based policies by government and non-government entities is seen in many areas. Consider species protection as an example. Commonly used policies used by private and/or non-private institutions include such market-based mechanisms as the direct purchase of habitat lands,20 “debt-for-nature” swaps, and the use of “offsets” as a part of land-use management. The use of market-based mechanisms as a part of government policy is not a simple matter, however. As exemplified, to some extent, in the currently popular rhetoric concerning “privatization,” critical policy design issues arise from questions that are institutional in nature — questions concerning individual incentives associated with a given policy design, the likely policy outcomes and a desire to avoid unintended consequences. The importance of such issues derive from the fact that in many, if not most, instances the characteristics of the institutions within which government policies must operate do not satisfy fundamental institutional features of the market paradigm. This argument is made immediately apparent by considering the basic assumptions of the (competitive) market paradigm which include: many buyers and sellers, perfect information, perfect mobility (ergo, perfect entry and exit), and well defined property rights. 19 As early examples of this debate, see Kneese’s (1962) classic work along with works by Crocker (1966) and Dales (1968). 20 Issues associated with the U.S. Army’s efforts to obtain habitat lands in markets are explored with experimental methods by McKee and Berrens (2000).

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Consider the following examples of instances in which characteristics of field institutions conflict with those assumed in the market paradigm. Western water markets are argued21 as being good examples of the use of markets to address water-based environmental and resource-related management or allocation problems. In many cases, however, it can be argued that almost none22 of the preconditions for competitive markets are met in the local water markets. Typically there is a single large buyer, often a municipality, and few sellers. Transactions costs may be substantial and factor mobility is hardly free due to the costs of moving the water to other users or uses. A second example is provided by a major problem facing agencies such as the Departments of Energy and Defense in their efforts to “privatize” the clean-up of toxic materials on federal lands. The request for proposal (RfP) institution that they must use is typically characterized by a single buyer and (in many cases) only two or three sellers (bidders).23 A third example involves the legal and legislative actions affecting environmental and resource allocation on tribal lands, which were based on the market paradigm. These have been argued to have likely failed Fifth Amendment provisions for “just compensation” due to the demonstrable lack of “perfect mobility” in the institutional environment in which the actions were taken.24 Within this debate on the use of markets, the potential contribution of experiment economics is immediately obvious. Assessments of policies that rely on market-based mechanisms will require analyses of the effects on agent behavior (and therefore policy outcomes) that arise in different (vis-`a-vis the market paradigm) institutions and, of course, this is what experimental economics is all about. There is one set of issues related to market-based environmental management that we wish to single-out for more detailed treatment in this section, viz., what we have learned to date from experimental assessments of markets for emissions rights and the implications of such lessons for future experimental research. The rationale for this choice is the opportunity that we are afforded to provide the reader with a particularly illuminating example of the importance and contemporary relevance of what we described in Section II as the fundamental methods issue in experimental economics: parallelism.

21 See

Anderson (1983) and Gardner (1985). as examples, Brajer et al. (1989). 23 See Dummer et al. (1998). 24 Cummings (1991). 22 See,

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Most experimental analyses of emissions markets have focused on the provisions of 1990 amendments to the Clean Air Act for tradable discharge permits (TDPs) for sulfur dioxide (SO2 ). Essential elements of these provisions and their implementation by the EPA can be briefly summarized as follows; more comprehensive summaries of markets for SO2 permits can be found in Cason and Plott (1996). Each permit gave the owner (initially, large producers of thermal electricity) the right to emit 1 ton of SO2 . Once used, the permit was exhausted. Permits were allocated (in the early 1990s) on an historical basis; power producers were each awarded an initial quantity of permits based on their 1985 emissions levels. Permits, once allocated, were fungible and could be traded (under conditions described below).25 Permits are bankable. The program was intended to allow for a systematic reduction in the level emissions through a planned reduction in the number of permits. The EPA established a “market” for emissions permits the institutional design for which was proposed by Hahn and Noll (1982). The essential features of this institution are the following. (i) sealed bids and asks for emissions permits are submitted to the EPA; (ii) bids received by the EPA are arrayed from highest to lowest bid (forming a demand curve); asks are arrayed from lowest to highest ask (forming a supply curve); and (iii) feasible trades involve buyers and sellers that are matched by the EPA. The highest bid is matched with the lowest ask for the first trade. The second-highest bid is then matched with the second-lowest ask for the second trade, and so on until feasible trades have been exhausted. Several policy questions concerning the EPA’s adopted market institution immediately arose, such as, will market power of participants be important; will there be sufficient trading in this newly introduced market for the program to accomplish its intended purpose (provide efficient prices); what is the effect of heterogeneity of production technologies on the volume of permit trades and the pricing? Interest in these questions concerning the efficacy of the EPA auction became escalated following experience with its initial operation in 1993. Relative to preauction expectation, the volume of trades in the auction was very small, and trading prices were much lower than expected. These 25 The EPA was concerned that there would be too little trading initially and mandated that each producer place 3% of their permit allocation in the EPA auction.

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anomalies presented an ideal invitation for the use of experiments to explain them, an invitation that was quickly taken up by a number of researchers. A substantial literature evolved in which results from emissions trading experiments were reported. Muller and Mestleman (1998) summarize this literature and pose the question: what have we learned from reported experimental research concerning emissions trading? Concern here is limited to Muller and Mestleman’s (1998) responses to this question that relate to the basic recurring theme of this chapter: responses that center on the issue of parallelism.26 The parallelism issue is most prominently seen in Cason and Plott’s (1996) effort to explain the anomalies seen in the 1993 EPA auction. The design of their experiments closely followed the Hahn–Noll institution.27 Cason and Plott’s (1996) results showed that under this institution the incentives of both buyers and sellers were to underreveal their true values, the results being a substantial reduction in the number of trades (relative to trades that would occur with an incentivecompatible mechanism) and permit prices that were “low” relative to a socially optimal price.28 Since these “results” are basically those seen in the 1993 EPA auction, Cason and Plott (1996) conclude that the EPA auction is not demand revealing and, consequently, that it provides incorrect information concerning abatement costs (the “anomalous” results observed in 1993). Of course, one of the basic ends sought with the use of an emissions market is price information — an approximation of competitive, market clearing prices — information that would be expected from a market institution that is incentive compatible. Such price information would then be expected to result in socially optimal levels of investment in pollution abatement technology and then efficient, least-cost compliance with standards established by the Clean Air Act. All else equal, Cason and Plott’s (1996) results suggests a failure in the EPA’s market and the attendant need to redesign the auction institution. 26 We thus ignore a line of experimental research that may be of interest to many readers: experiments focused on issues related to the potential affects of market power. See, as examples, Misiolek and Elder (1989), Brown-Kruse and Elliott (1990), and Brown-Kruse et al. (1995). 27 In Cason and Plott’s (1996) experiments, permits traded in the market had capital value and were bankable. 28 See Franciosi et al. (1993). It should be noted that these authors do find some degree of overbidding on inframarginal units.

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It turns out, however, that all else may not be equal. Muller and Mestleman (1998) note the EPA auction was not the only operating market in which one could trade emissions permits: at the Chicago Board of Trade, spot and futures markets for emissions permits had developed and constituted a substantial part of the total market for permits. Since the existence of this secondary market for permits was not a part of the institution used by Cason and Plott, an obvious parallelism question arises: what is the relevance of results from experiments in which subjects trade in a Hahn– Noll market institution for a “real world” field institution wherein agents may trade simultaneously in the (EPA) Hahn–Noll market and a secondary market in the Chicago Board of Trade?29 Intuition might suggest that the existence of the secondary market would affect the Hahn–Noll market in ways that are consistent with behavior witnessed in 1993: prices established in the secondary market would surely limit if not fix prices at which permits would trade in the EPA market; the source of supply (asks) for permits in the EPA market may have been substantially limited to those offered by the EPA (the only agent not operating in the secondary market), in which case the operation of the EPA market may have had little more effect that giving rise to a increase in the total (EPA and secondary) market’s supply of permits, thereby driving down permit prices. The incentive compatibility of the Hahn–Noll, and therefore now the EPA, market institution remains an open question, as does the weight of the parallelism critique of Cason and Plott’s study (Joskow et al., 1998). Resolution of these issues will require experimental explorations which incorporate multiple markets and interactions between technology choice and the market for emissions permits (see Ben-David et al., 1998). This issue serves to highlight the relevancy of parallelism as well as pointing to potentially productive lines of future research. An important set of recent criticisms of laboratory experiments has been promulgated by the proponents of field experiments as an alternative. As we have argued earlier, many of these criticisms are addressed by simply following the precepts of experimental construction enumerated

29 Still another source for a potentially important parallelism question related to Cason and Plott’s design concerns the initial allocation of permits to agents. Given that initial allocations were based on 1985 emissions levels, and that most power companies had necessarily reduced emissions between 1985 and the early 1990s at which time emission permits were allocated, the initial allocations may have been high, thereby leading to prices that were lower than expected. This subtlety is not captured in Cason and Plott’s experiments.

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by Smith (1982). However, for policy purposes, field experiments of the sorts discussed by List and Levitt (2007) are patently unsuitable for policy investigations since these experiments typically involve only rudimentary decision tasks with little or no context that would ensure parallelism with a policy relevant field setting. Many of the field experiments cited by List and Levitt investigate simple trading behavior; for example, they report on behavior in baseball trading card markets. While the underlying trading behavior has implications for many market settings, the policy relevance is not readily apparent. Other settings involve simple group decision making exercises with similarly limited policy angles. The inherent problem with such field experiments is the limited scope for institution construction within such naturally occurring settings as are available. Our position is that for the foreseeable future, policy investigations will require the controlled setting that is available in the laboratory. Here, the potential for experiments to inform policy discussions in very high indeed. References Alm, J, B Jackson and M McKee (1992a). Institutional uncertainty and taxpayer compliance. American Economic Review, 82(4), 1018–1026. Alm, J, B Jackson and M McKee (1992b). Estimating the determinants of taxpayer compliance with experimental data. National Tax Journal, 65(1), 107–114. Alm, J, B Jackson and M McKee (1993a). Fiscal exchange, collective decision institutions, and tax compliance. Journal of Economic Behavior and Organization, 22(3), 285–303. Alm, J, B Jackson and M McKee (2009). Getting the word out: Enforcement information dissemination and tax compliance behavior. Journal of Pubilc Economic, 93(3–4), 392–402. Alm, J, GH McClelland and WD Schulze (1992c). Why do people pay taxes? Journal of Public Economics, 48(1), 21–38. Alm, J, M Cronshaw and M McKee (1993b). Tax compliance with endogenous audit selection rules. Kyklos, 46(1), 27–45. Anderson, T (1983). Conflict or cooperation: The case for water markets, Unpublished MSS, Political Economy Research Center, Montana State University, Bozeman. Arrow, KJ (1987). Rationality of self and others in an economic system. In The Constrast between Economics and Psychology, R Hogarth and M Reder (eds.), Chicago: University of Chicago Press. Ben-David, S, D Brookshire, S Burness, M McKee and C Schmidt (1998). Heterogeneity, irreversible production choices and efficiency in emission permit markets. Journal of Environmental Economics and Management, 38(2), 176–194.

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Bjornstad, D, R Cummings and L Osborne (1997). A learning design for reducing hypothetical bias in the contingent valuation method. Environmental and Resource Economics, 10(3), 207–211. Blackburn, M, GW Harrison and EE Rutstr¨ om (1994). Statistical bias functions and informative hypothetical surveys. American Journal of Agricultural Economics, 76(5), 1084–1088. Bohara, A, M McKee, R Berrens, H Jenkins-Smith, C Silvia and D Brookshine (1998). The effects of total cost and group-size information on stated WTP: Open ended vs. dichotomous choice. Journal of Environmental Economics and Management, 35(2), 142–163. Brajer, V, A Church, R Cummings and P Farah (1989). The strengths and weaknesses of water markets as they affect water scarcity and sovereignty interests in the west. Natural Resources Journal, Spring, 489–509. Brown, TC, P Champ and RC Bishop (1996). Response formats and public good donations. Land Economics, 72(1), 152–166. Brown-Kruse, J and SR Elliott (1990). Strategic manipulation of pollution permit markets, manuscript, Laboratory for Economics and Psychology, University of Colorado, Boulder, CO. Brown-Kruse, J, SR Elliott and R Godby (1995). Strategic manipulation of pollution permit markets: An experimental approach, Working Paper 95-10, Department of Economics, McMaster University, Hamilton, Canada. Carson, RT, T Groves and MJ Machina (2000). Incentive and information properties of preference questions, Working Paper, Department of Economics, University of California, San Diego. Cason, T and C Plott (1996). EPA = s new emissions trading mechanism: A laboratory evaluation. Journal of Environmental Economics and Management, 30(2), 133–160. Cherry, TL, TD Crocker and JF Shogren (2003). Rationality spillovers. Journal of Environmental Economics and Management, 45, 63–84. Cherry, TL and JF Shogren (2007). Rationality crossovers. Journal of Economic Psychology, 28, 261–277. Cherry, TL, P Frykblom and JF Shogren (2004). Laboratory testbeds and nonmarket valuation: The case of bidding behavior in a second-price auction with an outside option. Environmental and Resource Economics, 29, 285–294. Chu, YP and RL Chu (1990). The subsistence of preference reversals in simplified and marketlike experimental settings. American Economic Review, 80, 902–911. Collins, J and C Vossler (2009). Incentive compatibility tests of choice experiment value elicitation questions. Journal of Environmental Economics and Management, 58(2), 226–235. Crocker, T (1966). The structuring of atmospheric pollution control systems. In The Economics of Air Pollution, H Wolozing (ed.), pp. 61–86. New York: W.W. Norton. Cummings, RG (1991). Legal and administrative uses of economic paradigms: A critique. Natural Resources Journal, 31(2), 463–473.

