Strategic Bargaining and Cooperation in Greenhouse Gas Mitigations: An Integrated Assessment Modeling Approach

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“Zili Yang’s book provides a clear explanation of important analytical tools that are crucial to understanding and analyzing a country’s incentive to control climate change. The book illustrates how game theory can be used to quantify the benefits of cooperation and to identify the strategies that could be implemented to enhance the effectiveness of a climate treaty. Yang carefully describes innovative methodologies and their applications to the study of climate change policy. This book should be used by all students and researchers who believe that a careful analysis of participation incentives is crucial to comprehend the future evolution of climate policy regimes.” —Carlo Carro, Department of Economics, University of Venice “Zili Yang has produced an impressive contribution to an under-studied but very important field—the strategic dimensions of climate change.” —Charles D. Kolstad, Department of Economics and Bren School of Environmental Science & Management, University of California, Santa Barbara “As a coauthor of RICE in 1996, Zili Yang is recognized as a pioneer in empirical multiregional climate modeling. One of the main virtues of his work is that it has allowed importing game-theoretic concepts in the economics and politics of climate science. In this book he presents the full background of the model, as well as extended developments since his early contribution. Thanks to the detailed presentation of his methodologies, the author definitely induces the reader to compare his results to other ones and thereby learn about fundamental aspects of how mankind might, or should, meet the challenge of climate change.” —Henry Tulkens, Professor of Economics and Public Finance, CORE, Université catholique de Louvain, Belgium

Strategic Bargaining and Cooperation in Greenhouse Gas Mitigations

Zili Yang is Associate Professor of Economics at SUNY Binghamton.

Strategic Bargaining and Cooperation in Greenhouse Gas Mitigations An Integrated Assessment Modeling Approach

Yang

The MIT Press Massachusetts Institute of Technology Cambridge, Massachusetts 02142 http://mitpress.mit.edu 978-0-262-24054-3

Zili Yang

Strategic Bargaining and Cooperation in Greenhouse Gas Mitigations An Integrated Assessment Modeling Approach Zili Yang The impact of climate change is widespread, affecting rich and poor countries and economies both large and small. Similarly, the study of climate change spans many disciplines, in both natural and social sciences. In environmental economics, leading methodologies include integrated assessment (IA) and game-theoretic modeling, which, despite their common premises, seldom intersect. In Strategic Bargaining and Cooperation in Greenhouse Gas Mitigations, Zili Yang connects these two important approaches by incorporating various game-theoretic solution concepts into a well-known integrated assessment model of climate change. This framework allows a more comprehensive analysis of cooperation and strategic interaction that can inform policy choices in greenhouse gas (GHG) mitigation. Yang draws on a wide range of findings from IA and game theory to offer an analysis that is accessible to scholars in both fields. He constructs a cooperative game of stock externality provision—the economic abstraction of climate change—within the IA framework of the influential RICE model (developed by William D. Nordhaus and Zili Yang in 1996). The game connects the solution of an optimal control problem of stock externality provision with the bargaining of GHG mitigation quotas among the regions in the RICE model. Yang then compares the results of both game-theoretic and conventional solutions of the RICE model from incentive and strategic perspectives and, through numerical analysis of the simulation results, demonstrates the superiority of gametheoretic solutions. He also applies the game-theoretic solutions of RICE to such policy-related concerns as unexpected shocks in economic/climate systems and redistribution and transfer issues in GHG mitigation policies. Yang’s innovative approach sheds new light on the behavioral aspects of IA modeling and provides game-theoretic modeling of climate change with a richer economic substance.

Strategic Bargaining and Cooperation in Greenhouse Gas Mitigations

Strategic Bargaining and Cooperation in Greenhouse Gas Mitigations An Integrated Assessment Modeling Approach

Zili Yang

The MIT Press Cambridge, Massachusetts London, England

( 2008 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. For information about special quantity discounts, please e-mail special_sales@mitpress .mit.edu This book was set in Palatino on 3B2 by Asco Typesetters, Hong Kong. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Yang, Zili, 1957– Strategic bargaining and cooperation in greenhouse gas mitigations : an integrated assessment modeling approach / Zili Yang. p. cm. Includes bibliographical references and index. ISBN 978-0-262-24054-3 (hbk. : alk. paper) 1. Greenhouse gas mitigation—Econometric models. 2. Climatic changes—Econometric models. I. Title. HC79.A4Y36 2008 2008017048 363.738 0 746—dc22 10 9 8

7 6 5

4 3 2 1

To Dean and Sean, who are my inspirations.

Contents

Preface

1

Introduction 1.1 1.2 1.3

2

2.3 2.4 2.5

5

Introduction 9 Efﬁcient Provision of Detrimental Externality and the Incentives 10 The Game of Stock Externality Provisions 13 The Solution Concepts of the Cooperative Game 23 Efﬁciency and Game-Theoretic Solutions 25

The RICE Model 3.1 3.2 3.3 3.4

4

1

Integrated Assessment of Climate Change 1 Game-Theoretic Analysis of Environmental Issues Motivation and Scope of This Research 6

Modeling Global Environmental Issues as a Cooperative Game of Stock Externality Provisions 9 2.1 2.2

3

ix

27

An Overview 27 Model Description and Calibration 28 Solution Concepts in the RICE Model 33 Solution Algorithms 36

Cooperative Game Solutions and Other Solutions in the RICE Model 53 4.1 4.2 4.3

Two Benchmarks: BaU and Open-Loop Nash Equilibrium Scenarios 53 Why Conventional Solutions Are Unﬁt for Analyzing International Cooperation 57 Cooperative Game Solutions of the RICE Model 70

viii

Contents

4.4 Synthesis 86 4.5 Side Payments toward Equalization of Mitigation Costs

5

6

Analysis of Game-Theoretic Solutions in RICE

103

5.1 Stability of the Grand Coalition 103 5.2 Sensitivity Analysis from a Strategic Perspective

108

98

Policy Applications of Game-Theoretic Solutions in RICE 133 6.1 Incentive Compatibilities of Unilateral Actions 133 6.2 Renegotiation of International Environmental Agreements (IEAs) 136 6.3 Distribution Analysis of GHG Mitigation Policies from Strategic Perspectives 150 6.4 The Second-Best Subcoalitions of GHG Mitigations 157

Epilogue: Further Research Directions Appendixes

161

165

1. The Description of the RICE Model 165 2. GAMS Code of the Example in Section 2.1 167 3. GAMS Codes of the RICE Model (Core Part) 168 Notes 179 References 183 Index 187

Preface

If one is asked what is the hottest research area on a hot subject today, the answer is not nuclear fusion, but global warming. Climate change is hot because its effects would be felt by millions of people, rich and poor, and by all economies, large and small; the research on climate change is also hot because it encompasses many disciplines in both the natural and social sciences. Climate change is a complicated issue on a global scale that has drawn the attention of scientists, engineers, and policymakers. Economic analyses of climate change have been a useful instrument for untangling the thorny socioeconomic facets of global warming. This research offers a fresh approach to the economics of climate change. In this study, economic modeling of climate change is incorporated with game-theoretic solution concepts. Such a connection bridges the two important research methodologies widely employed in the economics of climate change literature: integrated assessment (IA) modeling and game-theoretic modeling. My aim is to shed light on the behavioral aspects of IA modeling and provide game-theoretic modeling of climate change with richer economic substances. The new understanding derived from this research should help us to identify regions’ behavior toward international agreements on greenhouse gas mitigations and to rethink various policy issues regarding such agreements. The study is a comprehensive summary of my research activities in IA modeling. Many scholars in climate change research have inspired the work underlying this study. Foremost, I would like to express my sincere gratitude to William D. Nordhaus, whose guidance since my graduate student days has been invaluable. The RICE model envisioned and developed by him is a key component of this work. Carlo Carraro and Charles Kolstad have encouraged this research at

x

Preface

different stages. Intellectual exchanges with my former colleagues at MIT and Penn State have stimulated my thinking. These colleagues include Henry Jacoby, A. Denny Ellerman, Richard Eckaus, John Reilly, Richard Schmalensee, Ron Prinn, Andrei Sokolov, Chien Wang, and Adam Rose. I am very grateful to the Integrated Assessment of Climate Change Research Program at the U.S. Department of Energy. RICE model development associated with this study is supported by the program, under grant # DE-FG02–06ER64180. The hardware and software support from this grant made this research feasible. I also offer my appreciation to Dean Pierre Mileur of Harpur College, SUNY at Binghamton, for granting me a semester research leave so I could devote the time to writing this book and conducting intensive and time-consuming computer simulations. I would like to thank three anonymous referees for their suggestions on improving the manuscript. I also thank Philip Sirianni for help in preparing the manuscript. All remaining errors are my responsibility. Finally, I thank my wife, Julie X. Wang. Without her support over the years, the research described in this book would have been impossible.

1

1.1

Introduction

Integrated Assessment of Climate Change

Since the Industrial Revolution started the locomotive of economic growth worldwide, fossil fuels, along with other primary factors and technological innovations, have kept the economic engine running at a fast and steady speed. Rewinding history, it is hard to imagine civilization today without fossil fuels. While fossil fuels have brought prosperity to millions, they have also caused signiﬁcant environmental pollution problems at local and regional levels. Coal miners suffer from black lung disease; farmers endure the effects of acid rain; city dwellers inhale particulates in smog. Nevertheless, carbon dioxide (CO2 ) emissions produced during fossil fuel combustion seemed innocuous to most people until the past two decades when greenhouse effect and global warming became common terms. Carbon dioxide emission is the by-product of burning fossil fuels. By absorbing infrared radiation, CO2 concentration in the atmosphere, along with other so-called greenhouse gases (GHGs), traps heat near the earth’s surface. This greenhouse effect causes a global temperature increase. Today, scientists conclude that ‘‘most of the observed increase in global average temperatures since the mid-twentieth century is very likely due to the observed increase in anthropogenic GHG concentration’’ (Intergovernmental Panel on Climate Change (IPCC), 2007). In the Summary for Policy Makers of Working Group I in the most recent Fourth Assessment Report (AR4) of the IPCC, scientists conclude that by the end of the twenty-ﬁrst century, global surface temperature might increase by 1.8 C to 4.0 C, depending on different assumptions on social and economic drivers (IPCC, 2007). The consequences of such temperature increases are wide ranging: from rising sea level to extinction of certain fauna and ﬂora species, from reductions of

2

Chapter 1

agricultural outputs to increased weather-related disasters. The potential impacts of climate change are assessed in volume 3 of IPCC AR4. Shortly after natural scientists raised the climate change issue in the academic arena, economists began to examine the issue from a socioeconomic perspective. Early literature on this issue can be traced back to the late 1970s and early 1980s (Nordhaus 1977, 1982; Nordhaus and Yohe 1983). Since the 1990s, the amount of economic literature on climate change has increased exponentially. Economics of climate change is an important research topic in environmental economics. In Recent Developments in Environmental Economics (Hoel 2004), which claimed to include the ‘‘47 most important papers in environmental economics from 1993 to 2003,’’ eleven papers are on climate change directly. Climate change is a complicated research subject of global magnitude. Economic issues associated with it cannot be separated from the natural sciences. From the very beginning, the bulk of economic studies of climate change have been interactive with other science branches, such as climatology, ecology, regional sciences, and engineering. This multidisciplinary approach has gained prominence and formed a unique research ﬁeld: integrated assessment (IA) of climate change. Various models built for climate change research are labeled IA models, and most IA models are developed and employed by a group of scholars from several research ﬁelds, including economists. Famous and inﬂuential IA models have undergone continuous development and reﬁnement over the last decade, such as the IA models maintained by MIT (2007), Carnegie-Mellon (2007), Paciﬁc Northwest National Laboratory (2007), International Institute for Applied Systems Analysis (IIASA, Austria) (2007), Asian-Paciﬁc Integrated Model (AIM) group ( Japan) (2007), and National Institute for Public Health (RIVM, Netherlands) (2007),1 to name just a few by seniority. These IA models are buttressed by strong multidisciplinary research groups from these institutions. Economic components are part of them. In contrast to these comprehensive IA modeling endeavors, there are IA models developed by economists that are more focused on economic aspects of climate change. An incomplete list includes the MERGE model by Manne and Richels (1992), the CETA model by Peck and Teisberg (1992), the FUND model by Tol (1997), the G-Cubed model by McKibbin and Wilcoxen (1995), and the RICE model by Nordhaus and Yang (1996). Economic modeling has always been a crucial part, sometimes the central part, of IA modeling. Because climate change is a very compli-

Introduction

3

cated matter of long-lasting impacts and global scope, economic models in the IA framework exert the dimensionality and complexity of economic modeling to its limit. The methodologies of economic modeling in the IA framework include the following approaches: Computable general equilibrium (CGE) models, such as MIT’s EPPA model and Paciﬁc Northwest National Laboratory’s SGM model

•

Intertemporal optimization (nonlinear programming) models, such as the RICE model and the MERGE model

•

Scenario simulation models, such as Carnegie-Mellon’s ICAM model and RIVM’s IMAGE model

•

CGE models are set up on a database called the social accounting matrix (SAM). They allow great details of sectoral and regional disaggregations. In forecasting future GHG emissions and assessing GHG mitigation strategies, CGE models can offer much useful information. Modelers can build in ad hoc structures or ‘‘devices’’ in CGE models to analyze speciﬁc economic issues. One drawback of CGE modeling is that its dynamic feature is limited by data constraints. Usually, CGE models are either static or recursive dynamic. There is yet to be an operational forward-looking dynamic CGE model in the IA framework. Intertemporal, or dynamic, optimization models are limited in sectoral breakdowns due to the dimensionality constraint, but they are much more ﬂexible and powerful than CGE models in capturing economic agents’ decisions and responses to the future events. In addition, it is easier to build in economic functionality than in CGE models because of the treatment of intertemporal economic relationships. Furthermore, the model structure of dynamic optimization is more transparent than in the other two approaches. Scenario simulation models do not require time-consuming calculations to ﬁnd optimal solutions. The model itself is a set of calibrated economic relationships that does not involve any decision making or optimization behavior of economic agents when solving the model. Simulation models are a truthful reﬂection of modelers’ opinions and expertise in the economic relationships captured by the model. Such models can provide user-friendly interfaces and outputs. From an economic modeling perspective, CGE and dynamic optimization models are preferred to simulation models. The CGE models capture economies’ responses to market shocks, both prices and quantities. However, because climate damage will take place in the future,

4

Chapter 1

it is very difﬁcult to incorporate climate damage assessments in a recursive dynamic CGE model. In the IA framework with CGE economic models, climate impact models are separated from CGE models. On the other hand, dynamic optimization models are less ‘‘mechanical’’ and more ‘‘behavior bending’’ than CGE models. Dynamic optimization models can capture economic agents’ intertemporal decision-making processes. They can also incorporate GHG mitigation strategies and climate damage in a single modeling framework. Such strengths provide a platform for modeling economic agents’ behavior in an IA framework. Functions of economic models in the IA framework include but are not restricted to the following: ﬁrst, forecasting future GHG emission scenarios and serving as inputs to other components of IA modeling; second, analyzing cost-effectiveness of various GHG mitigation policies; and third, assessing a broad range of policy issues arising from climate change research. Parson and Fisher-Vanden (1997) and Kolstad (1998) provide detailed surveys and analyses of IA modeling. IA modeling has advanced since these two survey papers, but its functionality and focus in climate change research remain the same. In addition to IA modeling, economic analysis also permeates many derivative issues from climate change, from international trade to domestic taxation and from ecology to energy. Contributions by economists to IA modeling are fruitful. The research results appear frequently in mainstream economics literature as well as in multidisciplinary climate change research venues. These economic research outcomes culminate and are summarized in the IPCC’s Third Assessment Report (TAR), the series of IPCC Technical Reports, and most recently, IPCC’s Fourth Assessment Report (AR4). In Volume II (Impacts, Adaptation, and Vulnerability) and Volume III (Mitigation) of the assessment reports from 1990 to 2007, one can ﬁnd extensive work done by economists. All of these results are documented at the IPCC’s Web site (IPCC 2007). Many economists and research groups bring their research results into the formation of IPCC reports, which are accessible to a global audience. Economists also offer their comprehensive assessments and policy suggestions on climate change (for example, Stern 2007), either commissioned by the government or sanctioned on their own. Summarizing the achievements by economists in the IA framework would require an entire volume. I direct readers to the IPCC’s Web site to gain a more comprehensive view of the literature.

Introduction

1.2

5

Game-Theoretic Analysis of Environmental Issues

Today, climate change research is one of the major topics in environmental economics, a rich ﬁeld that attracts a plethora of research methodologies. Game theory, a pillar of modern microeconomics, ﬁnds broad applications and theoretic extensions in environmental economics. One of the central themes in environmental economics is market failure and free-riding behavior in pollution control. A famous early example of a game-theoretic application is the preference revelation mechanism in public good (bad) provision designed by Clarke (1971) and Groves (1973). Over the last two decades, game-theory models have been used to analyze various environmental problems from acid rain to preservation of commons. Two collections edited by Hanley and Folmer (1998) as well as Carraro and Fragnelli (2004) offer a spectrum of broad applications of game theory in environmental economics. A volume edited by Carraro and Siniscalco (1997) also indicates game theory’s application to environmental economics. In these works, speciﬁc environmental problems are examined in light of game-theoretic angles. Games are structured to address environmental problems. Game-theoretic solutions in this line of research are indicative rather than quantitative; the models are stylistic rather than realistic. Although it is still in early stages, game theory has been used to address several important aspects of climate change. In particular, scholars have made substantial progress on coalition formation theories with respect to international environmental agreements (IEAs). Notable publications include Chander and Tulkens 1995, 1997, on coalition theory; several contributions in Carraro 2003 on coalition formation theory with respect to global environmental issues; Barrett 1994; and Carraro and Siniscalco 1988 on self-enforcing coalitions. On the empirical side, Nordhaus and Yang (1996) ﬁrst introduced an open-loop noncooperative game solution concept into the RICE model, an inﬂuential IA model. Since then, several game-theoretic applications using the modiﬁed RICE model have appeared. For example, Eyckmans and Tulkens (2003) studied the coalition problem using a revised RICE model; Carraro, Eyckmans, and Finus (2006) examined optimal transfer problems in RICE; and Yang (2003b) investigated renegotiation proofness of incentive-compatible coalitions in RICE. In addition, Tol (2001) studied coalition issues inside the FUND model.

6

Chapter 1

After promulgation of the Kyoto Protocol in 1997, climate change became an important issue in international environmental politics. Negotiations, bargaining, and strategic posturing on implementing the Kyoto Protocol took place at intergovernmental and private-sector levels. The reality has prompted more active studies of climate change from a game-theoretic angle. Carraro and his research group at Fondazione Eni Enrico Mattei (FEEM 2007), Italy, along with environmental economists in other European countries, played a leading role in such research endeavors. The working paper series at FEEM offers a good view of research frontiers in the study of coalition theories applied to climate change and other environmental issues. Seminars and conferences devoted to the coalition theory applications in the environment and climate change have been held all over the world. However, game-theoretic analysis of climate change issues has yet to address the imperative policy issues directly. 1.3

Motivation and Scope of This Research

Integrated assessment of climate change is a leading research topic in environmental economics; game-theoretic modeling is a major research methodology of environmental economics. Despite the common premises, IA framework and game-theoretic modeling have limited intersection. We observe twin peaks in economic research on climate change, but no bridge connecting them. As more and more people realize the urgent need for international cooperation on GHG mitigation, both approaches expose their inadequacy in dealing with the issues. More comprehensive policy responses to climate change call for IA modeling from a game-theoretic perspective. Since its original release in 1996, the RICE model has been and continues to be an excellent platform for bridging IA modeling and game-theoretic solutions. RICE contains the ﬁrst documentation of noncooperative Nash equilibrium solutions in IA modeling. As described in the brief survey in the previous section, simpliﬁed RICE models have been used to study coalition and transfer issues extensively. Nevertheless, all of the previous studies of game-theoretic solutions in RICE are topic driven and fragmented. Scholars use the RICE model as an illustrative tool in their own research. The IA aspects of RICE and the potentials of its game-theoretic applications are not fully integrated. In this book, I adopt a comprehensive approach that com-

Introduction

7

bines IA modeling, noncooperative and cooperative game solutions, and policy analysis in the RICE framework. In the research framework here, I construct a cooperative game of stock externality provision—the economic abstraction of climate change—within the framework of the RICE model. This game connects the solution of an optimal control problem of stock externality provision with the bargaining of GHG mitigation quotas among the regions in the RICE model. By analyzing the properties of this game and regions’ incentives to join the grand coalition, I shed light on debates and policies of international cooperation in GHG mitigation. I hope to offer a new research angle to both IA modeling of climate change and applications of game-theoretic modeling in climate change. This study is the ﬁrst attempt at integrating the IA modeling and game-theoretic solutions comprehensively. It draws on a wide range of research results from scholars on both sides. Yet, the approach itself is accessible to both camps. The knowledge background of this research is based on well-established concepts in game theory and the abundantly documented IA modeling framework. The expositions in this research are self-contained and assume only basic knowledge of mathematical programming, the IA modeling of climate change, and cooperative game theory. The remaining chapters of this book are organized as follows. In chapter 2, I formulate the framework of stock externality provision as a social planner’s optimal control problem. Climate change is a special application for the general formulation. Then a cooperative game of providing stock externality is constructed as a bargaining process for shares in social welfare weights. Preparatory deﬁnitions and game-theoretic solutions in the context of the optimal control setting are introduced here. Solution concepts such as the Lindahl equilibrium and the Shapley value are deﬁned in the dynamic setting. The RICE model is described and reintroduced in chapter 3. I implement the cooperative game and its solution concepts in chapter 2 into the RICE framework. In addition, I develop and explain the numerical algorithms for solving game-theoretic solutions in RICE in detail. In particular, I discuss the iterative procedures and incentive checking designed for searching the core allocations, the Lindahl equilibrium and the Shapley value. These numerical algorithms are building blocks on the bridge that connects game-theoretic solution concepts with IA modeling.

