Annals of the International Society of Dynamic Games Volume 2

Series Editor TamerBa~ar

Editorial Board

Leonard D. Berkovitz, Purdue University P. Bernhard, INRIA, Sophia-Antipolis R. P. HanUiUiinen, Helsinki University of Technology Alain Haurie, University of Geneva N. N. Krasovskii, Academy of Sciences, Ekaterinburg George Leitmann, University of California, Berkeley G. J. Olsder, Delft University of Technology T. S. E. Raghavan, University of Illinois, Chicago Josef Shinar, Technion-Israel Institute of Technology B. Tolwinski, Operations Research Experts, Black Hawk, Colorado Klaus H. Well, Stuttgart University

Annals of the International Society of Dynamic Games

Control and Game-Theoretic Models of the Environment

CarIo Carraro Jerzy A. Filar Editors

Springer Science+Business Media, LLC

Carlo Carraro Universita degli Studi di Venezia Dipartimento de Economica Ca Fascari Venezia, Italy

Jerzy A. Filar School of Mathematics University of South Australia The Levels, SA 5095, Australia

Library of Congress Cataloging-in-Publication Data

Control and Game-Theoretic models of the environment I Carlo Carraro, Jerzy A. Filar, editors.

p. cm. -- (Annals of the International Society of Dynamic Games ; v.2)

Includes bibliographical references.

(hc : alk. paper ) 1. Environmental policy--Mathematical models. 2. Sustainable

development--Mathematical models. 3. Game theory. 4. Control theory. 1. Carraro, Carlo, 1957- II. Filar, Jerzy. III. Series. GEI70.G36 1995 95-22370 363.7'OOI5118--dc20 CIP

ISBN 978-1-4612-6917-5 ISBN 978-1-4612-0841-9 (eBook) DOI 10.1007/978-1-4612-0841-9

Printed on acid-free paper © Springer Science+Business Media New York 1995 Originally published byBirkhiiuser Boston in 1995

Softcover reprint ofthe hardcover Ist edition 1995

Copyright is not claimed for works ofU.S. Government employees. All rights reserved. No part of this publication may be reproduced, stored a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopy­ing, recording, or otherwise, without prior permission of the copyright owner.

Reformatted from authors' diskettes by Texniques, Inc., Brighton, MA

9 8 7 6 5 4 3 2 1

Table of Contents

Preface Carlo Carrara and Jerzy A. Filar . .................................... vii

Introduction: The Environmental Game O.J. Vrieze ......................................................... . xvii

Part 1: Models of Global Change and Sustainable Development

Differential Game Models of Global Environmental Management A. Haurie and G. Zaccour ............................................. 3

Sustainability and the Greenhouse Effect: Robustness Analysis of the Assimilation Function Herman Cesar and Aart de Zeeuw .................................... 25

Consumption of Renewable Environmental Assets, International Coordination and Time Preference Andrea Beltratti ...................................................... 47

Sustainable International Agreements on Greenhouse Warming­A Game Theory Study Veijo Kaitala and Matti Pohjola ...................................... 67

The Environmental Costs of Greenhouse Gas Emissions Michael Hoel and Ivar Isaksen ........................................ 89

Part 2: Environmental Taxes and Related Issues

Taxation and Environmental Innovation Carlo Carrara and Giorgio Topa ..................................... 109

Environmental Quality, Public Finance and Sustainable Growth Jenny E. Ligthart and Frederick van der Ploeg . ...................... 141

Environmental Pollution and Endogenous Growth: A Comparison Between Emission Taxes and Technological Standards Thierry Verdier ...................................................... 175

Rate-of-Return Regulation, Emission Charges and Behavior of Monopoly Anastasios Xepapadeas .............................................. 201

vi Table of Contents

Polluter's Capital Quality Standards and Subsidy-Tax Programs for Environmental Externalities: A Competitive Equilibrium Analysis Michele Moretto ..................................................... 231

Part 3: Pollution, Renewable Resources and Stability

The ESS Maximum Principle as a Tool for Modeling and Managing Biological Systems Thomas L. Vincent .................................................. 259

Pollution, Renewable Resources and Irreversibility OUi Tahvonen . ...................................................... 279

The Economic Management of High Seas Fishery Resources: Some Game Theoretic Aspects Veijo Kaitala and Gordon Munro .................................... 299

Pollution-Induced Business Cycles: A Game Theoretical Analysis David W.K. yeung .................................................. 319

Management of Effluent Discharges: A Dynamic Game Model Jacek B. Krawczyk ................................................... 337

Preface

This book collects some recent works on the application of dynamic game and control theory to the analysis of environmental problems. This collec­tion of papers is not the outcome of a conference or of a workshop. It is rather the result of a careful screening from among a number of contribu­tions that we have solicited across the world. In particular, we have been able to attract the work of some of the most prominent scholars in the field of dynamic analyses of the environment. Engineers, mathematicians and economists provide their views and analytical tools to better interpret the interactions between economic and environmental phenomena, thus achiev­ing, through this interdisciplinary effort, new and interesting results.

The goal of the book is more normative than descriptive. All papers include careful modelling of the dynamics of the main variables involved in the game between nature and economic agents and among economic agents themselves, as well-described in Vrieze's introductory chapter. Fur­thermore, all papers use this careful modelling framework to provide policy prescriptions to the public agencies authorized to regulate emission dy­namics. Several diverse problems are addressed: from global issues, such as the greenhouse effect or deforestation, to international ones, such as the management of fisheries, to local ones, for example, the control of effluent discharges. Moreover, pollution problems are not the only concern of this book. A correct, sustainable exploitation of renewable natural resources is also the objective of some analyses and policy recommendations (see, for example, the papers by Yeung and Kaitala-Munro). A common theme can be found throughout the book. There is recognition that an environmental problem and its interrelationships with economic activity and the dynamics of eco-systems are very complex and cannot be resolved with simple policy tools. Instead, it is necessary to use properly designed policy-mixtures in which several policy instruments are set at their "optimal" levels. Hence, policymakers both at the national and international level should debate over policy interventions which account for the interdependence between several tools and objectives in a dynamic framework, in order to prevent the economic and natural systems from taking unsustainable development paths.

Notice the importance of a dynamic analysis of environmental issues. The concern for future generations (explored in Beltratti's paper), the role of the assimilative capacities of nature (see Cesar and de Zeeuw's paper),

viii C. Carraro and J.A. Filar

the importance of irreversibilities (as in Tahvonen's work) and of capital stock's accumulation (cf. Xepapadeas' and Moretto's articles) are some of the elements that make it impossible not to use a dynamic framework when studying environmental problems. Moreover, the dynamic framework is often coupled with a game-theoretic one. This is a consequence of the natural context in which regulators' and polluters' decisions are recipro­cally interdependent. However, more subtle reasons are proposed in the book. For example, the interaction (game) among polluters may prevent traditional emission charges from being completely effective, thus calling for other policy tools (see, among others, Carraro-Topa's and Ligthart's papers). In a different way, interactions among governments at the inter­national level may offset, through the so-called leakage effects, unilateral efforts to reduce greenhouse gas emissions, thus calling for internationally coordinated policies (see, for example, the papers by Haurie-Zaccour and Kaitala-Pohjola).

The book is divided into three parts. The first part is devoted to mod­els of global change and sustainable development. It contains recent con­tributions in the field of dynamic modelling of global warming and global environmental management, including issues of sustainable development, international policy coordination and optimal growth with limited renew­able natural resources.

More precisely, in the paper "Differential Game Models of Global En­vironmental Management," A. Haurie and G. Zaccour deal with the mod­elling of economy-environment interactions for several countries which are assumed to behave competitively for the control of their own economic pro­cesses, but have to achieve jointly a common environmental management goal. In the first part of their paper, Haurie and Zaccour treat the modelling of issues related to process dynamics and players' interactions respectively. They discuss the control theoretic approach for representing the economic and pollution processes in a model of environment management, and also give a short discussion of a possible representation of uncertainty and risk in these models. The second part of the paper proposes a dynamic game modelling approach which combines the N-person equilibrium model with coupled constraints and the differential game model with active and pas­sive players. This approach enables the authors whose strategies are such to define a tax scheme which induces a set of equilibrium seeking players to achieve, in the long run, a global environmental goal.

The second paper of the first part, "Sustainability and the Greenhouse Effect: Robustness Analysis of the Assimilation Function" by H. Cesar and A. de Zeeuw, faces the issue of global environmental management and sus­tainable growth by focusing on the role played in dynamic environmental models by the assimilation function. In most existing optimal control mod­els with an environmental stock it is generally assumed that the assimilation

Preface ix

function of nature is linear. At the same time, however, there is quite some scientific uncertainty on the general form of this function outside a spe­cific range of values. In their paper, Cesar and de Zeeuw consider different (non-linear) specifications of the assimilation function in the case of the greenhouse effect. The optimal trajectories and the steady states are anal­ysed for the various functional forms. Slight variations in the assimilation function can result in a dramatic change in the steady state values. Besides, neither multiple equilibria nor the absence of non-zero production steady states can be excluded. Hence, this paper provides useful information on the sensitivity of dynamic environmental analyses on the specification of the assimilation function of nature.

A similar robustness analysis is carried out in "Consumption of Re­newable Environmental Assets, International Coordination and Time Pref­erence" by A. Beltratti, in which a two-country linear-quadratic model of depletion of a renewable resource is studied both in static and in dynamic terms. The model allows for negative consumption externalities through the action on the stock of the environmental good, which enters the util­ity function. The main result of the paper shows that the noncooperative solution of the dynamic model is characterised, in the steady state, by suboptimally low levels of environmental resources and consumption, thus providing incentives for international cooperation. The robustness of this conclusion is then checked with respect to several parameters defining the structure of the economy. In particular, attention is given to the rate of intertemporal time preference, a parameter which captures the preference for future generations' welfare of present governments. It is shown that the sub-optimality of the non-cooperative solution increases with the rate of time preference in a nonlinear way which depends on the other parameters of the economy.

The issue of policy coordination is also the object of the subsequent paper, "Sustainable International Agreements on Greenhouse Warming: A Game Theory Study" written by V. Kaitala and M. Pohjola. It is clear, and is well-described in O. Vrieze's introductory paper, that reducing dam­ages caused by climatic changes requires major international efforts. Many countries bear the view that the joint efforts should be undertaken under international agreements. The paper by Kaitala and Pohjola presents a dy­namic game theory model for an international environmental negotiation problem that may arise in the context of global climate change. The basic assumption is that countries differ in their vulnerability to global warming and that two coalitions will possibly be formed. One coalition may include countries that do not suffer from global warming, or where the damages are minor, and the other coalition may be joined by countries that will suffer as a result of global warming. The greenhouse problem is then modelled as an economic infinite-horizon differential game. Countries negotiate an

x C. Carrara and J.A. Filar

agreement among Pareto-efficient programs by allowing for transfer pay­ments which account for the existing international asymmetric effects of global warming. Transfer payment programs are designed in such a way that it is possible at any stage of the agreement to punish violations against cooperation and to discourage the other player from selfishly polluting the atmosphere. In this context, it is shown that the incentives for international cooperative control of global warming will become stronger and will occur with an increasing frequency.

The last paper of Part 1 of the book, "The Environmental Costs of Greenhouse Gas Emissions" by M. Hoel and I. Isaksen, deals with a crucial issue in environmental negotiations, namely the evaluation of the costs of global warming. An efficient, comprehensive climate policy should balance the cost of reducing emissions of each greenhouse gas against the environ­mental costs of the emissions of the gas. However, the assessment of these costs is generally a difficult task. In their paper, Hoel and Isaksen show how these environmental costs may be calculated using an optimal control dynamic model. This is first done for the traditional case in which, at any time, one is only concerned about the state of the climate. Then, a more general environmental cost function is considered, for which it is assumed that the rate of climate change is more important for the environment and the economy than the state of the climate. Besides providing a clear pre­sentation of the methodology, the paper also shows, through a numerical example, how the marginal costs of greenhouse gas emissions for both types of environmental cost function can be calculated.

The second part of the volume is devoted to the analysis of the in­teractions between technical progress, economic growth and environmental protection. The papers included in this second part are characterised by solid micro-foundations, and rely upon recent developments of industrial organisation theory. In particular, the robustness of traditional environ­mental policy prescriptions to imperfect competition is assessed in several papers. Therefore, this part of the book provides new insight into the optimal environmental policy-mix by showing how several tools may be ac­tivated in order to control polluting emissions without excessively damaging economic growth.

In the first paper, "Taxation and Environmental Innovation," C. Car­raro and G. Topa propose an industrial organisation dynamic model to analyse the effects of environmental taxation on firms, innovation activ­ity. A regulator is assumed to introduce an environmental tax. Firms may react both by changing output and by adopting anew, environmentally friendly technology. Conditions under which innovation is a firm's opti­mal choice are provided. The paper shows that firms innovation decisions are not simultaneous even when firms are identical (there exists diffusion). Moreover, firms have an incentive to delay the time of innovation, because

Preface xi

the new technology can only be achieved through costly R&D. Hence, there exists room for incentives that move firms to the socially optimal timing of innovation. These incentives have to account for the presence of asymmet­ric information (the regulator is assumed not to observe firms, innovation costs). In this context, Carraro and Topa show that there exists a family of contracts defined by a pair (time of innovation, innovation subsidy) as to induce firms to behave optimally. The proposed policy-mix (environ­mental tax and innovation subsidy) is shown to reduce emissions more, and to reduce output less, than environmental policies based on a single policy instrument.

A similar multiple instrument approach to emission control is proposed, within an endogenous growth model, by J. Ligthart in her "Environmen­tal Quality, Public Finance and Sustainable Growth." This paper extends theories of endogenous growth in order to deal with the optimal trade-off between economic growth and environmental quality in a meaningful fash­ion. Environmental quality is modelled in two different ways: (i) as a given stock which is damaged by a flow of pollution; and (ii) as a renewable re­source which is used as an input in production. After a brief discussion of pollution, taxation and the cost of funds, attention is focused on renewable resources in order to come to grips with the concept of sustainable growth. The government reduces the use of natural resources and improves environ­mental quality by imposing a levy on firms. Economic growth is boosted by productive government spending, but is hampered by distortionary taxes on income or capital. The first-best outcome can be sustained in a competitive market economy only if lump-sum taxes and subsidies are available. In gen­eral this is not the case, so the paper focuses on the setting of government policies in a second-best context, showing how different policy instruments have to be combined in order achieve an adequate control of environmental resources.

A careful comparison of environmental policy instruments in micro­founded endogenous growth models is also the objective of T. Verdier's paper "Environmental Pollution and Endogenous Growth: A Comparison Between Emission Taxes and Technological Standards." This paper devel­ops a model of endogenous growth with environmental pollution. Firms create, through R&D, new products and also design the environmental fea­tures of these products by choosing their output-emission ratios. Cleaner products are assumed to be more costly to develop than dirty products. Using an extension of the expanding variety product of Helpman and Gross­man, this paper investigates and compares the effects of emission taxes and technological standards. In particular, in the second best context where R&D subsidies are not possible, Verdier provides a welfare comparison of two instruments, emission taxes and technological standards, for a given pollution target that the policymaker wants to implement in the economy.

xii C. Carrara and J.A. Filar

Under certain conditions, it is shown that an emission tax, acting as an implicit R&D subsidy, may induce too much growth of the polluting indus­try compared to what is socially optimal. This effect can then reverse the usual cost effectiveness superiority of taxes over technological standards.

The problem of excessive growth induced by environmental policy is also analysed by A. Xepapadeas in his "Rate of Return Regulation, Emis­sion Charges and Behaviour of Monopoly." The well-known Averch-Johnson thesis indicates that the main result of rate-of-return regulation is overcap­italisation. By extending the model to include environmental externalities, this paper assumes that a regulated monopoly can adhere to environmental policy by undertaking investment in pollution abatement equipment, along with investment in output production. In this context, over- (or under-) capitalisation effects have a direct influence on the monopoly's emissions. Hence, it becomes crucial to verify whether optimal policy still induces over­capitalisation. Therefore, this paper analyses two related issues. The first is the direction and distribution of the effects of introducing rate-of-return regulation under a given environmental policy on investment in productive and pollution abatement equipment. The second is whether the regulated firm responds in the same manner as the unregulated firm to the introduc­tion of the above environmental policy.

Xepapadeas' paper analyses the impact of environmental policy on the level of the capital stock. By contrast, M. Moretto's paper "Polluter's Cap­ital Quality Standards and Subsidy-Tax Programs for the Environmental Externalities: A Competitive Equilibrium Analysis" focuses on the qual­ity of the capital stock. More precisely, the paper concentrates on the role of the physical features of the fixed assets in determining the extent of discharges. It considers the case where firms have access to a technol­ogy which allows them to regulate the quality of capital instantaneously, through a lump-sum maintenance expenditure which applies only when the state variable achieves a predetermined minimum quality standard. In a partial equilibrium framework (single firm and a long-run competitive in­dustry), the paper investigates the relationship between the optimal firm's barrier policy comprising the capital's minimum quality standard and the use of a subsidy/tax program for decreasing pollution emissions by those who generate externalities. Again, the main message is that the complexity of dynamic interactions between economic and environmental phenomena call for "sophisticated" policy measures in which several policy instruments are implemented.

The third part of the book contains several dynamic analyses of issues related to pollution activities in the presence of renewable resources and the stability of the economic and/or ecological system. The stability of a biological system facing damages induced by economic activity is indeed the object of T. Vincent's analysis in his "The ESS Maximum Principle

Preface xiii

as a Tool for Modelling and Managing Biological Systems." Ever since the advent of DDT and the discovery of mutant strains of mosquitoes immune to DDT, it has been public knowledge that ecosystems can and will evolve in response to human beings' efforts at control. While differential equations have been in common use as management models, it is uncommon to find any such models that attempt to capture the evolutionary game approach to modelling which should provide more realistic application. In order to include evolution into management models, and point out some areas of possible application, we are faced with two fundamental questions: what is evolving? And where is it evolving to? In the evolutionary game theory presented in Vincent's paper, the "what" are parameters in the differential game model associated with characteristics of the species that are clearly adaptive (such as sunlight conversion efficiency for plants or body length in animals). The "where" are the evolutionary stable strategies (ESS) to which these parameters can evolve. These strategies can be determined using the ESS maximum principle. This principle is extended here to include a wider class of models. The ESS maximum principle, when used with appropriate models, has the capacity to predict the evolutionary response of biological systems in response to human inputs. These inputs can include physiographic changes, harvesting, and the introduction or removal of new species and/or resources.

The stability of the ecosystem in the presence of irreversible pollution damages is also discussed in O. Tahvonen's paper "Pollution, Renewable Resources and Irreversibility." This study shows that irreversible pollution damage leads to nonconvexities in dynamic models. There may exist two locally optimal solutions: an optimal infinite horizon solution (sustainable) and an optimal finite horizon solution. In general, the choice between these optimality candidates must be made by comparing the present values of both policies. However, the study shows that there are special cases where the choice can be made on a priori grounds. In particular, it is shown that including the pollution problems in the renewable resource model changes the ordinary "optimal extinction" results.

Another ecosystem is analysed by V. Kaitala and G. Munro who fo­cus on "The Economic Management of High Seas Fishery Resources." This paper deals with the economic analysis of both "shared" and "straddling" fishery stock management. There are at least two important differences between the analysis of "shared" and "straddling" stocks. The paper des­ignates these differences as: 1) the problem of new entrants; 2) the number of participants or "players." In "shared" stock fisheries management, the number of coastal states, as joint owners of the resource, is fixed. In the case of straddling stocks, on the other hand, the existing Law of the Sea Convention allows, to some extent at least, hitherto non-participatory dis­tant water fishing nations to enter the high seas portion of a straddling

xiv G. Garraro and J.A. Filar

stock fishery. If unimpeded access is granted to new entrants, any attempt at cooperative management of a straddling stock may be undermined from the start. With respect to the number of participants or players, most models of economic management of "shared" fishery resources involve but two coastal states or "players." The assumption of bilateral exploitation of the relevant fishery resources proves to be a reasonable one in many real world cases of "shared" fishery resource management.

By contrast, in analysing the management of "straddling stocks," one cannot be content with the assumption that the resource is exploited by one coastal state and by only one distant water fishing nation. The typical "straddling" stock case is one in which a coastal state confronts two or more distant water fishing nations operating in the adjacent high seas. Moreover, the relevant set of distant water fishing nations may change through time. These facts greatly complicate matters and result in the analysis of "straddling" stock management being far more complex than the analysis of "shared" stock management. Given these differences between "shared" and "straddling" stock management, the paper by Kaitala and Munro first reviews briefly the economist's standard model of a fishery. Then it provides a detailed analysis of the case of "straddling" stock in which the coastal state does in fact confront but one distant water fishing nation and in which new entrants are effectively barred forever. This is the "straddling" stock case which most closely corresponds to the typical case of "shared" stock management applies with little or no modification. Next the paper relaxes the assumption of bilateral exploitation of the "straddling" stock, and allows for a situation in which the coastal state confronts three or more distant water fishing nations in the adjacent high seas. However, when the issue of new entrants is addressed, the paper does not even attempt to provide a full analysis. Rather, it provides an initial exploration of this issue, and lays out an agenda for future research.

Pollution management in the industrial sector is the objective of D. Ye­ung's analysis in the paper "Pollution-Induced Business Cycles: A Game Theoretic Analysis." In this paper, the industrial sector chooses the level of investment to maximise net revenue and the government imposes a tax and uses the tax proceeds for pollution abatement operations. The feedback of pollution on capital accumulation and the effect of the level of pollution on the natural rate of decay are incorporated in the model. The author solves for the (subgame perfect) feedback Nash equilibrium solution of the resulting differential game, and obtains explicitly the game equilibrium ac­cumulation dynamics of capital and pollution. Various properties of the equilibrium follow from this closed form solution. It is found that the game equilibrium output path exhibits continual oscillation about a long run equilibrium level. Moreover, when a constant rate of decay is introduced into the model, damped output cycles appear.

Preface xv

The dynamics of pollution resulting from the solution of a dy­namic game between government and polluters are also the outcome of J. Krawczyk's analysis of the management of effluents dumped into a stream by identifiable polluters. In his paper "Management of Effluent Discharges: A Dynamic Model," Krawczyk analyses the game between a Regional Coun­cil and some polluting firms, where the Regional Council imposes environ­mental levies on the polluters whose economic activity, otherwise beneficial for the region, results in pollution of the stream. The game is "played" in discrete time. In the game, polluters are the "followers," whereas the Council is the "leader." This formulation leads naturally to a Stackelberg solution concept for the game at hand. However, because of the obvious difficulties implied by this solution concept, in the paper an equilibrium is sought through the use of an applicable Decision Support Tool, wher­ever an analytical solution appears intractable. The polluters are supposed to be myopic and small; the Regional Council is interested in promoting production, collecting taxes, and in the clean environment. Moreover, the model of spread of the pollution within the stream allows for biodegrada­tion. The results suggest the possibility that adequate policy instruments can be found to manage effluent discharges in an optimal way.

This brief introduction to the papers contained in this volume is not meant to cover all issues analysed in the volume. Furthermore, this collec­tion of papers should be viewed as one of the first attempts to demonstrate that control and game theoretic techniques can be effectively applied to analyse important environmental problems. As such, we hope that the book will stimulate the reader to raise new questions, to discover new problems and methods, and to ask for more precise policy prescriptions. Of course, the entire subject of quantitative modelling of environmental problems is still in its infancy. Nonetheless it is a subject that will have profound impact on science, economy, society and ultimately on the environment in which we live. The chapters of this book demonstrate the complexity of the underlying problems; however, they also provide some answers, tools, and recommendations. They will certainly show the advances recently achieved by the theory of dynamic environmental games. Many of the papers will also prove to be useful for applied economists and policymakers. Finally, we believe that directions for future research are also clearly indicated, thus providing new stimuli for further advances in the understanding of the environment.

Carlo Carraro and Jerzy A. Filar

The Environmental Gamel

O.J. Vrieze

1. Introduction

In this contribution we want to motivate why game theory is indispensable to the understanding and tackling of the many current environmental prob­lems. The title of this contribution should be interpreted in two ways. On the one hand, there is the game of exhaustion that the population of our planet is playing against the environment. It will be explained in Section 2 why it is that, indeed, we can talk here about a game in the sense that there are two or more participants who, with a certain degree of freedom, may choose strategies in order to meet their goals. On the other hand the title refers to the many different types of games, induced by environmen­tal considerations, that occur between world regions or within regions or countries. All these games are based on the fact that there is a worldwide concern about the quality of the environment which, perhaps, can only be improved with the aid of unpopular measures. The main question that arises is: whose shoulders should bear the most heavy weights? In Section 3 this will be outlined in more detail. In both of the above interpretations of the title of this paper, game theory offers a natural framework to study the properties of the decision situation, as will be shown in the next two sections. In Section 4 we conclude that game theory can help our environ­ment to survive while, on the other hand, the survival of game theory can be strongly supported by the success of the environmental game.

2. The Game of Exhaustion

By the game of exhaustion, we mean the continuing interactions between homo sapiens and the environment. We restrict ourselves to those interac­tions that influence, at least partly, the course of development for the other participants. Obviously, environmental pollution is a prime example of this. At first glance, it appears that the interactions between human beings and their environment lack the characteristics of a game, in the sense that both

1 Editor's Comment: This paper is an invited commentary on one of the main themes of the volume: the relevance of game theory to environmental problems. Professor Vrieze has been active for a number of years in the areas of theoretical game theory, public health and environmental modelling.

xviii O.J. Vrieze

participants make moves. One might think that only human beings make moves. In my opinion this viewpoint is too narrow. Especially during the past fifty years there are enumerable examples that show that nature can and does strike back.

For instance, think of the depletion of the ozone layer, resulting in an increase of the incoming UV-radiation at the earth's surface. In fact this can be interpreted to be a very clever move by nature. The consequence of an increase in skin cancer frightens enough people to put pressure on industry to change to nature friendly methods of production. Thus the way that nature responds to the stresses that our modern society places on the environment can be seen as a strategy.

Of course, one could argue that the moves of nature are a deterministic reaction to human activities, governed by the rules of chemistry, physics, biology and ecology. The latter argument may be right; who knows? On the other hand the possible ways that nature can counteract are so abundant, and the exact outcome depends on so many tiny aspects, that nobody will ever be able to predict how nature will adapt in the coming century. The best thing one can hope for is that scientists will be able to provide us with probabilities, indicating the chances for certain future trends to emerge in response to future human attacks on the environment. Indeed, in this way the reaction of nature can be interpreted as a strategy choice out of an available set of strategies that fit within the rules of the game. It is not clear to me whether nature possesses its own payoff function in this game. And if it possesses one, what is the form of that function? And also for human beings, aggregated as the world population, the payoff function is not easily defined. Definitely, optimization of utility is the driving force. But can utility be identified with welfare, with social and economic stability, or with sustainable development?

Especially with respect to the above-mentioned criterion of sustainable development, a conflict is emerging between enjoying life now or taking care for a future where the quality of life can reach new heights. This aspect is far from being solved. During the recent past it looked as if short-term optimization of personal needs received the most attention. Fortunately, nowadays many people are willing to think beyond their lifespan. Con­ceiveably, this is inspired or even forced by the constitution of their genes.

What can game theorists contribute to the game described above? In the first instance, they should convince society that indeed a game is being played out: namely, that not only the strategy of the world population determines the future, but also the strategy, or a counteraction if you wish, of the environment, and that it is the combination of the two that will determine future life conditions. The next contribution of game theorists concerns the characterization of this game. Should it be described as a noncooperative game or as a bargaining game? Is it a leader-follower game

The Environmental Game xix

of the Stackelberg type and are differential games the appropriate settings? Or does this game fit into the concept of fuzzy games? And what about the role of information, say actual knowledge of the game data? Here we should mention the need for the gathering of reliable data. This will prove to be an enormous job in the coming years. In many decision situations the structure of the model is understood and says something about the types of data that are needed, but this is no guarantee that these data can be found. At a very early stage of the investigations the game theory researcher should seek clarification from the physicists, chemists, biologists, etc. as to what data they need and which level of uncertainty is acceptable. Only through such a close cooperation can a useful model be created.

In recent years, an enormous amount of mathematical effort has been put into game theoretic models with incomplete information. Ultimately, these might well be applicable. However, the present formulations of these models are far from being applicable. Thus there is a task for game theorists to bring the existing models closer to reality. This holds, by the way, not only for the models we are discussing here. In general, there is a tendency in game theory to analyse, very complex theoretical situations of conflict. Of course such work has to be done; however, not at the expense of neglecting the translation of the models to practical situations. The conclusion is that the game between homo sapiens and its environment offers an open field containing many interesting game theoretic questions, with a promising opportunity to support humans in their endeavor to play the game optimally.

3. The Society's Game for the Environment

As mentioned in the introduction, the society's game is based on two histor­ical features that have created the present situation. Firstly, not all coun­tries or world regions have contributed in the same way to the present state of pollution. Secondly, the present welfare status differs widely among the world population and therefore the willingness to contribute to a reduction of the pollutant emission rates also varies widely. That is to say, nobody freely wants to contribute: the developed countries want to maintain their welfare level and the developing countries want to reach the quality of life of the people in the industrialized countries. So, indeed here we have the situation of conflict which is so characteristic of game theory applications. At two levels one can distinguish game theoretic situations, namely the game between the developed countries and the developing countries, of­ten refered to as the "North-South game," and secondly the game between either the countries of the developed world regions or between those of the developing countries. This last class of games can be refined a bit further

xx O.J. Vrieze

in the sense that, within a country, there is a competition between indus­tries or companies, etc. Below we shall pay further attention to these two levels of conflict. We like to exhibit the game theoretic aspects by using the problem of the enhanced greenhouse effect as an example.

3.1. The North-South game

The present threats due to the greenhouse effect are caused by more than 75 countries. Energy production was the leading contributor. On the other hand, energy production created the present level of welfare. Hence, why should the developed countries reduce their emissions of greenhouse gases?

There are at least three reasons for a reduction of emissions, which can be called the practical, the ethical and the natural. The practical reason is that a continuation, or possibly even an increase, of energy consumption might cause great societal and economical worldwide instability, which ob­viously will impact on society. If the welfare gap between North and South is not narrowed in the coming century, the unfulfilled claim for a better world by the developing countries will lead to an explosion, with enormous drawbacks on everybody's welfare (utility functions). The ethical reason is the obvious one that all human beings on earth are the same, and that they have the same rights with respect to the use of the resources that the environment offers us. Thus, the developed countries are "obliged" to help the developing countries in creating an enjoyable world. The natural reason for emission reductions concerns the fact that the greenhouse effect, geophysically implied, presents itself at a mondial level. That is, green­house gases spread out through the atmosphere all over the world. Hence, production of greenhouse gases in one part of the world affects the climatic system all over the world. So countries will keep watching each other's emissions and will give warning signals when other countries behave too selfishly. Further, for the countries still in a developing phase, this natural property of greenhouse gases offers them some kind of a weapon: namely, if the rich countries are not willing to share their technology and other achievements, then the developing countries will start welfare-improving economic programs, no doubt based on energy production, which as ar­gued, may have adverse long-run effects on the living circumstances in the Northern countries.

What kind of a game is going on between North and South? With reference to the above considerations one could argue that it is noncoop­erative by its conditions. The strategy space of the North is especially bounded by political reasons to limit its emissions, as outlined above. At least three aspects of the strategies should be taken into account, namely the emission rate of greenhouse gases (i.e., the energy consumption), the effort to develop new technologies for efficient energy production and usage,

The Environmental Game xxi

and the willingness and extent of the Northern countries to support, either financially or technically, the Southern countries. The payoff functions will generally express the GNP or some utility function based on the level of welfare of a country, summed up over a certain time period.

The game theoretic solution of such a game concerns the existence of certain strategies for all the participants, which are acceptable in one or another way and which possess enough stability in order to expect that the players will stick to them. Obviously, the concept of an equilibrium point could serve the purpose. Again, of great importance for a practical application of such a game is the relation between the game model and reality. And again, the game theorist should make clear how the modelling should be performed and how the distance between the game model and the practical situation can be measured. I think that only then the deci­sion-makers will be willing to guide themselves by game theoretic models.

A relevant way to give politicians insight into the power of game theory concerns gaming sessions. In gaming sessions the participants fictitiously play the game in an accelerated time scale. They can observe how other players handle the game situations and what kind of long-run consequences their strategy choices will have. Evidently the above sketched game model can also be treated as a basis of a bargaining model or even as the input data for a cooperative game in characteristic function form, either with or with­out side payment. The bargaining model especially might be applicable, since in international negotiations participants often reveal disagreement strategies, even though it is not always obvious that they will turn to them in case of a disagreement.

A very important contribution of game theory concerns the investiga­tions of the differences in the solutions of the different game models (e.g., noncooperative, cooperative or bargaining). Informing the politicians which game conditions induce these differences might very well influence their at­titude towards the search for solution (compromise?) strategies. It might be clear that many game theoretic models could support the North-South dialogue. It is the responsibility of game theorists to take the initiative and to show the politicians the strength and value of this scientific theory of conflict.

Though the above analysis is inspired by the North-South problems, its applicability can, of course, be easily extended to situations with more than two players. In reality there exists a whole range of world regions, that have an increasing order of welfare status development. We think that this fact should not affect the usefulness of game models and that the model builders should take this aspect into account.

xxii O.J. Vrieze

3.2. Interregional cost allocation

Finally we would like to say a few words about what may be happening within a region or a country. Suppose that the limits on the emissions of greenhouse gases are set by international negotiations. The question is, how should a region manage to attain its goal and who should contribute what. In our opinion, this can best be described with the help of cooperative game theory. When we suppose that the internal rules of behaviour within a society are strong enough to force solidarity of the industry and the private markets with societal objectives, then meeting the international conditions of a society boils down to dividing the associated costs among the societal units. Application of cooperative game theory then asks for the values of the coalitions, that is the worth of subgroups of industry and other units of the society as well as their contribution to the emissions of greenhouse gases. As in the previous parts of this paper, here it will also be an enormous task to gather reliable data to develop applicable models. Nevertheless, we expect that in this area game theory might be of great help. There are already several examples of applications of game theory in practical cost allocation models.

For most of the solution concepts of cooperative games there are ax­iomatic characterizations available. The trend of these axiomatic charac­terizations is in the direction of the search for a mutually independent set of axioms that have a desired social or economic implication. Thus, by examining the axioms, a society might decide to accept or reject a cer­tain solution concept. At this point we should mention that a new type of model is needed, namely, dynamic cooperative games. Every year or, perhaps, for a period of five years, the countries have to make a decision on the cost allocation. Five years later this has to be done again, and so on. So the question is, how should such a situation be played: one co­operative game every five years, or one game over the whole time period? And furthermore, what is the relation between the summation of the solu­tions per period with the aggregate solution over the whole period? Even conceptually, it is not obvious how a dynamic cooperative game defined over "multiperiods," should be formulated and how a solution should be translated to the different periods.

4. Conclusions

In this paper we intended to clarify how the present concerns about en­vironmental pollution lead to both practical and abstract game theoretic questions. We illustrated our statements with the important example of the enhanced greenhouse effect. However, any other type of environmental waste problem could be used, though the impact might have quite a differ-

The Environmental Game xxiii

ent regional scale. It is our opinion that for policy makers game theoretic models could be of great value. One can not claim that policymakers are eagerly awaiting their arrival. However, that is only partially their fault. Game theorists should bring their models closer to practice and expose the politicians to the interpretations in which the politicians recognize their decision problems and which are close enough to reality in order to serve as a real support. Gaming sessions in an international setting might play an important role in enhancing the policymakers' understanding of game theory. Besides this aspect of applied game theory, we have mentioned that the present features of the environmental concerns give rise to new types of games like fuzzy games and dynamic cooperative games. Thus, it can be concluded that game theory can help the world and its population to survive and to move towards a sustainable development by analysing and solving environmental games. Moreover, environmental games can serve as a resource for game theorists to analyse undiscovered fields and thereby to enhance the beautiful field of game theory and contribute to its full blossoming.

Department of Mathematics, University of Limburg, Maastricht, The Netherlands

Part 1

Models of Global Change

and Sustainable Development

Differential Game Models of Global Environmental Management

A. Hauriel and G. Zaccour2

Abstract

This paper deals with the modeling of economy-environment in­teractions for several countries which are assumed to behave com­petitively for the control of their own economic processes but have to achieve jointly a common environmental management goal. The paper is organized in two parts treating of modeling issues related to process dynamics and players interactions respectively. In the first part we discuss the control theoretic approach for representing the economic and pollution processes in a model of environmental man­agement. We also give a short discussion of a possible representation of uncertainty and risk in these models. In part two we propose a dy­namic game modeling approach which combines two classic models, the N-person equilibrium model with coupled constraints proposed by Rosen and the differential game model with active and passive variables proposed by Brock. These models permit the definition of a tax scheme which induces a set of equilibrium seeking players to achieve, in the long run, a global environmental goal.

1. Introduction

Recent global economic trends show two apparently conflicting phenomena where the interdependence of countries increases, in particular in relation to environmental goals, while, at the same time, trade wars become more frequent. The achievement of global environmental goals, e.g., the con­trol of global warming cannot thus be represented as a purely cooperative game. A common goal has to be achieved by conflicting players. In this paper we propose a modeling framework which takes into account these two dimensions of international environmental management.

We define environmental management as the design of resource alloca­tion and pricing processes which lead to the efficient achievement of eco­nomic and environmental objectives. A theoretical perspective of environ­mental management can be obtained through an industrial organization point of view. The firms control dynamic processes (e.g., investment in

1 Research supported by FNRS-Switzerland, NSERC-Canada and FCAR-Quebec 2Research supported by NSERC-Canada

4 A. Haurie and G. Zaccour

different production or treatment capacities and/or accumulation of vari­ous forms of pollutants in the environment). The firms are also interacting and competing economic agents (e.g., different regions and/or countries exchanging resources and pollutants and competing on the international markets).

Differential games provide an attractive theoretical framework to study multi-agent dynamic interactions. The dynamic game structure seems par­ticularly relevant when one deals with global environmental changes like the greenhouse effect. Indeed the control of these global change processes passes through the coordination of policies to be implemented by a set of sovereign countries. A coordinating agency may be in charge of design­ing a globally efficient incentive mechanism. However, this agency should take into account the intrinsically noncooperative behaviour of the differ­ent "players" involved in a global economic competition. The aim of this paper is therefore to explore the possible use of concepts borrowed from the theory of differential games in order to develop operational models of global environmental management.

Two concepts are central in our developments: (i) the turnpike and (ii) the noncooperative equilibrium with coupled constraints. The first concept corresponds to the long run steady-state equilibrium; it is therefore clearly related to the idea of sustainable development. Thrnpikes have played a cen­tral role in the theory of optimal economic growth since the seminal model of Von Neuman [17]. Recently the turnpike concept has been extended to the case of uncertain system with jump Markov disturbances and to non­cooperative differential games (see [27]). The second central theme in this paper is the use of a solution concept, introduced by Rosen [33] under the name of normalized equilibrium, for a noncooperative game where the play­ers are bound to observe a common coupling constraint. The main thrust of this paper is to show that, by combining the characterization of equilib­ria proposed by Rosen with the long term optimality of turnpikes, we can provide a way to design asymptotically globally efficient emission taxes in a noncooperative world.

The paper is organized in two parts. The first part deals with the representation of global environmental management problems as optimal control or differential game problems. We relate this modeling approach to the currently very active domain of mathematical programming models of national energy-environment systems. We also show that the approach al­lows a modeling of uncertainty, in the form of a discrete event jump process which changes at random epochs the mode of the system. This description of uncertainty seems to be relevant to the environmental management con­text. The second part is dedicated to the study of the equilibrium concept introduced by Rosen, when different players share a common constraint. We show how, in the case of differential games with infinite horizon per-

Differential Game Models 5

formance criteria, the Rosen approach leads to an interesting design of a global tax scheme coordinating the competing players.

Part I Control Theoretic Models

In the first part of this paper we shall concentrate on the representation of the basic processes involved in global environmental management. They are of three categories: (i) capital and/or pollution accumulation processes, (ii) qualitative or modal system changes and, (iii) welfare accumulation pro­cesses. The resulting models are infinite horizon optimal control systems. We also propose a convenient way to take into account uncertainty in these models.

2. Economic and Environmental Accumulation Processes

In this section we consider different representations of the fundamental accumulation processes, for economic capital goods and for pollutants.

2.1 The capital accumulation process

A typical dynamic process in economics is the investment or capital ac­cumulation process. In a deterministic framework it is represented by a differential equation

k(t) = i(t) - j.tk(t) (1)

where k(t) is the stock of capital at time t, i(t) is the investment rate and j.t is the depreciation rate.

This capital accumulation process has been studied in depth in the optimal economic growth literature ([1)).

2.2 The Pollutant Accumulation Process

A typical pollution accumulation process can be represented, in a determin­istic framework, by the differential equation

S(t) = e(t) - vS(t) (2)

where S(t) is the stock of a pollutant accumulated at time t, e(t) is the emission rate of the pollutant at time t and v is the natural pollution elimination rate.

This formulation allows the modeler to take into account both the flow and accumulation effects of pollution.

6 A. Haurie and G. Zaccour

2.3 Representation of environmental constraints in an activity analysis model

We view the environment as a facility which is stocking the pollutants generated by economic activities. This facility can have a limited capacity or it may have a social cost which increases very sharply with the usage intensity. Then, in a way very similar to the other economic resources, the environmental resources can be incorporated in an economic production model.

Activity analysis models have recently been developed for different re­gions of the world, with the purpose of describing the long term energy­technology-environment choices. The basic structure of these models is a representation of a set of resources which are extracted, processed and con­sumed in activities. The choice of activity levels, compatible with resources availability and which optimize a global economic performance criterion is then formulated as a mathematical programming problem. A typical exam­ple of this approach is the MARKAL models (see e.g., [3]) developed under the aegis of ETSAP (an international committee of DE CD representatives).

2.3.1 A model with exogenous final demands

The following system is a general formulation in a continuous time opti­mal control setting of an activity analysis model of the same nature as MARKAL.

min 100 e-pt[a(t)i(t) + ,B(t)k(t) + f(t)q(t)]dt (3)

s.t.

i(t) - JLk(t) = k(t) (4)

Al(t)q(t) + A2(t)i(t) + A3(t)k(t) < h(t) (5)

B(t)q(t) = c(t). (6)

In this model, the variables are

k: the production capacities,

i: the investment rates,

q: the production levels.

Eq. (3) describes the total discounted (at a discount rate p) system cost composed of investment, maintenance and operation costs. Eq. (4) de­scribes the capacity accumulation process. Eq. (5) represents a set of struc­tural constraints on the production process (e.g., the interactions between the different processes involved in a refinery). Finally, Eq. (6) is a set

Differential Game Models 7

of consumption satisfaction constraints. The matrices AI, A2, A3, B(t) and the vector h(t) are exogenous parameters which describe the efficiency, input-output structure of the various activities involved in the process. The vector c(t) represents exogenously defined final demands to be satisfied by the production system. In energy models these demands are often called useful demands as they describe the energy services like transportation, space heating, industrial heat, etc. As a first approximation these useful demands are supposed to be price inelastic. The different energy forms are in competition for the satisfaction of this exogenous useful demand. The model describes then the energy substitution effect due to technology choices.

2.3.2 Environmental constraints

In order to include environmental management concern in this modeling framework we add to the constraints (4)-(6) the following ones describing pollution emission and accumulation:

e(t)

S(t)

E(t)q(t)

e(t) - vS(t).

(7)

(8)

The technical coefficients entering in the constraints (7) relate to emissions e(t) of various types of pollutants caused by production activity levels q(t). Eq. (8) is the pollutant accumulation process. In order to achieve global environmental objectives one may impose upper bounds on the total emis­sions or total accumulation of a whole set of pollutants:

e(t) < e(t)

S(t) < S(t).

(9)

(to)

As already mentioned, discrete time versions of the above model, e.g., the MARKAL models built for more than a dozen of countries, have already been used to perform large scale simulations of long range scenarios for acid gas or CO2 accumulation due to energy production and consumption (see [4] for more details).

2.3.3 Introduction of a market structure

Assume now that the final demands are endogenously defined through a demand law. The model introduced above can be adapted to different market structures.

Demand-supply equilibrium: We can represent a perfectly competitive market structure via a model similar to Eqs. (3)-(6) except that the final

8 A. Haurie and G. Zaccour

demand c(t) would now be price dependent. The market price is defined as the marginal cost of the production system (Le., is dictated by the supply curve of the producers). This approach has been implemented in the PIES model [30] and in a coupling between MEDEQ and MARKAL, two models for Quebec ([2]) describing demand and supply respectively.

Profit maximizing monopoly: If we assume a monopolistic market structure the basic economic model becomes

min 100 e-Pt[a(t)i(t) + ,8(t)k(t) + (-y(t) - p(t))q(t)]dt

s.t.

i(t) - JLk(t)

Al(t)q(t) + A2(t)i(t) + A3(t)k(t)

¢(t, q(t))

(11)

k(t) (12)

< h(t) (13) = p(t), (14)

where p(t) is now the market price of the final consumption goods and Eq. (14) is a description of the demand law.

Oligopoly: In the present study we are more particularly interested in the intermediate case where there are several suppliers on the market, let us say N players. The model refers to the system controlled by one of the players. The only difference with the previous formulation will be the replacement of Eq. (14) by Eq. (15)

¢(t, Q(t)) = p(t), (15)

where Q(t) = r:f=l Qj(t) is the total supply by the N players ofthe different goods on the market. The N players are thus linked through the price formation mechanism of the global market. They are assumed to play according to an open loop Nash equilibrium.

3. Randomness

Uncertainty and randomness pervade environmental management. In this section we discuss the possible representation of discrete events which trig­ger sudden random modifications of the system under consideration. These are economic and environmental events, e.g.,

• environmental catastrophes

• acquisition of knowledge on the natural processes involved

• date of introduction of new technologies

Differential Game Models 9

• changes in economies (booms vs. depression)

• changes in the political environment, etc ....

We will show how these qualitative or modal system changes can be mod­elled as a discrete event process superimposed on the economic and envi­ronmental accumulation processes described above.

3.1 Qualitative or modal system changes

A sudden change in production technology (e.g., the discovery of a feasi­ble nuclear fusion based energy production system) or a sudden change in the prevailing tastes of the consumers (represented e.g., by the political gains of the green parties) can be represented as discrete events. Another important category of discrete events can be associated with the accumula­tion of information. For example our knowledge of the consequences (e.g., global warming) of increases in atmospheric CO2 concentration is evolv­ing by steps as a consequence of vast research programs and installation of powerful data acquisition techniques (e.g., satellites). In the case of a technological breakthrough (e.g., nuclear fusion) there will be a change in production costs or in energy efficiency. In the case of a political change or of new scientific information there will be a modification of the prevailing constraints or welfare criteria, etc. We shall thus introduce a finite set E of different modes for the system into consideration. With each mode is associated a specific description of the system's dynamics. The dynamics will thus change with the prevailing mode for the system.

3.2 Combining accumulation and discrete event processes

In the previous sections we developed, as a motivating example, a linear model inspired from the MARKAL approach. In the rest of the paper we shall adopt a more general formulation with nonlinear state equations and payoff functions.

3.2.1 Multi-mode state equations

It will be convenient to adopt the general notations of control theoretic models. The economic and pollution accumulation processes can be repre­sented by state equations

= ff(x(t), u(t), t) E Ui(t, x(t))

(16) (17)

where Xi (t) E R is the state variable which describes the cumulated stock at time t of an economic good or environmental pollutant into consider­ation, x(t) E R n is the vector of all state variables which influence the

10 A. Haurie and G. Zaccour

accumulation process into consideration, u( t) E Ul (t, x( t)) c R m is the control vector which describes the possible actions that different agents in the system may have on this accumulation process. The constraint set on these controls may also be dependent on the parameter i whose role is now described. The functions if (.) and Ul (.) are supposed to satisfy the usual regularity assumptions (see [12]).

The superscript i E E, where E is a finite set, defines the mode of the system. When the mode changes the accumulation process also changes. This allows the modelers to introduce a set of different dynamics for the sys­tem under consideration. Modal changes will be represented as a stochastic process. This shall introduce uncertainty in the system. This uncertainty can be influenced by the controller decisions. A more formal description of this stochastic process is proposed below.

3.2.2 Mode switching as a controlled jump process

Let ~(-) be a stochastic process with values in E. A sample path for this process is a piecewise constant function of t. We assume that the process is characterized by jump rates

qkl(X(t), u(t)) . 1

11~o dt P[~(t + dt) = il~(t) = k, x(t), u(t)J,

i=/=kEE, (18)

qkk(X(t), u(t)) = - L qkl(X(t), u(t)). (19) i#

This description of the discrete event process is the simplest possible which retains the capability to model both state and control influences on the transition probabilities.

3.2.3 Hybrid systems with two time scales

The state of the system will thus be represented by the pair (x,~) E R n x E. The continuous part of the state describes an accumulation which corre­sponds usually to the fast part of the process. The discrete component ~ describes the modal system changes. It is usually a slow mode of the process. Therefore we assume that the modal switches will occur unfre­quently, thus leaving, in the average, enough time for the continuous state subsystem to reach a steady state.

3.3 Performance criteria

Each player (country) manages the economy in order to optimize some wel­fare. This welfare can depend on the utility derived from final consumption, or on the net cost of the economic activities. The welfare is also influenced

Differential Game Models 11

by the state of the environment. In several recent papers (e.g., [15]) one uses a combined welfare function with a term representing the damage due to environmental degradation.

3.3.1 Welfare and damage functions

Basically we shall represent the environmental management problem as the search for a maximum expected welfare subject to a constraint on the environmental damage. This damage constraint can be translated into a set of global constraints on emissions and accumulation of various pollutants.

3.3.2 Discounting

Discounting has been used in economic models as a representation of the decision maker time preference. The discount rate is then the marginal rate of substitution over time. Discounting may also be related to a rep­resentation of a random stopping time which dictates the termination of the process. More precisely the controlled process is supposed to last a random time T described as an exponentially distributed random variable with mean ~. The expected payoff of decision maker j is the expectation

(20)

where L j (-) is the instantaneous welfare function. The expression (20) can also be written as follows, once the distribution of the stopping time T is explicited: 100 e-ptLj(x(t), u(t))dt. (21)

This is precisely the infinite horizon discounted payoff functional.

3.4 A piecewise deterministic formalism

We have seen above that we can represent uncertainty as a discrete event jump process changing the modes of a fast moving control system. Recently several theoretical and computational developments have concerned the class of so-called piecewise deterministic systems. This theory is described in ([13], [31], [32], [6], [7], [29]). It deals with control systems with hybrid state and two time scales, corresponding to a deterministic fast dynamics (x variables) and a stochastic slow dynamics (~variable). It provides an operational framework, both for qualitative and quantitative analysis.

It is shown in [23], [29], that a piecewise deterministic framework leads to the consideration of a family of associated infinite horizon deterministic control problems involving, when we start the process in the discrete state

12 A. Haurie and G. Zaccour

~(O) = k, in addition to the x-state equations, the following extended payoff function

roo e-z(t){Lj(x(t), u(t)) + L qkl(X(t), u(t))Vl((x(t))}dt (22) io lEE-k

where the new state variable z introduced in place of the discount rate satisfies the following dynamics

i(t)

z(O) p - qkk(X(t), u(t))

0,

(23)

(24)

and the function Vl((x(t)) expresses the expected infinite horizon payoff when the ~ process jumps to discrete state i!, at time t, with current contin­uous state given by x(t). The composite performance criterion in Eq. (22) represents therefore the classical dynamic programming tradeoff between the optimization of the currently prevailing performance criterion, repre­sented by the function L](x(t), u(t)), and the the optimization of the ex­pected future conditions which will prevail after the next modal change represented by the term LlEE-k qkl(X(t), u(t))Vl(x(t)) (see [14] or [34]).

4. Turnpikes

When one deals with an infinite horizon optimal control problem having stationary state equations one can expect that, under general structural conditions, the optimal trajectories emanating from different initial states will tend to bunch together around a steady state optimal trajectory which has been called, in the economic literature, the Turnpike of the system. In the next part, which deals with the modeling of players interactions, we give a more formal characterization of the turnpike.

In [29] it has been shown that turnpikes can be also characterized for piecewise deterministic systems. In that context one may expect to have dif­ferent turnpikes associated with different modes of the system. Depending on the mode which prevails, the trajectory will be attracted to a different temporary steady state. When the mode changes, according to a stochas­tic jump process, the steady state is interrupted and the trajectory moves toward a new attractor. A numerical algorithm exploiting the turnpike property has been proposed in [7] for the optimal control case and in [27] for the differential game context. We refer the reader to these papers for a more detailed description of this control theoretic structure.

Differential Game Models 13

Part II Players and Interactions

In this second part of the paper we deal with the characterization of Nash­equilibria, under a joint environmental constraint. We basically view the environmental management game as follows:

• There are different countries, also called players, who behave non­cooperatively in economic competition;

• A global environmental constraint is imposed on all players together; it corresponds, e.g., to a global upper bound on CO2 concentration in earth atmosphere;

• An international agency (like UN) is in charge of coordinating the policies of the different countries; for this purpose it can levy emission taxes or emit tradeable emisson rights;

• Each country reacts dynamically and unilaterally to the agency coor­dinating actions;

• The aim of the agency is mainly to achieve a long run sustainable efficient development.

The global environmental constraint introduces a coupling between the dif­ferent players. By defining an appropriate common shadow price for the global environmental constraint associated with some weighting of the dif­ferent players the agency can induce the players to achieve the global envi­ronmental objective while playing an associated noncooperative game with decoupled constraints.

We shall also show that, by designing the tax scheme on the long run equilibrium defined by the turnpikes of the noncooperative open-loop games, the agency can achieve its long term objectives, even when the play­ers react dynamically (i.e., adapt their investment and emission schedules).

We shall first consider a deterministic model. Possible extensions to the stochastic framework indicated in previous sections will then be proposed along the lines of [27].

5. Equilibria with a Coupled Constraint Set

In this section we review the concept of normalized equilibrium proposed and characterized by Rosen [33]. In his seminal paper Rosen has studied the existence and uniqueness of equilibrium points for concave N-person games, where the players are subject to a coupled constraint. Such a constraint

14 A. Haurie and G. Zaccour

arises when each player j has an action set Uj but the N players together must satisfy a common constraint

(25)

We say that the constraint is coupled when U is a proper subset of the cartesian set U1 x ... Uj x ... UN.

Let cjJj (Ul, ... , Uj, ... , UN) be the payoff of Player j. By definition a coupled equilibrium is an N -tuple (ui, ... , uj, ... , uN) E U such that

cjJj(ui,··· ,uj, ... , uN) 2: cjJj(ui,·.·, Uj, ... ,uN)

VUj E Uj s.t. (ui, ... ,Uj, ... ,uN) E U,j = 1, ... ,N. (26)

In this definition, each player may only consider unilateral moves that keep the common constraint satisfied. This type of equilibrium is clearly relevant when one considers an international environmental management problem where a common cap on global damage is imposed, e.g., through an in­ternational agreement. A coupled equilibrium cannot be considered as a purely noncooperative solution concept since there must be an agreement to observe the common constraint.

If the constraint set U is defined by a set of inequality constraints and under appropriate regularity conditions, there is a vector Khun-Tucker multiplier )...1 associated with the constraint (25) for each player j. Rosen calls normalized equilibrium a coupled equilibrium such that the Kuhn­Tucker multipliers satisfy

j = 1, ... ,N, (27)

where )..0 2: 0 is a given vector and rj > 0, j = 1, ... , N are given weight numbers. This equilibrium is actually defined through the fixed point con­dition

where one defines

B(u*, u*, r) = maxB(u*, u, r) UEU

N

B(u*,u,r) = 2:=rjcjJj(ui, ... ,uj, ... ,uN). j=l

(28)

(29)

Therefore we see that there is a link between the concept of normalized equilibrium and the weighting of objectives as it occurs in vector valued optimization.

Under a strong concavity condition, Rosen has shown that there exists a unique normalized equilibrium associated with each positive weighting

Differential Game Models 15

vector r in a convex cone n. Furthermore he has shown that, under the same concavity conditions it will be usually true that the payoff of Player j will increase if one increases uniquely his (her) weighting rj, while keeping the other components of r constant.

6. Differential Games with Active and Passive Variables

In another seminal paper Brock [9] has studied a class of infinite hori­zon open-loop differential games (I HOLD G) with active, e.g., the capital stocks, and passive, e.g., the prices resulting from the market demand laws, variables. This type of model seems again very relevant to the study of inter­national environmental management. Consider N countries j = 1, ... , N. Assume that the international economic system is described by the follow­ing state equations, for j = 1, ... , N

Xj(t) = 1;(Xj(t), Uj(t)) (30)

Xj(O) xq 3

(31)

0 < 9j (Xj (t), Uj(t)), (32)

0 F(x(t),y(t)) (33)

0 > hk(x(t),y(t)), k=I, ... ,p (34)

where x = (Xj : j = 1, ... ,N). The payoff of country j is given by

Jj(Xj(')' y(.), Uj(')) = 100 e-Pjt Lj (Xj (t), y(t), Uj(t)) dt. (35)

In this system Xj E R nj is the active state variable of country j and Uj E Rmj is its control variable. y E R r is the passive variable. 1; : Rnj x Rmj -+ Rnj, 9j : Rnj x Rmj ~ Rqj, F : R n x Rr -+ Rr, hk : Rn x Rr ~ R, are sufficiently regular functions. Eqs. (30)-(32) define orthogonal constraints for the active variables of each country and Eq.(33), where F(·) takes its values in a space of same dimension as y, defines implicitly the passive variable as a consequence of the independent strategic choices of the N players (it could be e.g., a price resulting from the supply of each country). Eqs. (34) define a set of coupled constraints which may involve all the passive and active variables together (it could be e.g., a cap on global emission of a pollutant). Finally, Eq. (31) represent the initial conditions for the system.

We may thus say that Eqs. (30)-(32) and (31)-(35) represent the eco­nomic interactions of the N countries, whereas the coupled constraints (34) represent some common environmental goal.

Now assume that there exists a normalized equilibrium with weights r. Then there exists a common multiplier oX 0 (-) such that the coupled equi­librium is also the noncooperative open-loop equilibrium of the differential

16 A. Haurie and G. Zaccour

game

Equil. j=l, ... ,N 1000 [e-PjtLj(xj(t),y(t), Uj(t»

P AO(t) (36) - L ~hk(X(t),y(t»] dt

k=l J

s.t.

Xj(t) h(Xj(t), Uj(t» (37)

Xj(O) = xq (38) 3

0 < gj(Xj(t), Uj(t», (39)

0 = F(x(t),y(t» (40)

The payoff of country j is now given in Eq. (36) with a normalized shadow price for the common constraint. The common multipliers AZ(t) will then have to satisfy the global complementarity relations

AZ(t)hk(X(t),y(t» 0, k = 1, ... ,p

hk(X(t),y(t» < 0 k = 1, ... ,po

7. International Coordination as a Game with Coupled Constraints

(41)

(42)

Assume an economic system with N countries in interaction. Without regard to a global environmental damage the economic interaction is de­scribed as the following IHOLDG, where Equil. j =l, ... ,N refers to Nash open-loop solution,

Equil. j=l, ... ,N 1000 e-Pjt Lj (Xj (t), y(t), Uj(t» dt. (43)

s.t.

Xj(t) h (Xj (t), Uj(t» (44)

Xj(O) xq J

(45)

0 < g(Xj(t), Uj(t», (46)

0 F(x(t), y(t». (47)

This model is basically the type of game studied in [9], [26], [27]. It has been shown in these studies that under enough concavity a turnpike property holds for the infinite horizon open-loop equilibrium. A formal proof of this property can be found in [10] and [11].

Using the implicit programming formulation of [16], it is shown in [27] that the turnpike is the solution of the following static implicit equilibrium

Differential Game Models

problem

Equil. j =l, ... ,N

s.t.

o fJ(Xj, Uj) - pj(Xj - Xj)

o < g(Xj, Uj),

o = F(x,y)

17

(48)

(49)

(50)

(51)

where Xj, j = 1, ... , N is the solution of the equilibrium problem, Le., the turnpike level itself. Of course, when Pj = 0, j = 1, ... , N, this corresponds simply to the steady state equilibrium problem.

Introduce now a pollution accumulation process described by the fol­lowing equations

N

Sk(t) L ejk(t) - VkSk(t), k=l, ... ,p (52) j=l

Sk(O) = sg, k=l, ... ,p (53)

ejk(t) hjk(xj(t),y(t), Uj(t», j = 1, ... , N, k = 1, ... , p. (54)

This describes a situation where there are p types of pollutants which are emitted by each of the N countries according to the emission rate functions hjk (-) in (54).

If there is a recognition of a global damage due to the pollutant accu­mulations above an upper limit Sk, k = 1, ... ,p, an international agency can be charged to insure that, in the long run, the following inequalities will be satisfied

Sk(t) < Sk k = 1, ... ,p (55)

for t ~ e,

where e is a large value. Since the objective of the agency is in the long term, it is natural to consider that it imposes a set of coupled constraints on the implicit programming problem which defines the equilibrium turnpikes. In steady state the common environmental goal is described as follows:

N

Lhjk(Xj,y,Uj) < VSk k=l, ... ,p, j=l

(56)

which is in the form of Eq. (34). Actually this form is a little bit more general since we assume that the controls Uj are involved in the constraints. Notice also that these control variables are "separated" in the constraint, each Uj appearing uniquely in the function hj {). In order to achieve the

18 A. Haurie and G. Zaccour

global environmental constraint, in the long run, the agency can implement a normalized equilibrium with weighting r by defining a normalized effluent

tax >"jk = ~ for each pollutant k and each country j. The normalized J

steady state equilibrium path is then the solution of

Equil. j =l, ... ,N Lj(xj,y, Uj)

p >..0 L k-- -hjk(xj,y, Uj)

r· k=l J

s.t.

0 h(Xj, Uj) - pj(Xj - Xj)

0 < g(Xj, Uj),

0 F(x,y)

with the global complementarity condition

N

>..2(L hj'k(Xjl,y,Ujl) - VkSk) j'=l

N

0, k = 1, ... ,p

(57)

(58)

(59)

(60)

(61)

Lhj1k(Xj',y,Ujl)-VkSk < 0, k=I, ... ,p. (62) j'=l

Now we see that the agency has the possibility to design different tax schemes by varying the weighting r. These tax schemes will induce the players to satisfy the global environmental constraints in steady state. More precisely we make the following conjecture which is proven, for a special case, in [25].

Conjecture: Define a constant tax scheme which is designed for satisfy­ing the emission constraints in the implicit steady-state equilibrium prob­lem defined above by Eqs. (57)-(62). Consider the open-loop Nash equi­librium for the dynamic oligopoly model, without emission constraints but with emission taxed according to the constant tax scheme defined above. If the turnpike property holds true the Nash-equilibrium controls will drive the N -player system toward the steady-state equilibrium and thus satisfy the emission constraints in the long run.

Remark. We have been careful in the development of our model to keep a separation between the dynamics of the N players. Indeed the turn­pike property is more easily established in an open-loop game when such a separation holds (see [10], [11], [26]).

Differential Game Models 19

8. Extension to a Piecewise Open-Loop Setting

We consider finally the case where the system is subject to random modal changes, as described in Part 1. We may try to extend the theory developed in previous sections in a very similar way with the help of the two concepts of piecewise open-loop strategies and associated infinite horizon deterministic differential game.

Indeed it has been shown in [27] that, under conditions very similar to those used by Rosen in a static framework, a turnpike property holds for piecewise open-loop Nash equilibria in piecewise deterministic differential games. The design of a tax scheme inducing competing players to satisfy a long term environmental constraint is an order of magnitude more compli­cated than in the fully deterministic case. The long term objective of the agency, which is also the global binding constraint for the competing play­ers, has now to be formulated in probabilistic terms. The agency wants to achieve a random steady state which is acceptable. We can only conjecture, at this stage, that a piecewise constant tax scheme, where the effluent taxes would be adapted when the system's mode change, is possible to design in such a way that, in the long run, the average emissions satisfy the required constraint.

9. Conclusion

This paper has proposed an approach for the modeling of global environ­mental management issues which is based on the following prescriptions:

• model economic and pollution accumulation processes as infinite hori­zon controlled systems

• model uncertainty as a discrete-event jump process which changes the dynamics of the systems

• represent the interrelations of competing countries which have to sat­isfy a common global environmental constraint as a normalized equi­librium a la Rosen

• exploit the turnpike property of infinite horizon control systems to design a tax scheme which will achieve the long term common envi­ronmental objectives.

The present paper gives the directions of an agenda for future research. There are still several results to be formally proven. In particular the ex­tension of the approach to the piecewise deterministic control or differential game framework poses interesting theoretical questions. The other direc­tion of research which should be pursued deals with the computational is­sues. We have deliberately related this modeling to the MARKAL approach

20 A. Haurie and G. Zaccour

which has already produced a whole class of operational dynamic energy­environment models for OECD countries. There is also a natural link be­tween piecewise deterministic control systems and stochastic programming models which could be exploited to develop efficient numerical techniques using decomposition techniques in large scale mathematical programs (see e.g., [18]-[20]). Combining the computational efficiency of mathematical programming models with the game-theoretic concept used in the present paper could open the door to interesting applications in global environmen­tal management.

References

[1] Arrow KJ. and Kurz M., Public Investment, Rate of Return and Opti­mal Fiscal Policy, Johns Hopkins University Press, Baltimore, Mary­land, 1970

[2] Berger C., Haurie A., Loulou R., Lafrance G., Savard G., and Sur­prenant J-P., MEDEQ-MARKAL: un couplage entre deux modi'iles technico-economiques du systeme energetique du Quebec, RAIRO recherche operationnelle, vol. 21, pp. 21-50, 1987

[3] Berger C., Dubois R., Haurie A., Lessard E., Loulou R., and Waaub J-P., Canadian Markal: An Advanced Linear Programming System for Energy and Environmental Modelling, INFOR, vol. 30, no. 3, pp. 222-239, 1992

[4] Berger C., Loulou R., Soucy J.P., and Waaub J-P., CO2-Control in Quebec and Ontario, ETSAP lHASA meeting, Laxenburg, 1991

[5] Bertsekas D.P., Dynamic Programming: Deterministic and Stochastic Models, Prentice Hall, 1987

[6] Boukas E.K, Haurie A., and Michel P., An optimal control problem with a random stopping time, Journal of Optimization Theory and Applications, vol. 64, pp. 471-480, 1990

[7] Boukas E.K, Haurie A., and van Delft Ch., A turnpike improvement algorithm for piecewise deterministic control, Optimal Control Appli­cations and Methods, vol. 12, pp. 1-18, 1991

[8] Breton M., Filar J.A., Haurie A., and Schultz T.A., On the Com­putation of Equilibria in Discounted Stochastic Dynamic Garnes, in: Dynamic Games and Applications in Economics, Lect. Notes Econ. Math. Syst., vol. 265, pp. 64-87, 1986

Differential Game Models 21

[9] Brock W.A., Differential Games with Active and Passive Variables, in: Mathematical Economics and Game Theory: Essays in Honor of Os­kar Morgenstern, Henn and Moeschlin, eds., Springer-Verlag, Berlin, pp. 34-52, 1977

[10] Carlson D.A. and Haurie A., A turnpike theory for infinite horizon open-loop competitive processes, to appear SIAM journal on Control and Optimization, 1994

[11] Carlson D.A. and Haurie A., A turnpike theory for infinite horizon open-loop differential games with decoupled dynamics, submitted to Annals of the International Society of Dynamic Games, 1994

[12] Carlson D.A., Haurie A., and Leizarowitz A., Infinite Horizon Optimal Control: Deterministic and Stochastic Systems, Springer-Verlag, 1991

[13] Davis M.H.A., Piecewise deterministic Markov processes: A general class of non-diffusion stochastic models, J. of R. Stat. Soc., vol. 46, pp. 353-388, 1984

[14] Denardo E.V., Contractions mappings in the theory underlying dy­namic programming, SIAM Review, vol. 9, pp. 165-177, 1967

[15] van der Ploeg F. and de Zeuw A., International aspects of pollution control, Environmental and Resource Economics, vol. 2, pp. 117-139, 1992

[16] Feinstein C.D. and Luenberger D.G., Analysis of the asymptotic behaviour of optimal control trajectories: the implicit programming problem, SIAM J. Control Optim., vol. 19, pp. 561-585, 1981

[17] Gale D., The Theory of Linear Economic Models, McGraw-Hill, New York, 1960

[18] Goffin JL., Haurie A., and Vial JP., Decomposition and nondifferen­tiable optimization with the projective algorithm, Management Sci­ence, vol. 38, pp. 284-302, 1992a

[19) Goffin JL., Haurie A., Vial JP., and Zhu D.L., Using central prices in the decomposition of linear programs, 1993, European Journal of Operations Research, vol. 64, no. 3, pp. 393-409, 1992b

[20] Bahn 0., du Merle 0., Goffin JL., and Vial JP., Nondifferentiable and large scale optimization, Mathematical Programming, Series B, vol. 69, pp. 18-29, 1995

22 A. Haurie and G. Zaccour

[21] Halkin H., Necessary conditions for optimal control problems with infi­nite horizon, CORE Discussion Paper 7210, Also, 1974, Econometrica, vol. 42, no. 2, pp. 267-273, 1972

[22] Haurie A., Existence and global asysmptotic stability of optimal tra­jectories for a class of infinite-horizon, nonconvex systems, J. of Op­timization Theory and Applications, vol. 31, no. 4, pp. 42515-42533, 1980

[23]

[24]

Haurie A., Piecewise Deterministic Differential Games, in: Differential Games and Applications, T. Basar, P. Bernhard eds., Springer-Verlag, Lect. Notes in Control and Information Sciences vol. 119, pp. 114-127, 1989a

Haurie A., Duopole et Percees Technologiques: un Modele de Jeu Differentiel Deterministe par Morceaux, L 'Actualite Economique, Ecole des Hautes Etudes Commerciales, Montreal, vol. 65, no. 1, pp. 105-118, 1989b

[25] Haurie A., Environmental coordination in dynamic oligopolistic mar­kets, Group Decision and Negotiation, vol. 4, pp. 49-67, 1995

[26] Haurie A. and Leitmann G., On the global stability of equilibrium solutions for open-loop differential games, Large Scale Systems, vol. 6, pp. 107-122, 1984

[27] Haurie A. and Roche M., Turnpikes and computation of piecewise open-loop equilibria in stochastic differential games, Journal of Eco­nomic Dynamics and Control, vol. 18, pp. 317-344, 1993

[28] Haurie A., Smeers Y., and Zaccour G., S-adapted equilibria, JOTA, 1990

[29] Haurie A. and van Delft Ch., Turnpike properties for a class of piece­wise deterministic systems arising in manufacturing flow control, An­nals of Operations Research, vol. 29, pp. 351-374, 1991

[30] Hogan W.W. and Weyant J.P., Methods and Algorithms for En­ergy Model Composition: Optimization in a network of process odels, in: Energy Models and Studies, Lev, ed., North-Holland, Amsterdam, 1983a

[31] Rishel R., Control of systems with jump Markov disturbances, IEEE Trans. on Automatic Control, AC-20, pp. 241-244, 1975a

[32] Rishel R., Dynamic programming and minimum principles for systems with jump Markov disturbances, SIAM J. on Control, vol. 13, no. 2, pp. 338-371, 1975b

Differential Game Models 23

[33] Rosen J.B., Existence and uniqueness of equilibrium points for concave N-person games, Econometrica, vol. 33, pp. 520-534, 1965

[34] Whitt W., Representation and approximation of noncooperative sequential Games, SIAM J. Control, vol. 18, pp. 33-48, 1980

Department of Mathematics, Universite de Geneve, Geneve, Switzerland

Department of Mathematics, REC-Montreal, Montreal, Canada

Sustainability and the Greenhouse Effect: Robustness Analysis of the

Assimilation Function!

Herman Cesar and Aart de Zeeuw2

Abstract Optimal control models with an environmental stock highlight

the intertemporal trade-off between consumption and environmental quality. In these models it is generally assumed that the assimila­tion function of nature is linear (Nordhaus, 1982). At the same time there is quite some uncertainty on the general form of this function outside a specific range of values. In this paper, we look at differ­ent (non-linear) specifications of the assimilation function in the case of the Greenhouse Effect. The optimal trajectories and the steady states are analysed for the various functional forms. Slight variations in the assimilation function can result in a dramatic change in the steady state values. Besides, neither multiple equilibria nor the ab­sence of non-zero production steady states can be excluded. This will be shown with the use of simulations in a simple model of the Greenhouse Effect.

1. Introduction

As the term 'sustainability' is more and more in vogue, the ambiguity over its meaning is ever increasing. Pezzey (1989) lists some 60 definitions of sustainability, sustainable growth and sustainable development that exist in the literature. One of the most commonly used definitions of sustainabil­ity in environmental economics is that of 'the maintenance of the effective resource base'. In order to operationalise this concept, it is crucial to know the particular resource(s) of interest. Using the subdivision of resources by Barbier & Markandya (1990) into renewable, exhaustible and environmen­tal resources, sustainability can be defined for each of them respectively as:

1 A prelimilary version of this paper was presented at the ECOZOEK-dag, Tilburg, June 11, 1993. The authors want to thank Ruud de Mooij and Cees Withagen, as well as the two referees, for very useful comments. The usual caveats apply.

2Herman Cesar is assistant Professor in Environmental Economics at the Depart­ment of Economics, Tilburg University, The Netherlands; Aart de Zeeuw is Professor in Regional and Environmental Economics at the Department of Economics, Tilburg Uni­versity and Fellow of the Center of Economic Research, Tilburg, The Netherlands; Postal address: Department of Economics, Postbox 90153, 5000 LE Tilburg, The Netherlands

26 H. Cesar and A. de Zeeuw

• utilising renewable resources at rates less than or equal to the natural or managed rates of regeneration;

• optimising the efficiency with which exhaustible resources are used which is determined, among other things, by the rate at which re­newable resources can be substituted for exhaustibles and by technical progress;

• generating waste and pollution (Le., the environmental resources) at rates less than or equal to the rates at which they can be absorbed by the assimilative capacity of the environment.

Note that sustainability should be distinguished from sustainable develop­ment. The poldering of lakes in the Amsterdam region in the fifteenth century was not sustainable for the renewable resources in these lakes (fish and plants). However, it is difficult to claim that the poldering was there­fore bad for the sustainable development of the Netherlands. Hence, one has to be careful to really use the sustainability criterium on a species level3 . However, for the Greenhouse Effect and for many other problems of the environmental resource type, sustainability is a sine qua non. The reason is that production might go to zero for too elevated levels of 'pol­lution', as will be argued in the next section. Therefore, in the absense of sustainability of the resource use, sustainable development is not possible.

In this paper, Greenhouse gases are assumed to behave as other types of waste and pollution. Therefore, sustainability will be taken to imply that the long term flow of GHGs into the atmosphere should be (less than or) equal to the rate that can be absorbed by the assimilative capacity of the environment.

The crucial problem with this definition is that relatively little is known about the assimilation process of Greenhouse gases in the environment. Many authors, having stressed that biologists could not supply them with nice quantitative assimilation functions, take the linear specification as a computationally convenient proxy4. Counter-examples are, among others, Forster (1975), Dasgupta (1982), Barbier & Markandya (1990) and Pethig (1990), stressing the multiple equilibria problem that may result.

How the assimilative capacity of the environment is modeled depends very much on the ecosystem that is described and is in principle an em­pirical matter. Unfortunately, very little is known about the true process of assimilation, especially outside the range of values experienced in the recent past.

3 At a total stock of natural resources level, however, another problem is on the lurk: that is how to weigh the different stocks (see Pezzey, op. cit., p.15).

4 Another convenient approximation is to take the assimilation function as a constant term, as does Miiler (1991).

Sustainability and the Greenhouse Effect 27

Therefore, in this paper, different specifications ofthe assimilation func­tion will be considered. This gives an idea of the consequences of slight variations in the functional form for the steady state levels of pollution and consumption. It therefore implies a warning to researchers using de­terministic optimal control models in environmental economics to be very careful with any policy conclusions if the natural science knowledge of the assimilation function is poor. To illuminate this issue, the focus is on the most elementary Greenhouse Effect model that captures the intertemporal trade-off between consumption and pollution build-up (Nordhaus, 1982).

This model will be analysed in Section 2 in great detail. Also, a numer­ical specification of the model is given that is used for simulation exercises later on.

In Section 3, various specifications for the assimilation function are analysed. It will be shown that the steady state results differ greatly even with very small variations in this specification.

The papers ends with some tentative conclusions, and suggestions for future research.

2. The Rudimentary Model

2.1 General description

The model of Nordhaus (1982) is the most elementary one that captures the intertemporal trade-off between consumption and Greenhouse gas build-up. The economy is assumed to have fossil fuel as a sole productive input5 • The use of energy in the form of fossil fuel is, however, not only a 'good' but also a 'bad' in the sense that it enhances the Greenhouse Effect. It is assumed for the time being that there are no 'abatement' possibilities6 ,

unlike in Forster (1977) and many others. This means that the only way to reduce the Greenhouse Effect is to diminish fuel use and hence to decrease production.7The model of Nordhaus (1982) will be analysed in some detail because it serves as a benchmark case for comparison with the models with modified specifications of the assimilation function.

As elaborated in Cesar (1993), pollution can affect social welfare via a

5The world is treated throughout the paper as one economy. The strategic aspects that arise in a multi-country setting are ignored. For an elaboration on the strategic issues, see among others Van der Ploeg & de Zeeuw (1992) and Cesar (1993).

6 Abatement is written in quotations as there are very few real abatement possibilities in the case of the Greenhouse Effect. The main ways to reduce the build-up of GHGs is to economise on fuel or to introduce non-fossil energy sources. However, the term 'abate­ment"' seems to be used extensively in the optimal control literature on the Greenhouse Effect.

7In another paper (Cesar, 1993), investments in energy related technology will be introduced, which allows for a richer choice.

28 H. Cesar and A. de Zeeuw

'productivity' effect and via an 'environmental amenity' effect. Basically, the productivity effect of pollution refers to the situation with health risks of workers and other direct effects on production. The environmental amenity effect of pollution describes the case where pollution has a direct effect on social welfare.

Most models of the Greenhouse Effect specify the feedback of pollution on future welfare through amenity effects. This typifies the situations in which environmental quality is seen as a luxury good that does not influence consumption as such though it may affect the well-being of the individual in society. Generally it would, in our opinion, be appropriate to analyse both amenity and productive effects. However, Greenhouse gases (esp. carbon dioxide, methane and nitrous oxides) are benign8 . This is in contrast with most polluting gases that lead to nuisance, illness, retardation, etc. At the same time, Greenhouse gases affect the global climate in the long run, possibly leading to economy-wide disruptions. Therefore a deliberate choice is made for the assumption that the only effect of the build-up of Greenhouse gases is on productive capacity. This means that the feedbacks of the Greenhouse Effect are on production and not on amenity (social welfare) directly9. Finally, fossil fuels are assumed to be non-exhaustible. This is obviously not true. The rationale for using this simplification is the idea that environmental constraints will come much earlier in time than the physical fuel limits (e.g., there may be enough coal for more than a thousand years). A model that incorporates both the resource constraint and the environmental constraint is in Forster (1980) 10 .

These considerations can be formalised as follows: The social welfare function depends solely on consumption U (C) and is assumed to be increasing and strictly concave in C (that is U' (C) > 0 and UI/(C) < 0)).

Production, Y, is a function of energy use, F, and the build-up of the Greenhouse Effect, P (P of 'pollution'): Y = Y(F, P). Assume that Y(F, P) is a strictly concave function of F and P with continuous second derivatives and that it is increasing in F and decreasing in P: YF > 0, Yp < 0, Y FF < 0, Ypp < 0 and YFFYPP - YfiF 2: O. Besides, it is assumed that YPF < 011 .

As outlined above, there are no abatement possibilities nor ways to in­vest in energy efficiency. Hence, the only way to reduce the concentration of

8Note that CFCs (by destroying the ozone layer) can have very malign effects (e.g., skin cancer).

9In Cesar (1993), this choice is shown to be of key importance for the effect of a higher time preference for future consumption.

lOSee also Withagen (1993). lIThe reasoning behind this assumption is that Y(F, P) will be taken to be separable

into e(F).I(P), where e(F) is the energy function and I(P) is a Greenhouse damage function, with e' > 0, f' < 0 and e and I are both concave.

Sustainability and the Greenhouse Effect 29

CO2 and other GHGs is to use less fossil fuels by producing less. Therefore consumption equals production: C = Y(F, P).

The Greenhouse gas concentration is assumed to evolve over time ac­cording to the following process:

P = IF - a(P) (1)

This means that P increases over time due to fossil fuel use, F, with the coefficient I denoting the percentage of fossil fuel that will end up in the atmosphere. The stock of GHGs will gradually decay due to the assimilative capacity of nature: a(P).

In this section, a(P) is taken to be linear: a(P) = aP. This implies that the average atmospheric lifetime of GHGs is 1/ a 12 • In the next section, more realistic non-linear forms of the assimilation function will be analysed explicitly. Note that reforestation is not modeled here explicitly, though it would be an interesting extension, given that it opens the possibilities for the social planner to influence the assimilative capacity of nature directly13.