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Cummings, RG and LO Taylor (1999). Unbiased value estimates for environmental goods: A cheap talk design for the contingent valuation method. American Economic Review, 89(3), 649–655. Cummings, RG, S Elliott, GW Harrison and J Murphy (1997). Are hypothetical referenda incentive compatible? Journal of Political Economy, 105(3), 609–621. Cummings, RG, M McKee and LO Taylor (2001). To whisper in the ears of princes: Experimental economics and environmental policy. In Frontiers of Environmental Economics, L Gabel, H Folmer, S Gerking and A Rose (eds.), pp. 121–147. Cheltenham, UK: Edward Elgar Publishers. Dales, J (1968). Pollution, Property and Prices. Toronto, Canada: University of Toronto Press. Davis, DD and CA Holt (1993). Experimental Economics. Princeton, NJ: Princeton University Press. Diamond, P and J Hausman (1994). Contingent valuation: Is some number better than none?. Journal of Economic Perspectives, 8(4), 45–64. D¨ ummer, C, D Bjornstad and D Jones (1998). The regulatory environment guiding DOE=s Cleanup: Opportunities for flexibility, draft ms. (26 pp), The Joint Institute for Energy and Environment, Knoxville, TN, September 3. Dyer, D, J Kagel and D Levin (1989). A comparison of naive and experienced bidders in common value offer auctions: A laboratory analysis. Economic Journal, 99(1), 108–115. Evans, D (1997). The role of markets in reducing expected utility violations. Journal of Political Economy, 105, 622–636. Evans, MF, SM Gilpatric and M McKee (2008). Managerial incentives for compliance with environmental information disclosure programs. In Experimental Methods in Environmental Economics, T Cherry, S Kroll and J Shrogen (eds.), pp. 243–260. New York: Routledge Press. Fox, J, J Shogren, D Hayes and J Kleibenstein (1998). CVM-X: Calibrating contingent values with experimental auction markets. American Journal of Agricultural Economics, 80(3), 455–465. Franciosi, R, R Issac and D Pingry (1993). An experimental investigation of the Hahn–Noll revenue neutral auction for emissions licenses. Journal of Environmental Economics and Management, 24(1), 1–24. Gabel, HL and B Sinclair-Desagne (1993). Managerial incentives and environmental compliance. Journal of Environmental Economics and Management, 24(2), 229–240. Gardner, D (1985). Institutional impediments to efficient water allocation. Policy Studies Review, 5(3), 353–363. Hahn, R and R Noll (1982). Designing a market for tradable emissions permits. In Reform of Environmental Regulation, WA Magat (ed.), pp. 119–146. Cambridge, MA: Ballinger. Hamilton, JT (1995). Pollution as news: Media and stock market reactions to the toxics release inventory data. Journal of Environmental Economics and Management, 28(1), 98–113.

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Hanemann, WM (1994). Valuing the environment through contingent valuation. Journal of Economic Perspectives, 8(4), 19–44. Harrison, GW and J List (2004). Field experiments. Journal of Economic Literature, XLII(4), 1009–1055. Harrison, GW, RW Harstad and EE Rutstrom (1995). Experimental methods and the elicitation of values, Working Paper 95–11, Department of Economics, University of South Carolina. Joskow, P, R Schmalensee and E Bailey (1998). The market for sulfur dioxide emissions. American Economic Review, 88(5), 669–686. Kahneman, D, J Knetsch and RH Thaler (1990). Experimental tests of the endowment effect and the Coase theorem. Journal of Political Economy, 98, 1325–1348. Kneese, AV (1962). Water Pollution: Economic Aspects and Research Needs. Washington, D.C.: Resources for the Future, Inc. List, J and S Levitt (2007). What do laboratory experiments measuring social preferences tell us about the real world? Journal of Economic Perspectives, 21(2), 153–174. Loomis, J, A Gonzalez-Caban and R Gregory (1994). Substitutes and budget constraints in contingent valuation. Land Economics, 70(4), 499–506. Louviere, J (1996). Relating stated preference measures and models to choices in real markets: Calibration of CV responses. In The Contingent Valuation of Environmental Resources: Methodological Issues and Research Needs, DJ Bjornstad and JR Kahn (eds.), pp. 167–188. Cheltenham, UK and Brookfield, US: Edward Elgar Publishing Ltd. McEvoy, D (2010). Not it: Opting out of voluntary coalitions that provide a public good. Public Choice, 142(1), 9–23. McKee, M and R Berrens (2000). Balancing army and endangered species concerns: Green vs green. Environmental Management, 27, 123–134. Misiolek, WS and HW Elder (1989). Exclusionary manipulation of the market for pollution rights. Journal of Environmental Economics and Management, 16, 156–166. Murphy, JJ and JK Stranland (2006). Direct and market effects of enforcing emissions trading programs: An experimental analysis. Journal of Economic Behavior and Organisation, 61(2), 217–233. Muller, RA and S Mestelman (1998). What have we learned from emissions trading experiments? Managerial and Decision Economics, 19(4/5), 225–238. Neill, H (1995). The context for substitutes in CVM studies: Some empirical observations. Journal of Environmental Economics and Management, 29(3), 393–397. Neill, HR, RG Cummings and PT Ganderton (1994). Hypothetical surveys and real economic commitments. Land Economics, 70, 145–154. Pagan, J (1998). IRCA = s employer sanctions: An experiment on firm compliance and the implementation of a national ID. Journal of Economic Behavior and Organization, 34(1), 87–100.

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Parkhurst, GM and JF Shogren (2008). Smart subsidies for conservation. American Journal of Agricultural Economics, 90(5), 1192–1200. Plott, CR (1982). Industrial organization theory and experimental economics. Journal of Economic Literature, 20(4), 1485–1527. Plott, CR (1987). Dimensions of parallelism: Some policy applications of experimental methods. In Laboratory Experimentation in Economics: Six Points of View, AE Roth (ed.), pp. 193–219. New York, NY: Cambridge University Press. Portney, P (1994). The contingent valuation debate: Why economists should care. Journal of Economic Perspectives, 8(4), 3–17. Segerson, K and T Tietenberg (1992). The structure of penalties in environmental enforcement: An economic analysis. Journal of Environmental Economics and Management, 23(3), 179–200. Settle, C, TL Cherry and JF Shogren (2008). Rationality spillovers in yellowstone. In Environmental Economics, Experimental Methods, Cherry, Kroll and Shogren (eds.), Routledge UK. Shogren, J, SY Shin and DJ Hayes (1994). Resolving differences in willingness to pay and willingness to accept. American Economic Review, 84(3), 255–270. Smith, VK and C Mansfield (1998). Buying time: Real and hypothetical offers. Journal of Environmental Economics and Management, 36(2), 209–224. Smith, VL (1982). Microeconomic systems as an experimental science. American Economic Review, 72(4), 923–955. Smith, VL (1991). Rational choice: The contrast between economics and psychology. Journal of Political Economy, 99, 877–987. Smith, VL, G Suchanek and A Williams (1988). Bubbles, crashes, and endogenous expectations in experimental spot asset markets. Econometrica, 56(6), 1119–1151. Taylor, LO (1998). Incentive compatible referenda and the valuation of public goods. Agricultural and Resource Economics Review, October, 132–139. Taylor, LO, M McKee and S Laury (2001). Induced value tests of the referendum voting mechanism. Economics Letters, 71, April, 61–65. Tversky, A and D Kahneman (1974). Judgement under uncertainty: Heuristics and biases. Science, 185, 1124–1131. Vossler, C and M McKee (2006). Induced value tests of elicitation mechanisms for contingent valuation surveys. Environmental and Resource Economics, 36, 137–168. Wartick, M, S Madeo and C Vines (1999). Reward dominance in tax reporting experiments: The role of context. Journal of the American Taxation Association, 21(1), 20–31.

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Interdisciplinary Tools

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Chapter 11

Modeling the Economics of Ecosystem Services at Watershed Scale: A Spatial Model of Land Use Externalities and the Regulating Functions of Wetlands Silvio Simonit∗ and Charles Perrings† Arizona State University ∗ [email protected] † [email protected]

11.1.

The Problem to be Modeled

The regulating services are among the least understood but potentially most valuable services offered by ecosystems (MA, 2005). Among the most important examples of the regulating services described by the Millennium Assessment are those affecting water quality. In this context, wetlands play an especially important role through their absorption of nutrients. The services regulated in this way include sewage treatment, freshwater provision, and fiber (reed) production. But the main value of nutrient absorption by wetlands lies in the contribution it makes to the regulation of fish supplies. Inland fisheries — both capture fisheries and aquaculture — are of particular significance in developing countries, where they are often a primary source of protein for rural communities. While aquaculture

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production is rapidly increasing, capture fisheries are in decline worldwide. Wetlands have an important role in both, but the modeling problem addressed in this chapter relates to capture fisheries. There are two main reasons for the decline in capture fisheries. One is the fact that many fisheries are still effectively open access resources, which results in their overexploitation. The other is the effect of land-based pollution. Many freshwater and marine capture fisheries have been severely affected by sewage, nutrients, synthetic organic compounds, sediments, metals, radionuclides, oil/hydrocarbons, and polycyclic aromatic hydrocarbons (PAHs) (UN, 2004). The application of fertilizers in agriculture, fossil fuel burning, land clearance, and biomass burning are a major source of the nutrient load in freshwater, coastal, and estuarine systems (Oglesby, 1977; Nixon, 1988). Wetlands are one of the few natural buffers against the impact of land-based nutrient flows. In this chapter, we model the role of wetlands in mediating the negative externality generated by land-based activities on freshwater capture fisheries. We focus on water purification and waste treatment services provided through the effects of the biota on water pollution and filtration in inland waters. Regulating services of this kind have been interpreted as providing “insurance” — not by covering the financial consequences of loss, but by enabling the fishery to persist over a range of conditions (Griffin et al., 2009; Loreau et al., 2002). The value of the regulating services derives from the benefits they protect. In some instances, these benefits may be approximated by commodity prices. If the markets for commodities are reasonably complete, if property rights to all of the effects involved in their production and consumption are well defined, if the parties involved in a transaction have full information, and if wealth effects are small, then the prices at which commodities trade may be a good approximation of their social opportunity cost — their value to society. More often, the protected benefits are not captured in market prices. Markets for many regulating services simply do not exist and such market prices as do exist are unreliable indicators of value. In these circumstances, it is necessary to use other methods for the valuation of non-marketed ecosystem services (see Goulder and Kennedy, 1997; Barbier, 2007; Barbier et al., 2009; Heal et al., 2005; Pagiola et al., 2004; Smith and Pattanayak, 2002). Effective regulating services are the key to the sustainable use of ecosystems. A frequently cited measure of sustainability is the capacity of

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an ecosystem to continue to function over a range of conditions — generally approximated by the resilience of the system (the size of the shock needed to dislodge it from one state to another) (Kinzig et al., 2006; Walker and Meyers, 2004). Since the transition from one state to another generally occurs as a result of some variable or variables exceeding a threshold value, the value of the regulating services provided by any ecosystem lies in its capacity to reduce the likelihood that will happen. Considerable work has been done to model the various characteristics of wetlands by hydrologists, ecologists, fishery scientists, and limnologists among others (Lynne et al., 1981; Turner, 1982; Reckhow and Quian, 1994; Swallow, 1994; Carpenter and Cottingham, 1997; Carpenter et al., 1999; Janssen and Carpenter, 1999; Mitsch and Wang, 2000; Janssen, 2001; Trepel and Palmeri, 2002). In this chapter, we focus on the capacity of wetlands to reduce the likelihood that land-based nutrient flows will lead to the eutrophication of oligotrophic lake systems. We are less concerned with the role of wetlands as fish spawning grounds or nurseries for fry (Swallow, 1994; Barbier and Strand, 1997; Barbier and Sathirathai, 2002), than with their impact on the dynamics of a fishery affected by nutrient loads in open waters. In very many cases, changes in aquatic systems that follow from changes in nutrient loading due to land-based economic activities, are an externality of those activities. People whose activities on land damage aquatic systems ignore the consequences of their actions because there are no markets for the services provided by these systems. While it is recognized that internalization of such externalities requires an understanding of the interactions between terrestrial and aquatic activities, there are few attempts to model the problem. Early approaches either focused on simple correlations between changes in watersheds and changes in fisheries, or else identified the consequences for fisheries if the linkages were of varying strength (Ruitenbeek, 1989; Hodgson and Dickson, 1998). Limnologists have investigated the consequences for freshwater aquatic systems of changes in land use, vegetative cover, and fertilizer regimes within the watershed (Postel and Carpenter, 1997; Carpenter and Pace, 1997; Carpenter and Cottingham, 1997). More recently, economists have considered the theoretical problems posed by the interaction between users of lake and catchment resources (M¨ aler et al., 2003; Carpenter et al., 1999). Nevertheless, there are few attempts to model the interactions in real systems, or to estimate the value of land–water externalities of this kind.