8

Chapter 1

In chapter 4, I present, compare, and contrast the results of gametheoretic and conventional solutions of the RICE model from incentive and strategic perspectives. I examine the properties of different solutions in the context of integrated assessment. Through the numerical analysis of the simulation results, I clearly demonstrate the superiority of game-theoretic solutions over conventional solutions. In chapter 5, I explore the properties of game-theoretic solutions in RICE through sensitivity analysis from an incentive perspective. The issues include intertemporal stability of the grand coalition under the Lindahl social welfare weights, the range of solutions with the core properties or having the Lindahl equilibrium properties, and incentive reactions to false perception of climate change by individual regions. I apply the game-theoretic solutions of RICE to some policy-related issues in climate change in chapter 6. The difﬁculties confronted by unilateral actions such as the Kyoto Protocol are analyzed from an incentive angle. Sustainability of the Lindahl equilibrium solution under various unexpected shocks in economic/climate systems is examined. Redistribution and transfer issues in GHG mitigation policies are studied from game-theoretic perspectives. Furthermore, the second-best properties of subcoalitions of GHG mitigations are inspected. In the epilogue, I point out future research directions and topics employing the research methodologies in this book. In the appendixes, the algebraic description and parametric values of RICE are provided. In addition, the core part of the model codes in GAMS language is listed here. Large portions of the algorithm and output codes are not included, but they are available from the author on request.

2

2.1

Modeling Global Environmental Issues as a Cooperative Game of Stock Externality Provisions

Introduction

Chapter 1 brieﬂy surveyed two major approaches to economic analysis of climate change. One active approach is IA modeling in which economists either collaborate with natural scientists or incorporate research results from the natural sciences to analyze GHG mitigation costs and/or the impacts of climate change. Another approach is the gametheoretic modeling of strategic interactions of regions in dealing with climate change and the negotiation process of international environmental agreements on climate change. The literatures on both approaches are rich and abundant. They tackle one of the most complicated global environmental problems from fundamentally different perspectives. Consequently, the intersection of the two strands of literature is very limited. Besides the philosophical differences between the two schools of thought, which I will not argue here, the two approaches entail some fundamental difﬁculties in collaboration with each other. On the one hand, IA modeling has become more and more sophisticated. The scale, dimension, regional and sectoral details, and climatic science components of IA models have expanded greatly over the past decade. Some models need hours, if not days, to run on the computers. Such complexity prevents these IA models from being used in a game-theoretic framework. On the other hand, game theory, even applied game-theoretic modeling, relies on highly stylized and simple model structures. In the past, the insights game theories offer on environmental issues have been indicative rather than quantitative. Each of the two methodologies has its unique strength and weakness not shared by the other. Despite the disparities between them, it is imperative that IA models capture strategic decisions of regions and that

10

Chapter 2

the game theory models offer more concrete policy suggestions in climate change studies. In particular, negotiations on international treaties of GHG emission reductions call for such interactions. To do so, the gap between the two approaches should be bridged. Meeting at either side would be very difﬁcult; meeting halfway, IA models need to be rid of some of their complexities and the game-theoretic approach should embrace more involved numerical solutions. Judicious balance between essential features of IA modeling and game-theoretic solution concepts has to be considered. As stated in chapter 1, one of the aims of this study is to bridge these two approaches. Within the framework of the RICE model developed by Nordhaus and Yang (1996), I investigate various strategic and coalitional issues by seeking game-theoretic solutions of the model in this research. To establish my studies on a solid analytical ground, I set up an appropriate theoretical model and the cooperative game in this model. In subsequent sections, I introduce the economic intuitions behind the role of social welfare weight selection in efﬁcient provision of externality, the cooperative game of stock externality provision built on bargaining of social welfare weight shares, and the solution concepts associated with this game. 2.2

Efﬁcient Provision of Detrimental Externality and the Incentives

Externality is the common theme in both approaches we want to bridge. On the one hand, it is the economic abstraction of most environmental issues, including climate change; on the other hand, it is the background for many game-theory models, notably the mechanism design models of preference revelations. Naturally, characterizing climate change as an externality phenomenon is a key step to bridging the two approaches. A sufﬁcient condition for efﬁcient provision of externality is full internalization of external effects. From a modeling perspective, internalization of externality can be expressed as a social planner’s problem with multiple agents where private production and/or consumption have spillover effects on other agents. In such a problem, the social planner maximizes a social welfare function based on certain normative criteria subject to the externality and other speciﬁcations. In fact, efﬁcient provision of externality and/or public good (bad) has been thoroughly examined within a general equilibrium framework. The famous Samuelson rule (Samuelson 1954) and the existence

Modeling Global Environmental Issues as a Cooperative Game

11

of the Lindahl equilibrium (Lindahl 1919; Foley 1970) are cornerstones of the positive theory of externality. But examples of inefﬁcient market (Walrasian) equilibrium in the presence of externality enter microeconomics and public economics textbooks at all levels. The dichotomy of efﬁciency and inefﬁciency with respect to externality provision is conceptually clear to all well-trained economists. Characterization of a social welfare function (SWF) is a more complicated problem from a social choice perspective. Different speciﬁcation and functional forms of the SWF might have varied welfare implications. In empirical modeling, SWFs are often expressed as a weighted sum of individual agents’ utility functions. In a social planner’s problem with externality, all possible combinations of social welfare weights lead to the full internalization of the externality. Therefore, they all result in efﬁcient provision of externality. More important, different social welfare weights would have different impacts on agents. Selection of social welfare weights is a key to my studies here. I ﬁrst illustrate its role through a simple example of efﬁcient provision of externality. The example is a typical textbooklike illustration. In a tworegion economy, each region produces a private good xi . A by-product Z ¼ z1 þ z2 of private good production adversely affects both regions’ welfare. A social planner tries to internalize the externality by maximizing a weighted sum of regional welfare functions. In particular, the social planner faces the following problem: Max

fx1 ; x2 g

W ¼ ðj1 U 1 þ j2 U 2 Þ;

s:t: U i ¼ ei xiai ðz1 þ z2 Þ di ; zi ¼ ci xi2

i ¼ 1; 2:

j1 þ j2 ¼ 1;

ð2:1Þ

i ¼ 1; 2:

ð2:2Þ ð2:3Þ

In (2.1) through (2.3), ai , ci , di , and ei ði ¼ 1; 2Þ are parameters and they may differ across i. Here, side payments from one agent to the other are not assumed. Social welfare weights (j1 and j2 ) are the social planner’s marginal preference toward the agents. Giving some numerical values of the above parameters, I solve this constrained maximization problem numerically with different combinations of ji (the simple GAMS program is in appendix 2). For comparison, the Nash equilibrium of the above problem is also solved. The results are shown in table 2.1.

12

Chapter 2

Table 2.1 Numerical solutions of problem (2.1) x1

x2

z1

z2

U1

U2

j1 ¼ 0.1 j1 ¼ 0.3

0.275 0.696

2.854 2.377

0.053 0.339

2.444 1.695

1.010 0.480

3.972 3.605

3.474 2.668

j1 ¼ 0.5

1.087

1.871

0.828

1.050

1.553

2.888

2.220

j1 ¼ 0.7

1.509

1.319

1.593

0.522

2.348

1.679

2.147

j1 ¼ 0.9

2.034

0.615

2.895

0.114

2.880

0.560

Nash E.

2.693

3.349

5.077

3.366

1.117

0.172

W

2.536 n.a.

The simulation results indicate that although the social planner can fully internalize the external effects by solving (2.1) with an arbitrary set of social welfare weights, not every agent is always pleased with the social planner’s solution. In particular, an agent can be worse off than his Nash equilibrium position. For example, consider j1 ¼ 0:1 and 0.3 for agent 1 as well as j1 ¼ 0:9 for agent 2. In such cases, the worseoff agent would rather play a noncooperative game with the other than abide by the efﬁcient solution that internalizes the externality. The above results also indicate that the equal-weights ðj1 ¼ 0:5Þ solution does not correspond to the Nash bargaining outcome, namely, equal gains by both agents from the Nash equilibrium through cooperation. With heterogeneous or asymmetric agents, the implication of social welfare weight selection is much more complicated than the symmetric agent assumption often recited in highly stylized contexts. In essence, a social planner’s problem represents a command-andcontrol scheme. In modeling language, the social planner’s problem is often called the ‘‘cooperative solution’’ for convenience. However, to link a social planner’s problem with voluntary cooperation for providing externalities efﬁciently, additional incentive conditions must be met. The relationships between efﬁciency, incentives for cooperation, and the Nash equilibrium are shown in ﬁgure 2.1. Combinations of fðj1 ; j2 Þ j j1 þ j2 ¼ 1g map social welfare function W onto efﬁciency frontier CABD. The Nash equilibrium (N) is inefﬁcient and thus located inside the efﬁciency frontier. On segment CA, agent 1 is worse off than at the Nash equilibrium; on segment BD, agent 2 is worse off. Only on AB are both agents better off than their Nash equilibrium position. A social planner’s problem that corresponds to a plausible voluntary cooperation among the agents should map to AB.

Modeling Global Environmental Issues as a Cooperative Game

13

Figure 2.1 Efﬁciency vs. the Nash equilibrium

The above numerical example highlights the problem we are facing and indicates the modeling approach in this study. From a modeling perspective, I attempt to connect the internalization of externality through a social planner’s problem and the voluntary cooperation actions of such internalization. To make such connections, we need both efﬁciency and incentive criteria. The entire issue can be put into a few simple questions in the context of the above example: What is the incentive property of the social planner’s problem of externality provision under a given social welfare weight ðj1 ; j2 Þ?

•

What is the range of ðj1 ; j2 Þ that maps W onto segment AB in ﬁgure 2.1?

•

What is the most ‘‘desirable’’ allocation on the efﬁciency frontier from an incentive perspective?

•

To answer these questions, I call on game-theoretic methods. 2.3

The Game of Stock Externality Provisions

2.3.1 Background Textbook-type examples of external effects are not sufﬁcient to capture complicated climate change issues ( just as the ‘‘prisoner’s dilemma’’ problem cannot represent every strategic situation in noncooperative

14

Chapter 2

game theory). One of the most crucial characteristics of climate change is that it represents a special type of externality, namely, stock externality. A heuristic deﬁnition of stock externality is as follows. The externality is generated in production and/or consumption of private goods by agents as ﬂows over time. The cumulative stock level of it at time t affects other agents’ current and future utilities without their welfare being considered. Many environmental issues, climate change in particular, ﬁt this deﬁnition. Anthropogenic activities in regions, such as fossil fuel combustion and agricultural production, generate GHG emissions. Concentrations of GHGs in the atmosphere lead to temperature change, which in turn affects all regions’ welfare. Stock externality, by its deﬁnition, cannot be modeled in a static setting. Modeling a dynamic relationship in the economy is necessary to capture basic features of stock externality. Therefore, the ﬁrst assumption and major premise of this research is a dynamic framework. Naturally, the models of stock externality explored here are either optimal control problems or differential games. Departing from highly stylized game-theoretic models, I adopt an empirical dynamic IA model as the platform for game-theoretic modeling. The second assumption of the modeling framework employed here is its open-loop strategy structure. An open-loop strategy is one where the values of control variables ‘‘can be speciﬁed completely at the start of the planning period’’ (Holly and Hughes Hallett 1989, 58). In both cooperative and decentralized problems of externality provisions, agents decide their strategies (or externality provision proﬁle) over the entire time horizon at the very beginning. No feedbacks or reactions to unexpected shocks within the planning horizon are considered. From a game-theoretic angle, open-loop strategies are one-shot games in a dynamic framework. Mathematical models of externality provision with open-loop strategies are optimal control problems (in cooperative cases) or differential games (in the Nash equilibrium cases). I will elaborate on such models later in the book. Though open-loop strategies are static or one-shot from a gametheoretic perspective, they can still be treated as dynamic processes from a modeling perspective. The agents have more than one decision junction over time. At each junction, the agent makes the moves that affect the future according to the decisions made at the beginning. In addition, as a self-interested and rational decision-maker, the agent can and will make alternative moves at a junction that are different

Modeling Global Environmental Issues as a Cooperative Game

15

from what was planned at the beginning, if they are beneﬁcial. I consider such sequential decisions in section 6.2. 2.3.2 The Model As a ﬁrst step in IA modeling of climate change, I introduce a stylized model of efﬁcient provision of stock externalities that can be expressed as the following social planner’s problem: Max

fxi ðtÞg

V¼

N X

ji WiE

¼

i¼1

N ðy X i¼1

0

ji U i ðxi ðtÞ; BðtÞÞedt dt;

N X

ji ¼ N:

i¼1

ð2:4Þ s:t: bi ðtÞ ¼ Fi ðxi ðtÞÞ; B_ ðtÞ ¼

N X

bi ðtÞ sBðtÞ;

i ¼ 1; 2; . . . ; N: s > 0:

ð2:5Þ ð2:6Þ

i¼1

qU i qU i q2U i q2U i < 0; > 0; < 0; < 0; Fi0 > 0 qxi qB qxi2 qB 2 In the above system, the social planner maximizes the weighted sum of the present-value utilities of N agents (regions in international settings). U i , the instantaneous utility function of agent i, is a function of private good xi ðtÞ and stock of externalities BðtÞ faced by all agents. Each agent has a transformation function ðFi Þ between private good xi ðtÞ and ﬂow of externalities bi ðtÞ (equation 2.5). In this framework, the budget constraints of agents are implicit and endogenous. This transformation function ensures ﬁnite amounts of private good consumption (or supply) because of trade-offs between the private good and detrimental externality. The relationship between the ﬂows and stocks of the externality follows the motion equation given by (2.6). In this optimal control problem, xi ðtÞ, ﬂows of agents’ private goods, are the control variables. The other notations in the system are as follows: ji is the social welfare weight of agent i and is conﬁned on a simplex; d is the rate of pure time preference; and s is the decay rate of externality stocks. In addition, the derivatives of utility and transformation functions have the signs indicated above. The system is a model of detrimental stock externality. Thus this stylized model can be used to reﬂect environmental problems.

16

Chapter 2

If there is no social planner to internalize the stock externality as in (2.4), the agents in the economy would maximize their individual utility functions while treating the contribution of others to the externality as given. The equilibrium of decentralized utility maximization behaviors is the solution of the following differential game: Max

fxi ðtÞg

s:t:

WiM ¼

ðy

U i ðxi ðtÞ; BðtÞÞedt dt;

bi ðtÞ ¼ Fi ðxi ðtÞÞ;

B_ ðtÞ ¼ bi ðtÞ þ

i ¼ 1; 2; . . . ; N:

ð2:7Þ

0

N X

i ¼ 1; 2; . . . ; N:

bj ðtÞ sBðtÞ;

i ¼ 1; 2; . . . ; N:

ð2:5Þ s > 0:

ð2:8Þ

j0i

qU i qU i q2U i q2U i < 0; > 0; < 0; < 0; Fi0 > 0 qxi qB qxi2 qB 2 In the above system, each agent maximizes his own intertemporal welfare in current value form. He decides intertemporal trade-offs between production of the private good and generation of external effects while treating other agents’ external effects as given (equations 2.5 and 2.8). When every agent’s optimal solution paths are compatible or consistent with all other agents’ optimal paths, no agent has incentives to deviate from such solution paths. Therefore, the simultaneous solution or conjecture of N agents’ problems is the Nash equilibrium. As in system (2.4), the agents make the decision on the entire time path at the beginning. Therefore, the solution of this differential system is the open-loop Nash equilibrium. The above two systems capture the fundamental structure of stock externality provisions. As we will see later, the empirical IA model to be used for simulation in this study (i.e., the RICE model) is essentially the same model but with ‘‘ﬂesh and blood’’ or meaningful economic and environmental details. I will ﬁrst deﬁne the cooperative game and its solution concepts within the framework of the above systems prior to applying such an approach in a more complicated IA model. In addition to the game, we also want to know analytical properties of these two systems, especially the social planner’s problem. Dynamic relationships behind (2.5) and (2.6) can be very tricky. Efﬁciency conditions of system (2.4) hinge on how these dynamic relationships are speciﬁed. In the public economics literature, (2.5) is often

Modeling Global Environmental Issues as a Cooperative Game

17

speciﬁed as an aggregate transformation function between the sums of externality ﬂows and private goods. That is, instead of N transformation functions, we have ! N N X X bðtÞ ¼ bi ðtÞ ¼ F xi ðtÞ ¼ FðxðtÞÞ; or GðbðtÞ; xðtÞÞ ¼ 0 ð2:9Þ i¼1

i¼1

In (2.9), technologies for transforming private goods to externality are homogeneous across agents. In this research, I assume transformation functions are heterogeneous across agents—that is, F has an index for agents as in (2.5) instead of (2.9). The reason is that in climate change studies, regions (agents) have heterogeneous production technologies and different carbon intensities. Trade-offs between economic development and GHG emission reduction are regionally speciﬁc. In this framework, agents do not have any connections other than collective contribution to stock externality in (2.6) and the common external effects BðtÞ in everyone’s utility function. In particular, the framework does not have channels for unilateral transfers. Nevertheless, I will discuss the implications of allowing transfers but assuming (2.5) in most cases. Such an assumption is consistent with the RICE model to be introduced later and much more realistic than assumption (2.9). The efﬁciency conditions for solving system (2.4) bear symbolic similarities with the famous Samuelson rule (Samuelson 1954) for providing a public good efﬁciently. In Yang 2007, these conditions are derived under speciﬁcations (2.9), (2.5) without transfers, and (2.5) with transfers. Without deriving them again here, I will just list these efﬁciency conditions: (i) For homogeneous transformation function (2.9): " # N _ ðtÞ X l 1 qU i qU i ð2:10Þ ðs þ dÞ 0 ¼ qB ðtÞ qxi ðtÞ lðtÞ F i¼1 or Adjusted MRTx; b ¼

X

MRSx; B .

Here lðtÞ is the costate variable of motion equation (2.6). It can be obtained from the Hamiltonian of system (2.4). (ii) For heterogeneous transformation function (2.5) without allowing transfers:

18

"

Chapter 2

# N _ ðtÞ X qUi qUi l Fi0 ðxi ðtÞÞ ðd þ sÞ ¼ qBðtÞ qx ðtÞ lðtÞ i i¼1

ð2:11Þ

And (iii) for heterogeneous transformation function (2.5) allowing transfers: " # N _ ðtÞ X 1 qUi qUi l þ tri ðtÞ ðd þ sÞ 0 ¼ ð2:12Þ qBðtÞ qxi ðtÞ F lðtÞ i¼1 Here F 0 ¼ F10 ðx1 ðtÞÞ ¼ F20 ðx2 ðtÞÞ ¼ ¼ FN0 ðxN ðtÞÞ;

N X

tri ðtÞ ¼ 0

ð2:13Þ

i¼1

or

sðtÞMRTðtÞ ¼

X

MRSðtÞ.

Here, tri ðtÞ is a variable related to transfer or side payments as in Yang 2007. Condition (2.13) corresponds to ‘‘equalization of marginal rates of transformation’’ across agents or ‘‘equalization of marginal mitigation costs’’ across regions as used frequently in climate change policy debates. In the optimal solutions of system (2.4), control, state, and costate variables always satisfy (2.11) in the above sets of efﬁciency conditions. If transfers are allowed, Pareto improvements might be achieved by such transfers, and condition (2.12) would hold in such circumstances. A noteworthy observation from the above conditions is that they are invariant with respect to selection of fji g. Such an observation implies P that all points on simplex S ¼ fji j ji ¼ Ng map to optimal solutions in (2.4). It is just like the numerical example (2.1), where all combinations of ðj1 ; j2 Þ correspond to a social planner’s solution. Although fji g do not appear in these efﬁciency conditions, the choice of fji g affects the optimal path of state variable B ðtÞ, the optimal path of control variable xi ðtÞ, and the magnitudes or direction of transfer tri ðtÞ. More important, it affects agents’ incentives to cooperate (again, see the example in section 2.1). In fact, the cooperative game of providing stock externality as deﬁned in this research involves collective bargaining on choosing fji g: the share of each agent’s obligations of externality provision.