Assume the existence of a Central Planner, who seeks to maximise the discounted flow of social welfare, depending on consumption only14.15. Hence:

s.t.

max roo e-pt[U(C)]dt F 10

C = Y(F,P)

P IF-aP

where p is the discount rate.

p>O (2)

(3)

(4)

The model, as it stands here, is a so-called 'fixed infinite time free right hand endpoint optimal control model'. The necessary conditions for

12 Another way of saying this is that the half-life of GHGs is l/a. 13This could be added in the model in the following way by assuming reforestation R

with corresponding costs C(R): Consumption would then be equal to: C = Y(F, P)­C(R) and the evolution over time of the stock of Greenhouse gases: P = ,F- (R- aP where ( denotes the buffering capacity of reforestation R. See also Wit hagen & van der Ploeg (1991).

14This model can easily be decentralised. This allows to show that with appropriate Pigouvian taxation, a market economy can reach the social optimum (see Cesar, 1993).

15In a more recent model, Nordhaus models two state variables: the Greenhouse gas concentration (P) as well as temperature (T), where the latter is assumed to influence the economy. (Nordhaus, 1989). Tahvonen, von Storch & Xu (1992) take the same approach where, instead, T influences the economy (via the utility function). This specification makes it possible to incorporate the lag that seems to exist between emissions and actual temperature increase. This increase is ultimately the important factor for the economy. The specification with T is justified on the grounds that it might, in the end, be the speed of temperature changes that determines how badly ecosystems are affected.

30 H. Cesar and A. de Zeeuw

a solution to the above model, using Pontryagin's Maximum Principle16,

are: There exists a co-state function 'IjJ such that with the Hamiltonian de­

fined as17:

1t(F,P,'IjJ) = U[Y(F,P)) + 'IjJ(-yF - aP) (5)

the necessary conditions are given by Equation 4 and:

OJ-l of o1t oP

= U'YF +,,('IjJ = 0

U'Yp - a'IjJ = -"p + p"p or

(6)

"p = (p + a)'IjJ - U'Yp. (7)

Note that, given the concavity assumptions on Y(F, P) and on U(C), the necessary conditions for optimality are also sufficient.

Integrating Equation 7 and substituting the result in Equation 6 gives:

U'YF = -"( 100 U'[C(s))Yp[F(s),P(s))e-(p+a)(s-t) ds (8)

This equation denotes that at every point in time, the extra consumption due to an additional unit of fuel use equals the present value of the loss in output (in utility terms) due to the enhanced concentration of GHGs caused by this additional unit of fuel.

Assume limit conditions to hold for U(C) and Y(F, P), which prevent boundary solutions to occur1S• These conditions are:

lim U'(C) = 00 (9) C-+O

lim Yj..(F,P) = 00 (\:IP as long as Y(F, P) > 0) (10) F-+O lim Y~(F,P) = 0 (11)

p-+o

This allows us to focus on the interior solution(s) (with P > 0, F> 0, and C> 0) of the model. From the first order conditions given in Equation 6 the derivative of 1tF with respect to F, P and 'IjJ can be obtained:

o1tF of U'YF F + U"Y;' (12)

16Note that a deterministic setting is chosen throughout. This means that the open­loop solutions are optimal. In the case of uncertainty, feedback solutions should have been considered to allow for adaptive policies.

17It is also assumed that the transversality conditions are satisfied. In an infinite time horizon optimal control problem, this means that the present value of the costate approaches zero as time goes to infinity.

18Note that limit conditions are not the same as transversality conditions, discussed in the footnote above. Besides, the limit conditions are by no means necessary for optimality. They are assumed to hold here in order to concentrate on internal solutions (see Forster (1977) for an analysis of both boundary and internal solutions).

Sustainability and the Greenhouse Effect 31

(13)

'Y (14)

It is clear from the assumptions above, that °Zir < 0 and °l:l > 0 .

However, the sign of O~F is ambiguous. Here, it is assumed19 that oW' < o. The first order conditions above give the derived demand function for fossil fuel as implicit function of P and 'IjJ : F(P, 'IjJ), with:

F,p

_1{FP < 0 1{FF

_ 1{F,p > 0 1{FF

(15)

(16)

The signs of the derivatives of the demand function with respect to P and 'IjJ are as expected: both an increase in the Greenhouse gas concentration P and an increase in the valuation of the environmental damage 'IjJ push fossil fuel use F down (note that 'IjJ < 0).

Next, the dynamic system 4 - 7 will be analysed:

P 'YF- aP

~ 'IjJ(p + a) - U'[Y(F, P)]Yp[F, Pl.

In order to analyse the phase plane (P, 'IjJ), the derived demand function F(P, 'IjJ) is substituted into these differential equations of the state and the costate:

M(P,'IjJ)

N(P,'IjJ)

Then:

Mp 1{,pp M,p 1{,p,p N p = -1{pp N,p p -1{p,p

'YF(P,'IjJ) - aP = 0

'IjJ(p + a) - U'[Y(F(P, 'IjJ), P)lYp(F(P, 'IjJ), P) = o.

'YFp - a < 0 'Y~ >0 -U"(YFFp + Yp)Yp - U'(YPFFp + Ypp) > 0 p + a - U"YFF,pYp - U'YPFF,p > 0

Note that N p > 0, due to the assumptions on U(.) and Y(.) and N,p due to the assumption that O~F < o. This amounts to saying that the net effect of an increase of the GHG concentration on the marginal welfare

19 A sufficient condition for this to hold is to assume that Y(F,P) is separable in an energy function e(F) and a Greenhouse damage function f(P) as explained above and to further assume that U' + C.U" > 0, as is trivially the case for a specification of U(C) such as U(C) = 1~(7C1-(7.

32 H. Cesar and A. de Zeeuw

cost of the environmental degradation is negative. All the other inequalities follow also directly from the assumption made on the functions. Hence the slopes of the two loci are:

d1jJ I = - NN: < 0 (downward sloping) (17) dP .b=o 'Y

d1jJ I = - MM: > 0 (upward sloping) (18) dP P=o 'Y

The two loci are depicted in Figure 1:

p'" -t------------~----------------p

... 1V

1j! - 0

Figure 1: Phase Plane of (P,1jJ)

Stability Properties The stability properties can be determined by looking at the Jacobian matrix evaluated at the steady state (pocO, 1jJOO):

J = [ p -1tp", -1tpp ], (19) 1t",,,, 1t",p

hence, det J is:

det J = (p -1tp",)1t",p + 1tpp1t",,,, < o.

Sustainability and the Greenhouse Effect 33

This means that the equilibrium is a saddlepoint. Besides, as is clear from the slopes of the two loci, the equilibrium is unique, if the equilibrium exists.

Assume in Figure 1, that the economy initially is left of the point poo. The social planner will have to set 'l/J in its meaning of shadow value at its corresponding level on the optimal trajectory and follow this trajectory until the steady state is reached. In the deterministic setting of the model (no stochastic elements), there is no reason why the economy would ever get off the optimal path.

2.2 A numerical specification of the model

In the last section, a general formulation was given of a rudimentary econ­omy affected by the Greenhouse Effect. In this section, specific functional forms are chosen to enable a more detailed analysis of some of the features of this economy. To this end, the production function is now assumed to be separable of the form:

Y(F, P) = e(F)f(P) (20)

where e(F) is the energy function and f(P) the Greenhouse damage function. These functions as well as the specified social welfare function will be briefly described. The other functions, such as the decay function aP in P ,F - aP do not need to be specified any further2o .

The Energy Function

It is assumed that there are decreasing returns to fuel in a production func­tion in which all other elements (labour, physical capital, human capital) are fixed at the constant k. This function gives potential output in the absence of a Greenhouse problem as:

e(F) = kFo 0<8<1 (21)

The Greenhouse Damage Function

The function f(P) denotes the fraction of potential output that is not destroyed by the Greenhouse Effect so that the amount e(F)(1 - f(P» is lost. Therefore, a specification of the damage function is chosen so that f(P) is equal to 1 for P = 0 and declines concavely:

f(P) = p2 -: p2 P ~ P (22) p2

where P is taken such that it does not exceed P (see Figure 2).

20In the next section, however, non-linear forms ofthe decay function will be analysed.

34 H. Cesar and A. de Zeeuw

f (P)

P

conST

Figure 2: Greenhouse Damage Function f(P)

The Social Welfare Function

The social welfare function is taken to be:

U(C) = _1_C1- U

1-0" with 0 < 0" < 1 (23)

This is the well-known constant relative risk aversion utility function or, more aptly since there is no uncertainty in this model, the welfare function with constant elasticity of marginal utility.

With these specifications, the Central Planner solves the following sys-tem:

s.t.

max e-pt--C1-Udt 100 1

F 0 1-0"

C = kF8(P2;t2)

P = ,F - aP

p>O

This can be solved using the Pontryagin principle (see Section 2.1).

Numerical Values for the Parameters

In order to further analyse the model, values for the parameters a, " p, 8 and 0" have to be chosen.

a The parameter a indicates the percentage of the total concentration of GHGs absorbed by the environment per year. As indicated in Cesar (1993), this percentage is estimated at about 0.5%. Hence, a is taken to be 0.005.

Sustainability and the Greenhouse Effect 35

'Y The parameter 'Y denotes the fraction of fossil fuel use that ends up in the atmosphere. Scientists claim that about half of the anthropogenetic fossil fuel is dispersed in the atmosphere and the other half is absorbed by the oceans (Cesar, 1993). Therefore, 'Y is set to 0.5.

p The social discount rate (in real terms) is set to p = 0.03 in the bench­mark case with p = 0.01 and p = 0.06 as variations. The choice is, of course, rather arbitrary, and as is shown in Cesar (1993), the steady state values differ considerably with the various values of p.

8 The fossil fuel elasticity of production 8 is assumed to lie between zero and one. This parameter is set to 0.3. Admittedly, this value is arbitrary21. For instance, Gottinger (1992) has set 8 much higher {between 0.5 and 1.0)22.

(F The elasticity of marginal utility is set to 0.75. Note, however, that the steady state values of C, F and P are not influenced by the choice of (F. The only variable that changes is the costate 'IjJ.

Given the choice of the functional forms and the parameter values23 , the steady states and the trajectories of the relevant variables can be calculated. Welfare evaluations and phase-plane analysis are first elements towards an ultimate policy discussion on the basis of the model. Given the specification of the utility function as U{C), welfare evaluations are straightforwardly linked to the values of consumption. The level of steady state consumption for different parameter values is given in Cesar (1993). The phase diagram is depicted in Figure 324 •

Note that 'IjJ will increase in absolute terms over time if the initial level of the GHG-concentration (Po) is below the steady state (POO).

21Cesar (1993) presents a sensitivity analysis on variations of the parameter value of O.

22 Actually, Gottinger sets the product of 0 and u to 0.50 in one version and to 0.95 in another version of his model. Given that u ~ I, it is obvious that 0 is between 0.5 and 1.0.

23 Apart from the parameters described above, two more parameters have to be given values: k in the production function and P in the Greenhouse damage function. The constant k is set to 17.42257. The rationale behind this choice is that with this value, the rudimentary economy gives the same outcomes of the principle variables as the model in Cesar (1993) where k is replaced by a function of an energy-related technology variable. The constant P is set to 1000. This implies that with P currently at about 350 ppm, it is assumed that the GHG-concentration will not grow beyond a tippling of current values. Note that this is, of course, a quite arbitrary number. Increasing this number with 1% leads to an equiproportional rise in P and F and hence in a (1.01)0.3 -1 :::::: 0.3% increase in C.

24Note that this figure has the same structure as Figure I, which is not surprising.

36 H. Cesar and A. de Zeeuw

pOO p

/ Figure 3: Phase Plane of (P, 'ljJ) for numerical values in text

3. Sustainability

As stated in the introduction, there is quite some ambiguity over the mean­ing of the concept of sustainability. This concept will be operationalised for different specifications of the assimilation function of Greenhouse gases.

As elaborated in Section 1, the concept of sustainability is defined here as the requirement of generating pollution at rates "less than or equal to the rates at which they can be absorbed by the assimilative capacity of the environment" (Barbier & Markandya, 1990).

For Nordhaus' (1982) model of the Greenhouse Effect, sustainability implies that the long term flow of GHGs into the atmosphere (i.e., IF) is less than or equal to the rate at which it can be absorbed by the assimilative capacity of the environment (i.e., aP). This amounts to saying that in the evolution over time of the GHG concentration, P = IF-aP, sustainability implies that P is less than or equal to 0 in the long run. Note that in the previous calculations of the steady states (with P = 0, ¢ = 0), sustain­ability is automatically guaranteed. In these calculations, however, a very simple specification of the assimilative capacity of nature was considered. In the next subsection, attention is focused on more realistic functional forms to examine the implications of sustainability.

Sustainability and the Greenhouse Effect 37

3.1 The assimilative capacity of nature

As stated in Section 1, relatively little is known about the assimilation process of the environment. Ideally, the uncertainty with respect to the natural regeneration is modelled. This is, however, rather difficult, because it is not just the maximum carrying capacity of the environment that is uncertain, but also the whole process of assimilation. Therefore, the choice is made here to model different possible specifications of the regeneration function in a deterministic setting and to calculate how robust the results are for variations in this specification.

As a benchmark case, the assimilation function (aP with a = 0.05) of the previous section is taken. This numerical specification corresponds with an average atmospheric lifetime of GHGs of 200 years. This half-life holds approximately for current levels of the GHG concentration.

For higher levels of 'pollution', non-linearities in the assimilation func­tion cannot be ruled out (Arrhenius & Waltz, 1990). Therefore, different scenarios for possible future trends are elaborated here as depicted in Fig­ure 4. For low levels of the GHG concentration, the assimilation is assumed to be linear in P, with deviations from this trend at higher values of P. The assimilation function is referred to as a(P). The following scenarios are distinguished:

case 1 This is the standard case elaborated in the last sections. The as­similation function is a(P) = 0.005P.

case 2 In this scenario, the assimilation function is linear up to a certain point P, after which assimilation is constant. Hence: a(P) = 0.005P for P > P.

case 3 This case is more dramatic than the previous scenario. At the threshold level P, it is assumed that the ecosystem will break down (Barbier & Markandya, 1990). This means that a(P) « 0 for P > P.

case '50' to '53' Finally, a set of scenarios is considered, where the as­similation function is taken to be:

a(P) = 0.005P - aP3 (24)

where a has the value 50e-1O in case '50', 51e-1O in case '51', 52e-1O in case '52' and 53e-1O in case '53'. The rationale for this specification is that it is virtually identical to the linear assimilation function for small values of P and the differences between the four only appear for higher values of P. Note that the experience with the Greenhouse effect up till now is only with low levels of P. The values of the non-linear function as such are arbitrarily chosen. However, as will

38

Cl(P)

-Cl(P)

H. Cesar and A. de Zeeuw

cas. 1 (standard

can)

-----.-- cas~ 2

p

Figure 4: Assimilation for different specifications of o:(P)

be shown below, notwithstanding the extremely small differences in these cases, the corresponding steady state results are vastly different.

In the rest of this subsection, the consequences of the deviation from linearity for large values of P will be analysed for the cases described above. Note that the standard case (Case 1) has been discussed extensively in the previous section. Case 3 will be analysed first, as it is mathematically simpler than Case 2.

o:(P) collapses for P > P It is assumed throughout that the initial value of P, Po, is below P. The mathematical formulation is:

(P) = {O.005P for P ::; P 0: « 0 for P > P (25)

With this specification, substituting the assimilation function o:(P) for o:P in the model of the previous section amounts to adding the constraint P ::; P to this model and leaving the differential equation P = 'Y F - o:P

Sustain ability and the Greenhouse Effect 39

unchanged25 • The Lagrangian becomes in this case:

£(F, P,1jJ) = U[Y(F, P)] + 1jJ(-yF - aP) - v(P - P) (26)

This gives the same necessary conditions except for:

And hence:

With, additionally:

8£ , - = U Yp - a1jJ - v 8P

.,p = (p + a)1jJ - U'YP + v

v ~ 0 and v(P - P) = 0

(27)

(28)

(29)

To solve this model, when the initial stock Po is below P, assume that the constraint P :::; P becomes binding at time T. It is not optimal for P to bend back for T > T, (see Feichtinger & Hartl (1986; p.167-168) for an explanation), the following expression holds at time T: P = .,p = o. This means that the constrained steady state values (Joo, poo and ij;oo are reached at time T (c.f. Feichtinger & Hartl, 1986). Thus:

For t < T, V = 0 and .,p = (p + a)1jJ - U'Yp For t = T, V > 0 and '¢ = 0

This means that .,p will jump at t = T with the amount v, where v is26 :

The jump in .,p at t = T prevents the economy from 'shooting through' the steady state, as is illustrated in Figure 5:

Note that in order to reach the steady state under the constraint P :::; P, the optimal trajectory has to bend away from the unconstrained trajectory. Translating this for a decentralised economy, the government would have to set, in the constrained case, the Pigouvian taxes higher than in the

25 A linear assimilation function OtP combined with a Greenhouse damage function g(P), defined as:

g(P) = { f<:) for P:':; P for P > P

will give exactly the same mathematical solution as the model elaborated in the text. This specification is very similar to the one taken by, among others, Gottinger (1992). He takes f(P) = 1. In this case, the trajectory of the phase-plane of (P, 1f;) is above our path for small values of P but will fall more dramatically once P approaches P. The steady state value of 1f; in our case will lay above that of Gottinger.

26 As P is given, and as F will not jump at t = T (Feichtinger & Hartl (1986», F is known and hence, the values Coo and iiioo follow from the production function resp. Equation 6.

40 H. Cesar and A. de Zeeuw

P:P p

Figure 5: Phase Plane of (P,1/J) with constraint of P

unconstrained case in order to prevent the climate from collapsing. However instructive this may be, the basic problem with many real life environmental problems is that nothing can be said with any degree of certainty about when ecosystems might collapse. Note that the formulation with a(P) ~ 0 for P > P implies that the system is irreversible. Hence, once the ecosystem has collapsed, it is impossible for the economy to continue. In order to analyse this issue, however, stochastic optimal control methods have to be used.

a(P) becomes constant for P> P This case is less dramatic than the previous one, though much more difficult to work out. The assimilation function is:

a(P) = { 0.005~ for P ::; i!. 0.005P for P > P

(30)

In Figure 6, one possible situation is worked out. Note that the P = O-locus has a kink at P = P as P = "IF - aP becomes P = "IF - aP (the value of 1/J at this kink is referred to as 1/J2)' At the same time, the"j; = O-locus has

Sustainability and the Greenhouse Effect

steady state in the constrained case

p

Figure 6: Phase Plane of (P, 'I/J) with kinked assimilation function

41

a discontinuity at P = P as the left branch -J; = (p + a)'I/J - U'Yp becomes -J; = p'I/J - U'Yp for the right branch. The value of 'I/J at the intersection between the left branch and the vertical line P = P is referred to as 'l/Jl. Similarly, at the intersection with the right branch, the value of 'I/J is called 'l/J3. Hence, six possible situations can be distinguished27 :

1. 'l/Jl > 'l/J2 > 'l/J3;

2. 'l/J2 > 'l/Jl > 'l/J3;

3. 'l/Jl > 'l/J3 > 'l/J2;

4. 'l/J2 > 'l/J3 > 'l/Jl;

5. 'l/J3 > 'l/J2 > 'l/Jl;

6. 'l/J3 > 'l/Jl > 'l/J2;

27 Again the focus is on the case where P is smaller than the poo of the unconstrained case.

42 H. Cesar and A. de Zeeuw

The latter two cases have two equilibria, all the other cases have one equi­librium.

For simplicity, the focus is on Case 1.28 For Po :::; F, this case is basically the same as the assimilation function with collapse. For Po > F, the Lagrangian is:

£(F, P, 'I/J) = U[Y(F, P)] + 'l/JClF - aF) + w(P - F) (31)

This can be solved analogous to the previous problem with a jump at ~ of w = -U'Ypoo + (YI/Joo, so that the right branch of the trajectory ap­proaches the constrained steady state vertically, as is shown in Figure 6. This concludes the discussion on the constant assimilation case.

a(P) has an additional non-linear term The idea is here to extend the assimilation function with a non-linear term. For low values of P, this extra term is not visible. For higher values of P, however, this term will dominate, as is shown in Figure 7 for the cases '50', '51', '52' and '53' as defined above. Note that the ~ = O-locus is basically not visibly different for the four cases and is slightly steeper than in the standard case. The difference in the P = O-locus for the four cases is quite remarkabe for high values of P. The four scenarios will be discussed shortly:

case '50' This scenario has a unique steady state for P < ft, which is higher than poo in the standard case;

case '51' In this case there are two equilibria, one stable and one unstable. Assuming as before that the initial value of P is small enough, the steady state is the equilibrium with the lowest level of P. However, if the initial value of P is for some reason very large, there is no way to stop a collapse of the ecosystem from taking place;

case '52' This scenario has one equilibrium, that is a saddlepoint if reached from below and an unstable equilibrium if reached from above. This means that for initial values above p oo , the ecosystem will again break down;

case '53' In this case, there is no intersection of the two loci. This means that the economy will collapse. The steady state with zero fossil fuel use and hence zero production29 will, however, not be reached in finite time. This means that it is optimal for the economy to slowly cease to exist.

28Note that the other cases can become rather complicated. 29The equilibrium level of the concentration of Greenhouse gases is: P = (0.005 )0.5.

a53

It can easily be shown that this steady state is locally stable.

Sustainability and the Greenhouse Effect 43

p

- ~ = 0

~~ P = 0 53

Figure 7: Phase Plane of (P, 'ljJ) for different non-linear specifications of a:(P)

The different functional forms chosen here are, of course, arbitrary. The sole reason why they are elaborated is to show that the assimilation function may seem quite similar for low values of P in different scenarios but lead to surprisingly different results when the non-linear component starts to dominate. Economically, the result means that the long run levels of consumption, production and pollution depend crucially on the exact natural science knowledge of the environmental feedbacks.

4. Conclusions

There exists great uncertainty with respect to the actual assimilative capac­ity of nature for elevated levels of the concentration of Greenhouse gases.

At the same time, optimal control models of the Greenhouse Effect as­sume generally that the assimilation function of nature is linear (Nordhaus, 1990).

Given this uncertainty, the robustness of policy conclusions for changes in the specification of the natural regeneration function has been analysed in this paper in a deterministic optimal control setting.

The conclusion is that slight variations in the parameters of the assim­ilation function can have a dramatic impact on the steady state results.

44 H. Cesar and A. de Zeeuw

This implies a strong warning to researchers in environmental economics to use deterministic optimal control models for policy purposes when there is uncomplete knowledge about complex ecosystem interactions.

It also means that future research should concentrate on trying to get a better grip on the assimilation function for large values of pollution. In addition, it indicates that explicit modelling of uncertainty is extremely important.

References

[I] Arrhenius E. and T.W. Waltz, The Greenhouse Effect: Implications for Economic Development, World Bank Discussion Papers, No. 78, Washington D.C, (1990)

[2] Barbier E.B. and A. Markandya, The conditions for achieving envi­ronmentally sustainable development, European Economic Review 34, pp. 659-675, (1990)

[3] Cesar H.S.J., Control and Game Models for the Greenhouse Effect: Economics Essays on the Commedy and the Tragedy of the Commons, Springer-Verlag, (LNEMS nr. 416), Heidelberg, Germany, (1994)

[4] Dasgupta P., Control of Resources, Basil Blackwell, Oxford, (1982)

[5] Feichtinger G. and R.F. Hartl, Optimale Kontrolle Oekonomischer Prozesse, Walter de Gruyter, Berlin, (1986)

[6] Forster B.A., Optimal pollution control with a nonconstant exponen­tial rate of decay, Journal of Environmental Economics and Manage­ment, pp. 1-6, (1975).

[7] Forster B.A., On a One State Variable Optimal Control Problem: Con­sumption Pollution Trade-Offs, J.D. Pitchford and S.J. Turnovsky, eds., Applications of Control Theory to Economic Analysis, North Hol­land, Amsterdam, (1977)

[8] Forster B.A., Optimal energy use in a polluted environment, Journal of Environmental Economics and Management, 7, pp. 321-333, (1980)

[9] Gottinger H.W., "Economic Models of Optimal Energy Use inder Global Environmental Constraints", in: Conflicts and Cooperation in Managing Environmental Resources, R. Pethig, ed., Springer-Verlag, Berlin, (1990)

[10] Maler K-G., The Acid Rain Game II, Paper presented at the Autumn Workshop in Environmental Economics, Venice, Sept. 29-0ct. 5, 1991

Sustainability and the Greenhouse Effect 45

[11] Nordhaus W., How fast should we graze the global commons?, Amer­ican Economic Review, 72(2), pp. 242-246, (1982)

[12] Nordhaus W., The Economics of the Greenhouse Effect, Paper pre­pared for the 1989 Meetings of the International Energy Workshop and the MIT Symposium on Environment and Energy, August 1989

[13] Nordhaus W., To Slow or not to Slow: The Economics of the Green­house Effect, Cowles Foundation Discussion Paper, (1990)

[14] Pethig R., Optimal Pollution Control, Irreversibilities, and the Value of Future Information, Discussion Paper, No. 6-90, University of Siegen, (1990)

[15] Pezzey J., Economic Analysis of Sustainable Growth and Sustain­able Development, Environment Department Working Paper, No. 15, World Bank, (1989)

[16] Tahvonen 0., von Storch H., and Xu J., Optimal Control of CO2 Emis­sions, Paper Presented at the Annual Conference of the European As­sociation of Environmental and Resource Economists, Cracow, Poland, June 16-19, 1992

[17] Van der Ploeg F. and De Zeeuw A.J., International aspects of pollu­tion control, Environmental and Resource Economics, 3 pp. 117-139, (1992)

[18] Van der Ploeg F. and Withagen C., Pollution Control and the Ram­sey ProblemEnvironmental and Resource Economics, 1, pp. 215-236, (1991)

[19] Withagen C., Pollution and Exhaustibility of Fossil Fuels, Mimeo Eind­hoven University of Technology, (1993)

Consumption of Renewable Environmental Assets, International Coordination and Time Preference

Andrea Beltratti

Abstract

A two-country linear-quadratic model of depletion of a renewable resource is studied both in static and in dynamic terms. The model allows for negative consumption externalities through the action on the stock of the environmental good, which enters the utility function. It is shown that the noncooperative solution of the dynamic model is characterized, in the steady state, by suboptimally low levels of environment and consumption, and that such phenomenon increases with the rate of time preference in a nonlinear way which depends on the structure of the economy.

1. Introduction

The definition of sustainable development contained in the Report of the World Commission on Environment and Development [26] (" ... development that meets the needs of the present without compro­mising the ability of the future to meet their own needs") is essentially based on the role of stock variables as dynamic elements interconnecting the present and the future. Among these, the Report stresses the role of exhaustible and renewable environmental resources, especially when they are irreversible, and therefore there are permanent damages from an excess rate of harvest. Important examples are deforestation, desertification, and environmental capacity to absorb waste.

The definition points to the importance of the rate of time preference as a key element in the analysis of sustainability. In a forward-looking economy, rational agents compare current and future marginal utilities and productivities, and make decisions on the use ofresources which are strongly dependent on their rate of time preference. A first important issue is there­fore the analysis of models where agents deplete renewable resources on the basis of comparisons between the rate of time preference and the marginal productivity. There is a large literature on this problem, and on the ethical justification for discounting future utilities as a basis for choice of public investment, see Heal [11] and references contained therein.

48 A. Beltmtti

A number of papers consider the issue of sustainability of growth. The model studied in this paper is in spirit (but not in the analytical formu­lation) similar to that of Beltratti, Chichilnisky and Heal [1], who study the steady state of a closed economy exploiting renewable resources whose stock may affect both production and utility. Other papers are not focused on the specific issue of sustainability, but consider models that are very rele­vant from an analytic understanding of the preconditions for sustainability; among them we mention Tahvonen and Kuuluvainen [22] and [23], Michel and Rotillon [15], Van der Ploeg and Withagen [17]. See Smulders [21], and references therein, for sustainability in endogenous growth models.

Even though the definition of sustainability is given independently of any issue regarding international coordination, the Report contains a sub­stantial section devoted to the implementation of sustainability in a world populated by sovereign countries. Among such problems, the role of inter­national coordination is given an important place, especially due to ecolog­ical interdependence; for example Rogers [19] reports that more than 200 river basins, accounting for more than 50% of the land area of the earth, are shared by two or more countries. Often, commonality of use and ef­fects is associated with the natural environment, and indeed many problems connected with the common use of environmental resources have inspired economic analyses of international exploitation of resources. Prominent examples are the ozone layer, fisheries, water courses and climate change.

There have been few attempts to relate sustainability and international coordination in the context of formal models, apart from the general consid­erations raised by the Brundtland Report. Most of the models are static, see as an example Maler [14], even though there are a few strategic dy­namic models of exploitation of common resources, see Clark [4] and the references briefly discussed in the second section of this paper. Relating sustainability and international coordination seems however an interesting enterprise, because only a dynamic game-theoretic framework allows one to consider in a unified way aspects of excessive exploitation of renewable resources and aspects related to difficulties in implementing optimal paths in a decentralized world economy where the various countries follow non­cooperative strategies. For example, Hollick and Cooper [12] conjecture that "Optimal use of the commons requires some cutback in use, but that imposes costs on at least some users in the short run, until the commons recovers and settles into a long-run managed equilibrium". This clearly relates exploitation policies and long run equilibrium, but requires a more specific analysis as a condition for implementing corrective economic and environmental policies.

This paper relates sustainability to international coordination in the context of a simplified linear-quadratic model of consumption of a renew­able resource. Simplifications about variables and functional forms are

Consumption of Renewable Environmental Assets 49

common in the dynamic game literature concerned with the issue of an­alytical tractability of the models. In Section 3, the model is developed for a two-country world. Section 4 considers the static game restricted to the choice across steady states, while Section 5 studies the intertemporal problem. In both cases, the model is solved first for the case of a benevo­lent dictator, and then for the Nash equilibrium. It is shown that the static model is compatible with a representation of the tragedy of the commons in terms of welfare effects, but that overdepletion of the stock is not a neces­sary consequence of strategic interactions. In the dynamic version instead overdepletion of the stock of environmental assets is the outcome for a large range of the admissible parameters values. It is also shown, by means of numerical simulations, that the more impatient the players the more seri­ous the tragedy: patient countries take a deeper look into the future, and implement policies which are more sound in their use of natural resources, even in the context of a decentralized strategic equilibrium. Section 5 offers some concluding comments.

2. Dynamic Models of the Tragedy of the Commons

Clark[4] presents the basic structure of intertemporal models for studying the tragedy of the commons in a strategic framework. There are however other more recent papers on the subject, considering both symmetric and asymmetric players, and cast in both continuous and discrete time. The last distinction seems to be more important than one might think, even though the connection between the results obtained by the various models is not completely clear.

Galor [9] is an interesting early example of application of differential game theory to international policy coordination in the context of a North­South model. The two blocks are asymmetric; there is abundance of labor in the South, but the North has to import from the South raw materials. A result of global dynamic inefficiency of uncoordinated policies is shown with a simple specification of objective functions.

Dockner and Long [6] consider a continuous time model of emission and pollution. Emissions enter the utility function positively, while pollution has a negative impact on utility. Pollution increases over time because of emissions, but decreases spontaneously, at least partially. They show that fully coordinated pollution control achieves larger pollution than noncoop­erative pollution control, due to a free-rider problem. They also show that nonlinear Markov-perfect strategies, see for example Tsutsui and Mino [25] for an application to dynamic duopolistic competition, may support the Pareto-efficient steady state pollution stock.

Van der Ploeg and de Zeeuw [18] consider a similar model, both in a stock and in a flow version, and discuss at length differences between open-

50 A. Beltratti

loop and closed-loop information sets. In the open-loop strategy there is an infinite period of commitment, policies depend on the initial stocks and time, and countries stick to their policies, while in the closed-loop solution countries condition actions at each time t on the levels of the existing stock. Closed-loop is therefore a much more realistic equilibrium concept, and will be the only one considered here. However, closed-loop solutions require the use of Bellmann's dynamic programming rather than Pontryagin's maxi­mum principle, and can be implemented only in the context of models with simple specifications. In fact van der Ploeg and Lighthart [16] consider a more general model of sustainable growth with renewable resources and physical capital, displaying endogenous growth solutions, but limit them­selves to a comparison between the first-best and a second-best open-loop strategy.

Tornell and Velasco [24] consider a dynamic model of the tragedy of the commons, and apply it to the problem of capital flights. In their model util­ity is provided by consumption which depletes a resource with a constant marginal renewal rate. Different from most other models which are based on linear-quadratic specifications, they consider a more general isoelastic utility function (not including the stock) which provides consumption func­tions that are easy to analyze in terms of rates of growth. They do not analyze steady states, and define the tragedy of the commons in terms of rates of depletion of the common resource. They also show an interesting result for which introducing a second asset with lower productivity but well defined property rights may improve overall welfare. It seems essential to their results that there are constant returns to scale to all the assets.

Benhabib and Radner [2] also consider joint exploitation of a productive asset, again in the case where the asset does not enter the utility function, which is linear in consumption. They show that a trigger-strategy equilib­rium may exist depending on the discount rate, the detection technology and the initial conditions. They also introduce the concept of a switching equilibrium, according to which players follow inefficient depletion strate­gies up to a certain point, and then switch to an efficient trigger strategy. See also Rustichini [20] for analysis of existence of equilibria.

Dutta and Sundaram, in a series of papers, analyze the tragedy of the commons in the context of a discrete-time model. In [7] they show that Markov-perfect equilibria can be characterized by the opposite result of under-exploitation of the common resource; in [8] they show that there can be remarkable differences in terms of dynamics (cycles and chaos) between competitive and strategic models. An interesting topic for future research would be to connect results obtained in continuous-time and discrete-time models.

Consumption of Renewable Environmental Assets 51

3. A Two-Country Model of Depletion

One important aspect of sustainability of growth lies in the comparison be­tween the rates of harvesting and reproduction of the renewable resource. Many environmental problems like deforestation and water pollution can be imputed to an excessive use of the services of the resource, which sometimes can lead to extinction of the stock. This phenomenon can be considered only in the context of models where economic activity affects a natural resource. Variants of the so-called cake-eating model have been used fre­quently for this and other purposes, see Dasgupta and Heal [5] for the analysis of various versions where an exhaustible resource is used for pro­ducing consumption goods, and Krautkraemer [13] for including the stock of the resource into the utility function. This is important for an analysis of sustainable development, as generally environmental assets also provide direct utility.

The model considered in this paper develops a quadratic version of the problem of using a reproducible resource for providing consumption goods. While some of the growth models mentioned in the introduction, as well as those mentioned in this section, consider descriptions of the structure of the economic system involving the stock of environmental resources and the stock of capital, and sometimes also a third stock in the form of pollution, this paper considers a simplified framework which ignores both physical capital and pollution. The two basic blocks of the model are a description of preferences including both the flow of consumption and the stock of the resource, and a reproduction function of the stock which makes the resource a renewable one.

Formally, instantaneous utility at time t depends on the level of con­sumption of a produced good and on the existing stock of an environmental good, that is:

C2 f3A2 u(C, A) == U(C) + V(A) == aC -"2 + OA - -2- (1)

where C is consumption and A is the stock of the environmental asset. Quadratic utility is equivalent to assuming satiation from consumption and from the environment; a and ~ are the two bliss points for the flow and the stock for the two functions U(C) and V(C) defined above, and are usually taken to define the limits of applicability of the utility function. Here solutions will be looked for over the whole range of positive total utility; a declining but positive total utility may be justified by thinking of congestion effects of various kinds. It will be seen in the next section, after describing the consumption technology, that an optimal solution may well imply a level of the stock exceeding its bliss point.

Instantaneous utility is embedded in a forward-looking description of

52 A. Beltmtti

preferences of the standard type:

where 8 is the rate of time preference. In a two-country world, it is assumed that both countries derive con­

sumption from the same renewable resource, whose dynamics over time are given by:

(2)

Consumption by each country therefore depletes the common resource, even though the resource itself reproduces spontaneously. Such a dynamic equation represents the link existing between the economic and the ecolog­ical systems, and is at the heart of the discussion about sustainability of development, which can be interpreted as the requirement of maintaining the use of the resource within the boundary established by its natural re­production rate. In (2), at time t any level of total consumption which is equal to rAt is sustainable forever; lack of sustainability would result in­stead from a consumption rule dictating at any time t a flow of consumption larger than the flow coming from reproduction, rAt.

Such an extinction scenario will not be considered among the optimal solutions of the model, which will be analyzed mainly in terms of the per­manent steady state:

(3)

where the stock of the resource is proportional to consumption. The par­ticular level of the stock that is chosen depends on various parameters of technology and preferences, among which is important is of the rate of time preference. In the context 0 f the steady state therefore there is no problem of sustainability in a physical sense, even though the spontaneous actions of the two countries may drive the system to a level of the stock which is not optimal from the point of view of a benevolent dictator. The model is therefore used properly to compare welfare in various steady states rather than analyzing catastrophic scenarios of an unsustainable evolution.

Note that while countries may have an incentive to overconsume in the short run due to the positive rate of time preference and to the strategic nature of the problem, they also know that the flow of consumption and the stock of environment are complementary in the long run steady state, as, given the fixed reproduction rate, the steady state stock determines the steady state flow of consumption. There are therefore countervailing considerations about optimal exploitation of the resource.

Consumption of Renewable Environmental Assets 53

The model will be solved in the next section in terms of the steady state, that is imposing the condition that A = 0 and substituting the result in the objective function, ignoring any transitional dynamics. This is useful to set up the problem in familiar terms (for example it will be possible to derive standard reaction curves for the two countries), but cannot be used to assess the role of the rate of time preference which, as will be seen later, is an important determinant of the value of the variables in the steady state. In both cases (static and dynamic versions) the first-best solution achieved by an international planner is compared to the Nash equilibrium, which in the dynamic model is considered in its variant of Nash-Markov equilibrium.

4. Static Model

The static version of the model is obtained by considering the restriction according to which the stock is constant and consumption is proportional to the stock. This is equivalent to players choosing across different possi­ble steady states, ignoring the transitional dynamics due to differences in optimal choices and initial conditions.

4.1 The first best and the optimal choice of the stock

In the static model the planner maximizes the sum of instantaneous utilities by taking into account (3). Such a problem makes sense in the case of a quadratic utility function, which implies satiation in the levels of the goods, but would not make sense in the case of a utility function which implies non-satiation. In the latter case the solution would set both the flow of consumption and the stock of environment equal to infinity, given unboundedness of the linear reproduction function. Another case in which the model would make sense is when a utility function characterizing non­satiation were coupled with a bounded reproduction function, for example the logistic. The latter case is considered in the context of a closed economy in Beltratti, Chichilnisky and Heal ([I]) to derive the green golden rule.

Substituting the constraint into the objective function and maximizing with respect to the consumption flows:

maxCl,c2 (0{C1+ C2)- (C?;C~) +2B(C1;C2) _/1(C1;C2)2) (4)

the solution which is obtained for the case of symmetric consumption flows (C1=C2=C) is:

CSC = r{ra + 2B) r2 + 4/1

54

ASC = 2m+40 r2 + 4f3

A. Beltmtti

(5)

where the superscript se denotes the cooperative solution of the static model. The optimal stock is positively related to its marginal utility to the weight of consumption in utility, a, and is negatively related to the reproduction rate of the stock, as the larger the productivity the lower the stock which is needed to support a given level of consumption.

By considering (3), the objective function in equation (4) can also be written as a quadratic equation in terms of the stock as:

arA _ (r2:2) + 20A _ f3A2 =

= (ar + 20)A _ (r2 :4(3) A2 (6)

whose bliss point is not surprisingly equal to the optimal solution obtained in (5), which in turn is different from the bliss point for the function V(A), defined in (1), equal to ~. In general (see Figure 1) 2;2",+~~ > ~ when

0< 2af3. (7) r

This means that it is optimal to choose a stock corresponding to a direct utility lower than its bliss point when preferences are strongly tilted towards consumption rather than towards the stock as an environmental amenity and/or when its reproduction rate is low relative to other parameters. In this case the environment can be interpreted as a factor of production.

Conversely, when a is small relative to other parameters, it is optimal to choose a stock lower than the one maximizing V (A), as trying to reach the latter would involve a loss in utility, due to negative marginal utility of consumption, larger than the direct gain provided by the stock. Of course this would never happen in the following two cases: (a) if a quadratic utility for the stock were coupled with a function showing nonsatiation in consumption, as it would always be optimal to increase the stock at least to the level corresponding to its bliss point, or (b) if it were possible to dispose of the flow of consumption at no utility cost. Such a combination of parameters therefore seems to imply a less interpretable equilibrium where the stock of environment is kept low in order not to suffer any disutility from increased consumption.

4.2 The Nash solution and the static tragedy of the commons

The solution can be compared with the one obtained for the case where each country maximizes its own utility taking as given the consumption of

Consumption of Renewable Environmental Assets 55

the other country, but considering the effect of its own consumption on the

stock. For country one:

( C cr Il (C1 +C2) {3 (C1 +C2)2) maxc a l--+U --

1 2 r 2 r

fL(C, A), V(A)

f3

V(A)

2m +40

r2 + 4f3

Figure 1. Quadratic utility

fL(C,A)

A

The first order condition can be expressed directly in terms of the

reaction function: C _ ar2 + Or - {3C2

1 - r2 + (3 .

For country two:

C _ ar2 + Or - (3C1 2 - r2 + (3 .

The two reaction functions are downward sloping (see Figure 2), and

intersect at the point: C SN = r(m + 0)

r2 + 2(3

where SN means static and Nash, corresponding to a stock:

ASN = 2(m + 0) . r2 + 2(3

(8)

It is now important to define properly what is a tragedy of the com­

mons. If it is meant that ABc> ABN, then a simple comparison of equa­

tions (5) and (8) shows that the tragedy occurs in the static version of

the model if 0 > 2~f3, which is related to the condition described in (7).

56 A. Beltmtti

This case, in which the stock is low because of low marginal utility of con­sumption, is not particularly interesting. Under the more plausible set of values of the parameters, overdepletion of the stock does not occur. On the contrary, the two countries accumulate too much stock in the attempt to pick a high consumption steady state and ignoring the buildup of the stock on the part of the other country. As a result there is an excessive accumulation of the stock, which already belongs to the area of negative direct marginal utility. It can be concluded that a tragedy of the commons defined in terms of welfare always takes place in the model, regardless of the values of the various parameters, but that the precise manifestation of such coordination failure may involve either too much or too little stock. It is interesting to contrast this result with those obtained in the dynamic version of the model.

r(ra + 8)

r2 + 2f3

r(ra + 8)

r2 + 2f3 Figure 2. Reaction functions

5. The Dynamic Model

The dynamic version of the model is useful to take into account the role of the rate of time preference (it could also be used to incorporate explicitly the transitional dynamics towards the steady state, even though this issue will not be considered here). It is however considerably more burdensome from an algebraic point of view.

5.1 The first best

The dictator maximizing total welfare solves:

( (C~ + C~) 2) [ 1 H = a(C1 + C2 ) - 2 + 2BA - f3A + oX -C1 - C2 + rA .

Consumption of Renewable Environmental Assets 57

The optimal consumption rule is obtained from maximization with re­spect to consumption :

Ci = a - A, i = 1,2 (9)

and shows that consumption is a linear function of price. The price and the stock of the asset move over time according to the dynamic equations:

>. - 6A = -20 + 2{3A - Ar, (10)

A=rA-2(a-A). (11)

Moreover, the optimal solution must satisfy the transversality condi­tion:

lim e-6t AA = O. t--+oo

The steady state is characterized by the following variables:

CC _ r(a(r - 6) + 20) - r(r - 6) + 4{3 ,

AC = 2a(r-6)+40. r(r - 6) + 4{3

(12)

It is also possible to solve for the price as the present discounted value of future marginal utilities of the stock of environment:

A = 2100 e-(r-6)(0 - {3A)dt

according to which the price is large in anticipation of a stream of large marginal utilities of the stock, in turn due to a low level of the stock itself.

Stability of the model can be analyzed by considering the dynamic equations for the stock (11) and for the flow of consumption. The latter can be obtained by differentiating (9) with respect to time and then considering

(10). The matrix of the resulting system, [-;{3 6 __ 2 r ], may be used to

derive the stability requirement 6 < r + ~. Such a condition also implies that one must have 6 < r + ~ in order to have a positive stock in the steady state, see equation (12). However this is ensured if the parameters respect condition (7), since in this case ~ < ¥.

One interesting implication of the model is that there is a connection between (7) and the condition for a long-run negative relationship between the steady stock of environment and the rate of time preference: 8:6G

¢:}

Or < 2a{3. When this occurs, a more patient society can accumulate more

58 A. Beltratti

environment in the steady state and, given the technological relation of com­plementarity existing between consumption and environment across steady states, consume more. In this sense the model can take into account the importance of discounting as an element of discrimination against future generations in the definition of sustainable development.

5.1.1 The Nash-Markov equilibrium

Under the Nash-Markov equilibrium each country optimizes at every instant and chooses its actions taking as given the rules followed by the other country; strategies depend only on the current value of the state variables, and this makes the solution different from a repeated game, as the stock represents a memory of the past actions of the agents. The equilibrium is obtained by solving the Hamilton-Jacobi equation; for the first country:

8WI (A) = max [aCI- ~? +I1A-~A2+W{(A)[-CI-C2+rAl] (13)

where WI (A) is the value function of the first country, and W{(A) is the derivative of the value function with respect to the stock. To proceed, given the quadratic utility function, one can guess a quadratic value function of the type WI (A) = 'Y+cA + ~ A 2 , and then solve for the unknown coefficients. It is possible to show the following proposition:

Proposition 1 The value function is characterized by the following coeffi­cients:

>t' ~ ~ [~ - r ± (r -n 2 +313 (14)

N 11 - 2arJN c = ----'---:-:-

8 - r - 3rJN

N a 2 + 3(cN )2 - 4acN

'Y = 28

and the optimal stock is equal to:

(15)

Proof. By maximizing the right hand side of (13) with respect to Clone finds consumption:

Consumption of Renewable Environmental Assets 59

as a function of the available stock of resource and the parameters of the value function. Such consumption function can then be substituted back into (13), for both countries. In so doing one is left with a quadratic equation in the stock:

A2 (87] + 7]2 +!!. _ TJT - 27]2) + A [e (8 - 37] - r) + 2n7] - OJ + 2 2 2

+ (8" _ ~2 _ 3~2 + 2ne) = O. Such an equation holds regardless of the value of the stock only if the

terms in parenthesis are all equal to zero, and this gives three constraints which solve for the three unknown coefficients reported in the Proposition. Given the consumption rules, it is also possible to solve for the steady state stock. •

Let the two solutions to the quadratic equation (14) be denoted with 7]f > 7]f· The following proposition establishes the sign of such solutions:

Proposition 2 7]f > 0 > 7]f.

Proof. Given the stability condition, it is assumed that r > ~, so that

clearly 0 > 7]f. 7]f < 0 <=> [~- r + J (r - ~)2 + 3/3] < 0 <=> ~ - r <

-J (r - ~)2 + 3/3. As both terms of the last inequality are negative, the last condition is equivalent to:

which never holds since /3 > O. Therefore 7]f > 0 • Can one say something about the desired sign on the basis of economic

intuition? Looking at the value function, one would prefer to have a posi­tive value, since only in this case would a larger stock of resource imply a larger utility value. However looking at the equation that determines consumption as a function of the stock, one realizes that such a choice implies a perverse reaction function, as consumption of each country would be negatively related to the stock itself. The only possibility for having a model which yields sensible outcomes is to consider a case where 7]f < 0 to get the reaction function right, but at the same time eN + 7]f A > 0,

60 A. Beltmtti

as the last is a condition for having a positive marginal value of the stock, and of course A > O. It turns out that such conditions hold given some restrictions on the various parameters.

In order to derive such restrictions, note that:

Proposition 3 The stock can be rewritten as

AN = 2a(TJf +r-8)+20 r(r - 8) + 4(3 - 6(TJf)2 - 3TJfr + 28TJf

(16)

Proof. AN can be written in a way which is more directly com­parable to A C by considering the definition of TJf, i.e., 3TJf

[ ~ - r - J (r - ~) 2 + 3(3], squaring both sides and then solving for (3 to

get 4(3 = 12(TJf)2 + 8TJfr - 4TJf8. By using the last expression one can rewrite AN in the way described in equation (16) •

Some restrictions on the parameters have to be taken into account af­ter noting that the denominator of (16) is a quadratic function in TJN, which

takes the value 0 when TJN = f.J. [28 - 3r ± J(28 - 3r)2 + 96(3 + 24r(r - 8)] .

Depending on the value of the various parameters, one of these two numbers can be positive; this happens when (3 is large and when r - 8 is large.

Keeping in mind discontinuity of the solution at these two points, one can study the stock as a function of TJf, in order to verify that the solution is indeed positive. To this purpose, note that one can write

TJf = [~- i - ~] , where r == J (r - ~)2 + 3(3. As TJf approaches its upper bound, r -t 0, or, equivalently both (3 -t 0 and 8 -t 2r. However the restriction obtained by the stability analysis of the model, 8 < r + 4f!, implies that when (3 -t 0, 8 -t r, implying that the upper bound for TJf is reached as r -t ~. In this case one can easily show that:

2a ['28 - r.] 1· AN 6 2 l~ = 8 (ti.!:)

r-->2 3" 2

which is positive given that 8 -t r. The stock becomes equal to 0 when TJf = 8 - r - ~. It will therefore be

assumed that 112 [28 - 3r - 028 - 3r)2 + 96(3 + 24r(r - 8)] < 8 - r - ~, and the range 8 - r - ~ < TJf < ~ - ~ < 0 will be considered in what follows.

In comparing AN with AC one immediately sees that the numerator of the former is always lower than the numerator of the latter, so that a

Consumption of Renewable Environmental Assets 61

sufficient condition for having AN < AC is easily obtained by requiring that the denominator of AN be larger than the denominator of A C .

Proposition 4 A sufficient condition for the tragedy of the commons is that TJf > -~ + ~.

Proof. Immediate from a comparison of the two solutions: a sufficient condition for AN < AC is that -6(TJf)2 - 3TJfr + 28TJf > 0, which is equivalent to TJf > - ~ + ~ •

When 8 is small with respect to r one may well have that 8 - r - ~ < -~ +~, and this, due to the fact that the last proposition gives a condition which is only sufficient, means that there is a region to the left of -~ + ~ which needs to be checked for the existence of the tragedy of the commons. This will be done by means of simulations.

A few remarks are offered as comments for the results. First, it is of interest to point out that differences between first-best and

uncoordinated solutions cease if f3 = () = 0, in which case AN = A C = 2;. This shows the importance of specifying models including the stock in the utility function. This result of course depends on the choice of the specifica­tion of the quadratic utility function; Tornell and Velasco [24] for example obtain differences between coordinated and uncoordinated solutions also in the context of a model which does not consider the stock in the utility function, but they specify isoelastic preferences, and are concerned with a definition of the tragedy of the commons in terms of rates of depletion of the resource.

Second, contrary to what happens in the problem of choice across steady states, the stock of the environment in the coordinated solution is always larger than the stock in the Nash solution. Moreover, the more reasonable case from the static point of view implies an excess accumula­tion of stock in the static game, but a negative relationship between the rate of time preference and the optimal stock in the dynamic game.

Third, the problem of the commons can be even worse than described by Hollick and Cooper [12], since lack of coordination may in general affect the steady state of the model, and not only the short run. In the model studied in this paper for example, lack of coordination affects the long run solution of the model, so that the asset settles into a long-run equilibrium which is worse than the one obtained under full cooperation.

Fourth, numerical simulations show that the difference between cooper­ative and noncooperative solutions depends on the rate of time preference. The following table reports, for different values of 8 and r, and for a model where a = 10, () = 10 and f3 = 0.01, the ratio between the steady state

62 A. Beltratti

stock of the Nash solution and the steady state stock of the coordinated I . AN

so utlOn, F:

Table 1: Ratio between Nash-Markow and first-best steady state stocks

6 r 0.06 0.08 0.12 0.16 0.01 0.669 0.747 0.866 0.947 0.035 0.555 0.656 0.817 0.928 0.06 0.529 0.746 0.900 0.085 0.639 0.857 0.11 0.701 0.736 0.135 0.653 0.16 0.350

The results show that the lower the rate of time preference, the lower the ratio between cooperative and noncooperative stocks. In this sense, the Brundtland Report is correct in emphasizing the need for more intergener­ational equality, for example through a lower rate of time preference, as a necessary condition for sustainable growth. The results of this paper show that by decreasing the rate of time preference it may be possible also to decrease the problems caused by lack of coordination in the international exploitation of the commons. Finally, note a result which may well be specific to the model and to the parameters chosen, but may be worthy of further investigation: the negative effect of the rate of time preference is nonlinear, being significantly large only when it gets close to the up­per bound represented by the rate of reproduction of the environmental resource.

6. Conclusions

The paper has analyzed a model of joint exploitation of a common resource which enters the utility function. A linear quadratic specification is chosen for the sake of obtaining analytic solutions. It is shown that the model in the non-cooperative regime has stricter stability requirements than the model in the cooperative version; these are also the conditions under which a tragedy of the commons emerges. The stock in the non-cooperative steady state is lower than the stock in the cooperative steady state, as a result of an excess consumption of the environmental asset in the short run. Given the steady state complementarity between consumption and environment, the flow of consumption is also lower in the non-cooperative steady state. The difference between the efficient and the second-best solutions increases with the rate of time preference of the players; such an effect is nonlinear, being larger when the rate of time preference gets closer to its upper bound.

Consumption of Renewable Environmental Assets 63

Is such a model good for studying issues related to exploitation of par­ticular commons? As Hollick and Cooper [12] note, the issues vary some­what from one environmental asset to the other, so that it is unlikely that one model is useful for all situations. In particular, for the case of joint use of the atmosphere through emissions of CO2 gases, some additional features seem necessary, in particular the explicit introduction of uncertainty about the climatic model. Such uncertainty is likely to have major effects on how the various countries decide their attitude towards global management of emissions, especially given the current situation where a few of the players dispute the very existence of the climate change and scientific evidence is not conclusive. It is of interest to understand whether uncertainty increases or decreases the differences between first-best and noncooperative solutions. The extension of the model to include uncertainty seems also useful for the completion of a recent line of research on the formation of sub-coalitions of countries that decide to adopt a coordinated policy towards management of CO2 emissions. Such studies, see Carraro and Siniscalco [3] and Heal [10], adopt a static and certain framework which does not seem to be the ideal model for considering policies for restraining emissions.

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64 A. Beltratti

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[24J Tornell A. and Velasco A., The tragedy of the commons and economic growth: Why does capital flow from poor to rich countries?, Journal of Political Economy, 100, pp. 1208-1231, 1992

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[26J World Commission on Environment and Development, Our Common Future, Oxford University Press, Oxford, 1987

University of Torino and FEEM, September 1994

Sustainable International Agreements on Greenhouse Warming-

A Game Theory Study

Veijo Kaitala and Matti Pohjola

Abstract

Atmospheric concentrations of the greenhouse gases (e.g., C02) are increasing rapidly due to human economic and industrial activity. The thermal balance of the earth is changed resulting in overall cli­mate warming referred to as the "greenhouse effect". The greenhouse effect is a global issue. Reducing damages caused by climatic changes requires major international efforts. Many countries bear the view that the joint efforts should be undertaken under international agree­ments. We present a dynamic game theory model for an international environmental negotiation problem that may arise in the context of global climate change. Our game theory setting is based on the fact that the countries differ in their vulnerability to the global warming and that two coalitions will possibly be formed. One coalition may include countries that do not suffer from global warming, or where the damages are minor, and in the other coalition we may have coun­tries that suffer from the global warming. The greenhouse problem is modelled as an economic infinite-horizon differential game. The play­ers negotiate an agreement among Pareto efficient programs. Since the costs of one player will be reduced in cooperation while the costs of the other will increase, transfer payments may be used in negoti­ating an agreement. Transfer payment programs are designed such that it is possible at any stage of the agreement to punish violations against cooperation and to discourage the other player from selfishly polluting the atmosphere. The use of memory strategies in designing self-enforcing agreements is discussed. The main conclusion is that the incentives for international cooperative control of global warming will become stronger with an accelerating speed.

1. Introduction

The climate of the earth is determined by the balance between energy received from the sun and energy reradiated back into space. The ra­dioactively important gases, greenhouse gases (e.g., CO2 , nitrous oxide, methane), are transparent to incoming visible radiation but they absorb invisible thermal radiation. As an effect the thermal balance of the earth is

68 v. Kaitala and M. Pohjola

changed resulting in overall climate warming. This phenomenon is referred to as the "greenhouse effect" .

The greenhouse effect is a global issue. Reducing damages caused by climate change may require international efforts by most, if not all, nations. Many countries bear the view that the joint efforts should be undertaken under international agreements. Economists and environmental systems analysts warn, however, that "when the winners and losers have been iden­tified, there will be little interest on the part of the winners to alter their status in order to compensate the losers" (Glantz, 1988, cited by Ayres and Walter, 1991).

Climate is a common property. Emission of the greenhouse gases (e.g., CO2 ) from any country mixes with the climate and spreads effectively all over the world. Thus, global climate warming seems to be another expres­sion of the "tragedy of the commons", a concept used by Hardin (1968) to describe the inefficient use of common-property open-access resources.

In the absence of a World Government the only way out of the global common property dilemma seems to be through intergovernmental agree­ments (Barret, 1990). However, there is a problem that casts shade over global environmental cooperation for a better future. Global environmental agreements are difficult to achieve and to sustain because of the possibil­ities of cheating and free riding. In the case of cheating a country may negotiate, sign, and ratify an agreement and then breach it by leaving her obligations unfulfilled. A free rider country may benefit from environmen­tal cooperation among a set of other countries. Thus, the requirement that intergovernmental environmental agreements should be self-enforcing (Barret, 1990) or cooperative equilibria (Kaitala and Pohjola 1988, Munro, 1990) is becoming a strict necessity.

We study the problem of constructing self-enforcing environmental equilibria in the context of unidirectional externalities illustrated in the environmental economics literature, e.g., by upstream polluting countries and downstream suffering countries (Maler, 1990). The problem of unidi­rectional externalities may also materialize in the context of global climate warming. Countries that do not suffer from the global climate change still emit greenhouse gases contributing to the damages suffered by the rest of the world. Yet, it may be inaccurate to blame these countries for free riding if they do not benefit from the emission abatement of other countries, or for cheating if they do not accept an agreement. The purpose of the paper is to illustrate the use of dynamic game theory as a tool in predicting and analysing environmental policy problems arising at an international level (for other international or control and game theory aspects of environmen­tal problems, see e.g., Clemhout and Wan, 1994, Kaitala, 1986, Kaitala et al., 1992a, 1992b, Munro, 1990, Nordhaus 1991, Pethig, 1992, van der Ploeg and de Zeeuw, 1992, Uzawa, 1991).

Sustainable International Agreements 69

We study in this paper a dynamic two-player negotiation problem with unidirectional externalities. We assume that two coalitions of countries, differing in their vulnerability with respect to the global climate change, ne­gotiate on reducing the greenhouse gas emissions. In particular, we assume that one coalition includes the "vulnerable" countries and that the other coalition includes the "nonvulnerable" countries. The vulnerable countries demand that the nonvulnerable countries also reduce their emissions of the greenhouse gases.

The paper is set out as follows. In the next section we present a dy­namic model for the greenhouse game. We assume in particular that the greenhouse gases are stock pollutant's accumulating in the climate. In Sec­tion 3 we study international environmental negotiations characterized by unidirectional externalities. In Section 4 we study agreeable transfer pay­ment programs in dynamic environmental games. In Section 5 we briefly review an approach for using memory strategies in constructing credible and efficient self-enforcing agreements. An example is analysed in Section 6.

2. Greenhouse Game

We divide the countries in the world into two groups. In group 1 we have the "losers". These countries are vulnerable to the global warming suffering definite costs from it in the form of physical damages. In group 2 we have countries that are economically neutral with respect to the global warming. These countries do not suffer from the greenhouse effect. However, before entering an analysis of unidirectional externalities we first present a model for the dynamics of the greenhouse effect.

Let Q denote the deviation of the CO2 concentration from the 1990 level and let ei, i = 1,2, denote the amount of the CO2 emission of country i that contributes permanently to the global CO2 concentration. Further, let Gi(ei) and Di(Q) denote respectively the emission abatement costs and damage costs of player i. In the noncooperative CO2 abatement problem both players minimize their net costs given by

(1)

subject to

dQ ill = a(ei + e2) - (3Q, Q(O) = Qo = 0, (2)

for all t E [0,00), i = 1,2, and for all Q(t), where a and (3 are environmental parameters, and Pi is the discount rate of country i. We assume that Gi ,

i = 1,2, is a decreasing convex function satisfying Gi(ei) = 0 for some constant positive emission level ei. Vi is assumed to be an increasing con-

70 v. Kaitala and M. Pohjola

vex function in Q satisfying DI(O) = 0, and D2(Q) = 0 for all Q. Further, we assume that PI = P2 = p. Let a pair of feedback strategies (ei, e2) solve the game, and let Vi*(Q) denote the value of the noncooperative game for player i at Q. The noncooperative emissions of player 2 are e2(Q) = e2' for each Q from which it follows that V2*(Q) = 0 for each Q. For player 1 it is natural to assume that Vt (Q) > 0 for each Q.

In the cooperative CO2 abatement problem the players jointly minimize

J = 100 e-pt[CI(el) + C2(e2) + DI(Q)]dt (3)

subject to (2). Assume that a pair of feedback strategies (eJ', e~D provides a solution to the cooperative game. Clearly, cooperation entails positive emission abatement costs for each player. Thus, ViO(Q) > 0 for i = 1,2, where ViO(Q) denotes the value of the cooperative game for player i at Q.

We assume that the noncooperative and cooperative solutions discussed in the subsequent sections exist. An example dealing with a linear-quadratic specification is studied in Section 6.

3. Negotiations with Unidirectional Externalities

In this section we begin constructing self-enforcing environmental equilibria in the context of unidirectional externalities. The problem of unidirectional externalities arises if the countries that are vulnerable to the global climate change form a coalition to demand the nonvulnerable countries to abate their emissions. Recall that the nonvulnerable countries emit greenhouse gases contributing to the damages suffered by the rest of the world. It is then plausible that the nonvulnerable countries also form a coalition for the needs of the negotiation process.

We consider an asymmetric negotiation situation in who DI(Q) > 0 for Q > 0 and D2(Q) = 0 for all Q. In this case the noncooperative emissions of player 2 are e2 (Q) = e2' for each Q from which it follows that V2* (Q) = 0 for each Q. On the other hand, since environmental cooperation entails strictly positive abatement costs for player 2 the cooperative agreement is rational only from the collective point of view and lacks individual rationality. The latter property is true because environmental cooperation incurs costs to player 2, which then prefers noncooperation over cooperation. The general problem of collective rationality in common property problems is, however, that there does not exist a mechanism enabled to make the international agreements binding. Or as Munro (1986) puts it in context of international marine resource exploitation, "there is presumably no external body that will impose sanctions upon those breaching the agreement". The practice has also shown that exploitation of valuable resources is frequently accom­panied by conflicts of various degrees between the harvesters (e.g., Levhari

Sustainable International Agreements 71

and Mirman, 1980). For this reason, Kaitala and Pohjola (1988) concluded that the theory of cooperation under binding resource utilization programs does not provide a satisfactory means to treat current problems of conser­vation and utilization of shared or common resources. Thus, efforts should be made in developing self-enforcement mechanisms to be applied in global environmental problems.

Efficiency problems and cooperative equilibria in environmental dy­namic games have been studied in some detail in the context of interna­tional fisheries management models. Hiimiiliiinen and Kaitala (1982), for example, proposed that efficiency can be increased by changing the property rights of the fishery, e.g., by dividing the fishery into two exclusive zones. Due to the process of complete mixing of the gases in the atmosphere such a "privatisation of the atmosphere" has no effect. Hiimiiliiinen, Haurie and Kaitala (1984, 1985), on the other hand, constructed self-enforcing harvest programs which included threats of returning to full noncooperation by all the players if cheating occurs or if the agreement is not observed. Again, such a construction is not effective in the case of unidirectional externalities since one of the players is better off in noncooperation.

An applicable idea was proposed by Kaitala and Pohjola (1988). They showed that transfer payments are a feasible means for a player to buyout the other when joint resource exploitation prevents reaching the efficiency. We follow here our approach. In particular, we show that constructing a self-enforcing agreement in an environmental problem with unidirectional externalities can be divided into two steps. In the first step, we generate by the aid of transfer payment programs a set of collectively and individually rational solutions which are feasible at any time and any state. In the second step, we characterize a credible threat that is able to support or to enforce the agreement at any time and any state.

3.1 'Iransfer payment problem

In this section we reformulate the cooperative game problem as a side pay­ment problem. This enables one to characterize a class of equivalent co­operative solutions which also are acceptable for the nonsuffering player 2. The rationality behind this idea is that in order to negotiate an agreement with player 2 her costs must be compensated by player 1 that gains from cooperation.

Let Ti, i = 1,2, denote the side payment which either is a carbon tax (Ti > 0) or a carbon subsidy· (Ti < 0). The cooperative CO2 abatement problem is as follows:

(4)

72 v. Kaitala and M. Pohjola

subject to (2). Clearly, if no external environmental funds are used we have 71 = -72. The requirement of self-enforcement through time is given as

100 e-p;(s-t) [Ci(ei) + Di(Q) + 7dds :::; 100 e-Pi(s-t) [Ci (et) + Di(Q*)]ds

(5) for all t E [0,00), i = 1,2, and for all Q(t), where (ei,e;D and Q* de­note the noncooperative CO2 emission policies and the corresponding CO2

concentration trajectory.

as Note that V2*(Q) = 0 for each Q. Condition (5) can now be rewritten

100 e-P2 (s-t) [C2(e2) + 72]ds :::; V2*(Q) = 0

100 e-P1(s-t) [C1(e1) + D1(Q) + 7dds :::; vt(Q)

(6)

(7)

for all t E [0,00), and for all Q(t). In the case that player 2 agrees to cooperate a sufficient condition for (6) to hold is

(8)

for all t. Thus, player 2 is "bought out" by a constant or time varying side payment paid by player 1.

A refinement of the solutions concept requires, however, that a nego­tiation solution should be consistent. By this we mean that if one of the players at some later time point wants to renew the negotiation result and requires renegotiations in order to adjust the agreement to match the cur­rent changed conditions then the result from the renegotiations should be a continuation of the current prevailing agreement. In that case, no party suffers from a temptation to require renegotiations.

Unfortunately, there seems to be only one solution possessing the con­sistency property, namely

(9)

In this case, only the emission abatement costs of player 2 will be paid by player 1 but player 2 enjoys no additional share from the cost reduc­tion. Under this kind of an agreement no player will benefit from renegotia­tions unless the principle applied for sharing the cost reductions is changed. However, this transfer payment program fails to provide a real incentive for player 2 to cooperate with player 1 since under such an agreement she is indifferent between cooperation and noncooperation.

Sustainable International Agreements 73

Any other solution satisfying

- 72> C(e2)' (10)

seems to suffer from inconsistency. We next illustrate this problem in the context of the Nash bargaining solution.

3.2 Inconsistency of Nash cooperative agreements

We begin by considering the bargaining problem in the normal form. As­sume that the players agree to cooperate. Then, at time t = 0 and state Q(O) = Qo, the reduction in the costs of player 1 is Vj*(Qo) - Vt(Qo) and the increase in the costs of player 2 is V2* (Qo). It follows that the gain from cooperation is the cost reduction

(11)

When transfer payments are used the utilities are linearly transferable. Hence, the set of feasible solutions is convex, and the Nash bargaining scheme (Nash, 1950), or any other scheme (see e.g., Roth, 1979), can be used in determining a fair agreement between the countries. The reduction of the costs g(Qo) will be divided equally between the players (Nash, 1950, Munro, 1979), and the total costs for the players will be

Vi(Qo) = l/i*(Qo,ei,e;) - g(Qo)/2, i = 1,2. (12)

Thus, the cooperative agreement is as follows. The vulnerable player 1 first pays all the abatement costs of the nonvulnerable player 2 and then shares the excessive benefit with her. Thus, the incentive for player 1 to negotiate with player 2 is not the reduction of her costs under the assump­tion that player 2 behaves cooperatively but the potential benefits after player 1 has paid all the abatement costs of player 2. Since V2* = 0 the total costs for player 2 will always be negative, which means that player 2 will positively benefit from the agreement and will not suffer any costs at all.

At first sight, the Nash bargaining scheme seems to be consistent since it produces at any state or time the same solution: split the net cost reduc­tion equally between the players. However, unlike in the repeated games, problems will arise due to the evolution of the state in the dynamic games. Renegotiations will change the cooperative agreement although the share rule obtained by the application of the bargaining scheme will remain the same.

Consider again the agreement negotiated at Qo and t = O. Let {;o and (r respectively denote the cooperative and noncooperative trajectories. We have QO(O) = Q*(O) = Qo.

74 v. Kaitala and M. Pohjola

Consider next an arbitrary later time moment t' > O. Clearly, we have QO(t') =F Q*(t'). A continuation of the agreement requires that the value

(13)

is split in half. However, in the case that renegotiations will be carried out at QO(t') the players will split the value

(14)

which clearly differs from (13). Since the values of the cooperative and noncooperative games depend on the initial state and change along the cooperative trajectory, the value to be shared also changes along the tra­jectory. This seems to make it impossible to share the net cost reductions between the players in a consistent way. Every time that renegotiations are carried out, the absolute amounts of the side payments will change.

It seems impossible to develop efficient equilibrium agreements around consistent solutions. Thus, the solutions based on the memory strategies should eliminate both cheating and incentives for renegotiations. We next turn to characterize agreements around which efficient equilibria can be constructed.

4. Agreeable Transfer Payment Programs

We next consider agreeable transfer payment programs. The only require­ment that we pose in the subsequent study of cooperation is that an agree­ment must be agreeable to both parties. By this we mean that no player is obliged to breach the agreement during the realization of the agreement for the reason that noncooperation has become a more attractive alternative than cooperation.

We first review some basic ways to realize the transfer payment pro­grams. There are three different forms of lump sum transfer payments. First, the whole transfer payment can be paid at one time, after which it is expected that player 2 does not start polluting again. It is easy to see that this approach will produce a satisfactory result for the paying player only if the agreement is binding. Otherwise there is no guarantee that player 2 will invest the money into emission abatement and refrain from starting polluting again.

Second, the transfer payment can be paid as a continuous varying cash flow defined as the share from the total benefit flow enjoyed from coopera­tion. It allows the transfer payments to be paid as a share of the net cost reduction flow. It appears, however, that in cooperation the instantaneous net cost flow may exceed the instantaneous net cost flow in noncoopera­tion. This in turn means that if the net benefit flows are shared equally,

Sustainable International Agreements 75

then player 2 should suffer cost flow at the beginning of cooperation (see the example below). Since this may create some additional difficulties in reaching the agreement we shall consider yet another alternative.

We shall study in more detail the third option in which the transfer payment is paid as a continuous cash flow from player 1 to player 2. To illustrate the problems arising in the context of dynamic games we assume here for simplicity that the cash flow is constant in time. Assume also an additional property that the transfer payment program compensates the emission abatement costs flow of player 2 at each time moment.

We next characterize the set of potential transfer payment programs. A dynamic state trajectory is acceptable if it has the property that, given any intermediate value of the trajectory, there is no incentive for any player to apply noncooperative or threat policies (see below) given that the trajectory will be realized (Tolwinski et aI., 1986). Thus, acceptability is related to a particular trajectory, which in our case will be the cooperative carbon dioxide path Qo.

How can we characterize the set of the agreeable transfer payment pro­grams that can be used in agreements? Assume that the constant transfer payment along the agreement is 1"1 = -1"2 = T. The present values of the agreement to the players are

(15)

(16)

Since the transfer payment always covers the emission abatement costs of player 2 we have T > C2(e~} for any e~. Now, if the emission abatement costs of player 2 are decreasing in time then the minimum value for the transfer payment is given at the beginning of the game and is T min = C2(e~(O». On the other hand, if the emission abatement costs of player 2 are increasing along the cooperative trajectory then the minimum value can be defined as the least upper bound of the abatement costs along the cooperative trajectory. Formally, T min = sup C2 (e~), where e~ E e~ and e~ denotes the cooperative emission trajectory of player 2.

An upper bound for the transfer payment can be determined from the condition that the transfer payment must not exceed the benefit obtained by player 1. Formally,

Vi*(Q} - (vt(Q) + ~T} ~ 0 P1

(17)

for each Q E Qo. If (17) is true at Q we say that the transfer payment pro­gram is feasible for player 1 at Q. Assuming that d[Vt(Q}- Vt(Q})/dQ > 0

76 v. Kaitala and M. Pohjola

and d{r(t)/dt > 0, we get the result that all transfer payment programs that are feasible to player 1 at Q = 0 are then feasible to her at any state value along the cooperative trajectory {r. Thus, all such transfer payment programs are agreeable to player 1.

A condition for an agreement at Q = 0 is that the solution must be feasible for both players during the entire game. Existence of such coop­erative programs is not obvious in dynamic games and must be checked in each separate case. It may happen that along the evolution of the state trajectory the set of feasible transfer payments becomes larger or smaller, or may even change to a totally new set (for further discussion, see Kaitala and Pohjola, 1988). In this game under study it is possible to show that the set of feasible transfer payment programs becomes larger along an in­creasing trajectory and smaller along a decreasing trajectory. If the set of agreeable transfer payments at Q = 0 is not empty, we are able to construct efficient equilibrium agreements.

5. Efficient Equilibria with Transfer Payments

This section is devoted to a short discussion of game theory problems in constructing efficient equilibria by utilizing the collective inefficiency of non­cooperative feedback equilibria. The memory strategy approach is an al­ternative for negotiating credible international environmental agreements where binding agreements are not possible. Efficient equilibria have been studied in dynamic games literature only recently (e.g., Tolwinski et al., 1986, for reviews in economics, see Pohjola, 1986 , and in resource manage­ment, see Kaitala, 1986 ).

Assume that the player 1 communicates with player 2 proposing that player 2 should cooperate with player 1 and that one of the agreeable trans­fer payment programs should be realized. The players are assumed to use memory strategies, that is, at any time, the players know the current value of the state and in addition can recall some past information related to control policies, and perhaps to the evolution of the system state. Assum­ing memory to the players is essential since they are not able to tell on the basis of feedback information whether cheating has happened or whether the agreement has been observed. For example, it is natural to assume that information related to the evolution of the CO2 concentration and the realization of the transfer payment program will be memorized by the players. Note that memorizing past information differs crucially from de­layed information. If information, say, on the state trajectory, is obtained with a delay then it is not possible to use feedback strategies but it may still be possible to construct efficient memory strategies applying open-loop strategies in which the control functions are functions of time only.

A threat related to the dynamic agreement may be defined as a com-

Sustainable International Agreements 77

mitment to return to the Nash noncooperative mode of the game, if cooper­ative management of the climate fails for a certain detectable time period. Clearly, such a threat is unnecessarily strong. However, the purpose is to make the threat a technical one by guaranteeing that once a cooperative equilibrium is correctly constructed, the threat will never be realized. An implicit assumption here is that execution of the decisions and monitor­ing are accurate; and if errors in monitoring occur, the players are able in practice to tell monitoring errors from intentional deviations from the agree­ment. In the case that the Nash noncooperative solution is nonunique then the most efficient one capable of sustaining the agreement but producing the least collective harm will be chosen.

Introducing memory into the strategies enables the players to use the Nash feedback strategy pair as a threat to be applied as soon as a deviation from the cooperative agreement is detected. An approach in using mem­ory strategies in nonzero-sum dynamic games is based on Tolwinski et al. (1986), Haurie and Pohjola (1987), and Kaitala and Pohjola (1988). The steps for constructing the efficient memory equilibria are as follows:

(i) The players communicate with each other and agree on open-loop cooperative emission abatement policies and on a transfer payment pro­gram. The cooperative program must be agreeable, i.e., it has the property that no player will find noncooperation attractive at any time, if the coop­erative program is to be realized.

(ii) The players agree on some threat strategies to be applied, if coop­eration fails. In general, the only requirement imposed on the threat is that it must be credible. For this reason it is commonplace that formal studies use noncooperative equilibria as threats.

(iii) The players start by applying the agreed cooperative emission abatement policies.

(iv) If the agreement is observed, then cooperation is continued. (v) If cooperation fails, that is, if cheating is detected or if one player

requests renegotiations, then the players apply their threat strategies. There are two ways to deal with the detection of cheating. In the first

approach one may assume that it takes some time for the cheated person to detect that cheating has occurred. This construction is characterized by the 8-strategies where 8 is used to denote the delay in detection of cheat­ing (Tolwinski et al., 1986). In the other alternative of trigger-strategies the delay is made arbitrarily small (Tolwinski et al., 1986). Mathematical details and further properties can be found elsewhere (see Tolwinski et al., 1986, Haurie and Pohjola, 1987, Kaitala and Pohjola, 1988).

In a 8-strategy game, a player needs to compare two options: cheat a period of length 8 and then play noncooperation for the rest of the game, or cooperate. Thus, in 8-strategy games the punishments, that is, the realization of the threats, are delayed. It follows that 8-strategies, with

78 v. Kaitala and M. Pohjola

6 fixed, can be successful in constructing efficient memory equilibria but can also fail in providing equilibrium property of the agreement. The latter situation can arise, for example, if the time interval 6 and the discount rates Pi are large. Then, since an essential part of the benefit from cheating will be obtained during a short time period soon after the start of cheating, and since cheating will be detected only after a long time period, it can happen that a player will be better off by cheating than by cooperating.

On the other hand, cheating is not possible when trigger strategies can be used. In this case, cheating, or any deviation from the agreement, will be detected without delay. Hence, the only actual choices the players need to make are between cooperation and noncooperation.

6. Example

The atmospheric concentration of CO2 increased from 1981 to 1987 at an average rate of 3.0 Gt of C per year (Tans et al., 1990). The release of CO2 from fossil fuel burning is 5.3 Gt of C per year and the land use modification contributes 0.4 - 2.6 Gt of C per year. For the modelling purposes we assume that the annual emissions (all greenhouse gases are first transformed into equivalents of CO2 on the bases of their contribution to the global warming and then the CO2 equivalents are transformed into equivalents of C) are

em = L ef' = 8 billion tons C i

Two main sinks of anthroponic CO2 are the ocean uptake and CO2

input to the terrestrial ecosystems (e.g., photosynthesis by green plants). Tans et al. (1990) estimate that the global ocean sink is at most 1 Gt C per year. Thus, the share absorbed by the terrestrial ecosystems could be 2-4 Gt C per year. This view is in accordance with Nordhaus (1991) who estimated that a half of the anthropogenic carbon is removed by the natural sinks, oceans mainly, while the other half remains in the atmosphere. Thus, we have

a=0.5.

The parameter f3 is related to the atmospheric life time of the greenhouse gases. Nordhaus (1991) has estimated that

f3 = 0.005.

The current amount of C in the atmosphere at the moment is 700 billion tons C. Model (2) describes, however, the deviation from the current 1990 level, and thus, we take Qo = 0 billion tons C.

Sustainable International Agreements 79

Finally, we assume PI = P2 = 0.05

The estimates of the dynamics predict that if the emissions remain unchanged at the current levels the total increase of the greenhouse gases will eventually stabilize at the level of 800 billion tons C which corresponds to a situation where the greenhouse gas concentrations have doubled. It should be noted that the levels that will be reached in the game solutions below depend crucially on the functional forms of the damage.

To proceed, assume that the emission abatement cost functions of the two players are equal such that

Ci(ei) = ~Ci(ei - ef)2, i = 1,2, (18)

where Ci is a positive constant, and ef is the CO2 emission rate with no reductions carried out. We have

er = er = 4 billion tons C.

and CI = C2. A rough estimate (Nordhaus, 1991) proposes that the costs of removing all the emissions are

1000 billion US $/year

From this estimate of the global total costs we have 0.5cI(ei)2 + 0.5c2(er? = 1000 billion $ year-1 from which we get

Ci = 62.5 billion $ year- I (billion ton C)-2.

Assume further that DI = !dl Q2. Recall that D2 = 0, and V2* (Q) for all Q.

It can be shown by applying the Hamilton-Jacobi conditions that the value of noncooperation for player 1 is

(19)

where the coefficients fi, 'Yi, and JLi are given in the Appendix (see (25)­(27)). Further, the feedback noncooperative emission policy is given by (24) in the Appendix.

The cooperative value is also of the quadratic form and are given as

VO(Q) = Vt(Q) + V2'(Q) = ~fOQ2 + "'tQ + JLo (20)

where the coefficients fO, 'Yo, and JLo is given in the Appendix (see (35)­(37)). The cooperative policies can also be determined in a feedback form (34).

80 v. Kaitala and M. Pohjola

The incentive ~ V (Q) for cooperation at Q is given as the difference between of the total noncooperative and cooperative costs, that is,

~V(Q) = vt(Q) - VO(Q). (21)

It is obvious from the quadratic forms of the value functions that when the greenhouse effect proceeds and the concentrations of the gases increase then the incentive for greenhouse negotiations becomes more intense with an increasing speed.