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This study models land–water interactions in a particular catchment discharging into the Kenyan part of Lake Victoria. We aim to understand the consequences of the conversion of wetlands for the lake fishery. Two processes are at work. Nutrient enrichment has a positive effect on fishery productivity in nutrient-limited environments such as oligotrophic or mesotrophic lakes (Stockner and Shortreed, 1988; Melack, 1976a; Liang et al., 1981; Hoyer and Jones, 1983; Downing et al., 1990; Quir´ os, 1990; Gomes et al., 2002). However, there is also evidence that sustainable harvests of fish populations at upper trophic levels decline if the system becomes highly eutrophic (Beeton, 1969; Lee et al., 1991; Caddy, 1993). Excess nutrients affect fish productivity through changes both in the amount of food available (Bootsma and Hecky, 1993) and in the quality of the habitat (Hammer et al., 1993). Deoxygenated water boosts natural mortality of fish. Sedimentation negatively affects nursery grounds and may damage fish eggs. When combined with high fishing pressure, both effects can have a severe impact on fish stock biomass and fishery yields (Kemp et al., 2001). In Lake Victoria, fish production in all three riparian states has grown dramatically since the introduction of the Nile perch (Lates niloticus) in the early 1960s. In the Kenyan waters of the lake, output grew from around 17,000 tonnes per year in the 1960s to more than 200,000 tonnes in the early 1990s. During the 1980s, Lates catches increased exponentially rising in few years from virtually zero to almost 60% of total yield (Okemwa, 1984; CIFA, 1988; Ogari and Asila, 1990; Ogutu-Ohwayo, 1990; Ssentongo and Welcomme, 1985). However, from 1994 fish landings have been in sharp decline, mostly due to declining catches of Nile perch. By 1998, Nile perch landings were half of those at the beginning of the decade despite increased fishing effort. One factor in this has been eutrophication, caused by nutrient runoff from agricultural land, and from urban effluent on the lakeshore. It stems from two things. Nutrient loading from agricultural land stems from livestock and the application of fertilizers. However, there has also been a reduction in nutrient absorption by wetlands on the lake margins caused by the conversion of wetlands to other uses. A number of water quality analyses over the last decades show that Lake Victoria has progressively shifted from a mesotrophic to a eutrophic state. Increasing chlorophyll-a concentrations have been reported (Ochumba and Kibaara, 1989; Gophen et al., 1995; Kenyanya, 1999; Lung’ayia et al., 2000,

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2001) against baseline values provided by Talling in the 1960s (Talling, 1965; 1966) and Melack (1976b). A critical feature of the problem is that the value of the regulating function of wetlands depends both on landscape characteristics at the source of pollution, and resource use at the sink. Moreover, since each land user in a watershed produces a different level of externality depending on the physical characteristics of the land, we need to estimate the spatial distribution of pollution externalities in order to estimate the value of wetland functions in terms of the fishery it protects. One benefit of this is that it becomes possible to tailor solutions to the characteristics of the watershed.

11.2.

Model Specification

We model the relation between land use change, wetland area, water quality, and fish stock biomass. Although lack of long-term monitoring data make it difficult to calibrate a land use/water quality function on data for the particular system involved (Verschuren et al., 2002), we do have sufficient information on system structure and dynamics to be able to calibrate a model using data from a range of closely allied systems. The benefits protected by wetlands in this case are the downstream economic activities — a fishery, the value of which depends on water quality via its impact on fish biomass productivity. Nutrient load in the lake is influenced by both the extent and intensity of upstream agricultural and the buffering function of the wetlands. Since the effects of upstream activities are not reflected in market prices they are ignored by those responsible. That is, any costs/benefits imposed/conferred on other economic sectors are ignored by upstream land users. To estimate the impact of such externalities on the regulating function of wetlands, we model the interdependence between activities that affect or are affected by the characteristics of wetlands. Land allocation is described as follows: the watershed is defined as the sum of drainage land and wetlands, and its extent, in hectares, is given by L0 . Formally, the land is assumed to be allocated between two uses: agriculture, LA , and wetland conservation, LW . Land uses are assumed to be determined at the spatial unit of 1 ha. For each ith hectare within agriculture, land is either fallow, LFi , or in production, LCi , or both. Farming households manage soil productivity through rotation between tillage and fallow, while preserving the chosen level of agricultural intensity, β. This

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determines the ratio of LF to LC . It follows that L0 = LA + LW =

αL
0

[(1 − βi ) + βi ] + (1 − α)L0

(1)

i=1

with 0 ≤ α ≤ 1;

0 ≤ βi ≤ 1

where α is the proportion of the watershed in agriculture, such that LA = αL0 and LW = (1 − α)L0 ; and βi is the proportion of the ith hectare of agricultural land which is in “fallow”, such that LFi = βi and LCi = 1−βi . In other words, the choice over the share of land allocated to wetland conservation is expressed by α, while the choice of agricultural intensity is expressed by β. Land use externalities are modeled by introducing a nutrient runoff function for total phosphorus. This function describes nutrient flows from the catchment into the wetland. Phosphorus, unlike nitrogen, is not subject to leaching. In tropical soils, phosphorus is tightly bound by soil particles and runoff is represented almost entirely by soil erosion (Roy and Misra, 2003). The general form of the nutrient runoff function is as follows gt =

αL
0

git (Qit , Pit , βit , Ωi )

(2)

i=1

where phosphorus loading, gt , into the wetland is expressed in t P yr−1 and represents the sum of the loading from each hectare. Qit is a measure of soil phosphorus content in the top 20 cm of soil and expressed in kg P ha−1 ; Pit is phosphorus fertilizer application (kg P ha−1 yr−1 ); and Ωi is a long-term average for rainfall (mm yr−1 ). The specific form of the function estimated is αL
0 Si Ait (βit , Ωi )δθQit (3) gt = 10−3 i=1

Where αt and βit are the already defined indices of land allocation in the catchment, and production intensity for the ith hectare of agricultural land, respectively; Ait is a measure of soil erosion (t ha−1 yr−1 ); δ is a soil phosphorus enrichment factor; θ is the soil bulk density factor; 0 ≤ Si ≤ 1 is the sediment delivery ratio (accounting for sediment deposition within the basin). Soil phosphorus content depends on nutrient flows, crop production decisions, and soil dynamics. The general form of the soil phosphorus function is ∆Qit = Qi (Qit , Pit , βit , Ωi )

(4)

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and the specific form estimated here is Qit+1 = Qit + Pit (1 − βit ) + Φi (Ωi ) + χβit − ηyit (Qit , Pit , βit ) − Si Ait (βit )δθQit

(5)

yit = 10−3 (1 − βit )(¯ a + Pit (1 − ¯bPit ) + c¯Qit )

(6)

where

is the agricultural production function, with coefficients a¯, ¯b, and c¯. The variation in Qi is expressed in kg P ha−1 ; yit is crop productivity (t ha−1 yr−1 ) for the ith hectare; χ represents the nutrient build-up factor due to fallow rotation (kg P ha−1 yr−1 ); η expresses the phosphorus content in the harvested crops (kg P t−1 ); Φi is atmospheric deposition of phosphorus (kg P ha−1 yr−1 ) on the ith hectare; and the other parameters are defined as above. The soil dynamics in Eq. (5) represent a simplified nutrient balance approach (Smaling and Fresco, 1993; Smaling et al., 1996) where the initial natural stock of phosphorus concentration in the soil is increased by fertilizer application on tillage land (Lc ), nutrient build up due to fallow rotation, and atmospheric deposition. The nutrient stock is diminished by losses both through harvesting and soil erosion. Crop harvesting is accordingly an important component of nutrient dynamics. The regulating function of wetlands is modeled through nutrient retention. We follow those studies that have analyzed nutrient retention by wetlands as a function of wetland area and nutrient loads (Bystr¨ om, 1998; Dortch and Gerald, 1995). We adopt the general mass balance model for phosphorus presented by Kadlec and Knight (1996), with the choice of α, 0 ≤ α ≤ 1, fixing the extent of the wetland. The specific form of the function applied here is ∗

4

z(gt , αt ) = gt w(αt ) = gt e(−k(1−αt )L0 10

)/v

(7)

where zt is the nutrient outflow from the wetland to the main water body (t P yr−1 ); gt is nutrient inflow to the wetland (t P yr−1 ) and is determined by the nutrient runoff function (3); w(·) is nutrient retention per unit of nutrient loading, a function of wetland area; v is water inflow to the wetland (m3 ); and k is the constant areal removal rate (m yr−1 ) for phosphorus. Nutrient retention is the minimum rate achieved at the given wetland area. In other words, z(gt , αt ) is the maximum nutrient outflow to the water body associated with the given value of α.

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The impact of nutrient loading on the water body is modeled through the direct effects it has on the productivity of the lake fishery. Specifically, we focus on the direct effect on the growth of the fish population associated with the maximum nutrient outflows from the wetland. A change from oligotrophic to eutrophic status in the lake will be observed as a rapid decline in the productivity of the fishery. To capture this effect, we construct a bioeconomic model of the fishery that explicitly includes the impact of water quality. This is an approach that has been adopted in a number of studies (Ikeda and Yokoi, 1980; Fr´eon et al., 1993; Simonit and Perrings, 2005; Kasulo and Perrings, 2006). Following Simonit and Perrings (2005), we include a damage function that depends on nutrient loading from the watershed, D(zt ). The equation for fish stock dynamics in the presence of nutrient flows and harvesting is ∆X = X(Xt , D(zt )) −

n


h(Xt , Ejt )

(8)

j=1

which takes the specific form  
n Xt qE jt Xt ∆X = rX t D(zt ) − − K j=1

(9)

where Xt represents the fish stock biomass; D(zt ) ≤ 1 is the damage function in terms of nutrient loading zt ; X(·) and h(·) are the growth and harvesting functions, respectively; and r, K, q, and Et are the usual Gordon– Schaefer parameters (Gordon, 1954; Schaefer, 1954; 1957). More particularly, the damage function is represented by a negative quadratic function of chlorophyll-a concentration: D(zt ) = W (zt ) − cW 2 (zt )

(10)

where Wt is chlorophyll-a concentration (mg m−3 ) and c is an estimated coefficient. We accordingly assume that nutrient loading positively affects the growth of fish stocks up to a certain point, after which further increases in nutrient loading cause exponentially increasing losses. The environmental variable influences the growth rate at a given level of the stock (Simonit and Perrings, 2005). Changes in chlorophyll-a concentration are the result of nutrient loading from the catchment, partially buffered by the wetland. In Lake Victoria, there is evidence of nitrogen being the limiting nutrient in offshore waters