Modeling Global Environmental Issues as a Cooperative Game

19

Finally, the optimal stock externality level BðtÞ is higher in system (2.7) than in system (2.4), in the case of BðtÞ being a public ‘‘bad.’’ In other words, detrimental externalities, such as pollution, are always overprovided in the inefﬁcient Nash equilibrium, compared with an efﬁcient scheme. In this research, I do not investigate mathematical properties of systems (2.4) and (2.7) any further. I assume that they are ‘‘well behaved.’’ The unique solution paths for both systems exist, provided that the initial and transversality conditions are properly speciﬁed. 2.3.3 The Cooperative Game The cooperative game of stock externality provision is constructed within the framework of system (2.4). Whereas deﬁning the noncooperative game of stock externality provision is relatively straightforward, the cooperative game is more subtle. The noncooperative game is just the differential game (2.7) and its solution, but a solution of (2.4) is not necessarily a cooperative game solution. The intuition behind the cooperative game is in the example in section 2.1. We construct the cooperative game and its rules as follows. Formally, the cooperative game of providing stock externalities is characterized by a triplet Vðji ; xi ðtÞ; WiC Þ for i ¼ 1; 2; . . . ; N. Here ji and xi ðtÞ, as deﬁned in system (2.4), are decision variables for agent i. WiC is the payoff function for agent i. (Note that WiC is not weighted by ji . Superscript C stands for cooperation.) WiC is measured as agent i’s present value of lifetime utility in the optimal solution of system (2.4) under the negotiated outcome of fji g: Wic ¼

ðy 0

U i ðxi ðtÞ; B ðtÞÞedt dt

ð2:14Þ

This cooperative game is played in two stages. The ﬁrst stage is the bargaining. Agents negotiate their respective shares of contributions to stock externalities, namely, ji in system (2.4). Once the shares or social welfare weights are agreed on, agents form the grand coalition. The second stage is the action stage. All coalition members ask a ‘‘social planner’’ or ‘‘the UN’’ to solve the optimal control problem deﬁned by (2.4) through (2.6) with their respective social welfare weight ji negotiated in the ﬁrst stage. By the cooperation agreement, each agent will follow the efﬁcient paths of private goods fxi ðtÞg and the ﬂows of externalities fbi ðtÞg in system (2.4). Each member will then receive WiC

20

Chapter 2

payoff as determined by (2.14). Agents are fully aware of potential outcomes, both for themselves and for others, when they negotiate the social welfare weights fji g. That is, they know all parameters in system (2.4) and its solution structure. The cooperative game also speciﬁes the following penalty rule for potential violation of the above agreement. If an agent deviates from the optimal path determined above at a decision junction, all other agents will quit the grand coalition and play the noncooperative game deﬁned by (2.7) forever. We should point out that the solution of the social planner’s optimal control problem (2.4) with the negotiated fji g is the outcome of the cooperative game rather than the game itself. The deﬁnition of the game enables us to connect a ‘‘cooperative’’ optimal control problem with game-theoretic solution concepts. The optimal control problem (2.4) sets up the framework to play the game but is not the game itself. The weights of the social welfare function, fji g, play a signiﬁcant role in stock externality provisions. Given an arbitrary set of fji g, the solution proﬁle of system (2.4) is an efﬁcient path of stock externality provisions if transfers are not allowed. However, most efﬁcient paths are not likely to be the outcomes of the cooperative game deﬁned here, because they may not be compatible with some agents’ incentives for cooperation. This point is amply clear from the simple numerical example in section 2.1 and its solutions in table 2.1. This cooperative game affects the agents in two respects. First, it moves the aggregate stock externality to the efﬁcient level in (2.4) from the inefﬁcient level in (2.7), thus creating ‘‘win-win’’ potential for everyone. Second, one agent’s marginal gains in a cooperative scheme implies at least one other agent’s marginal losses. Such a symbolic zero-sum situation makes bargaining of contribution shares extremely important. To facilitate subsequent discussions, I rephrase some basic deﬁnitions in the context of the cooperative game deﬁned above. Deﬁnition 1 The characteristic function of the stock externality provision game The characteristic function vðSÞ, ðS H NÞ is the sum of the agents’ net utility gains from the agents’ utilities in the open-loop Nash equilibrium by joining the grand coalition deﬁned by the game. Formally, X X nðSÞ ¼ ni ¼ ðWiC WiM Þ ð2:15Þ iHS

iHS

Modeling Global Environmental Issues as a Cooperative Game

21

Here, superscript C represents an outcome from the solution of (2.4) and WiC is deﬁned as in (2.14); superscript M represents the solution of (2.7) and WiM is deﬁned as in (2.7). Furthermore, to assess whether a solution of system (2.4) is a credible outcome of the cooperative game, I restate individual rationality (IR) and incentive compatibility (IC) constraints in the context of systems (2.4) and (2.7). Evidently, agents can maintain the reservation utility levels by playing the open-loop Nash game as deﬁned in system (2.7). Therefore, the IR constraint can be deﬁned as Deﬁnition 2 Individual Rationality (IR) A grand coalition from the bargaining stage satisﬁes the individual rationality constraint if WiC b WiM ;

or

ni b 0;

Ei A N:

On the other hand, to satisfy the IC constraint, every region has to be better off in the grand coalition under fji g than under any other coalitions. These alternatives include the coalitions with a smaller number of agents (subcoalitions) and the grand coalition with other choices of fji g. Therefore, the IC constraint can be deﬁned as Deﬁnition 3 Incentive Compatibility (IC) A grand coalition from the bargaining stage satisﬁes the incentive compatibility constraint if WiC b Max Wib ;

Ei A N and b A B:

Here B is the set of all coalition possibilities and the deﬁnition of W is consistent with (2.14). Without side payments or transfers, simultaneous satisfaction of IR and IC is a necessary condition of a plausible bargaining outcome. Actually, IR and IC criteria are used to test the core properties of cooperative games here. In dealing with global climate change, a coalition is not politically feasible if it is not in the core. The only enforcement mechanism of an international environmental agreement is selfinterest. Thus, a cooperative suggestion without IR and IC conditions is not implemental. The cooperative game is deﬁned with respect to stock externality provision or climate change. From economic intuition, collaborations from more and more agents lead to Pareto improvements in externality provision. Namely, larger coalitions can always do better than smaller

22

Chapter 2

coalitions in dealing with stock externality. Using the terminology in cooperative game theory, the game of stock externality provision should possess the superadditive property, formally deﬁned as: Deﬁnition 4 Superadditivity A game is superadditive if for any disjoint coalitions R; S H N, the following property holds: If v R A VðRÞ and v S A VðSÞ; then ðv R ; v S Þ A VðR A SÞ: The cooperative game of stock externality provision is more complicated than many other games because of external effects. When we consider a subcoalition S H N, the actions of agents who do not joint the coalition (the agents in NnS) affect the coalition through the external effects. To describe such interactions and clarify the subsequent solution concepts, we have Deﬁnition 5 Hybrid Nash Equilibrium1 When agents in S H N form a coalition to solve system (2.4), they also play the open-loop Nash game (2.7), as a single player, with agents in NnS strategically. The outcome of this combined cooperative and noncooperative game is called the hybrid Nash equilibrium. As in most discussions about coalition formation, we are concerned with the stability of the grand coalition. That is, we seek to ascertain whether regions have any incentives to adhere to their initial commitments along the entire time path. Economic theories and real-life experiences tell us that ‘‘free-riding’’ tendencies hamper the implementation of efﬁcient provision of externalities. The properties of a coalition that can hold together are important for policymaking. Therefore, we need to formally deﬁne stability within our modeling framework. Deﬁnition 6 Stable Coalition An outcome from the cooperative game deﬁned above is a stable coalition if no agent has any incentive to deviate (or gain) from the original agreement on fji g under the penalty rule. The cooperative game of stock externality provision mentioned here is structured on a complicated dynamic system. I do not discuss detailed analytical properties of this game. They are beyond the scope of this study. Nevertheless, certain important issues need to be touched on here.

Modeling Global Environmental Issues as a Cooperative Game

23

In cooperative game theory, transferable utility (TU) games are examined much more extensively than nontransferable utility (NTU) games. TU games have more tractable properties than NTU games.2 As a practical matter, scholars assume that the cooperative game of stock externality provision is a TU game. Some empirical studies using the RICE model (the framework of this study) explicitly add a monetary metric term in the utility function (for example, Eyckmans and Tulkens 2003; Carraro, Eyckmans, and Finus 2006). To make the numerical simulations comparable to the previous results of RICE, I do not change any functional forms of the model, including the functional form of utility functions. Thus, the game deﬁned in the RICE model is an NTU game. A direct consequence of the NTU assumption is that a unit of ‘‘utils’’ of region A is not the same unit for region B. The ‘‘side-payment’’ option, which is actively debated in the policy arena, is therefore much more complicated in an NTU game than in a TU game. Because of the NTU assumption and other technical complications, I do not build in a side-payment mechanism in the RICE model used for this study. Nevertheless, the side-payment issues are treated as ex-post ‘‘side calculations’’ and discussed extensively in this research. Otherwise, the NTU or TU assumption does not have signiﬁcant effects on the numerical results of this study. 2.4

The Solution Concepts of the Cooperative Game

P Simplex S ¼ fji j ji ¼ Ng maps to a continuum of the solution set of system (2.4). After selecting fji g and solving the system, we can then analyze the properties of this solution. However, only part of the simplex is of interest to modelers and policymakers. In this study, I investigate a subset of the simplex for which the optimal control solutions are plausible outcomes of the cooperative game structured in this chapter. These solutions satisfy one or more familiar solution concepts listed below. 2.4.1 The Solutions with the Core Properties A solution of system (2.4) under a speciﬁc fji g is in the core (or has the core properties) if the grand coalition represented by this solution cannot be blocked (improved on) by any coalition S H N. If a solution of (2.4) is in the core, it is a possible outcome of the cooperative game, deﬁned in the previous section. Loosely speaking, having the core property is equivalent to satisfying the IC constraint

24

Chapter 2

in our framework. Our task here is to identify a subset of S ¼ P fji j ji ¼ Ng that maps the solution of system (2.4) to the core. 2.4.2 The Lindahl Equilibrium Because the central issue of this undertaking is stock externality, the Lindahl equilibrium (the most important pseudo-equilibrium concept in the public good and externality framework) should be introduced and implemented. The Lindahl equilibrium has been labeled a ‘‘thought experiment’’ (Atkinson and Stiglitz 1980), partially because of the difﬁculties in identifying it in reality. Here I follow the traditional deﬁnition of the Lindahl equilibrium and leave the identiﬁcation problem for the next chapter. I will follow the deﬁnition in the inﬂuential paper by Foley (1970). That is, there exists a set of ‘‘private prices’’ for the public good (externality in the context here). Such private prices for the public good reﬂect agents’ individual ‘‘willingness to pay’’ for the same amount of the public good. In the context of externality, each agent uses his own private price of externality to assess his demand for externality while taking other agents’ demands as given. The conjecture of all agents’ demands for externality is the Lindahl equilibrium. It satisﬁes the Samuelson rule as well as other alternative efﬁciency conditions, such as those presented in (2.11) and (2.12). In Foley 1970, the two fundamental theorems of welfare economics for an economy with public goods (or the Lindahl economy) are proven. It is parallel to the two fundamental welfare theorems with respect to the Walrasian equilibrium. According to the second welfare theorem for the Lindahl economy, the entire simplex S ¼ P fji j ji ¼ Ng corresponds to the Lindahl equilibrium subject to endowment transfers. However, we want to identify the Lindahl equilibrium without the endowment transfers as indicated in the ﬁrst welfare theorem of the Lindahl economy. The Lindahl equilibrium is in the core;3 consequently, identifying the Lindahl equilibrium is to ﬁnd a P particular fjiL g inside the subset of S ¼ fji j ji ¼ Ng with the core property. Despite its elusiveness, the Lindahl equilibrium is the most desirable outcome for externality provision.4 Without endowment transfers, the Lindahl equilibrium reﬂects the ‘‘willingness to pay’’ principle, which is consistent with incentives of agents. Although the Lindahl equilibrium is not a generic solution concept in the cooperative game, it is in the core associated with the game deﬁned in section 2.3. Thus the

Modeling Global Environmental Issues as a Cooperative Game

25

Lindahl equilibrium is a possible outcome of the cooperative game of stock externality provision, and it is the most interesting one. 2.4.3 The Shapley Value The Shapley value of the cooperative game deﬁned by system (2.4) is a vector of payoffs associated with a unique set of fjiS g. The Shapley value SðN; vÞ ¼ ðS1 ðN; vÞ; S2 ðN; vÞ; . . . ; SN ðN; vÞÞ satisﬁes Si ðN; vÞ Si ðNnf jg; vÞ ¼ Sj ðN; vÞ Sj ðNnfig; vÞ; X

Si ðN; vÞ ¼ vðNÞ

Ei; j A N:

ð2:16Þ ð2:17Þ

i

Here Si ðNnf jg; vÞ is the Shapley value of agent i in the cooperative game played by N 1 agents without player j. To apply the above definition to the grand coalition, we need to deﬁne Si inductively for all subsets of the coalitions. Deﬁning the Shapley value for an NTU game is complicated (Myerson 1991). Here we adopt a heuristic approach as in (2.16) and (2.17). The value deﬁnition used here only reﬂects equity consideration. Though the utilities are not transferable, the solution of (2.16) and (2.17) requires that each agent gains ‘‘equally’’ in its own utility measurement by joining in the grand coalition. In an empirical model where system (2.4) is clearly speciﬁed, (2.16) and (2.17) imply equal gains, measured in the common metric of the model, of every agent in the grand coalition by joining the other N 1 agents’ subcoalition. 2.4.4 The Open-Loop Nash Equilibrium (Benchmark) The open-loop Nash equilibrium is a noncooperative game solution concept. In our framework, it is the solution of system (2.7). Because it is used in deﬁning the value of the cooperative game (as in (2.11)), I enlist it with the cooperative game solutions in parallel. In addition, comparing noncooperative and cooperative game solutions measures the gap between inefﬁcient and efﬁcient outcomes in externality provision. The Nash equilibrium in fact is a benchmark for arrays of solutions investigated here. 2.5

Efﬁciency and Game-Theoretic Solutions

For system (2.4), the set of all efﬁcient solutions satisfy condition (2.11) if transfers are not allowed. But the cooperative game solutions of

26

Chapter 2

system (2.4), as deﬁned in section 2.3.3, are a subset of the efﬁcient solutions. In other words, the cooperative game solutions are furtherrestricted efﬁcient solutions based on incentives. For a large portion of this research, the efﬁciency criterion is (2.11) due to the absence of a transfer mechanism in our model. However, various transfer schemes, such as technological transfers and tradable emission permits, can be built into the IA modeling framework. Such schemes might result in possible Pareto improvements from condition (2.11) to condition (2.12). In essence, such Pareto improvements require the equalization of Fi0 ðxi Þ without changing B ðtÞ on the right-hand side of (2.11). The efﬁcient outcome of such equalization is (2.12).5 This condition is identical to the statement of ‘‘equalization of marginal mitigation costs across regions’’ frequently mentioned in the climate change literature. Studying the issues related to the equalization of marginal mitigation costs represents another dimension of IA modeling. If transfers are allowed, a complete process to achieve efﬁciency of system (2.4) should include two bargaining phases. The ﬁrst is the bargaining of fji g, which is structured explicitly in this study. In the climate change context, this phase involves negotiating the initial ‘‘quota’’ allowance or ‘‘permits’’ of GHG emissions that are incentive compatible. If the marginal mitigation costs are not equalized across regions under such initial quota arrangements (generally, they are not equal), regions will engage in the second-phase bargaining, namely, trading the emission permits through transfers. I should point out that a Pareto improvement through transfers, if any, can only start from incentive-compatible initial quotas agreed on in the ﬁrst-phase bargaining. Otherwise, the whole issue is a nonstarter. This prerequisite is often missed in policy debates. The subsequent studies in this book are based on an empirical IA model without transfers. It is possible to build many economic features into it, including transfer schemes. My goal here is to examine cooperative game solutions of the model or incentive aspects of model outcomes. Therefore, entire exercises are focused on the cooperative game of bargaining fji g, not on transfers. When discussing the potential Pareto improvements of side payments, I do not respecify the model to add side-payment channels. Rather, I conduct ex-post calculations to test whether there is room for further efﬁciency enhancement through side payments (see section 4.5).

3

3.1

The RICE Model

An Overview

The RICE (for Regional Integrated model of Climate and the Economy) model was originally developed by Nordhaus and Yang (1996). RICE is an integrated assessment (IA) model of climate change. It has played a remarkable role in integrated assessment of climate change over the last ten years. One can ﬁnd its voice in IPCC’s assessment reports over the past decade (SAR, TAR, and AR4). RICE is one among a plethora of IA models that have been built and maintained in recent years. I surveyed the functions of IA modeling in chapter 1. RICE is the most ﬂexible and transparent of the popular and inﬂuential IA models. Therefore, it is not surprising that RICE (along with DICE—its aggregate counterpart) has found a large number of external users and a wide range of applications from the classroom to the conference stage, and from policy assessments to game-theoretic applications. The evolution and development of RICE consists of three phases. The most widely cited RICE is the original version, which is documented in Nordhaus and Yang 1996. The original RICE is a natural extension of the famous DICE model (Nordhaus 1994). Instead of a single global economy as in DICE, RICE contains six regions. Regional disaggregation in RICE creates a signiﬁcant enrichment over DICE: it treats climate change as a stock externality phenomenon. Both original DICE and RICE are coded in GAMS language (Brooke, Kendrick, Meeraus, and Raman 2004). Following the original version, Nordhaus and Boyer (2000) developed a new version, labeled RICE-99. RICE-99 adopts a different modeling methodology from its predecessor. Its model structure, particularly the choice of control variables, has been changed. RICE-99

28

Chapter 3

introduces energy usage in the production function. In addition, it is coded in Excel instead of in GAMS. Beginning in 2002, Nordhaus and Yang initiated a new round of RICE model development. This ongoing project is not yet documented. The new RICE model, labeled RICE2007, combines the strengths of the two previous versions. The new RICE model has more detailed regional disaggregation, a shorter time step, and a longer time horizon, among other important revisions. The new RICE model, labeled RICE2007, returns to the GAMS platform. This research relies on a simpliﬁed version of RICE2007. The new RICE model, though simpler than most other IA models, is a model of stock externality provision, similar to what is discussed in chapter 2, with much needed empirical details. Because RICE is among the few IA models that treat climate change as a global stock externality phenomenon, the model is a convenient tool for game-theoretic modeling of climate change. In fact, a signiﬁcant portion of the growing literature based on DICE/RICE modeling follows a game-theoretic approach. An incomplete list includes Carraro, Eyckmans, and Finus 2006, Eyckmans and Tulkens 2003, and Yang 2003b. The empirical component of this research is entirely within the RICE framework. To lay down the empirical foundation, I will need to introduce the RICE model in more detail. In the remaining part of this chapter, I will describe the model structure to be adopted in the research, connections between the stylized model in chapter 2 and RICE, technical aspects of game-theoretic solution concepts in RICE, and the numerical algorithms for ﬁnding those solutions. 3.2

Model Description and Calibration

As an IA model, RICE is surprisingly simple. The complete mathematical expression of the RICE model used in this study can be on within a single page, as in appendix 1.1 With a group of twelve dynamic constraints and an objective function, RICE highlights the relationship between regional economic growth, GHG emissions and climate damage caused by such growth, and GHG mitigation strategies. And most important to this research: the RICE model treats climate change as stock externalities. The model used in this study is a simpliﬁed version of RICE2007, which is currently under development. The major difference between the model here and RICE2007 is the dimensionality of the model.