Assume that the transfer payment program is a constant payment flow from player 1 to player 2 such that the emission abatement costs flow of player 2 is always compensated, as discussed in Section 4.

It can be shown that when the initial value of the state is zero then the value of the cooperative state is strictly increasing for all t (see (41)) with the supremum given by (42)). It follows that the cooperative emission trajectory e2 is strictly decreasing in time. Correspondingly, the instanta­neous emission abatement costs of player 2, C2 (e2), are strictly increasing along the cooperative trajectory, and the minimum value for the transfer payment program is determined as

(22)

where c2 denotes the emission trajectory of player 2 along the cooperative state trajectory QO with the initial value Qo. Thus, in order to negotiate an agreeable transfer payment it is necessary to calculate the limit value which the pollutant accumulation process approaches under cooperation.

The minimum value of the possible transfer payment increases along the cooperative trajectory. What happens to the maximum value? Recall that the agreement must satisfy (17) along the cooperative trajectory. Thus the maximum value of the transfer payment is defined as

T!.JoUX = inf Pl[Vt(Q) - vt(Q)], QEQO

(23)

where QO is the cooperative trajectory defined above. It can be shown that the minimum value of Vt(Q)-Vt(Q) exists and is obtained at Q = o. Thus, TO'uX = pI[Vt(O) - Vt(O)]. This means that when pollutants accumulate in the atmosphere then the value of the state, Q, will increase and higher transfer payments can be used. However, if transfer payments higher than TO'uX are required at Q = 0 then player 1 is better off by noncooperating.

Tables 1 and 2 illustrate the example. Four different models with dif­ferent damage value d1 are analysed and compared. In Table 1 the values of the noncooperative and cooperative games at Qo = 0 as well as the in­centive ~ V(O) for cooperation (21) are given. The incentive ~ V(O) is to be shared by the players in the negotiations.

Sustainable International Agreements 81

Table 1: Game values at Qo = 0 (billion US $)

d1 Vt(O) V2*(0) Vt(O) V2O(0) ~V(O)

0.1 7500 0 5100 1100 1300 0.2 12400 0 6900 2300 3200 0.3 15900 0 7800 3300 4800 0.4 18500 0 8300 4000 6200

Table 2: Agreeable transfer payments (billion US $/year)

d1 QO -0 TOoin = Ci(ef) D1(QO) T.max ei 0

0.1 205 1.0 280 2100 120 0.2 117 0.6 370 1380 270 0.3 82 0.4 400 1010 400 0.4 63 0.3 430 800 510

It was shown above that when constant cash flows are used then the minimum transfer payment is determined by the conditions at which the cooperation stabilizes in the future. Thus, we next study the ultimate values of the game variables at infinity.

As the damage parameter d1 increases then the ultimate cooperative stock level of pollutants, QO (see eq. (42)), decreases as do the ultimate emission levels, ei, i = 1,2 (see Table 2). Correspondingly, the ultimate emission abatement costs, Ci(ef), increase. The damage flow, D1(QO), decreases although the damage parameter d1 increases. This is obviously due to the fact that under higher damage costs the cooperative pollutant stock level is allowed to increase much less than under lower damage costs. Thus, it follows from (22) that the minimum transfer payment is equal to Ci(ef). Although a smaller transfer payment may cover the cooperative emission abatement costs of player 2 at the beginning of the game this will not be the case later on.

Table 2 also gives the maximum constant transfer payments determined from (23). Thus, we see that when the damage costs are low, that is, d1 < 0.3, then the set of agreeable constant transfer payment programs is empty because the minimum transfer payment exceeds the maximum value. When the damage parameter increases then the set of agreeable

82 v. Kaitala and M. Pohjola

programs becomes nonempty. Thus, when the damage costs in our example are sufficiently high it is possible to design a constant transfer payment program such that no player is obliged to breach the agreement in the middle of the game. However, when the damage costs are low then the only possibility to design a transfer payment program is to agree on a time-variable transfer payment program in which the transfer payment flow increases in time (see e.g., Chander and Thlkens, 1992).

7. Conclusions

We have studied in this paper different aspects of international environ­mental agreements dealing in particular with global climate change. Our study proposes that the incentives for greenhouse negotiations and for in­ternational cooperative control of global warming will become more evident as the concentrations of the greenhouse gases continue to increase in the atmosphere. Thus, the research activity is expected to increase also in the field of environmental control and game theory.

We have shown that the design of cooperative programs for interna­tional environmental agreements may be a challenging and complicated task when we are dealing with stock pollutants and when the players are asymmetric with respect to emission volumes, emission abatement costs, or damages caused by the environmental change. It may happen that an agree­ment first satisfies the players but then becomes costly or unsatisfactory for one or several players. On the other hand, the set of possible agreements can become larger making renegotiations attractive at least to some players. As a whole, the dynamic properties of multilateral environmental agreements deserve to be analysed further. Research is needed in developing applicable schemes for environmental agreements and in developing theoretical and practical tools for making the agreements fair and self-enforcing. Finally, the role of environmental risks is missing almost completely in the control and game theory analyses of environmental agreements.

Acknowledgements. An earlier version of this study was presented at the Fifth International Symposium on Dynamic Games and Applications, Grimentz, July 15-17, 1992. The comments of an anonymous referee are greatly appreciated.

Sustainable International Agreements 83

Appendix

The appendix presents solutions to the linear quadratic game studied in Section 6.

Noncooperative greenhouse gas emissions

Applying the Hamilton-Jacobi conditions it can be shown that the nonco­operative CO2 emission policies are given as

* o:€i m 0: * . e· = --Q+ (e. - -T), ~ = 1,2, t Ci t Ci'

(24)

where

2 • 0 2 £* 2 -(p + 2,8 + 2°c;j) ± (p+ 2,8 + 2~)2 - 4~i (-di )

€:=----------------~--~-----------------• 2°2 Ci

(25)

€'!'o:(e~ + e"!'- - .f!.'V'!') * ' • J Cj IJ 'i = E*02 E~02 p+,8+ ...i--c · + T-

• J

(26)

J.L!= -- i +,:"o:(e'!l-+e'f!1---,!) /p ( 1 0:2(,*)2 0: )

• 2 Ci ,t J Cj J (27)

Note that when d2 = 0 then €2 ='2 = J.L2 = o. The differential equation for the noncooperative Q* becomes

dQ* = -A*Q* + B* dt '

(28)

where (29)

and B* (m 0: m 0: ) = 0: el - -,I + e2 - -,2

CI C2 (30)

The solution to the differential equation (28) is given as

Q*(t) = Q*(O)e-A*t + !: (1- e-A*t). (31)

Thus, the state trajectory converges to the value

1. Q*() Q-* B* 1m t= =-A. t->oo *

(32)

84 v. Kaitala and M. Pohjola

Cooperative global greenhouse

The cooperative CO2 abatement problem is posed as follows:

(33)

subject to (2). The Hamilton-Jacobi conditions give

(34)

where

(35)

(36)

° (10:2(,0)2 10:2('0)2 ° (m m)) / J..l = - - - - + "( 0: el + e2 p. 2 Cl 2 C2

(37)

The differential equation for the cooperative QO becomes

(38)

where

(39)

and 0: 0: BO = o:(em - _,,(0 + em - _,,(0).

1 Cl 2 c2 (40)

The solution to the differential equation (38) is given as

QO(t) = e-AOtQO(O) + ~: (1 _ e-AOt ). ( 41)

The state trajectory now approaches the limit value

1. QO() Q-o BO 1m t= =-A. t----+oo 0

(42)

Sustainable International Agreements 85

References

[1] Ayres R.U. and Walter J., The greenhouse effect: Damages, costs and abatement, Environmental and Resource Economics, 1, pp. 237-270, 1991.

[2] Barret S., The problem of global environmental protection, Oxford Review of Economic Policy, 6, pp. 68-79, 1990.

[3] Chander P. and Thlkens H., Theoretical foundations of negotiations and cost sharing in transfrontier pollution problem, European Eco­nomic Review, 36, pp. 488-499, 1992.

[4] Clemhout S., and Wan H. Jr., "The Non-Uniqueness of Markovian Strategy Equilibrium: The Case of Continuous Time Models for Non­Renewable Resources," in: Advances in Dynamic Games and Applica­tions, T. Basar and A. Haurie, eds., Birkhiiuser Boston, Cambridge, MA, pp. 339-355, 1994.

[5] Glantz M.H., "Societal Response to Regional Climate Change: Fore­casting by Agony," in: Workshop, M.H. Glantz, ed., Boulder CO, 1988.

[6] Hiimiiliiinen R.P., Haurie A., and Kaitala V., Bargaining on whales: A differential game with pareto optimal equilibria, Operations Research Letters, 3, pp. 5-11, 1984.

[7] Hiimiiliiinen R.P., Haurie A. , and Kaitala V., Equilibria and threats in a fishery management game, Optimal Control Applications and Meth­ods, 6, pp. 315-333, 1985.

[8] Hardin G., The Tragedy of Commons, Science, 162, pp. 1243-1248, 1968.

[9] Haurie A. and Pohjola M., Efficient equilibria in a game of capitalism, Journal of Economic Dynamics and Control, 11, pp. 65-78, 1987.

[10] Kaitala V., "Game theory models in fisheries management - A survey," in: Dynamic Games and Applications in Economics, Lecture Notes in Economics and Mathematical Systems, T. Basar, ed., Springer-Verlag, Berlin, pp. 252-266, 1986.

[11] Kaitala V., Equilibria in a stochastic resource management game under imperfect information, European Journal of Operations Research, 71, pp. 439-453, 1993.

[12] Kaitala, V. and Pohjola M., Optimal recovery of a shared resource stock: A differential game model with efficient memory equilibria, Nat­ural Resource Modelling, 3, pp. 91-119, 1988.

86 V. Kaitala and M. Pohjola

[13] Kaitala V., Pohjola M., and Tahvonen 0., Transboundary air pollution and soil acidification: A dynamic analysis of an acid rain game between Finland and the USSR, Environmental and Resource Economics, 2, pp. 161-181, 1992a.

[14] Kaitala V., M. Pohjola, and O. Tahvonen, An economic analysis of transboundary air pollution between Finland and the Soviet Union, Scandinavian Journal of Economics, 94, pp. 409-424, 1992b.

[15) Levhari D. and Mirman L.J., The great fish war: An example using a dynamic Cournot-Nash solution, The Bell Journal of Economics, 11, pp. 322-334, 1980.

[16) Mioiler K.-G., International environmental problems, Oxford Review of Economic Policy, 6, pp. 80-108, 1990.

[17) Munro G.R., The optimal management of trans boundary renewable resources, Canadian Journal of Economics, 12, pp. 355-376, 1979.

[18] Munro G.R., The management of shared fishery resources under ex­tended jurisdiction, Marine Resource Economics, 3, pp. 271-296, 1986.

[19] Munro G.R., The optimal management of transboundary fisheries: Game theoretic considerations, Natural Resource Modeling, 4, pp. 403-426,1990.

[20] Nash J., The bargaining problem, Econometrica, 18, pp. 155-162, 1950.

[21] Nordhaus W.D., To slow or not to slow: The economics of the green­house effect, The Economic Journal, 101, pp. 920-937, 1991.

[22] Pethig, R. (ed.), Conflicts and Cooperation in Managing Environmen­tal Resources, Springer-Verlag, Berlin, Heidelberg, 1992.

[23] Pohjola M., "Applications of Dynamic Game Theory to Macroeco­nomics," in: Dynamic Games and Applications in Economics, Lec­ture Notes in Economics and Mathematical Systems, T. Basar, ed., Springer-Verlag, Berlin, pp. 103-133, 1986.

[24] Roth A.E., Axiomatic Models of Bargaining, Springer-Verlag, Berlin, 1979.

[25] Tans P.P., Fung LY., and Takahashi T., Observational constraint on the global atmospheric CO2 budget, Science, 247, pp. 1431-1438, 1990.

Sustainable International Agreements 87

[26] Tolwinski B., Haurie A., and Leitmann G., Cooperative equilibria in differential games, Journal of Mathematical Analysis and Applications, 119, pp. 182-202, 1986.

[27] Uzawa H., "Global Warming Initiatives: The Pacific Rim," in: Global Warming. Economic Policy Responses, R. Dorbusch and J.M. Poterba, eds., Cambridge, Massachusetts, London, MIT Press, pp. 275-324, 1991.

[28] van der Ploeg F. and de Zeeuw A., International aspects of pollu­tion control, Environmental and Resource Economics, 2, pp. 117-139, 1992.

Systems Analysis Laboratory, Helsinki University of Technology, FIN-02150 Espoo, Finland

Helsinki School of Economics, FIN-OOlOO Helsinki, Finland

August 16, 1994

The Environmental Costs of Greenhouse Gas Emissions l

Michael Hoel and Ivar Isaksen

Abstract

An efficient, comprehensive climate policy should balance the cost of reducing emissions of each greenhouse gas (GHG) against the en­vironmental costs of the emissions of the gas. In this paper we show how these environmental costs may be calculated. ThIS is first done for the "traditional" case in whIch one at any time IS only concerned about the state of the climate. We next consIder a more general en­vironmental cost function, for which it is assumed that the rate of climate change is more important for the environment and the econ­omy than the state of the clImate. Finally, a numerical calculatIOn of marginal costs of GHG emIssions for both types of envIronmental cost functions is presented.

1. Introduction

An efficient, comprehensive, climate policy should balance the cost of reduc­ing emissions of each greenhouse gas (GHG) against the environmental costs of the emissions of the gas. In this paper we show how these environmental costs may be calculated. Although CO2 is the most important GHG, there are a number of other GHGs which are important for the development of the climate. If the marginal environmental cost of a particular GHG is, say, twice as large as the corresponding marginal cost for CO2 , one should abate this gas so much that the marginal cost of additional abatement is twice as large as the corresponding marginal cost of abating CO2 emissions. In the paper we show that the relative marginal environmental costs of different GHGs depend on a number of economic variables in addition to physical characteristics of the gases. This has previously been shown by Eckhaus (1992) and Schmalensee (1993). However, unlike these papers, we derive expressions for both absolute and relative marginal costs of GHG emissions for specific environmental cost functions. Moreover, we give a numerical calculation of these marginal costs. Numerical estimates derived from eco­nomic models have previously been given by Michaelis (1992). However, he does not specify an environmental cost function, but instead introduces

I Fmancial support from the Research CouncIl of Norway is gratefully acknowledged.

90 M. Hoel and I. Isaksen

an exogenous limit on a weighted average of the stock of greenhouse gases. Closer in spirit to the present paper are the work by Reilly and Richards (1993). The most important difference between this work and the present paper is that we use a more general environmental cost function (cf. Sec­tion 4), that our numerical model of climate change is given in more detail, and that we use different assumptions about the development of greenhouse gases than Reilly and Richards. The present paper is also closely related to Hoel and Isaksen (1994). The most important differences from this paper is that we now include the case of a more general environmental cost function (cf. Section 4), and that we use a longer time horizon in our numerical calculations.

We start by giving a brief discussion of what characterizes efficient emissions of GHGs (Section 2). In Section 3 we show how marginal envi­ronmental costs of GHG emissions may be calculated for the "traditional" case in which one at any time is only concerned about the state of the climate. This analysis is modified in Section 4, where we consider a more general environmental cost function. In this function, it is assumed that the rate of climate change is more important for the environment and the economy than the state of the climate. Finally, a numerical calculation of marginal costs of GHG emissions for both types of environmental cost functions is presented in Section 5.

2. Efficient Emissions of Greenhouse Gases

Let x denote the vector of GHG emissions at a specific time point (later denoted t = 0). Total income in the society is denoted by R(x), and the level of GHG emissions which maximizes this income is denoted by xo. The vector xo thus represents the emission level of GHG emissions in an economy which ignores all environmental effects, but which otherwise is efficient. Reducing GHG emissions below the levels given by xo is only possible at a cost, i.e., R declines as any x, is reduced below the level X,o· In other words, it is assumed that R, = aR/ax, < 0 in the relevant area of x. (Notice that the area of x for which all R, < 0 is not as simple as x < xo. We may e.g., have x? - x, "large" for all z > 1, while x~ - Xl is "small". In this case Rl > 0 if all Rh > 0 for x < xo.)

An efficient policy does not maximize R(x) without any consideration of the environmental impact. Efficiency instead requires that R(x) - C(x) is maximized, where C(x) is a (money) measure of the environmental costs caused by the emissions x. For C. = ac / ax, > 0 this leads to the following first order conditions to determine efficient emission levels:

R.(x) = C.(x). (1)

The Enmronmental Costs of Greenhouse Gas Emzsszons 91

If we knew the exact specifications of the functions R and C, we could thus in principle calculate efficient emission levels of all GHG gases. Even if this were the case, there would be several obstacles to designing a fully efficient climate policy. In the first place, an efficient policy would require an international agreement between all countries. Problems related to such agreements have been extensively discussed in the literature, see e.g., Bar­rett (1992), Carraro and Siniscalco (1993), and Bauer (1993). In addition to the problems discussed in this literature, there is a problem of monitor­ing emissions of several of the GHGs. To be able to enforce an agreement specifying emission levels from different countries, one must be able to mon­itor emissions from individual countries. This is probably not particularly difficult for CO2 (through information on fossil fuel consumption), but con­siderably more difficult for e.g., methane.

A second obstacle to designing a fully efficient climate policy is that for several GHGs, policy instruments to limit their use are rather limited. For CO2 emissions, a carbon tax is a usually considered an effective instrument for limiting emissions (see e.g Pearce, 1991). Several other GHGs, such as e.g., N20, CH4, and to some extent CFCs and HCFCs, are more difficult to regulate in a cost-effective manner. If one must rely on various types of direct regulation, which in most cases are not particularly cost-effective, the costs of reducing emissions are higher than what is implied by the function R in eq. (1). This should be taken into consideration when deciding how large emissions should be of the different GHGs.

Finally, the environmental costs, as measured by the function C, will for several GHGs include other environmental effects in addition to the effects on the climate. The most obvious example is the CFCs, which in addition to affecting the climate also affect the ozone layer. Similarly, taxes or other policy instruments directed towards the use of fossil fuels should take into account other effects of the fuel use in addition to CO2 emissions, such as e.g., emissions of NOxS02 and VOC (see e.g., Newbery (1992) and Hoel (1993».

The complications above are not discussed further in the present paper. Instead, we concentrate on estimating the marginal costs of the climate effects of emissions of the different GHGs. In other words, we calculate the part of the CIS in equation (1) which represents costs due to climate change. In addition to calculating the absolute levels of these marginal environmental costs we also calculate the relative marginal cost of each gas compared to CO2, i.e., C./C002.

3. The Marginal Cost of Climate Change

To find what the marginal environmental costs of GHG emissions are, we

92 M. Hoel and 1. Isaksen

must first describe the relationship between GHG emissions and the cli­mate development. The climate at time t is summarized by the increase in average global temperature above its preindustrial level, denoted by T(t). We assume that the damage of a climate change T(t) at time t has a (mon­etary) value equal to D(T(t), t), with DT = aD/aT> o. The marginal environmental cost of emitting GHG i at time zero is thus given by

roo -rt (()) aT(t) C, = 10 e DT T t ,t ax,(O)dt. (2)

The development of the climate depends on the development of the at­mospheric concentration of the GHGs. The atmospheric concentration of greenhouse gas i is denoted by S,(t)4. We measure atmospheric concen­tration in the same units as emissions per year. Transformation to more conventional measures such as "parts per million" is straightforward5 . The global average temperature is a lagged, increasing function of radiative forc­ing, which in turn depends on the atmospheric concentration of the GHGs. More precisely, we assume that this relationship is given by

t(t) = alA L h,(S.(t)) - T(t)] (3)

where h,(S,) is the increase in radiative forcing from GHG no. i since its preindustrial level (measured in W/m2 ). The functions h, vary between GHGs, see Houghton et al. (1990). In particular, the h-function for C02 is of type ALn(S)+B, while the h-functions for CH4 and N20 are of the type A(.jS) + B (where A and B are constant parameters which differ between the gases) 6. All CFC-gases have h-functions of the type AS, where the constant parameters A depend on which CFC gas we have.

The parameter A is the factor of proportionality between radiative forc­ing and the long-run temperature response. This factor of proportionality is uncertain. In our numerical analysis we have set A = 0.75, i.e., we have assumed that an increase of radiative forcing of 1W/m2 gives a long-run temperature increase equal to 0.75 degrees (celsius). This relation is based on the "best estimate" of climate sensitivity to radiative forcing as given by the 1992-report of the IPCC (Houghton et al., 1992).

4An exception is SC02(t), which stands for atmospheric concentration of C02 above its pre-industrial level

5The formula is as follows: 1 glgaton (= 1015 gram) emission of gas ~ is equivalent to (6.84/M.) ppm (parts per million) atmosphenc concentration, where M. IS the molecular weight of gas •. For instance, for C02 we have Mo = 44, so that 1 glgaton of C02 emission gives an increase in the atmosphenc concentration of C02 equal to 684/44 = 0 156 ppm

6For CH4 and N20 the relationship between atmosphenc concentrations and radiative forcmg are not additive as assumed m (3). However, thiS additive form IS a reasonable ap­prOXimation also for these gases, see Houghton et al (1992) for details In the numencal calculations in Section 5 we use the expression from Houghton et al. (1992).

The Enmronmental Costs of Greenhouse Gas Emlsszons 93

The parameter (1 represents the response time for the climate system. In the numerical analysis in Section 5, it is assumed that 1/(1 = 40 (years), Le., (1 = 0.025.

The differential equation (3) may be solved to give

T(t) = T(O)e-ut + (1)..ft e-U(t-T) L h.(S.(r»dr. (4) o •

Thrning next to the development of GHGs, we assume that they develop according to the following differential equations

(5)

The parameter 8. represents "natural depreciation" of GHG z. The type of "radioactive decay" assumed in (5) (Le., 8. constant) is a reasonable ap­proximation for most GHGs, at least as long as indirect effects are ignored. For C02, however, the process of removal of CO2 from the atmosphere is more complex than suggested by (5). Nevertheless, we shall use equation (5) for all greenhouse gases in this and the next section. In Section 5, the numerical analysis is based on a more complex and more correct description of the change of atmospheric concentration of CO2 and some of the other GHGs. In particular, it is in this section assumed that 8co2 decreases with time as suggested by the IPCC, see Houghton et al. (1992). Equations (5) may be solved to give

S.(r) = S.(0)e-6•T + 1T e-6.(T-f/x.('T])d'T]. (6)

From (5) and (6) it is clear that

8T(t) = (1).. t e-u(t-T)h~(S.(r»e-6.T dr 8x.(0) Jo

(7)

where h~(S.) = 8h./8S •. Inserting (7) into (2) we find

C. = (1)..100 e-(r+U)tDT(T(t),t){ 1t e(U-6.)Th~(S.(r»dr }dt. (8)

In our numerical calculations we specify

D(T(t), t) = A . T(t)a . eDtt (9)

so that (8) in this case may be written as

94 M. Hoel and I. Isaksen

In the next section we use the expressions above to calculate the marginal costs of climate change for 11 of the most important greenhouse gases. In addition, we calculate all relative GHG weights C./CC02'

4. Temperature Change Versus Temperature Level

In the previous section it was assumed that all that mattered for the envi­ronment at a specific time was the level of the temperature at that time, representing the state of the climate at that time. However, one could ar­gue that it is not only the state of the climate at any time which matters for the environment, but also how rapidly the climate is changing. The economy and the ecology can be expected to be able to adapt to a changed climate, as long as the rate of change is not too fast. For instance, under a sufficiently slow climate change natural systems may be able to acclima­tize or migrate to more favorable areas, while a rapid climate change may lead to extinction of vulnerable species. This point has been mentioned by e.g., Crosson (1989) and Peck and Teisberg (1992). Tahvonen (1993) has given an analysis of the general case in which the change in a stock pol­lutant as well as its level affects the environmental cost, and applied this general analysis to an example of global warming, cf. also Tahvonen et al. (1993). However, simply introducing into the damage function DO along with T and t does not in our opinion give a satisfactory description of the way a climate change affects the environment. It is not the current rate of temperature change at any particular moment (T) which is important, but rather the speed at which the climate has been changing over several decades. We therefore propose a somewhat different way of modeling the importance of the speed of climate change.

Let the damage function be D{K{t), t) instead of D{T{t), t), where the variable K{t)is defined by

(11)

and (3 is a non-negative parameter. It follows from (11) that

K{t) = J~oo T{r)dr = T{t) for (3 = o. (12)

In other words, the special case of (3 = 0 corresponds to the "usual" type of environmental cost function in which it is only the temperature level which matters. From (11) we can also derive (see Appendix 1)

lim K{t) = T{t) {3-+oo

(13)

The Enmronmental Costs of Greenhouse Gas Emzsswns 95

For the limiting case of 13 = 00 it is therefore only the current rate of tem­perature change which affects the environment. This is clearly an extreme case. More realistically, we would expect that the rate of change over sev­eral decades is what matters. If e.g., 13 = 0.1, the whole history of climate change affects the environment. However, in this case the current rate of change is e20{3 = e2 = 7.39 times as important as the rate of change 20 years earlier. Expressed alternatively, when 13 = 0.1, a 1 degree total tem­perature increase caused by a yearly increase of 0.05 degrees over 20 years is just as bad for the environment as a total temperature increase of 2.18 degrees caused by a yearly increase of 0.0435 degrees over 50 years.

In the numerical calculations in the next section, we shall consider the two cases of 13 = 0 and 13 = O.l.

Equation (11) may be rewritten as

K(t) = Koe-{3t + (1 + 13) fot e-{3(t-T)T(r)dr (14)

where

Ko = (1 + 13) [°00 e{3TT(r)dr. (15)

Instead of (2), we now get

[00 -rt oK(t) C, = Jo e DK(K(t), t) ox, (0) dt. (16)

With a specification of the damage function corresponding to (9) we thus get

C, = aA [00 e-(r-n)t[T(t)]a-10K(t) dt. (17) Jo ox, (0)

The term oK/ox. may be derived from (3), (4), (6) and (14). After some tedious calculations we obtain

oK(t) _ -(3t ox,(O) - lTAe I,(t) (18)

where

I.(t) = fot [e({3-6,)T h~ (8. (r)) - lTe({3-a)T {foT e(a-6,)7] h~ (8. (77), d77 } ]dr.

(19) From (18) it is easily verified that if 13 = 0, oK/ox. is equal to the expression for oT/ox. given by (7), which is a direct consequence of (12). Inserting (18) into (17) finally gives us

C. = lTAaA fooo e-(r+{3+a-n)t[K(t)]a-l I. (t)dt (20)

which coincides with (10) for the special case of 13 = o.

96 M. Hoel and I Isaksen

5. Numerical Analysis

In this paper no attempt is made to calculate optimal emission levels of different GHGs, as described by eq. (1). The analysis is limited to a calcu­lation of the marginal costs C. for an exogenously specified path of emissions of all GHGs. These time paths are as follows: for CO2 , the growth rate is assumed to be 1 percent below the growth rate of world gross product for the first 100 years, and 1.25 percent below the growth rate of world gross product for the remaining 300 years.

The growth rate of the world gross product is equal to the sum of the rate of population growth and per capita growth of the world gross product. The population growth in the 80's was 1.8%. We shall assume that popu­lation grows by 1.25% for the first 50 years, and that the growth rate then declines to 0.25% for the following 50 years. After the year 2090 population is assumed to be constant. These growth rates give a population of about 10 trillion in 2050, and about 11 trillion by 2100. These assumptions are roughly in line with other projections, see e.g., Houghton et al. (1990).

The average per capita growth of the world gross product was 1.2% in the 80's. We assume that the per capita growth will be 1.5% for the next 50 years, after which it declines to 1 % for the remaining 350 years our analysis covers.

In other words, we assume that the growth rate of world gross product is 2.75 percent for the first 50 years, 1.25 percent for the next 50 years, and 1 percent for the remaining 300 years of our analysis. CO2 emissions are thus assumed to grow by 1.75 percent the first 50 years, by 0.25 percent the next 50 years, and to decline by 0.25 percent a year after 2090. These assumptions for the development of CO2 emissions imply a slightly lower growth for the period 1990-2100 (0.9 percent on average) than one often sees in scenarios with no climate policy. Manne and Richels assume that the average yearly CO2 growth in such a scenario is 1.4 percent (1990-2100), while Peck and Teisberg (1992) and Cline (1992) assume average yearly CO2 growth rates of about 1.7 percent and 1.1 percent, respectively, for the period 1990-2100. Our somewhat conservative assumption about carbon emissions to some extent reflects a relatively modest assumption about the growth of world gross product, but also a somewhat more rapid decline in the ratio between CO2 emissions and gross product than several other studies 7 • Given our assumption about the development of world gross product, our assumption about CO2 emissions is consistent with moderate restrictions on CO2 emissions.

The yearly growth rates of emissions of CH4 and N2 0 are assumed to

7See e.g., Chne (1992, 1994) for an overview of assumptions made m several studies regardmg GDP growth and C02 emissions See also Manne and Rlchels (1994) for a discussion of assumptIOns about important parameters and vanables.

The Enmronmental Costs of Greenhouse Gas Emzsszons 97

be 0.8% and 0.3%, respectively, until 2020. This corresponds to the most recent observations of global growth rates of the compounds, see WMO (1992). CH4 and N20 are only partly affected by man made emissions, and the sources are poorly known, which makes control of emissions uncertain. We therefore assume continued growth also after year 2020, although at a slow rate (0.1% per year). The emissions of all other GHGs (chlorine compounds) are assumed to decline towards 2020, after which emissions are zero. This should be a reasonable assumption, since the "Montreal Protocol" requires a phaseout of ozone depleting substances over the next 20-30 years.

As mentioned earlier, we have assumed>' = 0.75 and f3 = 0.025. The parameter a in the damage function is assumed to have the value a = 1.5. The interpretation of this value for a is as follows: assume that a temperature increase of 3 degrees gives damages/costs corresponding to 2% of world GOP. The parameter "a" represents the curvature of the cost function for climate change. If the cost function for climate change was linear (i.e., a = 1),6 degrees temperature increase would give a cost which is twice as large as the cost of 3 degrees temperature increase, i.e 4% of world GOP. However, it can be argued that the cost of 6 degrees temperature increase is likely to be considerably higher than twice the cost of 3 degrees temperature increase. The assumption a = 1.5 means that we assume that a temperature increase of 6 degrees is assumed to cost 215 = 2.8 as much as the cost of 3 degrees temperature increase (Le., 5.7% of GOP for the example above). The assumption a = 1.5 is roughly in line with the estimates of Cline (1992).

The value of the parameter A depends on what f3 is. For the case of f3 = 0, i.e., only the climate level matters, we have set A = 115 billion 1990-dollars. Together with a = 1.5 this value means that a temperature increase of 3 degrees is assumed to cost approximately 2 percent of world GOP. This assumption is broadly in line with what is suggested by e.g., Cline (1992).

In our calculations, a temperature increase of 3 degrees (above prein­dustrial level) is reached in year 2086. With f3 = 0.1, K(2086) = 0.38. In order to have the same environmental damage in year 2086 in both cases (i.e., f3 = 0 and f3 = 0.1) we set A = 2587 for the case of f3 = 0.1.

Two important parameters in the determination of the marginal costs C. are 0: and r. Although we have implicitly assumed that these two pa­rameters are constant in order to simplify the expressions of the previous sections, we have let 0: and r depend on time in our numerical analysis. The parameter 0: expresses how the monetary damage of climate change develops over time for a constant climate. We follow e.g., Cline (1992) and Peck and Teisberg (1992) and assume that this damage is proportional to world gross product. The assumptions above about the development of the

98 M. Hoel and I. Isaksen

world gross product thus imply that we in our numerical analysis set 0:

equal to 2.75 percent for the first 50 years, 1.25 percent for the next 50 years, and 1 percent for the remaining 300 years of our analysis.

The appropriate interest rate r in a long-run analysis of the present type depends on how different generations are weighted together. A usual intertemporal objective function used in long-run dynamic analyses is

w = 100 e-pt N(t)u(c(t))dt (21)

where N(t) is population, u is a utility function, c(t) is per capita consump­tion, and p is a utility discount rate. With this objective function, it is well known that the appropriate discount rate r is given by

r = p+wg (22)

where w = u" c/u' is the elasticity of the marginal utility of per capita consumption and 9 is the growth rate of per capita consumption. We assume that this growth rate is equal to the growth rate of per capita world product, discussed above.

In a long-run analysis of the current type, the term w in (22) repre­sents society's attitude towards the distribution of consumption between generations. The more weight society gives to equity, the higher the value of w. The values used in economic analyses are often in the range 1-3. The logarithmic utility function, used by e.g., Peck and Teisberg (1992) and Nordhaus (1992), has w = 1. Scott (1989) has estimated w to be 1.5 for the United Kingdom, which is also the value used by Cline (1992), and which we use in our analysis.

Consider next the value of the term p in (22). This term represents discounting purely because of time. From an ethical point of view, it is difficult to defend a large value of p. If p = 0.03, for example (as used by e.g., Peck and Teisberg, 1992), and we assume that the time between two generations is 30 years, then each generation is given only 41 % of the weight of the previous generation. Even with p = 0.01, each generation is given only 74% of the weight of the previous generation. Equal weight to each generation implies that p = O. In the present analysis we use p = 0.01.

Table 1 summarizes our assumptions about the parameters affecting the marginal costs of climate change.

The results of the numerical analysis are given in Table 2. Consider first the marginal environmental cost of CO2 emissions. This cost is 23 or 16 dollars per ton of CO2 , depending on whether (3 = 0 or (3 = 0.1. This corresponds to 85 and 60 dollars per ton of carbon, respectively, which is equivalent to approximately 11 and 8 dollars per barrel, respectively. It is thus clear that the marginal environment costs of CO2 emissions are not strongly affected by the parameter (3, i.e., to what extent it is the change

The Enmronmental Costs of Greenhouse Gas Emzsszons 99

of the climate as opposed to the state of the climate which is important for the environment.

Table 1: Parameter values

1991-2040 2041-2090 2091 - 2390

A 0.75 0.75 0.75

a 0.025 0.025 0.025

a 1.5 1.5 1.5

A for {3 = 0 115 115 115 (bill. 1990$)

A for {3 = 0.1 2587 2587 2587 (bill. 1990$)

p 1% 1% 1%

w 1.5 1.5 1.5

n 1.25% 0.25% 0

g 1.5% 1.0% 1.0%

a=n+g 2.75% 1.25% 1.0%

r = p+wg 3.25% 2.5% 2.5%

The costs of some of the other GHGs are much more sensitive to {3 than CO2. This is particularly true for the GHGs with the shortest lifetimes, such as CH4 (methane), HCFC-22, HCH-134 and CH3 CCI3 . For these gases the environmental costs of emissions are much smaller for {3 = 0.1 than for {3 = O. To interpret the large sensitivity of {3 for the GHGs with short lifetimes, consider Figure 1. T ref stands for the temperature development in the reference scenario (drawn linearly to simplify the figure). T.o.C02 is the temperature development after an additional ton of CO2 emissions in year 1991, and T.o.CH4 is the temperature development after an additional ton of CH4 emissions in year 1991. Since CH4 has a shorter lifetime than C02, the difference between T.o.CH4 and Tref peaks earlier than the differ­ence between T.o.C02 and Tref. When {3 = 0, CC02 and CCH4 are given by the sums of the differences TC02 -Tref and TCH4 -T ref , respectively, mul­tiplied by the discounted marginal cost of the temperature for each year (cf. eq. (2)). CC02 and CCH4 are thus given by sums of variables which are positive for all years. To see what happens when {3 > 0, consider the limiting case of {3 = 00. In this case it is only the temperature change in

100 M. Hoel and I. Isaksen

Table 2: Absolute (1990$ per ton, i.e. 1990$ per 106 gram) and relative marginal costs of GHG emissions

Greenhouse LIfetIme C, C,/Cco2 GWP from

gas (= 1/8,) /3=0 /3 = 0 1 /3=0 /3 = 0 1 IPCC

CO2 120-300* 23 16 1 1 1

CH4 11 330 61 14 4 11

N20 150 4829 3485 207 211 270

CFC-11 60 63438 40655 2725 2465 3400

CFC-12 120 134561 93326 5780 5659 7100

CFC-13 100 59141 41178 2540 2497 4500

HCFC-22 16 41002 12290 1761 745 4200

HCF-134 15,6 25992 7604 1116 461 1200

CCl4 47 20196 11741 867 712 1300

CH3 CCl3 7 1974 138 85 8 100

CF4 1000 117676 92938 5054 5635 na

C2F6 1000 225116 177786 9669 10780 na

* The lIfetime of C02 18 assumed to Illcrease with time, from 120 years to more than 300

years ThiS IS III accordance with the suggestIOns by the IPCC, see Houghton et al (1990,

1992)

any year (from the previous year) which matters, cf. eq (13). CC02 and CCH4 are in this case given by the sums of the differences of the slopes of TC02 and TCH4 on the one hand and T ref on the other hand, multiplied by the discounted marginal cost of the temperature increase for each year (cf. eq. (16)). From Figure 1 it is clear that this sum for CO2 consists of positive terms until 2058, and of negative terms for the remaining years. For CH4, this sum consists of positive terms only until 2010, and of neg­ative terms for the remaining years. If we hypothetically assumed that the discounted marginal cost of temperature changes was the same for all years, the term CC02 would nevertheless be positive, since T6.C02_ T ref

in the end of our 400 year period is still 10 percent of its peak value. For T6.CH4_ Tref, on the other hand, the value falls to only 1 percent of its peak value already in 2202, and declines to zero before the end of our 400 year period. If the discounted marginal cost of climate change was constant, CCH4 would thus be zero (since the sum of the unweighted sum over all t of [T6.CH4(t)_ T6.CH4(t - 1)] _[Tref(t)_ Tref(t)] is equal to T6.CH4(2390)-Tref(2390) = 0). With f3 = 0.1, as in our example, it is not only the change of the climate which matters, so that CCH4 would be positive if the dis-

The Emnronmental Costs of Greenhouse Gas Em~sstOns 101

temperature

T ref

1991 2010 2058 year

Figure 1:

counted marginal cost of K was constant. It turns out that with f3 = 0.1, the discounted marginal cost of K is first rising (until 2025), then declin­ing. Moreover, 8K(t)/8xCH4(0) is positive for t < 2000, after which it turns negative. Since the weights for some of the negative terms (i. the discounted marginal cost of K) are higher than the weights for the positive terms, it is not a priori obvious that the weighted sum is positive. In our example it turns out that CCH4 (and all other C.) are positive. However, had we assumed p = 0 instead of p = 0.01, the marginal cost of emissions would have been negative for CH4, CFC-U, HCFC-22, CH3CCI3, CCl4 and HFC-134.

Table 2 also includes the "Global Warming Potential" (GWP) given by the IPCC (for a 100 year horizon) for each of the GHGs (except CF4 and C2F6)' The GWP of GHG z is defined as the integrated contribution to radiative forcmg over a hundred year period of an increase of current emissions of this gas relative to the correspomling contribution to radiative forcing of an increase of current emissions of CO2. Formally, we have

f100 e-li,'rh'(S (r))dr GWP. = Jo • •

f;oO e-lico2Th~02(SC02(r))dr (23)

For all gases except CH4, these GWPs are somewhat higher than the weights we calculate.

102 M. Hoel and 1. Isaksen

6. Conclusions

The relative weights of greenhouse gases depend on a number of economic assumptions, in addition to the physical properties of the different green­house gases. In particular, we have shown that the specification of the environmental damage function is important, especially for the greenhouse gases with a short lifetime. In our previous study (Hoel and Isaksen, 1994) we showed that it was particularly the weights of the short-lived greenhouse gases which depended on the curvature of the damage function as well as on the pure rate of discount p. Taken together, these results suggest that with our present state of knowledge about important economic variables, it is difficult to know how much one ought to abate the short-lived greenhouse gases.

Appendix A: Proof of Eq. (13)

Defining h(r) = T(r) - T(t) (A.l)

we may rewrite (11) as

K(t) = (1 + (J)T(t) [00 e-{3(t-T)dr + (1 + (J) [too e-{3(t-T)h(r)dr =

= 1+(JT(t)+(I+(J)jt e-{3(t-T)h(r)dr. (A.2) (J -00

Integrating by parts gives

K(t) = 1 + (JT(t) + 1 + (J h(t) _ 1 + (J jt e-{3(t-T)h(r)dr. (A.3) (J (J (J-oo

We know from (A.l) that h(t) = O. Assuming that the climate development is smooth, in the sense that the curve for T(t) has no kinks, will be finite for all t, cf. (A.l). It then follows that the last integral in (A.3) approaches zero as (J ~ 00. It is thus clear that

lim K(t) = T(t) {3--+00 (A.4)

which proves eq. (13).

Appendix B: The Numerical Calculation

In the numerical analysis, we use a climate model to make an independent calculation of the temperature development in the reference scenario. We

The Environmental Costs of Greenhouse Gas Em~sswns 103

also assume that the temperature in the initial year (1990) is 0.5 degrees Celsius above its preindustrial level. It is assumed that this temperature increase has taken place gradually over 100 years, i.e., a temperature in­crease equal to 0.005 degrees per year. From these assumptions we may calculate Ko from (15):

Ko = 0.005. 1 ; {3 . (I _ e-100,8). (B.l)

To calculate the development of K{t) we differentiate (14) and find that

K{t) = -(3K{t) + (I - (3)T(t). (B.2)

The corresponding expression in discrete time, which is used in our calcu­lations, is

K{t + 1) - K{t) = -(3K{t + 1) + {I + (3)[T(t + 1) - T{t)] (B.3)

or 1

K{t + 1) = 1 + (3K{t) + [T{t + 1) - T{t)]. (B.4)

Notice that this discrete time specification gives us

lim K{t + 1) = T{t + 1) - T{t) ,8-+00

(B.5)

which corresponds to eq. (13). The calculations of all C.s are based on (17). From our climate model we find how a specific increase in the emis­sions of each GHG in year 1991 affects the temperature in all years for the period 1991-2390. We thus obtain numbers for 8T{t)/8x.{0) for all t and i. Differentiating (B.4) gives us

8K{t + 1) 1 8K{t) [8T{t + 1) 8T{t) ] Qx.{O) = 1 + (3 . 8x.(0) + 8x.(0) - 8x.(0)

(B.6)

Inserting our numbers for all 8T{t)/8x.{0) into (B.6) and using 8K(0)/8x.{0) = 8T(0)/ 8x.(0) = 0 we obtain numbers for 8K{t)/8x.{0) for all t and i. Once all 8K(t)/8x.(0) are found, we calculate all C.s from (17).

References

[1] Barrett, S., "Self-Enforcing International Environmental Agreements", CSERGE Working paper GEC 92-34, University of East Anglia and University College London, (1992).

104 M. Hoel and I. Isaksen

[2] Bauer, A., "International Cooperation over Environmental Goods", mimeo, Volkswirtschaftliches Institut, University of Munich, {1993}.

[3] Carraro, C. and Siniscalco, D., "Strategies for the International Protec­tion of the Environment", Journal of Pubhc Economzcs 52, 309-328. North-Holland, {1993}.

[4] Cline, W., The Economics of Global Warming, Washington: Institute for International Economics, {1992}.

[5] Cline, W., "Socially Efficient Abatement of Carbon Emissions", in: Chmate Change and the Agenda for Research, T. Hanisch, ed., West View Press, Boulder, Colorado, {1994}.

[6] Crosson, P., "Climate change: problems of limits and policy re­sponses", in: Greenhouse Warmmg: Abatement and Adaptwn, Re­sources for the Future, Roesenberg et aI., eds., Washington, D.C., {1989}.

[7] Eckhaus, R., Comparing the effects of greenhouse gas emissions on global warming", The Energy Journal 13, 25-34, {1992}.

[8] Hoel, M., "Harmonization of carbon taxes in international climate agreements", Enmronmental and Resource Economzcs 3, 221-231, {1993}.

[9] Hoel, M. and Isaksen, I., "Efficient abatement of different greenhouse gases", in: Chmate Change and the Agenda for Research, T. Hanisch, ed., West View Press, Boulder, Colorado, {1994}.

[10] Houghton, J.T., Jenkins, G.J., and Ephraums, J.J., eds., Clzmate Change, The IPCC Sczentzjic Assessment, Cambridge University Press, (1990).

[11] Houghton, J.T., Callander, B.A., and Varney, S.K., eds., Climate Change 1992, The Supplementary Report to the IPCC Scientific As­sessment, Cambridge University Press, {1992}.

[12] Manne, A.S. and Richels, R.G., The costs of stabilizing global CO2

emissions: A probabilistic analysis based on expert judgements, The Energy Journal 15, 31-56, {1994}.

[13] Michaelis, P., Global warming: efficient policies in the case of multiple pollutants", Environmental and Resource Economzcs 2,61-78, {1992}.

[14] Newbery, D., Should carbon taxes be additional to other transport fuel taxes?, The Energy Journal 13, 47--60, {1992}.

The Envzronmental Costs of Greenhouse Gas Emzsszons 105

[15] Nordhaus, W., "The 'DICE' Model: Background and Structure of a Dynamic Integrated Climate Model of the Economics of Global Warm­ing", New Haven: Yale University, Mimeo, (1992).

[16] Pearce, D., "The role of carbon taxes in adjusting to global warming", The Economic Journal 101, 938-948, (1991).

[17] Peck, S.C. and Teisberg, T.J., "CETA: A Model for Carbon Emissions Trajectory Assessment", The Energy Journal 13, 55-77, (1992).

[18] Reilly, J .M. and Richards, K.R., Climate change damage and the trace gas index issue", Envzronmental and Resource Economzcs 3, 41-62, (1993).

[19] Schmalensee, R., Comparing greenhouse gases for policy purposes", The Energy Journal 14, 245-256, (1993).

[20] Scott, M.F., A New View of Economic Growth, Oxford: Clarendon Press, (1989).

[21] Tahvonen, 0., "Optimal Emission Abatement when Damage Depends on the Rate of Pollution Accumulation", Proceeding of the Environ­mental Economics Conference at Ulvon, June 10-13, (1993).

[22] Tahvonen, 0., von Storch, H., and von Storch, J., "Atmospheric CO2

Accumulation and Problems in Dynamically Efficient Emission Abate­ment", in G. Boero and Z.A. Silberston (eds.): Environmental Eco­nomics, Macmillan, London, (1994).

[23] WMO, Scientific Assessment of Ozone Depletion: 1991, WMO Global Ozone Research and Monitoring Project, Report No. 25, 1992.

Department of Economics, University of Oslo, and SNF (Center for Research in Economics and Business Admimstration), Oslo, Norway

CICERO (Center for Climate and Energy Research, Oslo, Norway) and Department of Geophysics, University of Oslo

Part 2

Environmental Taxes

and Related Issues

Taxation and Environmental Innovation 1

Carlo Carraro and Giorgio Topa

Abstract

This paper analyses the effects of environmental taxation on firms' innovation actIvity. A regulator is assumed to introduce an environmental tax FIrms may react both by changing output and by adopting a new, environment-friendly technology CondItIons under which the latter option is firms' optimal chOIce are provIded. The pa­per shows that firms' mnovation decIsIons are not SImultaneous even when firms are identIcal (there eXIsts diffusion). Moreover, firms have an incentive to delay the time of innovatIOn, because the new technology can only be achieved through costly R&D. Hence, there exists room for incentives that move firms to the socially-optImal tIm­ing of innovatIOn. These incentives have to account for the presence of asymmetric information (the regulator is assumed not to observe firms' innovation costs). The paper shows that there exists a famIly of contracts defined by a pair (time of innovation, mnovation subsIdy) such to induce firms to behave optImally The proposed polIcy-mix (environmental tax and innovation subsidy) IS shown to reduce emIS­sions more, and to reduce output less, than environmental polIcies based on a single policy mstrument.

1. Introduction

The paper deals with a crucial problem in environmental economics: can emission taxes reduce industrial pollution without negatively affecting out­put? What are the effects of environmental taxation on firms' innovation activity? More generally, can a regulator design a policy to induce firms to adopt less polluting technologies? Is a combination of emission taxes and innovation subsidies the optimal policy?

The paper answers these questions by studying two games:

(i) the innovation game among firms that decide whether and when to adopt a new, less polluting technology; this game enables us to anal­yse firms' R&D behaviour, and the consequent innovation diffusion process;

IThe authors are grateful to the FondazlOne EN! "Enrico MatteI" for financIal support and to DGXII, Environmental Programme.

110 C. Carrara and G. Tapa

(ii) the policy game between polluting firms and the regulator, who sets his policy instruments in order to induce the adoption of the new technology at a socially-optimal time.

With respect to the existing literature, the paper introduces several nov­elties. Environmental innovation has been studied in a seminal paper by Downing and White (1986). These authors examine the effectiveness of different policies in inducing an "environment-friendly" innovation by n identical firms. They consider a perfectly competitive market, and com­plete information for all agents: in particular, the government is assumed to know the production and abatement technology available to firms, and to be able to measure the amount of emissions discharged by each source. By contrast, our paper analyses the innovation strategies of oligopolistic firms, and explicitly introduces asymmetric information (the regulator im­perfectly knows firms' innovation technology).

Moreover, Downing-White (1986) assume that a new technology, pro­viding firms with lower abatement cost functions, becomes exogenously and instantaneously available. By contrast, we model the interaction betwen production and innovation by assuming that a less polluting technology can be achieved only through time-consuming and costly R&D. The incen­tive to innovate is provided by the government's policy.

Other papers on environmental innovation (e.g., Magat, 1979; Mendel­sohn, 1984; Milliman-Prince, 1989; Orr, 1976) are subject to similar critical remarks. Perfect competition and complete information are usually as­sumed; firms' innovation decisions are modelled in a simple way; no strate­gic behaviour is introduced.

In her recent survey, Reinganum (1989) reviews a large body of articles that explicitly consider the innovation process as a strategic decision taken by each firm, given its expectations of other firms' behaviour. In particular, this literature accounts for the relationship between the amount of R&D decided by the firm and the time to produce the invention: hence, the timing of innovation is the outcome of firms' strategic behaviour. As a consequence, it is possible to model the innovative activity as a race towards invention, where the final prize consists of a patent that allows the winner to enjoy a strategic advantage over other competitors in terms of reduced production costs, product differentiation, or superior quality. This race is best represented as a dynamic game among firms, in which each firm's optimal R&D strategy is jointly determined with the other firms' strategic variables. This is also the approach adopted in this paper.

A further step concerns the diffusion of innovation within the industry, once a new technology has been developed. The timing of adoption can differ across firms because of firms' heterogeneity (for instance, different risk aversions), but lags may arise even if all firms are identical. The model

TaxatIOn and EnV'tronmental Innovahon 111

proposed in this paper enables us to study the timing of adoption and its diffusion in the industry.

A game-theoretic framework is also necessary to analyse the relation­ship between the regulator and profit-maximising firms. The game is fur­ther complicated by the presence of asymmetric information. In most prac­tical situations, the government ignores the characteristics of firms' tech­nology, and/or is unable to observe their action, in terms of abatement effort.

As known (see Baron's 1989 survey), asymmetric information prevents the regulator from reaching a social optimum through traditional policies, such as taxes or standards, because firms use the informational advantage to maximize their own payoff. There exist however adequate incentives to deal with this problem (see Baron, 1985; Spulber, 1988). In the context of environmental innovation, firms may delay innovation by claiming that the necessary R&D activity is too costly. As shown in the paper, if the govern­ment cannot observe firms' abatement technology, firms actually succeed in delaying innovation, unless a new policy instrument is introduced. This instrument, an innovation subsidy, is devoted to the solution of the adverse selection problem: appropriate contracts can be offered by the regulator in order to "separate" firms, and provide the right incentive to timely innova­tion.

In the paper, the regulation problem is modelled as a two-stage game. In the first stage, the government sets its policy instruments (emission taxes and innovation subsidies); in the second stage, firms decide whether and when to innovate, and the level of output. The main conclusions of the paper are the following: at the equilibrium, emissions are lower both with respect to the status-quo (no taxation), and with respect to a situation in which emissions are taxed, but firms stick to the old technology. More­over, as expected, output levels are higher than in the pre-innovation state. Therefore, emission taxes, by inducing emission-reducing innovation, loosen the traditional trade-off between growth and environmental quality. As far as the pattern of adoption of the new technology is concerned, the paper shows that there is diffusion in the timing of adoption, even though firms are assumed to be identical and there is no uncertainty in the innovation process.

Moreover, optimal private times of adoption differ from optimal social ones, which are defined on the basis of a measure of total welfare. Firms postpone innovation with respect to the social optimum, in order to min­imize R&D costs. This conflict between government and industry, in the presence of asymmetric information about firms' innovative ability, can be regulated through the application of a direct revelation incentive scheme. Such a mechanism is defined as a menu of contracts, each composed of a pair {time of adoption, subsidy}, which is indexed with respect to a

112 C. Carraro and G. Topa

parameter summarizing technological abilities. Subsidies are such to cor­rectly separate different types of firms. We show that this mechanism en­ables the regulator to implement the socially-optimal dates of innovation, even in the presence of information asymmetries.

The article is organized as follows. The model is introduced in Section 2, where we specify firms' technology, the specific type of innovation we consider in this work, the taxation scheme imposed by the government, and the other relevant assumptions. Section 3 is devoted to the analysis of firms' behaviour, given the taxation scheme introduced by the regulator. In Section 4, welfare analysis is carried out; in particular, the socially­optimal taxation level is defined, and private and social times of innovation are compared. Finally, Section 5 focuses on the problem of asymmetric information, and on the properties of the optimal incentive mechanism. A final section summarizes the policy implications of our results.

2. The Model

Two identical firms compete a la Nash-Cournot in the same product mar­ket, where they offer a single homogeneous good. Both firms are subject to the same regulating environment, either because they are located in the same country, or because, if located in different countries, industry regula­tion is internationally coordinated2 • Time is assumed to flow continuously. Firms have complete information about market structure and competitors' technology. For simplicity's sake, the demand function is assumed to be linear: P(Q) = a - f3Q, where Q = ql + q2.

Before any environmental regulation is introduced, firms produce out­put using a single-product technology, defined as D, which is characterized by a fixed emission/output ratio k; in other words, emissions x" ~ = 1,2, are a linear function of firm i's output q, : x, = kq" k > o. Total emissions are X = Xl + X2 = kQ.

Firms are assumed to share the same technology. Constant returns to scale are assumed for simplicity's sake. The profit function when both firms use technology Dis:

IT, = [a - f3(q, + qJ) - c]· q, ~,J = 1,2,i i= j (1)

where c denotes the marginal cost. No pollution abatement is possible with technology D: firms can only reduce pollution by reducing output.

Firms can, however, adopt a different, more flexible technology, char­acterized by abatement possibilities, and a lower emission/output ratio.

2The problem of evaluatIng the profitability of InternatIOnal environmental agree­ments when technological innovation IS accounted for IS examIned In Carraro-Topa (1994). The stability ofsuch agreements IS analysed in Carraro-Simscalco (1993a, 1993b).

Taxatwn and Enmronmental Innovahon 113

This technology is not available to firms unless some R&D is carried out. Therefore, we assume that firms, by engaging in R&D activity, are able to develop and adopt a new technology, defined as F, which enables them to reduce the emission/output ratio.

Firms can develop this innovation within a time t from the beginning of research by spending a monetary amount pet); pet) is a deterministic, decreasing and convex function of t : it summarizes both R&D costs and adoption costs (adjustment of the productive processes and plants); pet) is a decreasing and convex function because the cost of innovation increases more rapidly as firms try to accelerate the time of innovation. Moreover, there is a constant flow of basic, freely-available scientific research that allows firms to reduce the costs of innovation as they delay its adoption.

Innovation is firm-specific: it can be patented, but cannot be sold; moreover, firms have no information about results obtained by other firms' R&D activity. Therefore, in order to obtain the new technology within time t, each firm must spend pet). In addition, we suppose that the two firms decide their own levels of R&D expenditure at the beginning of the inno­vation game, and cannot change their strategy over time (the investment in R&D is irreversible).

The adoption of the new technology by one of the two firms does not prevent the other from innovating as well: in other words, both firms share the same research objective (the new technology F) and the same innovation cost function pet), but autonomously decide the investment level. Firms can thus innovate at different times as well as simultaneously.

Let us define the new technology F: it is characterized by an emis­sion/output ratio k' ~ k; notice that k' is not constant, because F is a multi-product technology that enables firms to produce an abatement good a. jointly with output q •. Firm's emissions are therefore given by x. = kq. - a,; hence, the new emission/output ratio k' is defined by:

k' = kq. - a. < k q. -

Total emissions become: X = kQ-A, where A is the industry total emission abatement (A = al + a2).

The unit abatement cost is equal to d' == d/k. Using technology F, each firm can decide its optimal emission/output ratio by adjusting its abatement level a •.

If no environmental policy is introduced, firms adopt technology D. When the government introduces an emission tax, firms could be induced to invest in R&D in order to adopt the new technology. In this case, each firm chooses the initial investment pet), i.e., the time at which the innovation will be available, the abatement level a., and output q •.

Without loss of generality, we suppose that the government announces

114 C. Carraro and C. Topa

the adoption of emission taxes at time O. If the taxation scheme is properly designed, firms react by engaging in the innovation game, in which each decides whether or not to innovate and at what date. The government's taxation policy is the following: each firm is asked to pay a tax t(X) per unit of emission, where the unit tax is a function of total emissions X == Xl + X2.

The idea behind this taxation scheme is that the marginal and average tax should increase as the environmental problem becomes more important, i.e., as total emissions increase. The function t(X) is assumed to be linear: t(X) = (}X, where the parameter () > 0 is set by the government at a level that maximizes total welfare, given firms' behaviour in the second stage of the regulation game.

Total fiscal revenue is T == Tl + T2 = () X2; () is therefore the marginal taxation emission ratio. This particular specification of the emission tax strengthens the interdependence of the two firms' decisions: firm i's marginal tax is given by

dT. -d = 2(}x. + (}x3 > 0

X. (2)

where 2() is the slope of firm i's marginal tax with respect to its own emis­sions, whereas () is the slope of firm i's marginal tax with respect to the rival's emissions. As a consequence, under technology D, when firm i re­duces output as a reaction to emission taxes, it suffers from two kinds of negative externalities: first, given the shape of the reaction curves in a Nash-Cournot duopoly, a contraction in q. induces firm j to expand its out­put q3 and to increase its profits; secondly, the expansion in q, increases firm i's marginal tax, as implied by (2).

A further justification for this taxation scheme derives from the use (in Section 4) of an increasing and convex damage function in the government's welfare function; in other words, total damage from pollution increases more rapidly as total emissions X increase.

Finally, we assume that the government can measure, at no cost, each firm's emissions.

To conclude this section, let us define the firms' profit function. First, we consider the case in which both firms use technology D and are levied taxes Til i = 1,2:

n. = [0: - (3(q. + q,)]q. - (c + (}kX)q.

[0: - ((3 + (}k2 )(q. + q3) - c]q.

i =1= j, ~,J = 1,2

(3)

Consider now the case in which, following the introduction of the emission tax, both firms adopt the new technology F. As seen above, this technology enables each firm to abate a quantity a. of emissions at a unit cost d/k.

Taxatzon and Enmronmental InnovatIOn 115

When both firms adopt F, the profit function is:

lit = [a - (3(qt + q3)lq. - OX(kq. - a.) - cq. - d/k· a.

= [a - ((3 + Ok2)(q. + q3) + Ok(a. + a3) (4)

-clq. + [Ok(q, + %) - O(a. + a3) - d/kla. i ..J. •

f J, i,j=1,2

Emission taxes under technology F can therefore be interpreted as the opening of a second market (for the abatement good A). The duopoly game now takes place on both markets, which are not independent because the two goods are complementary (an increase in A has the same effect as an outward shift in the demand curve for Q, and viceversa).

Finally, consider the case in which one of the two firms has already adopted technology F, whereas the other still produces using technology D (in the presence of emission taxes). Suppose firm 2 has innovated; the profit functions are:

lil = [a - (3(ql + q2)lql - cql - OkXql (5)

= [a - ((3 + Ok2)(ql + q2) + Oka2 - clql

li2 [a - (3(ql + q2)]ql - OX(kq2 - a2) - Cq2 - d/k . a2

= [a - ((3 + Ok2)(ql + q2) + Oka2 - clq2 + (6)

+[Ok(ql + q2) - Oa2 - d/kla2

We label firms' equilibrium profits in each of the above cases in the following way: given the symmetry of the game, let ¢DD be firm i's profit when both firms use technology D with no emission taxes; ¢hD is profit when both firms use D subject to the taxation scheme T.; ¢}F is firm i's profit after both firms have innovated; ¢hF is profit when i sticks to the old technology while firm j adopts F; finally, ¢}D represents firm i's profit when it innovates whereas its rival still uses technology D3.

In the next section, we focus on the analysis of firms' behaviour in the second stage of the regulation game, i.e., after the government has announced the adoption of a taxation scheme on emissions. In particu­lar, we examine conditions under which firms are induced to innovate, the characteristics of the market equilibrium arising after innovation (in terms of output, emissions, price and profits), and the optimal private adoption dates for both firms.

3For the mdustry equilibnum profits we use a similar notation 'PDD are industry profits without taxes; 'PhD denote mdustry profits with taxes, when both firms use technology D, 'P~F represents the case m which both firms have mnovated and 'P~D are industry profits when one of the two firms has adopted F, while the other has not yet innovated.

116 C. Carraro and G. Topa

3. Firms' Behaviour in the Innovation Game

Given the tax imposed by the government in the first stage, firms engage in a dynamic game of innovation, deciding whether or not to adopt technology F and at what date. In order to determine the outcome of the innovation game, we need to compare firm's istantaneous profit flows under each of the following technological configurations:

(i) the government introduces the environmental tax. Both firms keep using technology D (this is the (DD/t) case - where "t" indicates the presence of the emission tax);

(ii) the tax is introduced and both firms use technology F ((FF It) case);

(iii) the tax is introduced: one firm produces according to D and the other according to F (the two cases (FD It) and (DF It) are symmetric because the two firms share the same technology). Moreover, as a benchmark case, we will compute the equilibrium variables when no tax is imposed, and both firms use technology D (DD case).

Equilibrium profit flows ¢DD, ¢hD' ¢~F' ¢~D' and ¢hF are deter­mined by computing the equilibrium of the game that takes place within the industry at each time period. Each firm decides its production level q and its abatement level a as the optimal strategies of a Nash-Cournot duopoly game.

In the Appendix, we present the equilibrium values for output, market price, profits, abatement and emissions levels in the three cases described above. The following Table shows the ordering of all variables in the four cases. Conditions A.l, A.3 and A.4 in the Appendix enable us to rank all quantities, prices and profits without ambiguity.

Let us provide a few comments on the results presented in Table 1:

• Remark 1: Output is highest in the case without taxation (DD), and is lowest when the governments tax emissions whereas firms use technology D: this is the standard environmental protection-output trade-off. However, total output Q rises as innovation spreads within the industry, thus making the impact of environmental policy on the output market less severe .

• Remark 2: Profits follow a similar pattern: the profit squeeze in­duced by the introduction of the tax is much lower when firms adopt the new technology4

4In ohgopoly, the effects of the environmental tax on firms' profits are not so obvious As shown in Carraro-Souberyran (1993), If market demand IS nonhnear and firms are not symmetric, It IS pOSSible that mtroducmg the tax raises the profits of some firms in the industry.

Taxatzon and Enmronmental Innovahon

Table 1: Ranking of equilibrium variables under alternative technological configuration

Output: A4 t A3 t A3 t t

qDD > qFD > qFF > qDF = qDD Q Qt A3 t A3 t DD> FF > QFD > QDD

Price: t A3 t A3 t

PDD > PFD > PFF > PDD

Abatement: t A3 t A3 t t t A3 t A3 t

aFD > a FF > a DF = aDD = aDD = 0 AFF > AFD > ADD = ADD = 0 Emissions:

Al t t A3 t A3 t Al t A3 t A3 t XDD > XDD = X DF > XFF > XFD XDD > X DD > X FD > X FF

Profits:

A. A 4 A.t t t t A 4 t t t 'i'DD > 'i'FD > 4>FF > 4>DF = 4>DD 4!DD > 4!FD > 4!FF > 4>DD

117

• Remark 3: Emissions are lowest in the (FF/t) case, implying that environmental innovation enables the governments to achieve a lower emission level than with the old technology, for any level of the tax rate () above () A3 •

• Remark 4: ()A3, the minimum tax rate which is necessary to induce firms to adopt the cleaner technology F, is determined by condition A.3 in the Appendix. The minimum tax rate is ()A3 == k2(;;~~-d) > 0, which is negatively correlated with the demand size and the emission output ratio k, and is positively correlated with the marginal cost c + d. Moreover, if () satisfies A.3, it is possible to show that the tax rate is such that:

Tax rate A 3 A3

tDD > tFD = tDF > tFF·

In order to characterize the innovation process in the industry, let us analyze the case in which one firm only innovates (the FD/t case). Notice that whoever innovates first gains substantially from innovation, exploiting the fact that the other firm has to reduce production in order to limit the burden of emission taxation. Production qj.. D is indeed larger than in all other cases; individual profit ¢j..D is very large, thus making industry profit CPj..D larger than in the (FF/t) case, even though the profit flow for the non-innovating firm remains at the ¢hD level. Moreover, residual emissions and the emission/output ratio are lower than in all other cases for the firm that innovates first.

118 C. Carrara and G. Topa

These considerations help us understand the equilibrium of the inno­vation game; in order to compute this equilibrium, we need to define firm i's monetary incentive to be the first innovator. The gain from innovating first is:

_ t t [Ok2(a - c) - ,B'dJ2 It = ¢FD - ¢DD = 4,B,B'Ok2 > 0,

whereas firm i's gain when it innovates after the other firm has innovated is:

_ t t [Ok2(a - c) - ,B'd)2 12 = ¢FF - ¢DF = 9,B,B'Ok2 > o.

It easy to see that It > 12 :

_ t t _5[20k2(a-c)-,B'd)2 It - h - ¢FD - ¢FF - 144,B,B'Ok2 > o.

This result is crucial to understand whether the pattern of adoption within the duopoly is simultaneous or diffused. Let us define firm 1 's in­tertemporal objective function (firm 2 is symmetric). The present value of innovation costs is p(t), which is assumed to be a continuous and twicely differentiable function for t E [0,00). The common discount rate is r; 71 and 72 are the adoption times for firm 1 and firm 2, respectively. We have:

"(1;( ) _{9}(71.72) if71~72 1 71,72 = 2( )·f gl 71, 72 1 71 > 72

where

and

Firm 1 's payoff is g} (71,72) given that it decides to innovate first; it receives g~ ( 71. 72) if firm 2 adopts technology F before firm 1. The function VI ( .) is continuous in 71 for a given 72, but it is not differentiable in 71 = 72. We also assume the following:

(a) p(t) ~ 0 Vt E [0,00); (b) p'(t) < 0 Vt E [0,00); limt-+oo p'(t) = 0; (c) - p'(O) > ¢~D - ¢bD; (d) p" (t) > r[¢~D - ¢bDJ . e-rt > 0, \It E [0,00).

(A. C)

Therefore, p( t) is a decreasing and convex function of t. The second part of (b) implies that firms do not postpone the adoption of F for an infinite

Taxatzon and Enmronmental Innovatwn 119

length of time; (e) rules out immediate adoption in t = 0, because in such case research and adjustment costs would be too high; finally, (d) makes both gK) and g~(-) locally concave in 1', for any given TJ , i,j = 1,2, i =I- j. Assumption A.C enables us to prove the following preliminary Lemma:

Lemma 1. There eX1,st a unique f E [0, 00) and a umque T E [0, 00) that respectwely maX'tmzze gl{.) and g2(-) wzth respect to f gwen T. In additwn, 0< f < T < 00.

Proof. Since gl (.) is continuous and strictly concave, there exists a unique f that maximizes it; f, which represents firm 1 's optimal adoption time when it adopts first, is defined by the following first order condition:

(4>bD - 4>~D) . e-rT - p'{;) = 0 (7)

Similarly, T - the optimal adoption time for firm 1 when it innovates after firm 2 - is the unique maximum of g2 (.) : it is defined by the first-order condition:

( ",t ",t) -rf '(-) 0 'l'DF - 'l'FD . e - p l' = . (8)

T> 0 derives from assumption A.C(e). Moreover, f < T because (4>}..D -4>bD) > (4>}..F - 4>bF); in other words, the gain 11 from adopting first is larger than the gain 12 from adopting later. Finally, T < 00 because

lim (4>bF - 4>~F) . e-rt - p'{t) < 0, '<11'2. t-+00

• • Remark 5: Condition (7) defines the first innovator's optimal adop­

tion time, balancing the discounted marginal cost 4>~D - 4>bD)' e-rT

of delaying innovation with the discounted marginal benefit -p'{f), which represents the cost reduction achieved by delaying innovation. The interpretation of equation 8 is analogous.

Let us now describe the innovation game. As already stated, each firm decides whether or not to innovate, and when, by choosing the R&D investment pet). In a deterministic context, this is equivalent to picking an adoption date 1'1 (where 1'1 = 00 means that the firm does not innovate at all).

Each firm's strategy space is then B, = [0,00), and a pure strategy for player i is a scalar T, E B •. Strategies are open-loop, because firms decide their own level of investment at t = 0 once and for all. The set of firm i's best responses to TJ is e.{TJ ) == {T. E B,I V.(T.,TJ ) ~ V.(T;,TJ ), '<IT' E B,}. The mapping e, : BJ --t B. is firm i's best-reply correspondence.

The Nash equilibrium of the game r = (Vb V2 , Bb B2 ) is defined by the strategy pair (Ti,T2) such that T,* E e(TJ ), Z,) = 1,2, i =I- j. The equilibrium is characterized by the following theorem:

120 C. Carrara and G. Topa

Theorem 1. Assumptions A.3 and AA are necessary and sufficient con­ditions for the existence of two Nash equilibria in pure strategies of the innovation game r :

( * *) (' -) d (** **) ('-) Tl,T2 = T,T an Tl,T2 = T,T

Proof. Given A.3 and A.4, firms find it profitable to abate, i.e., to adopt technology F (A.4 guarantees non-negative emissions). As for the optimal adoption times, the proof of Theorem 1 uses a series of Lemmas. The idea is to derive each firm's best-reply correspondence, and then find their two fixed points (Ti,T2) and (Ti*,T:;*).

Proof. g[(Tl,T2) - gr(Tl,T2) = (h - 12)(e-rTl - e-rT2 /r::; (~)o as Tl ::; (~)T2'

Lemma 3. gr(f, f) > gt(f, f), i = 1,2.

Proof. gt(f,f) > g[(f, f) = gr(f,f), where the inequality follows from Lemma 1 and the equality from Lemma 2.

Lemma 4. gt(f, f) > gr(f, f) i = 1,2.

Proof. gt(f, f) > gr(f, f) = gr(f, f), where the inequality follows from Lemma 1 and the equality from Lemma 2.

Proof. Let JL(f,f,f) = g[(f,T2) - gr(f,T2). By Lemma 3, JL(f,f,f) < O. By Lemma 4, JL( f, f, f) > 0. Since aJL/ aT2 = (h - 12 ) . e- rT2 > 0, the result follows by the intermediate value theorem and the monotonicity of JL in T2·

Lemma 6. Firm 1 's best reply correspondence is

{ f for T2 < f

6(T2)= {f,f} forT2=f f for T2 > f

Proof. The proof follows from the use of the definition of V1 (.) and of Lemmas 1 and 5 for each of the three intervals described above.

In order to prove Theorem 1, we only need to notice that firm 2's best reply - correspondence, by symmetry, is

for Tl < f for Tl = f

for Tl > f

Taxation and Environmental Innovation 121

It is then obvious that the best-reply correspondences intersect at two dis­tinct points, (f,f) and (f,f). •

We can therefore conclude that the non-cooperative innovation game is characterized by diffusion in adoption times. Given the structure of incentives It > 12, firms do not innovate simultaneously. The intuition behind this result is the following: both firms have an incentive to innovate first because It > 12 ; however, if both firms innovate at the same time, they lose the competitive advantage of being the first innovator, while paying the high R&D costs that enable firms to innovate sooner. One of the two firms thus prefers to save R&D costs, and innovates later. The structure of the innovation game is similar to the structure of a chicken game. If both firms choose their preferred strategy (to innovate first), they achieve a low benefit and pay large R&D costs. There are therefore two equilibria in which one firm innovates before the other one (notice that, as in a chicken game, the case in which the two firms innovate simultaneously, but later, is not an equilibrium because of the incentive It to innovate first) .

• Remark 6: Theorem 1 does not specify the identity of the first and second innovator. However, this is not of great importance because the two firms are identical. We can therefore define the equilibrium, privately optimal, adoption dates of the innovation game as (r( == f, r{ == f, where the superscript j = 1,2 indicates the firm that adopts first or second, rather than the identity of the firm.

Let us summarize the results proved in this section: the effects of emission taxes go beyond the usual emission reduction achieved through a contraction of firms' output. Environmental taxation, by making emis­sions costly, induces firms to change technology, switching to the more flexible production process F, which enables firms to choose the optimal emission/output ratio. The taxation scheme thus plays the role of an in­centive to technological innovation. As a consequence, output is larger and emissions are lower than in the case in which technology is fixed. How­ever, firms do not innovate simultaneously, and, as proved in the following sections, have an incentive to delay innovation (with respect to the socially­optimal adoption times).

4. Government's Optimal Policy

Having determined the firms' optimal behaviour in the second stage of the regulation game, we now turn to the analysis of the government's optimal policy in the first stage. The government decides whether or not to intro­duce emission taxes, and if it does, the optimal taxation level. Its strategy

122 C. Carraw and C. Tapa

space is defined as G == [0, (})5 and a pure strategy is a scalar () E G. The regulator's objective function at time t is the sum of the consumers'

surplus and industry profits6 .

W(t) = CS(t) + ~(t) (9)

Following a commonly used specification in environmental economics,7

the consumers' surplus is defined as the sum of net surplus from the con­sumption of Q and of tax revenues, minus monetary damages from pollu­tion:

Q* CS(t) = 10 P(Q)dQ - P(Q*) . Q* + t(X*) . X* - M(X*) (10)

where Q* and X* are the equilibrium values of industry output and total emissions computed in stage two of the game. Monetary damages M(X) are assumed to be a quadratic function of total emissions X:

M(X) = ~X2 2

(11)

The parameter A represents the shadow cost of emissions, or, alterna­tively, the shadow price of environmental quality. Formally, it should be derived from the maximization of consumers' utility subject to their bud­get constraint. Intuitively, a higher A shows the representative consumer's greater concern for pollution damages: A is thus an increasing function of the disutility from emissions. In this model, we take A as exogenously de­termined, and we focus the analysis on the impact of this "environmental sensitivity" on government's decisions.

Tax revenues do not appear in the expression for total welfare W (t) because they are pure transfers from firms to consumers. We thus have:

Q* W(t) = 10 P(Q)dQ - cQ* - d/k. A* - M(X*) (12)

Let us first compute the socially-optimal output and abatement levels: solving the usual first-order conditions, we gets:

- a-c-d Q = --:::---

f3 5The upper limit 0° on the marginal emission/taxation rate is supposed to be exoge­

nously determined by the legislative and institutional environment. 6We assume therefore that the government assigns equal weights to consumers' and

producers' interests. 7See, for example, Baron (1985), and Spulber (1988). 8Second-order conditions are always satisfied since W(t) is strictly concave.

Taxation and Environmental Innovation

A = Ak2(o: - c) - ((3 + Ak2)d (3Ak

123

where Q and A are non-negative if A.3 holds. Socially-optimal residual emissions are:

- - - d X = kQ - A = Ak > 0

It is easy to see that the socially-optimal abatement level is an in­creasing function of Aj on the contrary, socially-optimal emissions are a decreasing function of A. This result is quite intuitive: as consumers assign a greater value to environmental quality, the socially-optimal pollution level falls and the socially-optimal total abatement grows.

The next step is to compute the consumers' surplus CS(t), and total welfare W (t) for each of the different technology configurations defined in the previous section. The results are shown in the Appendix. Here we summarize their implications by ranking the values of the total welfare in the three cases in which the government introduces the tax. Define W(t I DD) == WbD' W(t IFF) = W}F and W(t I FD) = W}D' Then:

Lemma 7. A.3 is a sufficient condition for W}F > W}D > WbD > 0, V() E (A/3,2A].

Proof. The first two inequalities are proved by computing the differences (W}F - W}D) and (W}D - WbD) using condition A.3 as a sufficient condition, i.e., substituting 2(3' d for Ok2 (o:-c) into the resulting expressions. This enables us to prove that W}F > W}D > WbD' V() ~ 2A. In addition, WbD is positive for 0 > A/3, as can be seen from its definition. _

• Remark 7: Lemma 7 shows that, in the presence of environmental taxation, total welfare at time t rises as firms move from the D to the F technology: in other words, the diffusion of environmental innovation in the industry enables the regulator to increase total welfare.

Let us finally derive the government's optimal strategy: define the in­tertemporal welfare W as:

where

W = { wt if government introduces the tax WO if no taxation is imposed

W t == [71. wbD.e-rtdt+1T:1 W}D.e- rtdt+l°O W}r e- rtdt-p(Tl)-p(T2) 10 7"1 7"2

(13) and

W(t I DD) == WDD (14)

124 C. Carrara and G. Topa

Eq. (13) defines the intertemporal welfare when the government sets emission taxes such that the environmental innovation is adopted, i.e., () E A«(}) (as usual, 1'1 and 1'2 are the adoption dates of the first and of the second innovator respectively, and do not refer to the firm's identity). Eq. (14) defines the intertemporal welfare when no environmental taxation is introduced. The government's optimal strategy is determined by the following Theorem:

Theorem 2: Assume A.3 and A.4. If market demand satisfies 0: > c+3d, and firms are not myopic, then the optimal marginal taxation/emission rote iJ is strictly positive if and only if A E (AI. AA4], where

,Bd{ 4(0: - c) - 2d + V[7(O: - c) - 2d](0: - C - 2d)} --------------~~--~------------->O

2k2(0: - c)2

9,Bd 2k2(0: - c _ 3d) > A1.

When A belongs to this interval, the socially-optimal marginal emission­taxation rote is iJ = 2A/3.

Proof. See the Appendix .

• Remark 8: Condition 0: > c + 3d is necessary to determine the relative magnitude of AO and AA3 without ambiguity. If 0: is not larger than c+3d, however, W~F > WDD for A > A1 and A E (0, AO),

According to Theorem 2, the government finds it optimal to tax emissions if and only if consumers' valuation of environmental quality is sufficiently high (the upper limit A guarantees non-negative residual emissions). This result confirms what is intuitively obvious: consumers' valuation of a clean environment plays a crucial role in determining whether the government is willing to introduce restrictive environmental policies. Notice that the optimal emission tax scheme in which the marginal taxation-emission rate is () = 2A/3 determines an inefficient allocation of resources in terms of output Q, i.e., Q~F(iJ) < Q; however, it enables the regulator to achieve the socially-optimal level of pollution X, i.e., Xj..F(iJ) = X. The proof follows from obvious algebra.

Let us now analyise the socially-optimal innovation process, in order to determine whether: (i) diffusion is optimal; (ii) optimal private adoption times coincide with the socially-optimal ones. We need to modify assump­tion A. C in the following way:

(a) as in A.C; (b) as in A.C; (A.C')

Taxation and Environmental Innovation 125

(c) _p'(O) > WfrD - WbD; (d) p" (0) > r[WfrD - WbDl· e-rt .

As for assumption A.C, (e) implies that immediate innovation is non­optimal, whereas (d) guarantees local concavity of the intertemporal wel­fare W with respect to its arguments T1 and T2. Let (Tf, T!) be the equilib­rium socially-optimal adoption dates. Then, we prove the following Lemma, which parallels Lemma 1.

Lemma 8: There exists a unique pair (Tf, T!) that maximizes intertempo­ral welfare W; in addition, assuming A.3, we get: 0 < Tf < T! < 00, V() ~ 2A.

Proof. Socially-optimal adoption times (Tf, T!) are defined by the follow­ing first-order conditions:

(wt wt) _rr S '( s) 0 DD - FD· e 1 - P T1 = (15)

(w t wt) _rr s '( S) 0 FD - FF· e 2 - P T2 = (16)

Tf > 0 derives from condition A.C(e). Tf < Tf is proved by verifying that (WfrD - WbD) > (WfrF - WfrD)' \:j() S; 2A (this includes the specific case of () = 0.) Finally, T! < 00 because

1· (wt wt) _rrS '( s) 0 1m FD - FF· e 2 - P T2 < . 'T2-+CX>

• Therefore, the pattern of diffusion in adoption times, which was shown

to be optimal for the two firms, is actually socially-optimal. Let us now check whether social adoption dates differ from private ones. Emission taxes are set at their optimal level, i.e., () = 2A/3. The following theorem proves the result:

Theorem 3. If () = 0 = 2A/3, A.3 is a sufficient condition for Tf < Ti. In addition, T! < T{ if and only if:

A d ~~~~~~ < ~~--~ ,(3{3) + 2Ak2) 2k2(a - c)

(17)

where, == (13{3 + 4Ak2)/(5{3 + 4Ak2) > 1, VA ~ o.