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while phosphorus concentration seems to be the key element influencing phytoplankton growth in inshore waters (Mugidde, 2001). Since our analysis is limited to the Kenyan side of Lake Victoria, which is mostly represented by inshore waters, it is appropriate to consider phosphorus as the nutrient at issue. The decision problem can be modeled at several different levels. We distinguish between the private and social decisions involved in order to capture the externalities of activities that bear on the regulating services offered by wetlands. Farmers influence the downstream fishery through their decisions on land allocation (α), cropping intensity (β), and phosphorus fertilizer application (P ), subject to soil nutrient dynamics. The fishery sector takes these upstream externalities as given in its choice of fishing effort. Depending on the rules of access, fishers either maximize profit (if property rights are well defined or access is regulated), or equate total revenue and total cost (if property rights are not well defined or access is open. While we report the open access case, we are interested in the case where individual fishers choose the amount of fishing effort (E) that maximizes profit, taking into account the fish stock externality caused by nutrient loading (z); and where the fishery sector as a whole chooses the level of fishing effort that maximizes aggregate fishery profits. In the second case, the decision problem is to MaxE

∞
n
t=0 j=1

ρt πjF (Xt , Ejt ) =

∞
n


ρt (pqE jt Xt − cF Ejt )

(11)

t=0 j=1

subject to Eq. (8) and the initial level of the stock (X0 ). πjF represents the profit function for the jth fisherman; p is the price of fish; cF is the cost of fishing effort; ρ = 1/(1 + δ) is the discount factor; δ is the discount rate; and the other parameters are as defined in the Gordon–Schaefer model. The current value Hamiltonian for this problem ˜ = H

n


(πjF (Xt , Ejt ) + ρµt+1 (X(Xt , D(zt )) − h(Xt , Ejt )))

(12)

j=1

yields the following first-order necessary conditions  n 
˜ ∂πjF ∂h ∂H = − ρµt+1 =0 ∂Et ∂Ejt ∂Ejt j=1

(13)

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  n 
˜ ∂X ∂H ∂h ∂πjF = − ρµt+1 − − =0 ∂Xt ∂Xt ∂Xt ∂Xt j=1

  
n X t qEjt Xt  = 0 Xt+1 − Xt = rXt D(zt ) − − K j=1 

(14) (15)

By solving this system with (14) and (15) evaluated at the steady state we obtain the steady-state profit maximizing solution for fish stock X ∗ , yield Y ∗ , and fishing effort E ∗ . Table 11.1, reports the steady-state levels solutions to the problem where the fishery is under both open and regulated access. The results are compared with those of the standard Gordon– Schaefer model (i.e., with no environmental externality). Table 11.1: Maximum sustainable yield (MSY), open access (oa) and profit maximizing (*) steady-state solutions. MODEL 1: Standard Gordon–Schaefer model

MODEL 2: With environmental stressor

Xmsy =

K 2

Xmsy (W ) =

K [W (1 − cW )] 2

Ymsy =

rK 4

Ymsy (W ) =

rK [W 2 (1 − cW )2 ] 4

Emsy =

r 2q

Emsy (W ) =

r [W (1 − cW )] 2q

Xoa =

cF pq

Xoa (W ) =

cF pq

Yoa =

cF r(pqK − cF ) p2 q2 K

Yoa (W ) =

cF r(pqKW − pqKcW 2 − cF ) p2 q2 K

r(pqK − cF ) r(pqKW − pqKcW 2 − cF ) Eoa (W ) = 2 pq K pq 2 K "„ "„ « « K K δ cF δ cF X∗ = +1− X ∗ (W ) = + W − cW 2 − 4 pqK r 4 pqK r s„ s„ # # «2 « 8cF δ δ 2 8cF δ cF cF δ + + +1− + + W − cW 2 − + pqK r pqKr pqK r pqKr „ „ « ∗« ∗ X (W ) X Y ∗ = rX ∗ 1 − Y ∗ (W ) = rX ∗ (W )W 1 − − cW K KW

Eoa =

E∗ =

Y∗ qX ∗

E ∗ (W ) =

Y ∗ (W ) qX ∗ (W )

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The private and social problems in agriculture are similar. Farmers are assumed to maximize farm profits through choice of the level of phosphorus fertilizer, (Pi ), and the proportion of their land left fallow (βi ). Thus, the decision problem for the ith agricultural household is MaxPi ,βi

∞


ρt πiA (Pit , βit , Qit )

t=0

=

∞


ρt (pA (1 − βit )yit (Pit , Qit ) − cA Pit )

(16)

t=0

subject to soil nutrient dynamics, Eq. (4), to the intial level of phosphorus concentration in the soil (Qit=0 ); and where πiA represents the profit function for the ith farmer; pA is the price of harvested crop; cA is the unit cost of mineral fertilizer; and the other parameters and variables have already been defined. The current value Hamiltonian for this problem is ˜ = πiA (Pit , βit , Qit ) + ρλt+1 (Qi (Pit , βit , Qit )) H

(17)

which yields the following first-order conditions ˜ ∂πiA ∂H = + ρλt+1 dPit ∂Pit ˜ ∂H ∂πiA = + ρλt+1 dβit ∂βit

 

∂Qi ∂Pit ∂Qi ∂βit

 =0

(18)

=0

(19)



Qit+1 − Qit = Pit (1 − βit ) + Φi + χβit − ηyit − Si Ait θQit = 0   ˜ ∂H ∂πiA ∂Qi ρλt+1 − λt = − =− − ρλt+1 =0 dQit ∂Qit ∂Qit

(20) (21)

To get a measure of the value of a change in wetland extent for nutrient buffering, we need to identify the effect of interactions between the agricultural and fishery sectors along an optimal path. The social problem requires maximization of net benefits across sectors through choice of the level of fertilizer application P , wetland extent α, agricultural intensity β, and fishing effort E: MaxP,β,α,E

∞ αL n

0
t=0 i=1 j=1

ρt (πiA (Pit , βit , Qit , αt ) + πjF (Xt , Ejt ))

(22)

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subject to Eqs. (4) and (8). The current value Hamiltonian for the social problem is ˜ = H

αL n
0


 πiA + πjF + ρλt+1 Qi (Pit , βit , Qit )

i=1 j=1

+ ρµt+1 (X(Xt , D(zit )) − h(Xt , Ejt ))

(23)

which gives the following first-order conditions    αL
0  ∂πiA ˜ ∂X ∂Dt ∂zt ∂wt ∂H = + ρµt+1 git = 0 ∂αt ∂αt ∂Dt ∂zt ∂wt ∂αt i=1 αL
0  ∂πiA ˜ ∂H ∂Qi = + ρλt+1 ∂βt ∂β ∂β it it i=1    ∂X ∂Dt ∂zt ∂git + ρµt+1 wt = 0 ∂Dt ∂zt ∂git ∂βit αL
0  ∂πiA ˜ ∂Qi ∂H = + ρλt+1 ∂Pt ∂P ∂P it it i=1    ∂X ∂Dt ∂zt ∂git + ρµt+1 wt = 0 ∂Dt ∂zt ∂git ∂Pit  n 
˜ ∂πjF ∂h ∂H = − ρµt+1 =0 ∂Et ∂Ejt ∂Ejt j=1 αL
0  ∂πiA ˜ ∂H ∂Qi − = − ρλt+1 dQt ∂Qit ∂Qit i=1    ∂X ∂Dt ∂zt ∂git − ρµt+1 wt = 0 ∂Dt ∂zt ∂git ∂Qit

 n 
˜ ∂X ∂H ∂h ∂πjF ρµt+1 − µt = − = − ρµt+1 − − =0 ∂Xt ∂Xt ∂Xt ∂Xt j=1

(24)

(25)

(26) (27)

ρλt+1 − λt = −

(28) (29)

The right-hand terms in Eqs. (24)–(26) account for the external effects on the fishery sector due to marginal changes in α, β, and P , respectively. The externalities are evaluated at the optimal value of the costate variable µ, representing the shadow value of forgone fish production. Equation (25) indicates that the maximum social benefit is attained where the percentage

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of the area under fallow is such that marginal revenue is equal to the sum of both the marginal cost to the agricultural sector due to soil nutrient loss (on-site externality) and the marginal cost to the fishery (off-site externality). Equation (26) indicates that fertilizer should be applied up to the point at which the marginal benefits of improved soil nutrient balance are equal to the marginal cost to the fishery sector. The last term on the right-hand side of Eq. (24) is the value of the nutrient buffering function of the wetland at a given level of nutrient loading gt . Along an optimal trajectory, condition (24) implies that the marginal benefit of wetland depletion, measured in terms of the additional net benefits of agricultural production, should be equal to the marginal cost of wetland depletion measured in terms of the value of the forgone nutrient buffering services. Specifically, this is the value of the marginal impact of a change in wetland area on the growth of fish stocks, given nutrient inflows to the wetland. The marginal impact of a change in wetland area depends both on the effect of wetland extent on nutrient retention, (∂z/∂w)(∂w/∂αt ) = gt (∂w/∂αt ), and on the damage associated with changes in nutrient loading, ∂D/∂z. The value of this latter term depends on the state of the system. If the system is initially in an oligotrophic state, then the marginal impact of wetland depletion may be either positive or negative depending on the initial nutrient load. However, at the point where nutrient loads push the system from an oligotrophic into a eutrophic state, the marginal damage of a change in the extent of the wetland may be very high. If the system is initially in a eutrophic state (the damage has already been done), further changes in wetland extent may have little or no marginal impact on fish stocks. The value of this marginal impact — the discounted user cost, ρµt+1 , of the exploited fish stocks — depends on the solution to the optimal fish harvesting problem, which in turn depends on the market price of fish, access rights, the regulatory regime, and the fishers’ objectives.

11.3.

Regulating Functions and Land Use Externalities: The Case of Yala Wetlands

The approach may be used to identify and evaluate the forgone nutrient buffering function of either of the activities that impact nutrient loads in the fishery: wetland conversion or agriculture. To make matters concrete, we considered a proposal for the conversion of part of the Yala Swamp on the

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Fig. 11.1:

The Yala watershed.

Kenyan side of Lake Victoria. The Yala river discharges into Lake Victoria through the wetland (Fig. 11.1). Yala Swamp has already been significantly reduced by conversion for agricultural development since the mid-1960s, and an area of 9,200 ha from the remaining 15,200 ha of wetlands has been demarcated for “reclamation” in the Lake Basin Development Authority Five Years Plan 1989–1993 (Mwakubo et al., 2004). We took this proposal as the basis for the change in the extent of the wetland to be valued. To obtain an estimate of the damage function, (10), we linked nutrient loading to phytoplankton density (i.e., chlorophyll-a concentration), drawing on the phosphorus mass balance models first developed by Vollenweider (1968; 1969). Without the support of specific studies for Lake Victoria, we exploited the literature on chlorophyll/nutrient relationships in other freshwater ecosystems (Sakamoto, 1966; Megard, 1972; Dillon and Rigler, 1974; Jones and Bachmann, 1976; Schindler et al., 1978; Straˇskraba, 1980; Smith, 1982; Prairie et al., 1989). We selected the general equation proposed by Dillon and Rigler (1974) to describe the dependence of overall phytoplankton biomass on total phosphorus (Table 11.2). The flow of nutrients into the lake is described by Eq. (30). These are augmented by atmospheric deposition and nutrient loading from the rest of the Kenya basin of Lake Victoria. Since the effects of fertilizer application are highly sensitive to conditions in the watershed, we estimated the marginal impact on the growth of

Ait

L0 Qit=0

N

J

H

V

Our assumption

m3 yr−1 m yr−1 m3 yr−1 m3

1114∗ 106

0 5.3664∗ 1010

Ait = Ri Gi LSi Cit

309,027 30.25

1823

1460

12

COWI (2002)

t P yr−1

kg ha−1 yr−1

Research Tools in Natural Resource. . .

Modeling the Economics of Ecosystem Services at Watershed Scale (Continued)

GIS estimation based on Wischmeier and Smith’s (1978) USLE

GIS estimation Onyango (1994)

COWI (2002)

t P yr−1

ha kg P ha−1

Our assumption

m

Our assumption

Kadlec and Knight (1996)

m yr−1

12

Balirwa and Bugenyi (1988)

mg P m−3

P¯ (zt ) = (109 (z + J)) /(U + (s/H)V ) 16

Dillon and Rigler’s (1974) expression Vollenweider’s (1968; 1969) expression Chapra (1975)

Source

mg m−3

Units

Wt = 0.0731P¯t1.449

Value/Expression

9in x 6in

U

k

v

s

Chlorophyll-a concentration in the lake’s waters Phosphorus concentration in the lake’s waters Settling velocity for total phosphorus Water inflow to the Yala wetland Areal removal rate constant for phosphorus Hydraulic superficial outflow from the lake Average volume of the lake (Kenyan side) Mean depth of the lake (Kenyan side) Atmospheric deposition on the lake (Kenyan side) Superficial loading (Kenyan basin except Yala) Watershed extent Soil phosphorus concentration at t = 0 Soil erosion

Description

Values for the parameters used in the model.