The RICE Model

29

Instead of twelve regions as in RICE2007,2 I aggregate them into six regions. These six regions are: USA, OHI (Other High Income countries), EU (European Union), CHN (China), EEC (Eastern European Countries and the Former Soviet Union), and ROW (the Rest of the World). Here, the deﬁnition of USA, EU, and CHN are identical to RICE2007. USA and CHN are two sovereign countries. EU contains most Western European countries. OHI and Japan in RICE2007 are merged into a single region, OHI; EEC and Russia in RICE2007 are merged into EEC; the remaining ﬁve regions in RICE2007 are merged into ROW. The purpose of aggregation is to reduce the volumes of calculations without hampering the illustration of the main themes of this study. As we will see, calculation time and output volume increase exponentially with the number of regions. The basic structure of RICE includes four parts: 1. Objective function 2. Regional economic growth module 3. Carbon emission, concentration, and temperature change module (or ‘‘carbon cycle’’ module) 4. Economic-climate linkage The model can be formulated either as a social planner’s optimal control problem or as an open-loop differential game with regions as players. In the social planner’s optimal control problem, the objective function is a weighted sum of present values of intertemporal regional utility functions ((A-1) in appendix 1). The instantaneous utility function is the constant elasticity of substitution function with the elasticity of marginal utility equaling 1.3 The argument of the instantaneous utility function is regional per capita consumption. In contrast, the objective function of the open-loop differential game is a single region’s present value of intertemporal utility function ((A-14) in appendix 1). The economic module consists of equations (A-2) to (A-5). (A-2) is a Cobb-Douglas aggregate production function. The production function uses labor and capital as the two primary factors. The function is enhanced by AðtÞ, an exogenous and time-variant total factor productivity growth trend. (A-3) is the deﬁnition of real GDP. It is the gross output from (A-2) adjusted by a ‘‘climate factor’’ Wi ðtÞ. This Wi ðtÞ, deﬁned in (A-7), is the key term connecting climate and the economy. (A-4) is the real GDP identity. It reﬂects the trade-offs between current

30

Chapter 3

consumption and investment within the budget of ‘‘real GDP’’ Yi ðtÞ (not gross output Qi ðtÞ). (A-5) is the capital formation process. The climate module, consisting of (A-8) to (A-13), is a simpliﬁed ‘‘box model’’ of carbon cycle. The approach originated with Schneider and Thompson (1981) and was recalibrated by Nordhaus based on other carbon cycle models (see Nordhaus and Boyer 2000). The module captures dynamic processes of interactions among lower and upper oceans, and atmospheric carbon concentrations, radiative forcing of GHGs, and deep-ocean and atmospheric temperature changes. The module is a process of differential systems. The driver of this module is the aggregate GHG emission EðtÞ (the sum of regional GHG emissions Ei ðtÞ) generated from the economic module. EðtÞ affects GHG concentrations in three layers (MðtÞ, ML ðtÞ, MU ðtÞ); GHG concentrations affect radiatve forcing (A-(13)); radiative forcing leads to temperature change (A-(11)). The feedback from this module to the economic module is atmospheric temperature increase T1 ðtÞ. In sum, higher EðtÞ leads to higher T1 ðtÞ through this dynamic module. Economic and carbon cycle modules are linked through (A-3), (A-6), and (A-7) in the model. For each region, GHG emissions are declining proportionally with respect to GDP production, when there are no emission control efforts. This assumption is consistent with historical observations that long-run energy intensities (thus carbon intensities) reduce worldwide due to technological changes (Schmalensee, Stoker, and Judson 1998). In the model, by choosing GHG control rate m i ðtÞ ð0 < m i ðtÞ < 1Þ, a region can reduce GHG emissions from the nocontrol baseline in (A-6). When m i ðtÞ ¼ 0:1, it means 10 percent GHG emission reduction from the baseline, and so on. GHG mitigation ðm i ðtÞ > 0Þ reduces climate damage but is also costly. Such trade-offs are reﬂected by (A-7), which is the ratio of the GHG mitigation cost function and the climate damage function. Either temperature ðT1 ðtÞÞ increases or control cost increases ðm i ðtÞÞ lower the value of Wi ðtÞ. Because Wi ðtÞ is always less than 1, the real GDP deﬁned by (A-3) is always lower than gross output in (A-2). In the social planner’s problem (A-1) to (A-13), the control variables are Ii ðtÞ—the investment in private good production (GDP); and m i ðtÞ—the investment in public (or environmental) good contribution (GHG mitigation). To maximize the objective function, decisions have to be made on trade-offs between current and future consumptions ðIi ðtÞÞ as well as ﬂows of GHG mitigation costs and cumulative climate damages ðm i ðtÞÞ.

The RICE Model

31

Parallel to the social planner’s problem, the RICE model can be formulated as an open-loop differential game of providing climate externalities. This formulation maintains an identical model structure, except that each region maximizes its own objective function (A-14) while taking other regions’ emissions as given in (A-8 0 ). The solution of this noncooperative game is the open-loop Nash equilibrium. Having described the structure of RICE, it is easy to see the connection between the stylized model of stock externality provision in chapter 2 and RICE. Private good xi ðtÞ in systems (2.4) to (2.6) is regional GDP in RICE. It is endogenized in the economic module of RICE. Process (2.6) is the much more complicated climate module in RICE. The transformation relationship between private good and ﬂows of externality (2.5) is represented by the economic-climate linkage component in RICE. In particular, collective (or aggregate) GHG emissions from all regions affect every region’s welfare through temperature change. Therefore, we can claim that RICE is a model of stock externality provision. The model used in this research is recalibrated according to the most recent RICE model development by Nordhaus and Yang. The base year of the model is year 2000. The time horizon of the model is 250 years. In discrete form, the model is solved with ﬁve years per period, 50 periods in total. The initial values of all state variables and parameters, calibrated coefﬁcients of mitigation cost and climate damage functions, as well as coefﬁcients in exogenous trends are shown in appendix 3—the GAMS code for the core part of the RICE model used here. The calibrated baseline simulation results ðm i ðtÞ ¼ 0Þ of the RICE model used here are compatible with the consensus of IA modeling groups. In fact, the global GHG emission projection generated in this research sits in the middle range of IPCC TAR Working Group III’s compilation of the major IA modeling group’s projections (see IPCC 2001). I should indicate that the calibration of the model here does not reﬂect accurate predictions of exogenous trends in the regions. While they are consistent with the evolution of the RICE model development and compatible with moderate ranges projected by other IA models, the main purpose of calibration here is to characterize heterogeneous agents that are of interest from a game-theory perspective. Because the RICE model is used for game-theoretic simulation in this research, it would be helpful to highlight some important attributes

32

Chapter 3

Table 3.1 Summary of the regions USA

OHI

EU

CHN

EEC

ROW

Per capita income Population

High Low

High Low

High Low

Low High

Middle Low

Low High

Economic growth rate

Moderate

Moderate

Moderate

High

Moderate

High

GHG emissions

High

Low

Low

High

Moderate

High

Mitigation cost

High

High

High

Low

Low

Low

Climate damage

Moderate

Low

Low

High

Low

High

Decision making

Single

Almost single

Single

Single

Plural

Plural

of the regions (the players of the game) here. In essence, I specify the dynamic proﬁles of six heterogeneous regions (agents) in the model. Table 3.1 contains descriptive characterization of six regions. The characterizations are based on the calibration of the model. They affect the strategic behaviors of the regions. In particular, whether a region is a single decision maker in reality or not, is crucial in interpreting the simulation results. The United States and China are sovereign countries. Their actions in the negotiation of IEAs represent their national interests. European Union member countries have a unison voice in major international environmental issues. In the climate change context, the EU can be treated as a single decision maker too. OHI includes countries that are far apart geographically, such as Australia, Canada, and Japan. They may not always agree with each other on concrete climate change policies. Because OHI countries have similar income levels and shared environmental concerns, OHI can be treated as a single decision maker for our purposes here. Countries in EEC and ROW have much less in common. ROW in particular is not and will not be a single player in international environmental issues. Thus many economists would question the appropriateness of treating ROW as a single player in game-theoretic discussions. I share such concerns. As mentioned earlier, the assumption is mainly because of the dimensionality of simulations. ROW is simply a proxy. In subsequent analyses, I parameterize the ‘‘plural decision making’’ in ROW and investigate its impact on coalition formation (in sections 5.2.2 and 6.2).

The RICE Model

3.3

33

Solution Concepts in the RICE Model

Numerical solutions of the RICE model belong to four categories. One is the ‘‘business as usual (BaU)’’ baseline solution; the second is the inefﬁcient market equilibrium; the third is various policy-motivated second-best solutions; the fourth is various efﬁcient solutions in which climate externalities are fully internalized by a social planner. Of these four categories, the ﬁrst three are inefﬁcient solutions. Inefﬁcient Solutions The BaU solution assumes that m i ðtÞ ¼ 0 for all regions in the social planner’s problem (A-1). That is, regions take no GHG mitigation actions. In this solution, regions do not interact with one another because climate externality is not in the decision-making set. The BaU solution is unique. It offers the ‘‘ceilings’’ of regional and global GHG emissions under the given parametric assumptions of the model. The BaU solution is a useful benchmark or baseline for comparing with other solutions. The open-loop differential (noncooperative) game solution is the inefﬁcient market equilibrium. Conceptually, the solution is identical to the open-loop Nash equilibrium deﬁned in section 2.3. In the RICE model, each region decides its own GHG control rate m i ðtÞ, taking other regions’ control efforts as given. The conjecture of all six regions’ independent decision paths is the Nash equilibrium. Mathematically, the equilibrium is the solution of a differential equation system that is consistent with the six related optimal control problems as in (A-14). Because of its inefﬁciency from ‘‘free-riding,’’ the open-loop Nash equilibrium solution generates more global GHG emissions than the efﬁcient solutions. But for self-interests, regions would exercise low-level GHG mitigation efforts in the open-loop Nash equilibrium. As a result, the global GHG emission of the Nash equilibrium is lower than the BaU benchmark. Many GHG mitigation options have been put forward in policy debates and on the negotiation table. The binding terms in the Kyoto Protocol represent a comprehensive GHG mitigation policy regime. From a modeling perspective, policy targets, such as CO2 concentration ceilings and GHG emission quotas, are additional constraints or boundary conditions imposed on the system. To the original system, the optimal GHG mitigation policy solutions are the ‘‘second-best’’ solutions. For a given policy target, the solutions under different social

34

Chapter 3

welfare weights may lead to different implications. Furthermore, if the policy target is not consistent with the incentives of any region, the grand coalition cannot be agreed on without compensations. It is interesting to investigate the relationship between policy targets and its impact on coalition formation. In section 6.1, I will do so. Efﬁcient Solutions The solutions of system (A-1) to (A-13) under different social welfare weights are at the center of this research. In the absence of additional constraints, solution trajectories of state and control variables in the system are efﬁcient outcomes of the RICE model (after transfers, if necessary). Each point on simplex S ¼ P fji j ji ðtÞ ¼ 6g of social welfare weights maps to an efﬁcient solution of RICE.4 As pointed out in chapter 2, the selection of fji ðtÞg has strong positive and normative consequences. Furthermore, only a small subset of the simplex corresponds to plausible solutions of the cooperative game speciﬁed in chapter 2. Therefore, I focus on gametheoretic solutions and some popular efﬁcient solutions in IA modeling communities. Among the most frequently used social welfare weights in economic modeling are the so-called utilitarian weights ji ðtÞ 1 1. The utilitarian weight, or Benthamite weight, has a long-standing and unique position in welfare economics. Equal weight or ‘‘one man, one vote’’ is considered a fair arrangement in many situations. If the utilities are transferable, the equal weight assumption is innocuous and convenient (for its calculation simplicity) from a modeling perspective. However, the utilitarian weights can be misleading if they are applied to models with nontransferable utility functions. In particular, if it is used as a benchmark or starting point for economic analysis of stock externality provision, the results might be biased. Even with transferable utility functions, the normative impacts from wealth transfers might outweigh transfers necessitated by sales of emission permits. These two sources of transfers are very difﬁcult to separate in model construction. Finally, the incentive property of efﬁcient provision of stock externality under the utilitarian weights is questionable. If the agents are highly heterogeneous, the incentive property of the utilitarian outcome is even less desirable. Subsequent simulations in RICE will reveal the incentive properties of the utilitarian solution. Another important social welfare weight is the Negishi weight. The weight, named after Negishi (1960), connects a social planner’s opti-

The RICE Model

35

mum to the decentralized Walrasian equilibrium. With the Negishi weights, which are proportional to the inverse of marginal utilities of income for each region, the optimal solution of the social planner’s problem corresponds to the Walrasian equilibrium in the absence of externalities. In an economy without externalities, the solution of the social planner’s problem with the Negishi weights is a useful benchmark in which each region abides by its budget constraints. In an environment-economy model, such as RICE, the Negishi solution is also signiﬁcant because it is an efﬁcient outcome that does not require transfers. Therefore, it is an efﬁcient benchmark that separates efﬁciency considerations from redistribution concerns. Because of its welfare implications, the Negishi solution is sought in IA models, such as in Nordhaus and Yang 1996, Manne and Richels 1992 and Manne 2000. In Nordhaus and Yang 1996, a set of timevariant Negishi weights are identiﬁed and used to calculate efﬁcient global carbon taxes. I should note that the Negishi solution is not necessarily an incentive-compatible outcome in the presence of externalities. Some regions might not like their GHG mitigation obligations under the Negishi weights. Therefore, the Negishi solution is not likely an outcome from a cooperative game of externality provision. In this research, I seek the Negishi solution for the purpose of comparing and contrasting with the game-theoretic solutions. P The most interesting weights on simplex S ¼ fji j ji ¼ 6g are those plausible and credible candidates of game-theoretic solutions deﬁned in the previous chapter. Two important solution concepts of a cooperative game are the core and the Shapley value. Another related solution concept is the Lindahl equilibrium of the economy with stock externalities inside the core. Those solution concepts are to be sought in the RICE model. We really want to identify the allocation with the strongest ‘‘willingness to pay’’ by all regions (the Lindahl equilibrium) and the allocation that is ‘‘fair’’ and incentive compatible (the Shapley value). In RICE, all reasonable outcomes from the cooperative game deﬁned in the previous chapter should be in the core. Evidently, the core allocations are measured in continuum and the Lindahl equilibrium without endowment transfers is in the core. In this research, we want to identify a particular set of welfare weights that maps to the core of the cooperative game in RICE. Furthermore, we want to locate a particular point inside the core (namely, a particular fji g) that has the Lindahl

36

Chapter 3

equilibrium property without endowment transfers. By demonstrating that such welfare weights can be identiﬁed in the RICE model, we connect game-theoretic solution concepts and empirical IA models. The Shapley value occupies a unique position in the cooperative game solutions of international environmental problems. It is a ‘‘fair’’ benchmark of equal gains for each region by forming the global cooperation (or grand coalition in game-theory terms) under this particular arrangement. In the RICE model that consists of complicated dynamic systems and nontransferable utility functions, we try to identify the Shapley value in a heuristic way. The Shapley value solution in RICE is the unique solution of system (A-1) to (A-13) such that all regions have the same marginal utility gains by joining the coalition of the other ﬁve regions and forming the grand coalition. Therefore, identifying the Shapley value is to ﬁnd a set of fji g under which the solution possesses the above property. In this study, we will look for all solution concepts described above numerically in the RICE model. Except for the sensitivity studies in section 5.2.2, all solutions are obtained with the identical parametric assumptions in RICE. As we can see, all efﬁcient solutions are obtained by altering fji g in system (A-1) to (A-13) alone. 3.4

Solution Algorithms

The RICE model is coded in GAMS language. GAMS is a powerful high-level programming language for solving a variety of mathematical programming problems. Its simple syntax is transparent to economists and policymakers who have no extensive programming training. The numerical algorithms in this research are programmed and executed in GAMS.5 The BaU and utilitarian solutions are straightforward numerically. They are solved by calling a nonlinear programming (NLP) solver in GAMS. However, the other solution concepts require specially designed iterative algorithms inside GAMS. They are outlined below. Open-Loop Nash Equilibrium Procedure Finding the open-loop Nash equilibrium of the RICE model is to locate the paths of state and control variables that ﬁt the deﬁnition of Nash equilibrium, namely, to solve the system of differential equations deﬁned by system (A-14). It is an equilibrium outcome where each region has no incentive to adjust its paths further, given other regions’ decisions.

The RICE Model

37

The iterative procedure for getting the open-loop Nash equilibrium of RICE includes the following steps. First, a social planner’s problem is solved with an arbitrary m i ðtÞ ¼ m 0i b 0 to obtain initial paths of state and control variables of regions for subsequent iterations. Second, ﬁve out of six regions’ state and control variables are ﬁxed at the values in step 1. For example, all regions except for USA are ﬁxed. We then solve USA’s intertemporal optimization problem (A-14) to get USA’s optimal responses to other regions’ contributions to GHG emissions (as in A-8 0 ). After obtaining the solution, we ﬁx USA’s state and control variables from the solution of (A-14) and keep state and control variables of four out of the remaining ﬁve regions the same as in step 1 (say, all ﬁve except for OHI). We then solve problem (A-14) for OHI. Such a procedure of solving (A-14) is conducted on all six regions sequentially. Third, using the paths of state and control variables at the end of the second step as the new initial paths, we repeat the iterative procedure in step two until all regions’ solution paths of (A-14) are stabilized. That is, the iteration continues until no region will respond and adjust its own optimal paths any further, given other regions’ paths of state and control variables. This two-tier iteration leads to the openloop Nash equilibrium solution of the RICE model. This iterative procedure mimics the responses of regions to others’ contributions to stock externalities (GHG emissions) and convergence to the conjecture of the equilibrium. The robustness of the above procedure and the uniqueness of the open-loop Nash equilibrium have been proved by the following simple testing method. If we change the initial value of m 0i in step 1 and/or regional sequence in step 2, this two-tier iteration procedure converges to the same result. In addition, the algorithm has been robust during different phases of the RICE model development in which the initial conditions and parameters of the model are changed numerous times. The uniqueness of the Nash equilibrium has been tested in different versions of RICE. This heuristic algorithm for solving the noncooperative game solution is also crucial in solving the cooperative game solutions in the RICE model because it is the key component in solving the ‘‘hybrid’’ Nash equilibrium. This algorithm of allocating the open-loop Nash equilibrium is due to Nordhaus and Yang 1996. Time-Variant Negishi Solution Procedure The time-variant Negishi solution is a speciﬁc approach to obtaining efﬁcient global carbon taxes in Nordhaus and Yang 1996. The RICE model used in this study, as

38

Chapter 3

with the original RICE model, does not allow the trade of private goods (GDP) among the regions. Under the time-variant Negishi weights, there are no trade ﬂows even if they are allowed because the solution of the social planner’s problem (A-1) renders equalized shadow prices of the private good across regions in each period. In the Negishi solution, carbon taxes, expressed as the ratio of shadow prices of private good and public ‘‘bad’’ ﬂows, are equalized across regions in each period. In addition, this time-variant Negishi solution represents the Walrasian equilibrium in which regions live within their budgets in each period. No transfers can lead to any Pareto improvements. The procedure for identifying the Negishi weights is a tatonnement procedure. In this procedure, we ﬁrst solve a social planner’s problem with utilitarian weights to obtain paths of costate variables (or shadow prices) of capital. Then the social welfare weights are set at the ‘‘scaled’’ harmonic mean of regional shadow prices of capital in the utilitarian solution in each period. The solution of the social planner’s problem with ‘‘harmonic mean’’ weights is in the neighborhood of shadow price equalization of capital. After getting this initial set of time-variant weights, regional shadow prices of capital are compared. If a region’s shadow price of capital is relatively high, its weight is reduced and vice versa. Once the weights are adjusted, the social planner’s problem is re-solved under the new set of weights. This ‘‘ﬁne-tuning’’ process continues until the differences of regional shadow prices of capital converge to zero (or sufﬁciently small). In the end, the social welfare weights are proportional to the inverse of regional shadow prices of capitals in each period. The Core and the Lindahl Equilibrium Solution Procedure The algorithm for identifying the core allocations and the Lindahl equilibrium in RICE is much more complicated. It involves several procedures and requires extensive simulations. The entire algorithm is developed by the author for this research. As I discussed earlier, whether a social planner’s problem (A-1) is a plausible outcome of a cooperative game as deﬁned in chapter 2 depends on its having the core properties or not. If under a particular set of fji g, the grand coalition cannot be blocked by any subcoalitions, the social planner’s problem with this fji g has the core property. In essence, ﬁnding a cooperative game solution is to test the core property P of the solution of (A-1) on simplex S ¼ fji j ji ¼ 6g. Such a test in-

The RICE Model

39

volves repeatedly solving ‘‘hybrid’’ Nash equilibria (deﬁned in chapter 2) and extensive side calculations. In this six-region RICE model, there are 2 6 6 1 ¼ 57 possibilities of a single coalition strategically interacting with m ðm ¼ 0; 1; 2; 3; 4Þ single players (regions).6 We label these coalitional solutions as the ﬁrst-category coalitions. In addition, there are the following numbers of possible cases in which two coalitions (not a single coalition vs. regions) strategically interact with each other: six 5-member vs. 1member coalitions (they coincide with m ¼ 1 in the ﬁrst category coalitions); ﬁfteen 4-member vs. 2-member coalitions; and ﬁfteen 3-member vs. 3-member coalitions. We call these two-coalition cases the secondcategory coalitions. Finally, there are ﬁfteen possibilities of three 2member coalitions strategically interacting with one another. They are called the third-category coalitions. To facilitate the calculations, we use a 64 6 design matrix (see table 3.2) to characterize the ﬁrst-category coalitions. In this 6-region by 64scenario table, a ‘‘1’’ cell indicates that a region is in a coalition and a ‘‘0’’ cell means out. The top row (#1) of the table represents the grand coalition (all in) and the bottom row (#64) represents the Nash equilibrium (all out). Excluding the six shaded rows (they are 1-in and 5-outs) and row #64, the remaining 57 rows represent all possible ﬁrst-category coalitions. For example, row #10 in the design matrix represents a 4-region coalition {USA, OHI, CHN, EEC} and two noncoalitional regions (EU and ROW). Alternatively, this design matrix can be used in representing the second-category coalition scenarios. In this context, a ‘‘1’’ cell represents a region being in the ﬁrst coalition and a ‘‘0’’ in the second. I should note that there are duplicates when using a design matrix to represent the second-category coalitions. For example, row #8 and row #57 represent the same scenario. For a given set of fji g, we solve the entire 64 possible ‘‘hybrid’’ Nash equilibrium solutions (the ﬁrst category) for the RICE model, namely, all 64 scenarios represented by the design matrix. In a ‘‘hybrid’’ Nash equilibrium, regions inside the coalition cooperate with one another. Simultaneously, the coalition interacts strategically with other regions to achieve the open-loop Nash equilibrium among the coalition and nonparticipant regions. When inside the coalition, regional GHG mitigation share is based on fji g. For the second and third categories of coalition scenarios, we can deﬁne and solve the ‘‘hybrid’’ coalitional equilibrium in a similar way.

40

Chapter 3

Table 3.2 Design matrix No.

USA

OHI

EU

CHN

EEC

ROW

No.