Proof. Under condition A.C(d), it is easy to show, from the first order conditions that determine the {TP} and the {TS} pairs, that Tf < Ti iff ((WfrD - WbD) > (j/FD - 1>bD)' whereas T! < T{ iff (WfrF - WfrD) >

126 C. Carraro and G. Tapa

(4)~F - 4>bF)' The result thus follows from these comparisons, using as­sumption A.3 as a sufficient condition in the first one, and condition (17) in the second one. _

• Remark 9: Condition (17) can be interpreted in the following way: let us re-write (17) in terms of per-unit abatement cost d/k, keeping >. constant:

2>.k2(o: - c) d/k> ,(3(3 + 2>.k2) (17')

In other words, the social adoption time for the second innovator is earlier than the private one, if and only if the marginal abatement cost is sufficiently high, i.e., as abatement costs rise, firms are more reluc­tant to abate and therefore to innovate, and postpone the adoption of technology F.

We have thus shown that the social incentive to innovation is greater than the private one, i.e., the government would like to induce firms to ac­celerate their innovation process. In the next section, we define an incentive mechanism that induces the socially-optimal adoption pattern, in a context of asymmetric information on innovation costs.

5. Incentive Mechanisms with Asymmetric Information

In this section, the government's goal is to induce firms to innovate at the socially-optimal dates, accelerating their research and adoption processes with respect to the private optimum.

This problem is trivial in a context of complete and perfect information, because the government can use different instruments (from command and control to subsidy) to impose the optimal pair { T§, T§}. The problem is more complicated if the regulator cannot directly observe firms' behaviour, or if it ignores some characteristics of the firms that are subject to regulation.

We choose to model the presence of asymmetric information as a prob­lem of adverse selection: we suppose that the government cannot directly observe firms' innovation costs; more generally, the government ignores firm's innovative abilities, in terms of engineering and managerial resources. In order to introduce adverse selection, we re-define innovation costs as:

'l/Ji(t) == Wi • pet), i = 1,2 (18)

where Wi is a parameter that summarizes each firm's innovative ability. The support of Wi is fixed and equal for both firms, and is defined as n = [a, bJ, where a and b are positive constants. Greater values of Wi represent a lower efficiency in R&D and adoption processes. Therefore, firms are still

Taxation and Environmental Innovation 127

identical in terms of production technologies, but possess different degrees of innovative abilities.

The two Wi'S are independent realizations of the same random vari­able W : we assume that the distribution of W is common knowledge, i.e., the government ignores each firm's specific characteristics, but knows how Wi, i = 1,2, is distributed within the industry. The assumption that the Wi'S are independently distributed enables us to separate each firm's in­centive problem9 : were the two parameters correlated, the regulator would have to deal with a multi-agent adverse selection problem, which presents severe difficulties in terms of multiple equilibria and inefficiency of the in­centive mechanism 10.

The cost function p(t) is similar to the one described in the previ­ous sections. We modify assumption A.C in order to prevent firms from adopting at t = 0, and to guarantee the concavity of the intertemporal profit functions:

(a) as in A.C.; (b) as in A.C.; (c) -p'(O) > (W;'D - WbD)/a; (d) p" (0) > ~[W;'D - WbD]· e-rt •

(A. e")

The government's intertemporal welfare Wand firms' intertemporal payoffs [Vi(ri, rj)], i = 1,2 i =1= j, are the same as in the previous sections, with 'l/Ji (.) replacing Pi (.) (as we will see, the identities of the first and of the second innovator are not relevant for the incentive mechanism). Privately­and socially-optimal innovation dates are defined as in Lemmas 1 and 9, respectively. More precisely, using the following assumption:

A. H: The dimension b/a of the support n is strictly positive and has an upper limit: 0 < b/a::; 9/4. we have:

Lemma 10. Assumption A.H is a necessary and sufficient condition for diffusion to characterize the equilibrium outcomes of the innovation game between firms. In particular, 0 < r«wi) < r{(wj) < 00.

Proof. The proof coincides with that of Lemma 1. The privately optimal pair (r( (Wi), r{ (W j » is defined by the following first­order conditions which modify eqs. (7) and (8):

(¢bD - ¢~D)· e-rTi - Wip'(t)(r[) = 0

(¢bF - ¢~F)· e-rTt' - Wjp'(t)(r{) = 0 --~-------------------

9See Homstrom (1982). lOSee Demski and Sappington (1983) and Ma, Moore and Turnbull (1988).

(7')

(8')

128 C. Carram and C. Tapa

The proof of Lemma 1 clearly shows that diffusion occurs if and only if (1)~D -1>bD) > (b/a) . (1)~F -1>bF) : this is implied by b/a < 9/4, i.e by assumption A.H. •

Condition A.H simply means that differences in innovative efficiency within the industry cannot be too large for diffusion to occur. Notice that eqs. (7') and (8') imply that both Tnwd and T{(Wj) are an increasing function of Wi and Wj respectively. Therefore, as innovative efficiency de­clines, the optimal private strategy is to delay innovation, since the marginal benefit of delayed adoption increases with respect to the marginal cost rep­resented by foregone profits.

The socially-optimal innovation dates are defined by:

Lemma 11. A.3 and A.H are sufficient conditions for diffusion to characterize the socially-optimal pattern of innovation, for () ::; 2>', i. e., 0< Tf (Wi) < Tf (Wj) < 00, 'VB ::; 2>'.

Proof. The proof coincides with that of Lemma 9; the socially-optimal pair (Tf (Wi), Tf (Wj» is defined by the following first-order conditions, that replace conditions (15) and (16):

(wt wt) _rrl '( s) 0 DD - FD' e S - WiP Tl = (15')

(wt wt) _rr2 '( s) 0 DF - FF' e S - WjP T2 = (16')

• From eqs. (15')(16'), it is easy to show that the socially-optimal adop­

tion dates Tf (-) and Tf (-) are both an increasing function of Wi and Wj respectively. In other words, the government, when setting the optimal timing of environmental innovation, takes into account the firms' relative innovative ability. If the government knows that large R&D investments are necessary to accelerate the adoption of emission reducing technologies, the social optimal adoption dates may be delayed.

• Remark 10: The relationship between private and social optimal adoption times is still determined by Theorem 3, because its proof is not influenced by the presence of the efficiency parameters Wi and Wj. Hence, firms would like to postpone innovation (with respect to the socially-optimal innovation dates). This implies that the government has still an incentive to design an environmental policy-mix that will induce firms to accelerate environmental innovation (with respect to the private optimum).

Taxation and Environmental Innovation 129

It is now possible to characterize the optimal incentive mechanism which provides each firm with the necessary incentive to reveal its inno­vative ability, and to behave according to the regulator's objective func­tion. The government offers two distinct mechanisms, Ml and M2, aimed at regulating the firm that adopts earlier and the one that adopts later, respectively. Each mechanism is composed by a menu of contracts indexed with respect to Wi or Wj : each contract consists of an innovation date T that the government wants to achieve and by a monetary transfer or subsidy 8,

to induce firms to reveal their true characteristics; the two mechanisms are defined by:

Ml = {Tl(Wi), 81(Wi); VWi EO}

M2 = {T2(Wj),82(Wj); VWj EO}

(19)

(20)

where i :f j, i,j = 1,2. By reporting a certain level of efficiency Wi, the first (second) innovator chooses the particular contract indexed with respect to Wi (Wj).

Mechanisms Ml and M2 are designed in order to be implementable, that is, each contract must be such to induce the first (second) innovator to innovate at date Tl(Wi) (T2(Wj)), thus revealing its private information. In the model, we assume commitment: the government is able to credi­bly commit himself to the mechanisms announced at the beginning of the regulatory game.

The timing is different from standard adverse selection models. In the latter, the Agent sets his decision variable after choosing a single contract from the menu offered by the Principal. In our model, the government cannot sign a contract in advance with any of the two firms, because the identity of the first and of the second innovator will be known only in the second stage of the game. We therefore assume that the government credibly announces two distinct subsidy schedules 81 (-) and 82 (.) for the firm that adopts at date Tl (-) and T2 (-), respectively. Contracts will then be signed in the second stage.

Let us derive the mechanism Ml for the first innovator. His intertem­poral payoff is given by the function 91 (.) defined above, with 'l/Ji (.) replacing p(.), and with the addition of the subsidy:

This payoff function has the following properties:

891/8Tl = (¢~)D - ¢~D) . e-rT1 - WiP'(Tl);

891/881> 0;

130 C. Carrara and G. Topa

8gI/8wi = -p(rd·

Hence, glO is an increasing function of the subsidy; moreover, it satisfies the Spence-Mirrlees condition for separability:

(22)

Therefore we can apply Guesnerie-Laffont's (1984) Theorem 2 to our regulatory problem:

Theorem 4. Given the properties of the intertemporal profit function (23), any action profile rl(-) such that its derivative is non-negative (drddwi ::::: 0) is implementable via suitable compensatory transfers.

Proof. See Guesnerie-Laffont (1984). • Having determined the set of implementable mechanisms for the first

innovator, we derive the optimal mechanism Mh. Notice that transfers to firms are socially-indifferent, because the government equally weights consumers' surplus and firms' profits in his welfare function W. In addition, as stated above, the first innovator's socially-optimal innovation time rf (.) is an increasing function of the efficiency parameter Wi. Therefore, according to Theorem 4, it is implementable. We can then set ri(wi) = rf(wi).

As for the optimal subsidy Si(Wi), this is defined by the first-order condition for implementability:

ds"'!/dwi = - 8g1 . drl < 0 8rl dwi

(23)

and by the individual rationality constraint gl(wi) ::::: 0, VWi, where imple­mentability can be written as:

gl(wi,wi) gl(rl(wi), Sl(Wi),Wi; r2) ::::: gl(wi,wi)

= gl(rl(wi),Sl(Wi),wi;r2) VWi EO (24)

Integrating expression 23, we determine the information premium for the firm that innovates first. Notice that the information premium is a decreasing function of Wi, i.e., an increasing function of firm i's innova­tive ability. This is a standard result of incentive theory: a more efficient firm must receive a relatively larger premium to find it profitable to reveal its characteristics. This feature of the information premium is meant to counter-balance the natural incentive of more efficient firms to understate their ability, in order to receive a more favourable regulatory treatment.

Taxation and Environmental Innovation 131

The analysis is symmetric for the optimal mechanism M2 designed for the second innovator. The intertemporal payoff is:

= r'TI0,/,.t . e-rtdt + r'T20,/,.t . e-rtdt + Jo 'YDD hl0 'YFD

+ J.;:(.) if>~F' e-rtdt - wiP(rl(Wj» + 81 (Wj)(25)

with properties similar to those of gl (.) :

og; lor2 = (if>i>F - if>~F) . e-r'T2 - Wjp'(r2)j

og; I 082 > OJ

og; lowj = -p(r2).

Hence, g;(-) is also an increasing function of 82, and it satisfies the Spence-Mirrlees condition for separability. The optimal mechanism M2* therefore implements the socially optimal innovation time rf (-), because it is an increasing function of W2j 82, the optimal subsidy for the second innovator, is determined in a way similar to the one used to determine 8i.

Summing up, the optimal mechanisms Ml * and M2* that subsi­dize firms' innovation activity enable the government to separate different "types" of firms, and to achieve the socially-optimal pair of adoption dates (riS (,), rf (.». A policy-mix of emission taxes and innovation subsidies leads to a socially-optimal innovation process and emission control. However, there are still two inefficiencies: output is lower than the socially-optimal output, and the total subsidy is too high, because of the information pre­mium to be paid to the more efficient firm.

6. Conclusion

This paper has analysed the effects of an emission tax on firms' behaviour in an oligopolistic industry. The government is assumed to introduce a taxa­tion scheme that increasingly penalizes polluting emissions. Firms react by curbing output and, when the tax is properly designed, by investing in R&D so as to develop a new, cleaner production technology. The new technology enables firms to abate emissions, while increasing production and profits. Environmental innovation in the industry is shown to be diffuse. Morever, firms have an incentive to delay innovation with respect to socially-optimal adoption dates. The government's optimal policy is thus a pair (emission tax, innovation subsidy) that induces firms to anticipate the adoption of the environmental innovation. In the presence of asymmetric information, the optimal subsidy includes an information premium to separate different types of firms.

The paper provides some policy recommendations:

132 G. Garraro and G. Topa

• the effects of an emission tax cannot properly be evaluated without accounting for firms' R&D and investment strategies that lead to technological changes; the effects through technological changes are likely to be larger than direct effects achieved through output changes; hence, environmental taxation must be viewed and designed as a way of stimulating technological innovation,

• the timing of environmental innovation is relevant for a precise un­derstanding of environmental policy: the timing of adoption depends on firms' innovation costs; firms find it profitable to delay innova­tion, and to overstate innovation costs if information is asymmetric. Hence, environmental policy should combine stick and carrot, tax and subsidy, in order to prevent firms from underinvesting in R&D, thus postponing the adoption of the environment-friendly technology,

• the proposed policy-mix (environmental taxation and innovation sub­sidies) enables the government to achieve his main target, i.e., to reduce emissions without excessively penalizing industry profit and market share.

Appendix

1. The solution of the duopoly game under alternative technological configurations

Let us first compute, as a benchmark, the equilibrium of the duopoly game when no emission tax is imposed (DD case). In this case, the equilibrium values of output, market price, emission abatement, polluting emissions, and profit levels are given by (starred variables indicate equilibrium vari­ables): No taxation (DD).

Q* _ 2(0: - c) _ Q . - 3(3 = DD,

* (Q*) 0: + 2c P =p = -3- ==PDD;

at = ai = 0 == aDD;

k(o: - c) x* = x* = = x*· 1 2 3(3 - ,

X* - 2k(o: - c) = X . - 3(3 - DD,

ll* * o:-c_A.. 1 =ll2 = ~ ='I'DD;

2(0:-c)2 <PDD = 2</JDD = 9(3 ;

Taxation and Environmental Innovation 133

Ex-post emission/output ratio == (X/q)DD = k. Output is non-negative if and only if the following condition holds: A.I: a> c.

If the government introduces the taxation scheme described in Section 2, but both firms stick to the old technology, equilibrium values become: Emission Tax and DD-technology (DD/t).

Define (3' == (3 + () /2 . k 2 • Then

Q* _ 2(a - c)Qt - 3(3' DD

* _ P(Q*) _ 0(2(3 - 3(}k2 ) + 4(3c = t • P - - 6(3' -PDD,

ai = a~ = 0 == abD;

* _ * _ k(a - c) _ t • Xl - x2 - 3(3' = xDD'

X* = 2k(a - c) = xt 3(3' - DD

IT* - IT* _ (a - c)2 _,/..t • 1 - 2 - 9(3' = 'PDD'

<pt _ 2,/..t _ 2(a - c)2 . DD - 'PDD - 9(3' ,

Ex-post emission/output ratio == (x/q)bD = k. Output is non-negative if and only if condition A.I holds.

Let us now suppose that both firms move to the F technology as a reaction to the government's environmental policy. The equilibrium values are: Emission Tax and FF-technology (FF/t).

Q* _ 2(a - c - d) = Qt - 3(3 - FF

* _ P(Q*) _ a + 2( c + d) _ t . P - - 3(3 =PFF,

* (}k2 (a - c) - 2(3'd _ t al = a2* = 3(3(}k = aFF;

A* _ 2[(}k2(a - c) - 2(3'd] = At . - 3(3(}k - FF,

* * 2d_ t Xl = x2 = 3(}k = xFF;

X* 4d xt = 3(}k == FF;

* * (a - c - d)2 2d2 t ITl = IT2 = 9(3 + 9(}k2 == ¢FF;

<pt _ 2,/..t _ 2(a - c - d)2 2d2 . FF - 'PFF - 9(3 + 9(}P'

Ex-post emission/output ratio == (x/q) = (2(3d)/((}k(a - c - d)) < k. Output is non-negative if and only if:

134 C. Carraro and G. Topa

A.2: a> c+d. Notice that condition A.2 implies A.I. In addition, firms' emission

abatement a~F is strictly positive if and only if: A.3:0 > OA3 == (2,8d)/[k2(a - c - d)] > O. The economic interpretation of this condition is the following: the marginal tax rate () must be higher than 0 A3 in order to induce firms to abate a strictly positive amount of emissions. Therefore, there exists a "threshold" value of fiscal pressure below which firms do not abate. The threshold 0 A3 depends on demand and cost parameters in the following way:

The minimum marginal tax mte which is necessary to induce firms to inno­vate is a decreasing function of the demand level and of the emission/output mtio k, and an increasing function of abatement and production costs.

It might also be interesting to evaluate the effects of the emission tax on equilibrium profits and outputs; the following proposition summarizes our conclusions:

Firms' emission abatement a~F is an increasing function of the marginal tax mte 0; residual polluting emissions X~F' ex-post emission/output mtio (X/q)~F' and equilibrium firms' profit ¢~F are a decreasing function of O.

The above propositions can easily be proved by computing the derivatives of OA3 with respect to a, k, c, d, and of a~F' X~F' (x/q)~F' and ¢~F with respect to O.

It is interesting to observe that the minimum marginal tax rate required to induce strictly positive emission abatement falls as market demand in­creases: as the output market expands, taxation costs that penalize pol­luting emissions increase. Firms are thus induced to reduce the emis­sion/output ratio k by engaging in abatement activity, which becomes pos­sible by the adoption of technology F.

The last case to be considered is the one in which one firm adopts the less polluting technology, whereas the rival sticks to the old technology: Emission Tax and FD-technology (FD/t).

* Ok2(a-c)-2,8'd a-c_ t •

q2 = 4,8,8' + 3,8' = qFD'

Q* _ Ok2(a - c) - 2,8'd 2(a - c) = Qt . - 4,8,8' + 3,8' - FD'

* _ P(Q*) _ (4,8 + 30k2)a + (8,8 + 30k2)c + 6,8'd = t p - - 12,8' -PFD

ai =0;

Taxation and Environmental Innovation 135

* k(a - c) t * 6/3'd - Ok2(a - c) t xl = 3/3' == xDP; X2 = 6/3' Ok == XPD;

X* _ Ok2(a - c) + 6/3'd = xt . - 6/3'Ok - PD,

* _ (a - c)2 _ t. * [Ok2(a - c) - 2/3'd]2 (a - c)2 _ t . III - 9/3' = ¢>DP, 112 = 4/3/3'Ok2 + 9/3' = ¢>PD,

cJ>t _,/,.t ,/,.t _ [Ok2(a - c) - 2/3'd]2 2(a - c)2. PD - 'f'DP + 'f'PD - 4/3/3'Ok2 + 9/3' ,

Firm l's ex-post emission/output ratio == (x/q) = k; Firm 2's ex-post emission/output ratio ==

_ t 2/3[6/3'd - Ok2(a - c)] = (x/q)PD = 302k3(a _ c) _ 6/3'dOk + 4/30k(a _ c) < k.

In order to obtain non-negative polluting emissions X~D' we impose one last necessary and sufficient condition: A.4: 6/3'd ~ Ok2(a - c). In terms of 0, condition A.4 becomes

{ 0 < 0 - 6/3d

- A4 = k2 (a - c - 3d) O~O

if a> c+ 3d;

otherwise

i.e., when market demand is large (a > c + 3d), an upper limit on the marginal taxation-emission rate has to be imposed in order to get consistent results (otherwise the firm that innovates first wants to abate more than it pollutes, due to the constant returns to scale of the abatement technology).

Combining conditions A.3 and A.4, we get the feasibility region for the marginal taxation/emission rate:

A(O) = { (OA3, OA4] if a> c + 3d; ( 0 A3, 00 ) otherwise

2. Consumers' surplus and welfare under alternative technological configurations

Let us first consider the case in which no taxation is imposed by the gov­ernment: No taxation and DD-technology (DD).

136 C. Carrara and G. Topa

w _ 2(a - c)2(2/3 - Ak2) DD - 9/32 .

Notice that both CS'bD and WEw are decreasing functions of A: in partic­ular, CSDD > 0 iff A < /3/k2, whereas WDD > 0 iff A < 2/3/k2. Therefore, higher levels of the shadow price of environmental quality reduce consumer surplus and total welfare, through the increasing impact of damages from emissions.

Let us introduce the environmental tax. Three cases have to be studied: in the first one, neither firm changes technology, in the second one both firms adopt the cleaner technology, whereas in the third only one firm changes technology: Emission Tax and DD-technology (DD/t).

W t _ (a - c)2[4/3 + k2(30 - 2A)] DD - 9/3,2

Emission Tax and FF-technology (FF/t).

t [(a-c-d)2 (2d)2 ] CSFF = 2 9/3 + 30k . (0 - A) ;

W t = [4(a - c - d)2 (2d )2 . (30 _ 2A)] FF 9/3 + 30k .

Emission Tax and FD-technology (FD/t).

cst [Ok2(a - c) - 2/3'd]2 (a - c)[Ok2(a - c) - 2/3'd] - + + FD - 32/3/3,2 6/3,2

2/3(a - c)2 (0 - A)[Ok2(a - c) + 6/3'dj2 + 9/3,2 + 72(/3'Ok)2 ;

W;'D = CSJ.,D(O) + cI>~D(O).

3. Proof of Theorem 2

We proceed as follows: first, we prove a Lemma which shows that 0 = 2A/3 maximizes W t . Then, we compare W t (0) with WO, in order to determine whether or not the governments have an incentive to tax emissions. This comparison shows that wt(O) > WO holds in the interval (Af,A~4]' It is

Taxation and Environmental Innovation 137

then necessary to compare this interval with A(O) (expressed in terms of A), for the solution to be feasible.

Lemma A.I: If A.3 and A.4 hold, and firms are not myopic, the in­tertempoml joint welfare wt is maximized by 0 = 2A/3.

Proof. If A.3 and A.4 hold, the innovation game between the two firms results in both firms adopting technology F at different dates. Suppose further that firms are not myopic, Le., the discount factor 8 = 1/(1 + r) is sufficiently close to one. As the time horizon is infinite, this means that the maximization of W t boils down to the maximization of Wj,.F' which is the instantaneous welfare once both firms innovate. Therefore, o = argmaxOWj,.F = argmaxo{(60 - 2A)/02} is determined by the first­order condition: (2A - 30)/03 = O. This gives 0 = 2A/3 which is the unique candidate to be an absolute maximum of Wj,. F (it is actually a maximum since the second derivative is negative at 0 = 2A/3.) •

We still have to prove that there exists a non-empty interval of A such that wt(O) is greater than W O, Le., the two governments have an incentive to introduce the emission tax. Using the previous argument, we simply need to compare Wj,.F(O) and WDD. The inequality Wj,.F(O) - WDD > 0 is satisfied for the following values of A :

o < A < Ag U A > Af, (L.1)

where Af has been defined above and Af is given by

(3d{4(a - c) - 2d - J[7(a - c) - 2d](a - c - 2d)}

AC = --~----------~77--~~------------~ o - 2k2(a _ c)2 .

However, these two sub-intervals must be compatible with conditions A.3 and A.4 under which the two firms innovate. Once 0 is set at its optimal level 0, A.3 and A.4 impose the following restrictions on A :

(A.3' :)

(A.4' :) { \ < \C - 9(3d 'f 3d " - "A4 = 2k2 (a _ c _ 3d) 1 a > c + ; A :2: 0 otherwise

Assuming a > c + 3d, we get the following ranking:

138 C. Carrara and G. Topa

Combining conditions A.3' and A.4' with the two intervals L.1, we obtain: W}F > WDD if and only if>. E (>'f, >.f). •

References

[1] Baron, D.P., Regulation of prices and pollution under incomplete in­formation, Journal of Public Economics, 28, pp. 211-231, (1985).

[2] Baron, D.P., "Design of Regulatory Mechanisms and Institutions," in: Handbook of Industrial Organization, R. Schmalensee and R.D. Willig, eds., North Holland, Amsterdam, {1989}.

[3] Carraro, C. and Siniscalco, D., Strategies for the international protec­tion of the environment, Journal of Public Economics, 52, pp. 345-354, (1993a).

[4] Carraro, C. and Siniscalco, D., "Policy Coordination for Sustainabil­ity: Commitments, Transfers, and Linked Negotiations," in: The Eco­nomics of Sustainable Development, 1. Goldin and A. Winters, eds., Cambridge University Press, {1993b}.

[5] Carraro, C. and Soubeyran, A., "Environmental Taxation, Market Share and Profits in Oligopoly," in: Environmental Policy and Mar­ket Structure, C. Carraro, Y. Katsoulacos, and A. Xepapadeas, eds., Kluwer Academic Publishers, {1993}.

[6] Carraro, C. and Topa, G., "Should Environmental Innovation Pol­icy Be Internationally Coordinated?," in: Trade, Innovation, Envi­ronment, C. Carraro, ed., Kluwer Academic Publishers, Dordrecht, {1994}.

[7] Demski, J.S. and Sappington, D., Optimal incentive contracts with multiple agents, Journal of Economic Theory, 33, pp. 152-171, {1983}.

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Taxation and Environmental Innovation 139

[11] Ma, C.T., Moore, J., and Turnbull, J., Stopping agents from cheating, Journal of Economic Theory, 46, pp. 355-372, (1988).

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Department of Economics, University of Venice, GRETA, and Fondazione E. Mattei

Department of Economics, University of Chicago

Environmental Quality, Public Finance and Sustainable Growth

Jenny E. Ligthart and Frederick van der Ploeg!

Abstract

Theories of endogenous growth are extended to deal with the op­timal trade-off between economic growth and environmental quality in a meaningful fashion. Environmental quality is modelled in two different ways: (i) as a given stock which is damaged by a flow of pollution; and (ii) as a renewable resource which is used as an input in production. After a brief discussion of pollution, taxation and the cost of public funds, attention is focused on renewable resources in order to come to grips with the concept of sustainable growth. The government reduces the use of natural resources and improves envi­ronmental quality by imposing a levy on firms. Economic growth is boosted by productive government spending, but is hampered by dis­tortionary taxes on income or capital. The first-best outcome can be sustained in a competitive market economy, only if lump-sum taxes and subsidies are available. In general, this is not the case so that the paper focuses on the setting of government policies in a second-best context. Keywords: Sustainability, endogenous growth, renewable resources, productive government spending, distortionary taxes, resource taxes, public finance, cost of public funds, second best

JEL code: E60, H21, H41, Q28

1. Introduction

The pressing problem of environmental degradation raises the question of the ecological sustainability of continued economic growth. Are eco­nomic growth and maintenance of environmental quality compatible in the long run? Some pessimists claim that zero growth is necessary to avoid a "doomsday" scenario, i.e., these people argue that zero growth is a nec­essary (and probably not a sufficient) condition for preserving the natural environment and thus the quality (and, perhaps, even the existence) of life on earth. More optimistic observers argue that positive rates of economic

lThe authors are grateful to Lans Bovenberg, Caspar van Ewijk, Sjak Smulders, and Paul Tang for helpful discussions on earlier drafts of this paper.

142 J.E. Ligthart and F. van der Ploeg

growth may be compatible with the survival of our natural environment in which case one can speak of (ecological) sustainable growth. The issue of sustainability is not a very recent one but goes back to the "Limits to Growth" debate of the early 1970s initiated by Meadows and others. From then onwards the number of formal definitions of sustainability that ap­peared in the literature is mushrooming, each stressing a different aspect of sustainability (see Pezzey, 1989, for an overview). The idea of sustainable growth forms part of the much broader concept of sustainable development. The report "Our Common FUture" better known as the Brundtland Report, defines sustainable development in the following way:

Sustainable development is a process of change in which the ex­ploitation of resources, the direction of investments, the orienta­tion of technological development and institutional changes are made consistent with future as well as present needs. (WCED, 1987, p. 43).

The big question is how to make this definition operational. Here we do this by noting that the use of environmental resources is essential for the production process and thus for consumption and welfare. An operational definition of sustainable growth is then that the level of economic growth in production which ensures that "doomsday", in the sense that all natural resources are completely depleted, never occurs. Sustainable growth thus corresponds to any rate of economic growth that is compatible with a stable quality of the natural environment.

The existing literature on economic growth and the environment has almost exclusively dealt with environmental issues in neoclassical growth models (e.g., Forster, 1973; Keeler et al., 1971; Gruver, 1976; Brock, 1977; Becker, 1982; Van der Ploeg and Withagen, 1991; Tahvonen and Kuulu­vainen, 1991). The social optimal level of capital accumulation is deter­mined in a neoclassical framework of economic growth and pollution. In this literature capital accumulation and production inevitably generate pol­lution which causes a disutility to consumers. Pollution can be cleaned up by devoting resources to public abatement, naturally implying that less resources are available for private consumption. Forster (1973) concludes that the social optimum results in a lower capital intensity and a lower con­sumption level than under the Modified Golden Rule. However, in such a neoclassical context of traditional growth theory, the long-run growth rate is unaffected by pollution control. Van der Ploeg and Withagen (1991) ex­tend Forster (1973) and introduce Pigovian taxes (or equivalently the price of pollution permits) in the familiar Ramsey model in order to internalise environmental externalities. They conclude that the levy of Pigovian taxes results in a smaller capital stock and a lower consumption level than in a competitive market economy without government intervention. Tahvonen

Environmental Quality, Public Finance and Sustainable Growth 143

and Kuuluvainen (1991) study economic growth and pollution when renew­able natural resources are a factor of production. Here pollution negatively affects the reproductive capacity of the renewable resource.

The traditional neoclassical model of economic growth, as developed by Solow (1956) and others, is, however, not well suited to study the influence of environmental quality on the growth rate of output and consumption. In the steady state the growth rate of the economy is solely determined by exogenous factors, namely the sum of the rates of population growth and (labour-augmenting) technological progress. This crucial feature derives from the assumption of diminishing returns with respect to the accumula­tion of capital. As more and more capital is accumulated, less and less extra returns are generated implying that the rate of economic growth ultimately ceases, unless the quantity of non-produced inputs (e.g., labour, natural re­sources) continues to rise. The growth of non-reproducible factors is, in the traditional theories of economic growth, governed by exogenous forces (Le., a time trend). This compensates for diminishing returns and ensures that growth does not peter out. Thus, rather than explaining the rate of eco­nomic growth, the traditional theory assumes a long-run growth path, thus offering no scope for policy and environmental effects on long-run growth rates. This is why the traditional neoclassical theories of economic growth find it hard to come to grips with the crucial issue of sustainable growth.

A much more promising explanation of the rate of economic growth is provided by the recently developed theory of endogenous growth, often referred to as the new growth theory (see Van der Ploeg and Tang (1993), for a survey). In this theory the steady-state growth rate of the economy is no longer taken as given, but is endogenised to allow for the influence of economic variables (R&D, knowledge spill-overs, time preference, etcetera) and government policies (distortionary taxes, public spending on infras­tructure, etcetera). Due to the assumed absence of diminishing returns to a very broad measure of capital (including among other items knowledge, R&D, physical capital and infrastructure), the economy does not converge to an exogenous natural rate of economic growth but can grow forever. It is then reasonable to assume constant or even increasing returns, which generate an endogenous long-run rate of growth.

To really address the issue of sustainable growth properly, it is essen­tial to make use of the recent literature on endogenous growth. The main objective of this paper is thus to extend the new theories of growth to allow for the use of renewable resources as a factor of production and thus to give meaning to the concept of sustainable growth2 • This allows us to pose the question and analyse the optimal trade-off between economic growth, on

2Gradus and Smulders (1992) also build on the new theories of endogenous growth to analyse environmental issues However, they do not allow for renewable resources and thus cannot properly deal with the concept of sustainable growth.

144 J.E. Ligthart and F. van der Ploeg

the one hand, and environmental quality on the other hand. Attention is focused on how government policy should be set to ensure sustainability of economic growth in a world in which productive government spending, cap­ital accumulation and use of renewable resources in the production process give rise to a variety of (negative and positive) external effects.

Our endogenous growth model with renewable resources builds on the work of Arrow (1962), Barro (1990), Barro and Sala-i-Martin (1992), Romer (1986, 1987, 1990). We allow for knowledge spill-overs in production and in­clude productive government spending on the material or immaterial public infrastructure as a factor of production. The main idea is that government spending on infrastructure (e.g., schooling, R&D, highways and railways) increases the productivity of private capital and that this fully compen­sates for the diminishing returns induced by private capital accumulation. However, if government spending is financed by distortionary taxes, the government faces a trade-off between the negative impact on the rate of economic growth of distortionary taxes and the positive effect of produc­tive government spending on the growth rate.

Producers can invest in knowledge capital which has the character of a public good. In the absence of patent markets, accumulation of knowledge capital generates a positive externality. Investment programs undertaken by one firm expand the economy-wide knowledge base from which other firms can freely benefit. Thus, from a social perspective the benefits of investment are larger than the private benefits. Because firms do not inter­nalise these positive knowledge spill-overs, investments in knowledge capital are below the socially optimal level. The government can bring these bene­fits in line by providing a Pigovian subsidy on private investments in R&D, training, etcetera. Hence, not paying any attention to environmental issues leads one to conclude that in the absence of government intervention there is a too low rate of economic growth.

Environmental externalities, however, work in the opposite direction. The environment generally affects consumption as well as production. Con­sumers derive positive utility from consumption goods and environmental quality (proxied by the stock of renewable resources), while the harvest of renewable resources is used as a factor of production. There is a prob­lem of the common in the sense that each firm perceives the resource as a freely available production factor. Competition among firms may lead to over-exploitation of the common-property renewable resource causing a deterioration of environmental quality and lower social welfare. The mar­ket typically underestimates the social value of natural resources and the quality of the natural environment. Since producers pay too Iowa price for natural resource use in a competitive market economy, producers use too much of these resources and threaten sustainable development. In other words, environmental externalities imply that a competitive market econ-

Environmental Quality, Public Finance and Sustainable Growth 145

omy will grow too fast relative to the socially optimal outcome. The role of the government is then to levy a charge on the use of natural resources, so that private costs reflect social costs. The present paper focuses thus on two missing markets. Not only is there no market for natural resources, but there is no patent market either. Our model with renewable resources is closely related to papers by Bovenberg and Smulders (1993) and Van der Ploeg and Ligthart (1994). Both papers focus on the issue of sustainability in an endogenous growth model with renewable resources. Bovenberg and Smulders (1993) have a two-sector framework with a final good sector and a learning sector aimed at developing pollution-saving technology. They find that a shift towards greener preferences harms economic growth, increases the knowledge intensity of production (more pollution saving) and reduces the real interest rate when environmental quality enters utility but does not have a productive role. However, in contrast, when the environment is used as an input in production, the rate of economic growth and the real rate of interest may rise in reaction to a more ambitious environmental policy. Van der Ploeg and Ligthart (1994) study the desirability of international policy coordination in a two-country endogenous growth model in which there are international environmental externalities and international knowledge spill­overs (including positive spill-overs associated with productive government spending). It is shown that international policy coordination leads to an improvement in environmental quality and a lower rate of economic growth if the role of the knowledge spill-overs is insignificant. However, coordi­nating economic and environmental policies may harm the environment if there are, besides environmental externalities, large knowledge spill-overs. Our current paper is more concerned with productive government spend­ing as a growth catalyst and the use of natural resource levies as a way to protect environmental quality, and complements the results of Bovenberg and Smulders (1993) and Van der Ploeg and Ligthart (1994).

The command economy outcome in which the government directly al­locates resources yields a first-best outcome and is our point of reference. An important policy question is whether the first-best outcome can be sus­tained in a market economy. Two situations are considered, namely the case in which lump-sum taxes and subsidies are available to finance govern­ment spending and a situation in which they are not. In the former case, the command-optimum outcome can be sustained in a market economy if a suitable combination of Pigovian taxes and subsidies is employed.

When the government cannot use lump-sum taxes and subsidies, gov­ernment spending generally needs to be financed by an optimal mix of dis­tortionary taxes. Clearly, the public revenues obtained by Pigovian taxes on natural resource use are only by coincidence equal to the revenues required to finance productive government spending and capital subsidies. Thus, distortionary taxes have the dual task of raising revenue as well as cor-

146 J.E. Ligthart and F. van der Ploeg

recting environmental and production externalities which brings us within the domain of second-best economics. To minimise the excess burden of distortionary taxation, the government should choose an appropriate mix of subsidies on capital and levies on the use of natural resources.

Before we move on to study public finance in a model with renewable resources, we first look at second-best issues in a simple model in which a flow of pollution damages environmental quality. This section extends the work of Gradus and Smulders (1993) and Van Marrewijk et al. (1993), who study the relation between environmental protection and long-term growth in an endogenous growth model with pollution as a by-product of production, to allow for the effects of more environmental concern on the marginal costs of public funds and the national income share of government consumption.

The paper is structured as follows. Section 2 presents a simple model of endogenous growth with pollution as an inevitable by-product of pro­duction which causes disutility to consumers. In particular, we investigate the relationship between greener preferences, the rate of economic growth and the marginal cost of public funds. Section 3 describes a model in which renewable resources are used as input in production. We determine the first­best optimum and it is shown that the first-best outcome can be replicated in a competitive market economy given that the set of instruments of the government consists of lump-sum taxes and subsidies as well as Pigovian taxes and subsidies. We elaborate on the concept of sustainable growth and the transitional dynamics arising from shocks in preferences, technol­ogy and biological parameters. We show that sustainable growth implies sustainable use of renewable resources which means that the harvest rate of resources for production purposes matches the biological regeneration rate. Section 4 extends the model of Section 3 to allow for the case in which the government can only use distortionary taxes to finance public goods and is thus concerned with second-best economics. Section 5 concludes the paper.

2. Pollution, Public Finance and Endogenous Growth

In this section we study a simple endogenous growth model with a flow of pollution and unproductive government spending, Le., public consump­tion. We assume that agents live in a second-best world, which implies that lump-sum taxes and subsidies cannot be called upon to balance the gov­ernment budget. This brings us into the realms of public finance, Le., the setting of an optimal mix of distortionary tax rates to finance public goods (cf. Atkinson and Stiglitz, 1980; Auerbach, 1985; Bovenberg and van der Ploeg, 1994a, 1994b). We extend the work of Barro (1990) on endogenous growth and public finance to allow for the effects of environmental policy

Environmental Quality, Public Finance and Sustainable Growth 147

on economic growth. Van Marrewijk et al. (1993) already did some work in this direction, but did not consider the crucial role of the marginal cost of public funds in determining the optimal policy.

2.1 Preferences and production

Consider an economy with a large number of competitive firms which pro­duce a homogeneous final good. The production function of the represen­tative firm has the following very simple form:

(1)

where Yi and Ki denote output and capital of firm i, respectively, and A denotes economy-wide production efficiency. Production is characterised by constant returns to scale at the level of the individual firm as well as at the macro level. Firms maximise profits under perfect competition.

Omitting the wage bill, the instantaneous profits of the firm are given by

1000 [(1- r)Yi - KiJ exp[-18 r(s')ds'Jds, (2)

where Ki is investment undertaken by the firm and r is the market rate of interest. A dot above a variable denotes a time derivative. For simplicity, we ignore depreciation. The first-order condition for a maximum is

r = (1- r)A. (3)

Utility of household j (Uj) is, for simplicity, given by a quasi-linear specification:

Uj = 1000 [log[Cj(t)J +"'G log[G(t)J + "'EE(t)J exp( -(jt)dt, (4)

where Cj , G and E stand for private consumption, public consumption and environmental quality, respectively. Society attaches a weight "'G to public consumption and a weight "'E to environmental quality. Note that social welfare depends on private goods and the basket of publicly provided goods which consist of public consumption and environmental quality. Con­sumers have a rate of time preference (j and the elasticity of intertemporal substitution for private and public consumption is unity.

Household j takes the levels of public consumption and environmental quality as given, and maximises its utility (4) subject to its present value budget constraint:

100 cj (s)exp[-18 r(s')ds',Jds = Vj(t) + Hj(t), (5)

148 J.E. Ligthart and F. van der Ploeg

where Vj and Hj stand for non-human wealth and human wealth (the present value of after-tax wage income) of household j, respectively. House­holds maximise utility if the growth in their consumption equals the gap between the market rate of interest and the rate of time preference:

b - =r-O. C

(6)

For the moment, we do not consider renewable resources but assume a given stock of natural resources. Environmental quality is thus given by:

E=Eo-DY, (7)

where Eo is the initial level of environmental quality and D is the emission­output ratio. The level of environmental quality is negatively affected by the flow of pollution which is an inevitable by-product of production (e.g. noise, smoke, etcetera). Pollution is a negative external effect. Individual firms are too small to care about the pollution they generate. Of course, this implies that the decentralised market equilibrium is inefficient. Gov­ernment policy in the form of a tax on output is called upon to correct for environmental externalities. We assume that the government does not undertake abatement policies to clean up pollution.

2.2 The government budget and market equilibrium

The government finances its consumptive expenditures (G) with a tax on production (r):

G=rY. (8)

Note that, a tax on output acts as an implicit tax on capital so that similar results are obtained if the tax is replaced by a tax on capital. We assume that lump-sum taxes and subsidies are not available to ensure a balanced budget. Thus, the government has to employ taxes for the dual task of, on one hand, raising revenues to finance public consumption, and, on the other, internalising environmental externalities. Goods market equilibrium requires that the supply of goods equals the total demand for goods

Y= C+1 +G= C+K +G. (9)

2.3 Optimal public policy

From the goods market equilibrium condition, the government budget con­straint and the production function, we derive the growth rate of capital:

k C 7rK= K=(I-r)A- K· (10)

Environmental Quality, Public Finance and Sustainable Growth 149

Combining equations (3) and (6) gives the growth rate of private consump­tion

C 7re == 0 = (1 - r)A - 0, (11)

where the output-capital ratio is constant (y = Y/K = A). The rate of growth of private consumption is low if the tax rate is high and households are impatient (high 0) and is high if there is a high production efficiency of the economy. For balanced growth it is necessary that the capital stock and consumption grow at the same rate, i.e., 7rK = tre = 7r. This condition is satisfied if the consumption-capital ratio immediately jumps to the pure rate of time preference (c = 0/ K = 0). There are no transitional dynam­ics. Note that balanced growth implies a corresponding rate of growth in environmental damages.

To determine the optimal policy, the government maximises the social welfare function3 ,

u _ (1 + 1Ja)7r 1Ja log(r) _ 1JEDAKo - 02 + 0 0 - 7r + constant, (12)

with respect to the tax rate (r), subject to the Keynes-Ramsey rule

7r = (1 - r)A - O. (13)

This gives rise to the first-order condition

( 1 + 1Ja 1JE ) 1Ja - ~- (0_7r)2 DAKo A+ rO =0. (14)

We solve (14) for the modified Samuelson rule

'" 1JaO 01Ja ( 0 ) 2 ~ MRS == G = r A = MCPF == 1 + 1Ja -1JE 0 _ 7r DAKo, (15)

or alternatively, the optimal tax rule

Equation (15) corresponds to the modified Samuelson rule. This rule says that (the sum of) the marginal rate(s) of substitution between public con­sumption and private consumption, i.e., the ratio of the marginal social utility of public consumption (1Ja/G) and the marginal value of private in­come (1/0), should equal the marginal cost of public funds (MCPF). The

3For utility to be bounded, it is necessary to assume that the rate of time preference exceeds the rate of economic growth, i.e., B - 7r > o.

150 J.E. Ligthart and F. van der Ploeg

marginal cost of public funds is unity if "IE = 0 and TJG = O. Clearly, if the role of environmental externalities and public consumption is negligi­ble, there are no distortions or externalities present in the economy and the first-best outcome results. Of course, the optimal tax rate is zero in this case. A positive weight to public consumption (TJG > 0) raises the marginal cost of public funds above unity4, indicating that public goods are more costly than private goods. This distortion induces substitution away from public towards private consumption, thereby depressing the tax rate. How­ever, a positive weight to environmental quality ("IE > 0) depresses the marginal cost of public funds below unity (if TJG is sufficiently small), as taxation depresses growth in environmental damages and thus boosts the supply of environmental quality, thereby inducing substitution away from private towards public consumption.

Consider first the case without environmental externalities ("IE = 0). Given that () > T A, if more priority is given to public consumption (higher TJG), the national income share of public consumption and thus the optimal tax rate rises. This depresses the rate of economic growth. However, there is also a rise in the MCPF which depresses the demand for public consumption goods and thus attenuates the rise in the tax rate. A higher productivity of capital or more patient households (lower ()) leads to a lower optimal tax rate, a lower share of public consumption and a higher rate of economic growth.

The general case in which environmental externalities are present ("IE> 0) is depicted in Figure 1. The first equality of (15) corresponds to a negatively sloped demand curve (DD), which implies that a rise in the marginal cost of public funds leads to a decline in the demand for public consumption. Hence, the optimal tax rate and the national income share of public consumption fall. The last equality of (15) corresponds to a pos­itively sloped cost curve (CC), which implies that a rise in the tax rate raises the marginal cost of public funds. A higher tax on output depresses the rate of capital accumulation and economic growth. This depresses the tax base and raises the cost of public funds.

Consider a shift towards greener preferences ("IE rises). More concern with environmental damages shifts the cost curve (CC) downwards, thereby shifting the equilibrium from E to E'. The marginal cost of funds declines and the optimal tax rate rises, which in turn leads to a bigger share of public consumption in national income, less capital accumulation and a lower rate of economic growth. The decline in the rate of economic growth reduces the rate of growth in pollution damages, which improves environmental quality.

4This is the case if "Ia > "IE(9/(9 _11'»2 DAKo.

Environmental Quality, Public Finance and Sustainable Growth 151

c MCPF

c·

MCPF'

o

c C·

g' g=C/Y

Figure 1.

Now, consider what happens if society attaches less priority to public consumption ('f'/G declines). Both the demand curve (DD) and cost curve (CC) are shifted downward, thereby shifting the equilibrium from E to E". The decline in demand for public goods makes them relatively cheap which depresses the marginal cost of public funds. If the initial tax rate is not too large (i.e., T < ()/A), the optimal tax falls5 and the growth rates of the economy and pollution damages rise. In this sense, there is less provision of the two public goods, viz. public consumption and environmental quality.

2.4 Growth-promoting infrastructure versus public consumption and the environment

In this section we extend the analysis to allow for productive government spending as a growth catalyst. The production structure of the individual firm is given by a Cobb-Douglas specification (cf. Barro, 1990):

Yi = K i1- fJ (LiS)fJ . (17)

At the macro level this yields

Y=AK, ( S)~ A= -Y

(18)

51f TA > 8, the downward shift in the demand curve (DD) dominates the downward shift in the cost curve (CC) so that a reduction in ""0 induces a fall in T. Conversely, if () < T A, a fall in ""0 depresses T.

152 J.E. Ligthart and F. van der Ploeg

At the firm level there are decreasing returns to capital and constant re­tuns to scale in all individual inputs. However, at the macro-level there are increasing returns to scale, but constant returns in capital (K) and productive government spending (S). Labour supply is exogenous and can without loss of generality be ignored.

The government finances total public spending, which consists of pro­ductive government spending and public consumption by distortionary taxes on output:

G+S=7Y, (19)

with 9 == GjY and s == SjY and 7 == 9 + s. The goods market equilibrium with productive government spending is as follows:

Y=C+G+S+K. (20)

From the optimisation problem of the household and that of the firm, we derive the growth rate of private consumption:

6 ~ 7rc = C = (1- 7)A - (I = (1 - 7)(1 - {3)s>=iJ - (I. (21)

Thus, the growth rate of consumption is boosted by a higher national in­come share of productive government spending, whilst a higher national income share of public spending depresses the economic growth rate. In a balanced growth equilibrium capital, output, private consumption, public consumption, productive government spending and environmental damages all grow at the common rate 7r.

The optimisation problem for the government is to maximise social welfare

(l+1Je)7r log[((I+{3(1-7)s~)Ko] u= (12 + (I

+1Je log [(7 - s)s~] 1JEDs~ Ko

(I (I-7r (22)

subject to the growth rate of private consumption (21), where use has been made of the government budget constraint 9 = 7 - s. The first-order condition for the tax rate 7 yields

1JeC C1Je = = G gA

LMRS ==

MCPF (1 - {3) [1 + 1Je -1JE ((I ~ 7r) 2 DAKo 1 c

(I + {3, (23)

Environmental Quality, Public Finance and Sustainable Growth 153

and for the national income share of productive government spending (s) we get

(24)

The optimal national income share of productive government spending is less than the production share of productive government spending (/3). Clearly, Barro's (1990) result, i.e., s = /3, is modified, since public con­sumption crowds out productive government spending. In this sense there is a direct trade-off between public consumption and productive government spending. If there is no public consumption (Le., 9 = 0), equation (24) re­duces to s = /3.

Expression (24) and the government budget constraint yield the opti­mal tax rate

(25)

A shift towards greener preferences shifts the cost curve (CC) downwards. Hence, more environmental concern implies a higher tax rate (g and r both rise) which depresses the marginal cost of public funds and causes a shift from growth-promoting public spending towards public consumption. The shift from productive government spending to unproductive government spending leads to a decline of the economy's rate of growth and a consequent reduction in pollution. Thus, greener preferences lower both the marginal cost of public funds and the rate of growth of the economy.

3. Renewable Resources and Sustainable Growth: The First-Best Outcome

3.1 Renewable resources, productive government spending and spill-overs in production

Consider a closed economy in which firm i uses labour L i , capital K i ,

and renewable resources Ni (e.g. forests, grazing land, air, etcetera) to produce a homogeneous good Yi under perfect competition. The production structure of firm i is given by a Cobb-Douglas specification:

= AK~ K 1- o -{3 S{3 L~-o-'"Y N"! 'I. 'I. 2.'

a,/3,'Y> 0, a + /3 < 1, 'Y < 1- a, 'Y < 1 - /3, (26)

where K stands for the economy-wide capital stock. Production is charac­terised by decreasing returns with respect to capital at the firm level. At the aggregate level there are constant returns in capital and public spending, but increasing returns to scale in all factor inputs. The idea is that capital

154 J.E. Ligthart and F. van der Ploeg

is very broadly measured so as to include not only the stock of physical capital, but also the stock of ideas, R&D and human capital. Firms can thus also augment human capital through training and research. There are no effective patent markets, so that not all returns on investment in R&D, etc. can be internalised by individual firms. This shows up in the produc­tion function, which allows for spill-overs in production (Le., 0: + (3 < 1). Hence, the marginal productivity of capital of a particular firm is boosted by the stock of capital accumulated by other firms in the economy. In addition, the marginal productivity of private capital is boosted when the government allocates more resources to infrastructural goods (think of pub­lic R&D, public training programs, the legal system, roads, etcetera). Firms maximise profits which are given by

IIi = Y.: - WLi - (r + 8 + T)Ki - QNi, (27)

where w, 8, r+8+T and Q stand for the wage rate, the depreciation rate, the user cost of capital and the cost of (Le., the tax charged by the government on the use of) natural resources, respectively. The charge Q may also be interpreted as the price firms have to pay for pollution permits on the open market. In equilibrium profits are zero, because each firm faces constant returns in their own use of capital, employment and renewable resources. A positive share of labour requires that 'Y < 1 - 0:.

3.2 Household preferences and the quality of the environment

For simplicity, we ignore from now on the role of public consumption (G) in social welfare (Le., we set 'f/e = ° and drop public consumption G):

Uj = 100 (log(Cj(t)) + 'f/EE(t)) exp( -()t)dt, () > 0, 'f/E ;?: 0, (28)

Environmental quality (proxied by the stock of renewable resources) de­teriorates as more resources are used for productive purposes by firms in the economy, but in the absence of resource exhaustion the environment naturally rejuvenates itself according to a logistic growth function (e.g. Dasgupta and Heal, 1979):

E = ¢E(B - E) - N, ° < ¢ < (), B2 > 4N/¢. (29)

where B is a constant which stands for the carrying capacity of the popula­tion (Le., the renewable resource stock) in the absence of resource depletion. This specification allows for saturation in the growth of natural resources, that is the population increment (Le., the sustainable resource use) in­creases in the range (0, ~ B) while it decreases in the interval (~B, B). The

Environmental Quality, Public Finance and Sustainable Growth 155

maximum sustainable resource use (iB2ct» is thus attained when E = !B. For example, in a sea with a lot of plankton the fish population can grow easily because there is ample food available. However, as the fish population grows bigger, less plankton will be available for each individual fish. The special case of exhaustible natural resources corresponds to ct> = O. There is a problem of the common in the sense that as firms use more natural resources for productive purposes, the quality of the natural environment worsens and social welfare falls. Of course, use of natural resources also boosts production and consumption of marketable goods which increases social welfare.

Household j maximises utility (28) subject to its intertemporal budget constraint:

100 Cj(s) exp (-18

r(s')ds') ds = Vj(t) + Hj(t). (30)

Human wealth is the present value of after-tax wage income, that is

Hj(t) == 100 (w(s) - Tj(s)) exp (-18

r(s')ds') ds, (31)

where T j denotes lump-sum transfers from the government to household j. The budget constraint (30) assumes that the household cannot play Ponzi­games, i.e., its (non-human) debt cannot grow at a faster rate than the market rate of interest. Labour supply of each household is inelastic and fixed at unity (Lj = 1).

3.3 The government budget and market equilibrium

The government finances productive public spending (S) and (net) transfers to households (T) with a tax on capital (T) and a tax on natural resources (Q) and does not issue debt. One obtains similar results if one replaces the capital tax with a tax on output, but this is a bit more cumbersome as a tax on output acts as an implicit tax on both the use of capital and the use of natural resources. The government budget constraint is thus:

S+T=TK+QN. (32)

Labour market equilibrium requires L = 1. Capital market equilibrium requires that, as there is no government debt, V = K. We consider a closed economy, so that goods market equilibrium requires that the supply of goods equals demand for goods:

y = C + I + S = C + k + oK + S. (33)

156 J.E. Ligthart and F. van der Ploeg

3.4 Optimality conditions

Before competitive market outcomes are discussed, we analyse the first­best outcome. This is the outcome for a command economy in which the government can allocate resources directly and does not have to rely on tax instruments to influence the actions of private agents. It follows that the government internalises any externalities that result from spill-overs in production and the deterioration of environmental quality. Also, there are no excess burdens of taxation to consider. The social planner maximises social welfare (28) subject to the economy's budget constraint (33) and the resource constraint (29). This leads to the following Hamiltonian function:

H == log { C) + 'flEE + >. (AK1-.8 S.8 N'Y - C - 8K - S)

+JL (¢E{B - E) - N), (34)

where >. stands for the social value of capital and JL stands for the social value of the environment.

Necessary conditions for optimality are

He = Hs = HN = 0, HK = 0>' - '\, HE = OJL - p, (35)

where subscripts are partial derivatives of the Hamiltonian function. The transversality conditions associated with the backward-looking variables, K and E are

lim >.(t)K(t)e-Ot = 0, t-+oo

lim JL(t)E{t)e-Ot = o. t-+oo

(36)

(37)

From the first-order conditions it follows that the optimal national income share of productive government spending is given by s == SlY = f3 and the optimal level of private consumption by C = 1/>.. The growth rate of private consumption (7re) and the growth rate of the capital stock (7rK) are as follows:

where the output-capital ratio is given by

y == ~ = A (~).8 N'Y = (AN'Ys.8) ~ = (AN'Yf3.8) ~ . (40)

Equations (38) and (39) give rise to c = (r - O)c with c == C/K. There is balanced growth if private consumption, capital and output grow at the

Environmental Quality, Public Finance and Sustainable Growth 157

same rate, namely 'Ire = 'irK = 'lry == 'Ir. Clearly, to satisfy this condi­tion, c == C / K must immediately jump to the pure rate of time preference «(}) even if the output-capital ratio (y) changes over time. The rate of economic growth is high if the national income shares of productive gov­ernment spending and use of natural resources are high, households are patient (low ()), and the depreciation rate (8) is low. There is hysteresis in the sense that a society which starts off with a low initial capital stock ends up with a low level of private consumption.

The optimal use of renewable natural resources is given by:

(41)

with s = {3. The social planner allocates more natural resources to pro­duction if productive efficiency is high (due to high A or high s = (3), households are patient, or the social value of environmental quality (J-L) is low. Equations (29) and (41) give rise to:

(( )1-13 )~

E = 4>E(B - E) - (): (As13) , (42)

where the optimal national income share of productive government spend­ing is {3.

Given the assumption of constant returns to scale with respect to econ­omywide capital, the economy displays ever-increasing growth in output when there is growth in the use of natural resources. Clearly, this vio­lates Kaldor's stylised fact of a relatively stable capital-output ratio. It is also incompatible with sustainable growth as the quality of the environment must then eventually be completely destroyed. It follows that sustainable growth is only feasible if there is a steady level of the use of natural re­sources. This implies that the output-capital ratio is, in line with Kaldor's (1961) stylised facts, constant (see equation (40» while output, capital, government spending and national income grow at the same rate on a bal­anced growth path. Sustainable growth thus implies a steady level of the quality of the natural environment, so that use of natural resources in the production process (Le., the harvest rate) exactly matches natural resource rejuvenation (N = 4>E(B - E».

The social (money) value of environmental quality ('TIE/J-L) must equal the rate of time preference «(}) plus the natural depreciation rate of the stock of natural resources (4)(2E - B» minus the appreciation in the social value of the environment (Le., fJ,/J-L), hence:

I!. = () + 4>(2E _ B) _ 'TIE. J-L J-L

(43)

158 J.E. Ligthart and F. van der Ploeg

3.5 'I'ransitional dynamics

The socially optimal outcome follows from the state-space system (42)­(43) and is shown in Figure 2. Environmental quality is a predetermined variable, whereas the social value of environmental quality (and the use of natural resources) is a non-predetermined variable. There are two equi­libria: an unstable steady state (U) and a saddlepoint (E). The steady state E is a saddlepoint, because the determinant (6.) of the Jacobian of (42)-(43) at this point is negative (see Appendix):

N

6.=_('fIE¢)(2E-B)-2¢( I-f3 )N<O. (44) J.L I-f3-,

til)

IJO- f

'"""'=-------s· " - ,'- -------- - - - - - - - - - - - - - -- - --.':"--:>-:--~--~~'-'-- LtO-Iol'

. '.

" .' " ,

ISO-,M.

Figure 2.

The locus describing sustainable growth, called the iso-E locus (de­scribed by ¢E(B - E) = N), has a quadratic form. At a low level of the renewable resource stock, environmental quality increases permitting a high use of natural resources in production. At high levels of environmental quality the sustainable use of resources declines and the social value of the environment increases.

The locus describing a steady social value of environmental quality, i.e., the iso-J.L locus (described by (0 + ¢(2E - B»J.L = 'fiE), has a negative slope. Note, that:

( dJ.L ) < (dJ.L) < 0 < (dJ.L) . dE p=O dE 88 dE E=O

(45)

Before the transitional dynamics are discussed, it is useful to analyse the comparative statics of the steady state with respect to changes in ¢, e, 'fiE,

Environmental Quality, Public Finance and Sustainable Growth 159

A, Band s. Loglinearisation around the steady state of the state-space system yields:

~4E = (1 - (3)fiE - (311 - A - [(1 - (3)(2E - B)¢>6 - (1 - (3 - 'Y)] ¢ +(1- (3)(1- (6)0 + [(1- (3)¢>B6 + ~2(1- (3 - 'Y)] B, (46)

= _ e - + (~4 + 'Y6)(3) - + (~4 + 'Y6) A _ e (1 _ Oe )0 'Y .. l"'E 1 _ {3 9 1 _ {3 'Y .. l .. 3

+ b¢>6(2E + (2E - B)6)) ¢ + 'Y¢>B66B, (47)

= (1 - (3 - 'Y)6fiE + 2{30E611 + 2¢>E6A - [6(1 - (3 - 'Y)06 + (1 - (3)2¢>E] 0 - ¢>B66(1 - (3 - 'Y)B

-JL(1 - (3 - 'Y)6(2E + (2E - B)6)¢, (48)

where a tilde indicates a logarithmic deviation from the initial steady-state value and

~1 == (2: ~ :) > 0, 6 == (B ~ E) > 0, 6 == ~ = (0 + ¢>(21E _ B)) > 0,

~4 == (1 - (3 - 'Y)6 + 2(1 - (3)¢>E6 > 0.

(49)

(50)

(51)

To calculate the relative change in the output-capital ratio (47) use has been made of:

N = ¢-6E+6B, (52)

(1 - (3)y = A + 'YN = A + 'Y(¢ - ~lE + 6B). (53)

Hence, a lower basic growth parameter of environmental quality (¢» or a lower harvest rate (N) yields in the long run a better environmental quality (E). A high harvest rate boosts productivity and the rate of economic growth (see equation (38)).

An increase in the priority that society attaches to environmental quality relative to private consumption (higher "'E) leaves the sustainable growth locus unaffected, but shifts the iso-JL locus outwards. On impact the social value of environmental quality jumps upwards and it becomes socially optimal to use less natural resources in production (jump from the initial steady state E to P). It follows that on impact the productivity of capital and the growth rate of the economy fall. Over time environmen­tal quality improves gradually so environmental issues become less pressing and the social value of environmental quality falls (move from P along the

160 J.E. Ligthart and F. van der Ploeg

saddlepath SS to the final steady state E'). Natural resource use and economic growth thus gradually rise over time.

An increased concern with environmental quality induces in the short run as well as in the long run lower use of natural resources in production. In the long run environmental quality improves, but is insufficiently upgraded to warrant a greater use of natural resources in production. Hence, the fall in the short-run growth rate is greater than the fall in the long-run growth rate. The private component of social welfare falls while the public component of welfare increases. Note that a natural disaster destroying part of the environment induces exactly the same adjustment as an increase in the concern with environmental issues.

An increase in the carrying capacity of the population (higher B) shifts the iso-E curve downwards to the right and shifts the iso-p, locus to the right. On impact the social value of environmental quality jumps down so that it is socially optimal to use more natural resources in production. This increases the short-run growth rate of the economy. During the ad­justment process the social value of environmental quality declines further, more natural resources are harvested and environmental quality improves. Although the growth rate of the economy is higher and thus more resources are used in production, environmental quality improves in the final steady state. Namely, the higher carrying capacity increases the growth rate of the resource population for a given level of environmental quality and thus allows a higher sustainable resource use in production. Thus, in final steady state a higher growth rate of the economy is compatible with a better en­vironmental quality.

An increase in the efficiency of production arising from an economy­wide shock (higher A) or an increase in the national income share ofproduc­tive government spending (higher s) leaves the iso-p,locus unaffected, but shifts the iso-E locus upwards. The implementation effect of such a shock is more use of natural resources in production and thus a higher growth rate of the economy. The implementation effect is partially offset by the news effect, i.e., an immediate increase in the social value of environmental quality which reduces the use of natural resources in production and thus also the growth rate of the economy. As the economy moves along the sad­dlepath to the final steady state, the social value of environmental quality rises, the use of renewable resources falls and environmental quality deteri­orates. In the long-run equilibrium the output-capital ratio is higher, more renewable resources are used, a higher rate of economic growth is realised and environmental quality is worse than in the initial steady state.

Finally, an increase in the rate of time preference (higher 0) shifts both the iso-E and the iso-p, curve downwards. On impact the level of private consumption increases, more natural resources are used in production and environmental quality worsens. The latter effect leads to an increase in the

Environmental Quality, Public Finance and Sustainable Growth 161

short-run growth rate of the economy even though it is partially offset by the fall in savings. Thus, impatience increases the short run growth rate of the economy while environmental quality worsens (see equations (46)­(47)). In the long run society ends up with a lower rate of economic growth and an improvement in environmental quality. The short-run decline in environmental quality increases the growth rate of the natural resource stock which in addition to the decreased use of natural resources leads to an improvement of environmental quality.

3.6 Can the first-best outcome be sustained in a market economy?

An interesting question to ask is whether the first-best outcome can be sus­tained in a market economy. Two situations should be considered, namely a situation in which lump-sum taxes and subsidies are available to the gov­ernment and a situation in which they are not. Here attention is focused on the situation in which the government can resort to lump-sum taxes and subsidies. The tax rate on capital (T) and the tax rate on the use of nat­ural resources (Q) can be used to change relative prices in order to induce socially optimal behaviour.

Households maximise utility when the growth in their consumption equals:

6 - = r - (), C

(54)

so households postpone consumption and save when the market rate of in­terest exceeds the pure rate of time preference. Using (54) and the intertem­poral budget constraint of the household (30), one obtains the consumption function C = (}{V + H). Because the elasticity of intertemporal substitution is constant (in fact, unity), Engel curves are linear and aggregation across households is trivial.

Each firm maximises profits under perfect competition, hence sets the user cost of its capital, including the tax on capital (r + 8 + T) to the marginal product of its capital:

r+8 +T = ay. (55)

Substitution of (55) into (54) yields the growth rate of private consumption in a market economy:

6 1rc == C = ay - T - 8 - (). (56)

Comparing (56) with (38), it is clear that the after-tax marginal product of private capital (ay - T) may be lower than the social marginal value of cap­ital {{1 - (3)y) for two reasons. Firstly, knowledge spill-overs from one firm

162 J.E. Ligthart and F. van der Ploeg

to another are (in the absence of effective patent markets) not internalised in a market economy (i.e., a < 1- /3). Secondly, the government may need to levy a tax on capital (i.e., 7 > 0) in order to finance productive govern­ment spending which reduces the after-tax marginal product of capital and thus the growth rate of private consumption as well. Of course, if lump­sum taxes are available to finance public spending, the government would like to offer a subsidy on capital (7 < 0) in order to internalise production externalities in which case the second effect may off-set the first effect.

Each firm has to pay the government a levy (i.e., Q) for the use of nat­ural resources. Maximisation of profits requires that firms set the marginal product of natural resources bYIN) equal to the levy. The resulting de­mand for natural resources is in symmetric equilibrium given by:

(57)

Firms use more natural resources when productive efficiency (A), the capital stock or productive government spending is large, and the levy for the use of natural resources is small.

The first-best outcome can be sustained in a market economy if the government sets government policy in the following way:

1. the national income share of productive government spending is set to equal its coefficient in the production function (i.e., s = /3);

2. a subsidy on capital to ensure that the after-tax marginal product of private capital equals the social value of capital and the externalities associated with knowledge-spillovers in the production process are internalised (i.e., -7 = (1 - a - /3)y > 0); and

3. a levy on the use of natural resources is set to equal the social value of the environment scaled by the marginal utility of income (i.e., Q =

JlI>' = JlC = JlOK).

The levy on natural resources (Q) rises over time as the economy grows, because this is the only way in which the government can ensure that the use of natural resources is constant and economic growth is sustainable from an environmental point of view. Note that, instead of a levy on the use of natural resources, Q may just as well be interpreted as the price producers have to pay for pollution permits on the open market. This levy adjusts for the environmental externality while the capital subsidy internalises the knowledge spill-overs in production.

Revenues that are needed to finance the excess of subsidies on capital and productive government spending over and above revenues from taxes on the use of natural resources are obtained by levying lump-sum taxes

Environmental Quality, Public Finance and Sustainable Growth 163

from the private sector (i.e., -T = S - rK - QN). In fact, it is easy to show that the optimal national income share of transfers rises with the share of private capital and of natural resources in output:

T rK+QN-S t == Y = Y = 0: + "y - 1 < o. (58)

Since the importance of the use of renewable resources in production, ("f), is less than the combined effect of knowledge spill-overs and productive government spending in production, ( 1- 0: ), the government levies lump­sum taxes from the private sector (i.e., t > 0).

4. Optimal Taxation in a Second-Best Economy

Now consider a situation in which lump-sum taxes and subsidies cannot be called upon to ensure a balanced government budget. The contribution of Barro (1990) on public finance and endogenous growth and the discussion in Section 2 on public finance and pollution is further extended to allow for the use of renewable resources in the production process.

4.1 Optimality conditions

The government maximises social welfare (28) subject to private be­haviour (54)-(57), the government budget constraint (32) and goods market equilibrium (33). The government budget constraint (32) needs to be con­sidered, because lump-sum subsidies and taxes (T) cannot be used to ensure a balanced government budget. The policy instruments of the government are, in fact, the tax on capital (r), the levy on the use of natural resources (Q) and the level of productive government spending (S).

The current-value Hamiltonian function for the government is defined as:

H = IOg(C)+1}EE+>'((AKl-.BS.B)l~'" (~)~ -C-OK-S)

+1' (~E(B _ E) _ (~AK~-'SP) .~o)

+W ( 0: (AK"Y-.B s.B) l~" (~) ~ - r - 0 - (}) C

+x (rK + ("tAK1-.BS.BQ-"Y) l~'" - s) (59)

where >. stands for the marginal social utility of private income, w de­notes the social value of private consumption and X denotes the marginal

164 J.E. Ligthart and F. van der Ploeg

disutility of having to raise an additional unit of public funds. Necessary first-order conditions for the government are:

He = Ow+w= ~ -A+W(~)' (60)

Ha = (1 ~ 7) (~) (1 - (~) + a (~~) + 7 (~) ) -(A+X) = 0, (61)

H-r = XK-wC=O, (62)

HQ = -C27) (~) (A- ~ +7x+aw~) =0, (63)

HK = OA - >.

= (A- ~ +7X) (~ =~) y+a (~=~) wyc

+XT-6A, (64)

HE = OJ.L - jJ, = TJE + ¢(B - 2E)J.L. (65)

The marginal cost of public funds (X') is defined as the marginal disutility of raising an extra unit of public funds divided by the marginal private utility of private income (l/C), that is X' == XC. Thus, a higher marginal cost of public funds indicates that public goods become more valuable compared to private goods.

Levy on natural resources: When we use (62) and (63), we obtain an expression for the optimal levy on natural resources:

_Q ( J.L ) q = C = N + b + a)x' '

(66)

where A' == AC and q denotes the normalised resource levy. If the marginal cost of public funds (X') is zero, the optimal levy reduces to the one found for the first-best outcome (namely, Q = J.LIA = J.LC, as in Section 3). In a second-best context there is a downward bias in the optimal levy on natural resources. The point is that when the government must finance public spending by distortionary taxes on capital, the marginal cost of public funds rises and the government can thus afford less to internalise environmental externalities6 •

60£ course, we cannot say that the resource levy Q will be lower in the second-best

Environmental Quality, Public Finance and Sustainable Growth 165

Productive government spending: When we use equation (61) in com­bination with (62), we obtain an expression for the optimal national income share of productive government spending:

= f3 (A' + b + a)xI) f3

S \I I <. 1\ + X

(67)

If there is no excess burden of taxation and the marginal cost of public funds is zero, expression (67) reduces to the first-best outcome for the na­tional income share of productive government spending (i.e., s = f3). If public funds are scarce (i.e., X' > 0), the optimal national income share of productive government spending falls short of the first-best level. If envi­ronmental externalities b) are less important than production externalities and public goods (1- a), the second-best outcome introduces a downward bias in the national income share of productive government spending and the resulting rate of economic growth. Conversely, if use of renewable re­sources is relatively important in the production process (, > 1 - a), there is an upward bias in the national income share of productive government spending.

Subsidy on capital: When we combine (64) with (62), we obtain the following arbitrage equation:

(Xi) ,\ (1- f3)y + )..' (T + TSP) = (J + 8 - ~' (68)

where the Pigovian subsidy rate on capital is given by:

( 1- f3 ) [ (A'-~) 1 TSp == -ay+ 1-, ' --y:;- +a+, y > o. (69)

Equation (68) clearly shows that the first-best social marginal product of capital, i.e., (1- f3)y, appropriately modified for second-best considerations must equal the "social" user cost of capital (consisting of the pure rate of time preference plus the depreciation rate minus the rate of change in the social value of capital). The second-best social marginal product of capital is less than the first-best value, only to the extent that the second-best subsidy on capital (-T) exceeds the Pigovian component of the optimal subsidy on capital (TS P ).

Equation (69) shows that a higher levy on natural resources implies a lower Pigovian subsidy rate on capital. Hence, it is optimal for the govern­ment to employ a mix of tax instruments in the sense that a more ambitious

than in the first-best situation because J.L and>' will be different in these two situations as well.

166 J.E. Ligthart and F. van der Ploeg

environmental policy consists of both a higher resource levy and a smaller growth promoting (and thus polluting) capital subsidy. We can also ob­serve that an increase in the marginal cost of public funds (X') reduces the Pigovian component of the optimal subsidy on capital. When public goods become more valuable the government can afford less to subsidise capital.

4.2 Analysis of the balanced sustainable second-best growth path

In order to get an insight into the properties of the second-best outcome, the steady-state balanced growth path is analysed. Along a balanced and sustainable growth path the environmental variables E, Nand J.L are con­stant, while the economic variables C, K, Y and G grow at a common rate 7r = ay - T - 8 - () (as in equation (56)). It follows that A, wand X decline at the rate 7r, while y, g, T and TSp are constant on such a path. Clearly, to ensure the sustainability of the balanced growth path, the levy on the use of natural resources (Q) must rise at the rate of economic growth (7r). If the resource levy was constant, the stock of renewable resources would eventually be completely depleted.

We can use (65), (29) and (57) to obtain the following steady-state relationships:

N = ¢>E(B - E) = , (~) , () + ¢>(2E - B) = 'TIE,

J.L

where the consumption-capital ratio is given by

(70)

(71)

C c == K = (1 - a - s)y + T + () = (1 - a - ,)y + () > (), (72)

the output-capital ratio (y) is given by (40) and the normalised resource levy is q == QjC. The lower the social priority attached to environmental quality (1JE) and the higher the social cost of environmental quality (J.L), the lower the optimal stock and use of renewable resources. Note, that the consumption-capital ratio is greater than the pure rate of time preference (()), particularly if the share of labour (1 - a - ,) is high, because the importance of environmental externalities in production (!) is less than the combined effect of knowledge spill-overs and productive government spending in production (l-a).

The steady-state version of (68) can be written as

7r=aY-T-8-(}=(1-,6)y+(~:)(T+TSP)-8-(}, (73)

Environmental Quality, Public Finance and Sustainable Growth 167

which equates the market outcome for the rate of economic growth with the socially optimal rate of economic growth. Equation (73) can be used to obtain an expression for the optimal subsidy rate on capital

- T = C~I ~ XI) (1- a - ,8)y + (N ~ XI) TSp· (74)

The optimal subsidy rate on capital reduces to the first-best subsidy rate if the marginal cost of public funds is zero. If there is an excess burden associated with raising public funds and the marginal cost of public funds is positive, the Pigovian component of the capital subsidy negatively affects the second-best level of the optimal subsidy rate. In general, the optimal second-best subsidy rate on capital is a weighted average of the first-best subsidy rate and the Pigovian component of the second-best optimal sub­sidy rate on capital. In a second-best world the government sets a lower optimal subsidy rate on capital than in a first-best world.

The government budget constraint (32) can, with the aid of (58), be rewritten as:

T = (8 - 'Y)Y. (75)

Hence, the excess of public spending over the revenues from the resource levy must be financed by the capital tax. Upon substitution of (75) into (74) and solving for>..', we obtain:

A' = (;) (a + ; : ;~ 1 - 8) . (76)

The steady state of (60) yields together with (62) an expression for the marginal cost of public funds, namely:

X' = G) (1 - A'). (77)

The marginal social utility of private income (A) typically is less than the marginal private utility of private income (l/C), so that the marginal cost of public funds is positive.

Equations (38), (39) and (66)-(77) can be solved together for the vari­ables y, Q, g, TSp, N, E, 11", J-l, c, >..' and the marginal cost of public funds X' in terms of the parameters (), ¢Y, ""E, a, ,8, 6, 'Y, A and B.

4.3 Special case: Environmental externalities match production externalities

There is one very special case for which the second-best and first-best out­comes coincide, namely if the situation in which the usefulness of renewable resources in production ('Y) exactly matches the combined effects of knowl­edge spill-overs and productive government spending in production (I-a).

168 J.E. Ligthart and F. van der Ploeg

In that case, the share of labour is zero and the government does not need to levy additional taxes (Le., t = 0, see equation (58)). The marginal cost of public funds is zero, so that no distortions are introduced by moving from a first-best to a second-best situation.

This may be seen from equations (66), (67), (69), (72), (73), (76) and (77) which give q = /-L, S = (3, -T = TSp, TSp = -(1 - a - (3)y, c = (), 7r = (1- (3)y-li -(), and 1->" = X', respectively. The second-best outcome for this special case thus coincides with the first-best outcome.

4.4 Some numerical results for the general case

Although we have characterised some of the properties of the second-best outcome in Sections 4.2 and 4.3, it is useful to get a better feel for the re­sults by numerically solving them with plausible parameter values. We take the rate of time preference to be () = 0.02, the share of private capital, pro­ductive government spending and renewable resources to be, respectively, a = 0.2, (3 = 0.2 and 'Y = 0.6, the rate of depreciation of physical capi­tal to be li = 0.1, the biological rejuvenation rate to be ¢ = 0.01 and the biological potential to be B = 16.

Table 1 presents the numerical results when we raise the importance of environmental quality in social welfare in steps of five from 'fJE = 20 to 'fJE = 30. A shift towards greener preferences leads to a decline in the growth rate of the economy in both the first-best and the second-best world. Environmental quality increases, the social value of the environment rises, and the use of natural resources in production is less. Also, it is clear that greener preferences require a higher levy on the use of the environment whilst the government provides less subsidies on capital. When the first­best and second-best outcomes are compared it is clear that social welfare (W) and the growth rate of the economy are higher in the first-best case. The breakdown of social welfare in a public (WE) and a private (W p) com­ponent shows clearly that the private component is higher and the public component is lower in a first-best situation. Government spending is lower in a second-best world reflecting the scarcity of public funds. Also, in a second-best world the firms receive less Pigovian subsidies and a lower en­vironmental tax is levied. When a government has to finance government spending with distortionary taxes it is less able to internalise environmen­tal externalities. Clearly, the revenues from the natural resource levy cover both the capital subsidy and the cost of productive government spending. Note that the second-best subsidy on capital is approximately equal to the Pigovian subsidy implying a very small Ramsey component. The marginal

Environmental Quality, Public Finance and Sustainable Growth 169

cost of public funds (X') falls with a higher weight given to environmental quality (cf. Section 2).

Table 1: Numerical results for first-best and second-best case, parameters: {j = 0.1, a = (3 = 0.2, 'Y = 0.6, () = 0.02, A = 1, B = 16, if> = 0.01

'" 20 25 30

First-best W 1512.61 1754.74 2000.15 Wp 473.87 425.98 375.27 WE 1038.75 1328.76 1624.88

7r 0.22 0.20 0.18 E 10.93 11.56 12.11 N 0.55 0.51 0.47

J.L 23.25 23.70 24.22 T -0.26 -0.24 -0.23 q 23.25 23.70 24.22 c 0.02 0.02 0.02

Y 0.429 0.405 0.380 s 0.20 0.20 0.20

Second-best W 1323.62 1567.09 1815.96 Wp 253.78 199.38 144.83 WE 1069.84 1367.71 1671.13

7r 0.12 0.10 0.08 E 11.74 12.42 13.00 N 0.50 0.44 0.39

J.L 17.94 18.54 19.21 TSp 0.165117 0.151390 0.137230

T -0.165256 -0.151449 -0.137235 q 4.71 5.22 5.83

X' 4.75 4.44 4.12 c 0.096 0.089 0.082

Y 0.376 0.344 0.312 )..' 0.011 0.005 0.0006 s 0.16 0.16 0.16

5. Concluding Remarks

This paper investigates two types of endogenous growth models which in­corporate environmental quality. In the first model pollution is treated as a flow and it is assumed that the government cannot levy lump-sum taxes and subsidies, i.e., it considers a second-best world. We use this simple model to look at the marginal cost of public funds in relation to environmental care. The analysis shows that a more ambitious environmental policy reduces

170 J.E. Ligthart and F. van der Ploeg

the marginal cost of public funds, and boosts the optimal tax rate which in turn leads to an improved environmental quality and a lower rate of economic growth. Thus, there is a trade-off between economic growth and environmental concern. In addition, more environmental concern depresses the national income share of growth-promoting government spending but boosts the national income share of public consumption.

In the second growth model we look at renewable resources to come to grips with the concept of ecological sustainability. Sustainable growth requires a steady use of renewable resources along a balanced growth path. The rate of economic growth can be boosted by a higher national income share of productive government spending and by a greater use of renewable resources in production. Welfare maximisation requires a trade-off between the utility derived from a high rate of growth in private consumption and a better quality of the natural environment. When the government can resort to lump-sum taxes, the first-best outcome can be replicated in a de­centralised market economy through an appropriate subsidy rate on capital and a levy on the use of renewable resources. The excess of capital subsi­dies and productive spending over levies on the use of renewable resources is financed through lump-sum taxes. The sustainable rate of economic growth is higher if the rate of time preference is low or when the society attaches less priority to environmental quality. An increase in productivity or the national income share of productive government spending increases growth and the use of renewable resources in the short run, but depresses environmental quality.

In a second-best world the government faces the dual task of raising public funds and internalising externalities that result from missing mar­kets: (i) absence of a patent market and the resulting spill-overs in pro­duction; (ii) absence of a market for pollution permits and the resulting environmental externalities. For the comparison with the first-best out­come, it matters crucially that constant returns to scale at the firm level imply that environmental externalities in production are outweighed by the combined effect of knowledge spill-overs and productive government spend­ing in production. It then follows that the optimal national income share of productive government spending is below the first-best level while the op­timal ratio of private consumption to capital exceeds the first-best level. In a second-best world the government sets a lower optimal subsidy on capital and sets a lower levy on natural resources.