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P¯t

Wt

Variable

Table 11.2:

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355

Map (value for the ith hectare) Map (value for the ith hectare)

Mean annual rainfall

Soil erodibility factor

Length/slope factor

Land cover factor

Sediment delivery ratio

Intrinsic growth rate of the fish stock

Ωi

Gi

LSi

Cit

Si

r 0.201033

Si = 0.4724∗ (catchment area)−0.125 i —

0 ≤ Si ≤ 1

0 ≤ Cit ≤ 1

0 ≤ LSi ≤ 1

Research Tools in Natural Resource. . .

(Continued)

GIS estimation based on ILRI’s percent tree cover map and Wenner’s (1981) parameters for maize cultivation (0.7) and forest cover (0.01) GIS estimation based on ILRI’s digital elevation model and Vanoni’s (1975) expression Simonit and Perrings (2005)

GIS estimation based on ILRI’s soil classification map and Wischmeier and Smith’s (1978) soil parameters GIS estimation based on ILRI’s digital elevation model and Moore and Burch’s (1985) expression

GIS estimation based on Lufafa et al. (2003) expression ILRI’s rainfall distribution map

Source

356

0 ≤ Gi ≤ 1

mm yr−1

—

Units

9in x 6in

LS i = (upslope contributing ∗ area/22.13)0.4 (sin of slope angle in degrees/0.0896)1.3 Cit = 0.01βit +0.7(1−βit )

Ri = 47.5 + 0.38Ωi

Value/Expression

Rain erosivity factor

Description

(Continued)

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Ri

Variable

Table 11.2:

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Silvio Simonit and Charles Perrings

Soil phosphorus enrichment factor Maize crop productivity

δ

yit

η θ

Phosphorus build-up factor from forest fallow rotation Crop nutrient content Soil bulk factor

χ

Carrying capacity of the fish stock Impact coefficient for chlorophyll-a concentration Atmospheric deposition on Yala basin

Description

yit = 10−3 (1 − βit ) (924.6 + 60.806Pit + 2) 31.866Qit − 0.218Pit

1.5

9.4 0.000379

11

Φi = 0.053Ωi0.5

0.029214

53,082

Value/Expression

y (t ha−1 ); P , Q (kg P ha−1 )

—

kg P/t of harvest —

kg P ha−1 yr−1

kg P ha−1 yr−1

—

—

Units

Adapted from Mugunieri et al. (1997)

Stoorvogel and Smaling (1990). Estimated from Ministry of Agriculture (1987) Stocking (1984)

GIS estimation based on ILRI’s rainfall distribution map and Stoorvogel and Smaling’s (1990) expression Nye and Greenland (1960)

Simonit and Perrings (2005)

Simonit and Perrings (2005)

Source

9in x 6in

Φi

c

K

Variable

(Continued)

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Table 11.2:

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fish stocks from the nutrient loading from each ha of land in the watershed. The marginal impact of nutrient loading on the environmental variable W , chlorophyll-a concentration, is a function of nutrient loading (git ), wetland area, and a set of constants describing long-term average conditions in the lake:

1.449 −(k(1−αt )L0 104 )/v (P , β , Q , Ω )e + J + N g it it it it i Wt = (0.8034)1012 U + sV H (30) Where J is the long-term average atmospheric deposition on the lake’s surface (t P yr−1 ); N is the long-term average phosphorus loading from the Kenya basin of Lake Victoria other than Yala watershed (t P yr−1 ); V is the lake volume (m3 ); U is the hydraulic superficial outflow from the lake (m3 yr−1 ); H is the mean depth of the water body (m); and s is the settling velocity rate for phosphorus (m yr−1 ); and the other variables have already been defined. We next delineated the Yala watershed and estimated soil erosion across the watershed. Using the ArcGis 9.2 application and a digital elevation model (DEM) of Kenya based on the 1:250,000 contour map provided by the International Livestock Research Institute (ILRI), we defined the extent of the Yala watershed and its river network. We then estimated soil erosion and the associated sediment delivery that are key factors in determining nutrient loading. Using the universal soil loss equation approach (Wischmeier and Smith, 1978) and assuming no soil management to reduce soil erosion in the Yala basin, we obtained the soil erosion grid from map overlay of rain erosivity, soil erodibility, length–slope, and land cover factors, respectively. In this process, the rain erosivity layer is determined from a mean annual precipitation map (ILRI), adopting a functional relationship (Lufafa et al., 2003) estimated for a microcatchment study in Uganda which uses coefficients derived by regressing long-term rainfall against erosivity values determined by Moore (1979) for the Lake Victoria region. The length–slope grid is computed from the DEM using the approach developed by Moore and Burch (1985). This approach considers the upslope contributing area per unit width of contour (or rill). The soil erodibility layer was obtained from a soil characteristics map of Kenya (ILRI). The land cover grid was estimated by applying the land cover factor values — 0.01 and 0.7 for forest/savannah and maize cultivation respectively, reported by Wenner (1981) as representative for Kenya — to a digitalized map showing

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the percent of tree cover distribution in Kenya (ILRI). In other words, “fallow” land is approximated by tree cover, and the density of tree cover yields the spatial distribution of the parameter β in our general model. To account for the fraction of eroded soil that finally reaches the stream system or watershed outlet, we obtained a spatially distributed sediment delivery ratio (SDR) grid using the functional relationship proposed by Vanoni (1975). We then multiplied that by the soil erosion grid to obtain a sediment delivery map. This map estimates for each pixel the amount of sediments (as proportion of the eroded soil) that would be expected to reach the basin outlet. It therefore describes net soil erosion. Most of the parameters for the model were obtained from the literature (described in Table 11.2). The initial value of mean soil phosphorus concentration in the soil stock dynamics was taken from field experiments within the Yala basin (Onyango, 1994). Mean nutrient concentration in the soil depends on the specific soil bulk density for a given superficial layer of soil. We obtained the soil bulk factor by converting1 the soil bulk density of 1.32 g cm−3 reported for the Siaya District (Ministry of Agriculture, 1987) for the first 20 cm of topsoil layer. Since eroded soil is richer in nutrients than soil in situ, due to fine particles dislodged first in the process of erosion, an enrichment factor (Avnimelech and McHenry, 1984; Stocking, 1984) is generally used to multiply the estimated phosphorus content in eroded soil. Atmospheric deposition on land was estimated using an empirical function for sub-Saharan Africa available from the literature (Stoorvogel and Smaling, 1990; Smaling et al., 1993; Stoorvogel et al., 1993) together with the rainfall map (ILRI) expressing mean annual precipitation for the Yala watershed. Phosphorus build-up parameters for secondary forest fallow vegetation in Africa were taken from Nye and Greenland (1960). Stoorvogel and Smaling (1990) provide a wide range of data on nutrients content in harvested crops for sub-Saharan Africa. The agricultural production function was adapted2 from the maize production model for Western Kenya the 20 cm of topsoil layer, it gives a soil weight of 2640 t ha−1 , which is equivalent to a soil bulk factor θ = 1/2640 = 0.000379. To obtain the soil phosphorus concentration in terms of kg per tonne of soil we can multiply by θ any nutrient value expressed in kg ha−1 . 2 Their model estimates the yield response function to nitrogen and phosphorous applications at three experimental field stations in the Kisii District of Western Kenya. We transform the two variables model into a function of only one variable (phosphorus fertilizer), substituting nitrogen with phosphorus by assuming a fixed nitrogen/phosporus 1 Considering

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by Mugunieri et al. (1997). We assumed an average depth of 12 m and a volume of 5.3664∗1010 m3 for the Kenyan sector of Lake Victoria, and that the hydraulic outflow is fully compensated (U = 0) by inflowing water from the Ugandan and Tanzanian side of the lake with the same concentration of phosphorus. This assumption is known in the engineering literature as the continuously stirred tank reactor (Reckhow and Chapra, 1983) and implies that the lake’s waters are perfectly mixed. For the Yala watershed, there are no reliable estimates of phosphorus fertilizer consumption. Some regional studies indicate mean phosphate application rates of around 3–4 kg P ha−1 (Wanzala et al., 2001; Van den Bosch et al., 1998). Nevertheless, the observed soil phosphorus balance in the Yala basin is negative at −4 kg ha−1 yr−1 (Smaling et al., 1997). This indicates that the agricultural sector, as in most of Kenya, cannot sustain itself in the long run without either increasing fertilizer application (mineral or organic or both) or changing land use and soil management practices to reduce runoff. To estimate the value of the nutrient buffering function of the wetland under the proposal described above, we accordingly assumed sufficient nutrient phosphorus fertilizer application to maintain the soil phosphorus balance (i.e., we assumed sufficient fertilizer applications to assure the sustainability of agriculture). We then estimated the impact on nutrient absorption by the wetland with and without measures to reduce runoff. The required fertilizer varies for each hectare according to the natural conditions influencing soil erosion (site-specific rain erosivity, soil erodibility and length/slope factors, and the sediment delivery ratio) as well as the intensity of land use (βi ), which impacts both soil erosion through the land cover factor (Ci ), and soil phosphorus dynamics. The “sustainable” phosphorus application for each hectare was obtained by solving the soil nutrient balance equation (5) for P . The result is an expression which gives Pi as a function of βi and the net soil erosion or sediment yield Si Ai . We estimated the equation in GIS environment through the raster calculator function by using percent tree cover and sediment yield maps as input grids, and assuming the initial soil phosphorus concentration of Qit=0 = 30.25 kg P ha−1 for each ith hectare. The resulting map identifies the spatial distribution of the amount of phosphorus fertilizer (kg P ha−1 yr−1 ) required (Fig. 11.2). proportion of 2.55 as for the content of DAP fertilizer bags commonly used for maize cultivation in Kenya.

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Fig. 11.2: Phosphorus fertilizer required (kg ha−1 yr−1 ) for long-term soil nutrient balance in the Yala basin.

Given a constant nutrient inflow to the wetland, we then estimated the value of the nutrient buffering forgone by reducing the wetland extent by 60% (9200 ha). From (7), (9), and (10), and the expression for total lake phosphorus and conversion to chlorophyll-a concentration in Table 11.2, the  ∂D ∂z ∂w specific form of the general expression ρµt+1 ∂X ∂D ∂z ∂w ∂αt is    .449  w g 1.898  t t 13 wt gt 25 ρµt+1 rXt (0.1164)10 − (0.1871)10 c M M  wt gt kL0 10−4 × (31) M v where M =U+

sV H

Using (30), we derived the value of a change in wetland extent from the impact it has on the value of commercially exploited fish stocks. In the general case this depends on the harvesting strategy pursued by fishers, which varies with both access and management regimes. In the case of Kenya, where the fishery is managed to achieve maximum sustainable yield (MSY), the value of the buffering function of wetlands was calculated assuming catches at MSY: YMSY =

αL
0 i=1

YMSYi (Pit , βit ) =

rK  2 Wit (1 − cWit )2 4

(32)

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Using the 2006 value of fish (744.29 US$ t−1 ), a change in wetland area of 9,200 ha from an original 15,200 ha, and the parameter values given in Table 11.2, we calculated the annual impact of wetland reclamation in terms of loss to the MSY of the fishery. Since the MSY yield across the fishery also reflects a superficial loading from the rest of the Kenya watershed of Lake Victoria — 1823 t P yr−1 (COWI, 2002) — the change in nutrient load due to conversion of part of the Yala wetland is still small relative to the aggregate nutrient load. Note that this measures the impact of wetland reduction on the value of the variation of fish landings (∆YMSYi ), assuming fertilizer applications (Pi ) at the rate required to reach soil nutrient balance across the Yala catchment. If the wetland were to be reduced by 60%, the nutrient load “z” to the lake would increase from 34.3 t P yr−1 to 92.3 t P yr−1 , the size of the MSY fish catch would reduce from 190,778 t yr−1 to 188,112 t yr−1 , a loss of 2666 t yr−1 , equivalent to 1.98M US$. Dividing by the 9200 ha of wetland affected by “reclamation” yields a cost in term of forgone fishery production of 216 US$ ha−1 yr−1 . This measure of the value of the regulating services of wetlands was then used to assess the efficiency of reclamation, and to evaluate the feasibility of alternative mechanisms for delivering the same regulating services. Neglecting the cost of conversion, the additional yield of maize at the maximum potential yield of 5.7 t ha−1 yr−1 , if priced at 213 US$ t−1 , implies an annual “sustainable” revenue from conversion of 11.17M US$, or 1214 US$ ha−1 yr−1 . Since this requires a phosphorus application of 95 Kg P ha−1 at a cost of 257 US$ ha−1 , it implies a sustainable net benefit from conversion of 957 US$ ha−1 or 741 US$ ha−1 net of the fishery externality.