USA

OHI

EU

CHN

EEC

ROW

1

1

1

1

1

1

1

33

0

1

1

1

1

1

2

1

1

1

1

1

0

34

0

1

1

1

1

0

3

1

1

1

1

0

1

35

0

1

1

1

0

1

4

1

1

1

1

0

0

36

0

1

1

1

0

0

5

1

1

1

0

1

1

37

0

1

1

0

1

1

6

1

1

1

0

1

0

38

0

1

1

0

1

0

7

1

1

1

0

0

1

39

0

1

1

0

0

1

8

1

1

1

0

0

0

40

0

1

1

0

0

0

9

1

1

0

1

1

1

41

0

1

0

1

1

1

10

1

1

0

1

1

0

42

0

1

0

1

1

0

11

1

1

0

1

0

1

43

0

1

0

1

0

1

12

1

1

0

1

0

0

44

0

1

0

1

0

0

13

1

1

0

0

1

1

45

0

1

0

0

1

1

14

1

1

0

0

1

0

46

0

1

0

0

1

0

15

1

1

0

0

0

1

47

0

1

0

0

0

1

16

1

1

0

0

0

0

48

0

1

0

0

0

0

17

1

0

1

1

1

1

49

0

0

1

1

1

1

18

1

0

1

1

1

0

50

0

0

1

1

1

0

19

1

0

1

1

0

1

51

0

0

1

1

0

1

20

1

0

1

1

0

0

52

0

0

1

1

0

0

21

1

0

1

0

1

1

53

0

0

1

0

1

1

22

1

0

1

0

1

0

54

0

0

1

0

1

0

23

1

0

1

0

0

1

55

0

0

1

0

0

1

24

1

0

1

0

0

0

56

0

0

1

0

0

0

25

1

0

0

1

1

1

57

0

0

0

1

1

1

26

1

0

0

1

1

0

58

0

0

0

1

1

0

27

1

0

0

1

0

1

59

0

0

0

1

0

1

28

1

0

0

1

0

0

60

0

0

0

1

0

0

29

1

0

0

0

1

1

61

0

0

0

0

1

1

30

1

0

0

0

1

0

62

0

0

0

0

1

0

31

1

0

0

0

0

1

63

0

0

0

0

0

1

32

1

0

0

0

0

0

64

0

0

0

0

0

0

The RICE Model

41

Algorithmically, the ‘‘hybrid’’ Nash equilibrium calculation combines the optimal control problem and the differential game problem. In the above solution sequence, we intentionally keep the scenarios in rows 32, 48, 56, 60, 62, and 63 in the design matrix (shaded rows). These scenarios represent one region in a coalition and ﬁve regions that are out. Semantically, they are equivalent to row 64, namely, the open-loop Nash equilibrium. Therefore, the solutions from these seven scenarios should be the same if the algorithm is correct. The ‘‘hybrid’’ Nash equilibria are solved sequentially from row 1 to 64. The looping and branching of this sequence in GAMS are automatic. The numerical results from these seven rows are identical. This fact, again, proves the robustness of my algorithm. After 64 ‘‘hybrid’’ Nash equilibria are obtained, I conduct a series of incentive tests on 64 sets of solutions based on this fji g. First, to test individual rationality (IR) of the grand coalition under fji g (referring to the deﬁnition in chapter 2), I compare a region’s payoff (welfare) in a coalition with their Nash equilibrium payoffs. Only when a region’s payoff in a coalition is higher than its reservation payoff (i.e., the Nash equilibrium payoff) does this region have the incentive to join the coalition. These test results can be expressed symbolically in design matrix format. If a region is better off in a coalition than its Nash position, a ‘‘þ’’ sign is in the relevant cell; if it is worse off, a ‘‘’’ sign; if a region is out of the coalition, the cell is blank. In addition, if two payoffs are identical, such as in rows 32, 48, 56, 60, 62, 63, and 64, the cells are also blank. In sum, the signs in a row are the signs of differences between the payoffs of coalition members in this row and the payoffs of the last row (Nash equilibrium). To illustrate how to conduct such incentive checking, I present table 3.3 here. It contains regional welfare changes between coalitions and the Nash equilibrium (row 64) under the utilitarian weights.7 In table 3.3, incentives to join a coalition with ji ¼ 1 are clearly revealed. For example, four regions (USA, OHI, EU, and EEC) are worse off in the global coalition than in their Nash equilibrium positions (see signs in row 1); USA and OHI are worse off in the 5-member coalition (all but ROW) than playing the Nash game (see row 2), and so on. It is clear that a necessary condition for all coalitions under fji g having IR properties is that all entries in a table like table 3.3 show positive signs. In particular, all cooperative game solutions of system (A-1) (the grand coalition) should have the property such that all signs in row #1 are ‘‘þ.’’

42

Chapter 3

Table 3.3 Welfare changes from Nash equilibrium (utilitarian) No.

USA

OHI

EU

CHN

EEC

ROW

No.

1

þ

þ

2

þ

þ

þ

3

þ

4

þ

þ

5

6

þ

þ

þ

þ

7

8

þ

þ

þ

9

þ

10

þ

þ

11

þ

12

þ

13

14

þ

þ

þ

15

16

þ

þ

17

þ

þ

þ

þ

þ

þ

OHI

EU

CHN

EEC

ROW

33

þ

þ

34

þ

35

þ

36

þ

37

38

þ

þ

39

40

þ

41

þ

42

þ

43

þ

44

þ

45

46

þ

47

þ

þ

þ

þ

þ

þ

þ

48

þ

þ

18

þ

þ

19

þ

20

þ

21

22

þ

þ

23

24

þ

25

þ

26

þ

27

þ

28

þ

29

30

þ

þ

31

32

þ

USA

þ

49 50

þ

51 52

þ

þ

þ

þ

þ

þ

þ

þ

53

54

þ

þ

55

þ

þ

56 þ

þ

57

þ

58

þ

59

þ

þ

60 þ

61 62

þ

63 64

þ

The RICE Model

43

Another way to look at the incentives of regions is to check whether the optimal solution of system (A-1) under fji g can be blocked by any subcoalition or not. In table 3.4 (again, the utilitarian welfare weights) welfare changes between the grand coalition (row 1) and subcoalitions are recorded.8 If a region is better off in a smaller coalition, a ‘‘þ’’ sign is in the relevant cell; if it is worse off, a ‘‘’’ sign is recorded. Again, if a region is outside a coalition, the cell is blank. In this table, the grand coalition under the utilitarian weights can be blocked by at least 10 subcoalitions. They are rows #6, 14, 16, 22, 24, 30, 38, 40, 46, and 48. In these 10 rows, all subcoalition members have ‘‘þ’’ signs. For example, in row #6, all four regions are better off in the {USA, OHI, EU, EEC} coalition than in the grand coalition. For those subcoalitions containing mixed signs, the grand coalition might or might not be blocked by them. If the loser(s) (those with ‘‘’’ signs) can be fully compensated by the winner(s), the grand coalition is blocked; otherwise, it is not. In this study, I do not investigate different possibilities in such subcoalitions. Evidently, a sufﬁcient condition for a solution of the social planner’s problem (A-1) being a core allocation is that all entries in a table like table 3.4 show negative signs. Alternatively, if a model outcome has uniform ‘‘’’ signs in table 3.4 and has ‘‘þ’’ signs in the ﬁrst row in table 3.3, it is a core allocation. I should emphasize that the sufﬁcient condition involves the homogeneous ‘‘’’ signs in table 3.4. An individual row, such as row #33 in table 3.4, only indicates that all members in the subcoalition in that row are worse off than in the grand coalition, and nothing more. Finally, I identify the Lindahl equilibrium through an indirect searching procedure among the core allocations. The regional ‘‘private price’’ of stock externality in the Lindahl equilibrium is a concealed signal of the highest ‘‘willingness to pay’’ for the externality by this region, given other regions’ contributions. In RICE, such maximal ‘‘willingness to pay’’ can be treated as an outcome from the following dynamic conjecture. Starting from its Nash equilibrium position, a region is joined by other regions, one by one, to form larger and larger coalitions. To be a willing partner, the region’s payoffs have to increase (or at least be nondecreasing) as the coalition size expands. The same dynamic conjecture can also start from the grand coalition. A willing partner in the grand coalition would be strictly worse off if other regions drop off one after another. If a point on simplex S ¼ P fji j ji ¼ 6g possesses the property that such monotonic preference changes hold for all regions, then the point maps to one of the

44

Chapter 3

Table 3.4 Welfare changes from grand coalition (utilitarian) No.

USA

OHI

EU

CHN

EEC

ROW

1 2

þ

þ

þ

þ

3

þ

þ

4

þ

þ

þ

5

þ

þ

þ

þ

6

þ

þ

þ

þ

7

þ

þ

þ

8

þ

þ

þ

9

þ

þ

10

þ

þ

þ

11

þ

þ

12

þ

þ

13

þ

þ

þ

14

þ

þ

þ

15

þ

þ

16

þ

þ

17

þ

USA

OHI

EU

CHN

EEC

ROW

33

34

þ

þ

þ

35

36

þ

þ

37

þ

þ

þ

38

þ

þ

þ

39

þ

þ

40

þ

þ

41

42

þ

þ

43

44

þ

45

þ

þ

46

þ

þ

47

þ

48

þ

18

þ

þ

19

þ

20

þ

þ

21

þ

þ

þ

22

þ

þ

þ

23

þ

þ

24

þ

þ

25

þ

26

þ

þ

27

þ

28

þ

29

þ

þ

30

þ

þ

31

þ

32

No.

49 50

51 52

þ

þ

þ

53

þ

54

þ

þ

55

56

57

58

þ

59

60

61 62

63 64

The RICE Model

45

strongest ‘‘willingness to pay’’ allocations in the grand coalition. I test the hierarchical payoff changes of coalitions as explained above. If a set of weights fji g has those properties, it maps the solution of (A-1) to the allocation with the strongest incentives for all regions, namely, the Lindahl equilibrium. To ﬁnd the Lindahl weight fji g, I conduct the following meticulous checking on regional incentive changes. I measure regional welfare (or payoff) changes from an N-member coalition to ðN 1Þ-member coalitions, for N ¼ 6; 5; 4; 3, and for all six regions. The measurements of welfare changes are condensed in the symbolic tabulation as shown in table 3.5. In the table, there are four columns under each region. Here, the meaning of ‘‘þ,’’ ‘‘,’’ and blank are the same as in table 3.4: a ‘‘þ’’ sign indicates welfare increase; a ‘‘’’ sign indicates welfare decrease; a blank cell means outside the coalition or ‘‘not applicable.’’ Again, these signs are generated from welfare changes from N-member to ðN 1Þ member coalitions, unlike tables 3.3 and 3.4, where welfare changes are calculated against the Nash equilibrium and the grand coalition respectively. The reading of table 3.5 is less straightforward than that of tables 3.3 and 3.4. Nevertheless, interpretation of the table can be explained through the following examples. Row 24 represents the {USA, EU} coalition. This 2-member coalition can be formed by withdrawing from one of four 3-member coalitions. They are: {USA, OHI, EU}, {USA, EU, CHN}, {USA, EU, EEC}, and {USA, EU, ROW}. The entries in the four columns (from left to right) are welfare changes of USA and EU in the {USA, EU} coalition from the above four 3-member coalitions in the same order. In this utilitarian case, both USA and EU are better off by withdrawing from {USA, EU, CHN} (2nd cell) and {USA, EU, ROW} (4th cell); USA is also better off by withdrawing from {USA, EU, EEC} (3rd cell); otherwise, both regions are worse off (1st cell). As a second example, row 4 represents the 4-member coalition {USA, OHI, EU, CHN}. The two entries, again, from left to right, are welfare changes of those regions by withdrawing from two 5-member coalitions: {USA, OHI, EU, CHN, EEC} and {USA, OHI, EU, CHN, ROW}. CHN is worse off in both cases and the other three regions are better off in latter. Inductively, all entries in table 3.5 can be explained similarly. In the table, a ‘‘þ’’ sign in any entry implies some ‘‘compromises’’ in incentives to form the larger coalition: one or more regions can gain by defecting. Because the Lindahl equilibrium requires the strongest ‘‘willingness to pay’’ or incentives from all regions, it necessarily requires

46

Chapter 3

Table 3.5 Welfare changes between coalitions (utilitarian) No.

USA

OHI

EU

CHN

EEC

ROW

1 2

þ

þ

þ

3

þ

þ

4

þ

5

þ

6

þ

þ

7

þ

þ

þ

þ

þ

8

þ

þ þ

þ

þ

þ

þ

9

þ

10

þ

þ

þ

11

þ

þ

þ

þ

12

þ

þ þ

þ

13

þ

þ

þ

þ

þ

14

þ

þ þ

þ þ

þ þ

þ

þ

þ þ

þ

þ

þ þ

þ

þ þ

þ

þ

þ

þ

15

þ

þ þ

þ

þ þ

16

þ

þ þ

þ þ

17

þ

18

þ

þ

19

þ

þ

20

þ þ

þ

21

þ

þ

þ

þ

22

þ þ

þ þ

þ þ

23

þ

þ þ

þ

24

þ þ þ

þ þ

25

þ

þ

26

þ þ

þ

27

þ

þ þ

28

þ þ þ

29

þ

þ þ

þ

30

þ þ þ

þ þ

31

þ

þ þ þ

þ

32 33

34

þ

þ

35

þ

36

þ þ

þ

þ

The RICE Model

47

Table 3.5 (continued) No.

USA

OHI

EU

CHN

EEC

37

þ

þ

þ

38

þ þ

þ þ

þ þ

39

þ þ

þ

40

þ þ

þ þ

41

þ

42

þ þ

þ

43

þ þ

44

þ þ þ

45

þ þ

þ

46

þ þ þ

þ þ

47

þ þ þ

ROW

48 49

50

þ

þ

51

52

þ

53

þ

þ

54

þ þ

þ þ

55

þ

56 57

58

þ

þ

59

60 61 62 63 64

þ

48

Chapter 3

that all entries in a table like table 3.5 be negative (or have ‘‘’’ signs). That is, the grand coalition under the Lindahl social welfare weights preserve the strongest and consistent ‘‘willingness to pay’’ incentives. The RICE model is a complicated dynamic system. For such a system, it is very difﬁcult to further pin down analytical properties of the Lindahl equilibrium. The above searching procedure is a hierarchical and numerical procedure. It maps fji g to a point inside the core where it is compatible with the ‘‘willingness to pay’’ principle. It does not in any way guarantee the uniqueness of such an allocation. The homogeneity of signs in the three tables depends on the numerical precision of the calculation. Lying behind the signs are actually numerical values such as those shown in table 3.6. Marginal changes of the numerical values (caused by marginal shifts of fji g) in table 3.6 might or might not alter the signs in table 3.4. The Lindahl solution obtained through this searching algorithm is by no means the ‘‘best’’ result. In fact, when fji g is slightly perturbed, the ‘‘willingness to pay’’ property may still be preserved.9 For all practical purposes, we can call it a numerical solution of the Lindahl equilibrium. Or if one prefers, it is the representative allocation with the strongest ‘‘willingness to pay’’ from all regions inside the core. In summary, a sufﬁcient condition for a cooperative game solution of P RICE is that a point on simplex S ¼ fji j ji ¼ 6g maps to a core allocation. It is equivalent to stating that all entries of row #1 in table 3.3 are positive (IR condition) and all entries in table 3.4 are negative (IC condition).10 To narrow down our choices of the core allocations, the algorithm continues until all signs in table 3.3 are positive (we can harmlessly call it a strong IR condition). Furthermore, the Lindahl equilibrium also requires that all entries in table 3.5 be negative, in addition to (strong) IR and IC conditions. After setting up these incentive testing procedures and criteria, I use the relationship between fji g and the signs in the above three tables to search for the cooperative game solution. As it is elaborated extensively in the previous chapter, the larger the share in fji g, the lower the mitigation cost burden for a region in a social planner’s problem. In this iterative algorithm, I ﬁrst start from the problem (A-1) with an arbitrary set of fji g—for example, the utilitarian weights. After solving the entire spectrum of ‘‘hybrid’’ Nash equilibrium problems, I inspect the signs in the three tables. If a region has negative signs in table 3.3, and/or positive signs in table 3.4, and/or positive signs in table 3.5, its welfare weight, relative to other regions’, is increased on

The RICE Model

49

Table 3.6 Excerpt of table 3.3 in numerical expression No.

USA

OHI

EU

CHN

EEC

ROW

1 0.00001543 0.00000949

2

0.00005877

0.00001751

3

0.00000116

0.00000009 0.00000057 0.00000270

4

0.00005918

0.00001748

0.00001502 0.00001100

0.00000000

5

0.00001973

0.00000589

0.00000513

0.00000000

0.00000301 0.00000182

6

0.00006540

0.00001933

0.00001565

0.00000000

0.00000906

7

0.00002087

0.00000631

0.00000460

0.00000000

0.00000000 0.00000290

8

0.00006544

0.00001922

0.00001527

0.00000000

0.00000000

9

0.00000190

0.00000049

0.00000000 0.00000405 0.00000056 0.00000185

10

0.00005958

0.00001783

0.00000000 0.00001166

0.00000864

11

0.00000307

0.00000060

0.00000000 0.00000664

0.00000000 0.00000303

12

0.00006001

0.00001781

0.00000000 0.00001302

0.00000000

13

0.00002136

0.00000633

0.00000000

0.00000000

0.00000249 0.00000344

14

0.00006551

0.00001919

0.00000000

0.00000000

0.00000872

15

0.00002235

0.00000641

0.00000000

0.00000000

0.00000000 0.00000447

16

0.00006549

0.00001908

0.00000000

0.00000000

0.00000000

17

0.00000027

0.00000000 0.00000119 0.00000354 0.00000075 0.00000159

18

0.00005883

0.00000000

19

0.00000146

0.00000000 0.00000172 0.00000616

20

0.00005927

0.00000000

0.00001439 0.00001272

0.00000000

21

0.00001993

0.00000000

0.00000406

0.00000000

0.00000232 0.00000322

22

0.00006526

0.00000000

0.00001521

0.00000000

0.00000877

23

0.00002092

0.00000000

0.00000356

0.00000000

0.00000000 0.00000427

24

0.00006531

0.00000000

0.00001487

0.00000000

0.00000000

25

0.00000203

0.00000000

0.00000000 0.00000747 0.00000130 0.00000339

26

0.00005952

0.00000000

0.00000000 0.00001332

0.00000824

27

0.00000321

0.00000000

0.00000000 0.00000997

0.00000000 0.00000453

28

0.00005996

0.00000000

0.00000000 0.00001456

0.00000000

29

0.00002141

0.00000000

0.00000000

0.00000000

0.00000182 0.00000478

30

0.00006538

0.00000000

0.00000000

0.00000000

0.00000848

0.00001475 0.00001132

0.00000900

0.00000000

0.00000000 0.00000123

0.00000857

0.00000000

0.00000000

0.00000000

0.00000000

0.00000000

0.00000000

0.00000000

0.00000000

0.00000000 0.00000279 0.00000000

0.00000000

0.00000000

0.00000000

0.00000000

0.00000000

50

Chapter 3

Table 3.6 (continued) No.

USA

OHI

31

0.00002240

0.00000000

0.00000000

0.00000000

0.00000000 0.00000578

32

0.00000000

0.00000000

0.00000000

0.00000000

0.00000000

33

0.00000000 0.00000125 0.00000428 0.00001074 0.00000269 0.00000481

34

0.00000000

35

0.00000000 0.00000112 0.00000477 0.00001321

36

0.00000000

0.00001710

0.00001317 0.00001552

0.00000000

37

0.00000000

0.00000479

0.00000134

0.00000000

0.00000062 0.00000600

38

0.00000000

0.00001901

0.00001480

0.00000000

0.00000850

39

0.00000000

0.00000489

0.00000091

0.00000000

0.00000000 0.00000697

0.00001707

EU

CHN

0.00001343 0.00001434

EEC

0.00000775

ROW

0.00000000

0.00000000

0.00000000 0.00000594 0.00000000

0.00000000

the simplex. Under this new set of fji g, I repeat the whole process in the ﬁrst step. Based on the results from the three tables, further adjustments of fji g are made until all signs in table 3.3 are positive, and all signs in tables 3.4 and 3.5 are negative. From the above explanations one can sense that identifying core allocations (or ﬁnding possible cooperative game solutions) and the Lindahl equilibrium in an optimal control problem like (A-1) is time consuming and laborious. In subsequent studies, I execute the ‘‘hybrid’’ Nash equilibrium procedure and use the three tables for incentive checking repeatedly (most of them are not presented in this book). With the help of fast-improving computer technologies, it takes 12 minutes to execute a single round of optimization consisting of 64 ‘‘hybrid’’ Nash equilibrium solutions of the RICE model on a dual-core Xeon 3.0 GHz workstation. The Shapley Value Procedure In the RICE model, the Shapley value is a solution concept that depends on the outcomes of the model. Unless the results are checked with the deﬁnition, one cannot determine whether the solution is the Shapley value or not. Here, I use a guided ‘‘tatonnement,’’ or ‘‘trial-and-error’’ algorithm to search for the Shapley value of the RICE model. Schematic procedures consist of the following iterative steps. ð0Þ First, I pick an initial set of social welfare weights, fji g. By solving ð0Þ the ‘‘hybrid’’ Nash equilibria of the model with this set of weights, vi as deﬁned by (2.15) is obtained.