In future work we wish to extend the present analysis to allow for inter­dependencies in the global economy. A first step to analysing sustainable growth in the presence of a global common-property resource and inter­national spill-overs has been made in Van der Ploeg and Ligthart (1994), but more work needs to be done to allow for global linkage due to interna­tional trade in goods and assets as well. Another important issue to address

Environmental Quality, Public Finance and Sustainable Growth 171

is that of how tighter environmental policy may lead to capital flight (e.g. Bovenberg and van der Ploeg (1994b)) and how this affects growth, ecology and welfare.

A. Appendix

Linearisation of the dynamic system (42)-(43) around the steady state gives:

( ~) = (-¢(2E-B) JL 2¢JL

+ (lJ; ¢E 0 (2E - B)JL -¢JL -1

N (l-"Y-i3)A

o

(78)

¢ B 'f/ A ()

s

where variables now indicate deviations from the steady state. The deter­minant of the Jacobian of this system is given by (44). The comparative statics results (46)-(48) follow from applying Cramer's rule to the steady state of (78) and then converting the steady-state multipliers to elasticities.

References

[1] Arrow, K. J., The economic implications of learning-by-doing, Review of Economic Studies, 29, 155-173, (1962).

[2] Atkinson, A.B. and 8tiglitz, J.E., Lectures on Public Economics, McGraw-Hill, Maidenhead, England, (1980).

[3] Auerbach, A. J., "The Theory of Excess Burden and Optimal Tax­ation, 61-127," in: Handbook of Public Economics, Volume 1, A.J. Auerbach and Martin Feldstein, (eds.), North-Holland, Amsterdam, (1985).

[4] Barro, RJ., Government spending in a simple model of endogenous growth, Journal of Political Economy, 98, 8103-8125, (1990).

[5] Barro, RJ. and 8ala-i-Martin, X., Public finance in models of economic growth, Review of Economic Studies, 59, 645-661, (1992).

[6] Becker, RA., Intergenerational equity: the capital-environment trade­off, Journal of Environmental Economics and Management, 9, 165-185, (1982).

172 J.E. Ligthart and F. van der Ploeg

[7] Brock, W.A., "A Polluted Golden Age," in: Economics of Natural and Environmental Resources, Vernon Smith, (ed.), Gordon and Breach, (1977).

[8] Bovenberg, A.L. and Smulders S., "Environmental Quality and Pollu­tion Saving Technological Change: A Two-Sector Endogenous Growth Model," Center Discussion Paper 9321, Tilburg University, (1993).

[9] Bovenberg, A.L. and van der Ploeg, F., Environmental policy, public finance and the labour market in a second-best world, Journal of Public Economics, 55, 349-390, (1994a).

[10] Bovenberg, A.L. and van der Ploeg, F., Green policies in a small open economy, Scandanavian Journal of Economics, 96, 343-363, (1994b).

[11] Dasgupta, P. and Heal, G., Economic Theory and Exhaustible Re­sources, Cambridge University Press, Cambridge, (1979).

[12] Forster, B.A., Optimal capital accumulation in a polluted environment, Southern Economic Journal, 39, 544-547, (1973).

[13] Gradus, R. and Smulders, S., The trade-off between environmental care and long-term growth - pollution in three prototype growth models, Journal of Economics, 58, 25-51, (1993).

[14] Gruver, G., Optimal investment and pollution control in a neoclassi­cal growth context, Journal of Environmental Economics and Man­agement, 5, 165-177, (1976).

[15] Kaldor, N., "Capital Accumulation and Economic Growth," in: The Theory of Capital, F. Lutz, (ed.), MacMillan, London, (1961).

[16] Keeler, E., Spence, M., and Zeckhauser, R., The optimal control of pollution, Journal of Economic Theory, 4, 19-34, (1971).

[17] Meadows, D., The Limits to Growth - A Report for the Club of Rome Project on the Predicament of Mankind, Universe Books, New York, (1972).

[18] Pezzey, J., Economic Analysis of Sustainable Growth and Sustainable Development, Environment Department Working Paper, 15, World Bank, Washington D.C., (1989).

[19] Ramsey, F.P., A contribution to the theory of taxation, Economic Journal, 37, 47-61, (1927).

[20] Romer, P.M., Increasing returns and long-run growth, Journal of Po­litical Economy, 94,1002-1037, (1986).

Environmental Quality, Public Finance and Sustainable Growth 173

[21] Romer, P.M., Growth Based on Increasing Returns due to Specializa­tion, American Economic Review, Papers and Proceedings, 77, 56-62, (1987).

[22] Romer, P.M., Endogenous technological change, Journal of Political Economy, 98, S71-S102, (1990).

[23] Solow, R.M., A contribution to the theory of growth, Quarterly Journal of Economics, 70, 65-94, (1956).

[24] Tahvonen, O. and Kuuluvainen, J., Optimal growth with renewable resources and pollution, European Economic Review, 35, 650-661, (1991).

[25] van der Ploeg, F. and Ligthart, J.E., Sustainable Growth and Renew­able Resources in the Global Economy, in: Trade, Innovation, Envi­ronment, C. Carraro, (ed.), Kluwer Academic Press, (1994).

[26] van der Ploeg, F. and Tang, P.J.G., The macroeconomics of growth: an international perspective, Oxford Review of Economic Policy, 8, 15-28, (1993).

[27] van der Ploeg, F. and Withagen C., Pollution control and the Ram­sey problem, Environmental and Resource Economics, 1, 2, 215-236, (1991).

[28] van Marrewijk, C., van der Ploeg, F., and Verbeek, J., "Pollution, Abatement and Endogenous Growth: Is Growth Bad for the Environ­ment?," Working Paper, World Bank, Washington D.C., (1993).

[29] World Commission on Environment and Development, Our Common Future, Oxford University Press, Oxford, (1987).

Department of Macroeconomics, University of Amsterdam, Roetersstraat 11 1018 WB Amsterdam, The Netherlands

Environmental Pollution and Endogenous Growth:

A Comparison Between Emission Taxes and Technological Standards

Thierry Verdier l

Abstract

This paper develops a model of endogenous growth with environ­mental pollution. Firms create, through R&D, new products and also design the "cleanness" of these products by choosing their output­emission ratios. Cleaner products are assumed to be more costly to develop than dirty products. Using an extension of the expanding variety product of Helpman and Grossman, we investigate and com­pare the effects of emission taxes and technological standards. In particular, in the second best context where R&D subsidies are not possible, we make a welfare comparison of the two instruments for a given pollution target that the policymaker wants to implement in the economy. Under certain conditions, we show that an emission tax, acting as an implicit R&D subsidy, may induce too much growth of the polluting industry compared to what is socially optimal. This effect can then counteract the usual cost-effectiveness of taxes over technological standards.

1. Introduction

How does environmental conservation affect growth and economic devel­opment? Which kinds of policies should be used to satisfy simultaneously environmental preservation and steady growth of output per capita? Those are important questions that must be addressed as governments and people nowadays are getting increasingly concerned by environmental conservation and at the same time by steady growing standards of living. As a matter of fact one of the major problems faced at the Rio of Janeiro Conference on the Environment was the divergent views shared by governments on the most appropriate way to preserve the environment without impeding growth and economic development.

1 Financial support from the Fondation ENI Enrico Mattei is gratefully aknowledged. The views expressed in this paper are those of the author and do not necessarily represent those of the Fondation ENI Enrico Mattei. This research was initiated when the author was visiting MIT and Harvard University.

176 T. Verdier

Economists recently have made some progress in explaining and pre­dicting growth patterns of different economies. The so-called "endogenous growth" literature, with the works of Romer (1990), Lucas (1988), Aghion and Howitt (1992), Grossman and Helpman (1991a) and others, now pro­vides a convenient macroeconomic analytical framework to study the influ­ence of policies on growth performances. Yet only a few studies have tried to incorporate explicitly environmental concerns into this setting2 • The purpose of this paper is to draw upon the literature on endogenous growth to analyze some issues related to environmental conservation and economic growth. More precisely we will consider only one particular aspect of en­vironmental damage: the one concerning flow pollution by manufacturing firms3 . The focus of the analysis will be on the comparison between two instruments widely discussed in environmental policy debates: emission taxes and technological standards. The conventional wisdom is that emis­sion taxes are more efficient than technological standards or subsidies to regulate environmental pollution. This view has been somehow challenged when the polluting industry is not perfectly competitive (Besanko (198'7), when there is strategic trade in imperfectly competitive international mar­kets (Ulph (1992), Carraro and Siniscalco (1991), Verdier (1992)), or when there is strategic innovation and international trade (Carraro and Topa (1992), Ulph (1992)). All those previous studies however are partial equi­librium analyses and do not deal with the issue of the relative impact of environmental policy instruments on growth performances at the aggregate level of the economy.

Hung, Chang and Blackburn (1992) are certainly among the first to develop a general equilibrium endogenous growth model that incorporates environmental issues. Building on the work of Romer (1990), they consider a variety expanding type endogenous growth model where two types of differentiated products can be developed through R&D: "clean" products and "dirty" products. They show then that in a stable steady state, the economy can only produce one of the two types of goods ("clean" goods or "dirty" goods) and they make some comparisons of growth rates and wel­fare in the two possible long run steady states. While useful as a first step to understand issues concerning pollution and economic growth, in their model the fact that goods can only be totally "clean" or totally "dirty"

2There is a significant literature on growth and environment in a "nea-classical" setting (see notably Forster (1973), Dasgupta and Heal (1974), Krautkraemer (1985), Tahvonen and Kiiiiliivainen (1991». Much less has been done in an endogenous growth framework. A major exception is Hung, Chang and Blackburn (1992); see also more recently Michel (1993), Van der Ploeg and Ligthart (1993), Musu (1994) and Bovenberg and Smulders (1994).

3In particular we do not address the issue of the environment as a scarce factor of production. Evidently integrating this aspect in the setting of this paper would be an interesting line to develop in future research. See also Michel (1994) for an analysis with pollution considered as a stock.

Environmental Pollution and Endogenous Growth 177

makes it difficult to use for a comparison between emission taxes and tech­nological standards. In this respect, a model where firms can freely choose in the R&D stage the degree of "cleaness" of their product seems more appropriate. The model presented in this paper tries to fill this gap.

As is now quite usual in the endogenous growth literature, we consider a model where growth is defined as an expansion in product variety of a differentiated good supplied by monopolistic competition. Each product however will be characterized by a particular output-emission ratio. The novelty comes from the fact that, in the R&D stage, firms can choose the output-emission ratio of the new product introduced in the market. Design­ing cleaner products involves more resources spent in R&D. This feature naturally introduces a trade-off between growth of products variety and "cleaness" of the product developed. In this setting, we discuss the relative merits of emission taxes and technological standards (where a technological standard is defined by a fixed output-emission ratio that should be satisfied by all products in the economy). In this general equilibrium framework, we show first that small emission taxes need not reduce growth and may just, on the contrary, boost the number of products developed in the economy. The intuitive reason is that an emission tax increases the relative price of the manufactured goods, and decreases the demand and the quantity produced for those goods. This in turn, releases resources to be used in the R&D sector and consequently promotes growth. We then compare the growth performances of an emission tax and a technological standard which implement the same pollution target in the economy. We find that, as a natural extension of conventional wisdom, technological standards have a greater negative effect on economic growth than emission taxes. Finally we provide a welfare analysis of the two instruments, here again for a given pollution target. As was emphasized earlier in this introduction, this com­parison is not a trivial exercise as there are potentially many market failures (outside pollution) in the economy: imperfect competition, R&D market failures, spillover effects. In a second best world where the regulator can­not use all the necessary instruments to correct for the various distortions, it is not clear whether emission taxes should dominate technological stan­dards or not. We show that for not too binding pollution targets, emission taxes dominate technological standards, while for severe pollution targets technological standards may dominate emission taxes. The reason for this last result is simply that when pollution targets are severely binding, an emission tax induces too much growth of the industry compared to what would be socially optimal for the output-emission ratio induced by that tax rate. This may counteract the usual effect that emission taxes are more cost effective than technological standards. It creates the possibility that standards are welfare superior to emission taxes.

The plan of the paper is the following: in Section 2, we present the

178 T. Verdier

model and describe the equilibrium growth path. Section 3 and Section 4 discuss respectively the impact of emission taxes and technological stan­dards on equilibrium growth. Section 5 provides, for a given pollution target, a comparison between growth and economic performances under an emission tax and a technological standard. In Section 6, we focus on a welfare comparison of the two instruments. Finally, Section 7 offers some conclusions.

2. A Model of Endogenous Growth and Pollution

The basic set-up of the economy builds on the endogenous growth model with expanding product variety of Grossman and Helpman (1991a). There is one final differentiated environmentally unfriendly good supposed to gen­erate pollution emissions. Producers undertake two distincts activities: first, they make R&D and create blueprints for new varieties of differen­tiated products; second, they manufacture the products that have been developed previously. There is only one factor of production labor that is used for R&D and production. The consumption side is given by an in­finitely lived representative household maximizing an intertemporal utility function. The labor market is assumed to be perfectly competitive while the good market is characterized by monopolistic competition.

2.1 Consumer behavior

The representative consumer maximizes utility over an infinite horizon. His intertemporal preferences take the usual Dixit-Stiglitz form:

(1)

with:

u.,. = IOg{[ln x(j)Qdj]l/Q} - V(S.,.) (2)

where 0 < a < 1. n is the number of variety produced at time t. x(j) denotes consumption of brand j. St is the flow of pollution generated by the manufactured good at time t and V(·) is the disutility of pollution. We assume that V' (.) > 0 and V" (.) > O. It is a simple matter to see that a consumer, spending E at time t, maximizes instantaneous utility by purchasing:

(3)

where (T = 1/(1 - a) is the elasticity of substitution between any two

Environmental Pollution and Endogenous Growth 179

products and P is defined by:

(4)

is the consumption price index in the economy. The intertemporal max­imisation of (1) subject to an intertemporal budget constraint gives us the optimal path of spending E(t) satisfying the following usual Euler condi­tion:

B(t) E(t) = ret) - p (5)

with ret) the real rate of return on financial assets. It is convenient, just as in Grossman and Helpman (1991a), to impose a normalization of prices that makes nominal spendings constant through time. With E(t) = lVt, equation (5) gives us:

ret) = p \:It. (6)

2.2 Producers

a) Technology and pollution

Producers undertake two kinds of activities: R&D to create new products and the manufacturing of existing varieties. There is only one factor of production labor. The technology of production of differentiated products is with constant returns to scale, and after an appropriate choice of units, we may assume that one unit of good requires one unit of labor. The market for each manufactured product is characterized by monopolistic competition.

Each variety i of the manufactured good is polluting the environment and is characterized by a constant emission-output ratio k i so that the level of emission Si is given by:

(7)

where qi is the level of output of good i. We assume that, while doing R&D, a firm is able to choose its emission­

output ratio k i • More precisely, we suppose that in order to develop a new product with an output-emission ratio k at time t, a firm has to use an amount:

a(k) a(k, Kn(t)) = Kn(t) (8)

of labor where Kn(t) represents the stock of general knowledge capital available in the economy at that time. As in Romer (1990), Aghion and Howitt (1992), Grossman and Helpman (1991a and b) and other models of endogenous growth, we assume that R&D generates two products: one is a design of a new commodity. This blueprint gives appropriable benefits

180 T. Verdier

to the inventor in the form of a stream of monopoly profits. Secondly each research project contributes to a stock of basic knowledge Kn(t) that can be used by future innovators for the design of new products. This stock of knowledge cannot be appropriated by any innovator and can be considered as a public input into R&D. The formulation in equation (8) implies that as the stock of general knowledge increases, it becomes less costly in terms of labor to develop a new product. We restrict furthermore the specification by simply assuming that this stock of general knowledge Kn(t) is proportional to the number of products already developed at time t : net), so that we may pose:

Kn(t) = net). (9)

We assume also that a(k) is decreasing and convex in k. More precisely we suppose that there exists a level kn > 0 such that:

i) k E [0, kn]; ii) a'(k) < 0 and a"(k) :::; 0 for k E [0, kn]

and iii) lim a'(k) = -00. k->O

The preceding conditions capture the following two features: on the one hand, output-emission ratios technically are restricted into a range [0, kn] where kn can be considered as the "natural" output-emission ratio associ­ated with each variety adopted by firms without any environmental regula­tion. On the other hand, designing a product with an output-emission ratio smaller than kn involves a higher cost of R&D. This relationship captures the fact that the design of a "cleaner" product needs quite often more R&D and time to test and improve the product and make it satisfy the environ­mental constraints imposed by regulation. As we can already see, this will introduce a natural trade-off between growth and environmental quality of new products. Finally iii) assumes for convenience that it is infinitely costly to design a completely "clean" product.4

A convenient specification that will be often used in the sequel is the constant elasticity case:

a(k) = k-ry for k E [0, kn]; 1] > o. (10)

b) Regulation

In this paper we will be considering two possible intruments for the reg­ulation of pollution. The first one will be a tax T on pollution emissions

4This is done to avoid a cumbersome but straightforward discussion about an equi­librium corner solution of the emission-output ratio level at O.

Environmental Pollution and Endogenous Growth 181

which is rebated to the consumer as a lum-sump transfer; the second will be the imposition of a uniform technological standard on the emission-output ratio k that firms have to choose while developing products.

c) Profit maximization

We suppose that the government grants an infinitely lived patent to the original inventors of innovative products. So that firms manufacturing ex­isting varieties are monopolistically-competitive producers. Let us note w the wage rate at time t. Then facing a demand function like (3), the unique supplier of good i maximizes at each time t operating profits:

7ri = p(i)x(i) - C(i, ki)x(i)

where C( i, ki ) is equal to (w + Tki)x( i) if pollution is regulated by an emis­sion tax T and that the emission-output ratio of good i is ki' or C(i, k i )

is equal to wx(i) if pollution is regulated by technological standards. The solution of this profit maximization program gives simply the monopolis­tic price for good i : p(i) = C(i, ki)/o:. Using (3) (and suppressing time arguments for notational convenience) operating profits are equal to:

and:

(1 ) pt-u E ·th C(i, k i ) 7r' = - 0: -- WI p' =

t pl-u • 0:

P l-u -in l-Ud· - Pi ~. o

(11)

(12)

If we note Viet, ki ) the present value at time t of the infinite stream of profits accruing to a firm that produces a good i with output-emission ratio ki we get:

viet, ki ) = 100 e-[R(r)-R(t)] (1 - O:~i=~ki)l-U E(T)dT (13)

with:

R(t) = lot r(s)ds. (14)

d) Research and development of new products

We assume that firms may enter freely into R&D. At time t developing a new product i with an output-emission ratio k means an up-front cost of w(t)a(k)/n(t). The value of this product is viet, k). In this stage a firm has at most two decisions to take: first, to decide if it undertakes R&D to develop a new product. Second, when pollution is regulated by emission

182 T. Verdier

taxes, it chooses an output-emission ratio level for its product. The free entry condition gives:

(15)

with equality whenever n > o. This says that as long as there is positive growth, there should be a zero profit condition in the R&D sector and growth of new products ceases when the fixed cost of developing a new product exceeds the profit value of this product. Moreover when pollution is regulated by an emission tax, a firm developing a new variety also has to choose optimally its output-emission ratio k i in order to maximize the net value of this variety:

a(ki ) maxv(t, k i ) - w(t)-(-)

k, n t

which, using equation (13), gives the following condition:

(16)

-T1= e-[R(r)-R(t)] [Pir(ki )]-'" E(T)dT > w(t)a'(ki ). k:S; kn (17) t p}-'" - net) ,

with an equality in the first equation whenever k < kn . Given the nor­malization assumption E(t) = 1, equations (6), (14) and the symmetry assumption, firms charge the same price and choose the same output­emission ratio:

and one gets:

- T e-p(r-t) __ dT = w(t)-_· k:S; kn. 1= 1 a'(k)

t n(T)Pr net) , (18)

e) Market equilibrium

The market equilibrium in this economy is given by paths of prices and quantities such that: (1) all participants are optimizing, and (2) all markets (product, equity and labor) are clearing. Obviously because of monopolistic competition, demand equals supply in the product market; and because of Walras' law, the model is closed by simply looking at the equilibrium in the labor market: n 1

a(k)- + - = L n Pt

(19)

which simply states that total labor supply L has to be allocated be­tween labor demand emanating from the R&D sector per unit of time {a(k)ln}dnldt and the labor demand coming from manufacturing (given here by total output in the production sector lip).

Environmental Pollution and Endogenous Growth 183

3. Emission Taxes and Steady State Growth

In this section, we are interested in the study of the impact of an emission tax on the long run equilibrium growth rate of the economy. In a dynamic steady state, the number of products will grow at a constant rate 9 and prices p and W will remain constant5 • The value of a firm may be rewritten as:

1 1-0: v(t,k) = --(-) .

p+g n t

The free entry condition in R&D and equation (18) are restated as:

with equality as n > 0 and:

1-0: -- ~a(k)w p+g

__ 1_ o:T > a'(k)w. k < kn. p+gw+Tk - ,-

Similarly, the labor market clearing condition is given by:

0: a(k)g + w + Tk = L.

(20)

(21)

(22)

(23)

In the rest of the paper it will prove convenient to define the emission tax in labor units: u = T/w so that with positive growth, equations (21), (22) and (23) are equivalent to the following equations:

1- 0: -- =a(k)w p+g

~ < _1- 0: a'(k)k. k ~ kn l+uk - 0: a(k) ,

0: a(k)g + w(1 + uk) = L

(24)

(25)

(26)

Defining as in Grossman and Helpman (1991a), V as the inverse of the ag­gregate value of equity claims in this economy (Le., V = l/nv = l/a(k)w), we finally get the following set of equations:

(p + g) = (1 - o:)V

~ < _ 1 - 0: €(k). k < kn 1 + uk - 0: ,-

(27)

(28)

5It can be shown, along lines similar to Grossman and Helpman (1991a) that there are no transitional dynamics; along a rational expectation path, the economy jumps immediatly to the steady state with a constant growth rate.

184 T. Verdier

aV L g+ 1 +uk = a(k)" (29)

As it is simple to see, equation (28) determines the equilibrium level of output-emission ratio k*(u) as a function of the emission tax u. Similarly, equations (27) and (29) give, for a given k and a given u, the equilibrium steady state growth rate g*(u, k) and the inverse of aggregate equity value V*(u, k). Substituting the equilibrium value of k*(u) in those expressions provides g* and V* as functions of u. From this, we compute the equilibrium steady state values of the wage w*, the level of prices in the economy p* and quantity of the consumption good X* as:

w* = 1 . p* = w*; X* = a . V*a(k*)' a w*(l + uk*)

(30)

a) Equilibrium output-emission ratio

Let us look first at the determination of the equilibrium level of output­emission ratio k*(u) and let us make the following assumption:

AI) The elasticity f(k) = a'(k)k/a(k) is increasing in k and -f(kn) < a/(l - a).

This assumption ensures the existence of a unique output-emission ratio (smaller than kn) as a function of u 6 . Under assumption AI), we can represent equation (28) by Figure 1. Curve E represents the right-hand side - {(I - a)f(k)}/a of equation (28). This curve is decreasing in k. Curve F represents the left-hand side uk/(l + uk) of equation (28) and is increasing with k from 0 to an upper bound 1. The intersection characterizes the equilibrium output-emission ratio k* as a function of the tax rate u. It is clear that when u increases, curve F shifts up and the resulting equilibrium output-emission ratio decreases. As the emission tax gets bigger, it is more profitable for a firm to invest in the development of a "cleaner" product. More specifically, equation (28) may be restated as:

with:

uk = -(1- a)f(k) for u 2: un a + (1 - a)f(k)

(31)

(32)

(33)

6Moreover it rules out counter-intuitive comparative statics such as the fact that the equilibrium output-emission ratio k* increases with emission taxes.

Environmental Pollution and Endogenous Growth 185

uk

1 +uk ----------------------------------------

o k*(u) k

Figure 1.

that is, up to an emission tax un, firms do not do anything special to develop products more environmentally friendly and products are designed with the "natural" output-emission ratio kn. When the tax rate u is higher than un, it becomes profitable to design cleaner products and the equilibrium output­emission ratio varies inversely with the tax. Under the constant elasticity specification (equation (10)) for the a(·)and '" < a/(l - a), we get a very simple solution for the output-emission ratio k*(u) :

k*(u) = kn for u < un = (1- a)'fJ 2-- a-(1-a)",kn

(34)

k*(u) = (1 - a)'fJ ..!. for u ~ un. a - (1- a)", u

(35)

b) Growth rate and emission taxes

The determination of the growth rate 9 and the inverse of aggregate equity value V is given by equations (27) and (29) and can be plotted in Figure 2. Line A represents in the (g, V) space the no-arbitrage condition (27) while line B represents the labor market clearing condition (29). The intersection of the two lines gives us the equilibrium growth rate g*. Analytically, we find:

g*(u) = (1 _ a)(1 + 1uk*(u)) + a [ a(k:(U) (1- a)(1 + uk*(u)) - pal (36)

186

and:

v

V*

p

I-a

o g*

Figure 2.

L = a(k)

L

a(k)

T. Verdier

g

V*(u) = (1- a)(1 +luk*(u)) + a [a(k:(U) + p] (1 + uk*(u)). (37)

From this we can compute the equilibrium wage w*(u), prices p*(u), quan­tity produced X*(u) and pollution S*(u) in the steady state as:

1 w*(u) w*(u) . p*(u) = --; (38) = V*(u)a(k*(u))' a

a X*(u) = w*(u)(1 + uk*(u)); S*(u) = k*(u)x*(u). (39)

Of course, equation (36) is only valid for an interior solution of the system (ie. a positive growth rate g*). This is true if and only if: (1-a)(I+uk*(u)) > paa(k*(u))/L. Otherwise, growth is equal to zero and all resources are spent to produce the existing brands of the production good. In the sequel we assume quite naturally that:

Condition G: (1- a) > paa(kn)/L.

Condition G) ensures a positive growth rate when no environmental regu­lation is implemented in the economy7.

TIn the case of the constant elasticity function aU, this condition takes the following

Environmental Pollution and Endogenous Growth 187

Let us look now at the effect of emission taxes on growth. An inter­esting insight is the fact that, due to general equilibrium effects, one need not find that growth is reduced by an increase of emission taxes. Two opposite effects are at work. First there is the intuitive effect that as u is higher, firms choose to design "cleaner" products; this increases the la­bor cost of doing R&D and therefore dampens growth. On the other side, an emission tax increases the price that monopolistic firms charge for the manufactured goods; this reduces aggregate demand and output of that sector and consequently releases labor resources for the R&D sector. This general equilibrium effect positively affects the steady state growth rate. More precisely because of A1), it is straightforward to see that:

d(uk*(u)) = -f'(k)a(l- a) dk*(u) 0 du (a + (1 - a)f(k))2 du >. (40)

One can then fairly well analyse the effect of u on the growth rate by looking at equation (29). As u is increased, k* decreases, therefore a(k*) increases and the right-hand side of equation (29) decreases. This effect neg­atively affects the growth rate. At the same time however, because of (40), uk* (u) is increasing in u, therefore, for a fixed V, labor demand emanating from the manufacturing sector decreases. This effect releases resources for the R&D sector and, other things being equal, promotes growth. As shown in Figure 3, an increase in u does not affect line A but changes the slope as well as the intercept H of line B. The shift can be decomposed as before: the first effect translates the B curve downwards, negatively affecting growth. The second effect implies a clockwise rotation of the B curve around the point H, boosting equilibrium growth. In the constant elasticity specifi­cation, the two effects are clearly distinguished and we have the following proposition:

Proposition 1. When (10) is satisfied and 'r/ < a/(l- a) then the steady state growth rate g*(u) is increasing with u for u E [0, un) and is decreasing with u for u > un.8

Proof. For u $ un, k remains constant and therefore from equation (32) it is straightforward to see that g* is increasing with u. On the other hand, for u > un, uk remains constant and k is decreasing with u. From (32) again one finds easily that g* is decreasing with u. •

form: (1 - 0) > po(kn )-'1 / L. The condition for positive growth can be restated easily in terms of the emission tax rate:

(1 - 0)." [L (1 - 0) ] -1/'1 u < U C = - -~---''-:-

- 0 - (1 - 0)." po - (1 - 0)."

SIt is easy to see that, because of condition G, un < uc.

188 T. Verdier

V

B

° g* g*' H' H g

Figure 3.

When the emission tax is smaller than un, the output-emission ratio k remains constant and equal to kn. Only the positive effect is at work and growth is increasing with the tax rate in that range of variation. However, when u > un, a constant elasticity function a{·) implies that monopolistic prices do not change with u (uk remains constant). There remains then only the fact that a stricter environmental tax induces firms to design "cleaner goods" but in less quantity per unit of time. The growth rate is then a decreasing function of u.

c) Steady state output and pollution

One can also analyse in a similar fashion, the way emission taxes affect steady state output and pollution flows. Because of the ambiguous effect of taxes on the growth rate, one cannot generally predict the effect on the steady state output allocation and therefore on the pollution level9 • The constant elasticity case however gives again some useful insights.

Proposition 2. When (10) is satisfied and 'f/ < a/{1 - a); then:

i) steady state output X*{u) and pollution S*{u) are decreasing with u for u E [0, un];

ii) steady state output X*{u) is increasing with u for u > un,

9The only thing that can be said at this level of generality is that as the resource constraint can be rewritten as a(k*)g* + X* = L, then steady state output and pollution decrease whenever g* increases with the tax rate u.

190 T. Verdier

The growth rate, output and pollution levels can be easily deduced from this system:

L g! = (1- a) a(k!) - ap; X! = alL + paCk!)]; S! = k! X!. (45)

From this we have the straightforward proposition:

Proposition 3. The equilibrium growth rate g!(k!) is increasing with k!, output Xf(k!) is decreasing with k!, pollution SICk!) is increasing with k! if E(k) = a'(k)k/a(k) > (-1).

This proposition simply states that more restrictive environmental techno­logical standards (a reduction in kf) dampen growth, reallocate resources to production of existing products and may reduce pollution if the tech­nology of R&D is not too negatively sensitive to technological standards. Hence, contrary to the case of regulation by emission taxes, environmental technological standards cannot have a positive effect on growth.

5. Growth and Output Comparison Between Emission Taxes and Technological Standards

Let us now compare the economic performances between emission taxes and technological standards. Our point of departure will be to assume that the two regulation mechanisms are used to implement the same level of steady state pollution in the economy. More precisely, recalling equations (37),(38),(39) and (45), we have the following relationship between a tax rate u (and the resulting equilibrium output-emission ratio k*(u)) and a technological standard k! implementing the same pollution level S :

S = ak*[L + pa(k*)] = akf[L + a(k!)]. (1 - a)(1 + uk*) + a p

(46)

We still assume that AI) and 0 < -a'(k)k/a(k) < 1 are satisfied so that total pollution is an increasing function of k and for the emission tax, a decreasing function of u10• We can therefore invert equation (46) and define k!(S) and k*(S) = k*(u(S)) as respectively the output-emission ratios necessary to implement a pollution target S under a technological standard (an emission tax u(S)) mechanism. We have then the following proposition:

Proposition 4. Let u(S) and k! (S) be respectively an emission tax and a standard implementing the same pollution level S. Let k*(S), g*(S) and

lOTherefore pollution always decreases with more severe environmental policies (emis­sion taxes or technological standards).

Environmental Pollution and Endogenous Growth 189

iii) if"., < 1, then steady state pollution S*(u) is decreasing with u for u> un.

Proof. The first part i) of the proposition comes obviously from the fact that for u E [0, un], the equilibrium output-emission ratio kn does not change and g* increases; therefore from the resource constraint a(kn)g* + X* = L, an increase in the growth rate implies a decrease in manufactured output X*. Consequently S* = kn X* decreases also with u. Part ii) comes from the fact that the equilibrium steady state output X* in that case can be written as:

X* u _ a[L + pa(k*)] ( ) - (1 - a)(1 + uk*) + a·

(41)

For the constant elasticity case, uk*(u) stays constant with u. Therefore X*(u) depends on u simply through a(k*(u)). Hence it is increasing with u. Finally part iii) comes from:

S*(u) = ak*[L + pa(k*)] (1 - a)(1 + uk*) + a

(42)

From this and the fact that uk*(u) remains constant in that regime, it is simple to see that a sufficient condition for S* to be increasing in k* (or equivalently decreasing in u) is that k*a(k*) is increasing in k*. This can also be stated as "., < 1. •

Notice that we are not always sure that an increase in the emission tax rate u effectively reduces the level of pollution flows S*. While the equi­librium output-emission ratio k* is reduced as an optimal R&D response of firms, at the same time the total level of output X* may increase (case ii». Therefore the total level of pollution may be indeterminate without imposing a sufficient condition as in iii).

4. Technological Standards and Equilibrium Growth

In this section we consider the effect of technological standards on equilib­rium growth, output and pollution. The way we formulate technological standards is by assuming that the government imposes any product a fixed output-emission ratio kf. Hence, innovators no longer have this margin in which to adjust when developing a new product. The steady state equilib­rium equations are reduced to:

(p+ g) =

g+aV

(1- a)V L

a(kf)"

(43)

(44)

Environmental Pollution and Endogenous Growth 191

X*(8) (respectively gl (8) and Xl (8)) denote the steady state equilibrium output-emission ratio, growth rate and output levels under the emission tax (the technological standard). Then:

i) kl(8) < k*(8)j ii) XI(8) > X*(8)j iii) gl(8) < g*(8).

Proof. i) Note that:

Vu>OjVk*j ak*[L + pa(k*)] <ak*[L+pa(k*)]. (47) (1 - a)(1 + uk*) + a

Hence, from this and (46), it follows that:

ak/[L + pa(kl)] < ak*[L + pa(k*)] (48)

From 0 < -a'(k)k/a(k) < 1, the function ak[L + a(k)p] is increasing in k. Hence (48) implies that kl(8) < k*(8).

ii) From i) we get that a(kl(8» > a(k*(8». Moreover:

X*(8) = a[L+pa(k*(8))] j XI(8)=a[L+pa(kl(8»]. (49) (1 - a)(1 + uk*(8)) + a

Hence:

X*(8) = alL + pa(k*)] < a[L+pa(k*)] < a[L+pa(kl)] = Xl (8). (1 - a)(1 + uk*(8» + a

(50) Hence the result in ii) of the proposition.

iii) Finally from the resource constraint, i) and ii), we get:

I _ L - X I L - X* L - X* _ * 9 (8) - a(kl) < a(kl) < a(k*) - 9 (8). (51)

Hence the result announced in the proposition. •

Part i) of proposition 4 tells us the familiar result that technologi­cal standards are less cost effective than emissIon taxes to control pollu­tion. Under an emission tax, firms doing R&D have more margins within which to adjust when designing the "cleanness" of a new product. This means smaller costs of running research than under a technological stan­dard. Moreover, in a general equilibrium, under an emission tax, less re­sources are devoted to production and therefore more resources are available for the R&D sector. Both reasons make equilibrium growth faster under an emission tax than under a technological standard (point iii». Point ii) of

192 T. Verdier

the proposition says that the amount of resources allocated to production will be larger under the standard than under the tax scheme. This last result is reminiscent of the result of Besanko (1987) which shows that in an imperfect competition context, pollution regulation by a performance standard (or equivalently in this context an emission tax) leads to less in­dustry output than regulation by a technological standard. Besanko (1987) derived this result in a partial equilibrium framework with a fixed number of firms competing in Cournot fashion. Here this result is obtained in sim­ple dynamic general equilibrium with endogenous growth characterized by permanent entry of new products.

6. Welfare Comparisons Between an Emission Tax and a Technological Standard

So far we have only compared economic performances under the two envi­ronmental regulatory mechanisms. This section considers welfare compar­isons between the two instruments.

a) First best allocation

First of all, consider the first best allocation problem. The economy pre­sented here is one with potentially many market failures besides the obvious negative externality of pollution. First there is a static distortion that can arise due to the fact that the production sector is monopolistically com­petitive. In the present context however, because all firms face the same elasticity of demand and charge the same mark-up over marginal cost, rela­tive prices between varieties reflect relative marginal costs and therefore no static distortion emerges. Three other market failures are related to R&D activities. The first one arises from the fact that innovators do not fully take into account the increase of consumer surplus coming from the creation of a new product. This the so-called consumer-surplus effect (Grossman and Helpman (1991a)). The second is connected to the fact that innovators base their R&D decisions on considerations of private profitability and do not pay attention to the destructive effect on the profits of other firms (the profit destruction effect). Finally there is the fact that R&D activities pro­duce some basic public kowledge that can be used by all future innovators (the spillover effect). The consumer-surplus effect and the spillover effect suggest that the market equilibrium does not produce enough R&D and growth, while on the contrary the profit-destruction effect suggests that there is overinvestment in R&D generated by the market equilibrium. In fact it can be shown that with the Dixit-Stiglitz CES preferences for variety, the two effects, consumer-surplus effect and profit-destruction effect, can-

Environmental Pollution and Endogenous Growth 193

celout (see Grossman and Helpman (1991a) p. 82-83)11. This last remark leads to the conclusion that, in all, the social planner will have to correct only two market failures: the negative externality of pollution and the pos­itive spillover effect of R&D. More precisely the social planner problem can be written as:

100 [l-a ] max Ut = e-p(T-t) [--logn(r) +logX(r)] - V(kX(r)) dr (52) k,X,n t a

under the resource constraint:

a(k) :~g + X(t) = L Vt (53)

where we have used the fact that at each point of time firms should produce the same amount of good X/no It can be shown (see Appendix 1) that the optimal social growth gOP, production output xop and pollution flows sop

will be constant. This optimum can be decentralized by the use of an R&D subsidy and an emission tax as well as by a combination of an R&D subsidy and a technological standard. The optimal R&D subsidy with the technological standard however is larger than the optimal R&D subsidy with the emission tax scheme.

b) Second best comparison of environmental policies with fixed pollution targets

In this subsection we consider welfare comparison between emission taxes and technological standards when, for some reason, the governement cannot use simultaneously a R&D subsidy. The problem is not a trivial one because we are in a second best world. Therefore, there is no "a priori" straightforward comparison between emission taxes and technological stan­dards. As a matter of fact recent literature on environmental regulation with imperfect competitive markets or strategic behavior has given circum­stances under which technological standards may dominate emission taxes from a welfare point of view (see in particular Besanko (1987), Ulph (1992) or Verdier (1993)).

Following Section 5, we want here to compare emission taxes and tech­nological standards implementing a fixed pollution target level S. Given that for both instruments the market equilibrium is without transition and jumps immediately to a steady state we may at any point of time t compute

llThis result that consumer surplus effect and profit destruction effect cancel out is only true for a model of endogenous growth with expanding varieties. In an endogenous growth model with upgrading qualities (Aghion and Howitt (1992) or Grossman and Helpman (1992b» this result does not hold necessarily and one cannot say "on a priori grounds" whether equilibrium growth is higher or lower than socially optimal growth.

194 T. Verdier

the welfare function of the representative consumer as:

[1-0:] [1-0:]9 pUt = ~ logn(t) + ~ p + log X - V(S). (54)

Taking into account the resource constraint (ag + X = L), welfare at time o can be rewritten as:

[1 - 0:] [1 - 0:] 9 pUo = -0:- log no + ~ p + log(L - a(k)g) - V(S). (55)

Using notations from Proposition 3), a welfare comparison between emis­sion taxes and technological standards therefore involves a comparison be­tween:

W*(S) = 1 - 0: g*(S) + log[L - a(k*(S))g*(S)] (56) o:p

and:

with moreover:

kf(S)· Xf(S) = k*(S)· X*(S) = S. (58)

In the sequel, it will be useful to define the following function W(g, k) as:

1-0: W(g, k) = --g + log[L - a(k)g].

o:p (59)

It can be seen that this function W(g, k) is concave in 9 and increasing in k. Let us note:

gS(k) = argm:x W(g, k) = max {O, 1 ~ 0: [(1- 0:) a~) - o:p]} (60)

and:

WS(k) = W(gS(k), k) = 1- 0: gS(k) + log[L - a(k)g8(k)]. (61) o:p

In fact g8(k) and W8(k) are respectively the socially optimal growth rate and the optimal welfare level for a growing economy with a given output­emission ratio k that implies the cost a(k) of doing R&D in the economy. Notice that there is a limiting value kS (such that a(k8) = (1 - o:)L/o:p) under which the growth rate g8 is equal to zero and WS(k) is equal to log L. Notice also that because of (45) k 8 is the value of the output-emission ratio that gives a zero growth rate under a technological standard. Assumption AI) and the fact that 0 < -a(k)k/a(k) < 1 are supposed to be satisfied.

Environmental Pollution and Endogenous Growth 195

Finally let S8 be the pollution target that gives a zero optimal growth rate g8 when an emission tax is applied:

(62)

Then we have the following proposition:

Proposition 5. i) If the pollution target S is such that: g8{k*{S)) > g*{S), then emission taxes dominate technological standards (i.e., W*{S) > WI{S)).

ii) 38> 0 such that: VS E [SB-8, SB+8], technological standards dominate emission taxes (Le., W*(S) < WI(S)).

Proof. i) Consider a pollution target S that satisfies: gB{k*(S)) > g*(S). Then using proposition 4, we have that: g*(S) > gl(S). The two preced­ing inequalities coupled with the fact that the function W(g, k) is strictly concave in g, imply that g*(S) and gl(S) are on the increasing part of the function W(·, k*{S)). Hence:

WB(S) = W(g*(S), k*(S)) > W(gl (S), k*(S)). (63)

Moreover from proposition 4, kf (S) < k* (S). Therefore, taking into account the fact that W{g, k) is increasing in k, we get:

W{gl(S),k*(S)) > W{gl(S),kF{S)) = WI{S). (64)

Finally (63) and (64) give the result announced in i).

ii) By definition, for S equal to SB, the growth rate gB(S8) which maximizes W{g, k*(SB)) is equal to zero. Because for all kl (S) < k*{S), it is also clear by comparing (45) and (60) that the growth rate under the technological standard is also zero for the pollution target SB. Hence:

From this and the fact that the functions WI(S) and W*(S) are continuous in S, result ii) follows. •

Proposition 5 says that, depending on the range of pollution targets, emission taxes mayor may not dominate technological standards. With­out any environmental regulation, there is a "natural" level of pollution sn associated with the "natural" output-emission ratio kn. In that initial situation, it is clear that because firms do not internalize spillover effects of R&D, the equilibrium growth rate g*{sn) = gl (sn) is smaller than the growth rate g8(sn) that maximizes W(g, kn). Hence by continuity case i)

196 T. Verdier

is realized when the pollution target S is not too far from sn and when en­vironmental policy is not too severe. In that case emission taxes dominate technological standards because they are more cost effective and less pe­nalizing for growth. On the other hand when the pollution target becomes more binding (approaching SB) case ii) says that technological standards dominate emission taxes in a neighbourhood of SB. The reason is simply that emission taxes produce "too much growth" compared to what would be now socially optimal with an output-emision ratio k*(SB) induced by this policy. As a matter of fact, precisely at SB the growth rate gB(SB) which maximizes the welfare function W(g, k*(SB)) is equal to zero while the equilibrium growth rate with emission taxes is strictly positive. This divergence between the two growth rates generates a dynamic welfare loss of emission taxes which may outweight the cost effective gain they have compared to technological standards. In that case, technological standards dominate emissions taxes.

1. Conclusion

This paper considers a general equilibrium model of endogenous growth that incorporates explicitly some concerns about the environment. We have dis­cussed the comparison between emission taxes and technological standards. Basically the conclusion that comes from the analysis is complementary to what the existing literature on environmental pollution policy is saying in a static or partial equilibrium framework. The ranking of emission taxes and technological standards is not clear-cut once the regulator has a limited set of instruments preventing him to get into a first best world. Here we showed that the main circumstances under which technological standards are more likely to dominate emission taxes are cases where pollution targets are very constraining. Another potentially interesting result for policymakers is the fact that an emission tax to control environmental pollution need not be detrimental to economic growth.

Of course the model presented here is drastically simple. An area for fu­ture research would be to extend this analysis in various directions. First, rather than having an expanding variety growth model, one might con­sider rather a quality upgrading growth model in the spirit of Aghion and Howitt (1992) or Grossman and Helpman (1991). The important difference between the two frameworks is the fact that in the variety expansion model, socially optimal growth is always higher than equilibrium growth while this need not be the case in the quality model. This difference can have con­sequences for the dynamic welfare comparison between emission taxes and standards. Introducing physical or human capital accumulation besides product expansion would certainly increase the realism of the model. More

Environmental Pollution and Endogenous Growth 197

interestingly though, another avenue of research would be to allow for more than one factor of production and consider the impact of differential fac­tor intensity between R&D-abatements and production activities. In this respect a further extension would be to open the economy to trade and re­consider in this setting the hot debate about the international coordination of environmental policy and the implication for growth and development.

Appendix 1

The social planner problem can be recalled as written as:

max Ut = e-p(r-t) [--logn(r) + logX(r)] - V(kX(r)) dr (66) 100 [1-a ] k,X,n t a

under the resource constraint:

a(k) :~:~ + X(t) = 1 Vt. (67)

We have used the fact that at each point of time firms should produce the same amount of good X/no Let us define O(t) the multiplier of constraint (67). Then one can write the Hamiltonian H as:

The necessary and sufficient conditions that apply along an optimal trajec­tory with ongoing innovations are:

1 , () On (X) (69) = V kX k +a{k) X a'{k)

- a{k)2 O{L - X)n V'{kX)X (k) (70)

iJ = pO- e-a+O(L-X)] (71) an a(k)

lim O(t)n(t)e-pt = O. (72) t->oo

Defining the variable M(t) = O(t)n(t) and using (67), (71) can be rewritten as:

M I-a -=p-- (73) M aM

(72) and (73) give us M(t) = (1 - a)/(ap). From that and (69) and (70), one gets that the optimal k and X should be constant; therefore, from

198 T. Verdier

the resource constraint, the growth rate 9 should also be constant. The equations determining the social optimum can be rewritten as:

1 X

a'(k) 1 - a -----g

a(k) ap a(k)g + X

I-a v'(kX)k + apa(k) (X)

V'(kX)X (k)

L

(74)

(75)

(76)

This set of equations gives the first best levels of growth, output-emission ratio and manufacturing output gOP, kOP, XOP. Now it is easy to see that the market equilibrium conditions under an emission tax u and an R&D subsidy 1> are:

_ 1 - a a' ( k) k uk (77)

a a(k) l+uk

g+p (1 - a) X (1 + uk) aa(k) 1 -1> (78)

a(k)g + X L. (79)

Equations (77) and (78) determine then an optimal emission tax and R&D subsidy uOP, 1>°P that decentralize gOP, kOP, Xop as a market equilibrium.

Similarly the market equilibrium equations under an R&D subsidy and a technological standard are simply given by:

g+p =

a(k)g + X

(1- a) X

aa(k) 1 -1> L.

(80)

(81)

A technological standard equal to kOP and the R&D subsidy such that:

Op _ (1 - a) Xop 9 + P - aa(kop) 1 -1> (82)

will, given the resource constraint Xop = L - a(kOP)gOP, decentralize the first best. Furthermore it is clear from (78) and (82) that the optimal R&D subsidy 1>0P* under the technological standard has to be larger than the optimal R&D subsidy 1>0P under the emission tax scheme.

References

[1] Aghion P. and Howitt P., A model of growth through creative destruc­tion, Econometrica, 60, pp. 323-352, 1992.

Environmental Pollution and Endogenous Growth 199

[2] Besanko D., Performance versus design standards in the regulation of pollution, Journal of Public Economics, 34, pp. 19-44, 1987.

[3] Bovenberg A.L. and Smulders S., "Environmental Quality and Pollution-Saving Technological Change in a Two-Sector Endogenous Growth Model," ENI Enrico Mattei, wp n. 27.94, 1987.

[4] Dasgupta P. and Heal G., The optimal depletion of exhaustible re­sources, Review of Economic Studies, (Special Issue), pp. 3-27, 1974.

[5] Carraro C. and Siniscalco D., International competition and environ­mental innovation subsidy, Environmental Resource Economics, 1991.

[6] Carraro C. and Topa G., "Should Environmental Innovation Policy Be Internationally Coordinated?;" Paper presented at the Conference "The International Dimension of Environmental Policy" organized by the Fondazione ENI Enrico Mattei, 1992.

[7] Forster B.A., Optimal capital accumulation in a polluted environment, Southern Economic Journal, 39, pp. 544-547, 1973.

[8] Grossman G. and Helpman E., Innovation and Growth in the Global Economy, The MIT Press, Cambridge, MA., 1991a.

[9] Grossman G. and Helpman E., Quality ladders in the theory of growth, Review of Economic Studies, 58, pp. 43-61, 1991b.

[10] Hung V., Chang P., and Blackburn K., "Endogenous Growth, Environ­ment and R&D;" paper presented at the Conference "The International Dimension of Environmental Policy" organized by the Fondazione ENI Enrico Mattei, 1992.

[11] Krautkraemer J., Optimal growth, resource amenities and the preser­vation of natural environments, Review of Economic Studies, pp. 153-170,1985.

[12] Michel P., "Pollution and Growth: Towards the Ecological Paradise," ENI Enrico Mattei, wp n. 80.93, 1993.

[13] Musu I., "On Sustainable Endogenous Growth," ENI Enrico Mattei wp n. 11.94, 1994.

[14] Romer P., Endogenous technological change, Journal of Political Econ­omy, 98, pp. S71-S102, 1990.

[15] Thavonen O. and Kiiiiliivainen J., Optimal growth with renewable resources and pollution," European Economic Review, 35, pp. 650-661, 1991.

200 T. Verdier

[16] Ulph A., "The Choice of Environmental Policy Instruments and Strate­gic International Trade," in: Conflict and Cooperation in Managing Environmental Resources", R. Pethig, ed., Springer-Verlag, New York, 1992a.

[17] Ulph D., "Strategic Innovation and Strategic Environmental Policy;" paper presented at the Conference "The International Dimension of Environmental Policy" organized by the Fondazione ENI Enrico Mat­tei,1992b.

[18] Van der Ploeg F. and Ligthart J., "Sustainable Growth and Renewable Resources in the Global Economy," ENI Enrico Mattei, wp n. 26.93, 1993.

[19] Verdier T., "Strategic Trade and the Regulation of Pollution by Per­formance or Design Standards," ENI Enrico Mattei, wp n. 58.93, 1993.

DELTA ,CERAS Paris, CEPR London

Rate-of-Return Regulation, Emission Charges and Behavior of Monopoly

Anastasios Xepapadeas

Abstract

The well-known Averch-Johnson thesis indicated that the main result of rate-of-return regulation is overcapitalization. A regulated monopoly can adhere to environmental policy by undertaking in­vestment in pollution abatement equipment, along with investment in output production. In this context, over- or undercapitalization effects have a direct influence on the monopoly's emissions. This paper analyzes two related issues. The first is the direction and dis­tribution of the effects of introducing rate-of-return regulation under a given environmental policy, in the form of emission charges, on investment in productive and pollution abatement equipment. The second is whether the regulated firm responds in the same manner as the unregulated firm, to the introduction of the above environmental policy.

1. Introduction

Rate-of-return regulation is one of the major institutions of monopoly con­trol. The economic consequences of the attempt to control monopoly behav­ior through the regulation of its rate of return are primarily: (1) inefficiency in the firm's choice of productive inputs, (2) failure to control monopolistic reliance on price discrimination, and (3) failure to encourage technological change (Sherman 1989). Input inefficiency has received considerable at­tention since the appearance, in 1962, of the Averch-Johnson (A-J) model. The main finding is that when the allowed rate of return on capital ex­ceeds the actual cost of capital, the firm will invest more as compared to the unregulated case. This overcapitalization thesis or A-J effect has been examined in various contexts. Arguments have been put forward question­ing whether, in practice, regulators allow the rate of return to exceed the cost of capital (Joskow 1979). In the A-J original framework, however, proximity between the allowed return and the cost of capital leads to un­satisfactory results (Baumol and Klevoric 1970). This deficiency has led to alternative specification of the regulatory process (e.g., Bawa and Sibley 1980, Braeutigan and Panzar 1989). Another approach has been the exten­sion of the original static models (A verch and Johnson 1962, EI-Hodiri and

202 A. Xepapadeas

Takayama 1973, Takayama 1969) to a dynamic framework. In these exten­sions, EI-Hodiri and Takayama (1981) claim that the A-J effect holds in the dynamic case, whereas Peterson and Vander Weide (1976), Katayama and Abe (1989) suggest that undercapitalization might occur if the regulated firm cannot choose efficient labor inputs under the regulatory constraint. Dechert (1984) has also shown that the A-J effect does not occur when the firm operates under increasing returns to scale.

Monopolies like electric utilities, which are subjected to regulatory con­straints, generate pollutants that are discharged into the environment as a byproduct of their operation. Introduction of an environmental policy de­signed to control pollution by using emission charges - that is Pigouvian taxes - as an instrument, will induce firms to invest in emission-reducing technologies (for example, discharge abatement equipment). This stock of pollution abatement capital, in contrast to productive capital, will make it possible to reduce the total amount paid for emissions. Environmental innovation in the form of increasing abatement capital can therefore be re­garded as a defensive expenditure, which might be differentiated from pro­ductive capital by differences in installation, training costs, and so forth. Xepapadeas (1992) analyzes the investment behavior of an unregulated monopoly with respect to productive and abatement capital when emis­sion charges are imposed. Under rate-of-return regulation, however, there is an additional complication since the base for rate-of-return calculations includes both types of capital. In this context, two questions should be an­swered with respect to the regulated firm's behavior. The first is whether introduction of regulation under given environmental policy will produce A-J type effects in productive capital accumulation or in environmental in­novation or in both. The second is whether a regulated firm will respond to the introduction of environmental policy in the same manner as an unregu­lated firm, as far as investment in production and environmental protection are concerned. Answers to these questions might be of some importance to policy makers, since they will provide some indication of the effect of rate-of-return regulation on emissions. In a sense, environmental innova­tion might be accelerated if A-J type effects occur in the abatement sector, or decelerated if reverse type of effects occur.

The purpose of this paper is to examine the investment behavior of a rate-of-return regulated monopoly, under the assumption that the firm should follow a pre-specified environmental policy. This policy takes the form of emission charges per unit of pollutant emitted by the firm into the environment. Conditions for A-J type or reverse effects in productive or abatement capital, which relate to differences in adjustment costs associ­ated with investment in productive or abatement equipment, are formu­lated. Furthermore, the relative effects of introducing environmental policy on productive and abatement capital are compared in the regulated and

Rate-of-Return Regulation 203

the unregulated case. This issue is timely, especially in view of the much discussed introduction of a carbon tax in Europe, which would affect the investment behavior in the electricity sector, always a prime candidate for regulation.

The paper is organized as follows. Section 2 sets up an infinite horizon model for the regulated monopoly and specifies the rate-of-return constraint under emission charges and emission limits. Section 3 analyzes the firm's behavior under emission charges; optimality conditions are derived and the stability properties of the model are examined. In Section 4, the effects of regulation on the equilibrium productive and abatement capital are exam­ined by means of comparative static and comparative dynamic analyses. In the same context, the effects of introducing emission charges on equilibrium capital stocks are compared in the regulated and unregulated cases. The final section provides some concluding remarks.

2. A Model of the Regulated Firm

A profit-maximizing monopolist producing at each instant of time a sin­gle homogeneous output, using capital and labor as inputs, is considered. A single pollutant discharged into the environment is a byproduct of the output production process. The abatement of discharges involves addi­tional investment in pollution abatement equipment. Let k. = (kp, ka) denote capital employed in output production and pollution abatement respectively, at time t E [0,(0), with k. E Kp x Ka C R~, f = (lp,la) denoting labor employed in output production and pollution abatement re­spectively, at time t with 1 E Lp x La C R~. Thus the input set is defined as Y = Kp x Ka x Lp x La. Let q(t) denote output produced at time t with q E Q c R+ and e(t)denote net discharges into the environment at time t with e E E c R+. All sets defined above are assumed to be compact and convex and R+ = {J: ERn : Xi 2 0, i = 1, ... , n}.

The production structure of the firm is specified by using the following simplifying assumptions.

(A.I) Assuming the existence of an inverse demand function for the firm's output, the total revenue function is determined by a strictly concave twice differentiable and time invariant revenue function of kp(t), lp(t). This function is defined as:

Hereinafter subscripts associated with functions will denote partial derivatives, that is,

204 A. X epapadeas

(A.2) The firm's discharges into the environment are determined by a con­vex and time invariant emission function of capital and labor inputs in output production and pollution abatement. The emission function can be further specified by assuming that emissions generated from output production are determined according to a strictly convex twice differentiable function:

Emissions reductions, on the other hand, are determined according to a strictly concave twice differentiable abatement function:

Net emissions are therefore determined as:

Assumptions (A.l) and (A.2) introduce some simplifications into the production structure, which facilitate the subsequent analysis. Thus output production does not depend on abatement inputs, while the stock charac­teristics of environmental pollution are not considered. Given that the objective is private profit maximization, the adoption of a flow concept for pollution seems appropriate if the firm's technology is not affected by the ambient pollutant concentration. Stock effects might be required in a model where the objective is to maximize some welfare indicator.

(A.3) Net capital formation in the productive or abatement sector of the firm is defined as:

where I j (t) is gross capital formation in production abatement and 6 is the exponential depreciation rate assumed common, in order to simplify things, for both types of capital. It is further assumed that investment is "irreversible" (Arrow and Kurz 1970), that is:

I = (Ip(t), Ia(t)) ;::: o. (2)

(A.4) It is assumed that labor is a "flexible" input, but capital, both in production and abatement processes, is "quasi fixed". Installation of capital at a rate Ip(t) or Ia(t) results in a cost bIj + Cj(Ij), j = p, a where b is a common purchase price of a unit of capital equipment and Cj (Ij) is the full adjustment cost that includes installation costs, workers' training, and so forth. Since it is more expensive to acceler­ate the increase in capital stock, Cj(O) = 0, Gj(Ij) > 0, Gj'(Ij) > o.

Rate-oj-Return Regulation 205

Differences in the two adjustment cost functions can be regarded as reflecting differences in installation or training costs corresponding to productive or abatement capital.

(A.5) The available instrument for the implementation of environmental policy can take the form of exogenously determined emission charges. If we denote 7 as the tax per unit of emissions, then the total amount paid by the firm at each instant of time is:

(3)

Under the above assumptions, the instantaneous cash flow of the firm, when emission charges are used as an instrument of environmental policy, takes the form:

II = G(kp,lp) - I)wjlj + blj + Cj (lj )]- 7[S(kp, lp) - A(ka,la)] (4) j

where Wj is the wage rate for each type of labor. Environmental policy also determines the form of the regulatory con­

straint. If p > 0 denotes the maximum allowed rate of return on total capital, the constraint takes the forms:

G - " . W ·l· - 7(S - A) UJ J J <

kp + ka - p. (5)

The objective of the regulated monopoly is to maximize the present value of its cash flow over an infinite time horizon, subject to the regulatory and environmental constraints. The problem becomes:

(6)

subject to (1), (2), (5) and the non-negativity constraints:

(7)

where r > ° is the discount rate. By integrating (1) and using the corresponding initial condition, it

can be seen that Ij(7) ~ ° for all t E [0, t) implies kj(t) ~ 0 for all t E [0,00). Thus the non-negativity constraints on the state variables kj (t) are redundant (EI-Hodiri and Takayama 1981).

In the following sections we examine the input policy of the regulated monopoly as determined by the solutions of problem (6).

206

3. Input Policy under Emission Charges

3.1 Optimality conditions

The current value Lagrangean for problem (6) is defined as:

A. X epapadeas

L G(kp, lp) - ~)wjlj + bIj + Cj(Ij))- r[S(kp, lp)) j

+ L c/>j(Ij - okj ) + A[p(kp + ka ) - G(kp, lp) j

+ L wjlj + r(S(kp, lp) - A(ka, La))) j

Let (k*,1*, l*) be a solution to this problem. Then there exist continuous and piecewise continuously differentiable functions c/>j (t) and a piecewise continuous function A(t) such that, if we assume interior solutions for the control variables and e*(t) > 0, "It, the following conditions are satisfied with all expressions evaluated at the solution vector, along with the state equations:

(8a)

(8b)

(8c)

LA ;::: 0, A[p(kp + ka ) - G(kp, lp)

+ L wjlj + r[S(kp, lp) - A(ka, La))) = 0, A;::: 01 (9) j

(lOa)

(lOb)

The Arrow type transversality conditions (Arrow and Kurz 1970) are also

IFrom (8.a), (8.b) it can be seen that for>. i' 1, the rank condition is not satisfied for an effective regulatory constraint. The normality condition is, however, satisfied because the constraint function is convex in lp, la due to the assumptions on the G, S, A functions. Convexity of the constraint functions implies the satisfaction of the Arrow-Hurwicz-Uzawa condition (Takayama 1985).

Rate-a/-Return Regulation 207

assumed to be satisfied at the solution2 :

(11)

Before proceeding with analysis and comparisons of the optimal paths, some discussion is necessary about the implications of having a value of A = 1. Consider the following system:

It (k, l) = p(kp + ka) - G(kp, ka) + L wjlj + r[S(kp, lp) - A(ka, La)] j

h(k,l) = G,(kp, lp) - Wp - rS,(kp, lp)

13(k,l) = rA(ka,la) - Wa

and define the sets:

R = ((k,l): It(k,l) = O} V = ((k,l): fi(k,l) = 0, i = 1,2,3}

(12) (13)

The elements of set R determine the combinations of capital and labor inputs that satisfy the regulatory constraint for any given p. The elements of set V, on the other hand, are solutions of the non-linear system of It, h, 13 for any given p. Define the set F = R - V. In set F the regulated firm can choose inputs that satisfy the constraint without, however, being able to choose the efficient level of labor inputs. In set V the firm can satisfy both the rate-of-return constraint with equality and choose efficient labor inputs. As is shown in the appendix, under certain assumptions about the structure of the input set, the solution set or junction set V, is a compact subset of the input set Y containing more than one element. It follows from this that there exists a compact set K' such that for each k E K', the regulated firm can choose efficient labor inputs for any given p. Thus, for any (k, l) in the junction set V, AiL This result indicates that efficient choice of labor inputs, when the rate-of-return constraint is binding, can not be regarded as a limiting case satisfied for a unique value of capital input as has been shown to hold by Katayama and Abe (1989) for the case of a regulated utility with one type of capital. In a sense, the increase of the dimension of the input space by introducing abatement inputs provides the firm with greater flexibility so as to adjust its labor inputs in an efficient

2It is further assumed that the following jump conditions are satisfied (Kamien and Schwartz 1981, Seierstad and Sydsaeter 1987)

c/>p(r+) - c/>p(r-) = -(1 [p - (Gk - rSk)J

c/>a(r+) - c/>a(r-) = -(2[p - rAkJ

and (j ~ 0 ( = 0 if the regulatory constraint is ineffective). The jump condition is required because the regulatory constraint does not contain the control variables lp, la.

208 A. Xepapadeas

way. Under these conditions, three different types of paths for the control and the state variables can be distinguished:

i) Unregulated path, A(t) = 0,

ii) Regulated path with efficient choice of labor inputs A(t) > 0, A(t) ::J 1, (h;.,I) E V,

iii) Regulated path without efficient choice of labor inputs A(t) 1, (h;.,I) E F.

3.2 Stability analysis

Assume that A ::J 1, and also that the Hessians of G and A are negative definite and the Hessian of 8 is positive definite. Using the implicit function theorem, (8a) and (8b) can be solved for lp, la to obtain the short run labor input functions:

lj=lj(kj,wj,1'), j=p,a (14)

where, after differentiating totally (8a) and (8b), we obtain:

8lp 1'81k - G1k 8l Alk ak = G 8 ~ 0 as 1'81k ~ G1k , _a = -- > O. (15a)

p II - l' II aka rAil

alp = 1 < 0, ala = _1_ < O. (15b) awp Gil - 1'811 aWa rAil

alp _ 81 < 0 ala = _~ > 0 (15c) a1' - Gil - l' 811 'a1' rAil .

These results can be interpreted as follows. An increase in productive capital will not reduce labor input in production as long as the increase in the marginal revenue of labor resulting from adding one unit to the stock of productive capital, (Gkl), is no less than the increase in the marginal emission cost oflabor associated with the same change in productive capital, (1'8kl). On the other hand, an increase in abatement capital will always increase labor in abatement. As usual, an increase in the wage rates will reduce labor input. Finally, an increase in emission charges will reduce labor input in production and increase labor input in abatement. Since emissions in the short run are defined as:

we have by (15) that an increase in the emission charge will reduce the firm's emissions in the short run.

Rate-o/-Return Regulation 209

From (8c) the short-run demand for investment function is obtained as:

I j = 9j(¢j - b), gj = ~~, > 0, j = a,p (17) J

By substituting (14) and (16) into (1), (lOa) and (lOb), we obtain a system of four differential equations in the (k, ¢) space3 • Let;f denote the vector (k-k*,1!.-1!.*), where (k*,1!.*) denotes the equilibrium point defined

as usual by k:p = k:a = ~p = ~a = 0 and assume that as t ~ 00, A(t) ~ A < 00. From investment theory under adjustment costs it follows that a unique equilibrium point exists with k* > 0 by requiring (¢;, ¢~) > (C;(O), C~(O» (Takayama 1985). The differential system can be linearized around the equilibrium point to obtain the system

;i;. = J;f

where J is the Jacobian matrix evaluated at the equilibrium point.

with

J=

o

-0

o -(1- A)Oa

1 C" p

o (r + 0)

o

o I 1 C"

(rio)

a 8lp 8lp ) Op = ok (Gk - rSk) = (Gkk + Gkl) ok ) - r(Skk + Ski ok < 0

p p p

a ( ) 8la ) Oa = aka (rAk) = Akk + Akl aka < 0

according to the concavity-convexity assumptions made in (AI) and (A2). If A E [0,1] then the equilibrium point (k*, ¢*) is a local saddle point.

Using Dockner's theorem 3 (1985), we have for A E [0,1) :

det(J) = [(liaA)Oa-o(r+o)] [(I~A)op_o(r+o)] >0

K= [(l ia A)Oa- o(r+o)] + [(I~A)op_o(r+o)] <04

with

del(J) < (~)' 3¢> = (¢>p, ¢>k). 4For the definition of f<, see Dockner (1985).

210 A. X epapadeas

Therefore the matrix J has real roots, two of them being positive and two of them being negative, and the equilibrium point has the local saddle point property. This means that there is a two dimensional manifold con­taining the equilibrium point such that, if the system starts at the initial time on this manifold and at the neighborhood of the equilibrium point, it will approach the equilibrium point as t ---- 00.

When A = 1 relations (14) cannot be derived, but this does not cause any problem in local stability analysis since l does not enter the Jacobian of the system which is a non-singular matrix. For this case, det( J) = [8(r + 8)]2 > 0, K = -28(r + 8) < 0 and det(J) < (K/2)2, thus the conditions for a local saddle point are satisfied.

Therefore, provided that A does not exceed unity, both unregulated and regulated paths have the saddle point property. If A> 1, then det(J) could be positive, negative or zero, in which case the stability conditions are not satisfied. Solving (lOa) and (lOb) for A at the steady state and using (8c) we obtain:

A = cp - (Gk - TSk) = Ca - TAk p-(Gk-TSk) P-TAk

where Cj = (r + 8) (b + Cj (Ij )) is the rental price of capital services (cost of capital) in the productive or abatement sector. Thus, A < 1 implies that the regulated return is set above the user cost of capital in both sectors.

4. Comparative Analysis

In this section we examine, by means of comparative static and comparative dynamic analysis, the impact on the optimal level of inputs from introduc­ing rate-of-return regulation when emission charges are already in use, as well as the comparative effects with respect to a regulated or unregulated regime from introducing emission taxes. These effects can be analyzed by determining the effects from variations in p or T.

4.1 Comparative statics

Let pO be the rate of return corresponding to the unregulated path (A = 0). Rate-of-return regulation implies that the maximum allowed rate of return will be p < pO. The direction of change in production or abatement inputs can be determined by the corresponding derivatives with respect to p at the relevant neighborhood (Takayama 1985, EI-Hodiri and Takayama 1981). By taking derivatives with respect to T, the impact of emission charges with or without regulation can also be analyzed. All these derivatives can be obtained by comparative static analysis of the steady state of the dynamic system described by (8) through (11).

Rate-oj-Return Regulation 211

4.1.1 Efficient labor choice

Differentiate (8c) with respect to time to obtain:

(8c')

Use (8c) and (8c') to eliminate ¢j and ;Pj from (lOa) and (lOb). Then by using (1) at the steady state to eliminate I j and assuming efficient labor choice'\ E (0,1), we obtain, along with the regulatory constraint, the following system at the steady state:

o = (r + 6)b + ((r + 8)C~6kp -,\p - (1 - '\)(Gk - r8k)

o (r + 6)b + ((r + 8)C~ 6ka -,\p - (1 - '\)(Gk - r Ak)

o = p(kp + ka) - G + L Wjtj + r(8 - A) j

Differentiating totally with respect to kp, ka, '\, p, and r we obtain the fundamental system of comparative statics:

where

o Ya

-1/Ja

-(1- '\)8k ] [dP] (1 - '\)Ak . -(8 - A) dr

Yj = -(1 - '\)OJ + (r + 6)Cj'6 > 0, j = a,p

1/Jp = [(Gk - T8k) - p]

1/Ja = TAk - P

(18)

Using D = Yp1/J~ + Ya¢; > 0 to denote the determinant of the matrix, the relevant derivatives of the steady state are:

aka 1 ap = D [Ypk1/Ja - '\1/Jp(1/Jp -1/Ja)]

~~ = ~ [Ya1/Jpe - (1 - '\)(1/J~8k + 1/Jp1/Ja Ak)]

aka 1 [ 2 ] aT = D Yp1/Jae - (1 - '\)(1/JaAk + 1/Jp1/Ja 8k)

where k = kp + kaand e = 8 - A > 0, by the assumption of interior solution. To obtain some idea about the signs of these expressions, some observations are in order.

212 A. Xepapadeas

(a) By using (1Oa), (1Ob) in equilibrium, (¢p, ¢a = 0), we have tPP -1/Ja = (Cp - ca )/(I- oX) where (Cp, ca ) is the cost of capital in the respective sector as defined in Section 3.2.

(b) Using (5) to define p for the effective constraint, then tPP and tPa can be defined as follows:

( 0) .1. 1 (M cp - Ca k) 0 • 'f/p = k + 1 _ oX a < as cp :$ Ca

where M = -(G - Gkkp - Gllp) + T(S - Skkp - Sllp) - T(A - Akka -Alla) < 0 under strict concavity/convexity assumptions on G, Sand A.

( ii)

Using these results, some conclusions about the response of the equilib­rium capital stocks to rate-of-return regulation can be reached. The results are essentially in the form of sufficient conditions on the relation between the user cost of capital in each sector, so that over- or undercapitalization takes place.

Response to rate-of-return regulation, oX E (0,1)

(i) If Cp = Ca, then 8kp/8p, 8ka/8p < 0, overcapitalization in both sectors occurs when rate-of-return regulation is introduced, which is an A-J type of result. Labor's response is determined by the derivatives in (15). Abatement labor increases but the effect on productive labor is not straightforward. While abatement will increase, it is not clear what the effect will be on net emissions. (ii) If Cp =1= Ca , any type of effect might occur. It is possible for overcapital­ization to occur in one sector while undercapitalization occurs in the other, depending on the relative rental prices of capital in the two sectors.

Response to emission taxes, oX E [0, 1)

Turning to the effects of introducing emission taxes when rate-of-return regulation is present, it can be seen that if cp = Ca, then 8kp/8T < 0, but the sign of the derivative 8ka/8T cannot be determined a priori. This result introduces a deviation in the behavior of the regulated firm as compared to the unregulated one. The comparative statics for the unregulated firm results can be obtained by setting oX = 0 and ignoring the constraint in (17) as:

Rate-a/-Return Regulation 213

Table 1: Impact of regulation and emission charges on capital. Regulated firm, ..\ E (0,1).

Relation between user costs Impact of regulation Impact of emission

charges

8kp /8p 8ka/8p 8kp /8T 8ka/8T

cp = Ca or 'l/Ja = 'l/Jp <0 <0 <0 ?

'l/Jp > 0 >0 ? ? ?

Cp > Ca or 'l/Jp < 0 'l/Jp =0 >0 <0 <0 <0

'l/Jp < 0 ? ? <0 ?

'l/Ja > 0 ? >0 ? ?

cp < Ca or 'l/Jp < 0 'l/Ja =0 <0 >0 <0 >0

'l/Ja < 0 ? ? <0 ?

Unregulated firm, ..\ = 0

any relation <0 >0

between cP ' Ca

8kj /8p < 0 : overcapitalization, 8kj /8p > 0 : undercapitalization.

Therefore, while the unregulated firm responds to the introduction of emission charges by reducing the productive capital and increasing the abatement capital (see also Xepapadeas 1992), the regulated firm will re­duce productive capital but the effect on abatement capital is far more uncertain. This, of course, will affect the final emissions under unregu­lated or regulated regimes. The results implied by the comparative static derivatives are summarized in Table 1.

Although unambiguous comparisons are not always possible, it can be noticed that A-J type effects occur if the rental prices of capital are close to each other. If there are big differences in the user costs, then the sector with the high cost will tend to undercapitalize. This result can be explained by the fact that under rate-of-return regulation, the firm has to choose both the level and the composition of its capital base. When significant user cost differentials between productive and abatement capital exist, the firm might be able to achieve the profit-maximizing capital base by expanding the less

214 A. X epapadeas

expensive type of capital and contracting the more expensive, instead of expanding both of them. This observation could have some empirical sig­nificance since differences in the user costs reflect differences in adjustment costs.

The introduction of emission charges will most likely reduce produc­tive capital under regulation. The effects on abatement capital are more ambiguous. There are cases (c" > Ca , 'ljJp = 0) where emission charges un­der regulation reduce both types of capital. This ambiguity with respect to the abatement capital can be attributed to the effects of the regulatory constraint. An increase in emission taxes effectively increases the cost of productive capital and decreases the cost of abatement capital. Thus the unregulated firm will respond by reducing productive capital and increasing abatement capital. Under rate-of-return regulation, however, an increase in emission taxes might also reduce the profit-maximizing capital base. If by reducing productive capital, gross emissions have been reduced to a range such that further tax savings due to increased abatement are negligible, the lower capital base could be achieved by reducing both types of capital.

A

o

Figure 1. Equilibrium outside the junction set, ,\ = 1. ¢f = pd(r + 8), i = 1,2,3.

4.1.2 Non-efficient labor choice

Up to this point the analysis has been carried out under the assumption

Rate-oj-Return Regulation 215

that equilibrium inputs belong to the junction set V. In the remaining part of this section, this assumption is relaxed and the case where k ¢ K' and >. = 1 is considered. The optimality conditions for this case are derived from conditions (8) to (11) for>. = 1. In particular, we have:

4>j = b + Cj(Ij) (8d)

p(kp + ka) - G(kp, lp) + L wjlj + r[S(kp, lp) - A(ka, La)] = 0 (9') j

(10')

The Arrow type transversality conditions are also assumed to be satisfied. As has been shown in Section 3.2, the equilibrium point has the saddle

point property. It is not possible, however, to carry out comparative analy­sis of the steady state, since the Jacobian of the comparative static matrix (17) vanishes. In fact the equilibrium capital stock is obtained by (16) and (10) as:

which is independent of the functions G, S, A. Nevertheless, comparative static results can be obtained by using phase diagram analysis.

Response to rate-of-return regulation, >. = 1 In constructing the phase diagram in Figure 1, it is assumed that the kj = 0, ¢j = 0 loci are locally defined for k ¢ K' and furthermore that Gj' = {3j,

f3p > f3a, cp > Ca. These assumptions imply that the kp = 0 locus lies above the ka = 0 locus. For the unregulated case the ¢~ = 0 locus lies above the

¢~ = 0 locus, while for the regulated case the ¢y = 0 loci coincide with

4>: = 4>: = pj(r + 8). Unregulated equilibrium for productive and abatement capital takes

place at points El and E2 respectively. The responses of productive and abatement capital to the introduction of rate-of-return regulation depend on the position of the ¢y = 0 locus and are summarized in Table 2.

The results in this table indicate that capital's response to the regula­tion depends on the level of the regulated return and the relation between user costs of capital. High returns above A in Figure 1 lead to A-J type effects. For comparatively low returns below B, the shadow price of capi­tal is low, leading to contraction of the profit-maximizing capital base and under-capitalization for both types of capital. These results are comparable to those obtained by Katayama and Abe (1989). When the effects are in opposite directions, the sector with higher user costs tends to undercapital­ize if the return is not sufficiently high, while the sector with the lower user

216 A. Xepapadeas

Table 2: Table 1: Impact of regulation on capital, oX = 1, cp > Ca.

Example (Fig. 1)

p: ¢ = pj(r + 8) is kp ka Equilibrium point

¢

kp ka

¢>A O.C. O.C. ¢1 C D

¢=A N O.C. ¢=A E1 F

B<¢<A U.C. O.C. ¢2 G H

¢=B U.C. N ¢=B J E2

¢<B U.C. U.C. ¢3 M N

O.C. = overcapitalization; U.C. = undercapitalization. N = neutrality (no change to the unregulated case).

cost overcapitalizes if the return is not sufficiently low. This is a pattern of response similar to the case where oX E (0,1).

Response to emission taxes, oX = 1 The response to emission taxes when oX = 1 can be analyzed by differ­

entiating totally the regulatory constraint to obtain:

ak* -'!!.. ar

[p- (GZ,rSk)] S*

ak~ p- rAZ = :.-.--:--.!!:. ar A*

(19)

where kj is the equilibrium capital stock when oX = 1 and labor inputs take values in the set H: h(lp, ia I k;, k~) = O}. The signs of the derivatives in (18) can be determined as follows:

ak:5:5 p :5 Gk - rSk a: =~ 0 as p ~ G - rSk, or as r + 8 ~ r + 8 (19a)

with all expressions evaluated as k;. It follows from the last inequality that

akpjar is positive, zero or negative as the ¢: = 0 locus for oX = 1 lies below

or above the ¢~ = 0 locus at k;. In the same way,

aka :5 p :5 r Ak ( b) - => 0 as p = rAk or as -- > --. 19 ar - 'r+8-r+8

Rate-oJ-Return Regulation 217

Table 3: Impact of emission taxes on capital, >. = 1, Cp > Ca.

p: rP = p/(r + 8) is akp/ar aka/or

rP>A <0 >0

rP=A =0 >0

B<rP<A >0 >0

rP=B >0 =0

rP<B >0 <0

That is, aka/r, is positive, zero or negative as the ¢;; = 0 locus for>. = 1 lies above or below the ¢~ = 0 locus at k~.

These results can be clarified with reference to Figure 1 where, for equilibrium at C, it holds that the ¢R = 0 locus lies above the ¢~ = 0 locus at k; and thus akp/r < o. The rest of the equilibria in Figure 1 can be analyzed in the same way. All the possible results are summarized in Table 3.

It seems that "normal" responses in accordance with the unregulated case can be expected at relatively high returns. At low returns, a possible contraction of the profit-maximizing capital base might lead to abnormal responses.

It should be noted that changes of the assumptions about the relation of user costs in production or environmental regulation will change the re­sponses under regulation or emission taxes. There is a pattern, however, indicating that high returns lead to A-J type effects and to "normal" re­sponses when emission taxes increase, and also that the sector with higher user cost tends to undercapitalize if the regulated return is not sufficiently high.

Finally, of interest might be the possibility of having unregulated equi­librium in the junction set but regulated equilibrium outside the junction set. Figure 2 illustrates this possibility for the case of abatement capital. It is assumed that Ka = [8, T] and K~ = (8, U]. Unregulated equilibrium takes place at point E. If all points in [8, T] were in the junction set, reg­ulated equilibrium for A-J type effects would take place at E'. This is not, however, possible. In (U, T], >. = 1 and the equilibrium point depends on the position of the rPa = p / (r + 8). To determine the position of rPa, suppose that E' was feasible with k~ as the equilibrium capital stock. The following

218 A. Xepapadeas

inequalities are satisfied at the equilibrium point, k~.

For A = 1, we obtain by dividing the last inequality by (r + 8) :

p rAk 4>a = r + 8 > r + 8·

This implies that 4>a will be above B. If the regulated return is sufficiently high so that 4>a is above C (e.g., 4>1), then A-J type of effects will take place. A possible optimal path is depicted by (DFGH). If the regulated return is sufficiently low so that 4>a is below C (e.g., 4>2), then overcapitalization is not possible. The optimal path goes along (DF) for an initial period, then at the boundary of the junction set, switches to the unregulated path (JE). The effect of this case is neutrality. Similar analysis can be carried out for productive capital.

o I . I tP~ = 0

s u k!

Figure 2. Unregulated equilibrium in the junction set. Regulated equilib­rium in the junction set 4>f = pd(r + 8) i = 1,2.

In general, the effects of having the equilibrium point moving outside the junction set when rate-of-return regulation is introduced, depend on the structure of the junction set and the level of the regulated return. High returns will tend to produce A-J type effects5 while low returns might result in undercapitalization or neutrality. In the latter case, the optimal path switches from the regulated to the unregulated path.

5Ifthe upper bounds of Kj and Kj are close, then high returns will result in neutrality. The optimal path will switch from the regulated path to the unregulated path.