11.4.

Using the Model for Policy Experiments

The modeling problem addressed in this chapter requires the integration of models of nutrient retention in wetlands and the dynamics of a freshwater fishery. Globally, the Millennium Ecosystem Assessment concluded that the regulatory services of wetlands are declining, largely because of conversion for agriculture, forestry, industry, and domestic dwellings (MA, 2005). To understand the impact of conversion in the particular setting of the Yala swamp, we modeled the interactions between activities that depend on the regulating services of the wetland, and the activities that cause a change in those services. The problem for policy in this case, as in many others, is that the value of regulating ecosystem services is rarely calculated and

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almost never factored into decision making. It is important to understand the value of such services not only to assess the return to public investments that change those services, but also to identify and evaluate potentially corrective policies. We have already observed that the value of the buffering function of wetlands is sensitive to conditions both on- and off-shore. It is accordingly possible to identify the upstream conditions associated with changes in that value. In the case of Yala, for example, it is possible to identify the contribution of each hectare of land in the catchment to the nutrient load entering the wetland, and to use this information to identify policy options. To illustrate, summing loads across the catchment, the baseline phosphorus loading “g” into the wetland is 176.1 t yr−1 . Using the GIS raster calculator with the percent tree cover (Fig. 11.4(a)) and phosphorus fertilizer (Fig. 11.2) maps as input grids, along with the price of fish (744.29 US$ t−1 ), we obtained the spatial distribution of the nutrient loading externality described in Fig. 11.3(a). Note that the nutrient loading externality ranges from (virtually) zero for sites that experience minimum soil erosion, to US$ 907 ha−1 yr−1 for sites that lose a significant amount of sediment annually. Note also that the spatial distribution of the nutrient loading externality changes as a result of the wetland conversion (Fig. 11.3(b)). Areas that previously had no significant downstream externality now have an impact on the fishery, potentially up to 2465 US$ ha−1 yr−1 for some sites. There are two ways to use the spatially distributed externality data. First, if the externality is assigned by source (and not by the converted area), we obtain an estimate of the additional damage done across the catchment as a result of the wetland reduction (Fig. 11.3(b)). In particular, by taking the difference between the two grids, we can isolate the spatially distributed value of the forgone wetland buffering function as a result of “reclamation.” This represents a spatially distributed wetland “reclamation externality”, which varies between farmers according to the use and the initial physical characteristics of the land. Second, if the full value of the externality was assigned to the converted area (Fig. 11.3(c)), then the damage associated with the conversion would be 216 US$ ha−1 yr−1 . Given that net revenues from maize could potentially reach a maximum of 957 US$ ha−1 yr−1 , it is likely that conversion would still be upheld as a rational use of resources, even taking the loss of regulating services into account. Of course, the benefit–cost ratio of conversion does depend on the marginal impact of wetland conversion on the fishery. At the value of fish used here, the additional nutrient loading associated with wetland

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Fig. 11.3: Spatial distribution of nutrient loading externality before wetland reclamation (a) and after wetland reclamation, with externality assigned by source (b) or by the converted area (c).

conversion would have to lead to a reduction in MSY of not less than 6.4% (4.4 times the projected reduction) before it compromised the decision. Nevertheless, the loss of wetlands does have a cost, and the subsidiary question is therefore whether there are alternative ways to obtain the services of wetlands at lower cost. Information on the spatial distribution of the externality means that we can evaluate the effect of alternative methods of delivering the same services. Specifically, if a change in land management practice is able to reduce nutrient loads, it would be worth considering whether the cost-effectiveness of this is a solution. Recent interest in the potential for payments for ecosystem services is relevant here (Bulte et al.,

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2008; Engel et al., 2008; Ferraro and Kiss, 2002; Ferraro and Pattanayak, 2006; Wunder, 2007; Wunder et al., 2008). We considered whether there exists a set of payments that might be funded out of the gains from wetland conversion that would deliver the same benefits to the fishery as the converted wetland (Fig. 11.4). Figure 11.4(b) shows the land cover (in terms of percent tree cover) required in order to neutralize the increased nutrient loading into the lake due to wetland reclamation. By changing land cover, it is possible to decrease loading “g” such that loading “z” remains at the same value as before wetland reclamation. Summing across the watershed, the baseline phosphorus loading “g” under the current situation is 176.1 t P yr−1 , which could be reduced to 65.4 t P yr−1 by changing land cover according to Fig. 11.4(b). The value of 176.1 t P yr−1 is greater than COWI’s (2002) estimate of 102 t P yr−1 due to the fact that we impose an agricultural sustainability constraint–requiring fertilizer applications to be consistent with reaching soil nutrient balance. Figure 11.5 shows the payments to upstream farmers that would be warranted if they changed agricultural intensity on their plots according to Fig. 11.4(b). In other words, this is

Fig. 11.4: Current distribution of percent tree cover (a) and percent tree cover required to compensating for reduced wetland buffering function (b).

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Fig. 11.5: Spatial distribution of payment for ecosystem services to farmers for reducing agricultural intensity by changing percent tree cover.

the distribution of payments implied by a scheme to substitute the regulating services offered by tree cover in arable lands for the nutrient buffering services lost through wetland conversion. The total cost of the payments that would compensate farmers for on-farm nutrient buffering services is 3.86M US$ yr−1 , or 35% of the total gains from wetland conversion to crop production. It would in principle be feasible, therefore, to implement a system of payments for ecosystem services that would be effective in offsetting the loss of nutrient buffering from wetland conversion. 11.5.

Concluding Remarks

While this example is illustrative only, it indicates the value of the capacity to model environmentally mediated interactions between sectors in a coupled socio-ecological system. The specific modeling problem addressed in this study is the integration of nutrient retention in wetlands and the dynamics of a freshwater fishery. The effect of changes in nutrient loading induces changes in fish biomass that depends on the initial level of loading and the initial state of the system. In general, a reduction of nutrient loads when the lake is in a eutrophic state will lead to an increase in fish biomass. Similarly, the nutrient retention function of wetlands is reversible. If the wetland area or biomass is increased, nutrient retention will also increase. It is also substitutable, in that erosion control and nutrient retention at the point of fertilizer application can substitute for end-of-stream buffering in the wetland. The model enables us to estimate the value of the nutrient absorption externalities of wetland conversion, to project the consequences of alternative wetland conversion strategies, and to identify alternative sources of the same services. Wetlands do provide a number of other

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ecosystem services, but in this case nutrient absorption and its effects on the commercial fishery in the lake may in fact be the most important. The more general problem addressed is that the value of the regulating ecosystem services may be uncovered only by understanding the contribution they make to the production of the things that people care about directly — the MA’s provisioning and cultural services. That is, the value of the regulating services derives from the value of the provisioning or cultural services they protect. Since the sustainability of the provisioning or cultural services depends on the capacity of the system to deliver those services over a range of environmental conditions, the value of the regulating services will vary both with the value of the protected service and the variability of environmental conditions. In this study, we have taken the measure of this service to be the minimum nutrient retention provided, given prevailing climatic, soil, agricultural, and institutional conditions. Other measures — based on acceptable risk or confidence — might also be appropriate. In all cases though, it is possible to identify the change in value associated with a change in conditions. Since the regulating services are among the least understood of the ecosystem services, while being among the most important to the sustainability of the coupled system, the approach described here may be of some value in other systems where regulating functions involve similar mechanisms.

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Chapter 12

Geographical Information Systems Models and Spatial Data Analysis Robert P. Haining∗ and Jane Law† ∗

University of Cambridge [email protected]

†

12.1.

University of Waterloo [email protected]

Introduction

The term “model” is used generically to denote a simplification or abstraction of some usually complex set of objects, events, or processes. A model is the outcome of a process that selects and emphasizes relevant features. This chapter is about the models that arise when using geographical information systems (GISs) and the added value that comes from applying the tools and methods of spatial data analysis. Such are the gains that can be achieved by bringing together GISs and spatial data analysis that the label “Geographical Information Science” is often used to refer to this marriage of technology and methodology when it is applied to good-quality data in order to address well-defined questions or test hypotheses (Goodchild and Haining, 2004). Models are embedded into the design and application of GISs in several different places. First, the term model appears in relation to different conceptual models of the real world. The first stage in the process of digitally capturing the infinite complexity of an area’s geography is to conceptualize the attributes that are distributed across the earth’s surface either in

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terms of discrete objects (e.g., location of factories) or continuous fields (e.g., land cover map). Second, the term model appears in relation to different representations of fields and objects. Both discrete objects and continuous fields can be represented in terms of points, lines, or polygons (Haining, 2003, pp. 43–46; Longley et al., 2005, pp. 74–77). A third use of the term model is in relation to the data model that is employed in a GIS. This is dependent on the use to which the GIS is to be put and the phenomena under study and their characteristics. The data model specifies for the particular application how space and time are to be represented, which attributes are to be recorded, and what operational definitions will be adopted for the purpose of taking measurements. Whilst these options may be presented in the language of choice, in practice the analyst is often dependent on secondary data perhaps collected by public agencies (the police, health service) and subject to disclosure restrictions to preserve confidentiality so that they come in an aggregated form that has been decided by the agency. The output from this data modeling process is usually a set of tables stored as files or databases — for example shape files, a digital vector storage format for storing geometric location and associated attribute information and which has links to the spatial data matrix (Haining, 2003, pp. 54–67). The .shp file stores the feature geometry and the .dbf file the data on the attributes associated with each spatial object. Adjacencies and hence “neighbors” can be calculated even though the geometric information contained in the shape files is not explicitly topological. Longley et al. (2005, pp. 179–192) provide an overview of the geographic data models used in a GIS. The choice of data model defines the types of geographic operations that can be performed because it is the data model that supports map display, querying, editing, and analysis. The data model constitutes the template into which specific aspects of the earth’s surface is fitted (Longley et al., 2005, p. 364). The fourth type of model may be descriptive, explanatory, predictive, or for policy or planning purposes and for want of a better umbrella term this fourth type of model will be called a process model. This term is not ideal since process involves time and usually some notion of causation, whereas most of the models to be covered here are static and cannot be interpreted in a causal sense. Included are models that may express our ideas about process and how spatial patterns originate and develop, as well as models that express our ideas about how attributes are related to one another at a point in time. Within a GIS, models may be statistical or mathematical, static,

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or dynamic. They may be written as closed form expressions that can be manipulated to yield predictions or expectations or they may depend on simulation techniques. Models may be defined at the level of individual agents or for spatially aggregated groups. The range of possible models is large but the sorts of models that are important in relation to a GIS are those where location and spatial relationships are fundamental to the structure of the model and its outcomes. The purpose of GIS models is to describe and explain geographical variation and to do this, the locations of the elements within the model and their spatial relations to each other are critical. The models through which we seek to understand the world and to understand geographical variation are, in this sense, spatial (involving relationships across space) as well as geographical (where things are). Spatial data analysis can be conveniently classified into two activities. The first involves the reduction of spatial pattern to a few statistical summaries. These summaries may be numerical, graphical, or cartographical. The second involves comparing observed patterns against a set of expectations or theory about how the pattern originated or developed. The first of these activities can be broadly associated with what is termed “exploratory spatial data analysis” or ESDA. The second of these activities is associated with making inferences — using models to estimate parameters of interest and test hypotheses — and is sometimes referred to as “confirmatory spatial data analysis” or CSDA. In the sections that follow we shall consider both the ESDA and CSDA methods and their relationship to GIS models. However, before we can proceed, in the next section we review the main types of spatial data that are encountered and how they map into statistical theory.

12.2.