The RICE Model

51

Second, I calculate the interim Shapley value according to its deﬁnition: " # X ð0Þ 1 ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ ð0Þ S1 ðN; v Þ ¼ v ðNÞ Si1 ðNnf1g; v Þ þ S1 ðNnfig; v Þ N i01 ð3:1Þ ð0Þ

ð0Þ

ð0Þ

ð0Þ

Si ðN; vð0Þ Þ ¼ S1 ðN; vð0Þ Þ þ Si ðNnf1g; vð0Þ Þ S1 ðN j fig; vð0Þ Þ i 0 1 ði ¼ 2; 3; 4; 5; 6; and N ¼ 6Þ ð0Þ

ð0Þ

ð0Þ

ð3:2Þ ð0Þ

Third, I compare vi and Si . If vi > Si , region i’s payoff with ð0Þ fji g is larger than it deserves under the Shapley value, and vice ð0Þ ð0Þ versa. In such a case, I reduce ji and increase those jj ’s with the reverse inequalities. I re-solve the entire spectrum of ‘‘hybrid’’ Nash equið1Þ libria of the model under the new set of social welfare weights, fji g ð1Þ ð1Þ ðkÞ ðkÞ to get vi and Si . Finally, the procedure repeats until vi ¼ Si , for 11 all i, with sufﬁcient numerical precision. When the procedure stops, I note the social welfare weights in the ﬁnal iteration as the Shapley value weights fjS g. In the Shapley value searching procedure, I do not check IR and IC conditions during iterations. According to cooperative game theory, unless the game is convex, there is no guarantee that the Shapley value is deﬁnitely in the core (see Moulin 1988). It is very difﬁcult to test whether the cooperative game deﬁned in chapter 2 is convex or not in RICE. Using the three tables, I check the core property of the obtained numerical Shapley value solution under jS at the end. Because the searching algorithm relies on hierarchical iterations, I use the same ‘‘hybrid’’ coalitional equilibrium procedure in the Lindahl equilibrium searching. The difference here is that the iterative adjustment criterion is based on the above formulas (3.1) and (3.2), not the signs in the three incentive checking tables. The procedure here is equally time consuming. Side-Payment Calculations Different regions have heterogeneous mitigation costs. Without transfer channels, GHG mitigation costs are not equal across regions. As a result, the cooperative game solution of RICE corresponds to efﬁciency condition (2.11) in chapter 2. If transfers are allowed, the solutions would correspond to efﬁciency condition (2.12). As I indicated earlier, (2.12) is a potential Pareto improvement over (2.11).

52

Chapter 3

Modeling an endogenous transfer scheme is doable in RICE as an IA modeling task. However, it would make the model more complicated and deﬂect the cooperative game approach used in this study. The game is the bargaining of initial GHG emission quotas and collective actions on the agreement. If quota trading or side payments were to be implemented, they are the ‘‘third stage’’ (in contrast to the two stages of the cooperative game deﬁned in chapter 2) of strategic interactions among regions. Because a mechanism of side payments is not modeled in RICE, we simply conduct ex-post (ofﬂine) calculations of marginal GHG mitigation costs of regions under various solutions. Then, we assess the magnitudes of possible side payments to achieve equalization of mitigation costs under different solutions.

4

Cooperative Game Solutions and Other Solutions in the RICE Model

4.1 Two Benchmarks: BaU and Open-Loop Nash Equilibrium Scenarios I begin the presentation of simulation results with the most basic scenarios. The business-as-usual (BaU) scenario offers an overview of the scope and range of RICE model outputs. I report regional GDP and GHG emissions (unit is GCE: gigaton of carbon equivalent) in the BaU case with m i ðtÞ 1 0. The trends are plotted in ﬁgures 4.1 and 4.2 respectively.1 The BaU scenario establishes the boundary and scales of state variables in the RICE model. It offers the upper limit of global GHG emissions over time as well. Therefore, it also provides the upper bound of potential temperature increase over time. These results are largely taken as assumptions for further simulations. Experts may have different opinions on the mean values of economic forecasting and the future GHG emissions. IPCC (2007) presents a spectrum of predictions from different IA models based on various assumptions and economic drivers. This research does not investigate them any further. Figure 4.1 is a long-run prediction of regional GDP growth. It is easy to notice that ROW is assumed to have the highest GDP growth potential. USA maintains the highest GDP level throughout the time horizon. On the other hand, the projection on China’s growth potential is on the conservative side; despite this we assume that China has the highest total factor productivity growth. In sum, ﬁgure 4.1 is a scenario of diverse growths. Because regional GHG emissions are declining at different proportionalities to GDP levels (equation (A-6)), ﬁgure 4.2 shows similar patterns as in ﬁgure 4.1, with some signiﬁcant shifts, for regional GHG emissions. RICE predicts that ROW is and will remain the largest GHG emission bloc. In addition, China will be a signiﬁcant

54

Figure 4.1 Regional GDP growth trend (BaU)

Figure 4.2 Regional GHG emissions (BaU)

Chapter 4

Cooperative Game Solutions in the RICE Model

55

Figure 4.3 GHG control rates (Nash)

GHG emitter. All other regions’ GHG emissions grow at much slower paces. Again, different IA models might have very diverse predictions about economic growths, energy intensities, and GHG emissions. I will keep the same assumptions in all solutions in this chapter for consistency purposes. In particular, only social welfare weights are altered in solving the different social planner’s problems. Besides the BaU benchmark, the open-loop Nash equilibrium is also a useful benchmark. Some information from the open-loop Nash equilibrium solution is provided in ﬁgures 4.3 and 4.4. They are regional GHG control rates m i ðtÞ and carbon taxes (¼ shadow price of domestic GHG emissions/shadow price of capital). In this noncooperative and strategic setting, GHG control efforts by regions are minimal. The results are consistent with the well-established economic principle that pollution (a detrimental external effect) is overgenerated and undercontrolled in the decentralized market equilibrium. In this laissez-faire picture, ROW exerts the highest GHG control rates (see ﬁgure 4.3). This result does not appear to be compatible with reality. No one expects that the less developed countries will voluntarily control their GHG emissions outside the realm of international agreements, at least in the foreseeable future. However, the result is consistent with the assumptions made in RICE, where ROW is a proxy single region. Speciﬁcally, self-motivated GHG control rates are mainly

56

Chapter 4

Figure 4.4 Carbon taxes (Nash)

determined by a region’s mitigation cost and climate damage functions. In an open-loop strategic setting, other regions’ actions also affect domestic decisions. Because ROW is the largest contributor of GHG emissions and is the most vulnerable region to climate damages, it should pay more attention to GHG emissions if it is a single decision maker (factually not), just as the model predicted. Carbon taxes, shown in ﬁgure 4.4, reﬂect regions’ ‘‘willingness to pay’’ for GHG mitigation under a noncooperative and strategic environment. They are consistent with regional GHG control rates. They are based on internal trade-offs of GHG mitigation costs and climate damage caused by the emissions, taking other regions’ behavior as given. ROW has the highest carbon taxes, at around $10 to $80 per ton of carbon in current value terms. EEC, which is mainly the former Soviet Union, is the lowest. It is lower than $3 per ton of carbon throughout the time horizon. Again, the pattern of carbon taxes is consistent with the assumptions made in the model. For example, CHN’s GHG control rates are in the middle range among the six regions but its carbon taxes are much lower than the others (with the exception of EEC). This reﬂects the assumption that it is relatively cheap to reduce GHG emission in CHN. One major condition for the model results to hold true in reality is that each region perceives its own (and others’) mitigation cost and climate damage correctly. If they do so and act rationally, the model

Cooperative Game Solutions in the RICE Model

57

results prevail. ROW is a single region in the model. It makes uniﬁed decisions on behalf of numerous countries in the bloc. While the assumption is a modeling necessity, it is an oversimpliﬁcation of the real world. In particular, these heterogeneous countries as a whole do not perceive the future damage in a reasonable way. Therefore, in reality, we do not observe the behavior from ROW (as a single bloc) as predicted by the model. Minding this caveat, we continue our analyses under this assumption. In fact, this assumption is innocuous in most of our studies. In solving a social planner’s problem (A-1), stock externalities are fully internalized. Breakdowns of regions do not matter. If mitigation cost and climate damage estimates are correct, the model outputs reﬂect efﬁcient paths of ROW, as a collective aggregate of many countries. Nevertheless, ROW outcomes in the open-loop Nash equilibrium are biased.2 The open-loop Nash equilibrium is the benchmark for subsequent analysis. On the one hand, it contrasts noncooperative game results with those from the cooperative game, such as the Lindahl equilibrium and the Shapley value solutions; on the other hand, it compares inefﬁcient market equilibrium results with various efﬁcient outcomes from the social planner’s problem (A-1). As the benchmark, the Nash equilibrium has the lowest aggregate GHG control rates (measured as the sum of regional control rates and weighed by regional GHG emissions) and the highest global GHG emissions (except for BaU with m i ðtÞ 1 0). To offer a clear picture, we plot global GHG emission paths of BaU and the Nash equilibrium in ﬁgure 4.5. This ﬁgure will be juxtaposed with other scenarios subsequently. 4.2 Why Conventional Solutions Are Unﬁt for Analyzing International Cooperation The utilitarian and Negishi social welfare weights are widely used in economic models that are formulated as social planner’s problems. Most IA modelers that utilize regionally disaggregated IA models to ﬁnd globally efﬁcient solutions adopt either type of social welfare weights. The utilitarian weights, for their computational convenience and other reasons, are particularly popular. As we will see, modeling results with these assumptions deserve scrutiny. One should be especially careful in applying those assumptions to policy analysis of international cooperation in GHG emission mitigation because of their incentive properties.

58

Chapter 4

Figure 4.5 Global GHG emissions

The RICE model solution under the utilitarian weights ðji 1 1Þ is the outcome of a simple (relative to other solution concepts in this study) optimal control problem. Obtaining a numerical solution is straightforward. I present the key relevant results in ﬁgures 4.6 through 4.8. These results are quite revealing. Under the utilitarian weights, we observe a clear dichotomy between the ‘‘North’’ or ‘‘rich’’ (USA, OHI, EU) and the ‘‘South’’ or ‘‘poor’’ (CHN, EEC, ROW). The ‘‘social planner’’ favors the ‘‘South’’ and asks the ‘‘North’’ to pay a large portion of the GHG mitigation cost. Both GHG control rates and carbon taxes of the ‘‘North’’ are extremely high. They are required to reduce 30 percent to 40 percent of their baseline GHG emissions. The GHG control rates of the ‘‘South’’ are substantially lower than those of the North (see ﬁgure 4.6). To do so, the ‘‘North’’ has to impose carbon taxes at the thousands of dollars per ton of carbon level. In contrast, the carbon taxes of the ‘‘South’’ are small fractions of the North’s (see ﬁgure 4.7). As a result, GHG emissions of USA, OHI, and EU are under stringent control, while ROW and CHN maintain high GHG emission paths (see ﬁgure 4.8). This picture depicts a pattern similar to what the Kyoto Protocol prescribes (before allowing any carbon permit trading), although policy regimes are totally different. The reasons for such striking results are very simple. The instantaneous utility function in RICE is in the form L i ðtÞ logðC i ðtÞ=L i ðtÞÞ.

Cooperative Game Solutions in the RICE Model

Figure 4.6 GHG control rates (utilitarian)

Figure 4.7 Carbon taxes (utilitarian)

59

60

Chapter 4

Figure 4.8 Regional GHG emissions (utilitarian)

When ji 1 1, factor L i ðtÞ before the logarithm function in the utility function ensures ‘‘one man, one vote’’—an essential principle of utilitarianism. The social planner does not weigh any person differently. Because ROW and CHN have large populations (together they count for more than 80 percent of global population), their de facto weights in a six-region model are very high. From the simple model in chapter 3 we learned that in an efﬁcient scheme of stock externality provision, namely, a social planner’s problem, the higher the welfare weights, the lower the mitigation burdens, ceteris paribus. Therefore, the GHG mitigation burdens shouldered by ROW and CHN are much lower in a social planner’s problem weighed solely by population. Accepting all other assumptions of the RICE model, the weights alone can explain these controversial results. The Negishi weights offer another set of polarizing and ‘‘fair’’ (for some) results. The model solution under the Negishi weights is the ‘‘equilibrium’’ with respect to the private good. Because shadow prices of capital are equalized across regions in each period under the timevariant Negishi weights, there are no incentives for capital ﬂows across regions, even if trade of the private good is allowed. This efﬁcient outcome, where each region lives within its own budget constraint, is the Walrasian equilibrium in absence of external effects. Relevant results from the Negishi solutions are summarized in table 4.1 and ﬁgures 4.9 through 4.11. Table 4.1 contains the values of

Cooperative Game Solutions in the RICE Model

61

Table 4.1 Selected values of time-variant Negishi weights Periods

USA

OHI

EU

CHN

EEC

ROW

1 5

2.0764 2.2599

2.3727 2.1223

1.2562 1.3070

0.0580 0.0633

0.1620 0.1536

0.0748 0.0940

10

2.0877

2.2055

1.3509

0.0783

0.1660

0.1117

15

1.9189

2.3123

1.3856

0.0881

0.1742

0.1209

20

1.7849

2.4052

1.4142

0.0933

0.1784

0.1241

25

1.6809

2.4836

1.4378

0.0949

0.1796

0.1233

30

1.5999

2.5496

1.4573

0.0942

0.1788

0.1201

Figure 4.9 GHG control rates (Negishi)

Negishi weights in selected periods. The Negishi weights of the North regions are much higher than 1, the utilitarian benchmark. The South regions are much lower than 1. The social planner ‘‘favors’’ the North over the South in GHG emission mitigations under the Negishi weights. With such weights, GHG control rates of the North are lower than the South (see ﬁgure 4.9) and regional carbon taxes are equalized (ﬁgure 4.10). Regional GHG emissions under the Negishi weights are shown in ﬁgure 4.11. The results, again, are predictable from the relationship between social welfare weights and GHG mitigation burden sharing. Regions with lower weights are required to pay more for GHG mitigations.

62

Figure 4.10 Carbon taxes (Negishi)

Figure 4.11 Regional GHG emissions (Negishi)

Chapter 4

Cooperative Game Solutions in the RICE Model

63

With the Negishi weights, opportunity costs of marginal mitigation efforts are equal across regions. Therefore, it is not possible to ﬁnd a region where a unit of GHG emission can be reduced more cheaply than in other places. The same model under the utilitarian and Negishi weights might lead to diametrically different results. If policymakers argue with one another based on different assumptions of social welfare weights, they would never achieve consensus. One can argue that ‘‘prince’’ and ‘‘pauper’’ should have an equal say in global environmental issues; others might insist that the ‘‘rich’’ should not transfer a penny to the ‘‘poor’’ for handling common environmental concerns. In sum, the distributional gap that separates utilitarian and Walrasian (Negishi) approaches is too great in dealing with global environmental problems. As I have argued, only global cooperation can achieve efﬁciency in providing stock externalities. Whether the utilitarian and the Negishi solutions can be adopted as starting points for international negotiation is a legitimate question to ask. The question can also be asked from a more technical angle. I have elaborated extensively on the connection between the cooperative game of stock externality provision and the feasibility of international cooperation in the previous chapter. A cooperative solution (a social planner’s problem) is a basis for forming a coalition only when it meets incentive criteria. Otherwise, it is a nonstarter. Therefore, let us inspect individual rationality (IR) and incentive compatibility (IC) of the utilitarian and Negishi outcomes. The incentive checking results for the utilitarian weights are in chapter 3 (tables 3.3 through 3.5), where I use them to illustrate incentive checking procedures. To save space, I do not present them again; instead, I refer the reader to the tables in chapter 3 directly. Table 3.3 shows that four regions (USA, OHI, EU, and EEC) prefer the Nash equilibrium to the grand coalition (see row 1). Thus the grand coalition under the utilitarian weights does not meet the basic IR condition. Furthermore, USA is willing to form ﬁve subcoalitions under the utilitarian weights. They include {USA, OHI, EU, EEC} (row 6), {USA, OHI, EU} (row 8), {USA, OHI, EEC} (row 14), {USA, OHI} (row 16), and {USA, EEC} (row 30). No other coalition options are attractive to USA, and USA prefers the Nash equilibrium to all of these. The OHI situation is similar to USA. All of OHI’s preferred coalitions are with USA. Choices for EU and EEC are mixed. While they do not like the grand coalition, they prefer a dozen or so subcoalitions to the Nash equilibrium. Finally, CHN and ROW are clear winners in the grand coalition

64

Chapter 4

under the utilitarian weights. In addition, ROW prefers being in any coalitions, rather than interacting strategically with other regions. CHN is almost the same as ROW except for the {CHN, ROW} (row 59) coalition, in which it is a loser. Table 3.4 reveals blocking opportunities for the grand coalition under the utilitarian weights. The grand coalition can be blocked by eleven ﬁrst-category subcoalitions (rows 6, 8, 14, 16, 22, 24, 30, 38, 40, 46, and 54). Clearly, the utilitarian solution is not in the core. Four out of six regions do not like the initial GHG mitigation quota under the utilitarian weights. No international agreement can be reached based on such an arrangement. Or, the utilitarian outcome cannot be a cooperative game solution. Table 3.5 demonstrates a mixed picture too. There are numerous opportunities for payoff improvements by retreating from a large coalition and forming a smaller one. The result indicates that it is very difﬁcult to forge a coalition of GHG mitigation at any level based on the utilitarian principle. The utilitarian weights certainly do not lead to the Lindahl equilibrium without endowment transfers. The incentive checking reinforces the conclusions one can draw from the dichotomized solution paths of state and control variables (ﬁgures 4.6 through 4.8). That is, the utilitarian outcome is radical and infeasible institutionally. One might suggest that we start from the utilitarian solution and then use a transfer scheme to achieve consensus for global cooperation. The thorny points are: Who should pay whom? What are the magnitudes of such transfers? One superﬁcial and paradoxical argument is that because the North is an ‘‘unwilling’’ partner here, the South should ‘‘bribe’’ the North into the grand coalition. In addition, because the South has much more at stake from climate change, they should pay extra in the form of transfers from the utilitarian baseline. The argument seems valid. Regrettably, it is established on the false premise of a utilitarian rule. Another suggestion of transfers is in the opposite direction. Because marginal GHG mitigation costs in ROW are much lower than they are in USA, OHI, and EU, these North regions should shift partial mitigation burden to ROW. Meanwhile, they would compensate ROW for their extra efforts. Several policy options currently on the table, such as joint implementation ( JI) and clean development mechanism (CDM), are based on such thinking. Again, the magnitude of such technological transfers would be misleading if the calculations are based on the utilitarian baseline. In an unequal world, the marginal utility of a dol-

Cooperative Game Solutions in the RICE Model

65

lar (or income) is higher in poor regions. Under the utilitarian weights, the social planner would like to see monies ﬂowing from ‘‘rich’’ to ‘‘poor.’’ Those are solely social welfare–enhancing and redistributional measures. In an economic-environmental model, private sectors, aggregate or disaggregate, are dominantly larger than environmental impacts. Therefore, endogenous capital ﬂows generated from a social planner’s problem with utilitarian weights are largely for redistribution purposes. It is not surprising that IA models based on utilitarian weights can render large amounts of transfers. The implication of utilitarian weights on transfer issues is ambiguous. Both sides (rich to poor or poor to rich) can argue their polarizing positions unyieldingly. The discussions have not touched on the issue of nontransferable utilities. If a dollar in USA is not a dollar in ROW, transfer issues are further complicated. Our general conclusion here is that because the utilitarian arrangement is incompatible with the incentives of a majority of regions, policy debates about international cooperation of GHG mitigation under this assumption are usually futile. The Negishi weights pose another set of interesting incentive problems. Tables 4.2 through 4.4 present the information. First and foremost, the Negishi solution is not in the core. In table 4.2, CHN and EEC do not like the grand coalition. They prefer all subcoalitions to the grand coalition. In particular, the grand coalition is blocked by the {CHN, EEC} subcoalition (row 58). Table 4.3 shows that CHN and EEC prefer the Nash equilibrium to all coalitions except for the {CHN, EEC} subcoalition, where CHN is worse off than the Nash equilibrium (row #58). In addition, ROW’s incentives to join subcoalitions are mixed. CHN and EEC have many welfare-improving opportunities by withdrawing from the grand coalition under the Negishi weights. The results in table 4.4 indicate that the Negishi weights do not correspond to the Lindahl equilibrium, though they map to the Walrasian equilibrium in the absence of externalities. The Negishi outcome does not give any reasons for transfers. Because carbon taxes are equalized across regions, policymakers cannot ﬁnd a more cost-effective place to shift GHG mitigation burdens. If there is no redistribution agenda, the second fundamental theorem of welfare economics prevents any social welfare-improving transfers here. However, this seemingly ideal outcome is not ﬁt for international cooperation for GHG mitigation because it is incentive incompatible. The incentive properties of the Negishi outcome can be explored by examining regional carbon taxes. In the Nash equilibrium (ﬁgure 4.4),

66

Chapter 4

Table 4.2 Welfare changes from grand coalition (Negishi) No.