Rate-oj-Return Regulation

4.2 Steady state emissions response to regulation and environmental policy

Steady state emissions are defined as:

219

where kj, lj take their steady state values. Therefore emissions response to regulation or environmental policy is determined by the sign of the following derivatives:

8eOC = (8S 8kp + 8S atp ) _ (8A 8ka + 8A ata )

8p 8kp 8p atp 8p 8ka 8p ata 8p

8eoc = (8S 8kp + 8S atp ) _ (8A 8ka + 8A ata ).

8T 8kp 8T atp 8T 8ka 8T ata 8T

In these expressions, the derivatives 8kj j8p, 8kj j8T j = p, a have already been determined in the comparative static analysis. Labor's response at the steady state, on the other hand, is determined by the following long-run derivatives:

at· at· 8k· atj _ atj 8kj atj . _ 8; = 8~ 8;' 8T - 8kj 8T + 8T' J - p, a.

It should be noted that the changes in the return p have only long-run effects on labor, through adjustments in the stocks of capital. Changes in the emission charge have both short-run direct effects and long-run effects, through changes in the stocks of capital, on labor. Since the results of the previous sections indicated the possibility of A-J or reverse effects, the direction of the change of eoc can not be determined a priori. For example, with reference to Table 1, if cp > Ca and Gk = TSk, ('ljJp = 0), that is the emission charge is sufficiently high, then introduction of regulation will increase emissions, while an increase in the emission tax under regulation will have ambiguous results on emissions since both types of capital will be reduced. Comparing these results with those of Xepapadeas (1992), we have that while without regulation an increase in emission charge is expected to reduce emissions, the presence of rate-of-return regulation disturbs things and the actual outcome depends largely on the parameters of the model. Emission taxes under regulation might even increase emissions.

4.3 Comparative dynamics

The impact of introducing regulation or emission charges on the optimal solution can be analyzed by using comparative dynamic analysis. Two approaches will be used: the first introduced by Oniki (1973), examines the

220 A. Xepapadeas

effects of changes of p or T on the entire optimal path of the capital stocksj the second uses the dynamic envelope theorem developed by Caputo (1990) and LaFrance and Barney (1991) to examine the behavior of cumulative functions when p and T change.

Oniki's method uses phase plane analysis. Therefore in order to apply the graphic method, the problem should be reduced to a two-dimensional one. Suppose we concentrate on the abatement sector. Solution of (lOa) and (1) after using (8c) to eliminate rpp, and assuming G';' = f3jto simplify things, will result in solutions kp = kp(tj >., p, T), Ip = Ip(tj >., p, T). Substi­tute these in (9), for>. > 0, and solve for>. to obtain>. = >.(tj p, T, ka ). The function for >. can be substituted into the dynamic system for the abate­ment sector to obtain solutions ka = ka(tj p, T), Ia = Ia(tj p, T). If these solutions are inserted back into the system for the abatement sector, the following identities are obtained:

where >. == >.(tj p, T, ka(tj p, T»

ka == Ia(tj p, T) - 8ka(tj p, T)

(20a)

(20b)

Considering perturbations in p near pO, the following variational differ­ential equation system is obtained by differentiating system (20)

. 1 (Ia)p = (r + 8) (Ia)p + f3a {(TAk - p)(>'p + >'ka(ka)p) - (1- >')Oa(ka)p - >.}

(21a) (ka)p = (Ia)p - 8(ka)p (21b)

where

Since ka(O) = k~ is fixed, we have for the perturbed initial condition

(21c)

Assuming positive capital stocks and investment at the steady state, the perturbed terminal conditions are determined from Table 1, by taking into account that dlj = 8dkj in equilibrium. For example, we have

(k:)p (I;:»p < 0 for cp > Ca, 'l/Jp = 0,

(k:)p (I;:»p > 0 for cp < Ca, 'l/Ja = o. (21d)

Rate-oj-Return Regulation 221

Using the sign restrictions for the parameters of (21a) and (21b) and that 8)..j8p < 0 at the relevant neighborhood of p, the optimal perturbed arcs corresponding to the terminal conditions (21d) are shown in Figure 3.

It can be seen from the diagrams that whether overcapitalization or un­dercapitalization occurs, there might be an initial period where investment will respond in the opposite way. For example, for the case in Figure 3, al­though overcapitalization occurs due to regulation (Figure 3a), investment might be reduced at some initial period. Making alternative assumptions about the perturbed terminal conditions, all the cases of Table 1 can be examined.

(a) (b)

Figure 3. Optimal perturbed arcs from changes in p. (a) Ca < Cp, 'l/Jp = 0 (b) Ca > cp , 'l/Ja = O.

The effects of introducing emission taxes can be analyzed in a similar way by considering perturbations in r. Differentiating the system (20) with respect to r, the following variational system is obtained:

. 1 (Ia)'T = (ra+8)(Ia)'T+ {Ja {(rAk-p)(A'T+Aka(ka)'T)-(I-A)na(ka)'T-(I-A)Ak}

(22a)

(ka)'T = (Ia)'T - 8(ka)'T (22b)

(22c)

Perturbed terminal conditions are determined according to Table 1, thus

(k';')'T > 0 for 'l/Jp < 0, 'l/Ja = O. (22d)

The unregulated case in the presence of emission taxes can be examined if we set ).. = 0 in (20) or (22) with perturbed boundary conditions

222 A. Xepapadeas

Optimal perturbed arcs for the variational system (22) and the corre­sponding unregulated case are shown in Figure 4.

(a) (b)

Figure 4. Optimal perturbed arcs from changes in T. (a) Regulated firm Ca < Cp, 1/Ja = o. (b) Unregulated firm.

It can be seen from the diagrams that although the regulated firm in­creases abatement capital, at some initial time period abatement investment might be reduced.

The effects of perturbation in p when A = 1 can be analyzed in a similar way by setting A = 1 in system (21)6.

To apply the dynamic envelope theorem 7 , we define the optimal paths for If and r/>, obtained as solution of the corresponding differential sys­tem. The -optimal paths are functions of the parameters of problem (6), that is, kj(t;!!.), r/>j(t;!!.) where!!. is the vector of parameters of (6), !!. = (wp ,Wa ,b,T,p,k2,k?,r,8). The open loop solutions for labor and in­vestment inputs are defined as:

I j = gj(r/>jCt,!!.)) == g;(t,!!.).

Substituting into (4), the maximum instantaneous profit function is ob­tained as II*(t;!!.), then the optimal value function is defined as:

To analyze effects of changes in p and T, the dynamic envelope theorem is used to obtain:

6For >.. = 1, it is not possible to examine the effects from changes in T.

7See also Xepapadeas (1992) for an application of this method in the analysis of the dynamic behavior of the firm.

Rate-oJ-Return Regulation 223

(i) Effects from changes in p

{)J*(v) roo {)(e-rt £*) roo {)p- = io 8p dt= io e-rt),(tj~)[k;(tj~)+k~(tj~)]dt

where £* is the current value Lagrangean evaluated along the optimal path, and ),( T,~) is the solution for)' as described in the previous section. Thus, the partial derivative of the optimal value function with respect to the rate­of-return is the cumulative discounted demand for total capital (productive and abatement) weighted by the multiplier), E (0,1]. If ), = 1, that is, we have inefficient labor choice, the cumulative discounted demand for capital takes its maximum value.

(ii) Effects from changes in T

{)J;;~) = 100 8(e;;£*) dt = 100 e-rt(l_ ),(tj~))[S(k;(tj~),l;(tj~))-

-A(k~(tj~),l~(t,~))] :::; o. Thus the partial derivative of the optimal value function with respect to emission charges is the negative of the cumulative discounted emission func­tion weighted by (1 - ),). It is interesting to note that if), = 1, that is, the firm is not constrained to choose the efficient labor input, but any level of labor input that satisfies the regulatory constraint, then {)J* I {)T = o. An increase in the emission charges will not affect the present value of the profits for the firm.

5. Summary and Conclusions

A monopoly whose behavior is controlled by rate-of-return regulation could be additionally subjected to environmental regulation in the form of emis­sion taxes, if its productive activities result in ambient pollution. The purpose of this paper is to analyze the effects of this joint regulatory frame­work on the investment decisions of the firm, with respect to productive and pollution abatement equipment. Specifically, the impact of rate-of-return regulation on investment when the regulated firm has to follow a prespec­ified environmental policy is analyzed, and also the effects of introducing environmental policy, in the form of emission charges, on regulated and unregulated regimes are compared.

The results obtained indicate that the effect on the firm's investment policy in productive and pollution abatement equipment seems to depend on three main factors: the differences between adjustment costs in the productive and abatement sectorj the possibilities that the regulated firm has in choosing efficient labor inputsj and the level of the regulated return. The combination of these factors could result in A-J or reverse type effects in one or both types of investment.

224 A. X epapadeas

Assuming that the firm can choose efficient labor inputs in the junction set, the effects of regulation have been shown to depend largely on the adjustment cost differentials between productive and abatement capital. In general overcapitalization tends to occur in the sector with relatively lower adjustment costs. If there are no cost differences, A-J type effects occur in both sectors.

Comparing the effects of emission charges on unregulated and regu­lated regimes, it was shown that, although productive capital is likely to be reduced as a result of the policy in both regimes, there is a difference in responses with respect to abatement capital. The unregulated firm will increase abatement capital; the regulated firm's behavior will depend, how­ever, on adjustment cost differentials.

When the firm chooses capital inputs outside the junction set, then the response depends mainly on the level of the regulated return. High returns support overcapitalization while low returns support undercapitalization. Furthermore, the sector with the higher adjustment costs tends to under­capitalize if the returns are not high enough. For low returns the response to an increase in emission taxes might be "abnormal" as compared to the regulated case.

The significance of these results lies in the fact that they reveal that un­der certain circumstances, the objectives of environmental policy might be reversed by the presence of a binding rate-of-return regulatory constraint. Consider the case where dlp/dkp > 0 in (15). If investment in productive capital is characterized by high adjustment costs and the regulatory return has an intermediate value, then net emissions will be reduced as a result of regulation (Tables 1, 2 and Figure 1). Net emissions might, however, increase if investment in abatement capital is characterized by high adjust­ment costs. A similar pattern of responses prevails when the regulated firm is subjected to emission taxes. Under certain circumstances, emission taxes could result in increased net emissions.

Although the circumstances under which the various effects described appear mainly an empirical issue, especially where adjustment costs are concerned, the analysis indicates that the presence of rate-of-return regula­tion could affect the investment policy of a firm in a way such that objectives of environmental policy formulated in terms of restricted emissions could be impeded.

In the context of the present analysis, areas of further research could include analysis of other types of regulation, like the mark-up regulation or the price cap regulation, on the behavior of the firm with respect to the introduction of emission-reducing technologies.

Rate-oJ-Return Regulation 225

Appendix

In this appendix, we show that the junction set V is a compact subset of the input space Y containing more than one element. It follows that there exists a compact set K' such that for each k E K', the regulated firm can choose efficient labor inputs when the regulatory constraint is effective.

Let Y = Y' x La with Y' = Kp x Ka X Lp. Since K j and Lj , j = p,a are compact sets, Y and Y' are also compact. For each l~ E La the system of Ii (kp, ka, lp, l~) i = 1,2,3 is a non-linear system of three equations with three unknowns (kp, ka, lp). The solution of this system, provided it exists, for any given l~ determines the junction set V. Assume that inputs in set Y' satisfy the boundedness restrictions" (kp, ka , lp) 11< B for a finite bound B and that Y' is homeomorphic to the unit disk D3 = {;r E R3 :11 ;r II:::: I}. That is, there exists a one-to-one mapping, " of Y' onto D3 , such that, and ,-I are continuous. Solutions of the system of Ii can be regarded as zeroes, that is equilibria, of the continuous vector field I : y f-' R 3 . Assume that the vector field points in on the boundary of Y':-This assumption implies that the elements of the junction set do not correspond to extremely high or low values of the inputs in space Y'. Under the assumption made above, the following proposition can be shown to hold: (1) The vector field I has, in general, an odd (hence non-zero) number of equilibria. This means that the system of Ii has, in general, an odd number of solutions.

Proof. Let yO = (k~, k~, l~) be a point which is a zero of the vector field I (or the system of equations Ii that is li(yO,I~) = 0, i = 1,2,3. The index of yO is defined as:

+1 if det(-J(yO)) > 0

-1 if det(-J(yo)) < O.

where J(yO) is the Jacobian matrix of the vector field at yO (Milnor 1965, Varian 1983). By the Poincare-Hopf theorem, the sum of indexes of the different zeroes of I equals the Euler characteristic of Y'. Since Y' is home­omorphic to the unit disk, its Euler characteristic is X(Y') = +1 (Mas-Colell 1985). If we denote with I(y!) the index at the point y!, m = 1, ... , M where L has a zero, the Poincare-Hopf theorem means that

M

L I(y~) = +l. m=l

So, as long as zero is a regular value (that is, the Jacobian at y! does not vanish), there exists an odd number of zeroes for the vector field L •

226 A. X epapadeas

It should be noticed that a uniqueness result cannot be established. For uniqueness, it is required that the negative of the Jacobian determinant at all equilibria be positive. If this requirement is satisfied, the index at all equilibria is + 1 and the sum of the indexes at all equilibria is + 1. It follows, therefore, that there can be only one equilibrium. For this problem, the negative of the Jacobian determinant at equilibrium can be written as:

p - G2 + r S - kO p - r A - kO det( _Jo) = - G?k - rSPk 0

o rA?k

-G? +wp +rSp G~ - rSR

o

= -rA?dp + (G2 + rSZ)(G~ - rSR)]

since -G? + wp + rSp = 0, and at all equilibria due to the assumptions on A, G, S functions. Thus, for regular equilibria

> > det(1 _Jo I) < 0 as p - G2 + rS - kO < O.

However, as shown in Section 4.1.1, the sign of this term depends on the relative rental price of capital in the two sectors. Therefore, an invariant sign for det( -JO) cannot be established.

(2) For any l~ E La there exist upper hemicontinuous correspondences Fl(l~) = {kp }, F2(l~) = {ka}, F3(1~) = {tal. The image of La under Fi ,

i = 1,2,3 is a compact set.

Proof. The existence of correspondences Fi(I~) can be established by us­ing the implicit correspondence theorem (Mas-Colell 1985). Since La is compact and Fi upper hemicontinuous, the image of La under Fi defined as

is a compact set. • From propositions (1) and (2), it follows that the junction set defined

as V = Fl(La) x F2(La) x F3(La) x La is a compact subset of Y. Define the compact set K' = Ft(La) x F2(La)' For any k E K'

the regulated firm can choose efficient labor inputs in the set, defined as F3(la) = {ta} under binding rate-of-return constraint.

This result can be compared to the standard A-J problem when no abatement inputs are involved. Making assumptions similar to the ones made above about the structure of the input set, the efficient capital/labor combinations when the regulatory constraint is effective are determined at a zero of the vector field

h (k, I) = pk - G(k, I) + wI = 0

Rate-oj-Return Regulation 227

h(k,l) = G,(k,l) - w = o. So for any return p, the junction set is defined as the combination of (k, l) that solves the above system. From the second equation we obtain, us­ing the implicit function theorem, k = h(l, w). Substituting in the first equation, we obtain:

0= ph(l, w) - G(h(l, w), l) + wl = !tel, p)

with

°fzl = ph' - (Gkh' + G,) + w = h'(p - Gk) =1= 0

since G, = w by the optimality condition, and p - Gk > 0 due to the strict concavity of the G function. Therefore, using the implicit function theorem, we have that for any p = pO, there is a point (kO, lO) that satisfies both the regulatory constraint !t and the efficient labor condition h. This equilib­rium point is the unique junction point (kO, lo) determined by Katayama and Abe (1989). For this point only, it holds that A =1= 1.

References

[1] K.J Arrow and M. Kurz, Public Investment, the Rate of Return, and Optimal Fiscal Policy, Johns Hopkins Press, Baltimore, 1970

[2] H. Averch and L.L. Johnson, Behavior of the firm under regulatory constraint, American Economic Review, 52, 1052-1069, 1962

[3] R.R. Braeutigan and J.C. Panzar, Diversification incentives under "price-based" and "cost-based" regulation, Rand Journal of Eco­nomics, 20, 373-391, 1989

[4] W.J. Baumol and K. Klevoric, Input choices and rate-of-return regu­lation: An overview of the discussion, Bell Journal of Economics, 1, 162-190, 1970

[5] W.J. Baumol and W. Oates, "The Theory of Environmental Policy," Cambridge University Press, 1988

[6] V.S. Bawa and D.S. Sibley, Dynamic behavior of a firm subject to stochastic regulatory review, International Economic Review, 21, 627-642, 1980

[7] R.M. Caputo, How to do comparative dynamics on the back of an envelope in optimal control theory, Journal of Economic Dynamics and Control, 14, 655-683, 1990

228 A. Xepapadeas

[8) W.D. Dechert, Has the Averch-Johnson effect been theoretically justi­fied?, Journal of Economic Dynamics and Control, 8, 1-17, 1984

[9) E. Dockner, "Local stability analysis in optimal control problems with two state variables," in: Optimal Control Theory and Economic Anal­ysis 2, G. Feichtinger, ed., North-Holland, 1985

[10) M.A. El-Hodiri and A. Takayama, Behavior of the firm under regula­tory constraint: Clarifications, American Economic Review, 63, 235-239, 1973

[11) M.A. El-Hodiri and A. Takayama, Dynamic behavior of the firm with adjustment costs under regulatory constraint, Journal of Economic Dynamics and Control, 3, 29-41, 1981

[12) P. Joskow, Inflation and environmental concern: Structural change in the process of public utility regulation, Law and Economics" 17, 219-237, 1979

[13) M.1. Kamien and N.L. Schwartz, Dynamic Optimization: The Calcu­lus of Variation and Optimal Control in Economics and Management, North-Holland, 1981

[14) S. Katayama and F. Abe, Optimal investment policy of the regulated firm, Journal of Economic Dynamics and Control, 13, 533-552, 1989

[15) J.T. LaFrance and L.D. Barney, The envelope theorem in dynamic optimization, Journal of Economic Dynamics and Control, 15, 355-385, 1991

[16] A. Mas-Colell, The Theory of General Economic Equilibrium: A Dif­ferential Approach, Cambridge University Press, 1985

[17] J. Milnor, Topology from a Differentiable Viewpoint, University Press of Virginia, 1965

[18] H. Oniki, Comparative dynamics (sensitivity analysis) in optimal con­trol theory, Journal of Economic Theory, 6, 265-283, 1973

[19] D.W. Peterson and J.H. Van der Weide, A note on the optimal in­vestment policy of the regulated firm, Atlantic Economic Journal, 4, 51-56, 1976

[20] A. Seierstad and K. Sydsaeter, Optimal Control Theory with Economic Applications, North-Holland, 1987

[21) R. Sherman, The Regulation of Monopoly, Cambridge University Press, 1989

Rate-oj-Return Regulation 229

[22) A. Takayama, Behavior of the firm under regulatory constraint, Amer­ican Economic Review, 59, 255-260, 1969

[23) A. Takayama, Mathematical Economics, Cambridge University Press, 1985

[24) H. Varian, "Dynamical systems with applications to economics," in: Handbook of Mathematical Economics Vol. I, K.J. Arrow and M.D. Intriligator, eds., North-Holland, 1981

[25) A. Xepapadeas, Environmental policy, adjustment costs and behavior of the firm, Journal of Environmental Economics and Management, 23, 258-275, 1992

Department of Economics, University of Crete, Crete, Greece

Polluter's Capital Quality Standards and Subsidy-Tax Programs

for Environmental Externalities: A Competitive Equilibrium Analysis

Michele Moretto!

Abstract

The paper concentrates on the role of the physical features of the fixed assets in determining the extent of discharges. It considers the case where the firms have access to a technology which allows one to regulate the quality of capital instantaneously through a lump-sum maintenance expenditure which applies only when the state variable hits a predetermined minimum quality standard. In a partial equilib­rium framework (single firm and a long-run competitive industry) the paper investigates the relationship between the optimal firm's bar­rier policy comprising the capital's minimum quality standard and the use of a subsidy/tax program for decreasing pollution emissions by those who generate externalities.

1. Introduction

There is a basic consensus in the technical literature about the fact that the amount of pollution emissions not only depends on the use of pollutant inputs that are intrinsically liable to be dispersed in the environment, but also on the quality of physical capital used in production. In agriculture, for example, the extent of emissions (e.g., losses of nutrients and pesticides) and, consequently, the required abatement measures, are often significantly correlated to the physical features of the site where polluting activities take place (e.g., the soil's water retention capacity).

This dependency may occur in the context of traditional point source problems as well as in that of non-point source (NPS) problems where in­dividual emissions cannot generally be monitored at reasonable costs, or inferred from observation of ambient pollutant concentrations. To over­come this lack of observability a line of research, often referred to as an

1 A preliminary version of this work was supported by the Foundation ENI-Enrico­Mattei as part of the research project "Energy and Environment: Markets and Policies" . I am grateful to Cesare Dosi, Nunzio Cappuccio, Diego Lubian and participants at sem­inars at University of Padva and Foundation ENI-Enrico-Mattei for helpful comments. The usual disclaimer applies.

232 M. Moretto

"indirect approach" (Dosi and Moretto, 1993), suggests the possibility of basing regulatory policies on estimated emissions as determined by avail­able bio-physical models (Griffin and Bromley, 1982; ShortIe and Dunn, 1986; Dosi and Moretto, 1992, 1993, 1994; Moretto and Graham-Tomasi, 1994).

Most of the available models, besides production patterns and man­agement practices, explicitly account for the role played by the physical characteristics of the fixed assets in determining the extent of pollutant run-offs, so that, other things being equal, estimated emissions vary across firms according to their capita1's physical endowments.

Following a previous line of research (Dosi and Moretto, 1992, 1993, 1994), in this paper we concentrate on the capital's physical characteristics ("capital quality") which, besides affecting production possibilities, accord­ing to the selected bio-physical model, are considered key parameters for estimating pollutant emissions at source. The evolutionary pattern of the state variable, capital quality, is assumed to be affected by stochastic ex­ogenous shocks which make future capital quality status uncertain with an increasing variance.

In addition, we allow the firms access to a technology for regulating the quality of physical capital through lump-sum maintenance expendi­ture, which applies only when the state variable hits a predetermined lower barrier. Thus, in our model, the term "barrier policy" refers to the firm's decision with regard to the capita1's minimum quality standard at which lump-sum maintenance expenditures come into effect.

Within this framework, we investigate the efficiency of a subsidy/tax program as an instrument of environmental policy for an individual firm as well as for a long-run competitive industry. In particular we explore the role played by the optimal firm's barrier policy comprising the capital's minimum quality standard in supporting the fiscal policy for decreasing pollution emissions. The type of subsidy considered involves a payment to the firm based on current reductions in its emissions, estimated through the selected bio-physical model, against a benchmark level of emissions announced by the authority. The pure Pigouvian tax system emerges as a limit case when the benchmark level is set at zero.

The problem has been extensively analyzed in the literature in a partial equilibrium framework. Although in the short-run subsidies and taxes are considered as substitutes, in the long run, where the entry and exit process is permitted, the use of subsidies can lead to inefficiencies (Baumol and Oates, 1985).

In the next section we present the basic model and notation. Section 3 is devoted to the solution of the individual firm's dynamic optimization problem and conditions under which fees and subsidies do not influence the optimal barrier policy. In the same section we also explore how the

Polluter's Capital Quality Standards 233

lump-sum maintenance expenditure and the subsidy/tax system adopted may determine whether or not the firm continues its operations. It will be shown that the pure tax system encourages exit, which may lead to a lower level of emissions for each firm actually in business.

Next, in Section 4, we deal with the steady-state equilibrium of a com­petitive industry as a whole. By allowing heterogeneity of firms regarding their physical characteristics a process of selection takes place. The less productive firms leave the industry, while new entrants come in.

The importance of the firm's capital quality standard and the bench­mark pollution level in determining the effect of the fiscal programs are evident. The Pigouvian tax program increases the selection effect. Fewer firms are induced to enter and remain in the industry, with a consequent fall in the total level of emissions.

Moreover, an increase in the marginal cost of regulation increases the selection effect and reduces the capital quality standard, so that if the capital quality standard chosen by the firms is not sufficiently large to ensure profits for all the firms actually in business the number of firms deciding to enter and remain in the industry falls, with a consequent fall in the total level of emissions. On the other hand, if the capital quality standard chosen is able to ensure profits for all the firms actually in business, more firms may enter increasing the total level of emissions. Section 5 summarizes the results.

2. The Basic Model

A. Capital Quality, the Firm's Marketable Output,

and Estimated Pollutant Emissions

Environmental damages are considered as being dependent on the flows of pollutant emissions at source which, although not directly observable, can be estimated through bio-physical models. According to the model which has been granted with "political" legitimacy, emissions are a function of the firm capital's physical characteristics.

Zt = Z(Ot), (1)

where Ot represents an index for the firm's capital quality at time t.

Assumption 1. Indicating with dOt > 0 an improvement in the capital's overall quality, Z(O) holds the following properties: Z'(O) < OJ Z"(O) > OJ Z(oo) = OJ Z'(O) = -00 and Z'(oo) = o.

As well as affecting the extent of pollutant emissions, the capital quality also affects the firm's production possibilities

(2)

234 M. Moretto

where Qt represents the firm's marketable output at time t.

Assumption 2. Q(O) holds the following properties:

Q'(O) > OJ Q"(O) < OJ Q(O) = OJ Q'(O) = 00 and Q'(oo) = O.

B. The Subsidy/Tax Scheme

Following the definition given by Baumol and Oates (1989, p. 214), the type of subsidy with which we will be concerned involves a payment to the firm based on the reduction in its output of a pollutant or in some other sort of damage to the environment. In particular, taking Zt to be the level of the firm's emissions at time t, and Z to be the benchmark against which improvement is to be measured, the subsidy payment can be described by a relationship where the payments to the firm increase with the amount by which it decreases its emissions2 •

For the sake of simplicity, we will assume a linear relationship, with the subsidy payment per unit of emissions being constant over time, so that the payment to the firm becomes

(3)

where v and Z are constant3 •

This will be positive, if and only if Z > Zt, that is, the benchmark emission level is set higher than the firm's current level of emissions under the subsidy program. On the contrary, if Z < Zt, the firm will receive a negative subsidy (pay a tax) proportionate to the deviation between the benchmark emissions level and the actual level. Hereafter (3) will be men­tioned interchangeably as a subsidy or subsidy/tax scheme4 •

If, in the above scheme, we set Z = 0, we get the pure tax system with the firm paying a fixed Pigouvian tax v per unit of emissions.

2In the traditional literature of environmental economics both the subsidy per-unit of emissions and the tax per-unit of emissions are set equal to the marginal social evaluation of the environmental damage, at the point where this coincides with the marginal private benefit of emissions. In this literature, typically, taxes and subsidies correspond to two different bargaining processes relative to different assignments of property rights regarding environmental resources. The subsidies simulate a situation in which the property rights are assigned to polluters, while taxes simulate a situation in which the property rights are assigned to the victims of the pollution. For a detailed analysis of the feasibility of the two fiscal policies see Musu, (1991).

3The fact that Z is assumed to be constant over time is not relevant for the results obtained in this paper; it reflects the hypothesis whereby the regulator is not interested in changing his benchmark value for the firm's capital quality over time.

4We do not consider the case of a pure subsidy system where the firm receives a subsidy as long as Z > Zt and zero otherwise.

Polluter's Capital Quality Standards 235

The fundamental difference between programs of Pigouvian taxes and subsidies is immediate. With taxes, we need to specify only the tax rate, but a system that also involves subsidies requires that we specify values for two parameters: the unit subsidy v and the benchmark level of emissions z.

Moreover, since we have assumed that the flows of pollutant emissions can be estimated through the bio-physical model (1), specifying the bench­mark level Z implies specifying a benchmark value for the firm's capital quality index 0. This allows us to describe the subsidy payment by the re­lationship between 0 and the current value Ot. In particular, the subsidy (3) will be positive as long as the benchmark capital quality 0 is lower than the current value Ot.

C. The Dynamics of Capital Quality

The state variable capital quality, Ot, follows an Ito's process, regulated with a lower barrier S ~ o. At Ot = S, an infinitesimal control dLt is applied to Ot and gives it a "push" upward. Overall, the stochastic differential equation for Ot is

0(0) = 00 . (4)

/1 > 0 stands for the capital quality's constant depreciation rate and 0-for the intensity of fluctuations. {Wt } is a standard Wiener process (or Brownian motion) with E{dWt } = 0 and E{(dWt}2} = dt. {Lt } is a right­continuous, non-negative and non-decreasing stochastic process, which in­creases only when Ot = S (Harrison,1985, p.23)5.

Apart from {Ltl, by the standard theory of the Brownian motion, the variable In(Ot) is normally distributed with mean In(Oo) - (/1 + (1/2)0-2)t and variance 0-2t. Using the properties of lognormal distributions, we ob­tain E(Ot; 00 ) = 00 exp( -/1t), thus -/1 is the negative trend in the capital quality's growth rate.

Control dL is applied only when Ot hits S, so that a minimum amount of regulation is exercised to keep the state variable from going below the range S ::; 0 < 00. The increment dLt is operated at a cost det, which is given by a cost function that we assume to be linear.

5We define the regulated process {Ot} by the relationship Ot = Xtl t , where {Xt} is a geometric Brownian motion with stochastic differential dXt = J.LXtdt + uXtdz and initial value S ~ 00 < 00. Moreover, considering the arguments in Harrison (1985, p. 22) and in particular Proposition 6, we can identify {It} as a process defined as It = sup{X7-!S}, with 0 ~ T < t, and 10 = 1, which increases only when Ot = Sand S is positive. Now, applying Ito's Lemma to Ot, we get dOt = ItdX +Xtdl = J.LXtltdt+uXtltdz +Xtlt(dlt/lt). Finally assuming dLt = Ot(dlt/lt) we obtain (4) in the text.

236 M. Moretto

Assumption 3. dCt = C(dLt ) = bdLt , b> 0, where b is the marginal control cost constant over time. The cost {Ct }

is also a right-continuous, non-negative and non-decreasing process, which increases only when {Ltl does.

Since a restriction in the admissibility range of {Otl involves a reduction in the expected level of emissions, the lump-sum maintenance expenditure dCt takes on the sense of a sunk abatement cost.

D. The Firm's Problem

The firm wishes to maximize its market value, i.e. its discounted expected cash flow over the planning horizon [0,00).

Setting the marketable output price constant and equal to p, the market value of a competitve firm subjected to the subsidy/tax scheme (3) is given by

FV(Oo; S) = Eo{ 100 e-rt [(PQ(Ot)+v(Z(O)-Z(ot}) )dt-dCt] 0(0) = 00 },

(5) where S :::; Ot < 00, and r is a constant discount rate.

It is worth noting that the value function (5) can be taken to represent the firm's market value in the general case encompassing all three of the relevant possibilities. That is, a subsidy/tax program, a pure tax program, or the absence of both. In fact, setting Z = 0 (Le. {) = 00) we have the pure tax case, with the firm having vZ(Ot) deduced from its current profits and paying the tax rate v per-unit of emissions. Setting v = 0, on the other hand, we obtain the case with neither taxes nor subsidies.

For any barrier S, the firm's problem consists of maximizing the value (5) under the constraint described by equation (4), and then optimizing with respect to S.

3. The Firm's Dynamic Optimization Problem

A. Problem Set Up

Let's begin by analyzing the relative effects of the pure tax and the sub­sidy /tax schemes on the equilibrium of an individual firm.

From (5), the firm's market value under the two types of fiscal incentives differs only by the constant term vZ(O). The subsidy program may be interpreted as equivalent to a tax on pollution, vZ(O), plus a lump-sum subsidy given by the constant vZ(O). This allows us to conduct the analysis by referring to the subsidy/tax case alone, and analyzing the Pigouvian system as a particular case when Z = o.

Polluter's Capital Quality Standards 237

Because a lump-sum subsidy does not affect the firm's behavior, as will be seen later on, the choice between a pure tax and a subsidy/tax policy does not influence any of the firm's decisions except the convenience of staying in business or not.

In order to solve the problem (5), we need to characterize the two constraints (I) and (2). To keep it mathematically tractable, we specify the bio-physical model and the production function as

Assumption 4.

Z(lh) = zB;"/

Q(Bt ) = qBf,

'f/ > 0 and z > 0;

O<v<1 q>O

According to assumptions 3 and 4, the firm's market value function at time t becomes

(6)

where

The firm maximizes (6) subject to equation (4), and Bt is given. As usual in this maximization, the sample path of {Wtl is assumed to con­tain all the information relevant to the firm's problem, and Ed·} denotes conditional expectation taken at time t over the distribution of the {Wt} and {Bt } processes. While {Wtl is exogenous to the firm's problem, the admissible range for {Bt } is determined by the optimal lower barrier for the capital quality.

B. The Firm's Market Value

Since the current cash flow pattern is independent of the control dLt , the aim is to evaluate F(Bt ; 8) for given 8 and, then, to identify the lower barrier, 8*, which maximizes the firm's market value (6) 6.

Proposition 1. Within the zone of no intervention (8 < B < 00), the process {Btl moves onward on its own, and the expected change in the

60ptimal regulation of Brownian motion is a topic which is finding several economic applications. Examples are the work of Bentolila and Bertola (1990) dealing with em­ployment decisions with hiring and firing costs; Dixit (1989), on entry and exit decisions in foreign markets; Pindick (1988), dealing with irreversible investment decisions of a firm facing uncertain demand or costs.

238 M. Moretto

value function F, brought about by the pay-off flow (pqBr - vzB;''1) and the effect of discounting, satisfies the following differential equation

(7)

where Poo and Po are partial derivatives with respect to B. The proof follows directly by applying the results obtained by Harrison

(1985, chapter V) for regulated Brownian motions. For a similar result, see also Dosi and Moretto (1994).

Equation (7) is a linear differential equation in P. The general solution may be expressed as the sum of two parts: the solution of the homogeneous equation (complementary function) and a particular solution. That is,

V(Bt ) stands for a particular solution of (7), while pI and p 2 represent two solutions of the associated homogeneous equation and hI and h2 are two constants to be determined.

The particular solution A convenient particular solution of (7) is constituted by the expected

discounted flow given by (6) ignoring the cost of regulation and then the barrier S.

Et{ 100 e-r(r-t) [(pqB~ - VZB;7J)dr] I B(t) = Bt}, (9)

0< Bt < 00

From the concavity of the pay-off function and the properties of the lognormal process {Bt}, it is easy to prove the following proposition (Dosi and Moretto, 1994).

Proposition 2. As long as r > f.l", + (1/2)",(", + 1)(1, the particular solution (9) is bounded and takes the form

(10)

where7

M = q >0 r + f.lv - (1/2)v(v - 1)(12

N = z r - f.l", - (1/2)",(", + 1)(12 > O.

7To consider the term vZ as a tax on pollution, with v being the per-unit tax rate, N should be positive, which occurs if the denominator is also positive.

Polluter's Capital Quality Standards 239

The complementary function Omitting [pqOr -vzOt '1l] from (7) we obtain the associated homogeneous

function, which appears to be a differential equation of Euler type. To solve it we can guess a functional form of type of and check by substitution if it works. The solution is

(11)

Kl and K2 are respectively the positive and negative roots of the quadratic equation

with

1 2 2 ( 1) 2"a K - J-t + 2" K - r = 0

Kl = :2[(J-t+~a2)+ (J-t+~a2r +2ra2] >0,

K2 = :2 [ (J-t + ~a2) - (J-t + ~a2 ) 2 + 2ra2] > 0,

and hl' h2 are two constants to be determined.

The firm's market value for given S

(12)

By adding (10) and (11), equation (8), denoting the firm's market value for a given barrier S, can be rewritten as

F(Ot; S, hl, h2) = pMOr -vNOt '1l + (hl(S)Of1 +h2(S)Of2 ) , S:$ Ot < 00.

(13) The first term on the right-hand side represents the firm's market value

when no control is exercised, while the term in brackets, i.e., the comple­mentary function, may be interpreted as the value of the option of intro­ducing a control (Dixit, 1991). To determine the constant hl it can be noted that if Ot is very high, there is little probability that the lower barrier will be reached in a finite time-span. Therefore, the option of activating a barrier should be nearly worthless. For this, we need to set hl = O.

The remaining constant h2 can be determined by introducing a bound­ary condition on the general solution (13). Since F(Ot; S) is defined as the expected value of an integral of bounded cash flows, its pattern cannot be discontinuous except when Ot hits S, and instantaneous control is exer­cised. However, in the case where the regulator dLt is of a small magnitude, and the cost dCt is positive but infinitely small, the following "smoothing pasting condition" must hold

Fo(S) = b. (14)

240 M. Moretto

Whatever the value of S, the firm pays bdS to exercise control and obtain an increment in its market value equal to Fo(S)dS. 8

Written in terms of the functional (13) and eliminating the subscript for convenience, condition (14) yields

h(S) = ~[bSl-K - pMvsv - K - VN'f/S-'1- K]. (15)

Substituting (15) in (14), we get

F(Ot; S) = pMOf - vNO;-'1 + ~ [bS1- K - pMvsv - K - VN-T]S-'1- K] Of.

(16) Equation (16) represents the market value of a firm which exercises a

control, at a marginal cost b, whenever Ot hits S. Since by fixing a lower boundary S the firm should do better, the constant h(S) must be positive. Obviously, if no control is exercised, the firm's market value is given only by the particular solution (12).

Finally, adding the lump-sum subsidy given by the constant vZ and rearranging, we obtain

(17)

= pMOf - vNO;-'1

+ ~ [bS1- K - pMvsv - K - VN'f/S-'1- K] Of + ~vzB-'1.

Note that in the general solution (17) the particular integral V(Ot) is increasing in 0 and concave, while the option to introduce the control is decreasing in 0 and convex. Thus, under the condition that ('f/ + K) < 0, FV(Ot; S) turns out to be convex for 0 near zero and concave for 0 near +00. In addition, as there is no reason why the term (l/r)Z - (N/z)Zt should be positive, the firm's market value for given S may be negative.

The general shape of (17) is as shown in Figure 1. This raises the question of whether it is legitimate or not to describe

the component vZ as a lump-sum payment, for it may influence the firm's decision about continuation or cessation of operations. For example, Bau­mol and Oates (1989, p. 216) point out that the payment vZ should not in principle be contingent upon the firm's decision to stay in business. That is, this payment should not have any direct effect on the firm's set of de­cisions to establish an incentive scheme identical to that of the Pigouvian

8Condition (14) states that if F(et ; S) were not continuous and smooth at the critical point et = S, the firm could do better by using the difference formed by the two terms in (14) to improve its market value. A heuristic derivation of this condition can be found in Dixit (1991).

Polluter's Capital Quality Standards 241

I ..-

s : ....... / ()V ()

........ . .. -......... J ............. .

Figure 1: Market value function with S < (}v.

tax. Therefore, the benchmark level would lose any implication in terms of reduction in pollutant emissions, and the lump-sum subsidy should be paid not only to those who continue polluting activities but also to any potential polluters.

To avoid these complications we follow the suggestion that the subsidy payments in any period t are limited to firms that are actually in business in that period.

The following section is devoted to deriving for an incumbent firm the optimal lower barrier S and investigating the relationship between such a barrier and the firm's exit rule.

C. Conditions for Optimal Control and the Firm's Exit Rule

Since S appears only in the constant h(S), optimizing FV((}t; S) with respect to S is equivalent to finding a stationary value for h. This gives the following condition

Foo(S*) = O. (18)

In the literature dealing with instantaneous regulation of Brownian motion, equation (18) is referred to as "super contact condition" (Dumas, 1991), and it is none other than the natural extension of (14) interpreted

242 M. Moretto

as the first order condition for determining the firm's optimal barrier, 8. From (18), after some tedious manipulation, we obtain

Foo(8*) = -K(8*)K-1hS(8*) == (19)

= (8*)-1 (b(K - 1) + (1/ - K)l/pM(8*t-1

(1] + K)1]VN(8*)-71-1) = O.

Moreover, since we are looking for a positive barrier, the sufficient condition for a local maximum becomes

Fooo(8*) = -K(8*)K-1hss (8*) == (20)

(8*)-2 (b(K - 1)(1/ - 1) + (1] + K)(1] + 1/)1]VN(8*)-71-1) < 0

which is always satisfied if (1] + K) < 09 .

A simple diagram adapted from Dixit (1991) gives a clear illustration of the solution. As a consequence of the convexity/concavity of FV(Ot; 8) the general appearance of Fo(O; 8) is as shown in Figure 2. Then to solve the problem, we must adjust the constant h until Fo(Ot; 8) becomes tan­gent to the horizontal line at b, and the respective point of tangency defines 8*. Furthermore, since the shape of Fo(Ot; 8) is independent of the bench­mark Z(O), the optimal choice for 8 will be independent of whatever fiscal incentive is adopted.

Let us now turn to the exit policy. In every period a firm actually in business makes the decision whether to leave the industry or stay.

As there are no fixed costs the value of exiting the industry should be worthless. However, since the operative profits net of the tax on pollution vZ(O) for some 0 can, and will, be negative exit may take place. Therefore the term vZ(O) plays the role of an oportunity cost the firm must pay every period to stay in the industry. The optimal decision of an incumbent firm is given by

Wi(Ot; 8*) = max{O, FV(Ot; 8*)} (21)

where the value of outside opportunities is normalized at zero.

9It can be noticed that conditions (22) and (23) are equal to conditions (33) and (34) in Dosi and Moretto (1994). In Dosi and Moretto an agency finds it convenient to grant the firm a subsidy to induce it to renounce its own barrier and accept a higher one. The subsidy is proportionate to the maintenance expenditure, afforded whenever the capital quality hits the lower barrier prescribed by the agency and evaluated in accordance with a social damage function. On the contrary, in our work the subsidy is proportionate to the estimated level of emissions per unit of time. Therefore, the firm's lower barrier will coincide with the agency's in Dosi and Moretto every time the damage function is proportionate to the estimated level of emissions with the coefficient of proportionality interpreted as the per-unit subsidy rate.

Polluter's Capital Quality Standards 243

b -- - -- - - - --.;·;:·:-:i·············· ... ......: .. ~ ...... .

/ I I ..•••••..• :' I I ....... .

: I I ...... .

:' I I . ..... - ....

.: I I :' I I

s ()

Figure 2: Marginal value function with 8 < BV.

The definition (21) assumes that the firm makes the exit decision after observing realization of the current period's capital quality Bt .

As a consequence of the fact that, for the range of interest, the value function FV(Bt; 8) is increasing in B, for any 8, p, v and b, if an optimal exit rule exists it will be given by a reservation capital quality BV (8*) = inf{B: Wi(B; 8*) > O} > 8*, so that the firm with Bt < BV(8*), will exit in period t lO • Obviously if BV(8*) < 8*, the effective reservation becomes 8* and the firm will always decide to stay in the industryll.

lOr£ the value of outside opportunities is a positive function of () (with or without exit costs), the exit trigger reservation capital quality is associated to an optimal stopping time and, in general, higher than (), i.e., the firm finds it convenient to stay in business for a shorter time (see Dixit 1989 and Moretto, 1995b). Moreover, if the exit rule is related to the accumulated maintenance expenditure the exit trigger value will be always lower than S (see Moretto, 1995a).

11 By the convexity/concavity of FV «(}j S), in order of 0 < S* < (}v* (S* > (}v* it must be the case that FV(S*) < 0 (FV(S*) > 0) and FIJIJ(S*) = O. Substituting the latter into the former we get

FV(S*) = ~ (K - v)(TJ + v)pM(S*)v-l + (1 + K + TJ)b) + ~vZ. TJK r

Since the first term on the r.h.s. does not have a definitive sign, FV(S*) can be either positive or negative.

244 M. Moretto

In addition, since a reduction in the benchmark level Z moves the value function downwards and to the right the optimal exit rule implies an increase of the reservation value ()V{S*). In the case of pure tax where Z = 0 (Le. () = 00), we obtain a reservation value higher than the one with subsidy: ()V{S*) < ()T{S*).

The same remarks hold in cases where the control is not exercised (Le. h = 0), and the process {()tl is free to fluctuate between zero and infinite with boundaries which are non-attracting.

As, in order to exercise the option of regulating capital quality, h must be positive {Le. the firm finds it profitable, in terms of higher expected cash flow, to restrict the admissibility range of ()t downward), the exit decision will be in general improved. That is, ()V{S*) ~ ()T{S*) ~ ~ ()V{h = 0) ~ ()T{h = 0).

These results may be summarized in the following proposition.

Proposition 3.

a) For the individual firm, the choice between a pure tax and a sub­sidy ftax system, to induce a decrease in pollution emissions, does not affect the decision regarding the capital's optimal quality standard.

b) The pure tax system increases the reservation value for the firm to continue its operations. However, a firm can reduce this reservation value by applying a lump-sum maintenance expenditure to regulate the capital quality.

c) A higher reservation value implies a reduction in the expected level of emissions.

First of all it should be stressed that, since the choice of S* is indepen­dent of the fiscal system applied, there is nothing that will guarantee that S* > if in order to receive a subsidy and keep the current level of emissions below the level planned by the authority. As a consequence there will be periods in which emissions will be below the benchmark level and the firm will receive a subsidy and periods when the emissions will be above this benchmark level with the firm paying a tax per unit of emissions.

The first part of Proposition 3 recalls Proposition 1, Chapter 14, of Baumol and Oates (1989). With a marginal subsidy or tax rate constant over time, the firm's choice of optimal standard (the lower barrier) will in no way be influenced by the fiscal system adopted. At most the fiscal program will affect the firm's decision whether or not to stay in business. This will be depend not only on the standard laid down by the authority but also on maintenance expenditure.

Generally speaking, the pure tax system increases the reservation value, which involves a higher probability that a firm may find itself with a nega­tive market value and be forced to leave the industry. On the other hand,

Polluter's Capital Quality Standards 245

since polluting emissions are a decreasing function of the firm's specific capital quality, a higher reservation value also implies a reduction of the expected level of emissions for each firm in the industry.

Only if 8* ~ ()T(8*) > ()"(8*) do both fiscal incentives give the same expected level of emissions.

As a result, the amount of total industry emissions based on individ­ual firms' performances may turn out to be higher under the subsidy/tax system than under the pure tax system. This is the well known result by Baumol and Oates (1989, Ch. 14, Proposition 2): in a model where pollut­ing emissions are a fixed and rising function of the level of industry output, subsidies backfire in reducing total industry emissions.

To deal with this issue in the next section we analyze a competitive industry where the firm's specific capital quality, accounting for the source of uncertainty, is the cause of a substantial amount of resources allocation across firms. Heterogeneity of firms generates a selection process: the less productive firms leave the industry, while new entrants come in.

By allowing for entry and exit, a system of taxes will in general give a different level of optimal capital quality standard than the subsidy program. The firm's market value will then be different under the two fiscal incentives, not only for the lump-sum payment vZ but also for the value of the option of introducing the regulation. This will bring about differences in the size of the industry and in the level of emissions.

The second part of Proposition 3 also has important consequences for pollution emissions. As we have shown, if the firm has access to a technology which allows it to regulate the quality of the physical assets instantaneously through lump-sum maintenance expenditure, the reservation value for both fiscal policies decreases and the exit selection will be ameliorated. However, if on the one hand this implies a higher expected cash flow, on the other hand it increases the expected level of emissions.

We conclude the section regarding the behavior of a single firm by analyzing the cases in which the unit subsidy rate v and the output price p are set at zero, and making some comparative analysis. Setting v = 0 we obtain the case with neither taxes nor subsidies. Solving (19) for 8, the optimal capital quality standard becomes

* [ b(K - 1) ] 1/(1/-1) * 8 (v=O)= pMv(K-1) <8, (22)

while the second order condition is always satisfied. Moreover, since now F(()t; 8*(v = 0» is always positive, whatever the value of 8, the firm will never exit the industry.

As could be expected, with neither taxes nor subsidies, the firm will set a value for the capital's quality standard which is lower than the one it would have chosen had there been a fiscal policy. Since the admissibil-

246 M. Moretto

ity range for {Ot} is higher, it follows that the expected flow of pollutant emissions will be also higher.

Finally, it is interesting to note that even when the output price is set at zero, the option of imposing a regulation on S has a positive value. Then, S*(p = 0) corresponds to the optimal barrier to minimize the loss attributable to the tax on pollution vZ(O). Solving (19) for S, the optimal quality standard of capital becomes

* _ _ [ b(K - 1) ] -1/(7)+1)

S (p - 0) - vN1](K + 1]) , (23)

while the second order condition is always satisfied. Let us then consider the comparative statics of S* and FV(Ot; S*) with

respect to b, p, and (1. Totally differentiating (19) we derive the effect of an increase in the marginal cost of regulation

dS* = _ F(}(}b(S*) < o. db F(}(}(} (S*)

(24)

Higher values of b reduce the value of the option of introducing the regulation, which, in turn, reduces the firm's market value: dFv /db < O.

Opposite consequences derive from an increase in the output price p

dS* = _ F(}(}p(S*) > O. dp F(}(}(} (S*)

(25)

Higher values of p increase both the value of the option and current profits, and hence the firm's market value: dFv / dp > 012 .

Finally, if (12 ____ 0, K ____ 00. Thus we verify that in the absence of uncertainty the option to regulate the quality of capital becomes worthless and h goes to zero.

4. A Competitive Industry

A. Long-run Results

In this section we will be concerned with the stationary asymptotic equilib­rium of a competitive industry where the output price is constant, the firms' market values depend on a specific capital quality drawn from an asymp­totic distributions and entry and exit is based on a selection process13 .

12 An increase in the per-unit subsidy rate v yields dS* / dv > o. As pointed out in the text, the firm deduces vZ(llt) from its current profits and pays the tax rate v per-unit of emissions. It will be a motivation to raise the barrier as high as possible reducing the expected value of emissions and hence of the fees. Unfortunately there is also a reduction of the current profits which makes the effect on FV((h; S) unpredictable.

13The idea of a competitive industry equilibrium model where firm size is explained by the firm's specific shocks may be attributed to Jovanovic (1982). Hopenhayn (1992a, b) also analyzes a model of entry and exit based on selection. The model presented in this paper relies on Hopenhayn (1992a).

Polluter's Capital Quality Standards 247

Since the capital quality process {Od is regulated to stay within the interval [B,oo) with infinite as a non attracting boundary (i.e. it is never absorbed), this poses a steady-state, ergodic distribution G(Oj B) with den­sity g(Oj B) > 0, which is a transformation of a truncated exponential dis­tribution. g( OJ B) is jointly continuous, increasing in B and decreasing in 0 (Bentolila and Bertola 1990). Moreover, by G(Oj B) as the unique station­ary distribution of {Ot}, under our assumptions also {FV(Otj B)} converges in distribution to a steady-state value function FV(Otj B). FV(Oj B) stands for the firm's total value as the discounted flow of its profits when the qual­ity of capital is drawn from G«(}j B) and no reallocation of resources across firms has taken place up to this point.

The industry consists of a continuum of identical potential entrants and incumbent firms which produce a single homogeneous product with a market value expressed by PV(Oj B). Firms behave competitively, taking prices as given and there is no restriction to entry. Aggregate demand is given by the inverse demand function D(Q), where Q is the aggregate output of firms in the industry.

Assumption 5. D(Q) is continuous, strictly decreasing and D(oo) = o.

Assumption 6. The firm's specific capital quality 0 is drawn indepen­dently from G(Oj B). G(Oj B) is common knowledge.

By the time-invariant and deterministic rule (19), in the long run the capital's optimal quality standard chosen by each incumbent can be seen as a function of p with, from (25), B'(p) > O.

Conditional on p the exit rule (21) now becomes

Wi(O, B(P),p) = max{Oj FV(O, B(p),p)}. (26)

On the other hand, we assume that there is an unlimited number of potential entrants, and each bears an entry cost ce > 0 which is sunk after entry. For a potential entrant, the expected discounted market value is given by

We(B(P),p) = fsoo PV(Oj B(P),p)g(Oj B(P))dO - ce = (27)

= (1-')') [(K-V)(l-')'-K+V)PMB(Pt_ (1 - ')' - K)K (1 - ')' - v)

(K + 1])(1- ')' - K + 1]) VNB(P)-'1] + !vziJ-'1 _ ce, (1-')'+1]) r

where expectation is taken over 0 making use of g(Oj B) = (1-')')0"(-2/8"(-1, and')' = -2/-l/q2 < O.

In definition (27) it is implicitly assumed that all the information that the entrants possess prior to observing the realization of their capital quality

248 M. Moretto

is given by the steady-state distribution G(Oj S). Each prospective entrant knows this distribution, but can only discover its actual capital quality by making a non recoverable entry investment. After observing a bad realiza­tion an entrant may always exit the industry.

Entry requires We(S(p),p) 2:: 0, while free entry implies We(S(P),p) ::; 0, with no entry if the inequality is strict. Equilibrium must be such that We(S*(p*),p*) = 0.

Heterogeneity of firms implies a process of selection. The timing of the decision can be described as follows: incumbent firms observe their capital quality realization and decide whether to stay in the industry or leave. The potential entrants who decide to enter the industry draw the quality of their physical assets from G(Oj S). The firms that decide to stay in the industry produce the quantity Q(O) and the price is determined competitively to equate aggregate demand.

Total industry output will be given by the sum of the output of all firms in the industry. Since the firms are of measure zero and production depends on capital quality realization, aggregating the supply functions of all firms yields

Q(S,'IjJ) = J Q(O)'IjJ(dOjS). (28)

'IjJ(.j S) is a measure over the firms' capital quality which summarizes the mass of the firms that have decided to remain in the industry. The measure of the total size of the industry is given by H = 'IjJ([S, +oo)j S), while for any set of capital quality [S, OJ, 'IjJ([O, S]j S) is the mass of firms with quality S ::; 0 ::; O. 'IjJ(.j S) is called the state of the industry (Hopenhayn, 1992a).

Recalling that S = S(p), substituting in (28) we obtain Q(S(p), 'IjJ) which gives the aggregate supply in the industry when the price is p and the measure of firms is 'IjJ. Q(S(p), 'IjJ) is linearly increasing in 'ljJ. The price p is, therefore, an equilibrium price for 'ljJ if p = D[Q(S(p), 'Ij!)].

Since within our asymptotic configuration we ruled out entry and exit paths and the random evolution of capital quality {Otl, all the effects of uncertainty are summarized by the value function FV(Oj S) and the capital quality distribution G(Oj S). The long-run equilibrium for this industry can be represented by a vector (p*, OV* ,H*, 'IjJ*) where H* > 0, such that14 :

14The stationary equilibrium developed by Jovanovic (1982) and Hopenhayn (1992a, b) corresponds to a steady-state analysis of a dynamic system driven by the evolution of a productivity shock. The probability distribution for these shocks in each period depends on an initial distribution, the conditional distribution of a Markov process independent across firms, and the entry and exit rules. On this basis if a steady-state exists, this implies stationary distributions for firm size, profits and firm value. Concentrating on the long run distribution function G and on the value function F the industry equilibrium in the text may be compared to the non ex post uncertainty case treated by Hopenhayn (1992a).

Polluter's Capital Quality Standards

i) p* = D[Q(S*(p*), 'Ij;*)];

ii) Wi [OV' ; S*(p*),p*] = 0;

iii) We[S*(p*),p*] = 0;

iv) 'Ij;* is the measure for (H*, ov·).

249

Condition i) is the market clearing condition; ii) represents the optimal exit rule; iii) is the free entry zero profit condition and iv) the state of industry.

Proposition 4. Under our assumption a unique stationary equilibrium exists.

Proof. By the fact that FV(O; S(p),p) and g((}; S(p)) are continuous and increasing in p, it is easy to check that We(S(P),p) is also continuous and increasing in p. Therefore, under the assumption that We(S(O),O) < 0 (which is always satisfied in the pure tax scheme), there is a unique value of p for which We(S*(p*),p*) = O. Let p* be the unique solu­tion to We[S*(p*),p*] = 0 with S(p) given by (19), and (}v· = inf{() : Wi((}; S*(p*),p*) > O}. Since FV(O; S(p),p), for the range of interest, is in­creasing in () it follows that Ov· is the unique solution to FV((}; S(p*),p*) = O. To define the measure 'Ij; we must distinguish the case in which (}v· > S*(p*) from the case where (}v· :::; S*(p*). If (}v· > S*(p*), 'Ij; is the mea­sure with total mass H* and distribution given by the conditional of G on [(}v', 00), that is

'Ij;*([OV' 0)] = H* G(O; S*(p*)) - G((}v*; S*(p*». , I-G((}v*;S*(p*))

However, if Ov· :::; S*(p*), the effective reservation value becomes S*(p*) and the measure reduces to 'Ij;**([S*(p*), OJ) = H**G(O; S*(p*». For S*(p*) ~ ov*, the incumbents always make positive profits so all the firms that are already in the industry remain there. Moreover, by the fact that D[Q(S*(p*), 'Ij;)] is continuous and strictly decreasing, there is a unique value H, for each case, satisfying these conditions _

Proposition 4 and (27) imply that if ov* > S*(p*)

ce - ~vzO-7j[1 - G(Ov*)] = 100 F(O; S*(p*),p*)g(O, S*(p*»d(}, (29) r ov*

Q('Ij;*) = roo Q((})'Ij;*(dO;S*(p*)) = 1-, H*(Ov*)v. (30) lov* 1 - , - v

250 M. Moretto

While, if ()v* ::; S* (p*)

1 - 100 ce - -vz()-TJ = F((); S*(p*),p*)g((), S*(p*»d(),

r S* (29')

Q(1fJ**) = fOO Q(()1fJ**(d(); S*(p*» = 1- l' H**(S*(p*»V. k* l-I'-v

(30')

The l.h.s. of (29) and (29') represents the expected cost of entry net of the lump-sum subsidy borne by all the entrants in the industry. The equations (30) and (30') are the total output in both cases.

B. Comparative Statics Results

Suppose we now want to determine how the optimal competitive equilib­rium responds to changes in the benchmark emissions level Z. From (27), a decrease in Z (i.e., an increase in 0) leads to an increase in p* and hence an increase in S*. To sign the effect of an increase in p* on ()v* we first show that the slope of FV as a function of () increases for all () > S. Taking the derivative of Fo((); S) with respect to p yields,

d;: = VM[()V-l _ SV-K()K-l]

which is always positive for () > S. This means that after an increase of p* and S*(p*) the new market value FV must intersect with the old one to the right of ()v* to maintain the condition that We(S*(p*), P*) = O. This implies that d()v* jdp > O.

Applying this result to the case of a pure tax scheme where Z = 0, in accordance with the result obtained for the individual firm, both the reservation capital quality and the optimal quality standard increase. That is, sv* ::; ST* and ()v* ::; ()T* .

Moreover, by the fact that D(Q) is continuous and strictly decreasing and d()v* jdp* > 0 (dS*jdp* > 0), a reduction of the benchmark emissions level Z tends to reduce the total mass of firms in the industry. That is H*T ::; H*v (H**T ::; H**V).

Interpreting the application of a subsidy as a reduction of the oppor­tunity cost of remaining in the industry, the selection process shows that a lower barrier to exit can be associated with a lower mass of firms in the industry15.

As we have noted in the previous section, Proposition 3 does not enable us to reach unambiguous conclusions about the relative desirability of tax

15Hopenhayn (1992a) finds that when a higher fixed cost corresponds to a higher opportunity cost of remaining in the industry, a lower barrier to exit can be associated with higher average q values.

Polluter's Capital Quality Standards 251

or subsidy/tax programs in terms of reduction of pollution emissions. This statement becomes more precise in a competitive industry framework with selection.

Proposition 5. In a competitive industry, where polluting emissions are a decreasing function of the firm's specific capital quality, a decrease in the benchmark level of emissions leads, other things being equal, to a lower amount of ex ante total industry emissions.

Proof. Since an increase in jj leads to an increase in p*, to evaluate the effect that a decrease in Z has on the total industry emissions it is sufficient to consider the derivative with respect to p*. Like total industry output, total industry emissions will be given by the sum of the emissions of all the firms in the industry:

Z("p*) = [00 Z«(})"p*(d(}j S*(P*» = 1- '1 H*«(}V')-'1, if (}V' > S*(P*), 18v ' 1 - '1 + 7]

(31)

Z("p**) = [00 Z«(})"p**(d(}j S*(p*» = 1- '1 H**(S*(p*»-'1, (31') k, 1-'1+7]

if (}V' ::; S*(p*).

Taking the derivative with respect to p* yields

dZ Z(nl.*) [ 1 dH* 1 d(}V'] 'f (}V' S*(P*) dp* = 0/ H* dp* - 7] (}v' dp* < 0, I > , (32)

dZ = Z("p**) [_1_ . dH** _ ~. dS*] 0 if (}V' < S*(P*). dp* H** dp* 7] S* dp* < , - (32')

Recalling that both (}V' and S* are increasing in p* and H is decreasing in p*, (32') and (32') are negative. This proves the proposition. _

From Proposition 5, the pure tax system where Z = 0 yields ex ante total industry emissions lower than those that would occur under the sub­sidy/tax system. That is: Z(ST',(}T') < Z(SV',(}V').

A reduction in the benchmark level by the authority induces a higher zero profit price for the entrants which, in turn, increases the value of the regulation and of the selection effect. A greater number of firms leave the industry with a consequent fall in the total level of emissions.

Another important result of Proposition 3 concerns the question whether it is convenient for the firm to regulate the admissibility range for the capital quality (). Bearing a cost b per unit of regulation a single firm is able to reduce the reservation capital quality to continue its operation. This reduction occurs with both the fiscal systems analysed.

252 M. Moretto

However, a lower level of the reservation value is associated with higher expected pollution emissions. Therefore, for an industry as a whole the role played by the firm's capital quality standard appears crucial for the possibility of extending this result.

Proposition 6. In a competitive industry, where polluting emissions are a decreasing function of the firm's specific capital quality, an increase in the marginal cost of regulation leads, other things being equal, to a lower level of total industry emissions if 8*(P*) < ()v·. If 8* (p*) 2:: ()v', the effect cannot in general be predicted, and may go in either direction. However, if a marginal addition to the cost of regulation increases the mass of the industry, this will be accompanied by an increase in total industry emissions.

Proof. Let 8 = 8(p, b) as a function of p and b. Using the implicit function formula on (28) it is easy to check that dp* / db > 0, which, by the positive relationship between ()v· and p*, makes d()v' / db > O. Now we need to establish the sign between Hand b; in doing so we distinguish two cases:

a) If 8*(P*) < ()v', applying the implicit function formula on the market clearing condition i), we obtain

dH*

db (33)

where f* = f(Q*) < 0 is the elasticity of the demand function evaluated at the equilibrium output Q*. Taking account of (33) and specializing (32) for b we get

(34)

b) If 8*(P*) < ()v· the total industry output is given by (31'). Applying again the implicit function formula on i), we obtain

1It _ v 'f)* (88* 1It + 88*)] db fIT ~ 8p* -dl} (J{)

---- =----------~--~------~~ db 1 'f)*

?IF

dH**

~ [e;',b - ? (A;e;',b + Ab)] = *

~~ (33')

where f** = e(Q**) < 0 is the elasticity ofthe demand function evaluated at the equilibrium output Q**. A; and Ab are the partial elasticities of 8 with

Polluter's Capital Quality Standards 253

Table 1:

Equilibrium Decreases of the Marginal increase of the values benchmark level cost of regulation b.

of emissionZ S* < ()v* S* ~ ()v*

p* increase increase increase

S* increase decrease decrease ()v* increase increase increase

H* decrease decrease

H** decrease ? / increase Z* decrease decrease

Z** decrease ? / increase

respect to p and b respectively, and e;* ,b is the elasticity of the equilibrium price p with respect to b. From (27) we get

e - w; ~.2.. _ AI; p*,b - we S* A* A* . s s s

Multiplying both sides by AS and adding AI;, it is immediately clear that A;e;* +AI; is negative. The sign of (33') is in general nonpredictable. Finally, specializing (32) for b we get

dZ Z(ol.**) [ 1 dH** 1 (A* c* A*)] db = 'f' H** --;u;- - 'Tl"b S'>p* ,b + b • (34')

If dH** / db is positive (34') is also positive. This concludes the proof. • Table 1 summarizes the results of Propositions 5 and 6. In general an increase in b involves a reduction in total industry emis­

sions. As shown in Table 1, an increase in b has a positive effect on ()v* and a negative effect on S*. The selection effect becomes more stringent while the value of the option to regulate the capital quality decreases. A lower mass of firms remains in the industry, each of which generates a lower ex­pected level of pollution emissions. If, as it was done by Dosi and Moretto (1994), we interpret a reduction of b as a subsidy to induce the firms to set a higher capital quality standard for reducing emissions, from the above results this appears to be counterproductive and only an increase in the cost of regulation is able to reduce the level of emissions.

The possibility that a contrary effect may occur when S* 2: ()v* does not contradict the results obtained so far. Given that S* and ()v* move in

254 M. Moretto

opposite directions when b increases, if S* is the effective reservation value a conflict arises. As long as the capital quality standard remains higher than the hypothetical reservation value, the selection effect becomes less stringent and hence there is an increase in each firm's expected pollution emissions. The capital quality standard ensures a positive market value for all incumbents, which increases the mass of firms and also total industry emissions.

When the increase in b is large enough to reverse the disequality S* ;::::: (}v* the above result applies and the total amount of industry emissions is reduced.

5. Summary

In this paper we have concentrated on the role of the physical features of the fixed assets used in production in determining the extent of (NPS) discharges. We have assumed that the state variable "capital quality" is subject to a Brownian motion which may be regulated with a lump-sum, linear maintenance expenditure, and controls come into effect only when the state variable hits a lower barrier. We have explored the symmetry existing between a simple Pigouvian program and a more complicated subsidy/tax program, which relies on a benchmark level of emissions announced by the authority, in influencing the firm's decision about optimal capital quality standard and pollution emissions.

Both the equilibrium of an individual firm and that of a long-run com­petitive industry are analyzed. In the latter, the heterogeneity of firms gives rise to a selection process by which the less productive firms leave the industry and new entrants come in.

We show that for the single firm a Pigouvian system or a subsidy/tax system do not influence the decision regarding the capital's optimal quality standard. However, the tax system increases the reservation value for the firm to continue its operations which, in turn, involves a general reduction in the expected level of emissions from the firms actually in business.

This result is also valid for a competitive industry as a whole. Inter­preting the establishment of a subsidy as a reduction of the opportunity cost of remaining in the industry, a cut in this subsidy is also seen as a reduction in the exit barrier for potential entrants. This, in turn, increases the selection effect. A smaller number of firms are induced to enter and remain in the industry with a consequent fall in the total level of emissions.

Finally, if a reduction in the marginal cost of regulation is considered, as in Dosi and Moretto (1994), a subsidy to induce the firms to set a higher capital quality standard for reducing emissions, the entry and exit process reduces the selection effect and increases the expected pollution emissions

Polluter's Capital Quality Standards 255

for each firm. In particular if the capital quality standard chosen is not sufficiently large to ensure a positive market value for all incumbents, a reduction in the cost of regulation leads to an increase in industry emissions.

References

[1] Bentolila S. and Bertola G., Firing costs and labour demand: How bad is eurosclerosis ?", Review of Economic Studies, 57, pp. 381-402, (1990).

[2] Baumol W.J. and Oates W.E., The Theory of Environmental Policy, second edition, Cambridge: Cambridge University Press, (1989).

[3] Dixit A., Entry and exit decisions under uncertainty", Journal of Po­litical Economy, 97, pp. 62tH>38, (1989).

[4] Dixit A., A simplified treatment of the theory of optimal regulation of Brownian motion", Journal of Economic Dynamic and Control, 15, pp. 657-673, (1991).

[5] Dosi C. and Moretto M., "Interventi di politica ambientale in con­dizioni di informazione asimmetrica: il caso dell'inquinamento da sor­genti diffuse" , in: Economia dell'informazione e economia pubblica, G. Muraro, ed., Bologna: II Mulino, pp. 219-244, (1992).

[6] Dosi C. and Moretto M., "NPS Pollution, Soil Quality, and the Choice of the Time Profile for Environmental Fees", in: Theory, Modelling and Experience in the Management of Nonpoint Source Pollution, C. Russell and J.F. Shogren, eds., Kluwer Academic Publisher, Boston, pp. 91-121, (1993).

[7] Dosi C. and Moretto M., "NPS Environmental Externalities and Polluter's Site Quality Standards under Incomplete Information", in: NPS Pollution Control: Issues and Analysis, C. Dosi and T. Graham-Tomasi, eds., Kluwer Academic Publisher, Boston, pp. 107-136, (1994).

[8] Dumas B., Super contact and related optimality conditions, Journal of Economic Dynamics and Control, 15, pp. 675-685, (1991).

[9] Griffin R. and Bromley D., "Agricultural Runoff as a Nonpoint Exter­nality", American Journal of Agricultural Economics, 64, pp. 547-552, (1982).

[10] Harrison J.M., Brownian Motion and Stochastic Flow Systems, John Wiley and Sons, New York, (1985).

256 M. Moretto

[11] Hopenhayn H.A., Exit, selection, and the value of firms, Journal of Economic Dynamics and Control, 16, pp. 621-653, (1992a).

[12] Hopenhayn H.A., Entry, Exit, and Firm Dynamics in Long Run Equi­librium, Econometrica, 60, pp. 1127-1150, (1992b).

[13] Jovanovic B., Selection and the evolution of industry, Econometrica, 50, pp. 649-670, (1982).

[14] Moretto M., "Controllo ottimo stocastico, processi regolati e optimal stopping," Rivista Internazionale di Scienze Economiche e Commer­ciali, (1995a), forthcoming.

[15] Moretto M., "Firm Specific Shocks and Entry-Exit Timing," Quaderno no. 37, Department of Economics, Univeristy of Padva, (1995b).

[16] Moretto M. and Graham-Tomasi T., "Control of Nonpoint Source Pol­lution in a Spatial Setting: A Simplified Approach", Nota di Lavoro n.l, Foundation EN! Enrico Mattei, Milan, (1994).

[17] Musu I., "Environmental Subsidies: Types, Purpose, Effects", Nota di Lavoro n.4, University of Venice, (1991).

[18] Pindyck R., Irreversible investment, capacity choice, and the value of the firm, American Economic Review, 78, pp. 969-985, (1988).

[19] ShortIe J.S. and Dunn J.W., The relative efficiency of agricultural source water pollution control policies, American Journal of Agricul­tuml Economics, 68, pp. 668-677, (1986).

Department of Economics, University of Padva, and Foundation ENI-Enrico Mattei, 35123 Padva, Italy

Part 3

Pollution, Renewable Resources

and Stability

The ESS Maximum Principle as a Tool for Modeling

and Managing Biological Systems

Thomas L. Vincent

Abstract

Ever since the advent of DDT and the discovery of mutant strains of mosquitoes immune to DDT, it has been public knowledge that ecosystems can and will evolve in response to our efforts at control. While differential equations have been in common use as manage­ment models, it is uncommon to find any such models that attempt to capture the evolutionary potential of the species being managed. Here, we will present an evolutionary game approach to modeling which should provide more realistic management models, and point out some areas of possible application. In order to include evolu­tion into management models, we are faced with two fundamental questions: What is evolving? and Where is it evolving to? In the evolutionary game theory presented here, the "what" are parame­ters in the differential game model associated with characteristics of the species that are clearly adaptive (such as sunlight conversion efficiency for plants or body length in animals), which we will call strategies. The "where" is the evolutionarily stable strategy (ESS) to which these parameters can evolve. These strategies can be deter­mined using the ESS maximum principle. This principle is extended here to include a wider class of models. The ESS maximum principle when used with appropriate models, has the capacity to predict the evolutionary response of biological systems subject to human inputs. These inputs can include physiographic changes, harvesting, and the introduction or removal of new species and/or resources. Results are discussed in terms of some typical managed ecosystems.

1. Introduction

In Maynard Smith's words, "An 'ESS' or 'evolutionarily stable strategy' is a strategy such that, if all members of a population adopt it, then no mutant strategy could invade the population under the influence of natural selection" [1, p. 204]. The concept of an ESS is useful in providing an idealized state (which may, in turn, be changing with time) toward which individual members oCa biological community will tend to evolve as a result of the natural selection process.

260 T.L. Vincent

There is an extensive literature on translating the ESS concept into a mathematical setting [2], [3]. This includes our own work see [4]-[8] on the development of an evolutionary game theory for determining ESS strategies for both differential and difference equation models.

Since the fitness of each individual organism in a biological community may be affected by the strategies of all other individuals, the essential el­ement of a "game" exists. This game is an evolutionary game where the individual organisms (players) inherit their survival strategies (phenotypic characteristics) from a continuous play of the game from generation to gen­eration. The evolutionary game includes both ecological and evolutionary processes. It is the ecological process involving the interaction between individuals and the environment that determines fitness. It is the evolu­tionary process involving population dynamics that translates the fitness of an individual into changes in the number and frequency of individuals using a particular strategy. Through appropriate models, the evolutionary game may be given a mathematical setting. Most commonly, the strategies are assumed to be constants associated with certain adaptive parameters in the model. The ESS is a particular constant (or constants) that provides the stability property described by Maynard Smith.

The ESS concept, as well as the particular approach we have taken to determining ESS strategies, has a wide range of applications, not only in biology, but in economics [9] and social sciences as well [10]. Any dy­namical process where the "fitness" of a given individual in a population is determined by the strategies used by all individuals has the potential for strategies evolving with time. In fields other than biology, individuals are usually thought of as being free to make their own choices, however we will take the biological view here, where the strategy dynamic is based on the fact that individuals inherit their strategies.

Our ESS maximum principle and the use of a strategy dynamic based on the fitness generating function are tools which may be used to find ESS strategies. The ESS maximum principle, which provides an interesting link between evolutionary stability and optimization, also provides a set of nec­essary conditions that may be used to determine ESS strategies directly. Alternately, ESS strategies may be determined by seeking an equilibrium solution to the differential equations modeling both the strategy dynamics and the population dynamics. The focus of the work presented here is to extend the class of models for which our methods apply and to demonstrate how managing an ecosystem may effect the ESS. The ESS maximum princi­ple and the strategy dynamics based on the fitness generating function are presented here for a model which has been extended to include resources that the species may be consuming. This development will be followed by a summary of some previous results involving human induced evolu-

The EBB Maximum Principle 261

tion, along with some new results involving chemotherapy and a consumer resource system.

2. Population Dynamics

A group of individuals is said to be evolutionarily identical [11] if they share the same strategy set and if the ecological consequences of using strategies from this set are the same for all individuals of the group (in the parlance of what follows, two individuals will be evolutionarily identical if they have the same G-function). A group of evolutionarily identical individuals is said to be of the same species if they can interbreed and are all using the same strategy.

Let Xi be the density of individuals of species i. The strategies used by individuals of species Xi are given by u i with the possibility, through evolution, for u i = u j , j =f. i. In this paper, crossbreeding between species is not allowed, but there can be a distribution of strategies within each species, and u i will then simply characterize all the strategies used by individuals of species i (e.g., the mean). The various strategies used within a species will be called phenotypes.

Consider a community of Nx species, all of which are evolutionarily identical. Assume that the environment is fixed and we are interested in modeling the dynamics of the population density. Here we will express the dynamics in terms of differential equations, so "density" more properly refers to species biomass rather than species numbers. Difference equations are more appropriate in the latter case. However the development given here may be easily extended to difference equations models [7]. Assume that the dynamics of a given species can be described by

Xi = Xi (t)HdU, x(t), r(t), k] i = 1, ... ,Nx (1)

where the dot denotes differentiation with respect to time, Xi is the popu­lation density of species i, Hi is the fitness function of species i,

x = [Xl,'" ,XNJ is the vector of population densities for all Nx species in the community under consideration, and

U - [ I Nx] - U , ... ,u

is the matrix of strategies currently used by all the species in the com­munity. The strategies u i may be either scalar, u i = Ui, or vector, u i = [ui, ... , u}y..lT. We impose the requirement that each strategy u i

262 T.L. Vincent

be an element of the same constraint set U,

where RNu is the Nu dimensional set of real numbers. Note that Hi is allowed here to be a function of a number of other

variables,

and parameters,

k= [kl, ... ,kNk ].

This represents a further generalization of our previous formulation. The additional variables can include other quantities affecting fitness, such as nutrients or other resources. The parameters can include any fixed quan­tity that will affect fitness, such as efficiencies, interaction coefficients, and growth rate coefficients. It is assumed that the resources can be expressed in terms of differential equations of the form

Tj = Fj[(U, x(t), r(t), c] j = 1, ... ,Nr (2)

where the variables are as previously defined and

c = [CI, ... , CNJ are the constants associated with the resource model, such as mineralization rates and light decay factors. Henceforth, when the meaning is clear, we will drop the (t) from variables.

3. Fitness Generating Function

We have previously introduced the notation of a fitness generating func­tion which we call a G-function for short [5], [12], [13]. The G-function is required for the development of the ESS maximum principle, which gives necessary conditions for an ESS.

Definition 1 (G- function) A function G (u, U, x, r, k) is said to be a G­function for all species in a community sharing the same strategy set if

G(ui, U,x,r,k) = Hi(U,x,r,k)

for all of the indices i corresponding to the Nx species.

(3)

The ESS Maximum Principle 263

We will assume here that the community can be described in terms of a single G-function. In a more general setting, a given community could be composed of more than one group of evolutionarily identical individu­als, each with their own G-function. This more general setting has been discussed in detail elsewhere 111], 114). Note that, when the dynamics of two individuals can be described by the same G-function, they must be evolutionarily identical, as choosing the same strategy will have the same consequences for both.

The G-function has the property that the fitness of an individual, Hi, using one of the strategies of the matrix U is obtained when u is replaced by that individual's strategy. It follows that the fitness of an individual then depends on its "choice" for u. In terms of the G-function, the system dynamics (1) may be written as

(4)

We need to distinguish between strategies that are ESS and those that are not. Let the composite of the first strategies of U be called a coalition matrix, U c = lu1 , ... , UU), where (1 ~ 1. Let the composite of the remaining N x strategies be designated by the matrix Urn = luu+1, ... ,UN X). The total density of all individuals in the community is given by

We will assume that the population dynamics are such that N remains finite for all time t.

Definition 2 (ESS) A coalition matrix Uc is said to be an EBB if, for all other strategies in Urn and all initial frequencies xi(O)/N satisfying the conditions

1. Xi(O) > 0 for i = 1. .. (1

2. Xi(O) ~ 0 for i = (1 + 1. .. N x

3. 1 - € < {L: xikO) ,i = 1 ... (1 }

0< € < 1,

the dynamics given by (1) yields

< 1 for some € in the interval

lim ~ Xi (t) = 1 t-+oo~ N

with

i=1

1· Xi(t) O· 1 1m -N > ,z = ... (1. t-+oo

264 T.L. Vincent

The ESS may be a coalition of any number of strategies all with non­zero initial densities. The definition is local in the sense that {xi(O)/N,i = 1, ... ,a} is allowed to be arbitrarily close to 1. If the definition holds for € close to 1 then the ESS would be a global attractor in frequency space. Note that any number of non ESS strategies are also allowed. Since

u N",

LXi+ L Xi =N, i=l i=u+l

the dynamics for the non-ESS individuals must obey

r 0 Xi(t) = 0 t~~~ N .

i=u+l

This definition is consistent with Maynard Smith's verbal definition given earlier. Species identified by the strategies in the coalition matrix U c

will persist through time no matter how many other arbitrary strategies are introduced by Urn. Moreover, individuals using strategies in Urn will die out with time.

In defining the ESS for matrix games, Maynard Smith focused on the optimality of the strategy by requiring the ESS strategy, when common, to have a fitness higher than that of the rare alternative strategy. However, to provide an ESS definition applicable to the broader class of continuous games, the focus needs to be on the stability of strategy frequencies making up the ESS. This is done by requiring continuous stability [15], [16] in the definition of an ESS coalition. That is, we require that the ESS increase in frequency when common and that, eventually, all rare "mutant" strategies decrease in frequency with respect to changes in both initial frequencies and strategy [17].

4. The ESS Maximum Principle

A measure of how well any given group of individuals is doing at time t is certainly given by their corresponding population size, Xi' However, if we wish to measure how well one group is doing relative to all others, then the frequency of those individuals as defined by

Xi Pi =-

N (5)

is the proper measure to use. Likewise, we can define the total number of all individuals using coalition strategies,

The EBB Maximum Principle

and other strategies, No:

N m = LXi, i=o-+l

and their corresponding frequencies in the population,

Ne

and

Pe= N

Nm Pm= N'

265

Using these definitions, we obtain the following dynamical equations [7]:

where

i = 1, ... ,Nx

Pe = Pe(jle - fI)

Pm = Pm(fIm - fI)

0-

- "Pi He = L..J -Hi(U,x,r,k) i=l Pe

No: - "Pi Hm = L..J -Hi (U,x,r,k)

i=o-+l Pm

No:

fI = LPiHi(U,x,r,k) =PefIe + PmfIm. i=l

Note that Xi and Pi are related through (5) above.

(6)

(7)

(8)

(9)

Lemma 1 ffue is an EBB such that {Pe(t)} is monotone increasing for all t ~ tm ~ 0, then fIe > fIm for all t ~ tm > O.