Types of Spatial Data

What contribution can spatial data analysis make to the data models and process models found in a GIS or used in GIS-based applications? We examine four main types of spatial data for which exploratory and confirmatory statistical methodology has been developed. We base our classification on Cressie’s, taken from his 1993 book Statistics for Spatial Data. — Point data. These are data on an attribute that varies continuously over the area of study. In practice, measurements on the attribute have been taken at a finite number of discrete, fixed, locations (points or blocks)

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so that the data constitute a sample from the surface. If the attribute is considered a random variable, then various assumptions are made about the random field (or random process) responsible for the data values. Typically, these assumptions include that the covariance (or autocovariance) between the random variables at two locations depends only the distance between them and possibly the direction. If the assumptions include a joint distributional model, then this allows likelihood inference (Banerjee et al., 2004, pp. 6–7). Where the random variable assumption is made, Cressie (1993, p. 8) refers to these data as “geostatistical data,” presumably because of the correspondence with the data frequently encountered in traditional geostatistics. Recently however, geostatistical theory has been extended to cover area or regional data which we consider next (see, for example Goovaerts, 2006). — Area or regional data. This is data where the area of study is either exhaustively partitioned into smaller geographical areas or regions (e.g., census tracts) or takes the form of discrete points (e.g., retail outlets in a city). The attribute data associated with areas are typically aggregated into counts or averages. The spatial aggregation may be arbitrary (e.g., pixels on a remotely sensed image) or be based on some aggregation principle (e.g., see http://www.statistics.gov.uk/geography/census geog.asp for the design of spatial units used in the 2001 UK census). If the attribute is considered a random variable, spatial association can be introduced using a graph or neighborhood structure which is based on the locational arrangement of the areas or points. Ripley (1981, p. 78) uses the term “regional and lattice data” while Cressie (1993, p. 8) refers to it as “lattice data” but this latter term is not one we use here. Models can be specified in which the random variable observed at one location is expressed as a function of the same variable at other, usually nearby, locations. A rather different and more recent modeling approach for capturing spatial association has been through spatially varying random effects (Besag et al., 1991). — Point process data. In this case, the locations of the points that make up the spatial point pattern are the outcome of a random process. In the most general form of this type of data there may be covariate data attached to each point (e.g., either a tree is diseased or not) and the term marked point process is used in this case (Banerjee et al., 2004, pp. 8–9). Inference with these data may employ Ripley’s K function (Ripley, 1981, pp. 158–164) which can be used to test, for example, whether patterns are random or more clustered or more regular than random (see, for

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example, Haining, 2003, pp. 247–250). Since the theoretical value for the K function is known for some spatial point process models, this can be used to suggest a plausible model for the data (see Diggle, 2003). — Object data. In the case of this data, regular or irregular polygons are distributed across a study area. In some cases the polygons form discrete shapes with spaces between them (e.g., like grains and pores in a sedimentary rock), in others they provide a continuous mosaic-like coverage (e.g., like plant communities populating a region). The Boolean model provides an important and flexible class of models for carrying out statistical inference for this type of data (Cressie, 1993, p. 736). A random point process is specified for the location of each object (e.g., a homogeneous point process model) and, independent of this, a random set process is specified for the physical extent of each object (e.g., disks of random radius). These four data types can be mapped across to the field and object views through which, as we have noted, a GIS conceptualizes the real world. The point data type corresponds to samples from a field variable because the underlying reality is continuous. The point process model corresponds to a point-object variable whose location is conceptualized as the outcome of some spatial stochastic process. The area or regional data type corresponds to geographical aggregates of objects. The object data type may correspond to polygons from an object view or polygons constructed from a field view, for example one in which areas are segmented into homogeneous subregions such as areas covered by the same land use type. The above data types essentially capture a static representation of attributes across a study area. However, there are other types of data that are important in GIS models and which reflect dynamic or space–time events. Interaction data are counts of people or goods flowing between the nodes of a network. Motion or movement data identify the positions of individuals or groups at a sequence of points in time — for example, the movement of people over different retail areas in a central business district. We do not have space to consider these data types here. If the GIS analyst is unwilling to make statistically based assumptions about his or her data, then it is not possible to move beyond descriptive summaries of spatial data — useful although these may be. Assumptions are needed about how the data have been generated (the data-generation model) if inferences are to be drawn. Without such assumptions, the spatial analyst is often limited to descriptive summaries and estimating quantities

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based purely on geometric properties of the spatial units. For example, areal interpolation which involves linking different datasets that have been collected on different (incompatible) spatial frameworks must be undertaken using techniques that stress the geometrical relationships between the source and target spatial units and will give no indication of the size of likely error (Gotway and Young, 2002). The same comment applies to many of the techniques for spatial interpolation which estimate the attribute value at a site where no data have been collected using observations on the attribute made at nearby locations (see, for example Haining, 2003, pp. 152–178). Spatial sampling theory is also based on assumptions about the statistical properties of the surface to be sampled.

12.3.

Spatial Data and Non-Classical Inference

If we are willing to make statistical assumptions about the data in a GIS (i.e., specify a data-generating model) then inference becomes possible. Statistical inference is concerned with drawing conclusions about populations on the basis of sample evidence. It is central to CSDA and involves the following activities: modeling spatial patterns and relationships between variables, estimating model parameters, hypothesis testing, and spatial interpolation. This section examines some of the fundamental properties of spatial data, how these properties are characterized statistically, and the types of inference that can then be undertaken. However, it is not inference in the “classical” sense, namely inference based on a data-generating model that assumes independent and identically distributed (iid) sample observations. Recourse to that basis for inference in the case of spatial data can seriously mislead because it produces overestimates of how much statistical information is contained in the sample data. This in turn leads to underestimates of parameter standard errors which in turn leads to inflated type I errors when undertaking tests of hypotheses (Haining, 2003, pp. 275–278). The special nature of the inference problem in the case of spatial data can be demonstrated with reference to three remarks, the first two of which are strong arguments, the third less so. First, spatial data values close together in space are more alike than those further apart. This is sometimes referred to in the GIS literature as Tobler’s First Law of Geography (see, for example Longley et al., 2005). The presence of “spatial structure” means data values (unless they are sufficiently far apart) will not be

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independent. A data value on an attribute measured at one location carries statistical information about the value of the same attribute at nearby locations. Second, given the heterogeneous nature of the earth’s surface, inferences drawn from a set of data collected in one area are only valid for that area. If the aim is to generalize to other areas, the experimental design must be extended to these other areas. Third, in many GIS applications the analyst has access to a complete enumeration of cases — no sample is taken. This may be true, although it may also be an illusion as in the case of census data where in most applications we are rarely interested in the population of an area as it existed on the night the census was taken. Inference with spatial data is possible in the same way it is possible with classical data by postulating a data-generating model — but it is not the model of classical statistics based on iid observations. Spatial models are required that allow data values to be spatially structured or spatially correlated. That structure might be captured through the model’s mean (e.g., a trend surface model) or through its second-order variation (autocovariance) or both. The mean might comprise explanatory variables as in the instance of regression modeling. For an introductory review of such models, see for example Haining (2003, pp. 289–378). In the case of geostatistics, it is the semivariogram that is used to describe the spatial structure in a dataset and which plays a central role in kriging — a model-based approach to spatial interpolation which yields point estimates of unknown values as well as estimates of error. A more limited form of inference, often used when classifying map pattern, involves randomly permuting the observations (see, for example, Longley et al., 2005, p. 361). The actual spatial arrangement of values observed on a map and quantified using, for example, the Moran statistic, is benchmarked against the distribution of Moran values obtained as a result of randomly permuting the given set of observations 99 or 999 (or more) times. Banerjee et al. (2004) identified the following inference problems associated with different types of spatial data. In the case of point data: what can be inferred about the spatial process that has generated the partial realization and what can be inferred about the attribute values at locations where no data have been collected (the spatial interpolation problem)? In the case of area or regional data: is there a spatial pattern in the data, are there relationships among the variables (for example can the variation in an attribute be explained by other variables as in a regression model), if

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data are to be transferred from one set of spatial units to another what can be inferred about the values on the new units? In the case of point process data: are the point locations clustered and in a marked point pattern is there clustering of the marked points? In the following sections, we shall discuss ESDA and CSDA in the context of these questions. ESDA provides a starting point for statistical analysis which may or may not proceed to the fitting of models and carrying out inference (CSDA). We shall review likelihood function and Bayesian inference. The latter has become popular in part because of the availability, comparatively recently, of software to fit different types of Bayesian models. 12.4.

Exploratory Spatial Data Analysis

ESDA comprises techniques for exploring spatial data — summarizing spatial properties of the data, detecting patterns, formulating hypotheses, and identifying unusual cases. A common query takes the form: “where are those cases (e.g., that lie above the mean or the median) on the map?” Techniques are visual and numerically resistant (Haining, 2003, p. 182). In the case of point data, Banerjee et al. (2004, p. 39) propose the “so-called first law of geostatistics” as the underlying data model for ESDA. Data are conceptualized as partitioned into a mean term and an error term — the mean corresponds to first-order behavior and the error captures secondorder behavior through an autocovariance function. The mean may take the form of a trend, while the autocovariance represents spatial association after extraction of the trend. In the case of ESDA applied to area or regional data Haining et al. (1998) adapted the Tukey (1977) “smooth and rough” data model: Spatial data = Spatial smooth + Spatial rough Spatial smooth includes trend and second-order variation. The latter includes both general clustering, which is a global-map property referring to a general tendency for areas that are close together in geographic space to have similar attribute values, as well as localized clusters of high values. Spatial rough includes individual outliers (extreme values in either a distributional sense or given the values in adjacent areas) and uncorrelated white noise. It can be thought of as the variation in the data that is left over

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after extracting the spatially smooth element. Data models are important because they indicate the types of data properties the analyst ought to look for and therefore suggest the sorts of quantitative tools that might be appropriate. A number of ESDA tools are available for identifying these different data elements when analyzing a single variable. The choropleth map is a tool of fundamental importance in suggesting the presence of trend and second-order variation in area data while contour mapping or threedimensional “perspective” plots are used on point data (Banerjee et al., 2004, pp. 40–41). Map smoothing techniques based either on mean or median smoothing and using moving windows may help reveal structure in the mapped values (Kafadar, 1994). There are many forms of smoothing both linear and non-linear but the basic idea is to move a window of given size over the map centring it on each area or data point in turn. The size of the window determines the degree of smoothing. Large windows improve the precision of the statistic but increase the risk of bias and oversmoothing because information is being borrowed from locations further away. Small windows reduce the risk of bias but may not improve precision much because little information is being borrowed. More formally, trend surface fitting based on a bivariate polynomial in the two coordinates defining the location of each area can be used to identify trend. Other quantitative techniques for trend detection include median polishing (Cressie, 1993) and row and column boxplots (Diggle and Ribeiro, 2002). If data are available on other variables that are thought to be associated with the variable of interest, then scatterplots and added variable plots may be useful in suggesting which ones are most important in explaining the variation in the mean. Second-order variation is examined using the residuals after mean variation has been removed. In the case of point data, the empirical semivariogram is used to explore second-order variation and the directional semivariogram in contour form for detecting the presence or otherwise of anisotropy (Banerjee et al., 2004, pp. 42–48; Isaacs and Srivastava, 1989, pp. 149–151; Haining, 1990, pp. 284–286). Ripley’s K function (Ripley, 1977), which is based on counting the number of point objects within areas of varying size, is used to test for pattern in a mapped point process at a range of distance scales. Testing takes the form of comparing the empirical function against what would be expected under a random point pattern with significant departures (segments of the function lying outside the

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Monte Carlo generated envelope) indicating either clustering or uniformity in the point pattern at the corresponding distances. Diggle and Chetwynd (1991) extend the methodology to explore for clustering in a marked point process. In the case of area data the Moran scatterplot, the plot of each attribute value against the average of its neighbors can be informative not only in identifying second-order variation but also in suggesting the presence of spatial outliers and localized clusters (Haining, 1990). Moran’s I statistic can be used to test the null hypothesis of no spatial autocorrelation against the non-specific alternative that there is spatial structure and can be calculated for different orders of spatial separation (Cliff and Ord, 1981; Haining, 1990). This test, however, is sensitive to variation in the level of support as arises for example when analyzing data on per capita rates (e.g., disease incidence rates) where areas vary in population size. Modifications to the original Moran test have been suggested (Oden, 1995; Assun¸c˜ao and Reis, 1999). A number of tests have been proposed to detect the presence of localized clusters on a map — of interest for example when looking for individual disease clusters or crime hotspots. A widely used technique is the scan test (Kulldorff, 1997) because it is based on the likelihood ratio statistic and avoids the problem of multiple testing that bedevils this area of analysis. However, it only detects the presence of the most important cluster and is conservative in detecting the presence of other clusters. Other exploratory techniques for detecting localized clusters have been proposed (e.g., Besag and Newell, 1991; Getis and Ord, 1992; 1995; Anselin, 1995). The presence of localized clusters is one aspect of a wider concern with spatial heterogeneity — when map structure is different in different map segments (Fotheringham et al., 1997). Geographically weighted regression is used to explore for heterogeneity in the relationship between a dependent variable and a set of explanatory variables (Brunsdon et al., 1996). Implementing these techniques within software that creates both numerical summaries and graphical displays enhances ESDA (Wise et al., 1999). Ideally, the different display windows are linked so that the analyst can highlight cases (brush) in one display window (e.g., a graphics window showing a scatter plot or a box plot) and see the corresponding cases highlighted in other graphics windows (e.g., a map window). Brushing and dynamic brushing (where the highlighted area is moved by the user over the map and as it is moved the highlighting in the other windows is updated) are among the many visual techniques available to support ESDA (Haining, 2003, pp. 188–225).