USA

OHI

EU

CHN

EEC

ROW

1 2

þ

þ

3

4

þ

5

þ

6

þ

7

8

9

þ

þ

10

þ

þ

11

þ

12

þ

13

þ

14

þ

15

16

17

USA

OHI

EU

CHN

EEC

ROW

33

þ

þ

34

þ

þ

35

þ

36

þ

37

þ

38

þ

39

40

41

þ

þ

42

þ

þ

43

þ

44

þ

45

þ

46

þ

47

48

þ

þ þ

18

þ

19

þ

20

þ

21

þ

22

þ

23

24

25

þ

þ

26

þ

þ

27

þ

28

þ

29

þ

30

þ

31

32

No.

49 50

51 52

þ

þ

þ

þ

þ

þ

53

þ

54

þ

55

56

57

þ

þ

58

þ

þ

59

þ

60

61 62

63 64

þ

Cooperative Game Solutions in the RICE Model

67

Table 4.3 Welfare changes from Nash equilibrium (Negishi) No.

USA

OHI

EU

CHN

EEC ROW

1

þ

þ

þ

2

þ

þ

þ

3

þ

þ

þ

4

þ

þ

þ

5

þ

þ

þ

6

þ

þ

þ

7

þ

þ

þ

8

þ

þ

þ

9

þ

þ

þ

þ

þ

þ

12

þ

þ

13

þ

þ

þ

þ

þ

þ

16

þ

þ

17

þ

EEC

ROW

33

þ

þ

þ

34

þ

þ

35

þ

þ

36

þ

þ

37

þ

þ

38

þ

þ

39

þ

þ

40

þ

þ

þ

41

þ

42

þ

þ

43

þ

44

þ

þ

45

þ

46

þ

47

þ

þ

USA

þ

þ

þ

þ

þ

þ

48 þ

18

þ

þ

19

þ

þ

20

þ

þ

21

þ

þ

22

þ

þ

23

þ

þ

24

þ

þ

25

þ

26

þ

27

þ

28

þ

29

þ

30

þ

31

þ

32

CHN

10

14

EU

þ

11

15

OHI

þ

No.

þ

49 50

þ

51 52

þ

þ

þ

þ

þ

þ

þ

þ

53

þ

54

þ

55

þ

þ

þ

56 þ

þ

57

58

þ

59

þ

þ

60 þ

61 62

þ

63 64

þ

68

Chapter 4

Table 4.4 Welfare changes between coalitions (Negishi) No.

USA

OHI

EU

CHN

EEC

ROW

1 2

þ

3

4

5

6

7

8

9

10

þ

þ

11

þ

12

þ

þ

13

þ

þ

14

þ

þ þ

þ þ

þ

þ

þ

þ

þ þ

þ

15

16

17

18

þ

þ

19

þ

20

þ

þ

21

þ

þ

22

þ

þ þ

23

24

25

þ

þ

þ

þ

26

þ

þ þ

þ

þ þ

27

þ

þ

28

þ

þ þ

29

þ

þ þ

30

þ

þ þ þ

31

þ

þ

þ þ

þ

þ

þ

32 33

34

þ

þ

35

þ

36

þ

þ

þ

þ þ

þ

Cooperative Game Solutions in the RICE Model

69

Table 4.4 (continued) No.

USA

OHI

EU

CHN

EEC

37

þ

þ

38

þ

þ þ

39

40

41

þ

þ

þ

þ

42

þ

þ þ

þ

þ þ

43

þ

þ

44

þ

þ þ

45

þ

þ þ

46

þ

þ þ þ

47

ROW

þ

þ

þ

þ

þ

48 49

þ

þ

þ

þ

50

þ

þ þ

þ

þ þ

51

þ

þ

52

þ

þ þ

53

þ

þ þ

54

þ

þ þ þ

55

56 57

þ

þ þ

þ

þ þ

58

þ

þ þ þ

þ

þ þ þ

59

þ

þ þ

60 61 62 63 64

þ

þ þ þ

70

Chapter 4

CHN and EEC have the lowest carbon taxes. Their ‘‘willingness to pay’’ for GHG mitigation, reﬂected in the Nash equilibrium, is lower than others. To raise their carbon taxes to the regionally equalized level in the Negishi outcome, their welfare suffers, as shown in table 4.3. The two regions with the lowest carbon taxes in the Nash equilibrium, namely, CHN and EEC, do not like the Negishi outcome. In RICE, CHN is calibrated as a high energy (and high carbon) intensity economy. Comparable GHG mitigation would cost less in CHN at lower quantities. To equalize carbon taxes across regions, it requires CHN to reduce its GHG emissions substantially. EEC is the least vulnerable region to climate damage due to its latitude. Equalized carbon taxes in the Negishi outcome are too costly for these two regions. In other words, they do not want to pay as much as other regions because they believe that either mitigation costs are too high or climate damage is not serious. Therefore, they are the losers in the Negishi scheme and there is no compensation mechanism for their losses. In summary, the two most popular social welfare weights in IA modeling have intrinsic defects in their interpretation. They do not meet basic IR and IC conditions needed for forming a feasible international coalition. If a model assumes nontransferable utility, the utilitarian and Negishi weights are nonstarters for analyzing international cooperation issues. Even under the strong assumption of transferable utility, calculations of magnitude and directions of transfers can be misleading. The analysis here shows that a solution of the social planner’s optimal control problem (often mislabeled as a cooperative solution) does not necessarily represent a cooperative game solution. Applying IA models to the analysis of international cooperation of GHG mitigation, modelers have to seek more appropriate solution concepts that are consistent with incentive compatibility of all regions. That is, we should seek cooperative game solutions as deﬁned in chapter 2.3 4.3

Cooperative Game Solutions of the RICE Model

4.3.1 The Lindahl Equilibrium Using the algorithm described in chapter 3, I identify a numerical solution of the Lindahl equilibrium. It is the solution of (A-1) under a set of social welfare weights that has sweeping and homogeneous signs in all three incentive checking tables, namely, the three tables corresponding to tables 4.3, 4.4, and 4.5 for the Negishi weights. For this Lindahl

Cooperative Game Solutions in the RICE Model

71

Figure 4.12 GHG control rates (Lindahl)

equilibrium solution, all entries in a table like table 4.3 have ‘‘’’ signs; all entries in the table like table 4.4 have ‘‘þ’’ signs; and all entries in the table like table 4.5 have ‘‘’’ signs. Because of their sign homogeneity, I will not print them to avoid redundancy. The Lindahl social welfare weights are j L ¼ ðj USA L ; jOHI L ; jEU L ; jCHN L ; jEEC L ; jROW L Þ ¼ ð1:574; 1:805; 1:094; 0:202; 1:229; 0:096Þ. Some key results are plotted in ﬁgures 4.12 through 4.14. As ﬁgure 4.12 shows, regional GHG control rates (control variables) range from 4 percent to 8 percent in the beginning periods and 7 percent to 16 percent in the peak periods across regions. They are more conservative (lower) and clustered than the utilitarian and Negishi scenarios. Because of assumptions of heterogeneous regional growths and carbon intensities, these very different control rate patterns across scenarios do not change emission level rankings (comparing ﬁgures 4.6, 4.9, and 4.13). Naturally, the higher the control rate, the more the GHG emission reductions from the BaU emission benchmark in ﬁgure 4.2. Carbon tax patterns in ﬁgure 4.14 are very interesting. For USA, OHI, and EU, carbon tax levels are much lower than their utilitarian contributions: from $1,500 to $2,500 per ton of carbon down to $200 to $350 per ton of carbon. They are more in line with the uniﬁed Negishi carbon taxes. The ranking of their ‘‘willingness to pay’’ for carbon taxes

72

Chapter 4

Figure 4.13 Carbon taxes (Lindahl)

is a good reﬂection of their attitude toward the Kyoto Protocol, coincidentally or not. CHN and EEC have the lowest carbon taxes in the Nash equilibrium. They are the most unwilling partners in the Negishi scheme. Here, to encourage their participation, they pay the least and much less than their Negishi dues (again, comparing ﬁgure 4.10 and ﬁgure 4.14). ROW’s high GHG control rates and high carbon taxes are due to the ‘‘single decision maker’’ assumption and its calibrations. The most interesting aspect of the Lindahl outcome is its incentive compatibilities. We can recite the cooperative game structure outlined in chapter 2 verbatim in the context of the RICE model and obtain this set of results. The cooperative game is played as a one-shot bargaining process of social welfare weights. Once fjiL g are agreed on, the social planner’s problem (A-1) with this set of weights is solved. Regions follow the optimal paths determined by (A-1). The regional payoffs in this grand coalition are higher than with any other coalitional arrangement. The outcomes from fjiL g are the most desirable to all regions. These strong results are exactly what the homogeneous signs in the three incentive checking tables (not shown) reveal. The arrangement is efﬁcient because further adjusting fji g will not lead to a Pareto improvement. Regions also cannot ﬁnd any other strategic positions, namely, forming coalitions other than the grand coalition under fjiL g, to be better off.

Cooperative Game Solutions in the RICE Model

73

Figure 4.14 Regional GHG emissions (Lindahl)

Under the parametric assumptions of the model, if regions have perfect foresight about the future (open-loop structure) and collaborate rationally with one another in the bargaining process, they should be able to come to the collective decision on j L and the optimal paths of control and state variables in (A-1) with this j L . Most important, the outcome is consistent with every region’s incentives. Most anomalies in the utilitarian and Negishi scenarios are overcome in the Lindahl outcome. Though the Lindahl equilibrium solution has the most desirable properties for a cooperative outcome, we must not forget that we are dealing with a complicated stock externality provision problem. The key questions include: How strong are the incentives that bind regions together? Is there a ‘‘self-enforcing’’ agreement in the presence of externalities? These questions are especially imperative in the context of IEAs on GHG mitigations. Wherever there are externalities, free-riding behaviors are ubiquitous. Free-riders expect to be gaining from others’ collective actions and hope to avoid being caught. In this research I am not concerned with the mechanism design aspect of externality issues, as in some highly stylized models in which agents simply do not want to freeride. Instead, I want to investigate the likelihood of holding the grand coalition in a more realistic IA model. In chapter 2, I speciﬁed the following game rule: if a region defects, other regions will play Nash

74

Chapter 4

thereafter. With this simplistic and ‘‘harsh’’ rule, the IR condition can hold the grand coalition together. However, such drastic reactions are not consistent with international protocol and probably not realistic. Therefore, I want to see what else can bind the grand coalition together without threatening to dissolve the grand coalition completely. I have created a table that is a mirror reﬂection of table 4.2 for this Lindahl solution. Table 4.5 shows regional welfare changes between the grand coalition and staying outside the coalition while strategically interacting with the coalition. If a region gains by defecting from the grand coalition and staying out of any coalition, a ‘‘þ’’ sign is recorded in the cell; if it loses, a ‘‘’’ sign is in the cell; if a region is in a ﬁrstcategory subcoalition, the cell is blank. That is, table 4.5 reports regional welfare changes between the ﬁrst row and ‘‘0’’ cells in the design matrix (table 3.2). The results are very interesting. If the other ﬁve regions remain in the coalition and only one region defects, the defector always gains. This can be seen from rows 2, 3, 5, 9, 17, and 33 of table 4.5. The phenomena seem consistent with free-riding behavior. Here, a single freerider gains from others’ continuing cooperation and nonretaliation. However, this result is established on the assumption that other agents stick to their original commitments in the grand coalition and do not react to the defection. Here, the defector interacts with the 5-member coalition strategically. The defector tries to beneﬁt from others’ collective efforts; coalition members also adjust their control variables to accommodate the defection. But the cooperation rule inside the 5member coalition remains the same (unaltered social welfare weights). Whether a defector could actually gain from such a strategic situation is not a sure thing. In the RICE model, a single defector will gain from the situation in which the other ﬁve regions stick with their welfare weight share in j L and adjust their control variable paths. When more than one region defects simultaneously, payoff-gaining opportunities disappear rapidly along with the number of defectors. First of all, no defector, except for ROW in row 20, can gain if there are more than two defectors simultaneously. Second, if there are two defectors, at least one is a loser. The grand coalition with the Lindahl weights is much more stable than it looks on the surface. A region would have second thoughts about defection if it knows that other regions might have similar motives. If they put the intention into action, they would end up losing together. Such a coalition is very stable and nearly ‘‘self-enforcing,’’ though not completely worry-free. We

Cooperative Game Solutions in the RICE Model

75

Table 4.5 Welfare change from grand coalition at outside (Lindahl) No.

USA

OHI

EU

CHN

EEC ROW

No.

USA

33

þ

þ

34

35

þ

þ

36

37

38

39

40

41

þ

42

43

44

45

46

47

48

49

þ

1 2 3

þ

4

5

þ

6

7

þ

8

þ

þ

9 10

11

þ

12

13

þ

14

15

16

þ

þ

þ

17 18

19

20

21

þ

22

23

24

25

26

27

28

29

30

31

32

þ

þ

OHI

EU

CHN

EEC

ROW

þ

50

51

52

53

54

55

56

57

58

59

60

61

62

63

64

76

Chapter 4

should be quite content with the Lindahl results, knowing that this is a game of stock externality provision where ‘‘free-riding’’ should be ubiquitous! In contrast to the Lindahl equilibrium, we report the same welfare changes for the utilitarian weights in table 4.6. In this table, ‘‘þ’’ signs outnumber ‘‘’’ signs 2 to 1. The distribution of the signs is along the region line. Apparently, four regions do not like the grand coalition and want to defect all the way (this outcome does not and cannot come from the cooperative game). The table also provides a counterexample that a single defector (CHN or ROW here) is worse off all the time. Here, two regions have no desire to free-ride in an efﬁcient scheme of externality provision. Table 4.6 further demonstrates that the utilitarian results are irrational from an incentive perspective. One important observation (one should draw from common sense) is that in obtaining the core allocations even the Lindahl equilibrium does not require unilateral transfers in a cooperative scheme, if stock externalities cause damage to all regions. However, it is in all regions’ interests to provide externalities efﬁciently through equalization of mitigation costs. Such an efﬁcient outcome, as indicated in chapter 2, requires transfers. 4.3.2 The Shapley Value Another important solution concept of the cooperative game is the Shapley value. As indicated earlier, the Shapley value in RICE is deﬁned heuristically. The algorithm introduced in chapter 3 incorporates the formal deﬁnition of the Shapley value and the measurement of welfare changes in the RICE model. The calculation of the Shapley value is to ﬁnd a j S on the simplex such that every region has equal gains (measured in ‘‘utils’’) by joining the grand coalition from outside the 5-member subcoalition. The calculation results are contained in table 4.7. The Shapley social welfare weights are in the ﬁrst row. The remaining ﬁve rows are various numerical results of the Shapley value calculations. The Shapley value vector, calculated in original units of the model’s utility functions (in ‘‘utils’’), is in the second row. For clear reading, I rescale the vector to sum to six in the third row, and rescale the Shapley value in percentage to show each region’s relative share in the fourth row. Finally, as in any numerical exercises, I report the numerical precision measurements when I stopped iteration procedures in the ﬁfth and sixth ðmÞ rows. They are the absolute and relative differences between vi and ðmÞ Si as in (3.1) and (3.2) when I terminate the iterations at round m.

Cooperative Game Solutions in the RICE Model

77

Table 4.6 Welfare change from grand coalition at outside (utilitarian) No.

USA

OHI

EU

CHN

EEC ROW

No.

USA

33

þ

34

þ

35

þ

þ

36

þ

þ

37

þ

38

þ

39

þ

þ

40

þ

þ

41

þ

þ

42

þ

þ

43

þ

þ

þ

44

þ

þ

þ

45

þ

þ

46

þ

þ

47

þ

þ

þ

48

þ

þ

þ

49

þ

1 2 3

þ

4

þ

5

6

7

þ

8

þ

þ

9 10

þ

11

þ

þ

12

þ

þ

13

þ

14

þ

15

þ

þ

16

þ

þ

þ

17 18

þ

19

þ

þ

20

þ

þ

21

þ

22

þ

23

þ

þ

24

þ

þ

25

þ

þ

26

þ

þ

27

þ

þ

þ

28

þ

þ

þ

29

þ

þ

30

þ

þ

31

þ

þ

þ

þ

þ

þ

32

þ

OHI

EU

CHN

EEC

ROW

þ

50

þ

þ

51

þ

þ

þ

52

þ

þ

þ

53

þ

þ

54

þ

þ

55

þ

þ

þ

56

þ

þ

þ

57

þ

þ

þ

58

þ

þ

þ

59

þ

þ

þ

þ

60

þ

þ

þ

þ

61

þ

þ

þ

62

þ

þ

þ

63

þ

þ

þ

þ

64

þ

þ

þ

þ

78

Chapter 4

Table 4.7 Shapley value calculation results USA Weight Shapley value

2.1264 0.00519

OHI 1.7569 0.00257

EU 0.9302 0.00392

CHN 0.1431 0.00782

EEC 0.9523 0.00281

ROW 0.0911 0.03381

Scaled Shapley value

0.5549

0.2745

0.4192

0.8365

0.3001

3.6148

Shapley value share (%)

9.2479

4.5749

6.9863

13.9423

5.0009

60.2477

Absolute error (0.0001)

1.94

0.74

1.29

0.63

0.75

0

Relative error (%)

3.743

2.863

3.292

0.811

2.679

0.0003

Judging from the relationship between social welfare weights and GHG mitigation obligations, USA is the most favored region here because it has the largest share of social welfare weights. ROW’s weight is the lowest among the six regions and also the lowest across the scenarios we already investigated. In contrast, ROW has a dominating share in the Shapley value vector, which is almost 1.5 times as large as all other ﬁve regions together; next in the ranking is CHN and then USA, with OHI at the bottom and EEC nearby. The ranking in the Shapley value vector here is consistent with economic interpretation of the Shapley value (to be elaborated shortly). Finally, given the complexity of the model and the time-consuming nature of calculations, we should be content with iteration accuracies. Final rounds of searching were based on adjusting the fourth digits of the social welfare weights and absolute errors at the 104 level. Marginal gains from continuing the incremental searching in a small neighborhood in 5dimensional space are very limited. Optimal paths of regional GHG emission control rates, carbon taxes, and GHG emissions of the Shapley value solution are plotted in ﬁgures 4.15 through 4.17. EEC’s GHG control rate and carbon tax are lowest among the regions. To gain equally with other regions, EEC, which suffers the lowest climate damage, would pay the least for the grand coalition. ROW’s GHG emission level and control rate are the highest. This outcome is consistent with the calibration of the model. For a region with the highest baseline GHG emissions, ROW has to maintain the highest GHG control rates to offer ‘‘spillover’’ gains to other regions. In addition, its control rates are sufﬁciently high such that other regions might have equal gains by joining the grand coalition. Another noteworthy observation is that EU has the highest carbon tax and USA has the lowest among the North. Again, the ranking pattern is consistent with their revealed ‘‘willingness to pay’’ in the post-Kyoto period.

Cooperative Game Solutions in the RICE Model

Figure 4.15 GHG control rates (Shapley)

Figure 4.16 Carbon taxes (Shapley)

79

80

Chapter 4

Figure 4.17 Regional GHG emissions (Shapley)

The Shapley value outcomes bear resemblances to those of the Lindahl equilibrium (comparing the two sets of graphs of GHG emission control rates and carbon taxes). Therefore, the logical step to take here is to check IR and IC conditions of the Shapley outcome. The results are shown in tables 4.8 through 4.10. Table 4.8 contains ‘‘’’ signs only. In the Shapley value solution, no region can gain anything by defecting from the grand coalition and forming a subcoalition. That is, the grand coalition under the Shapley weights cannot be blocked by any subcoalitions. Clearly, the Shapley value is in the core. In table 4.9, all signs in the ﬁrst row are positive. All regions are better in the grand coalition than in playing the open-loop noncooperative Nash game. Thus the Shapley value solution meets the IR condition. These results suggest that the numerical solution of the Shapley value can be generated by the cooperative game deﬁned in chapter 2. There are mixed signs in tables 4.9 and 4.10. Evidently, the Shapley value solution might run into difﬁculties in coalition formation due to those signs. Alternatively, the incentive properties of the Shapley value solution are not as strong as the Lindahl equilibrium. The Shapley value solution here is a heuristic solution from a normative perspective. The algorithm is based on one property of the Shapley value. In fact, the outcome is consistent with the strategic posture of regions in international negotiation and bargaining of climate treaties,

Cooperative Game Solutions in the RICE Model

81

Table 4.8 Welfare change from grand coalition (Shapley) No.

USA

OHI

EU

CHN

EEC ROW

OHI

EU

CHN

EEC

ROW

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

1 2

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

USA

48

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

No.

49 50

51 52

53

54

55

56

57

58

59

60

61 62

63 64

82

Chapter 4

Table 4.9 Welfare change from Nash equilibrium (Shapley) No.

USA

OHI

EU

CHN

EEC

ROW

No.