Proof. It follows from (7) that fIe > fI for all t > tm . Thus fIe > PefIe + PmfIm. Since Pm > 0 and Pe + Pm = 1, it follows that fIe > fIm ,which proves the lemma. -

We will now assume that, under strategies ue and um , nontrivial equi­librium solutions exist for the coalition members and resources given by

Hi[U,x*,r*,k] = 0 i = 1, ... , (J

Fj[U,x*,r*,k] =0 i = 1, ... ,Nr

266 T.L. Vincent

and x; =0 i = a + 1, ... , N x •

Furthermore, assume that there exists a neighborhood of x* and r*, such that for any x and r in this neighborhood in the limit as t ---+ 00, x ---+ x* and r ---+ r*, that is, x* and r* are asymptotically stable.

Theorem 1 (ESS Maximum Principle) Let G(u, U, x, r, k) be the fit­ness generating function for the system. If u C is an EBB such that {Pc (t)} is monotone increasing for all t ~ tm > 0 and if x* and r* are asymptotically stable equilibrium points as defined above, then G (u, U, x* , r* , k) must take on a maximum with respect to u E U at u 1 ... u"'.

Proof. If u C is an ESS subject to any number of other strategies, then, in particular, it must be an ESS subject to a single alternate strategy. For a single alternate strategy, it follows from (9) that, in the limit as t ---+ 00,

fIm = H".+l (U, x*, r*, k), (10)

Since x* and r* are equilibrium points, it follows from (6) and (7) that

i=1, ... ,Nx • (11)

Substituting (10) and (11) into the condition of the lemma yields

i = 1, ... ,0'

which, from the definition of G, implies

G(ui U x* r* k) > G(U".+l U x* r* k) , , , ,- "" i = 1, ... ,0'

which proves the theorem. • It should be noted that a similar principle is available for difference

equation models [5], [7]. However, the difference equation version given in these references does not include resource variables. In applying the ESS maximum principle, it is useful to work in terms of a G*-function defined as follows:

G*(u) == G(u, U,x*,r*,k)

where U, x*, and r* are fixed. Assuming that the strategies are unbounded, the above theorem is equivalent to requiring that, for i = 1, ... , a,

- =0 [8G*] 8u u=u i

The EBB Maximum Principle 267

and, for scalar u,

with the following equilibrium condition on population and resource den­sity:

i = 1, ... ,0"

F(U,x*,r*,k) = 0 j = 1, ... ,Nr •

5. Strategy Dynamics

Dynamics for the mean strategy may be obtained from the basic assump­tion that within each species, a variation in strategies exists. Assuming a symmetric distribution in strategies with a small variance, we have previ­ously obtained this dynamic for a class of difference and differential equation models [6], [8]. AS5ume that individuals of species i, as identified by Xi, can be subdivided into 2m + 1 phenotypes, Yij, where j = -m, ... , 0, ... , m. That is,

m

Xi = L Yij'

j=-m

(12)

The strategies of the phenotypes are denoted by vij • They are assumed to be distributed according to

(13)

where j = -m, .. . , 0, ... , m and 8ui is an incremental change in strategy. According to this arrangement, the phenotype YiO is actually "playing" the strategy u i , which in turn characterizes all individuals of the species Xi. For example, if the mean strategy for a given species is u i = 10, with 8u i = 0.1, then the strategies of the phenotypes Yi(-2), Yi(-l), YiO, Yil, and Yi2 will be 9.8, 9.9, 10, 10.1, and 10.2. The phenotypes form a cluster of strategies about u i , that will allow u i to change with time. The actual breeding process between phenotypes of the same species is left undefined; however, it must be such that it maintains (13) with time.

It follows from (12) that the rate of change of species Xi is given by

m

Xi = L Yij.

j=-m

268 T.L. Vincent

Using the G-function to evaluate Yij(t + 1) we obtain

m

Xi = L YijG(vij , U, x, r, k), (14) j=-m

where U and x have now been expanded to include the subdivided strategies and populations. If we denote the frequency of individuals of species i and phenotype j by

(15)

it then follows from (15) that (14) may be written as

where Gi is a mean G-function defined by

m

Gi = L %G(vij , U,x,r,k). j=-m

Because of the distribution of population densities that now exists among the phenotypes that make up a given species, there will also be a corresponding distribution of strategies within a species. This distribu­tion will result in strategy dynamics for the mean strategy used by a given species. The mean strategy at time t is defined by

Individuals Yij are playing v ij , and we wish to track how the mean ui

changes as Yij changes with time. As the individual strategies do not change with time, we have

If we assume that qij is symmetrically distributed about YiO it can be shown (8) that u i = ui . Furthermore, if j8ui is small and qij is symmetric, then it can be shown (8) that

The EBB Maximum Principle 269

For the scalar case (Nu = 1, u i = Ui), if we define

m

17; = L qij(jDui? j=-m

as the variance of the strategies used by the phenotypes about the mean strategy Ui, we obtain the following expression for the strategy dynamic:

(16)

The coefficient of aG / au scales the rate of evolutionary change. This co­efficient has much in common with the way heritability and genetic and phenotypic variances scale evolutionary rates in quantitative genetic mod­els [18).

Fisher's fundamental theorem of natural selection, which states, "The rate of increase in fitness of any organism at any time is equal to its genetic variance in fitness at that time" [191, may be interpreted in terms of the G­function as follows: for a given x, r, k, and U, G plotted versus u represents an "adaptive landscape" [201. Here, the adaptive landscape, as defined by the G-function, is pliable and readily changes shape with population density and strategies of the various species. The slope of the landscape at any point represents the change in fitness for a given change in strategy. Hence, the greater the genetic variance, the greater the change in fitness and, by (16), the more rapidly an organism will evolve toward an ESS.

5.1 Evolution via strategy dynamics

As long as 17; is sufficiently small and Qij is symmetric, (4) and (16) are appropriate equations for determining population and strategy dynamics with scalar strategies. The magnitude of the variance will control how rapidly the strategies change. Clearly, if 17[ = 0, then no change in strategy is possible. If a species ultimately evolves to an ESS under this system of equations, then the ESS maximum principle must be satisfied with the resultant strategy. By solving (4) and (16) together, we not only obtain the population density and strategy but, as time goes on, all surviving species may represent an ESS coalition candidate. This candidate can then be tested using the ESS maximum principle. Provided an ESS exists, we should be able to find it using strategy dynamics, along with the ESS maximum principle.

270

6. Human Induced Evolution

6.1 Harvesting

T.L. Vincent

Whether intentional or accidental, harvesting and cropping of prey species by humans introduce a new selective pressure on an ecosystem. Body size, for example, is one such evolutionary trait that may be brought under selec­tion. It seems to be the inclination of humans to harvest the largest individ­uals, and in so doing, put evolutionary pressure on smaller sizes. We have examined this phenomenon in previous studies using the ESS maximum principle with difference equation models (with no resource equations). In [5], we examine the evolutionary response of "harvesting" a Lotka-Volterra type of system. We show how harvesting provides selective pressure for a change in evolutionary traits. For example, the use of traps or nets that selectively collect larger individuals can result in directional selection for smaller individuals which we show will result in a new ESS with a smaller yield for the harvester. Harvesting techniques that are designed to be most effective against individuals using the ESS strategy can result in disrup­tive selection. Insecticides are an example of this. They are designed to be effective against insect traits that actually occur, not those which may occur. The evolutionary result of this type of disruptive selection on a Lotka-Volterra type of system is shown to produce a new ESS that is a coalition of two phenotypes.

6.2 Ecosystem management

In [11] and [21], using multiple G-functions, we examine the evolutionary response of a prey-predator ecosystem to the addition or removal of species or phenotypes. We show the importance of ecological parameters such as niche breadth in the makeup of the ESS. Under a broad niche breadth, only a single prey phenotype and a single predator phenotype will coexist at the ESS. Under a narrower niche breadth the ESS may be composed of more than one phenotype for each species and the ecological and evolutionary results of removing a phenotype become complex and interesting. For ex­ample, consider the case when the ESS is composed of two prey phenotypes and a single predator phenotype. We show that the forced removal of one of the prey phenotypes will result in a new ecological community in which the remaining prey increases in number with a small decrease in the preda­tor species. Without further knowledge of the system, an ecologist might conclude that the two prey species are intense competitors and that the re­moved prey phenotype is relatively unimportant to the predator. However, it is the predator who, through disruptive selection, allows for the coex­istence of two prey phenotypes. Over evolutionary time, without further removal, a new prey phenotype will evolve, and the system will eventu-

The ESS Maximum Principle 271

ally return to the ESS. However, if "management" dictates that the one prey phenotype not be allowed, both the prey and predator will evolve new strategies without ever establishing a new ESS. The forced removal of the predator species can have dire consequences. The immediate consequence will be the increase in the number of both prey species. This would suggest that the predator is unimportant in the organization of this community or to the coexistence of the two prey species. This is true only in "ecological" time. In the absence of the predator, the two prey phenotypes will evolve toward a common strategy. In other words, the predator is evolutionar­ily necessary for the maintenance of the two prey phenotypes, but is not ecologically necessary for their coexistence. This just gives a hint of the complexity involved in managing such a system when many phenotypes are maintained by cross coupling between trophic levels [21]. The absence of a species or phenotype may not appear to be significant over ecological time, however, over an evolutionary time scale, they will generally be vital. In­sofar as species are evolutionary products of their environment, evolution will not preserve a species in the absence of its environment. This provides a strong plea for preserving pristine environments. In their absence, many species will not be evolutionarily stable and can be expected to die out or evolve toward new forms. Given the proper models, all this complexity can be studied using the ESS maximum principle. Moreover, byemploy­ing strategy dynamics, the evolutionary process that ultimately drives the system toward the ESS is made evident.

6.3 Chemotherapy

It has been previously pointed out by Coldman and Goldie that "much ex­perimental evidence has accrued that [cancer] cells which display inheritable resistance are the cause of treatment failure" [22]. They also developed a model that incorporates this resistance effect. In [8], this problem was ex­amined using a form of the Lotka-Volterra competition model that has been extensively studied in an evolutionary setting [4], [6], [23]-[27]. The reader may want to consult the literature for more appropriate models [28]. The following scenario should be thought of as a "simulation" that, to some ex­tent, seems to mimic what happens in the treatment of cancer with drugs. In this case again, there are no resources (Nr = 0). Using the notation of equation (1), the Hi fitness function for a given species i, is given by

R N",

Hi(U,x,k) = R - K( .) LO:(Ui,Uj)Xj, U z j=l

where Nx is the total number of species currently in the community, R is the intrinsic rate of growth common to all species, K(Ui) is the carrying capacity of the species i and 0:( Ui, Uj) is the competitive effect of species j

272 T.L. Vincent

using scalar strategy Uj on the fitness of individuals of species i. The above fitness may be expressed in terms of the following G-function:

R N",

G(u,U,x,k) = R- K(u) ~a(u,Uj)Xj'

The particular form of the model used has

a(U,Ui) = l+exp [(U-;;;J3)2]_exp [-:a;]

(17)

Note that, for this G-function, k = [R, K m, ak, ao".B] . We have previously shown [6] that, by varying the environmental parameter ak, this system can have an ESS coalition of 1 or more. Let us suppose that healthy cells are identified by a positive value for the evolutionary parameter U and that cancer cells are identified by a negative value for u. For "small" values of ak we have shown that a single strategy ESS exists with a positive value for u. For example, using the parameters [R = 0.25,Km = 100,ak = ao: = .B = 2], the ESS maximum principle predicts an ESS coalition of 1 with [Ul = 1.213, xi = 83.199]. As ak increases (e.g. environmental changes), we have shown that the ESS will change from a positive coalition of 1 to a coalition of 2, one strategy positive and one strategy negative. In other words, cancer cells now have an opportunity to "evade". This process may be illustrated dynamically using (4) and (16), where the normal cells are started at their ESS values and the cancer cells are introduced at a small population with a negative strategy [Xl = 5, U2 = -1] and ak = 3.1294. The system was run until equilibrium was reached [Ul = 3.1294, U2 = -0.2397, xi = 51.062, x2 = 39.283]. This solution satisfies the ESS maximum principle, so a new ESS coalition of 2 is obtained by this process. Assume that, at this point, treatment is started using cell-specific drugs. In particular, assume that treatment is provided by simply adding a "harvesting" term to (17) so that the G-function becomes

R N x [ ( _)2] G(u, U, x, k) = R - K(u) ~ a(u, Uj)Xj - kh exp -0.5 u:;,. U , (18)

where kh is a term expressing the level of drug dosage, u = -0.2397 is the identified cancer cell strategy at which the drug is most effective and ak is the variance in effectiveness. Starting with the equilibrium conditions above along with the parameters [R = 0.25, Km = 100, ao: = .B = 2, ak =

3.53553, ah = 1, kh = 0.5] and integrating (4) and (16) it is found that

The ESS Maximum Principle 273

chemotherapy is effective initially, but the cancer cells ultimately recover as they evolve to a new strategy [Ul = 3.1705, U2 = -2.6793, xi = 51.001, x2 = 39.3231]. The net effect is that rather than curing the cancer, the cell specific drug caused the cancer to evolve to a new form.

6.4 Consumer resource example

There is an interesting class of problems where the "game" aspect of the interaction between the species is indirect, through the consumption of resources rather than direct, through the G-function, as in the above ex­amples. There is an extensive literature on consumer resource modeling [29]-[31], and one of the simpler models [32] will be borrowed here in or­der to illustrate some possible consequences of human induced evolution on such models. It is assumed that there are two resources, rl and r2, whose dynamics are given by

(19)

(20)

where Tl and T2 represent the constant supply of nutrients rl and r2, Ul and U2 represent the percentage of total effort each consumer (e.g., plants) Xl

and X2 spend consuming resources rl and r2, respectively. The consumers Xi are assumed to satisfy differential equations of the form

(21)

where el and e2 are nutrient values associated with the resources rl and r2. The fitness of individual consumers is easily expressed in terms of the fitness function

or, alternately, G(u,r,k) = (elrl-e2r2)u+e2r2

where the strategy u is constrained by

o ::; u ::; 1.

(22)

In this example, k = [et, e2]. Since the G-function is linear in u, it follows from the ESS maximum principle that, as long as the switching function 'l/J defined by

'IjJ = elrr - e2r~

is non-zero there will be a single-strategy ESS (coalition of one) given by

{ 1 if'IjJ > 0 u= 0 if'IjJ<O

274 T.L. Vincent

The more interesting case corresponds to 1/1 = O. In this case, the G-function is "flat" which allows for any number of species to coexist [32]. In this example, for a certain relationship between the supply point [T1' T2] and gradients associated with the resource equations, we will obtain 1/1 = O. Thus a very diverse ecosystem could collapse into a monosystem under improper management practice.

In order to illustrate the rich structure of this simple example and to also illustrate the danger of exploiting such a system, consider the following scenario: suppose there are two resources on an island that is about to be invaded from the mainland by one or more plants. Assume that the mainland plants all have strategies with u ~ 0.4. For this example, assume that two different invader types, U1 = 0.4 and U2 = 0.1, arrive in small numbers with Xl = 0.1, X2 = 0.2, e1 = 0.3, e2 = 0.7, and m = 0.7. The resources are defined by (19) and (20) above, with T1 = 10 and T2 = 5. Before the invasion starts, the resources are at equilibrium so that r1 (0) =

T1 and r2(0) = T2. If the invading plants do not have the ability to evolve (0'1 = 0'2 =

0), then ecological (O'i = 0) and evolutionary consequences (O'i =I 0) are identical and the dynamical equations (19), (20), and (21) will produce the equilibrium solution xi = 7.25, xi = 0, ri = 2.56, and ri = 0.93. That is, only the first plant survives. Note that in this case 1/1 = 0.117. If however the invading plants can evolve (e.g., O'~ = O'~ = 0.02) then ecological and evolutionary consequences can differ. From (16), (19), (20), and (21) the following equilibrium solution is obtained: xi = 5.54, xi = 1. 75, ri = 2.33, ri = 1.0, U1 = 0.52, and U2 = 0.22. In this case, the invading plants evolve into two new types that can coexist. Note that in this case 1/1 = O. A similar result is obtained under any number of invading plants. That is, with O'i =I 0, only one plant will survive and, with O'i = 0, it is possible to have a community of many plant types.

Continuing with the example of two plant types at the equilibrium conditions obtained above, now consider human intervention in the system, through harvesting of resource 1, so that the supply of nutrients T1 is re­duced from T1 = 10 to T1 = 2. This will change both the ecological and the evolutionary potentials of the system. Assuming again that O'~ = O'~ = 0.02, equations (16), (19), (20), and (21) produce the following equilibrium so­lutions: xi = 0, xi = 3.95, ri = 2.0, ri = 1.0, and U1 = U2 = 0 with 1/1 = -0.1. Thus, harvesting the first resource at such a level will result in a "collapse" of the two-species community of plants to a single plant type which will specialize on the second resource. This same (albeit, more dramatic) result will be obtained if this same level of harvesting is initiated on a community of many plant types.

The EBB Maximum Principle 275

7. Discussion

The modeling challenge is to identify strategies and to understand how these strategies influence fitness. Once a G-function has been determined that will provide the fitness of any individual in the ecosystem, the methods presented here may be used to predict the course of evolution as a function of the management programs used. While it may be obvious that harvesting large fish will produce selective pressures for smaller sizes, or that treating a cancer with chemotherapy will tend to produce resistant strains, it is not obvious what the new size(s) will be or what the new strain(s) will be. Given a proper model, this theory can make these predictions provided an ESS as defined here exists. It should be noted that if the strategy dynamic as given by (16) is used to find a strategy that satisfies the ESS maximum principle, then this is a valid result, even if some other process is actually used by the organisms to arrive at the ESS. That is, we need to distinguish between necessary conditions, which must be satisfied for an ESS to exist, and questions associated with how a population might actually attain an ESS. The ESS maximum principle only addresses the question of existence. Equation (16) represents one way in which a population might attain the ESS. Provided an ESS exists and the ecosystem has some mechanism for actually attaining it, either (16), along with the system dynamics, or the ESS maximum principle can be used to find it.

What we have shown here is that the ESS coalition solution may be greatly affected by management practices. This includes not only changes in the evolutionary response of individual species, but the makeup of an entire biological system itself in terms of the number of coexisting species at one or more trophic levels.

References

[1) J. Maynard Smith, Evolution and the Theory of Games, Cambridge University Press, Cambridge, 1982.

(2) S.E. Riechert and P. Hammerstein, Game theory in the ecological con­text, Annual Review Ecological System, vol. 14, pp. 377-409, 1983.

(3) W.G.S. Hines, Evolutionary stable strategies: A review of basic theory, Theoretical Population Biology, vol. 31, pp. 195-272, 1987.

(4) T.L. Vincent and J.S. Brown, "An evolutionary response to harvest­ing," Modeling and Management of Resources under Uncertainty, Lec­ture Notes in Biomathematics, no. 72, pp. 83-99, Heidelberg; Springer­Verlag, 1987.

276 T.L. Vincent

[5] J.S. Brown and T.L. Vincent, A theory for the evolutionary game, Theoretical Population Biology, vol. 31, pp. 140-166, 1987.

[6] T.L Vincent, Y. Cohen, and J.S. Brown, Evolution via strategy dy­namics, Theoretical Population Biology, vol. 44, pp. 149--176, 1993.

[7] T.L. Vincent and M.E. Fisher, Evolutionarily stable strategies in dif­ferential and difference equation models, Evolutionary Ecology, vol. 2, pp. 321-337, 1988.

[8] T.L. Vincent, "An Evolutionary Game Theory for Differential Equa­tion Models with Reference to Ecosystem Management," in: Advances in Dynamic Games and Applications, Tamer Ba§ar and Alain Haurie, eds., Birkhauser Boston, pp. 356-374, 1994.

[9] D. Friedman, Evolutionary games in economics, Econometrica, vol. 211, pp. 637-666, 1991.

[10] R. Axelrod and W.D. Hamilton, The evolution of cooperation, Science, vol. 211, pp. 1390-1396, 1981.

[11] T.L. Vincent and J.S. Brown, The evolutionary response to a changing environment, Applied Mathematics and Computation, vol. 32, pp. 185-206, 1989.

[12] T.L. Vincent and J.S. Brown, Stability in an evolutionary game, The­oretical Population Biology, vol. 26, pp. 408-427, 1984.

[13] T.L. Vincent and J.S. Brown, The evolution of ESS theory," Annual Review of Ecology and Systematics, vol. 19, pp. 423-443, 1988.

[14] J.S. Brown and T.L. Vincent, Predator-prey coevolution as an evolu­tionary game, Lecture Notes in Biomathematics, vol. 73, pp. 83-101, Springer-Verlag, Heidelberg, 1987.

[15] I. Eshel, Evolution and continuous stability, Journal of Theoretical Biology, vol. 103, pp. 99-111, 1983.

[16] I. Eschel and U. Motro, Kin selection and strong evolutionary stability of mutual help, Theoretical Population Biology, vol. 19, pp. 420-433, 1981.

[17] S. Lessard, Evolutionary stability: One concept, several meanings, Theoretical Population Biology, vol. 37, pp. 159-170, 1990.

[18] M.L. Taper and T.J. Case, Quantitative genetic models for the coevo­lution of character displacement, Ecology, vol. 66, pp. 355-371, 1985.

The EBB Maximum Principle 277

[19] R.A. Fisher, The Genetical Theory of Natural Selection, Oxford; Clarendon Press, 1930.

[20] S. Wright, Evolution and Genetics of Populations, Vol. 3, Experimental Results and Evolutionary Deductions, University of Chicago Press, Chicago, Illinois, 1977.

[21] J. S. Brown and T.L. Vincent, Organization of predator-prey com­munities as an evolutionary game, Evolution, vol. 46, pp. 269-1283, 1992.

[22] A.J. Coldman and J.B. Goldie, A model for the resistance of tumor cells to cancer chemotherapeutic agents, Mathematical Biosciences, vol. 65, pp. 292-307, 1983.

[23] T.J. Case, Coevolution in resource-limited competition communities, Theoretical Population Biology, vol. 21, pp. 69-91, 1982.

[24] J.D. Rummel and J. Roughgarden, Some differences between invasion­structured and coevolution-structured competitive communities: A preliminary theoretical analysis, Oikos, vol. 41, pp. 477-486, 1983.

[25] J.D. Rummel and J. Roughgarden, A theory of faunal buildup for competition communities, Evolution, vol. 39, pp. 1009-1033, 1985.

[26] J. S. Brown and T.L. Vincent, Coevolution as an evolutionary game, Evolution, vol. 41 pp. 66-79, 1987.

[27] T.L. Vincent, "Strategy dynamics and the ESS," Dynamics of Com­plex Interconnected Biological Systems, pp. 236-249, Birkhaiiser, New York,1990.

[28] G.W. Swan, Optimization of human cancer radiotherapy, Lecture Notes in Biomathematics, No. 42., Springer-Verlag, Berlin, 1981.

[29] D. Tilman, "Resources: A graphical-mechanistic approach to compe­tition and predation," The American Naturalist, vol. 116, pp. 362-393, 1980.

[30] D. Tilman, "Mechanisms of plant competition for nutrients: The ele­ments of a predictive theory of competition." In Perspectives on Plant Competition, J. Grace and D. Tilman, eds., Academic Press, 1990.

[31] P.A. Abrams, The nonlinearity of competitive effects in models of com­petition for essential resources, Theoretical Population Biology, vol. 32, pp. 50-65, 1987.

278 T.L. Vincent

[32] D.L. Scheel, T.L.S. Vincent, J.S. Brown, and T.L. Vincent, "Trade­offs and coexistence in consumer-resource models, Part II: Evolution," MS, (in preparation).

Aerospace and Mechanical Engineering, University of Arizona, Thcson, Arizona 85721

Pollution, Renewable Resources and Irreversibility

OlIi Tahvonen

Abstract

The study investigates irreversible pollution damage in the con­text of renewable resources. It is shown that irreversible pollution damage leads to nonconvexities in dynamic models. There may exist two locally optimal solutions: an optimal infinite horizon solution (sustainable) and an optimal finite horizon solution. In general, the choice between these optimality candidates must be made by com­paring the present values of both policies. However, the study shows that there are special cases where the choice can be made on a priori grounds. Including the pollution problem in the renewable resource model changes the ordinary "optimal extinction" results.

1. Introduction and Overview

Antropogenic emissions may cause different kinds of irreversible effects. If the emissions in question accumulate in the environment, high concentra­tion levels may affect the rate of decay of the pollution stock, which may finally decrease to zero, implying that the accumulation of pollution is ir­reversible. However, as noted by Cruver (1976), emissions accumulation captures only a part of the basic dynamics of pollution. This is because the effects of pollution may last long after the pollution itself is gone, in­dependently of the stock/flow characteristics of emissions. The pollutant concentration may be high enough to damage the environment to the point where it is impossible for some natural populations to regenerate and may thus cause extinctions.

Studing the economic problems of irreversible pollution damage calls for dynamic models. However, the dynamic pollution control models intro­duced by Keeler et al. (1972), Plourde (1972), and Smith (1972), and later studied e.g., by Conrad (1992), do not include the possibility of irreversible pollution damage. This is partly due to the assumption of "constant ex­ponential rate of decay" which means that the natural assimilation of the pollution stock is linearly and positively related to the level of pollution stock. Thus, the accumulation of pollution is always reversible. Because pollution damage is assumed to be a function of the accumulated pollution stock, the level of damage is also reversible.

280 O. Tahvonen

This shortcoming in pollution control models was noted by Forster (1975) (see also Strom 1972). Instead of a constant exponential rate of de­cay, they assumed that the assimilation function satisfies an inverse U-shape form. This means that when the level of pollution reaches some high criti­cal level the decay of the pollution stock is zero. The assimilation function is assumed to be strictly concave below the critical level and assimilation is zero above the critical level. Forster found the possibility of multiple steady states and that it may be optimal to accumulate the pollution stock above the critical level (e.g., if the rate of discount is high enough). However, the formulation by Forster includes mathematical difficulties. This is due to the assimilation function, which is not concave (nor is it convex). In addition, it is not differentiable at the critical stock level. These features imply that straightforward application of the maximum principle is not possible. The fact that these kinds of nonconvexity problems are inherent in pollution problems was first noted by Starret (1972).

This paper specifies the irreversibility and nonconvexity problem in a slightly different form than Forster (1975). We consider situations where pollution affects the regeneration rate of a natural biological population (e.g., a fish stock). With high pollution levels, the growth rate of the popu­lation may finally become zero or negative. This may imply extinction after a period of time. Once this irreversible loss has occurred, increasing the pollution level further does not cause additional damage. This means that the problem includes the nonconvexity problem noted by Starrett (1972) and an irreversibility problem analogous to that studied by Forster (1975).

The model will include the rate of harvest (in addition to the rate of emissions) as an endogenous control variable. Thus the model comes close to the renewable resource models studied e.g., by Clark (1976). In these models it may be optimal to harvest the population to extinction if the rate of discount is higher than the maximum intrinsic growth rate (given that the costs of harvest are independent of population size) or if the minimum population size is above zero and the initial population size is close to the minimum population size (Clark 1972, Lewis and Schmalensee 1977, Cropper 1988).

Adding the pollution problem to this model changes the extinction or irreversibility results cited above. The minimum viable population size is assumed to be above zero, so it is not surprising that with low initial popula­tion levels it is optimal to harvest and pollute the population to extinction even though an optimal sustainable policy exists. However, the analysis shows that although an optimal sustainable policy exists (Le., the rate of discount is not too high to imply the nonexistence of a steady state), it may be nonoptimal, independently of the initial level of the resource stock. In these cases there always exists an optimal harvest and emission abate­ment policy which exhausts the stock in finite time. In other words, when

Pollution, Renewable Resources and Irreversibility 281

the carrying capacity of the natural population is costly to maintain, the population is not safe from extinction even though the maximum intrinsic growth rate exceeds the rate of discount and the initial level of the popu­lation is high. In these cases, the nonoptimality of the sustainable policy follows from the nonconvexity properties of the pollution problem. It may yield higher net benefits to harvest and pollute the population to extinction and stop emission abatement than to apply a sustainable harvesting policy and to maintain the carrying capacity by costly emission abatement.

In general, the choice between sustaining and extinction policies must be made by comparing the present value of these policies. Studying marginal benefits and costs does not reveal the globally optimal solution. The nonoptimality of sustainable policy is most obvious when the rate of discount is high, when the growth of the population is very pollution sensi­tive, and when emission control costs are high compared to the net benefits from resource harvesting.

From the point of view of optimal control theory, the paper utilizes the sufficiency theorem for free final time problems developed by Seierstad (1988). In several cases it enables us to find the globally optimal solution when both finite and infinite horizon optimality candidates exist.

The paper is organized as follows. The beginning of Section 2 specifies the model and the assumptions used in the analysis. Next, the infinite and finite time optimality candidates are studied by using phase diagrams and the sufficiency theorem for free final time optimal control problems. Section 3 concludes the paper.

2. Pollution Control and Renewable Resource Harvesting

Renewable resource models are often interpreted as models of the problems facing a sole owner of a resource stock (see e.g., Clark 1976). Here, the model also includes the pollution problem and the solution to the model defines both optimal emission control and renewable resource harvesting policies. It is thus natural to interpret the model as an optimization prob­lem of a public authority or a private resource owner who has exclusive property rights in terms of both the harvest and emission levels affecting the population.

The optimization problem we will study is as follows:

max W = loT [B(h) - C(e)]e-6tdt e,h,T,x(T)

s.t. : ± = f(x, e) - hj

h~O e ~ OJ

x(O) = Xo > OJ

282 O. Tahvonen

x(T) ;::: 0,

where h (a control variable) is the rate of harvest, e is the rate of emissions, x (the state variable) is the size of the renewable resource stock, e-lit is the discount factor, B(h) gives the net benefits of harvest, G(e) denotes emission abatement costs, and F( x, e) is the growth function of the resource stockl .

The natural production function F(x,e) satisfies the following condi­tions:

FEC2 j

Fe(x, e) < 0, Fee < 0 for all e ;::: OJ for all e < eo there are ~ (,1<e ;::: 0) and xe (xe :5 xe) implying that F(~,e) = F(xe,e) = 0, A(l) and if e = eo, then ~ = Xej

Fxx < 0, Fxe < 0, Fxx . Fee - (Fxe)2 ;::: O.

A(l) means that function F is twice continuously differentiable and the marginal (negative) impact of pollution increases with the level of pollution. Note that with all pollution levels below eo there are two stationary states of the population: ~ is unstable, and xe is stable. xe is the carrying capacity of the environment. If pollution is high enough, i.e., e > eo, it follows that F(x, e) < 0 for all levels of x, and the size of the population declines towards zero (comp. Kahn & Kemp 1985). Net benefits from the harvest B E C2 are assumed to satisfy:

BEC2 j

B'(h) > 0, B" < OJ A(2) limh-+o B'(h) < 00, B(O) = 0, B'(h*) = O.

According to A(2) the net benefits from harvesting are a rising and concave function of the harvest rate. However, it is assumed that there is some high harvest level h* which implies that marginal net benefits are finally zero.

The emission abatement cost function G(e) has the following properties:

GEC2 j

G'(e) < 0, Gil > OJ for e < e*j G(e) = G'(e) = 0 for e ;::: e*j A(3) lime-+o G(e) > -00.

G can represent the cost function of a single firm, or an aggregate cost

lThis formulation is close to the work presented in Tahvonen (1989). The model takes pollution as a flow variable because nonconvexity problems are difficult to study without the aim of phase diagrams. A similar model with stock pollution is studied in Tahvonen (1991). However, this paper concentrates on optimal infinite time solutions. In a general equilibrium framework the problem is studied in Tahvonen & Kuuluvainen (1993).

Pollution, Renewable Resources and Irreversibility 283

function of several polluters. According to A(3) the marginal control costs are assumed to increase with reductions in the emission level. The emission level e* is assumed to follow if no resources are used for abatement. It is thus the private optimum for the use of waste disposal services. It follows from the private decisions concerning the output and recycling levels when firms do not take external effects into account.

Let us apply some definitions before investigating the main properties of the model. We call a solution which maintains the stock forever and satisfies all necessary and sufficient infinite time optimality conditions an "optimal infinite time solution". Accordingly, by "optimal finite time solution" we refer to a path which is the optimal one among all those paths that lead to exhaustion of the resource in finite time.

We will first derive the necessary conditions for optimality. Then we study optimal infinite and finite time solutions. After this we compare optimal finite and infinite time solutions, i.e., we try to find globally optimal solutions. Finally, we study the comparative static properties of the steady state.

Write first the current value Hamiltonian function2 :

7t = B{h) - C{e) + cp[F{x, e) - h], (1)

where cp is the costate variable. If the optimal T is infinite, the necessary conditions for optimal solu­

tions are (Seierstad and Sydsreter 1987, Theorem 3.12):

7th = B'{h) - cp :::; 0 h;?: 0 and h . 7th = 0, (2)

7te = -C'{e) + cpFe{x, e) :::; 0, e ;?: 0 and e . 7te = 0 (3)

rp = cp[8 - Fx{x, e)], (4)

x = F{x,e) - h, (5)

together with x{O) = Xo and limt-too x{t) ;?: o. If the optimal T is finite, the necessary conditions include, in addition (Theorem 2.11 in Seierstad and Sydsreter 1987):

B{h) - C{e) + cp[F{x, e) - h]lt=T = OJ

cp{T) ;?: 0, x{T) ;?: 0, cp{T) . x(T) o. (6)

(7)

According to equation (3), marginal user costs determine the optimal rate of emission abatement. Emissions are optimally controlled if marginal control

2In general the Hamiltonian function is written 1-£ = <po [B(h) -G(e») +<p[F(x, e) -h), where it must hold that (<po, <p) i= O. Let us show that rPo i= o. Suppose that <Po = o. Then <pet) i= 0 for t E [0, T). Assume that <pet) > o. Then by (2) and (3) h = 0 and e = 0 for all t E [0, T). This cannot be the optimal solution. If <pet) < 0, we have by (4) that <peT) < 0 and a contradiction with (7). Thus we use <po = 1.

284 O. Tahvonen

costs equal the value of the marginal effect of emissions on the productivity of the resource stock.

Optimal infinite time (sustainable) solutions

Proposition 1. With all Xo ;::: ~, an optimal infinite time policy is to approach a steady state which exists if Fx(~, 0) > 8 holds and which is a unique saddle point and defined by (i) Fx(xoc" eoo ) = 8, (ii) F(xeo, eoo ) = hoo, (iii) b'(hoo ) = C'(eoo )/ Fe (xinfty, eoo ).

Proof. To show PI let us analyse the extremum dynamics in x - rp state space. By the implicit function theorem, the harvest rate and the emission level can be defined as functions of x and rp from equations (2) and (3). When e, h > 0 we have:

h = h(rp), h'(rp) < 0,

e = e(x, rp), ex < 0, erp < O.

(8)

(9)

Equations (4) and (5) can be written as a pair of autonomous differential equations in x and rp :

:i; = F[x,e(x,rp)]- h(rp),

cp rp{8 - Fx[x,e(x,rp)]}.

(10)

(11)

Let us first study the properties of curve :i; = O. (10) now defines rp as a function of x, and the derivative of this curve is defined by

(12)

where xe implies Fx[xe, e(xe, rp)] = 0 on curve :i; = 0 [curve Fx is rising in rp - x space by A(I)]. Thus, the slope of curve :i; = 0 is negative for x:::; xe and is, in general, indeterminate for x> xe. However, when x -t Xo from below, it must hold true that h -t 0 (or h=O) and e -t 0 for :i; = O. By A(I), A(2) and A(3), and by (2) and (3), this holds only if rp -t rp2 when x -t xo, where rp2 is defined to be the lowest value of cp which implies h(rp) = e(xo, rp) = O. Similarly, rpi is the lowest value of rp which implies that h(rp) = e(~,rp) = O. Hence, curve:i; = 0 may be depicted e.g., as in Figure 1. a:i;/arp > 0 implies that the sign of:i; = 0 is positive (negative) above (below) the :i; = 0 curve.

From equation (11) we can derive:

(13)

Pollution, Renewable Resources and Irreversibility 285

'PI L III

. 'PI (T")

~J IV

. 'P2 (T" )

o L ___ --'-~_-'--_--L-__ --.J~ ________ --+ X

X'" x'" o Xo

Figure 1: The phase diagram in x - cp state space.

because (j - Fx = 0 on the '-P = 0 curve. The sign of the numerator can be shown to be positive by the concavity assumptions in A(l). The level at which the curve approaches the x = 0 axis depends on whether

> (j <: Fx(O,e*).1f < holds, the curve starts from the cp = 0 axis. Otherwise it starts from the origin or has a positive cp-intersection. By the assumption Fx(~, 0) > {j with cp sufficiently high the curve exists and intersects with curve ± = 0 above stock level {fo. Thus, the steady state inevitably exists.

The stability properties of the equilibrium may be verified by consid­ering the Jacobian matrix of the system (10) and (11) evaluated at the equilibrium point xoo , CPoo :

Je = [ (Fx + Feex Fee<p - hI] - Fxx + Fexex)CP -cpFexe<p

Computing yields

6. = -(Fx + Feex)cpFexex + (FeE<p - h')(Fxx + Fexex)CP < 0

and

286 o. Tahvonen

where 6. = Det(Je ).

When restricting 8 ~ 0, it follows that tr(Je )2 > 46.. This implies that the equilibrium is a saddle point. 6. < 0 because Fx = 8 ~ 0 at the steady state. If more than one equilibrium exists, some of them must be other than saddle points. Multiple equilibria thus imply a contradiction, and the steady state must be unique.

Let us study the sufficiency properties of the saddle point paths (r, w) (see Figure 1) using the Mangasarian sufficiency theorem for infinite horizon problems (Mangasarian 1966, Seierstad and Sydsreter Theorem 3.13). Note that the control set is convex, the derivatives of functions B, C, R, and F are continuous, the saddle point paths satisfy the necessary conditions (2)­(5) and the Hamiltonian is concave in (x, e, h) due to A(I), A(2), and A(3). In addition, saddle point paths satisfy

(14)

for all admissible x(t) [XO(t) denotes the path toward the saddle point], because cp(t)x(t) ~ 0 and limt-HXl e-6t cp(t)xO(t) = 0 on the saddle point paths. Note that although we have not applied restriction x(t) ~ 0 explic­itly, the initial size Xo > 0 and the terminal size restriction limt---+CXl x(t) ~ 0 together with A(I) imply that x(t) < 0 is not admissible. Hence paths w and r are the optimal infinite time solutions.

The steady state conditions in Pl(b) follow directly from (2)-(5). •

Steady state condition (i) is similar to the usual modified golden-rule equa­tion in growth theory and in models of renewable resources when stock effects are not present (see e.g., Plourde 1970, Berck 1976). There is a slight difference, however. This is because the marginal growth rate which must equal the rate of discount is determined also by the rate of emis­sions. Hence, at the steady state the marginal growth rate augmented by reductions in the emission level must equal the rate of discount. Condition (ii) simply states that the growth rate of the stock at the optimal level of emissions must equal the rate of harvest. Because there are two control variables, equation (iii) is also needed to define the equilibrium.

The dynamics of the model can be presented in x - h state space also. h can be solved from (2) and <p, cp and e can be eliminated by (4), (2) and (9) to get:

x = F{x,e[x,cp(h)]} - h, (15)

h = -cp(h) { 8 - Fx{x, e[x, cp(h)]} } /( _B"), (16)

dh/dxlx=o _ [ Fx + ~eex ] { ~ 0 for x E (~o, xel

(17) = Feerpcp - 1 < 0 for x E (xe, xol

Pollution, Renewable Resources and Irreversibility 287

(18)

Curve h = 0 declines when h < h* (Figure 2). At level h* it is a horizontal line because <p(h*) = O. The declining curve starts at stock level x"" and

e* when h --+ 0 it follows that x --+ x"" if e[x"", <p(0)] = O. (Otherwise the h = 0

o 0 curve approaches some x < x"".)

o

h h=O T

h*~--__

o ~L-_________ -L ___ J-_____ X

x"" e* xU) x"" o

Figure 2: The phase diagram in x - h state space.

Curve :i; = 0 is initially rising but, in general, indeterminate when x E (Xe, xo). The level of h must approach zero when x --+ Xo from below and x --+ !fa from above.

Because oh/ox = <p(h)(Fxx + Fexex)/(C" - B") < 0 [by A(2.1)], it follows that h < 0(> 0) at the right (left) side of curve h = O. Analogously :i; < 0(> 0) above (under) the :i; = 0 curve.

If Xo > xoo «), the optimal infinite time solution is path T(W). Op­timality is assured because these paths satify the Mangasarian sufficiency theorem, as shown in Pl.

Optimal finite time solutions

The paths toward the steady state are, in general, optimal only among the solutions that maintain the resource forever. In other words, the optimality is conditional on the length of the planning horizon. We will next show that a finite time policy satisfying all the necessary conditions for optimality may also exist.

288 O. Tahvonen

Proposition 2. Although Fx(xQ, 0) > 8 holds, an optimal finite time policy may also exist. Along the optimal finite time path e < e* and 0 < h < h* until the resource is completely exhausted.

Proof. Paths in region I have x < 0 and cp < 0 (see again Figure 1). At these paths x = 0 axis is reached in finite time because x is bounded away from zero. These paths must hence satisfy (from 6):

1i(cp, 0) = B[h(cp)]- C[e(O, cp)] + cp{F[O, e(O, cp)]- h(cpnlt=T = 0, (19)

which is the Hamiltonian at the moment T. Differentiation of (19) with respect to cp yields 1irp = :i; < O. The negative sign holds at region I. Equation 1i(cp,O) cannot hold with cp(T) = 0 because this would imply 1i(cp,O) = B(h*) > 0 (See A(2)). Nor can (19) hold if cp(T) - CP3, where CP3 implies h(cp) = e(O,cp) = O. This would imply 1i(CP3,0) < O. Because 1i(cp,O) is monotonic with respect to cp and 1i(cp, 0) < 0, a value of cp which satisfies 1i(cp, 0) must exist. Denote this by cpO(TO).

The path with cpO(TO) satisfies all necessary conditions for the optimal finite time solutions. Because cpO (TO) is above zero and Fe(O, e) < 0 by A(l), it follows that emissions are controlled until the population is completely dissipated. Analogously, the rate of harvest is above zero but under the uncontrolled harvest rate at the terminal date. _

Above we have shown that there is a finite time path at least with low initial stock levels which satisfies all necessary conditions for optimality. Let us next study the problem of the sufficiency of this optimality candidate.

Proposition 3. Given that an infinite time solution is not globally op­timal the necessary conditions for an optimal finite time solution are also sufficient.

Proof. We will apply the sufficiency theorem of Seierstad (1988) for free final time problems (see also Seierstad and Sydsreter 1987 theorem 2.13)3.

3 According to the sufficiency theorem for free final time problems by Seierstad and Sydsreter (1987), an admissible triple [XTO (t), eTO (t), hTO (t») defined on [0, TO) is opti­mal if: (i) For each T E [TI' T2), where 0 :::; TI :::; T2, there exists an admissible triple [XT(t), eT(t), hT(t») defined on [0, T) with associated adjoint function 'PT(t9 which sat­isfies the Arrow sufficiency theorem (Theorem 2.5 in Seierstad and Sydsreter 1987), (ii) eT(t) and hT(t) belong to fixed bounded subsets of associated control sets for all t and T, (iii) XT(t) is continuous in T and {'P(T) : T E [T!, T2)} is bounded, (iv) and in addition it must hold that the function

Pollution, Renewable Resources and Irreversibility 289

Let us first note that because 1i is concave in (x, e, h), all paths that satisfy (2)-(5) and (7) must also satisfy the Arrow sufficiency theorem for fixed terminal time problems (see Theorem 2.5 Seierstad and Sydsreter 1987). We will find that the sufficiency properties of the candidate for an optimal finite time solution depend on the initial size of the stock and whether the path satifying the necessary conditions intersects with the x = o axis below or above the path radiating from the saddle point equilibrium (the path with CP# in Figure 1). This leads us to study four different cases.

Case 1. Assume first that Xo :s: &l. In this case there are no admissible infinite time solutions. Here Tl = 0 and T2 = f, where f is the length of the time period where the population disappears, although e = h = 0 for all t E [0, f] (see note 2). Denote accordingly by f the terminal date when the stock is exhausted by applying e = e* and h = h* for all t E [0, f]. Hence Tl = 0 < f < TO < f = T2 , where TO is the candidate for optimal T with associated cpo. When T E [0, fJ, cp(t) = 0 for all t E [0, T) by (4) and (7). x(T) 2:' 0 and e = e* and h = h* for all t E [0, f]. In addition, cp(T) = 0 implies F(T) = B(h*) - C(h*) > 0, where F(T) is the value of the Hamiltonian at the terminal date as a function of the length of the planning horizon. Consider problems with T E [f, TO]. (4) and (7) imply that cp(t) > 0 for all t E [0, T]. (7) is satisfied because x(T) = o. These paths exist below path (, assuming that cp'l(TO) =} F(T) = 0 in Figure 1, because ax/acp > 0 and the paths with different T cannot intersect. Because 1i<p(Cp, 0) < 0, F(T) > 0 when T E (f, TO) as required. Consider T E (TO, f). These paths exist between the path with CPl (T) (where CPl =}

e = h = 0 at x = 0) and the path cpJ.(TO). Along these paths F(T) < 0 clearly holds. Now we have shown that when Xo < J:.o an admissible triple [XT(t), eT(t), hT(t)] with associated adjoint function CPT(t) satisfying the Arrow sufficiency theorem exists with all T E [0, f]. Because conditions (ii)-(iv) (see note 3) are satisfied we can conclude that the path which satisfies condition (6), in addition to the other necessary conditions, is the optimal finite time path and the globally optimal solution.

Case 2. Assume next that Xo 2:' 2<.0 and CP2(TO) :s: CP#, where CP2(TO) =}

F(T) = o. When T E [0, TO) the solutions satisfy (i)-(iii) and F(T) > o. If Xo 2:' Xoo the solution paths with T E (TO, 00) exist above path I but under the path with cp#. These paths have cp(T) > CP2(TO) and hence F(T) < o. When Xo < Xoo and T E (TO,oo), some [or all if CP# = cp2(TO)) of the solution paths exist above the path with cp#. Because all of them exist above path I in all these cases F(T) < 0 as required. Thus, path I is an

has the property that a TO E [TI' T2J exists such that F(T) :::: 0 for T ::; TO if Tl < TO, F(T) ::; 0 for T :::: TO if T2 > TO.

290 O. Tahvonen

optimal finite time solution and also the globally optimal solution.

Case 3. Assume that CPI(TO) > CP#, where CPI(TO) =} F(T) = 0, and Xo ~ Xl. When Xo > Xl no finite time path which satisfies the necessary conditions for optimality exists. The paths with T E [0, 00) exist below the finite time optimality candidate, i.e., the path ,. All these paths have F(T) > 0, which at once implies that the globally optimal solutions are now the saddle point paths. When Xo = Xl we have F(T) ~ 0 for all T E [0, 00)

and saddle point path w is again optimal.

Case 4. Finally, let us assume that CPI (TO > CP# and J!.o < Xo < Xl.

Denote by T+ the extinction date along path , when Xo < Xl but which has :i; > 0 in the beginning. When T E [0, T+) the analysis does not differ from those considered above, i.e., when T E [0, TO) F(T) > 0 and when T E (TO,T+) F(T) < 0 as required in (iv). However, when T E [T+,oo) F(T) ~ 0 and condition (iv) in the sufficiency theorem is not satisfied. But consider again any of the paths with T E [T+, 00). If some of these paths dominate the corresponding path with lower T which starts from the left side of the :i; = 0 curve, the path itself must be dominated by the infinite time solution, i.e., by the saddle point path. This follows because if it pays to increase the stock level at the beginning of a finite time solution, it must pay to repeat this choice when the initial level of the stock is again reached. This implies that it pays to repeat this cycle indefinitely when compared to the path which leads to extinction. But this means that a finite time solution cannot be globally optimal. Thus in this case, we will not consider finite time solutions at all because we prefer a globally optimal solution. In other words, if the solution with cpO (TO) dominates the optimal infinite time solution, it must also dominate the solutions with T E [T+, 00),

because they are dominated by the optimal infinite time solution. This establishes the optimality of the path with cpo (TO) among the finite

time solutions, given that it is not optimal to maintain the stock forever .

• In Case 4 above, the choice between the saddle point path and the path which satisfies the necessary conditions for an optimal finite time solution must be made by computing the present value of both paths. There are two local optima for the time horizon (see remark 8 in Seierstad 1988). Note that if the infinite time path is globally optimal (with a given initial stock level), an optimal finite time solution does not exist. However, given that Xo < J!.o, an optimal infinite time path always exists, even though it would be globally optimal to exhaust the resource in finite time.

Pollution, Renewable Resources and Irreversibility 291

Special cases

We next investigate three special cases in which the choice between infinite and finite time solutions can be made without present value computation. In other words, we try to find global optimality a priori.

Proposition 4. Although Fx(~, 0) > 8 holds, there is always a high enough discount rote which implies that at the optimal steady state emission control costs exceed the net harvesting benefits and furthermore that it is optimal to exhaust the resource in finite time with all Xo.

Proof. Note first that curve U(x, cp) = B(hcp)]- C[e(x, cp)] = 0 declines in x-cp space and intersects the x = 0 axis with finite cpo Assume that 8 = 81. With this rate of discount the steady state Xoe1 is under the U = 0 curve and the choice between finite and infinite time solutions must be made, in general, by comparing the present value of both paths. An increase in the discount rate moves the cj; = 0 curve to the left by A(1). Note that curve x = 0 must exist above curve U = 0 when x -+ ~ from above. When 8 -+ Fx(~, 0) from below it inevitably holds that the equilibrium will rise above curve U = 0 because hoe -+ 0 and eoe -+ O. Assume that 82 [< Fx(~, OJ) is such a level of discount which implies that the equilibrium (Xoe2) exists above U=O curve.

Figure 3: The global optimality of finite time solution when U (hoe, eoe ) < O.

Define cp# as the value of the shadow price on the path which radiates from this saddle point equilibrium when the path intersects the x = 0 axis.

292 O. Tahvonen

Next we show that if 8 = 82 a finite time path satisfying the necessary conditions for optimality must exist with all Xo because cpo (TO) < cp#. To see this differentiate 'H with respect to time and use (2)-(5) to get: ii = xcp8. Consider now the path radiating from the saddle point equilibrium. Near the equilibrium 'H < 0 because U < 0 and x < O. By ii = xcp8, the sign of ii is negative along this path and thus 'H(O, cp#) < O. Because 'Hcp( cp, 0) < 0 it follows that cpO(TO) < cp# and a finite time path which satisfies all the necessary conditions must exist with all Xo. The sufficiency of this finite time path follows because it satisfies the sufficiency theorem of Seierstad (Case 2 in PI). •

The nonoptimality of the infinite time path can also be shown by another argument. Note first that a feasible finite time path exists with all Xo yielding positive present value benefits (e.g., put cp = 0 for all t E [0, T]). Assume that 8 = 82 and Xo > X oo2. Because (CPoo, Xoo2) is above U = 0 curve, U < 0 at the steady state. Infinite time path T can in the beginning yield a positive contribution to the present value. However, a stock size (X3

in Figure 3) must exist after which the infinite time strategy yields negative present value benefits. Hence at stock level X3 it is better (and feasible, although not optimal) to jump to some finite time path which always yields a positive present value. An analogous reasoning applies when x < X oo2.

Hence an infinite time policy cannot be globally optimal in this case. Note also that exhaustion is optimal even if 8 = 0, given that the

optimal steady state exists above U = 0 curve. In these cases the infinite time policy would become too expensive in terms of high pollution control costs and low harvesting benefits. A more traditional extinction outcome is described in the following.

Proposition 5. If Fx(~o,O) > 8 does not hold, the stock will be exhausted in finite time with all Xo.

Proof. The optimal infinite time path is l/ (Figure 4). It yields nega­tive present value after intersecting the U = 0 curve and is dominated by discontinuous finite time paths.

There is a path which satisfies all necessary finite time conditions with all xo, i.e., path ( with cpO(TO). This path also satisfies the sufficiency theorem of Seierstad, because all finite time paths with T < TO (T > TO) must exist below (above) path (. •

An opposite case to PI(£) and PI(g) is demonstrated in Proposition 6, i.e., a case where an infinite time policy may be considered optimal on a priori grounds.

Pollution, Renewable Resources and Irreversibility 293

'1'. lJ

'1'" (TO)

~---U=O

o x

Figure 4: The global optimality of finite time solution when Fx(!foO) < 6.

Proposition 6. If 6 = 0 and net harvesting benefits exceed the emission control costs at the optimal steady state, it follows that with Xo > !fo an optimal finite time path does not exist and it is optimal to maintain the resource indefinitely.

Proof. We show that 6 = 0 and U(xoo,CPoo) > 0 imply that cpO(TO) > cp#. When 6 = 0 the optimization problem under consideration is autonomous and by (5) and (20) it follows that 1t(x, cp) = 0 for all 0 ::; t ::; T. At the path radiating from the equilibrium 1t(x, cp) > 0 because U(xoo, CPoo) > 0 by assumption. Hence 1t(0, cp#) > 0 and cpO(TO) > cp# in this case. Consider Xo > Xl in Figure 5. At these stock levels no finite time path satisfying the necessary conditions exists. By P3 (Case 3) it is then optimal to maintain the resource indefinitely. At initial stock levels !fo < Xo ::; Xl a finite time path satisfying the necessary conditions exists. It, however, yields a finite stream of benefits. 6 = 0 and U(hoo, eoo ) > 0 together imply that an optimal infinite time path yields an infinite stream of benefits. This means that the finite time path cannot satisfy sufficiency conditions. Thus the infinite time path is always globally optimal and no optimal finite time path exists if Xo > !fo. The globally optimal solutions with Xo > !fo are the saddle point paths wand T. •

294 o. Tahvonen

'P- cr-) 'l'U~--~

()

xCI)

--__ u=O

Figure 5: The global optimality of infinite time solutions.

Comparative statics

x

Let us finally consider the comparative static characteristics of the steady state. Denote a shift in the net marginal benefits from harvesting, by D..£ and a shift in the marginal emission control costs accordingly by D..K-.

Proposition 7. The comparative statics of the steady state are given by the following table:

variables parameters

+ + + +

+ +

Table 1: Comparative statics of steady state.

These derivatives follow directly form (2)-(5). •

Pollution, Renewable Resources and Irreversibility 295

The sign of aeoo/a{) is undetermined because an increase in the rate of dis­count implies lower resource stock level and thus lower effects of emissions on the growth of the stock (A(l)). At the same time higher {) implies higher marginal value of the resource implying lower emissions. The other effects of an increase in the discount rate are the same as in models without pol­lution (e.g., Berck 1976). As the weight of present benefits increases, the marginal productivity of the population stock has to be increased and thus the steady state stock level and harvest rate decrease. The shadow price of the stock increases due to the increase in marginal net benefits of a lower harvest rate.

A rise in harvesting net benefits implies changes in stock and harvest levels due to the interaction of emission control and harvesting activity. As the value of the population as a source of renewable raw material in­creases, the (eqUilibrium) harvest rate rises, the emission rate declines, and the shadow price and the stock level increase. Similarly, when marginal emission control costs rise, the value of the environment as a sink for waste increases, the harvest rate declines, the stock level decreases and the shadow price increases.

3. Conclusions

In renewable resource models without endogenous pollution, the assump­tion that the minimum viable population size equals zero together with the property that the growth potential of the species exceeds the rate of dis­count guarantee that it is optimal to maintain the resource forever (Berck 1976, Theorem 2.1 and Clark 1976). Lewis and Schmalensee (1977) and Cropper (1988, Proposition 1) show that if the minimum viable population size is greater than zero, an infinite time solution cannot be optimal if the initial population level is sufficiently close to the minimum viable popula­tion level, even though the rate of discount is not too "high". Nor is the global optimality of an infinite time solution guaranteed by a "low" rate of discount in our model. This is the case because, in addition to a positive rate of discount and a nonzero minimum viable population level, there are pollution control costs. These costs may e.g., simply exceed (or be close enough) the net harvesting benefits at the optimal steady state. It is also noticeable that in contrast to the model where pollution is exogenous, the exhaustion of the resource will always take place in finite time. In contrast to this, we also showed that there are cases where at least a high initial stock size guarantees the optimality of an infinite time policy.

In general, the analysis suggests that in pollution problems the ques­tions of irreversibility and nonconvexities are closely interrelated. This means that the decision whether it is optimal to prevent an irreversible

296 O. Tahvonen

pollution damage may require a comparison of the net benefits of an op­timal sustainable policy and an optimal irreversible policy. As shown, in some cases the choice is easier because an optimal sustainable or irreversible policy may not exist.

To study these questions, Seierstad's (1988) sufficiency theorem for free final time problems can be applied. Renewable resource studies sometimes apply the sufficiency theorem of Mangasarian (1966) for fixed time problems although these models have endogenous final time (see Cropper et al. 1979 and Cropper 1988).

References

[1] Berck, P., Natural Resources in a Competitive Economy, (dissertation), M.LT., (1976).

[2] Clark, C.W., Profit maximization and the extinction of species, Jour­nal of Political Economy, 81, 950-961, (1973).

[3] Clark, C.W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources, John Wiley and Sons Inc., New York, (1976).

[4] Conrad, J.M., Stopping rules and the control of stock pollutants, Nat­ural Resource Modeling, 6, 315-327, (1992).

[5] Cropper, M.L., A note on the extinction of renewable resources, Jour­nal of Environmental Economics and Management, 15, 64-70, (1988).

[6] Cropper, M.L., Lee, D.R., and Pannu, S.S., "The optimal extinction of a renewable natural resource", Journal of Environmental Economics and Management, 6, 341-348, (1979).

[7] Forster, B.A. A note on economic growth and environmental quality, Swedish Journal of Economics, 74, 281-285, (1972).

[8] Forster, B.A., Optimal pollution control with a nonconstant exponen­tial rate of decay, Journal of Environmental Economics and Manage­ment, 2, 1-6, (1975).

[9] Gruver, G., Optimal investment in pollution control capital in a neo­classical growth context, Journal of Environmental Economics and Management, 5, 165-177, (1976).

[10] Kahn, J.R. and Kemp, W.M., Economic losses associated with the degradation of an ecosystem: the case of submerged aquatic vege­tation in Chesapeake Bay, Journal of Environmental Economics and Management, 12, 246-263, (1985).

Pollution, Renewable Resources and Irreversibility 297

[11] Keeler, E., Spence, M., and Zeckhauser, R., The optimal control of pollution, Journal of Economic Theory, 4, 19--34, (1972).

[12] Lewis, T.R, and Schmalensee, R., Non-convexity and optimal exhaus­tion of renewble resources, International Economic Review, 18, 535-552, (1977).

[13] Mangasarian, 0., Sufficient conditions for the optimal control of non­linear systems, SIAM J. Control, 4, 139-152, (1966).

[14] Plourde, G.C., A simple model of replenishable resource exploitation, American Economic Review, 60, 518-522, (1970).

[15] Plourde, G.C., A model of waste accumulation and disposal, Canadian Journal of Economics, 5, 119-125, (1972).

[16] Seierstad, A., Sufficient conditions in free final time optimal control problems, SIAM J. Control and Optimization, 26, 155-167, (1988).

[17] Seierstad, A. and Sydsreter, K., Optimal Control Theory with Eco­nomic Applications, North-Holland, New York, (1987).

[18] Smith, V.L., Dynamics of waste accumulation: disposal versus recy­cling", Quarterly Journal of Economics, 86, 600--616, (1972).

[19] Starrett, D. A., Fundamental non-convexities in the theory of exter­nalities, Journal of Economic Theory, 4, 110--120, (1972).

[20] Strom, S., Dynamics of pollution and waste treatment activities, Mem­orandum from Institute of Economics University of Oslo, (1972).

[21] Tahvonen, 0., On the dynamics of renewable resource harvesting and optimal pollution control, Acta Academiae Oeconomicae Helsingiensis, A:67, Helsinki School of Economics, (dissertation), (1989).

[22] Tahvonen, 0., On the dynamics of renewable resource harvesting and pollution control, Resource and Environmental Economics, 1,97-117, (1991 ).

[23] Tahvonen, 0., and Kuuluvainen, J., Economic growth, pollution, and renewable resources, Journal of Environmental Economics and Man­agement, 24, 101-118, (1993).

University of Oulu, SF-90570 Oulu, Finland

The Economic Management of High Seas Fishery Resources: Some Game Theoretic Aspects!

Veijo Kaitala and Gordon Munro

1. Introduction

At the urging of the 1992 United Nations Conference on the Environment and Development, held in Rio de Janeiro, the United Nations announced in December 1992 the establishment of an intergovernmental U.N. Conference on Highly Migratory and Straddling Stocks, which was scheduled to hold its first full session in July 1993. The conference is to address a critical issue in the management of transboundary fishery resources, namely the management of fishery resources to be found both within the coastal state 200 mile Exclusive Economic Zone (EEZ) and the adjacent high seas. At the close of the U.N. Third Conference on the Law of the Sea in December 1982, the issue had appeared to be of minor importance. One decade later, the issue had come to be seen as a threat to the New Law of the Sea itself (Kaitala and Munro 1993).

The term highly migratory stocks refers primarily to tuna, which, be­cause of their highly migratory nature, move between coastal state EEZs and the remaining high seas. Straddling stocks refer essentially to all other fishery stocks found in both the EEZ and the adjacent high seas. 2

For certain historical reasons, the U.N. continues to make a distinction between highly migratory and straddling stocks. For analytical purposes, however, the two can be combined. We shall do just that and refer hereafter to the combined stocks as "straddling stocks broadly defined."

The fact that the issue of management of straddling stocks broadly defined stands as a threat to the New Law of the Sea itself arises in part from the additional fact that the articles of the Law of the Sea Conven-

IThe authors express their thanks to the Beijer International Institute of Ecological Economics for its hospitality. The support of the Yrjo Jahnsson Foundation to V.K. and the support of the U.B.C. Sustainable Development Research Institute to G.M. are greatly appreciated. The authors express as well their appreciation for the useful comments of two anonymous referees.

2With the exception of anadromous stocks, e.g.. salmon. High seas fishing of such stocks was taken seriously at the U.N. Third Conference on the Law of the Seas. Directed high seas fisheries focused on such stocks are deemed to be illegal (Kaitala and Munro 1993).

300 v. Kaitala and G. Munro

tion pertaining to high seas fisheries management are vague and imprecise. Those portions of straddling stocks to be found in the adjacent high seas are exploited by fleets of coastal states and so-called distant water fishing nations. The aforementioned articles are particularly vague on the division of rights, duties and responsibilities between coastal states and distant wa­ter fishing nations with respect to the high seas portion of straddling stocks (Kaitala and Munro 1993).

While the management of straddling stocks was considered to be of minor importance at the close of the U.N. Third Conference on the Law of the Sea, there was at the time a trans boundary fishery management issue that was considered to be of major importance. This was the management of fishery resources which crossed the boundary of one coastal state EEZ into that of another. The economic analysis of the management of what are now commonly referred to as "shared" fishery resources is now reasonably well developed (see the survey article: Munro 1990). This economic analysis rests upon a blend of the economist's standard dynamic model of a fishery confined to a single EEZ and the theory of dynamic games.

It seems appropriate that we should commence by asking how far the aforementioned analysis will carry us in examining the second transbound­ary fishery management issue, namely, that of managing straddling stocks.

The economic analysis of "shared" fishery stock management will not, of course, carry us all the way. Indeed, we can, without difficulty, list at least two important differences between the analysis of "shared" and "straddling" stocks. We shall designate these differences as: 1) the problem of new entrants; 2) the number of participants or "players."

In "shared" stock fisheries management, the number of coastal states as joint owners of the resource is fixed. In the case of straddling stocks, on the other hand, the existing Law of the Sea Convention allows, to some extent at least, hitherto non-participatory distant water fishing nations to enter the high seas portion of a straddling stock fishery. If unimpeded access is granted to "new entrants," any attempt at cooperative management of a straddling stock may be undermined from the start.

With respect to the number of participants or players, all of the mod­els of economic management of "shared" fishery resources, of which these authors are aware, involve but two coastal states or "players." The as­sumption of bilateral exploitation of the relevant fishery resources proves to be a reasonable one in many real world cases of "shared" fishery resource management. Moreover, it has proven possible to apply the model on an ad hoc basis to cases in which three or more coastal states share the fishery resource (Munro 1990).

In analysing the management of "straddling stocks," on the other hand, one cannot be content with the assumption that the resource is exploited by one coastal state and by only one distant water fishing nation. The

The Economic Management of High Seas Fishery Resources 301

typical "straddling" stock case is one in which a coastal state confronts two or more distant water fishing nations operating in the adjacent high seas. Moreover, the relevant set of distant water fishing nations may change through time. These facts greatly complicate matters and result in the analysis of "straddling" stock management being far more complex than the analysis of "shared" stock management.

In light of these complexities, we introduce a simplifying assumption. We assume throughout that all cooperative agreements are binding in na­ture (Munro, 1979). To relax this assumption would complicate matters considerably and is beyond the scope of this paper (see, for example, Kaitala and Pohjola, 1988).

With these differences between "shared" and "straddling" stock man­agement kept in mind, we proceed as follows. First, we review briefly the economist's standard model of a fishery confined to the waters of a single EEZ. The model provides the foundation for all that is to follow. We then go on to examine the case of a "straddling" stock in which the coastal state does in fact confront but one distant water fishing nation and in which new entrants are effectively barred forever. This is the "straddling" stock case which most closely corresponds to that of the typical case of "shared" stock management. Not surprisingly, we find that the received analysis of "shared" stock management applies with little or no modification.

Next we relax the assumption of bilateral exploitation of the "strad­dling" stock and allow for a situation in which the coastal state confronts three or more distant water fishing nations in the adjacent high seas. In so doing, we touch upon the new entrants problem. Such are the complications introduced, that this paper will not even attempt to provide a full analysis. Rather, this section of the paper will constitute an initial exploration of the issue, which will, in turn, be seen to layout an agenda for future research.

2. The Basic Economic Model of the Fishery

The economic analysis of "shared" stock management consists, as we have already noted, of a blend of the economist's basic dynamic model of a fishery confined to the waters of a single state, plus the theory of dynamic games. Since the aforementioned basic economic model of the fishery will be seen to provide the foundation for the analysis of the management of all transboundary fishery resources, it is appropriate that we should commence with a brief review of the model (see for example Munro and Scott 1985).

The starting point for economists in their analysis of fisheries manage­ment is the common property nature of capture fishery resources. It is argued that, if a commercially valuable fishery resource is exploited on an open access basis in which fishermen are permitted to compete in an unhin-

302 V. Kaitala and G. Munro

dered, unregulated manner, the fishery resource will invariably be subject to excessive depletion from society's point of view. The open access out­come is then contrasted with the optimal exploitation of the resource from society's point of view, that would presumably occur if the fishery were controlled by an all powerful social manager.

We describe the resource dynamics, in the absence of harvesting, by the following extremely simple deterministic differential equation model

dx/dt = F(x), x(O) = xo (1)

where x(t) is the non-negative state variable representing the fishery re­source, or biomass, at time t, while F(.) is the growth function of the biomass. We assume that F(x) is concave in x such that F(O) = F(K) = 0 for some K > 0 and F(x) > 0 for x E (0, K). The biomass K is the carrying capacity, or the natural equilibrium level, of the resource.

We now introduce harvesting and suppose that we have the following harvest production function for the coastal state fishery:

hc(t) = Ec(t)x(t), (2)

where Ec is fishing effort, the flow of labor and capital services devoted to harvesting fish. It is assumed that Ec is of the feedback form, such that Ec(t) = Ec(x(t)). Furthermore, we assume that Ec(x(t)) E [0, Ecax).

With the presence of harvesting, the resource dynamics can be de­scribed as:

dx/dt = F(x) - Ecx. (3)

We assume that the resource growth is affected only by the stock itself and harvesting.

Now, let us introduce prices and costs. We assume that both the de­mand for harvested fish and the supply of fishing effort are perfectly elastic. We thus have the price of fish P = p constant and the unit cost of coastal state fishing effort C = Cc constant.

At any time t, the net revenue from the fishery, or resource rent, will be given by:

1r = (px - cc)E. (4)

The objective of management, from society's point of view, is seen as that of maximizing the present value of the net revenue, or resource rent, from the fishery. This can be expressed as:

(5)

s.t. (3), where r is the social rate of discount.

The Economic Management of High Seas Fishery Resources 303

Consider next the optimal strategy for the exploitation of the resource. The problem (3),(5) has a unique optimal solution in which the harvesting strategy Ec(x) is discontinuous in the state variable (Clark, 1990). There is an optimal, steady state, resource stock level xc' which is determined by the following equation (see e.g., Clark, 1990):

F'( *) cc(xc )F(xc) _ Xc - * - r, p-cc(xd

(6)

where cc(xc) = cc/xc. Alternatively, we can re-express (6) as

~ dd {p - cc(xc)}F(xc) = p - cc(xc). r x

(7)

Given that the capital employed in harvesting is perfectly malleable, the optimal approach path is the most rapid one. Hence denoting the optimal effort rate of Ec(t) we have:

for x(t) > Xc for x(t) = Xc for x(t) < xc.

(8)

Thus, optimal resource depletion and optimal resource recovery are compo­nents of the same optimal harvesting rule in which the initial phase depends on the resource level. This solution has been thoroughly analysed in fish­eries economics literature (see e.g., Clark 1990, Munro and Scott 1985, Kaitala and Pohjola 1988).

By way of contrast, if the fishery is not managed by a social manager, or the equivalent, but is rather an open-access, competitive fishery, the resource will almost certainly be driven below xc. The individual fishermen have no incentive to conserve the resource. Rather they will discount the future wholly. That, however, is the same as setting r = 00. Return to (7). If r = 00, then we have:

p - cc(xc) = o. (9)

Exploitation continues until resource rent from the fishery is fully dissi­pated. The stock level Xc is referred to by economists as bionomic equi­librium.

Thus Xc = xc' if and only if r = 00. Since r is almost certainly much less than infinity, we can, with confidence, conclude that Xc < Xc and that open access conditions will definitely lead to overexploitation of the resource from society's point of view.

304 v. Kaitala and G. Munro

3. Bilateral Exploitation of Straddling Stocks

We now turn to consider our first example of the management of straddling stocks. It is assumed that there is one relevant coastal state that is con­fronting but one distant water fishing nation in the high seas adjacent to the EEZ. It is assumed further that prospective new entrants to the fishery are effectively barred forever. Finally, it is assumed that a large enough portion of the stock is to be found in the adjacent high seas such that the distant water fishing nation can deplete the resource heavily.

The example is unquestionably extreme. It will, however, set the stage for more realistic examples to follow in the next section. Moreover, the ex­ample will show the limitations to the application of the analysis of "shared" fishery resources to the straddling stock problem.

If there is but one distant water fishing nation present and if it is protected from new entrants indefinitely, then it can be argued that the distant water fishing nation has de facto, if not de jure, property rights to the resource which it shares with the coastal state. Hence it should come as no surprise that the analysis of "shared" fishery resources applies with little or no modification.

The economics of "shared" fishery resources has been studied in detail (e.g., Clark 1980; Kaitala and Pohjola 1988; Munro 1979; for reviews see: Kaitala 1985, 1986; Munro 1990). In these studies it is common to allow for differences between the "players" which will lead in turn to differences in perceived optimal management strategies. In this paper we shall select one such difference for our examples, namely that arising from disparities in fishing effort costs. It will be supposed that, where such disparities exist, barriers to flows of inputs are sufficient to maintain the disparities.

3.1 Noncooperative exploitation of straddling stocks

We suppose initially that there is no cooperation between coastal state and distant water fishing nation in the exploitation of the straddling stock. Such noncooperative exploitation can be modelled by applying game theoretic solutions such as the Nash noncooperative feedback solution. The analysis of this solution concept to linear fishery models was first presented by Clark (1980).

As in the Clark model, the resource dynamics are described by a de­terministic differential equation model:

dy/dt = G(y) - Ec(t)y - ED(t)y, y(O) = Yo (10)

where y(t)is the non-negative state variable representing the level of the straddling stock at time t, G(·) is the growth function of the stock, and Ec(t) and ED(t), respectively, are the fishing efforts of the coastal state

The Economic Management of High Seas Fishery Resources 305

and the distant water fishing nation. We assume that G(y) is concave in y such that G(O) = G(K) = 0 for some K > 0 and G(y) > 0 for y E (0, K). The stock size K is the carrying capacity of the resource.

Fishing effort Ei, i = C, D, and the catch hi are related to each other by the bilinear relation as defined in (2). The fishing efforts Ei are assumed to be of the feedback form such that Ei(t) = Ei(y(t)). Furthermore, we assume that Ei(y(t)) E [O,E;nBXj.

It is supposed that each "player" i, i = C, D will attempt to maximize the present value of its share of the stream of the net economic returns from the fishery over time:

(11)

s.t. (10), where r is the common social rate of discount of C and D, p is the price of harvested fish and Ci, i = C, D is the unit cost of fishing effort. We allow for the possibility that Co =I- CD.

The Nash noncooperative equilibrium strategies can be used in pre­dicting the players' behavior in the absence of cooperation (e.g., Basar and Olsder 1982). The non-cooperative strategies are characterized by the equilibrium property meaning that no nation (player) is tempted to deviate unilaterally from applying these strategies if the other nation applies her equilibrium strategy. When the strategies are functions of the stock level, or the state variable, then the strategies are called feedback strategies.

The straddling stock game (10), (11) has an equilibrium solution in which the strategies Ei(Y) are discontinuous in the state variable (Clark 1980). In order to characterize the noncooperative equilibrium solutions, the concepts of optimal resource stock level and bionomic equilibrium dis­cussed in Section 2 must be brought to bear.

Suppose for the moment that the fishery was entirely under coastal state control. The optimal resource stock level, as seen from the coastal state perspective, Yc would be given by the following equation:

F'(yc) - c'dYc)G(:c) = r p-cc(yd

(12)

where cc(yc) = co/Yc. Conversely, bionomic equilibrium at which the net economic returns from the fishery are reduced to zero will be given by the following equation

p- cc(yc) = 0 (13)

which can be expressed as: Yc = cc/p. (14)

If the resource was owned exclusively by the distant water fishing na­tion, there would be a corresponding optimal stock level and a correspond-

306 V. Kaitala and G. Munro

ing bionomic equilibrium:

Yv and YD'

Both the optimal stock levels and bionomic equilibrium are clearly de­pendent upon unit fishing effort costs and the price. If it should be the case that Cc < CD, then we would have

Ye < Yv (15)

and (16)

Let us suppose, for the sake of argument, that it is indeed the case that Cc < CD' Then a Nash noncooperative feedback equilibrium solution for the high seas fishery game can be presented as follows (see Clark 1980):

{ Em

= y > min {Ye, YD} c ,

Eg (y) = G(y)/y, . {* CO} (17) Y = mm YC'YD

0, Y < min{Ye'YD}

E~(y) = {

Emax Y>Yv , (18)

0, Y:::; YV'

Let it be supposed that Ye < Yv, and that Yo > Ye' The resource will initially be harvested at the maximum rate. The resource will eventually be reduced to Ye and harvested thereafter on a sustainable basis. Since Ye < YV' D will have been driven out of the fishery, once the steady state equilibrium has been achieved.

Now let it be supposed that, while Ye < Yv, Yo < Ye' The resource has been subject to overexploitation in the past. No harvesting will occur until the resource stock has achieved the level Y = ye. Thereafter, the re­source will be harvested on a sustainable basis. D will have no opportunity whatsoever to engage in harvesting.

It thus becomes obvious that if Ye < YV' no cooperation will be nec­essary. D will be driven out of the fishery. The resource will come fully under the control of the coastal state.3

If, on the other hand, Ye > Yv, then an incentive for cooperation will exist. The distant water fishing nation will eventually be driven out of the fishery. The resource will be stabilized at Y = YV' Since, however,

3For the sake of completeness, we should note that, if we find that y > Yv' it could pay the coastal state to engage in short term cooperation, e.g. by buying out the distant water fleet, as the resource is reduced to the level Yv' The analysis of temporary cooperative coalitions is, however, omitted in this paper.

The Economic Management of High Seas Fishery Resources 307

y'D < Ye, the outcome is distinctly sub-optimal from the point of view of the coastal state - hence the incentive for cooperation exists.

If we should have CD < Ce, rather than CD > Ce, then a similar set of arguments will hold, except that the distant water fishing nation is now the relatively more efficient of the two. An incentive for cooperation will exist ifYD < Yo'

If CD = Ce, then, under noncooperation, the resource will be driven down to a common bionomic equilibrium, Y = Yoo. The incentive for coop­eration will be transparent.4

3.2 Cooperative management of straddling stocks

To suppose that C and D are prepared to engage in cooperative manage­ment of the resource is to suppose that the two are prepared to negotiate with each other for the purpose of establishing a Pareto efficient agreement. With such an agreement, one "player" cannot be made better off under an alternate agreement except at the expense of the other. An agreement is Pareto inefficient if there exists another agreement such that, either both players can be made better off, or one player can be made better off without damaging the position of the other. Let it be noted in passing, for future reference, that one class of Pareto efficient solutions includes the use of transfer payments.

The formal definition of a Pareto efficient agreement is as follows (Leit­mann 1974). A pair of control functions (Ee, ED) constitutes a Pareto efficient agreement at a given initial state Yo if and only if for any other pair (Ee, ED) either

or for i = 1 or 2,

where Ji(y, Ee, ED) denotes the value of the fishery at the stock level Y when the nations apply their strategies (Ee , ED)'

There remains the problem of determining which among the many Pareto efficient agreements will prevail. The issue is complicated by the possibility of differing goals of management. If, to continue with our pre­vious example, it should be the case that Ce i- CD, then the two potential co-managers of the resource will take different views on management.

40verexploitation of the resource under noncooperation is, we concede, not a general result. Under special circumstances, it is possible that noncooperation will, in fact, lead to "underexploitation" in terms of the social optimum. See, in particular, Dutta and Sundaram (1993) and Fischer and Mirman (1992).

308 V. Kaitala and C. Munro

Suppose that Co < CD, then other things being equal, we shall have Yo < Yv' The distant water fishing nation will prove to be more conserva­tionist than the coastal state.

Munro (1979) has demonstrated that, in the context of the manage­ment of "shared" fishery resources, the Pareto efficient resolution of the differences in management objectives is greatly simplified if side or trans­fer payments are allowed. The implication of side or transfer payments is that a "player's" return from the fishery is not dependent solely upon the harvest share of the "player's" fleet.

The outcome is such that equal weight is given to the management preferences of the two players. A harvest program is selected which will maximize the global net economic return from the fishery. In so doing, the management preferences of the low cost "player" in fact become domi­nant (Munro 1979). The low cost player compensates the high cost player through transfer payments. Bargaining takes place over the division of the total return from the fishery.5

We see no reason why transfer payments should not be introduced into cooperative management arrangements pertaining to the cooperative man­agement of straddling stocks. Indeed, there is good reason to suspect that it would prove to be difficult to achieve stable cooperative management agree­ments if transfer payments were absent in the bargaining process (Kaitala and Munro 1993).

Let us continue with our example and suppose that Co < CD. Hence in a cooperative management regime with transfer payments, the management preferences of the coastal state will be dominant. Let us therefore denote the present value of the global net returns from the fishery, at a given stock level y and in following a particular harvest strategy, by wc(y}. Let

5 Let us admit to the need for a qualification. If the harvest shares are not constrained, but are rather considered part of a single overall bargaining package, then it will certainly be true that, with transfer payments, the management preferences of the lower cost fishing nation will prevail. Indeed, given the linearity of our model, we can predict with assurance that the low cost fishing nation will, in effect, buyout the high cost fishing nation (Munro 1979).

Harvest shares may not be unconstrained, however. In many, if not most, cooperative fisheries arrangements in the real world, bargaining takes place over the harvest shares before any bargaining over the management regime occurs. More often than not, the harvest shares are determined on the basis of an agreed upon formula. The harvest shares remain fixed thereafter.

It can be shown that, if the harvest shares are predetermined, and then made invariant over time, one cannot say a priori which "player's" management preferences will prevail, given the existence of transfer payments (Munro, 1979).

This qualification could be of more than passing interest if the coastal state, instead of being the low cost harvester, was rather the high cost harvester. Given the attitude of most coastal states towards their management rights with the Exclusive Economic Zone, it is reasonable to suppose that the typical coastal state would not tolerate a complete surrender of its management rights with respect to a straddling stock.

The Economic Management of High Seas Fishery Resources 309

respective shares of the aforementioned net returns of the coastal state and distant water fishing nation under an agreement with transfer payments be denoted by wg (y) and wf} (y) respectively. We have:

wc(y) = wg(y) + wf}(y). (19)

The shares defined by (19) are Pareto efficient. If one "player" re­ceives more, the other "player" must receive less. In addition, when trans­fer payments are used the set of feasible solutions becomes convex. The convexity of the Pareto frontier makes it possible for one to apply a bar­gaining scheme (e.g., Nash 1950, Roth 1979) in order to identify one of the nonunique Pareto optimal cooperative solutions to represent the coopera­tive agreement between the harvesting countries.