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For applications of some of the techniques in this section in the context of economics, see for example: Le Gallo and Ertur (2003), Bateman et al. (2005, pp. 74–77), and Picone et al. (2009). 12.5. 12.5.1.

Confirmatory Spatial Data Analysis Likelihood-based inference

For much of the last 30 to 40 years, the likelihood function (or close variants of it) has been used for model fitting and model selection in the case of area data. However, the maximum likelihood estimate can in some cases be difficult to compute, especially in large samples, and the desirable properties associated with maximum likelihood estimation in the classical (iid) case are difficult to prove in the case of spatial models. An inspection of early modeling texts (e.g., Upton and Fingleton, 1985; Anselin, 1988; Haining, 1990) illustrates this. The basis for parameter estimation in likelihood theory is maximizing the logarithm of the likelihood function with respect to the unknown parameters. Hypothesis testing in the case of nested models, where one model is obtained by imposing constraints on a more general model, is based on the likelihood ratio which is the ratio of two likelihoods derived from competing hypotheses about the value of the unknown parameter. In more complicated, non-nested, modeling situations Akaike’s information criterion is used for model selection as it takes into account model complexity. We illustrate these points with reference to the first order simultaneous spatial autoregressive model (SAR). This model has figured prominently in the regional science and economics literature both as a model for describing certain types of spatial variation (see, for example the spatial competition models in Haining, (1983; 1984)) and as a model for spatially correlated errors in regression (Anselin, 1988). See also Dubin et al. (1999) and Anselin et al. (2004). The first-order SAR model for the variable Y measured at i = 1, . . . , n locations is given by Y (i) = µ(i) + ρ

n


w(i, j) [Y (j) − µ(j)] + e(i)

(1)

j=1

where w(i, j) equals 0 if i = j and if sites i and j are not “neighbors” and positive if they are. For example, in the case of a map divided into areas,

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two sites may be defined as “neighbors” if they share a common boundary but many other criteria can be, and have been, invoked (Haining, 2003, pp. 79–87). The weights are often defined so that row sums are standardized: n j=1 w(i, j) = 1. This introduces the restriction that ρ < 1. A positive value of ρ indicates positive spatial dependence between adjacent areas. The terms {e(i)} denote iid normal random variables with mean 0 and variance σ 2 . It follows that the distribution of the {Y (i)} is multivariate normal with the expected value of Y (i), E[Y (i)] = µ(i) for all i. Using the matrix W to denote the n-by-n set of {w(i, j)} then the autocovariance matrix for the distribution is given by Cov[(Y − µ), (Y − µ)T ] = σ 2 (I − ρW)−1 (I − ρWT )−1

(2)

where I is the n-by-n identity matrix and Y and µ are vectors of length n. The log likelihood of this model is given by −

1 n ln 2πσ 2 + ln |(I − ρWT )(I − ρW)| 2 2 1 − 2 (Y − µ)T (I − ρWT )(I − ρW)(Y − µ) 2σ

(3)

where |·| denotes the determinant and T denotes matrix or vector transpose. The maximum likelihood estimator for ρ cannot be written in closed form; so, numerical methods are needed to compute it. Evaluating the determinant term is a problem for very large matrices although the sparcity of the matrix can be exploited. As a result, a number of other non-likelihood estimation methods have been developed including pseudolikelihood and coding approaches (Besag, 1975). Note that maximum likelihood estimation and least squares estimation of ρ are not identical and while the former can be seriously biased in small samples the latter is also biased and may not be consistent (Whittle, 1954). For a fuller discussion of estimation methods, see Cressie (1991, pp. 458–498). Two widely used models that incorporate the SAR model are the regression model with spatially correlated errors and the regression model with a spatially lagged dependent variable (see Haining, 2003, pp. 313–314). The mean {µ(i)} is expressed in terms of a set of covariates ({Xj }j=1,...,k ) and corresponding regression parameters ({βj }j=1,...,k ) and intercept (β0 ). The regression model with spatially correlated errors is written Y (i) = β0 + β1 X1 (i) + β2 X2 (i)+, · · ·, +βk Xk (i) + u(i)

(4)

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where {u(i)} is assumed multivariate normal with mean zero and autocovariance defined by (2). The regression model with a spatially lagged dependent variable is written Y (i) = β0 + β1 X1 (i) + β2 X2 (i)+, · · ·, +βk Xk (i) + ρ

n


w(i, j)Y (j) + e(i)

j=1

(5) The inference theory for these two models can be found in Ord (1975). The Moran test for spatial autocorrelation, adapted by Cliff and Ord (1973) for the irregular areal partitions encountered in quantitative geography, approximates the likelihood ratio test for ρ
= 0 against the null hypothesis that ρ = 0 in the first order (normal) SAR model. The theory has been extended to testing for spatial autocorrelation in the residuals from a classical (least squares) regression where the errors are assumed to be iid (Cliff and Ord 1973; 1981; Brandsma and Ketellapper, 1979). There are other models for spatial variation that geographers have used on regional data, notably the conditional spatial autoregressive model and related “auto” models (Besag, 1974) and the moving average model (Haining, 2003, p. 302). In the case of all these models, likelihood inference tends to be problematic because of the bilateral nature of spatial interaction where an event in area i can depend on events in any direction from i. (By contrast time series events display unilateral dependency relationships, i.e., events at time t depend on events at t − 1 or before but not on time t + 1.) This in turn is because spatial models for describing spatial structure in regional data have tended to be written down in a form in which the observed variable (Y in the case of (1)) at a location i is expressed as a function of itself at other nearby locations (for details of some of the complications this causes see Haining, 2003, p. 301). Some Bayesian approaches to spatial modeling, which we examine next, have taken a rather different approach to modeling spatial variation. In conclusion, we noted in the previous section that the empirical semivariogram is an important tool for exploring second-order variation in a set of data. However, the empirical semivariogram cannot be used directly for spatial interpolation because it is not necessarily conditionally negative-definite. The solution is to fit a valid semivariogram model to the empirical semivariogram — selected from the set of theoretical semivariograms that do have the required property. If the normal assumption holds

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for the data, then maximum likelihood methods are sometimes used or restricted maximum likelihood (Cressie, 1991, pp. 90–94). Because spatial variation is modeled in geostatistics through the semivariogram rather than through the specification of interaction models the problems noted in the previous paragraph are avoided. We do not review these here but for more comparative comments see Haining et al. (2010). For applications on economic topics see, for example Gillen et al. (2001) and Addink (1999). 12.5.2.

Bayesian approaches

Bayesian inference consists of three probability components: the data likelihood and the prior and posterior distributions for the unknown model parameters. Let y denote the observed data and θ the vector of unknown model parameters. The prior distribution p(θ) expresses our beliefs and uncertainties about θ. Bayesian inference integrates the prior distribution with the data likelihood which specifies the probability of the data given the true value of θ. This integration, following from Bayes theorem, yields the posterior distribution — the probability distribution for θ given the data. The posterior distribution expresses the (revised) uncertainty about θ after taking into account the data. Often a vague (also known as a non-informative or a flat or a diffuse) prior is used so that the data play the dominant role in determining the posterior distribution. For further reading on Bayesian statistics, see Pollard (1986), Berger (1980), and Lee (2004). An early use of Bayesian inference with geographical data is given by Hepple (1974) but it was in the 1990s that interest among geographers and regional scientists in Bayesian approaches to spatial modeling began to develop. The work of Hepple and LeSage are particularly noteworthy. Hepple (1995) considers non-nested model comparisons in terms of weights (the W matrix) and spatial specifications while Hepple (2004), which includes a good basic introduction to Bayesian inference, extends the approach to both nested and non-nested spatial models. Le Sage (1997, 2000) considers Bayesian inference for the SAR model. A recent area for Bayesian inference has been in the modeling of counts for small areas. For the purpose of illustration we use the Poisson distribution although the same principle can be applied to fit the binomial (count data) or Bernoulli (binary data) distributions (see, for example Law and Haining, 2004).

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Let Y (i) be a count variable for area i, such as the number of cases of a disease or the number of burglaries. Covariates, {X(i )} = {X1 (i), X2 (i), . . . , Xk (i)}, are hypothesized to explain variation in the outcome variable. Provided that the count in each area is sufficiently small and the {Y (i)} are mutually independent, then each Y (i) may be assumed Poisson distributed with parameter λ(i). It follows that the expected value E[Y (i)] = λ(i). We assume λ(i) = E(i)R(i) where E(i) is the expected count that can be calculated under the assumption that the events are occurring at random across the study region, and R(i) is the area-specific relative risk in i. Taking the natural logarithm log λ(i) = log E(i) + log R(i)

(6)

R(i) is predicted using the {X(i)}: log R(i) = α + β1 X1 (i) + β2 X2 (i)+, · · ·, +βk Xk (i), where α, β1 , β2 , . . . , βk are the usual regression parameters. This yields the usual Poisson generalized linear model where log λ(i) = log E(i) + α + β1 X1 (i) + β2 X2 (i)+, · · ·, +βk Xk (i)

(7)

with log E(i) an offset term (McCullagh and Nelder, 1983, p. 138). To allow for the presence of spatial structure in the relative risks that maybe the result of spillover effects between adjacent regions or to unmeasured or even unknown area level covariates and/or data errors that are spatially autocorrelated, a spatially structured random effects term, S(i), is added to (7): log λ(i) = log E(i) + α + β1 X1 (i) + β2 X2 (i)+, · · ·, +βk Xk (i) + S(i)

(8)

The spatial random effects term may also model any extra-Poisson variation or overdispersion (Haining et al., 2009). S(i) can be specified in different ways. The most commonly used spatial random effects models are the proper conditional autoregressive (CAR), intrinsic conditional autoregressive (ICAR), and the convolution models. We discuss these briefly below. In the case of the proper CAR model Prob{S(i)|S(j) = s(j), j =
i, j a neighbor of i }  
w(i, j)s(j), ω(i)2  ∼ N γ j=i

(9)

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The conditional distribution of S(i) given its neighbors {S(j)} is normal  with mean γ j=i w(i, j)s(j) and variance ω(i)2 . This variance parameter controls the amount of variability in {S(i)}. γ is a spatial dependency or spatial interaction parameter (similar to ρ in (4) and (2)) which is to be estimated. γ must lie between (1/χmin) and (1/χmax ) where χmin and χmax are the smallest and largest eigenvalues of the contiguity matrix W = {w(i, j)}. With small area data, w(i, j) is usually assigned a value of one if i and j are neighbors and zero otherwise. The ICAR model is a special case of (9) where γ = 1. The distribution  (9) becomes: N (s(i), ω 2 /m(i)), where s(i) = j=i w(i, j)s(j)/m(i) and  m(i) = j w(i, j) is the number of neighbors of area i. When an unstructured random effect term, U (i), is added to an ICAR model this defines a convolution model, sometimes referred to as a convolution prior (Besag et al., 1991). The convolution model combines both spatial and non-spatial structured random effect terms and is the most robust model among those with spatial structure (Lawson et al., 2000; Spiegelhalter et al., 2002). For further information on these and other approaches to modeling spatial random effects, see Mollie (1996) and Wakefield et al. (2000). With the advance of computation technology, spatial random effect models, which were previously intractable, can now be fitted in WinBUGS using Markov chain Monte Carlo (MCMC) which simulates posterior distributions. BUGS stands for “Bayesian inference Using Gibbs Sampling”. WinBUGS can be downloaded free from http://www.mrc-bsu. cam.ac.uk/bugs/. Figure 12.1 shows a typical programming code in WinBUGS for a Poisson model with a convolution spatial random effect (S(i; γ = 1) + U(i)) and one covariate X1 . U (i), is normally distributed with mean zero and precision (=1/variance) prec.u. S(i), is specified by the ICAR distribution (car.normal in winBUGS) with precision parameter prec.s. Both prec.u and prec.s have the gamma distribution (a, b) for priors. A common choice for a and b are 0.5 and 0.0005, respectively, which provides a plausible range for relative risks (Kelsall and Wakefield, 1999, but see also Spiegelhalter et al., 2004). Sensitivity analysis to ensure that estimates of model parameters are consistent while using different noninformative priors is necessary. The intercept term, α, is assigned an improper uniform prior, dflat(), as required by WinBUGS for the ICAR (car.normal) distribution, due to the inclusion of a sum-to-zero constraint on the spatial random effects. The prior specified for the regression

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Geographical Information Systems Models and Spatial Data Analysis model { for (i in 1 : N) { is the total no. of small areas

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#N O[i] ~ dpois(lambda[i]) log(lambda[i])