The bargaining is summarized as follows. Let the initial level of the straddling stock be fixed. A strategical assumption in the negotiation mod­els is that threat strategies will be applied if the negotiations prove to be unsuccessful. The threat strategies are defined here as the Nash nonco­operative strategies (16)-(17). Application of these strategies yields na­tion i a net economic return that is denoted by Ji (Yo, E~, E~). The pair (Jc(yO, E~, E~), JD(YO, E~, E~)) is referred to as the threat point of the cooperative game.

Each player expects a share at least equal to its threat point return. Otherwise, there is no reason for it to accept the agreement since it will be better off by refusing to cooperate. Thus, the net economic returns to be received by the coastal and distant water fishing nations respectively must be at least Jc(yo,E~,E~) and JD(Yo,E~,E~). It follows that the best result that the coastal nation C can expect is the net economic return wc(yo) - JD(Yo, E~, E~) and the worst is Jc(yo, E~, E~). For the distant water fishing nation D these values become wc(yo) - Jc(yO, E~, E~) and JD(YO, E~, E~), respectively.

The excess net economic return e(yo) obtained from cooperation at Yo is independent of the way it is shared and is defined as

e(yo) = wc(yo) - Jc(yo,E~,E~) - JD(Yo,E~,E~). (20)

An application of the Nash bargaining scheme (Nash 1950, 1953), for example, in negotiations with transfer payments divides e(yo) between the two players. Thus cooperation is expected to yield the following result for the two players (Munro 1972):

Jc(Yo) = Jc(Yo,E~,E~)+ae(yo) (21)

J1(yo) = JD(Yo,E~,E~) + (1- a)e(yo) (22)

where a = 1/2, and Jt(yo), i = C, D, denotes the value of the fishery for nation -i at Yo under the transfer payment agreement. The equal division of

310 v. K aitala and G. Munro

e(yo) between the two players follows because the use of transfer payments has the effect of making the bargaining problem symmetric.

While an incentive to establish a cooperative management agreement exists, it is true that cooperation may be jeopardized. There is, for example, the threat of cheating. It is also true, however, that there exist means to guard against such threats to cooperation. These means are well understood in fisheries economics and need not detain us further here.

4. Multilateral Management of Straddling Stocks

We now turn to the much more realistic case in which the coastal state confronts more than one distant water fishing nation. Such will be the complexities arising from multilateral management that we shall not be able to do much more in this section than put forth a series of conjectures. Analysis, far beyond the scope of this paper, will be required to bring forth solutions.

In addressing the possibility of more than one distant water fishing nation, we are compelled to attempt to address the issue of new entrants, i.e., new distant water fishing nations entering the high seas portion of the straddling stock fisheries. What can be said without hesitation is that, if there is unrestricted access for new entrants, achieving meaningful cooper­ative resource management will be virtually hopeless (Kaitala and Munro 1993).

As the Law of the Sea Convention now stands, it seems apparent that new entrants cannot in fact be barred entirely. A proposal, which Canada intends to bring forth to the U.N. Conference on Highly Migratory and Straddling Stocks, would appear to provide a framework within which there would be at least some hope of establishing an effective cooperative resource management regime. In essence, the Canadian proposal would enable the existing management coalition to declare the straddling stock fishery to be fully utilized. A prospective new entrant would be informed that it could gain access to the coalition, but only if an existing member chose to withdraw.6 This raises the obvious possibility that an existing member of the coalition might be persuaded to transfer its membership to a new entrant - for a price, i.e., an existing member of the coalition may sell off its membership.

In this section we shall, in order to simplify the discussion, suppose that new entrants are either barred outright or that they are to be admit­ted only if an existing member of the coalition chooses to withdraw. We shall suppose further that there are but three members of the coalition, one

6Robert Applebaum, Department of Fisheries and Oceans, Canada, personal communication.

The Economic Management of High Seas Fishery Resources 311

coastal state confronting two distant water fishing nations. While a distant water fishing nation may depart to be replaced by another, the number of coalition members will remain at three. Finally, as in the previous sec­tion, we shall suppose that the three are identical, except in terms of their harvesting costs.

While the size of the coalition is, by assumption, small, important prob­lems related to the management of the resource and to the coalition remain. First, the obvious threat of noncooperation exists. Secondly, during the bargaining over resource management, it will have to be asked whether the three should be treated as distinct and equal, or whether it would make more sense to think of the distant water fishing nations as a sub coalition acting like a single player. Alternatively we must consider the possibility of other strategic subcoalitions emerging during the negotiations. We must also obviously consider the possibility of one original distant water fishing nation transferring its membership in the coalition to a prospective new entrant. What impact would such a transfer of membership have? Could the mere threat of such a transfer influence the negotiations within the coalition? These questions remain to be answered.

Let us consider first the consequences of noncooperation.

4.1 Noncooperation

The resource dynamics continue to be described by (10), as two compo­nents: ED = EDI + ED2 , where Dl and D2 denote the two initial distant water fishing nations.

Let it be supposed for the sake of argument that Cc < CD I < CD2 • Let us suppose further that:

(23)

A straightforward result is that, when the nations act independently, the Nash cooperative feedback equilibrium solution is such that the resource will be depleted in a most rapid approach manner until the level Y[)l has been reached (Kaitala 1989). That is to say:

Ef(y) ={

{ EcRX , Y > Y[)l

= G(y)/y, y = y[)l

0, y < Y[JI

EiRX , Y > yr'

0, y:::; Yr', i = D1,D2 •

(24)

(25)

Thus, as in the two "player" case, the straddling stock resource will be

312 V. Kaitala and G. Munro

subject to overexploit at ion if noncooperation prevails. Indeed, the outcome is virtually identical to that of an unregulated open access fishery.

4.2 Cooperative agreements

We now turn to an examination of the cooperative management of the straddling stock by the three players. We continue to assume that cooper­ative agreements upon being achieved are binding, and that side payments, or transfers, among the three constitute a feasible policy instrument.

At first glance, we would anticipate that the lowest cost "player," C would dominate the management of the resource. Indeed, given the nature of our model, we could expect that C would effectively buyout Dl and D 2. The cooperative agreement would be focused on the sharing of the total net returns from the fishery among the three.

Let wc(y(O)) denote the present value of the net economic return from the fishery at stock level y, upon following a harvest strategy prescribed by C. Let wg(y(O), wgl(y(O» and wg2(y(O», respectively, denote the shares of C, D 1 , D2 of the aforementioned global net economic return from the fishery under a cooperative agreement. We have:

wc(y(O» = wg(y(O» + wgl (y(O» + wg2 (y(O». (26)

The shares defined by (26) are Pareto efficient. If one "player" receives more, then at least one of the other two "players" will receive less.

While this seems straightforward enough, there are in fact at least four different alternative arrangements for the management coalition which must be considered. The alternatives are:

1. Subcoalitions among the players are not feasible. Furthermore, new entrants are barred forever, hence it is not possible for Dl or for D2 to transfer its membership to a distant water fishing nation currently outside the coalition [6]. We do not give serious consideration to the possibility of the coastal state transferring its membership.

2. Subcoalitions are not feasible, but the transfer of membership is fea­sible and becomes an essential part of the bargaining process.

3. The reverse to 2. Transfer of membership is infeasible, while the establishment of coalitions is feasible.

4. Both the transfer of membership and the establishment of coalitions are feasible.

Let us consider each of the alternative arrangements and their implications in turn.

The Economic Management of High Seas Fishery Resources 313

No sub coalitions with nontransferable membership

Consider now the first alternative 1) in which we have three strictly indepen­dent "players," none of whom have the option of transferring membership in the coalition. Since subcoalitions are deemed to be infeasible, we need to assume that a failure to achieve a cooperative agreement will result in the application of a fully noncooperative solution, in which all three play­ers play against one another. Since each player can expect to receive at least its threat point payoff, i.e., the payoff arising from the solution to a noncooperative game, the global net returns from the fishery to be shared are equal to: wc(y(O)), minus the sum of the threat point payoffs. Denote the net returns to be shared as e(y(O)), and we have:

e(y(O)) = wc(y(O)) - L Ji (y(O)),Eg,EZ1 ,EZ2 )· (27) i=e,Dl ,D2

There seems to be no reason why all the requirements for an appli­cation of the Nash bargaining scheme to reach a fair agreement would not be satisfied under this management arrangement. A straightfor­ward application of the Nash bargaining scheme gives the result that, under the transfer payment regime, the economic benefits from coopera­tion, that is, e(y(O)), will be divided equally among the three countries. Thus, the cooperative net revenue that nation i will receive is equal to e(y(o))/3 + Ji(y(O),E~,E~l,E~2). This will be the case even if the two distant water fishing nations bear completely different harvesting costs. For example, CD l may be only slightly greater than Ce, while CD2 »ce. The idea of equal shares rests upon the assumption that, if anyone of the three "players" refuses to cooperate, then cooperation breaks down entirely. Thus each of the "players" can be seen to be making an equal contribution to cooperation, and to the economic benefits which arise therefrom.

No subcoalitions with transferable membership

With alternative 2), we retain the assumption of no subcoalitions, but allow for the possibility of one player transferring its membership to a prospective new entrant. Let it be supposed that there is in fact but one prospective new entrant, D3 • We shall suppose further that, if a transfer of membership is to take place, it must be done before the commencement of the cooperative management program.

It is reasonable to suppose that neither Dl nor D2 would even consider transferring its membership unless it hoped to gain thereby. Let it be supposed that an incentive for transfer arises from the fishing effort costs of D3 , in relation to those of Dl and D2 • Specifically, let it be supposed that:

314 v. Kaitala and G. Munro

Thus, while Dl would have no incentive to contemplate a transfer of mem­bership, D2 could indeed have such an incentive. The difference in harvest­ing costs holds out the promise to D2 of a profitable sale of membership.

Indeed, we can now argue that the very possibility of effecting a transfer of membership will enhance the bargaining power of D2 in relation to C and D1• Consider the following: if D2 simply ignores the presence of D3 and negotiates a cooperative arrangement with C and D1 , D2 can expect to receive:

1 N N N 3 e(y(O» + JD2(Y(O), Ec ,ED1 ,ED2 )

= 1 wc(y(O» - 1 Ei=C,Dl,D2 Ji(y(O), E~, EK, E~)

+JD2(y(O),E~, EK, E~2)

= 1 wc(y(O» -1 Ei=C,Dl Ji(y(O), E~, E~l' E~2)

+~ JD2 (y(O), E~, E~l' E~2)· (28)

If, on the other hand, D2 were to transfer its membership to D3 , then D3 would receive (ignoring its payment to D2 ) the following:

1 ,",IN N N 2 N N N) 3 wc(y(O»- ~ 3 Ji (y(O), Ec , ED1 , E D2 )+3 JD3(Y(O), Ec , E D1 , ED3 . i=C,Dl,

(29) Thus the difference is:

which is what C and Dl combined would stand to lose if in fact D2 were to transfer its membership to D3 .

If C and Dl are indifferent as to whether side payments are made to D2 or D3 , then it follows that simply by threatening to transfer its mem­bership to D3 , D2 could increase its payoff from the cooperative agreement. Presumably, D2 would have to make a payment to D3 in order to ensure that the transfer threat was credible. If D2 's subnegotiations with D3 could be kept separate from the main negotiations, a not unlikely outcome would be one in which D2 saw its payoff from the negotiations increased, over and above what it would have enjoyed had it refused to consider transferring membership, by the following amount:

By implication, D3 would receive an amount equal to (31).

The Economic Management of High Seas Fishery Resources 315

In any event, if D2 did succeed in increasing its payoff in this fashion, it would have done so by using the threat power of another nation, D 3 , on its own behalf. Whether in fact this outcome can be supported by a game theoretic foundation remains to be seen.

The possibility of transferring a membership to another nation thus in­troduces several different subproblems which must be studied in analysing the alternative outcomes of the negotiations on the joint management of straddling stocks. A major conclusion of our discussion here is that the possibility of one player transferring a membership to one outside the orig­inal coalition makes the negotiation problem into one among four players, instead of three. Thus, the prospective new entrant may influence the man­agement negotiations and indeed may enjoy part of the net return from the fishery, even if it does not formally acquire membership in the coalition.

Subcoalitions with nontransferable membership

We now consider alternative 3 in which membership is nontransferable, but in which members of the overall management coalition can form subcoali­tions. When the three were assumed to act independently, a necessary condition for the achievement of a cooperation agreement was that each player receive no less than its threat point payoff. That remains a nec­essary condition. Now, however, we add a further necessary condition, namely that the agreement be subcoalition proof. That is to say, under a cooperative agreement, a subcoalition must receive a payoff at least as big as it would have received had the members of the coalition cooperated with one another, but had refused to cooperate with the third member of the overall coalition.

Consider first the most obvious subcoalition, namely Dl plus D2. If C refuses to cooperate, D1and D2 have the option of acting independently. The resource will ultimately be driven down to y'fjl and both will be forced out of the fishery. Alternatively Dl and D2 can form a subcoalition, e.g., the lower cost Dl could in effect buyout D2. Ultimately, the resource would still be driven down to y'fjl. Whichever option offers the greatest returns for the subcoalition will be chosen.

Another possible, but not particularly likely, coalition consists of C plus D2 • The resource would, as in the previous example, be driven down to y'fjl.

The third, and most interesting, sub coalition is between C and the lowest cost distant water fishing nation D1 • Suppose that the sub coalition is formed, but then proves unable to cooperate with D2 • The resource will, in this case, be stabilized at y'fj2' not y'fjl. If y'fj2 » y'fjl' the joint payoff to C and Dl may be considerably greater than it would be in the absence of the subcoalition.

316 V. Kaitala and G. Munro

Thus, the possibility of subcoalitions makes the negotiation set smaller than the negotiation set without sub coalitions. That is to say, the set of cooperative solutions from which the agreement should be chosen will be a proper subset of the set of cooperative solutions in the absence of subcoalitions.

Furthermore, we note as well that the possibility of subcoalitions should be taken into account when determining the respective shares of C, D 1 , and D2 in the cooperative management of the resource. The development of the full solution to this problem is, however, beyond the scope of this paper.7

Subcoalitions with transferable membership

Alternative 4, in which both subcoalitions and transfers of membership are possible, is obviously the most complex. Here we do no more than sketch an example of a subnegotiation problem that could be expected to arise in this context.

Return to our discussion of alternative 2 in which subcoalitions are not feasible, but in which transfers of membership are. We showed how D2 could use the threat of transferring membership to D3 to extract a larger share of the net economic return from C and D 1 . With C and Dl acting independently, their defenses were deemed to be weak. If they could form a subcoalition, however, they would be in a position to negotiate with D3 directly, making offers and counteroffers, resulting both in a reduction in the price that D3 is prepared to offer D2 for membership and in a reduction of D2 's position as a player. D2 's position would become essentially passive in which it could accept or reject D3 's offers. The authors are unaware of any existing solution to this problem in the context of dynamic cooperative games.

5. Conclusions

Over the past several years, a new and important transboundary fisheries management issue has arisen under the New Law of the Sea. The issue con­cerns the management of fishery resources to be found in both the Exclu­sive Economic Zone and the adjacent high seas, what we term "straddling stocks broadly defined." This new transboundary fisheries management is­sue, which is now the focus of a major U.N. intergovernmental conference, can be compared with the older form of transboundary fisheries manage­ment issue, the management of fishery resources "shared" by two or more coastal states.

In this paper, we make a first attempt in analysing the economic man-

7For alternative approaches to the solution of such game theory problems, see Mesterton-Gibbons 1992.

The Economic Management of High Seas Fishery Resources 317

agement of "straddling stocks." In so doing, we bring to bear the theory of dynamic cooperative and noncooperative games.

We commence by asking how far we can proceed in understanding the management of "straddling stocks" by applying the now well established economic analysis of the management of "shared" fishery resources. The answer is not very far. While the consequences of noncooperation are seen to be much the same in both cases, cooperative management of "straddling" stocks proves to be far more complex than the management of "shared" stocks.

As a consequence of its complexity, this paper must be seen as no more than an introduction to a very difficult subject. Even with the introduction of rather restrictive assumptions, we are forced to leave many questions unanswered.

Much future research, and many additional papers, will be required to provide a complete analysis of the issue.

References

[1] Basar, P. and G.J. Olsder, Dynamic Noncooperative Game Theory, New York, Academic Press, (1982).

[2] Clark, C.W., "Restricted Access to Common-Property Fishery Re­sources," in: Dynamic Optimization and Mathematical Economics, P. Liu, ed., New York, Plenum Press, 117-132, (1980).

[3] Clark, C.W., Mathematical Bioeconomics: The Optimal Management of Renewable Resources, 2nd edition, Wiley-Interscience, New York, (1990).

[4] Dutta, P.K. and R.K. Sundram, How different can strategic models be?, Journal of Economic Theory, vol. 60,41-61, (1993).

[5] Fischer, R.D. and L.J. Mirman, Strategic dynamic interaction: Fish wars, Journal of Economic Dynamics and Control, vol. 16, 267-287, (1992).

[6] Kaitala, V., Game Theory Models of Dynamic Bargaining and Con­tracting in Fisheries Management, Helsinki University of Technology, Institute of Mathematics, Systems Research Reports All, (1985).

[7] Kaitala, V., "Game Theory Models in Fisheries Management: A Sur­vey," in: Dynamic Games and Applications in Economics, T. Basar, ed., Lecture Note in Economics and Mathematical Systems, Berlin, Springer-Verlag, 252-266, (1986).

318 V. Kaitala and G. Munro

[8] Kaitala, V., Nonuniqueness of No-Memory Feedback Equilibria in a Fishery Resource Game, Automatica, vol. 25, 587-592, (1989).

[9] Kaitala, V. and G.R. Munro, "The Management of High Seas Fish­eries," paper prepared for the International Conference on Fisheries Economics, Solstrand Fjord, Norway, (1993).

[10] Kaitala, V. and M. Pohjola, Optimal recovery of a shared resource stock: A differential game with efficient memory equilibria, Natural Resource Modeling, vol. 3, 91-119, (1988).

[11] Leitman, G., Cooperative and Noncooperative Many Player Differential Games, Springer-Verlag, Vienna, (1974).

[12] Mesterton-Gibbons, M., An Introduction to Game Theoretic Modeling, Redwood City, Addison-Wesley, (1992).

[13] Munro, G., The optimal management of transboundary renewable re­sources, Canadian Journal of Economics, vol. 12, 355-376, (1979).

[14] Munro, G., The optimal management oftransboundary fisheries: game theoretic considerations, Natural Resource Modeling, vol. 4, 403-426, (1990).

[15] Munro, G. and A.D. Scott, "The Economics of Fisheries Management," in: Handbook of Natural Resource and Energy Economics, vol. 2, A.V. Kneese and J.L. Sweeney, eds., Amsterdam, North Holland, 623-676, (1985).

[16] Nash, J., "The Bargaining Problem," Econometrica, vol. 18, 155-162, (1950).

[17] Nash, J., Two person cooperative games, Econometrica, vol. 21, 128-140, (1953).

[18] Roth, A.E., Axiomatic Models of Bargaining, Springer-Verlag, Berlin, (1979).

The Beijer International Institute of Ecological Economics The Royal Swedish Academy of Sciences Box 50005,8-10405 Stockholm, Sweden Fax: (46) 8-152464 Tel: (46) 8-6739500

University of British Columbia Department of Economics

#997 - 1873 East Mall, Vancouver, B.C. Canada V6T 1Z1 Fax: 604-822-5915 Tel: 604-822-2876

Pollution-Induced Business Cycles: A Game Theoretical Analysis

David W.K. Yeungl

Abstract

This paper presents a differential game of pollution management. The industrial sector chooses the level of investment to maximize net revenue and the government imposes a tax and uses the tax proceeds for pollution abatement operations. The feedback of pollution on capital accumulation and the effect of the level of pollution on the natural rate of decay are incorporated in the model. We solve for the (subgame perfect) feedback Nash equilibrium solution of the game, and obtain explicitly the game equilibrium accumulation dynamics of capital and pollution. Various properties of the equilibrium follow from this closed form solution. It is found that the game equilibrium output path exhibits continual oscillation about a long run equilib­rium level. Finally, when we allow a constant rate of decay, damped output cycles appear. Key words: Differential game, feedback Nash equilibrium, pollution-induced business cycles.

1. Introduction

Though physical waste from industrial production has been a major and continual source of pollution since the industrial revolution, the first for­mal economic study concerning pollution appeared in 1932 in Pigou's The Economics of Welfare. In the past three decades, increasing attention has been given to physical waste that accompanies production and consump­tion. D'Arge and Kogiku (1972), Plourde (1972), Forster (1975), Das­gupta (1982), Lin (1987), Hartl (1988), and Plourde and Yeung (1989) have studied the issue in an optimal control framework. Recently, con­cerns over strategic reactions in the competing situation between policy makers and pollution generating agents lead to the development of game theoretical analysis in pollution control (for examples, see Misiolek (1988), Yao (1988), Millerman and Prince (1989), Yeung (1992), and Yeung and Cheung (1993)). In this paper, we develop a differential game between a policy maker (the government) and the industrial sector. The industrial

IThe author would like to thank two anonymous referees for their extremely helpful comments.

320 D. W.K. Yeung

sector chooses an investment strategy to maximize net revenue from output produced for consumption. The government values output and consump­tion positively but pollution negatively. Its objective is to tax industrial output and spend the proceeds on pollution abatement, to maximize a so­cial welfare function containing consumption and pollution as arguments. Two dynamic processes - one for capital accumulation and the other for pollution build up - are considered in the game.

Capital accumulation depends on investment by the industrial sector and the physical depreciation of capital stock. While existing economic studies leave out the direct effect of pollution on capital accumulation, environmental studies show that pollution does have adverse effects (like corrosion) on engineering materials (see Huang (1992)). Raghu and Hsieh (1989) find that structures constructed on and adjacent to chromium waste sites underwent significant structural distress. Effects such as tilting of walls, buckling of floor slabs, and flaking of mortar from masonry were discovered. The experiments of Ferenbaugh et al. (1992) indicate that sulfur-oxidising bacteria have the potential to break down Sulphex (a mix­ture of elemental sulfur and plasticizers used in road construction) paving roads. To capture this phenomenon in the game considered below, we specify a positive relationship between capital depreciation and the level of pollution. Pollution build-up is determined by the amount of indus­trial output, the level of pollution abatement and the natural rate of decay of pollutants. Recent ecological research shows that biogeochemical feed­backs in pollutants result in more rapid environmental deterioration than is predicted (see Schimel (1990)). Climate scientists have long professed the presence of natural amplification - positive biogeochemical and ecological feedbacks - of global warming (see Leggett (1990)). Evidently the rate of self-purification of many pollutants is affected by the amount of waste load in the atmosphere. To model this, we follow Forster (1975) and hypothesize that the natural rate of pollution decay is negatively related to the level of pollution present. We solve the game for its feedback Nash equilibrium, and derive a set of state-dependent feedback Nash equilibrium strategies. The resulting time paths of capital and pollution accumulation are obtained explicitly. In particular, it is found that the level of output would oscillate around a long run equilibrium. At the same time, there is also a pollution cycle.

The paper is organized as follows. Section 2 presents the game model. The feedback Nash equilibrium solution of the game is obtained in Section 3. The game equilibrium time paths of capital and pollution which lead to closed cycles are derived in Section 4. Section 5 provides a model variation which allows the rate of decay to be constant and obtains damped cycles of output and pollution. Section 6 concludes.

Pollution-Induced Business Cycles 321

2. The Game Model

Consider a game model with two decision makers - the government and the industrial sector. The industrial sector uses capital to produce a homo­geneous product and pollution is generated in the process of production. The government acts as a regulator and levies a pollution tax on output and uses the tax proceeds for pollution abatement. The industrial sector maxi­mizes the present value of net profits by choosing the amount of investment (via choosing a saving rate); or

max {'XO m [(1- 7r(s»RK(s)! _ r(s)P(s)] e-rsds (1) ".(s) io

where K(s) is capital stock at time s, P(s) is the level of pollution, RK(s)! is the production function, r(s) is the pollution tax rate, 7r(s) is the saving rate, 7r( s )RK (s)! is the amount of output saved for investment purposes, r is the discount rate, m is the unit price of output which is normalized to 1 for notational clarity.

Capital accumulation is governed by the dynamics: 2

K(s) = [7r(s)RK(s)!]! K(s)! - a(:~:D! K(s) (2)

K(O) Ko

where (7rRK!)!K! is an Uzawa's type of installation function which con-I

verts investment into capital stock, and a(~~:D 2" is the depreciation rate of capital stock.

Capital depreciation exhibits dependence on the ratio of pollution to capital. This reflects the adverse effects of pollution on capital for reasons mentioned previously.

The government on the other hand levies a pollution tax (by choos­ing a tax rate indexed on the level of pollution) and uses it for pollution abatement to maximize an objective function which is positively related to the amount of output left for consumption and negatively to the level of pollution:

max roo {1'[(1- 7r(s»RK(s)! _ r(s)P(s)] - wP(s)!} e-rsds (3) r(s) io

where I' and ware the government's welfare weights for consumption output and pollution.

2(1) and (2) share a common feature with the standard growth model with savings for investment purposes in that they require a positive initial capital stock. It is assumed that there exists an initial capital stock, which may be in the form of land, natural waterways or waterfalls.

322 D. W.K. Yeung

The pollution build-up dynamics are specified as3

p(s) =

P(O) =

[ 1] 1 1 1 (n)~ gP(s)'i RK(s)'i - b(7(S)P(S»'iP(s)'i - P(s) P(s)

Po. (4)

[gP(s)t] is the amount of pollution created when one unit of output is produced. Note that the higher the current level of pollution, the higher the amount of pollution created per unit of output. The result reflects the recent discovery of biogeochemical feedbacks and interactions of pollutants (see Schimel (1990». b(7P)~P~ represents a pollution clean-up function the output of which depends on the amount of spending on pollution abate­ment 7 P and the level of pollution. The constant b indicates the efficiency

1

of the process. The natural rate of decay of pollutants (/(s») 'i decreases as the level of pollution rises. This follows from Forster's (1975) argument that the rate of self-purification declines as the amount of pollution increases.

Finally, the duration of the game is [0,00), the state space [K(s),P(s)] E R2, the control spaces n(s) E II S;;;; Rand 7(S) E r S;;;; R where II is the set of all feasible saving rates and r is the set of all feasible tax rates.

3. Feedback Nash Equilibrium Solution

To avoid the problem of time inconsistency, we consider a (subgame per­fect) feedback Nash equilibrium solution of the game described in the above section. A feedback saving strategy (respectively tax strategy) is a decision rule that depends on time and the current states only -n(s) = ¢(K(s), P(s), s) (respectively 7(S) = tp(K(s), P(s), s».

Definition 1. A pair of feedback strategies {n*(s) = ¢*(K(s),P(s),s), 7*(S) = tp*(K(s), P(s), s)} constitutes a feedback Nash equilibrium solu­tion for the game (1)-(4), if there exist functionals V(K(t), P(t), t) and U(K(t), P(t), t) which satisfy the following conditions (see Basar and Ols­der (1982»:

V(K(t), P(t), t) = It [(1 - ¢*(K*(s), P*(s), s»RK'(s)~ -tp*(K*(s), P*(s), s)P*(s)] e-rsds

~ It~ [(1 - ¢(K(s), P(s), s»RK(s)t

-tp*(K(s), P(s), s)P(s)]e-rsds

3Note that for any initial positive pollution po. P(s) will remain positive in finite time s. It is assumed that there exists an initial (perhaps very low) level of pollution by nature itself.

Pollution-Induced Business Cycles 323

'V q,(K(s), P(s), s) in the feasible set of1l"(s), which satisfies the accumula­tion dynamics (2) and (4);

U(K(t), P(t), t) = hoo {,[(I - q,*(K*(s), P*(s), s))RK*(s)!

cp*(K*(s), P*(s), s)P*(s)] - wP*(s)!} e-rsds

~ hoo {,[(I - q,(K(s), P(s), s))RK(s)!

- cp*(K(s), P(s), s)P(s)] - wP(s)! } e-rsds

'V cp(K(s), P(s), s) in the feasible set ofr(s), which satisfies the accumula­tion dynamics (2) and (4); and where

](*(s)

P*(s)

K*(O)

1 1 1 P* (s) 1

= [q,*(K*(s), P*(s),s)RK*(s)?l)2 K*(s)?l - a(K*(s))?l K*(s)

1 1 1 1 = [gP*(s)?l]RK*(s)?l - b[cp*(K*(s), P*(s), s)P*(s)]?l P*(s)?l

( p~ S ) ) ! P* ( s )

= Ko and P*(O) = Po

• One salient feature of the concept of feedback Nash equilib­

rium introduced in Definition 1 is that if a pair of strategies {q,*(K(s), P(s), s), cp*(K(s),P(s),s)} provides a feedback Nash equilib­rium solution to the game with duration [0,00], its restriction to the time interval [t,oo] for t > 0 provides a feedback Nash equilibrium solution to the subgame defined on the shorter time interval [t,oo], with initial states taken as {K(t), P(t)}. So far, only a countable number of games have been identified to have a feedback Nash equilibrium solution.4 In the sequel, we will obtain the feedback Nash equilibrium strategies q,*(K(s),P(s),s) and cp*(K(s), P(s), s).

The value functions V(K(t), P(t), t) and U(K(t), P(t), t), vis-a-vis dy­namic programming techniques, have to satisfy the following Hamilton­Jacobi-Bellman equations (see details in Basar and Olsder (1982) pp. 284-287):

-Vi = max {[(I - 1I"(t))RK(t)! - r(t)P(t)]e-rt '/r(t)

[ 1 1 3 1 1] +VK 1I"(t)?lR?lK(t)4 - aP(t)?lK(t)?l

4For some examples of games with a feedback Nash equilibrium, see Reynolds (1987), Fershtman (1987), Sorger (1989), Yeung (1989), Jorgensen and Sorger (1990), and Yeung and Cheung (1993). A list of examples of such games can be found in Yeung (1994).

324 D. W.K. Yeung

+ Vp gP(t)2RK(t)2 - br(t)2P(t) - OP(t)2 [ 1 Ill]}

-Ut = max {,[(l- 7r(t))RK(t)! - r(t)P(t)]e-rt - wP(t)! e-rt r(t)

[ 1 1 3 1 1] +UK 7r(t)2R2K(t)4 - aP(t)2K(t)2

+ Up gP(t)2RK(t)2 - br(t)2P(t) - OP(t)2 . [ 1 Ill]} (5)

Performing maximization of the expressions inside the curly brackets gives:

7r(t) =

r(t)! =

(6)

(7)

Substituting (6) and (7) into (5), we obtain a pair of partial differential equations:

-vt = e-rt [RK! _ YkKe2rt _ b2 U; Pe2rt ] 4 41'

+VK [.!:fKert -aP!K!]

+Vp [gP!RK! + ~~Upert - OP!]

-Ut = e-rt [,RK! - ,Yj-Ke2rt - ~~U~Pe2rt]

+UK [.!:fKert - aP!K!]

Proposition 1. System (5') admits a solution:

lIt V(K(t), P(t), t) = [AIK(t)2 + C1P(t)2 + Dlle-r

U(K(t), P(t), t) = [A2K(t)! + C2P(t)! + D2le-rt

where

Al 4rR

agR+4r2

C1 -2aR

agR+ 4r2

A2 4,rR - 2gRw

agR + 4r2

C2 -(2a,R + 4rw)

agR + 4r2

(5')

Pollution-Induced Business Cycles 325

(8)

Proof. See the Appendix. • Upon evaluating VK and Up from V and U in Proposition 1, we obtain

a set of state-dependent feedback Nash equilibrium strategies:

7r*(t) = ¢*(K(t), P(t), t) = 1iRA~ K(t)-! = (ag~:~r2)2 K(t)-! (9)

r*(t) = cp*(K(t) P(t) t) = b2 C2p(t)-1 = b2 (2a"),R+4rw)2 p(t)-1. , , 16")'2 2 16")'2 (agR+4r 2 )2

Remark. The feedback Nash equilibrium is subgame perfect (Selten (1975», in which the pair of solution strategies in (9) constitutes an equilibrium for every possible subgame starting at t E [0,00).

4. The Game Equilibrium Dynamics:

Pollution-Induced Business Cycles

To characterize capital accumulation and pollution build-up in game equi­librium, we substitute (9) into (2) and (4):

K(s)

p(s)

K(O)

1 1 1 1 = 4"A1K(s)2 - aP(s)2K(S)2

1 1 (b2 ) 1 gRK(s)2P(S)2 + 4, C2 - n P(s)2

Ko and P(O) = Po.

Solving this pair of non-linear differentials gives:

K(s) = [h1..!!:..- cosys + h2..!!:..- sinys - (~C2 _ n) /9R] 2 2y 2y 4,

P(s) [ h1 sin ys - h2 cos ys + :~] 2

where

y JagR 4 '

2y 1 (b2 ) / h1 = ~ K~ + 4, - n 2y agR and

h2 = A1 - p.! 4a o·

(10)

(11)

326 D. W.K. Yeung

To obtain the movements of output Q(s) = RK(s)!, we perform the following variable transformations: Q(s) = RK(s)! and pes) = P(s)!. The resultant dynamics of Q( s) and p( s) are:

( Q(s)) (0 pes) = ~g

aR) (Q( )) ( & ) -- S 8R

02 peS) + (~~G2 - ~). (12)

The solution of (12) is:

Q(s) = hI~:COSYS+h2~:sinYS-(!~G2-n)/9 (13)

( ) h · h Al P s = I sm ys - 2 cos ys + 4a (14)

with values of y, hI and h2 being the same as those in (11). A geometric representation of the solution paths (13) and (14) can be

shown in a phase space. Since the characteristic roots of the 2 x 2 matrix in (12) are ±i..jagR, there is a vortex. Figure 1 below demonstrates the movements of Q(s) and pes).

p

p=o

~ r-----~--------+---------+_------

r L. o~------------~---------------------- Q

- (*C1 - n)/g

Figure 1: Phase diagram characterizing the movements of Q and p.

Pollution-Induced Business Cycles 327

Proposition 2. There exists a business cycle with amplitude of fluctuation

H = [RKg + (!~ _ 0) / g] 2 + [~: (:~ _ PO!) ] 2,

cyclical period 211" /y and output fluctuating around a long run equilibrium level -(!~C2 - O)/g.

Proof. Note that H can be expressed as V(h1 ~~)2 + (h2~~)2. We can then obtain a value Q such that:

h1 aR COSQ=~ and

H

Hence (13) can be written as:

h aR • 2 2y SlDQ=~.

Q(s) = H cos Q cos ys + HsinQsinys - (!~ C2 - 0) / g.

Making use of the fact that cos 01 cos O2 + sin O2 sin O2 = cos( 01 - O2), we have

Q(s) = H COS(ys - Q) - (!~ C2 - 0) / 9

which is a modified cosine function of time s with amplitude H, a cyclical period 211"/y and fluctuations around -(~~C2 - O)/g. •

Note that the amplitude of fluctuation depends on the initial values Ko 1 2 1 A

and Po. When RKg equals -(~I'C2 - O)/g and Pl equals Ta"' the am-1. 1.

plitude H = o. As the initial point {RK ~ , P02} moves further away from

{-(!~C2 - O)/g, * }, H increases. In similar fashion one can obtain a

pollution cycle from (14). Moreover, since the business cycle is solved ex­plicitly, effects of parametric changes can be obtained readily. For instance:

Pl An increase in the pollution emission parameter produces more fre­quent output fluctuations and a lower level of average output.

Proof. Since ~ > 0, an increase in 9 reduces the length of the period of a cycle 211" /y and hence makes fluctuations more frequent.

The average value of output AQ = - (~~C2 - 0) / g, and

oAQ _ _ b2 (2a"{R + 4rw)(2aRg + 4r2) _ ~ < o. og - 4"{ (ag2R + 4gr2) g2·

328 D. w.K. Yeung

5. Constant Rate of Pollution Decay and Damped Cycles

In this section, we examine the case when we relax the assumption that the rate of decay of pollutants is affected by the level of pollution. Consider the situation where there is a constant rate of decay. The pollution build-up dynamics (4) then become:

p(s) = [gP(s)]! RK(s)! - b[T(S)P(S)]! P(s)! - OP(s) (4')

The Hamilton-Jacobi-Bellman equations for the game (1), (2), (3) and (4') are:

-yt = e-rt [RK~ _ YIKe2rt _ b2 U; Pe2rt ] 4 4')'

+VK [.!::fKert - aP~K~]

+Vp [gP~RK~ + g~Upert - OP]

-Ut = e-rt ['YRK~ - 'YfKe2rt - !~U~Pe2rt]

+UK [.!::fKert - aP~K~]

+Up [gP~RK~ + g~Upert - OP] . (15)

Following the analysis used in Proposition 1, we can readily prove the fol­lowing proposition.

Proposition 3. The value functions satisfying (15) are:

where

V(K(t), P(t), t) = [AIK(t)! + C1P(t)! + D1]e-rt

U(K(t), P(t), t) [A2K(t) ~ + C2P(t) ~ + D2]e-rt

R(r + ¥) -=---=--'---:--"'-'--::::-:-::- > 0 [~+r(r+ ¥)]

-!aR --=----"----:,..,-::- < 0 [~+r(r+¥)] R[(r+ ¥h - ~] [~+r(r+ ¥)] -[~ +wr]

2 < 0 [~+r(r+ ¥)]

Pollution-Induced Business Cycles 329

• The feedback Nash equilibrium strategies are:

1r*(t) = ¢*(K(t), P(t), t) = I~RA~K(t)-!

r*(t) = <p*(K(t), P(t), t) = 1:~2 CiP(t)-l. (16)

Substituting these equilibrium strategies in the dynamical equations (2) and (4'), we obtain

K(s)

p(s)

1 1 1 I = 4AlK(s)2 - aP(s)2K(S)2

I 1 b2 1 = gRK(s)2P(S)2 + "4'YC2P(S)2 - OP(S).

Once again, let Q(s) = RK(s)! and p(s) = P(s)!, and obtain:

( Q(S») (0 _aR) (Q(s») (¥) p(s) = !g -¥ p(s) + ~~C2 .

The characteristic roots of the 2 x 2 matrix in (18) are

(a) Underdamped, Cycles

_~ ±.; (~)2 _ agR

2

(17)

(18)

(19)

In the case when (~)2 < agR, the roots are complex and there is a stable focus or an underdamped cycle. The moments of output and the square-root of pollution stock are indicated in Figure 2.

Output oscillates about a long run equilibrium level 1~~ - 4~9 C2 and the amplitude of fluctuation dies down as according to:

Q(s) = e-!js [Hh2c + hl~) coscs - ~ (hlc - h2~) sincs]

+ (AIr! _ b2C2) 4ag 4"(g

330

p

p=o /

I

D. W.K. Yeung

Figure 2: Phase diagram characterizing the movements of Q and p in the case of an underdamped cycle.

where

(20)

The pollution dynamics follow an underdamped cycle according to

(b) Overdamped and Critically Damped Cycles

In the case when (~)2 > agR, we have a pair of negative and real roots. We have a stable node with overdamped motion. The movements

Pollution-Induced Business Cycles 331

of Q and p are shown in Figure 3. In particular, there is at most one relative extremum and output will move towards the long run steady state A10 b2 C 5 4ag - 4-yg 2·

p

/

p=o

/ /

/ t -

* t------~!--------- Q = 0

/ O~--~-------------------------Q

- b2 C 4g-y 2

Figure 3: Phase diagram characterizing the movements of Q and p in the case of an overdamped or critically damped cycle.

The solution path of output can be obtained as

2( 0) ),s 2( 0) ),s (A 10 b2C2 ) Q(s) = - Al + - hIe 1 + - A2 + - h2e 2 + -- - --9 2 9 2 4ag 4,g where

Al = (-~ + J(~)2 -agR) /2

A2 = ( - ~ - J (~) 2 - agR) /2 5For more details on damped motion, see Nitecki and Guterman (1986).

(22)

332 D. w.K. Yeung

hi =

The pollution path follows

P(s) = [P(sW = [hi eA18 + h2eA28 + :~ r (23)

In the case when (~)2 = agR, we have a double root -!,t. There is a critically damped cycle. The dynamical behaviour is similar to that in the case of an overdamped cycle. The corresponding output path and pollution path are respectively:

Q(s)

P(s)

where

hi = 1

p''i _.& o 4a

gRKj _ po!n _ A1n _ b2 G 2 4 i6a 8')' 2·

6. Concluding Remarks

Despite the voluminous research on business cycles, there appears to be nothing resembling a consensus among researchers and policy makers about the sources of such cyclical behaviour. This reflects the diverse nature of the causes of fluctuations in aggregate economic activity. Pure economic theories - such as the Keynesian and monetarist paradigms, and the ratio­nal expectations and neo-classical approaches (see Belongia and Garfinkel (1992) for descriptions of these theories) - have occupied the limelight in providing an explanation. This paper presents a new explanation for the formation of business cycles, namely, the feedback and interaction among economic activities and the pollution stock. In particular, it presents a differential game of pollution management which incorporates the feedback of pollution on capital accumulation and the effect of the level of pollution on the natural rate of decay. We solve for the (subgame perfect) feedback

Pollution-Induced Business Cycles 333

Nash equilibrium solution of the game, and obtain explicitly the game equi­librium accumulation dynamics of capital and pollution. It is found that the game equilibrium output path exhibits continual oscillation about a long run equilibrium level. Finally, when we allow a constant rate of decay, damped output cycles appear.

Appendix

Proof of Proposition 1

Differentiate V and U in Proposition 1 to obtain VK, Vp, UK and Up. Upon substituting them into (8), cancelling common multiples and rearranging common terms, we obtain:

rA1K! + rC1P! + rD1

( R)! a ! A2 b2 2 b2 C fl C = R+g"2C1 K2 - "2A1P2 + it - 16,2C2 + 8,C1 2 -"2 1

and (A.I)

- ('VR + flBC2)K! - (f!.A2 + w)p! - ,A~ + £C22 + A,A2 - flC2 - I 2 2 16 16, 8 2

For (A.I) to hold, the following equations must be satisfied:

rA1 = R + g11C1

rC1 =

rD1 =

rA2 =

rC2 =

-%A1 A2 b2 2 b2 fl it - 16,2C2 + 8,C1C2 - "2 C1

,R + 9.f}C2

-(%A2 + w) rD2 = _~ - £C2 + A,A2 + b2 C2 _ flC

16 16, 2 8 8, 2 2 2

Solving (A.2.I) and (A.2.2) simultaneously yields

Al = 4rR agR + 4r2

-2aR and C1 = R 4 2· ag + r

Solving (A.2.4) and (A.2.5) simultaneously yields

A - 4,rR - 2gRw d C _ -(2a,R + 4rw) 2- an 2- .

agR + 4r2 agR + 4r2

(A.2.I)

(A.2.2)

(A.2.3)

(A.2.4)

(A.2.5)

(A.2.6)

(A.3)

(A.4)

With AI, A2 , C1 and C2 given as in (A.3) and (A.4), we can obtain

(A.5)

334 D. W.K. Yeung

A!, A2, CI, C2, Dl and D2 in (A.3)-(A.5) are indeed the unique solution of (A.2). The unique solvability of (A.2) leads to a unique solution (as that in Proposition 1) for system (A.l). For similar results in a dynamic game setting, see Basar and Olsder (1982) pp. 253-254. •

References

[1] Basar, T. and G.J. Olsder, Dynamic Noncooperative Game Theory, Academic Press, New York, (1982).

[2] Belongia, M.T. and M.R. Garfinkel, eds., The Business Cycle: Theo­ries and Evidence,Kluwer Academic Publishers, London, (1992).

[3] d'Arge, R.C. and K.C. Kogiku, Economic growth and environment, Review of Economic Studies, 40, 61-78, (1972).

[4] Dasgupta, P., "Environmental Management Under Uncertainty, in: Explorations in Natural Resource Economics, V.K Smith and J.V. Krutilla, eds., Johns Hopkins University Press, Baltimore, (1982).

[5] Ferenbaugh, R.W., E.S. Gladney, L.F. Soholt, KA. Lyall, M.K Wallwork-Barber, and L.E. Hersman, Environmental interactions of sulphlex pavement, Environmental Pollution, 16, 141-145, (1992).

[6] Forster, B.A., Optimal pollution control with a nonconstant exponen­tial rate of decay, Journal of Environmental Economics and Manage­ment, 2, 1-6, (1975).

[7] Hartl, R.F., The control of environmental pollution and optimal invest­ment and employment decisions: a comment, Optimal Control Appli­cations and Methods, 9, 337-339, (1988).

[8] Huang, H., Studies of acid rain in the eastern United States: A review, International Journal of Environmental Studies, 41, 267-275, (1992).

[9] Jorgensen, S. and G. Sorger, Nash equilibria in a problem of opti­mal fishery management, Journal of Optimization Theory and Appli­cations, 64, 293-310, (1990).

[10] Leggett, J., "The Nature of the Greenhouse Threat," in: Global Warm­ing: The Greenpeace Report, J. Leggett, ed., pp. 14-43. Oxford Uni­versity Press, Oxford, (1990).

[11] Lin, W.T., The control of environmental pollution and optimal invest­ment and employment decisions, Optimal Control Applications and Methods, 8, 21-36, (1987).

Pollution-Induced Business Cycles 335

[12] Millerman, S.R. and P. Prince, Firm incentives to promote technolog­ical change in pollution control, Journal of Environmental Economics and Management, 17, 247-265, (1989).

[13] Misiolek, W.S., Pollution control through price incentives: the role of rent seeking costs in monopoly markets, Journal of Environmental Economics and Management, 15, 1-8, (1988).

[14] Nitecki, H.Z. and M.M. Guterman, Differential Equations with Linear Algebra, Saunders College Publishing, New York, (1986).

[15] Pigou, A.C., The Economics of Welfare, 4th ed., Macmillan, London, (1932).

[16] Plourde, C.G., A model of waste accumulation and disposal, Canadian Journal of Economics, 5, 119-125, (1972).

[17] Plourde, C.G. and D. Yeung, A model of industrial pollution in a stochastic environment, Journal of Environmental Economics and Management, 16, 91-105, (1989).

[18] Raghu, D. and H. Hsieh, Origin, properties and disposal problems of chromium ore residue, International Journal of Environmental Studies, 34, 227-235, (1989).

[19] Reynolds, S.S., Capacity investment, preemption and commitment in an infinite horizon model, International Economic Review, 28, 69-88, (1987).

[20] Schimel, D., "Biogeochemical Feedbacks in the Earth System," in: Global Warming: The Greenpeace Report, J. Leggett, ed., pp. 68-82, Oxford University Press, (1990).

[21] Selton, R., Reexamination of the perfectness concept for equilibrium points in extensive games, International Journal of Game Theory, 4, 25-55, (1975).

[22] Sorger, S., Competitive dynamic advertising: A modification of the case game, Journal of Economic Dynamic and Control, 13, 55-80, (1989).

[23] Uzawa, H., Time preference and the Penrose effect in a two-class model of economic growth, Journal of Political Economy, 77, 628-652, (1969).

[24] Yao, D.A., Strategic responses to automobile emissions control: A game theoretic analysis, Journal of Environmental Economics and Management, 15, 419-438, (1988).

336 D. W.K. Yeung

[25] Yeung, D., A class of differential games with state-dependent closed­loop feedback solutions, Journal of Optimization Theory and Applica­tions, 62, 165-174, (1989).

[261 Yeung, D., A differential game of industrial pollution management, Annals of Operations Research, 37, 297-311, (1992).

[27] Yeung, D., A lemma on differential games with a feedback Nash equi­librium, Journal of Optimization Theory and Applications, 82(1), 181-188, (1994).

[28] Yeung, D. and M. Cheung, Capital accumulation subject to pollution control: A differential game with a feedback Nash equilibrium, Annals of Dynamic Games, 289-360, (1994).

School of Economics, Centre of Urban Planning & Environmental Management The University of Hong Kong, Pokfulam Road, Hong Kong TEL: (852) 859-1035 FAX: (852) 559-1697 E-mail: WKYEUNG@HKUCC.HKU.HK

Management of Effluent Discharges: A Dynamic Game Modell

Jacek B. Krawczyk

Abstract

This paper is concerned with the problem of the management of effluent dumped into a stream by identifiable polluters. The problem involves a Regional Council which imposes environmental levies on the polluters whose economic activity, otherwise beneficial for the region, results in pollution of the stream. The model for the problem of effluent management is formulated as a dynamic game between the Regional Council and the polluters. The game is "played" in discrete time. The players in the game are the polluters ("followers") and the Council (the "leader"). This formulation leads naturally to a Stack­elberg concept of the solution for the game at hand. Because of the obvious difficulties implied by this solution concept, an equilibrium will be sought through the use of an applicable Decision Support Tool wherever an analytical solution appears intractable.

The polluters are supposed to be myopic and small; and the Re­gional Council is interested in promoting production, collecting taxes, and maintaining a clean environment. The model of spread of the pollution within the stream allows for advection and biodegradation.

1. Introduction

The problem of managing effluent is omnipresent. Practically every farm, factory and human settlement is producing liquid waste which is eventu­ally dumped into a more or less distant river, lake or sea. These can cope with effluent quite well (e.g., by "neutralising" it by dilution) until cer­tain critical concentration levels of environmentally unfriendly substances, present in the emissions, are exceeded. Concentrations above those levels can cause environmental damage like an epidemic, extinction of a species, destruction of recreational or spiritual value of an area etc. Keeping that damage minimal (in some sense) through managing effluent is therefore an acute necessity which has been recognised in literature; see [13], [4), [12), [14).

The cumulative effects of pollutants on the environment, and of human activities on production and pollution imply the use of a dynamic model in

1 Research supported by VUW GSBGM and WRC.

338 J.B. Kmwczyk

the effluent management process. The dynamic aspect of pollution control has been studied, among others, in [14], [4], and [5].

In general, the water pollution can be a point-source where identify­ing effluent emission points is possible2 (cf. e.g., [4]), or distributed-source (cf. e.g., [6]), in any other case3 .

Note that the above categorisation differs slightly from that which dis­tinguishes between the point-source pollution from an identifiable polluter, and the non-point-source pollution not traceable to the polluter (cf. [14], [12]). If there was an efficient method of curbing pollution in the "non­traceable" case it would be attractive for the authority to use this method as it would spare them the cost of monitoring how much an agent pol­lutes. However, the common conclusion which one can find in literature, cf. [14], [12], is that the minimum, or satisfactory, concentration of a pol­lutant can be achieved if "suspected" polluters are taxed indiscriminately 4

once the pollutant critical levels have been exceeded, apparently irrespec­tive of the producers, actual emission or waste abatement. Such a solution to the pollution problem implies "group responsibility" and can be difficult to legislate in countries with strong farmers' lobbies. In this paper, we concentrate on the (identifiable) point-source effluent management prob­lem and thus prepare a base for a piece of legislation according to which polluters would be taxed depending on how much they pollute.

Another important distinction between the sources of effluent is whether it is an effect

a. of an economic activity of an agent, or

b. of a non-economic activity of a community.

While the effluent from b. might, in some areas, be more intensive than a., it is difficult5 to think that it would be controlled through a commercial mechanism. In this article we look for economic instruments and will deal with case a.

Individually monitored or not, indiscriminately levied or not, effluent producers are potentially in conflict with each other. The cause of the conflict can be economic competition, and/or global common constraints on the amount of effluent tolerated by the environment. On the other hand, the

2Which implies that monitoring of a single polluter is possible. 3Here, one can distinguish between the distributed-source pollution where it can be

traced to a polluter, and where it cannot. 4 Actually, for the stochastic case, (14) proposes to differentiate among polluters by

making the levy amount dependent on the second derivative of a polluter value function. This does not seem to be easily computable, hence not applicable.

5Though not impossible, think of penalties which a community would pay for not having installed a waste neutralisation plant.

Management of Effluent Discharyes 339

producers depend on some regional authority whose aim is to negotiate, and legislate, agreements on admissible pollution and abatement policies of each polluter. Features such as the conflicting interests and the possibility of negotiations naturally suggest a game theoretical approach towards solving the effluent management problem. Existing results in environment control, obtained through dynamic games, cf. [101, [14], do not concern point-source water pollution. This paper is, to the author's knowledge, one of the first to try a dynamic game model for management of point-source water pollution (cf. [6]).

The approach presented in this paper attempts to be consistent with the way in which negotiations concerning improvement of the water quality of a certain stream polluted by farmers are conducted. It assumes that if agreement about setting the emission levels is possible, it would be im­plemented by the Regional Council through the fees and charges levied on the polluters using the stream. The model for the effluent management problem is formulated as a (non-smooth) dynamic game between a leader and followers. The game is "played" in discrete time over a year but can be extended to an infinite time horizon. The players in the game are the pol­luters ("followers") and the Regional Council ("leader"). This formulation leads naturally to a Stackelberg concept of solution for the game at hand. In general, finding a Stackelberg equilibrium for a dynamic game is diffi­cult (cf. [1]). We will recommend the use of a Decision Support Tool (DST) similar to the one used in [81, should an analytical solution be unavailable.

The paper is organised as follows. First, in Section 2, the assumptions under which the model can function, are formulated. Next, in Section 3, the polluters' problem is discussed and modelled. In Section 4, The Re­gional Council problem is formulated. The possibility of solution to the resulting game problem is discussed in Section 5. In Section 6, a hierarchi­cal optimisation problem of controlling a fictitious farm to environmentally acceptable standards is solved. Finally, the paper ends with conclusions and directions for future research.

2. Assumptions

Before we formulate the mathematical model of the problem at hand we draw up a list of assumptions about the physical situation which we model.

A. In the modelled area, the farms are "small" but "dangerous".

This means that we expect a polluter to be able to abate his6 pollution entirely e.g., by buying the manure removal services rather than investing

6Without prejudice against either gender an anonymous polluter will be referred as he and the Regional Council as they.

340 J.B. Kmwczyk

in a physical abatement capital. On the other hand, an existing pollution sediment pond is almost full so that it will begin to leak into the river by the end of a production period if no abatement takes place.

B. In the modelled area, the polluters are "myopic".

This means that we expect a polluter to maximise his yearly profit rather than a sum of discounted profits over an optimisation horizon. That be­haviour can be perceived as practical in a situation when the Regional Council adjusts the levies each year and the polluters are left in uncertainty concerning the amount of future levies (for some results on how uncertainty impacts on repetitive control see [9]).

C. The Regional Council is interested in the economic prosperity of their region.

D. The Regional Council taxes the pollution emission ''proportionally''.

Here, "proportionally" means that a polluter will be levied according to how much he contributes to pollution at a critical spot, rather than to how much effluent he injects into the environment.

E. The stream flow is fast so that the pollution diffusion process can be neglected.

F. The Regional Council is interested in maintaining an avemge pollu­tant's concentmtion level within acceptable bounds.

This means that the Regional Council's environmental concern is to avoid prolonged exposure of the environment to the pollution.

In the subsequent sections, we will build a model subject to the above assumptions.

3. The Polluters' Problem

In this section we model the polluters' economic response to a given level of an environmental levy.

3.1 Pollution production

We suppose that each polluter makes decisions that control his emission levels (as well as the market output, abatement etc.). Assume that the i-th polluter produces a pollutant within a production period (a month, say) t, t = 1,2, ... , in the amount a1ri where 1ri is the polluter's (live) stock and where a > 0 can be interpreted as a "technological" coefficient, e.g., nl kg of manure per n2 kg of stock per month.

Management of Effluent Discharges 341

The pollution in the sediment pond will change from one period to the other according to the following equation:

y;(t+1) = y;(t) + CX7r~t) _ KY) - Q~t), y;(0) - given (1)

where KY) ~ 0 is abatement "effort" (e.g., manure removal), Q~t) is effluent

and y;(0) is the original contents of the pond.

In general, effluent Q~t) (here, the leak of the manure into the stream

from the i-th farm), calculated for the same period as 7r~t), is a function of pollution Y;(t), abatement K1t) and the size of the pond fi. We will suppose that this function is of the following aggregate form:

Q1t) = max(O, Y;(t) + CX7r}t) - KIt) - fi). (2)

The abatement effort KY), 0 ~ KIt) ~ min(fi, Y;(t» will be how much pollution a polluter decides to neutralise. Note that equation (2) and the above constraint make the pollution production model non-smooth.

3.2 Pollution transport

Suppose that we are interested in keeping track of the Ammonia-N7 con­centration at a given section of the stream. Let

A - -qi (7ri' Yii r) = CXA Qi(7ri, Yii r)

be the concentration of the A mmonia-N emissions released to the stream by the i-th polluter at timeS r, r E [ro, rll and CXA is the Ammonia-N contents in the effluent intensity Qi(7ri, Y, r).

The level of Ammonia-N concentration Ci(r, Xi) at time T, r E [TO, Tl], and at the chosen section which is Xi meters from where the i-th polluter is dumping his waste waters, can be described by the following partial differential equation (cf. [3]i compare [11]):

aCi(T,Xi) = -~Ci(T,Xi) _! aCi(r, Xi) aXi v v aT

(3)

71n this paper we will implicitly assume that curbing Ammonia-N emission is satis­factory. Actually, there are a dozen or so water quality measures like Suspended Solids, Conductivity, Enterococci, Biological Oxygen Demand etc., concentration of Ammonia­N being just one of them. Bearing in mind that all the measures are correlated quite strongly, choosing one of them should not be regarded as limiting the results obtained in the sequel. In particular, Ammonia-N usually dominates the total dissolved inorganic nitrogen so it appears to be a highly significant measure of water quality, see [7].

sl.e., in a "micro" time scale as opposed to a "macro" time scale t = 0,1,2 .... More generally, the variables q;, Cj, Qi etc. refer to one (production) period t, t = 0,1,2, .... For notational simplicity we drop the index t in this section. The variables will be indexed with t when we will consider the intertemporal character of the effluent management problem.

342 J.B. Krawczyk

where A is the biodegradation rate and v is the stream velocity. Here, following Assumption E, v is large relative to the diffusion coefficient and we can neglect diffusion, cf. [3).

The boundary conditions for equation (3) are: Ci(O,Xi) = 0, Ci(T,O) = qt(7ri, iii; T) (and Ci(T, 00) = 0). The constants A and v can depend on the season. The solution to (3) is:

(4)

As we have N polluters, each Xi meters distant from the chosen section of the stream, the total concentration of the pollutant C( T) at the section will be

N

C(T) = L Ci(T, Xi). (5) i=l

As said in Assumption F we are interested in an average level of pollution (per production period). Assuming that v is large enough to make ~ small when compared with a production period T == [T~, Tn, we can easily compute the average concentration Ci(Qi,Xi) of pollution originated from the i-th polluter within T as:

-~x· J"7"I A (V !!Oi.) d Q e v' "T qi 7ri,.li;T- v T -~x· i (6) Ci(Qi,Xi) = 0 =aAev'-.

Tl - TO T

Also, the total average concentration at a given stream section can be as­sessed:

N

c(Ql.Xl. ... ,Qi,Xi, ... ,QN,XN) = LCi(Qi,Xi). (7) i=l

3.3 Profit maximisation

A one-period polluter's profit can be defined as

g~t)(/l; u~t), Kft» = p. u~t) - fi(7r;t» - d· K?) - /l(c(t» Ci(Q(t), Xi) (8)

where u~t) is the "other" control9 of the i-th polluter, which can be inter­preted as sales (at the end of period t) and is related to the average stock 7ri through the state equation:

7r~O) given. (9)

We expect the sales to be positive; however, as long as 7r}t) remain non­negative, a negative u~t) could be tolerated and interpreted as the cost to the

9The first is K~t) • •

Management of Effluent Discharges 343

polluter of purchasing reproductive material 10. The (exogenous) produc­tion price is p, li(-) is the i-th polluter's cost function, d is the cost of a unit of abatement, v is stock reproduction/decay rate and J.L(c) is the leader's decision rule on how much tax on pollution has to be applied in order to keep the river environmentally sound. Note that through accepting (8) as the i-th polluter's profit function we follow Assumption A which limits our interest to farmers who do not consider investing in a fixed capital.

If we assume that a production period [TO, Ttl corresponds to a month, the i-th polluter's annual problem consists of maximisation of:

11

~ ~t)( . ~t) K~t») + W( P2») ~ g. J.L, u. , • 11". (10) t=O

. h (0) (11) (0) (11) ( (12»). WIt respect to U i , ... , ui and Ki , ... , Ki , where W 1I"i IS a final state function.

The choice of W(1I"~12») depends on the policy which the Regional Coun­cil wants to implement through the environmental levies. The Regional Council does not expect farmers to dramatically change their fixed capital, or to liquidate their stock, unless the farm's production was found to be extremely environmentally unfriendly. Therefore we will model W(1I"~12») as

W(1I"P2») = _p(1I"(12) _ 11"(0»)2 _1/JJ.L(c(12») Ci(Q(12), Xi) (11)

p > 0,1/J > O. The first term models a polluter'S preference for not changing his "fixed" capital (e.g., buildings). The second term reflects the fact that, should a farmer feel obliged to consider winding down his business, he would have to allow for the next-year's first-month's environmental tax in the current year's budget. Parameter 1/J is the Regional Council policy instrument. Through setting 1/J < 1, the Regional Council will encourage farmers to stay in business; legislating 1/J > 1 would make diminishing stock more attractive.

For notational compactness we introduce a new symbol IIi to denote all i-th polluter's decision variables as

(12)

h - [ (0) (11)1' - [(0) (11)1' were Ui = ui , ..• ,ui and Ki = Ki , ... ,Ki . Note now that through the levy rule J.L(c) , polluter j-th's decision II j has an impact on the i-th polluter's action IIi. We will allow for that dependence in the i-th polluter's profit by noting it as

11

Gi (J.L;IIl, ... II i , .. IIN) == Lg;t)(J.L;u~t),K?») + W(1I"~12»). (13) t=o

lOHere, we assume that the sale price equals the purchase price.

344 J.B. Krawczyk

Let us assume that !i(1I"i) and p,(c) were chosen such that the following Nash equilibrium exists: l1

(14)

If agents behave rationally they will choose ITi , given p, and other exoge­nous parameters. In the result, the maximising sales and abatement ef­fort become functions of p: Ui(p,) and Ki(p,). Subsequently, the levels of stock 1r~t)(p,), pollution "production" 1i(t)(p,) and effluent Q~t)(p,) can be computed for t = 0,1 ... 11. This, through (6) and (7) and because of As­sumption B, allows us to establish the relationship between the polluter's decision variables and the pollution. This relationship is the i-th polluter's reaction function to the levy rule p.

Note that, even for one polluter, the problem (14) is a rather difficult optimisation problem as g?) are nonsmooth functions of K i . Moreover, because of the lack of the discount factor, the actions taken in month four, say, will probably be interchangeable with those of month five. This will make the maximum of Gi "flat" and difficult to compute.

4. The Regional Council's Problem

The Regional Council, according to Assumptions C and D, is interested in the clean environment, promoting production, and collecting revenue; they also have to maintain the pollution monitors. This objective, for one production period (for the time being, we omit t), can be modelled by a function12

N

h(c; TIl. TI2 ... TIN; p,) = -r/>c2 + L ((PUi + P,Ci) . (15) i=l

As previously noted, TIi is the i-th follower's decision, and p, is the decision rule of the Regional Council; coefficient ( represents a "local" tax rate (could be part of GST13). Expression rj>c2 is (obviously) non-negative and represents the cost (within a production period) which the Regional Council has to incur in order to "clean the mess" (or "face" it), resulting from the

llFor an N-person non-zero sum infinite game to have a Nash equilibrium in pure strategies, strict convexity of G; in TIi for every TIj j = 1,2, ... , i-I, i + 1, ... , N is required, and TIi has to be from a compact set, cf. (2).

12Which represents certain revenue of the Regional Council. The revenue should also cover the monitors' maintenance cost mN where m is a monitor maintenance cost which is independent of the problem's decision variables. In fact, the Regional Council's ob­jective is multicriterial and could be modelled as in (8); in that context, (15) can be interpreted as an attempt of scalarisation of the "true" multidimensional objective.

13Goods and Services Tax.

Management of Effluent Discharges 345

average pollution level c. It also captures the fact that small concentrations are less costly to deal with than large concentrations. Note that, because concentration levels cannot be negative, the adopted cost function will not punish the Regional Council for over-abating the pollution. 14

Now, suppose that the followers behave rationally i.e., they use a solu­tion to (14). Substituting it in (15) defines

(16)

Assume that the Regional Council is engaged in long-term planning and that J.t is their stationary levy rulel5 :

(17)

which means that the Regional Council sets the levy for period t depending on the average pollution level. (The pollution level, in turn, is a function of the polluters' state and decision variables (see (14), (7), (6))).

The Regional Council's objective function can be defined as

00

J(c(O) , J.t) = L btii(c(t) , J.t) (18) t=O

where 0 < b < 1 is a discount factor; c(O) and c(t) are the initial, and at time t, average concentrations of the pollutant, respectively. Hence the Regional Council optimisation problem can be defined as follows:

(19)

5. The Decision Support Tool

The effluent management problem consists of the Regional Council finding an optimal levy strategy while the polluters are maximising their profits. Using the notation and formulae from the previous section, the problem defines a game as follows:

argmaxJ(c(O);J.t) } (a) = argmaxGi(J.t,llt, ... lli, ... llN), i=1..N (b) =

(20)

14However, if a critical level c was given, below which the pollutant's concentration levels were considered harmless, the penalty function should have the following form:

I/>(c - c)~

where (z)+ means max{O,z}. 15Because the strategy is stationary i.e." the same for each t, we do not need to

distinguish between a decision rule at t, and the whole sequence of such rules.

346 J.B. Kmwczyk

Note that (20) is a hierarchical game problem. One can look for a solution to this game under a Stackelberg solution concept (cf. [2]).

An analytical solution to (20) will not very often exist16 . We recom­mend a satisfactory solution to this game which will be computed through a Decision Support Tool (cf. [8]). It will be easier to interpret and apply by the Regional Council decisionmakers than other solutions to this problem. The DST will function in the following way:

I. The Regional Council sets up an environmental levy rule ,./ (see (17)) which places a price on the emission levels. Typically, as this price increases production output and pollution both decrease.

II. The Regional Council solves the followers' problem (20b) which sim­ulates a polluter's reaction to the levies imposed.

The Regional Council examines the (simulated) results. In particular, indices h(c,J-l), and J(c(O);J-l1 ) are to be computed. As the indices are only aggregated measures of the Regional Council objectives, answers to the following questions have also to be considered: has concentra­tion been confined to acceptable limits? by how much has the output decreased? will this have an impact on the region's employment sit­uation? etc.

H answers to the above questions are not satisfactory, i.e., the Re­gional Council cannot accept the trade-off between the economic activity and conservation implied by the levy rule, then the rule is modified17 and the Regional Council returns to step I.

III. Otherwise the levy rule is saved as an element of a set T; the corre­sponding values of index h, and J, are also saved.

The Regional Council returns to step I and repeats the steps I-III until the set T contains a few elements.

IV. The Pareto set P is created from T by eliminating J-l for which the corresponding Js are dominated.

V. The Regional Council selects one levy rule from P which will be en­forced.

We suppose that between two extreme situations: i. no levies, maximal production, uncontrolled pollution, and ii. high levies, no production, minimal pollution

16Even if a solution to (20) exists it might not be unique. In particular, if (14) admits more than one equilibrium, the leader's strategies will range between so called pessimistic and optimistic Stackelberg solutions, d. [1).

171£ the rule (17) was chosen linear (or affine), the slope (and/or the intercept) would be modified.

Management of Effluent Discharges 347

there will be at least one which will be acceptable to the Regional Council; hence the above "algorithm" should have an element to converge to. In general, many satisfactory solutions may exist each with a different trade­off between economic activity and the use of the environment. The solution which will be enforced will typically be chosen from the Pareto set using some voting procedure, or simply common sense.

A solution which is arrived at through the Decision Support Tool will be called satisfying. Note that the satisfying solution possesses a Stackelberg solution property which is the follower's best reply to the leader's policy. Investigation of whether the Regional Council's policy p, maximises their criterion (18) globally may be of lesser importance for a real-life decision­making process. Indeed, the Regional Council's actual problem is multicri­terial and choosing p, via the DST takes care of this fact.

6. The Numerical Example

We will illustrate how the DST introduced in Section 5 should be used for solving a problem of type (20).

In this section, we assume that the "lower" optimisation level is con­stituted by one follower18 only. In other words, we restrict our interest to the hierarchical component of game (20).

6.1 Meanwhile, "back on the farm" ...

We are concerned with the efHuent discharge from a farm and in this case have initially chosen a pig farm to model the process. The operation of this farm is assumed to occur over a period of 12T i.e., twelve months, with monthly sales, checks or controls. In order to illustrate the process, early figures were based on statistical averages for pig farms.

We assume that the farmer holds pigs up to an average weight of around 60 kg and has an alternative at the end of a month either to maintain the stock or to sell. Suppose that this farm begins with approximately 330 pigs (for sale). Most farms have about 10-12 sows per 100 pigs so this farm would have about 30-35, on which we base the reproduction rate v. Each sow can wean (in ideal conditions) around 20 pigs a year, in this case 600-700 pigs (36,000-42,000 kg). So if we start19 with 11"(0) =20,000kg, and maintain a constant number of pigs through sales each month, we should have produced close to 40,000 kg over the year. The rate v = 1.167 allows this.

We run a few scenarios of sales {u(t) H~o to examine what income can be

18Single, or a multi follower which could be an aggregate. 19Since now N = 1 we drop the index i from all variables.

348 J.B. Kmwczyk

associated with a farm this size, still without an environmental levy. Figure 1 shows an "optimal" and a plausible sales scenario; the corresponding stock quantities are in Figure 2.

sales

7000 sceosrio

6000

5000

4000

JOOO optimal

2000

1000

l , 8

Figure 1: An "optimal" sales and a plausible-sales-scenario (u(t) in kg per

month).

stock

scenario

19000

1BOOO

17000

16000

1500~O t===::;::==~=====i===;===::;,;;;:o ==:::;"'2

Figure 2: An "optimal" stock and a plausible-sales-scenario stock (1r(t) in kg per month).

Management of Effluent Discharges 349

As introduced in Section 3, y(t) represents the production of the ma­nure each month. With a technological coefficient of 0:' = .5 each pig is discharging half of its weight in effluent each month. The effluent Q(t) de­pends on the abatement effort K(t); let d = $.I/kg. The average pollution contribution of each farmer over a month is simply the amount emitted (Q) multiplied by the exponential dispersion rate (e-~X) and divided by the time frame T = 30 days (see (6)).

pollutantsconcentration

,0

Figure 3: Average non-abated pollutant's concentration corresponding to the "optimal" stock, to a stock scenario; and the partially abated pollution (c(t) in ~ per month).

A distance of x = 1000 metres away from the pollution source was chosen for the test section of the river which has a linear velocity of v=1 m/s. From these parameters a biodegradation rate of A = .0001 seemed reasonable, based on resulting pollution figures. Subject to these parameter values, the size of the manure sediment pond Y = 300,000 kg and its initial contents Yo = 295,000 kg (see Assumption A), the pollution concentration levels corresponding to the "optimal" and a "plausible" sales scenarios were computed (see Figure 3 above).

The farmer's revenue is assumed to come only from the sale of stock, that is, the price (p) multiplied by the us (sales). The farmer's cost function is assumed to be quadratic20 - the usual convex shape, and a function of the stock only - in the form

(21)

20Compare footnote (11).

350 J.B. Kmwczyk

The parameters al = .6667.10-5 , a2 = -.26667, a3 = 14,000 (p = .002 and 'If; = .5 for the final state function, see (10)) were chosen so as to obtain realistic costs each month for this size farm). The incomes generated were $62,647 and $41,404 for the two sales scenarios, respectively.

Assume now that the farmer, for altruistic reasons, decides to abate the pollution in the amount of $500 per month. This will result in diminishing his income by

12 months x 500$/month = $6,000.

The resulting pollution concentration is shown in Figure 3 (together with concentrations resulting from no abatement efforts). We assume that the neutralisation of one unit of Y costs d =$.1 . Therefore the $500 spent on abatement per month can correspond to the removal of 5,000 kg of manure (per month). The $6,000 (per year) polluter's effort toward the pollution abatement, which diminishes his profit by the same amount, is a guideline for us of how to design the environmental rule J-l. If we accept the pollution levels which result from the $500/month abatement, for the rule to be an incentive to abate the pollution, this farmer's environmental tax should exceed $6,000 in the event of no abatement.

6.2 ... and in the Regional Council

In the event of no abatement by the farmers the Regional Council is faced with the necessity of "cleaning the mess" in order not to forgo tourist incomes, avert an epidemic disaster etc. If the abatement of one unit of pollution (Y) costs d to the farmer, we will assume that the cost to the Regional Council of neutralising one unit of the "pure" pollutant c has to exceed

d· T· e~x D=---

QA

for environmentally significant levels of c. We will arbitrarily assume that the pollutant's level ~=.25 kg/month is critical for the environment and calibrate ¢C2 in the following manner:

(22)

which results in 4> = 16D.

Furthermore, assume that ( = .075 (7.5%). In this numerical example, we will not compute the long-term Regional

Council objective function J. The polluter is expected to keep the twelfth month's stock close to the zero-th month's one (p > 0), therefore the next year's solution should not differ too much from the current one's. Con­sequently, h( c, J-l ) (or h( c, J-l) / (1 - 0), 0 - discount factor), can also be a measure of the Regional Council's long term objective.

Management of Effiuent Discharges 351

Having set all the problem parameters we can proceed with the appli­cation of the Decision Support Tool.

In Step I, we have to propose a particular form of the levy rule. Let the pollution tax be proporiionaZ 21 to the amount of pollutant

/-l(c)=M·c (23)

where M (in $/ ~) is the Regional Council's decision variable. Now we can compute the value of a one-year Regional Council objective function h(c,/-l), and the follower's profit, for a given value of M.

The h's values corresponding to the farmer's "optimal" (still without abatement) and "plausible" sales scenarios, for M chosen at the level of 1000, are given in Table 1. This table shows how the levy mechanism could work in general terms: threatened by the tax, the follower will choose to abate, which will improve his, and the leader's, objective function values.

Table 1: Leader's and follower's objective values.

$0.0 $6,000

(-50,736; 62,647) (-10,214; 56,647) (-40,919; 52,149) (-7,806; 53,787)

Table 2: Leader's and optimal follower's objective values.

I abatement· II $0 ° I $0 ° I $0 ° I $1

M=O (-50,736; 62,647) - - -M-5oo - (-50,182; 62,123) - -

M - 2,000 - - (-47,556; 60,569) -M - 5,000 - - - (-43,016; 57,532)

I abatement· " $1888 I , $4345 I , $5579 " ,

I M -7,500 (-26,332; 55,369) -- " l M - 10,000 - (-12,123; 54,262) - II

I M - 12,500 - - (-6,702; 53,492) II

In Step II the follower is expected to solve his problem (14) optimally. Table 2 shows the follower's profits computed as the results of his optimal responses to various M, and the corresponding leader's objective values.

210ther decision rules can and should be discussed. Note that the constant tax rule (i.e., just JL(c) = M) would lead to the polluter's profit (8) being linear in K, with all consequences of this fact like the bang-bang control, which cannot be excluded a priori.

352 J.B. Kmwczyk

The annual abatement cost is now determined by the product sum:

where krJ is the optimal abatement given M. The optimal abatement time profiles are shown in Figure 4.

abatement

M12S00 5000

1,----~H10000 __ 1/

4000

3000

2000 M7500

10

Figure 4: Optimal abatement (K(t) in kg per month).

As stated in footnote (12), and in step III of the DST, the Regional Council's problem is multicriteria!. Hence h( c, J.L) is an aggregate measure of the plethora of indices. Consequently, the decisionmaker will usually also want to know what the follower's reaction profiles corresponding to each M are. (The comparison of a few sets of the profiles with the corresponding h( c, J.L)s can teach the decision maker how to interpret the different values of h(c, J.L).) In that sense, Figures 5, 6 and 7 complement Table 2 by show­ing the pollutant's concentration, and the sales and stock, which are the consumer's optimal replies to the leader's decisions on M.

The set T of Step III is defined through Table 2 and Figures 4-7. As it contains the non-dominated solutions only, it is identical with P (Step IV).

It is interesting to see how the polluter's behaviour will be modified de­pending on the environmental tax. For small M, the reduction in pollution (see Figure 5) is not large and achieved only through reducing the stock (see Figure 7). For 500 < M ::; 5,000 the pollution diminishes visibly, but still only by reduction of stock. And finally, for M > 5,000, the abatement is becoming intensive, whereas the stock is kept approximately constant.

Management of Effluent Discharges 353

pollutantsconcentration

MOandMSOO

MSOOO

Kl000Q

10

Figure 5: Average pollutant's concentration (c(t) in ~ per month).

The apparently irregular pattern of optimal sales and stock for M ~ 2,000 (see Figures 6 and 7) requires comments. The follower's objective function is "flat" in u(t) for ts from the middle of the year; in other words, months in the middle of the year are indistinguishable for the farmer, and for the optimisation routine. Therefore, whichever value of u(t) the routine was picking up as optimal depended to a great extent on the starting point.

Finally, we get to know the solution that the Regional Council may want to enforce in Step V. The M = 7,500 solution seems to be a secure candidate. It gives the Council a "good" index and also guarantees that the pollutant's concentration is kept below the limit of .25 kg/month (Figure 5). In (22), this level was assumed to be critical. In this sense, the M = 7,500 is a satisfying solution.

7. Concluding Remarks

In this paper, we presented a comprehensive model of effluent management which resulted in a hierarchical game with a Nash equilibrium at the lower level. In the numerical example, we concentrated on the hierarchical aspect of the game and examined the interactions between the leader and a fol­lower. We showed that this game can be solved by arriving at a satisfactory solution obtained through the use of a Decision Support Tool.

Further research should include a study of a numerical procedure which would handle more than one follower, and a relaxation of some of the As­sumptions. In particular, modelling farmers as non-myopic decisionmakers,

354 J.B. Krawczyk

sales

Figure 6: Optimal sales (u(t) in kg per month).

stock

Figure 7: Optimal stock (n(t) in kg per month).

i.e., allowing them to invest in a fixed capital (e.g., abatement, buildings) would be interesting, and computationally challenging. However, the use­fulness of the solution to a myopic farmer's problem, as presented in this paper, extends to many intertemporal optimisers who consider expansion of their abatement capital (but not of any other fixed capital like build­ings, as this would destroy a "stationary" cost function (21)). Our farmer, given a digging company quote for an additional pond and provided with the information on his yearly profit with the existing pond, and with the new pond added (both from the DST), will be able to make an intelligent22

decision as to whether or not to build the new pond.

22To the extent to which he can correctly predict the future discount/interest rate.

Management of Effluent Discharges 355

References

[1] BlUiar T., "Time consistency and robustness of equilibria in non­cooperative dynamic games," in: Dynamic Policy Games in Eco­nomics, Ploeg van der F. and A. J. de Zeeuw, eds., North Holland, Amsterdam, 1989.

[2] BlUiar T. and G. K. Olsder, Dynamic Non-cooperative Game Theory, Academic Press, New York, 1982.

[3] Domenico P.A. and F.W. Schwarz, Physical and Chemical Hydrogeol­ogy, J. Wiley & Sons, New York, 1990.

[4] Beavis, B. and I.M. Dobbs, The dynamics of optimal environmental regulation," Journal of Economic Dynamics and Control, 10, pp. 415-423,1986.

[5] Clemhout, S. and H.Y. Wan Jr., Dynamic common property resources and environmental problems," JOTA, vol. 46, no. 4, pp. 471-481, 1985.

[6] Haurie, A. and J.B. Krawczyk, "A Game Theoretic Model of River Basin Environmental Management," Proceedings of the Sixth Inter­national Symposium on Dynamic Games and Applications, St. Jovite, Quebec, 1994.

[7] Kingett Mitchell & Associates, eds., "Assessment of the effects of the discharge of wastewater from the Waikanae sewage treatment plant on the environment," Prepared for Kapiti Coast District Council, Envi­ronmental Consultants, September 1992.

[8] Krawczyk, J.B., Controlling a dam to environmentally acceptable stan­dards through the use of a decision support tool, Environmental fj

Resource Economics, 5(3), 1995, (in press).

[9] Krawczyk J.B. and G. Karacaoglu, On repetitive control and the be­haviour of a middle-aged consumer, European Journal of Operational Research, vol. 66, no. 1, pp. 89-99, April 1993.

[10] Martin, W.E., R.H. Patrick, and B. Tolwinski, A dynamic game of a transboundary pollutant with asymmetric players, Journal of Envi­ronmental Economics and Management, vol. 24, pp. 1-12, 1993.

[11] Padgett, W.J., A stochastic model for stream pollution, Mathematical Biosciences, 25, pp. 309-317, 1975.

[12] Sergerson, K., Uncertainty and incentives for non-point pollution control, Journal of Environmental Economics and Management, 15, pp. 87-98, 1988.

356 J.B. Krawczyk

[13] Tietenberg, T.R., Economic instruments for environmental regula­tion," Oxford Review of Economic Policy, vol. 6, no. 1, pp. 17-31, 1990.

[14] Xepapadeas, A.P., Environmental Policy Design and Dynamic Non point-Source Pollution, Journal of Environmental Economics and Management, vol. 23, no. 1, pp. 22-39, 1992.

Faculty of Commerce and Administration, Victoria University of Wellington PO Box 600 Wellington, New Zealand

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