1

þ

þ

þ

þ

þ

þ

2

þ

þ

þ

þ

3

þ

þ

þ

þ

4

þ

þ

þ

5

þ

þ

þ

þ

6

þ

þ

þ

þ

7

þ

þ

þ

8

þ

þ

þ

9

þ

þ

þ

þ

10

þ

þ

þ

11

þ

þ

þ

12

þ

þ

13

þ

þ

þ

14

þ

þ

þ

15

þ

þ

16

þ

17

þ

þ

þ

þ

þ

þ

OHI

EU

CHN

EEC

ROW

33

þ

þ

þ

þ

þ

34

þ

þ

þ

þ

35

þ

þ

þ

36

þ

þ

þ

37

þ

þ

þ

38

þ

þ

þ

39

þ

þ

40

þ

þ

41

þ

þ

þ

42

þ

þ

þ

43

þ

þ

44

þ

45

þ

þ

46

þ

þ

47

þ

þ

þ

þ

þ

þ

þ

þ

48 þ

þ

þ þ

18

þ

þ

19

þ

þ

þ

20

þ

þ

21

þ

þ

þ

22

þ

þ

þ

23

þ

þ

24

þ

25

þ

þ

þ

26

þ

þ

27

þ

þ

28

þ

29

þ

þ

30

þ

31

þ

32

þ

USA

þ

49 50

þ

51 52

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

53

þ

þ

54

þ

þ

55

þ

þ

þ

56 þ

þ

57

þ

þ

58

þ

þ

59

þ

þ

60 þ

61 62

63 64

þ

þ

Cooperative Game Solutions in the RICE Model

83

Table 4.10 Welfare changes between coalitions (Shapley) No.

USA

OHI

EU

CHN

EEC

ROW

1 2

3

4

5

6

7

8

9

10

11

12 13

14

15

16

17

18

19

20

21

22

23

24

25

26

þ

27

28

þ

29

30

31

32 33

34

þ

35

36

þ

84

Chapter 4

Table 4.10 (continued) No.

USA

OHI

EU

CHN

EEC

37

38

39

40

41

42

þ

43

44

þ

45

46

47

ROW þ

þ

þ

þ

48 49

50

þ

51

52

þ

53

54

55

þ

þ

56 57

58

þ

þ

59

60 61 62 63 64

þ

Cooperative Game Solutions in the RICE Model

85

under the parametric assumptions of RICE. The issue here is how do we interpret this numerical result and what are its implications. First, the Shapley value can be interpreted as relative contributions by regions to potential climate damages. Calculated Shapley value share ranking is: ROW (60.25%), CHN (13.94%), USA (9.25%), EU (6.98%), EEC (5.00%), and OHI (4.57%). In the BaU scenario, the ranking of GHG emission levels at T ¼ 30, from high to low, is: ROW, CHN, USA, EU, EEC, and OHI (see ﬁgure 4.2). In the model calibration, a unit of temperature increase causes different levels of damage across regions. The ranking of climate damage, from high to low, is: ROW, CHN, EU (¼ OHI), USA, and EEC (see appendix 2, GAMS code of RICE). While there is one instance of crossover among regions in damage speciﬁcation, all other rankings are consistent with the Shapley value share ranking. In all three rankings, ROW ranks on the top and the runner-up is a distant second. In the ‘‘market’’ of detrimental stock externalities, the six regions can be viewed as oligopolistic suppliers. The Shapley value share of each region is an indicator of its market power. This interpretation is similar to that of Wolak and Kolstad 1988, in which the Shapley value is used to measure monopoly powers of coal suppliers. Here, ROW has the dominant ‘‘market power’’ and CHN is an important player. Second, from an externality provision perspective, the Shapley value can be treated as relative bargaining power in the negotiation of costeffective GHG mitigation treaties. The largest contributor and most severe sufferer from GHG emission should have the most powerful voice in GHG mitigation treaties. This interpretation is similar to that of Shapley and Shubik 1954, in which the Shapley value is used to explain members’ power in a committee. In any grand coalitions of GHG mitigation, ROW and CHN are power players. Under both interpretations, ROW is a dominant power.4 ROW and CHN together have a near 75 percent Shapley value share. One direct implication of this Shapley value solution is that without ROW and CHN’s participation, any GHG mitigation cooperation would not be effective. Such a situation would be like a world crude oil market without OPEC’s participation. Among North regions, USA is the most powerful one by having the largest Shapley value share. If USA stays out of a coalition, then cooperation is also ineffective. From a normative perspective, the Shapley value solution is a ‘‘fair’’ outcome with incentive compatibility. USA has high GHG emission levels and relatively low climate damage. The Shapley value weight of

86

Chapter 4

Table 4.11 Comparison of social welfare weights USA

OHI

EU

CHN

EEC

ROW

Utilitarian Negishi*

1 2.0958

1 2.2633

1 1.3184

1 0.0675

1 0.1587

1 0.0963

Lindahl

1.574

1.805

1.094

0.202

1.229

0.096

Shapley

2.1264

1.7569

0.9302

0.1431

0.9523

0.0911

* They are 50-period discounted averages of time-variant weights.

USA is much higher than any other solution concepts we have investigated. This ensures a lower GHG mitigation burden for USA. Since ROW is the highest GHG emitter and suffers the most from climate damage, its Shapley value weight is the lowest among all scenarios. On balance, the cooperative game played out this way is fair to all regions. In summary, the two cooperative game solutions (i.e., the Lindahl equilibrium and the Shapley value), offer new insights into IA modeling. Desirable properties of these two solution concepts have long been proven theoretically. Empirically, as I have shown here, they demonstrate better qualities in incentives: the key to international agreements on GHG mitigation. The difﬁculties in employing these two superior solution concepts in IA modeling are to ﬁnd them. In this research, I show that ﬁnding such solutions is doable, though difﬁcult. 4.4

Synthesis

After presenting conventional and game-theoretic solutions of RICE separately, I juxtapose the results from these solution concepts and compare them with one another in this section. Much of the painstaking numerical work is to identify the points on the simplex of social P welfare weights, namely, selecting a vector j on S ¼ fji j ji ¼ 6g. The weight searching results are summarized in table 4.11. Fundamentally diverse principles lead to the different points on the simplex. The utilitarian weight is based on a simple ‘‘one man, one vote’’ idea. The Negishi weight maps to the Walrasian equilibrium with respect to the private good in the model. The Lindahl weight requires the strongest incentive constraints and each region has the maximal ‘‘willingness to pay’’ for its GHG mitigation contribution. The Shapley value weight guarantees that each region has equal gains by joining the

Cooperative Game Solutions in the RICE Model

87

Figure 4.18 Global GHG emissions

grand coalition of six. Finally, the former two are conventional modeling solutions and the latter two are cooperative game solutions. Using the utilitarian weight as a numerical benchmark ðji 1 1Þ, USA and OHI are consistently weighted more heavily in other solution concepts; CHN and ROW are consistently lighter. Under Lindahl and Shapley value weights, EU and EEC are near their utilitarian benchmarks. The reasons and consequences of such numerical diversities in weights have been analyzed in the previous section. It is interesting to see the impacts of these diverse social welfare weights on aggregate GHG emissions. In ﬁgure 4.18, global GHG emissions in these four solution concepts, along with the BaU and Nash equilibrium, are plotted in one graph. An immediate observation is that the GHG emission paths of the Negishi, Lindahl, and Shapley value weights are almost indistinguishable from one another. In addition, they lie roughly in the middle between the Nash equilibrium and utilitarian paths. Though not plotted, similar patterns appear in global GHG concentration and temperature changes. The utilitarian solution results in the most stringent GHG emission reduction. This outcome is very revealing. All four solution concepts in sections 4.2 and 4.3 are efﬁcient. But aggregate paths of optimal solutions of state variables are not invariant to the weights selected. The problem of optimal provision of stock externality does not have the unique and

88

Chapter 4

invariant solution path for the externality ex ante. The intuition behind this conclusion is apparent: if more weights were placed on the regions with the most severe climate damages, it would result in more stringent, yet efﬁcient GHG emission paths. For example, the weight j ¼ ð0:2; 0:2; 0:2; 0:2; 0:2; 5Þ would render an efﬁcient global GHG emission path much lower than the utilitarian path. In fact, I plot the aggregate GHG emission path under the above weight in ﬁgure 4.18 for illustration. It is labeled as ‘‘extreme’’ in the graph. Another conclusion from ﬁgure 4.18 is that in order to maintain incentive compatibilities across all regions, less drastic mitigation measures are necessary. The two cooperative game solutions allow more GHG emissions than the utilitarian solution. To keep less vulnerable regions in the coalition, their welfare has to be considered. If the GHG mitigation target is too stringent, everyone has to pay more, even the regions that are least responsible for GHG emissions. Game-theoretic solutions of IA models add another dimension to the solution requirement, namely, the incentive requirement. From an incentive perspective, draconian GHG mitigation measures are not recommended. Because they are inconsistent with major regions’ incentive to join, global cooperation (the grand coalition) is not enforceable. After examining aggregate outcomes, I inspect regional outcomes. Since each region presents itself as a unique and speciﬁc case, I go over them one by one. The regional GHG control rates are control

Figure 4.19 GHG control rates (USA)

Cooperative Game Solutions in the RICE Model

89

variables with respect to the externality, and the carbon taxes are calculated from costate variables of the constraint condition of the externality. I focus on regional paths of these two variables across scenarios. The GHG emission control rates and carbon taxes in different solutions are plotted separately by region in ﬁgures 4.19 to 4.30. In each graph, the paths of the four efﬁcient solution concepts and the Nash equilibrium are plotted. Both GHG emission control rates and carbon taxes of the Nash equilibrium are consistently at the bottoms across regions. These Nash equilibrium trajectories serve as the lower bound for comparing and contrasting with other efﬁcient solutions. USA For USA (see ﬁgures 4.19 and 4.20), both GHG control rates and carbon taxes divide signiﬁcantly across solution concepts due to the utilitarian outcome. In the utilitarian solution, USA has to control more than a quarter of its GHG emissions and pay over a thousand dollars per ton of carbon as carbon taxes. From incentive checking tables 3.3 to 3.5, such heavy burdens are totally unacceptable to USA. If the utilitarian solution is used as a baseline for international cooperation, no transfer scheme can bring USA into the alliance. The initial GHG mitigation quota for USA is too high. In addition, GHG control rates in the two cooperative game solutions and the Negishi solution for USA are very close. The Lindahl

Figure 4.20 Carbon taxes (USA)

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Figure 4.21 GHG control rates (OHI)

equilibrium almost coincides with the Negishi outcome in later periods. By nearly doubling USA’s welfare weights from utilitarian to Lindahl, its mitigation burden reduces disproportionately (comparing carbon taxes in ﬁgure 4.20). This phenomenon indicates strong nonlinearity of the RICE model. It also proves, though indirectly, the irrational nature of the utilitarian solution from USA’s perspective. Finally, USA shares a lighter burden in the Shapley value solution than in the Lindahl equilibrium. The Shapley value solution takes into account that USA has high GHG emission levels and low climate damage among all regions. To allow USA to have the same gains as others, further concession to USA, namely, giving USA higher social welfare weight, is needed. OHI and EU The patterns of GHG control rates and carbon taxes of OHI and EU are very similar to each other (see ﬁgures 4.21 to 4.24). They both endure very high GHG control rates and expensive carbon taxes in the utilitarian outcome, just like USA. In contrast to USA, the two cooperative game solutions (Lindahl and Shapley value) for OHI and EU nearly coincide. Such phenomena are shown in ﬁgures 4.21 and 4.23 as well as in table 4.11. This result indicates that incentives and fairness are not in conﬂict for these two regions. It also implies that OHI and EU are more likely to take a proactive attitude

Cooperative Game Solutions in the RICE Model

Figure 4.22 Carbon taxes (OHI)

Figure 4.23 GHG control rates (EU)

91

92

Chapter 4

Figure 4.24 Carbon taxes (EU)

in international negotiations of GHG mitigations. This point seems consistent with the attitudes and positions of these two regions in preKyoto and post-Kyoto negotiations of international cooperation for GHG mitigations. The Negishi social welfare weights of the three developed regions are slightly higher than their respective Lindahl and Shapley value weights (see table 4.11). Because the Negishi outcome is strongly opposed by CHN and EEC, the three developed regions would not defend the Negishi position. Small sacriﬁces from the Negishi position can bring in strong incentives for the grand coalition. On the other hand, efﬁcient global GHG emission levels of the Negishi solution and cooperative game solutions are very close. In either the Lindahl or the Shapley case, small sacriﬁces or moderate increases in GHG mitigation efforts from the Negishi position by the three developed regions can make the three developing regions willing to participate in global coalitions. CHN CHN is an important player in the mitigation bargaining game and a signiﬁcant GHG emitter. In ﬁgures 4.25 and 4.26, the GHG control rate and carbon tax patterns are the reversals of those of the North regions. GHG control rates and carbon taxes in the Negishi solution

Cooperative Game Solutions in the RICE Model

Figure 4.25 GHG control rates (CHN)

Figure 4.26 Carbon taxes (CHN)

93

94

Chapter 4

Figure 4.27 GHG control rates (EEC)

are the highest among the four efﬁcient solutions. CHN’s per capita income is low. Higher marginal utilities of income lead to a very low social welfare weight for CHN in the Negishi scheme. The low social welfare weight, in turn, leads to a high GHG mitigation contribution. As a result, the heavy GHG mitigation burden of CHN is beyond its tolerance in the Negishi solution. For CHN, the utilitarian solution is the most favored efﬁcient solution for its large population base. In the Lindahl equilibrium, CHN’s GHG control rates and carbon taxes are very close to the utilitarian outcome. In this efﬁcient outcome with the strongest incentive to form the grand coalition, the other ﬁve regions are content with such a low threshold for GHG mitigation contribution from CHN. In several rounds of international negotiations on climate change, developed countries showed such a propensity toward China and other developing countries. However, whether it is wise to ask for no obligations from CHN at all, such as the Kyoto Protocol prescription, is debatable. Finally, CHN might be least satisﬁed with the Shapley value solution on the equality of ‘‘equal gains.’’ Because the utility function is of the form LðtÞ LogðCðtÞ=LðtÞÞ, its scale is larger owing to population. CHN, with large LðtÞ, is thus intrinsically at a disadvantage in a crossregion comparison of ‘‘util’’ changes. To achieve such equal gains,

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95

Figure 4.28 Carbon taxes (EEC)

CHN’s GHG control rates and carbon taxes in the Shapley value solution are relatively high. EEC EEC is an interesting case. In ﬁgures 4.27 and 4.28, conventional solutions and cooperative game solutions are far apart. GHG control rates and carbon taxes are much higher in both utilitarian and Negishi solutions. Because these two conventional solutions are based on distributional rules, regions’ attitudes toward GHG emission and the impacts of climate change on regions are not deciding factors. EEC does not gain in the utilitarian case on population grounds or in the Negishi case on income grounds. An interesting pattern here is that although EEC’s Negishi weight is further away from its utilitarian weight (0.1587 vs. 1) than its cooperative game solution weights (1.13 and 0.9486 vs. 1), its GHG control rates and carbon tax paths under the Negishi weights are closer to the utilitarian paths than the Lindahl and Shapley value outcomes. This observation indicates that other regions’ actions might have a stronger inﬂuence on domestic control efforts than social welfare weights. Comparing with other regions, EEC’s control rates and carbon taxes are moderate even in the utilitarian and Negishi outcomes. Nevertheless, such moderate efforts are not compatible with the incentives of EEC (see tables 3.2, 3.3, 4.2, and 4.3). To induce EEC into the grand coalition with IR and IC constraints, the other members can only require very light GHG mitigation efforts

96

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Figure 4.29 GHG control rates (ROW)

from EEC. The major reason is that EEC’s climate damage is very low. Mitigation costs can easily exceed internal beneﬁts for EEC, even taking into account other regions’ contributions. Therefore, in either Lindahl or Shapley value solutions, EEC has a large share of social welfare weights and low GHG control rates. ROW ROW is the most controversial case. A partial source of potential controversies is from the model calibration. In RICE, ROW is a huge group of many countries, including India and OPEC. It represents the majority of the world’s population and the lowest per capita GDP; it contributes to most of global GHG emissions in the future. Optimistic prediction on ROW’s long-run economic growth is accompanied by the pessimistic forecast of its GHG emissions (see ﬁgures 4.1 and 4.2). Disproportionately large sizes in all aspects make ROW unique. In the context of forming a global coalition, the assumption that ROW acts as a single entity is a highly questionable premise. To make logical arguments, I accept all assumptions on ROW here and only revise them in the discussions in chapter 5. In ﬁgures 4.29 and 4.30, the paths of ROW’s GHG control rates and carbon taxes are a reversal to EEC: they are higher in the two game-theoretic solutions than in the other two efﬁcient solutions. The utilitarian outcome is the most favored by ROW. ROW’s population

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97

Figure 4.30 Carbon taxes (ROW)

base is the deciding factor for the outcome. ROW has the highest de facto weights and thus the lowest GHG mitigation obligations. Since it is not compatible with the incentives of many other regions, this outcome is not feasible. As a heavy GHG emitter and the most severe sufferer of climate change, ROW should pay more than in the utilitarian scheme. ROW’s GHG control rates and carbon taxes are highest in the two cooperative game solutions. Although stringent, the Lindahl equilibrium and Shapley value solutions are fully compatible with ROW’s incentives. ROW (if it behaves like a single decision maker) would participate in either regime voluntarily. This result is mainly due to ROW’s vulnerability to climate change damages and dominantly large GHG emissions. Another interesting observation is that although the control rates and carbon taxes are lower in the Negishi solution than in the cooperative game solutions, ROW still prefers the latter. This phenomenon shows that a region might be willing to pay more if it thinks that the other regions contribute fairly. The Negishi outcome is not a terrible choice for ROW either: it is better off here than in the Nash equilibrium (see table 4.3). Assessing all four efﬁcient solutions, ROW should have the least incentive obstacles in cooperating with other regions (see the signs of ROW columns in the incentive checking tables for the utilitarian and Negishi outcomes).

98

Chapter 4

After inspecting the model solutions region by region, we can conclude that incentive compatibility is one of the most crucial factors in ﬁnding feasible policies for international cooperation on GHG mitigation. If we want to use IA models to assess GHG mitigation IEAs, the game-theoretic approaches are key links between modeling and reality. Our practices here show that IA models, if properly speciﬁed, are capable of tackling such tasks. A necessary condition for meeting IR and IC conditions is that the solution paths are in the core of the cooperative game played by the regions. The Lindahl equilibrium, on the other hand, is a sufﬁcient condition for meeting IR and IC conditions. The Lindahl equilibrium solution obtained here possesses the most desirable incentive properties for international cooperation. Across all regions, the Lindahl equilibrium solution offers moderate GHG control rates and reasonable carbon taxes that are acceptable to all. In contrast with game-theoretic solutions, I further reveal the inappropriateness of the utilitarian and Negishi solutions in addressing policy issues. 4.5

Side Payments toward Equalization of Mitigation Costs

The various cooperative game solutions discussed above are consistent with regions’ participation incentives. If parametric assumptions hold, regions should be able to form a grand coalition with the selected social welfare weights. If there is no transfer scheme, the solutions we chose are ﬁnal. There would be no further Pareto improvement possibilities. The original RICE model has been set up as such. The cooperative game is to decide each region’s GHG mitigation obligation or ‘‘quota.’’ Once the ‘‘quotas’’ are agreed on, all regions fulﬁll their obligations domestically. The private goods in RICE are the regional GDPs. They are all denominated by constant US$. Regional GHG mitigation costs are also measured by units of carbon per US$. In the regional transformation function between private good and GHG emission (equation (A-6)), GHG mitigation costs are measured with the same monetary unit. If one region incurs higher marginal GHG mitigation costs to meet its quota than others, it is possible to further improve regional welfare by shifting the mitigation burden from such a high-cost region to lowcost regions, accompanied with full compensation. The shifting can continue until marginal GHG mitigation costs are equalized across regions. The aggregate GHG emissions would be the same with or without transfers.

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99

Figure 4.31 Marginal GHG mitigation costs (Negishi)

Although such a transfer scheme is not modeled in RICE, we can still examine its effects through side calculations after model solutions. By calculating marginal GHG mitigation cost paths of regions, we can measure the wedge between high and low marginal costs and the room for further welfare gains through ‘‘quota trading.’’ In the RICE model, marginal GHG mitigation costs are given by the following formulas (they are in discrete form in the GAMS simulation): dðCCi ðtÞÞ b1; i m i ðtÞ b2; i 1 ðb2; i þ ð1 b2; i Þm i ðtÞÞ ¼ dm i ðtÞ si ðtÞð1 m i ðtÞÞ 2

ð4:1Þ

ðHere CCðtÞ denotes unit mitigation cost in $/TCE.Þ First we examine a scenario in which side payments are not needed for achieving equalization of marginal mitigation costs. This is the Negishi outcome. After solving the model under the Negishi weights, a side calculation of (4.1) produces ﬁgure 4.31. Marginal GHG mitigation costs of the six regions are very close (but not identical). The reasons for the slight deviation from the ‘‘equalization of marginal cost’’ criterion in ﬁgure 4.31 are mainly numerical. Discrete, instead of continuous, functional forms contribute to numerical errors; numerical errors in calculating Negishi weights also propagate into the system and lead to additional errors in side calculations.5 But the numerical results are close enough, particularly in contrast to other outcomes.

100

Chapter 4

Figure 4.32 Marginal GHG mitigation costs (Lindahl)

The marginal GHG mitigation costs of the Lindahl equilibrium are plotted in ﬁgure 4.32. We can observe moderate differences in marginal mitigation costs among the regions. As I have emphasized before, the Lindahl equilibrium embodies the strongest incentives from regions. For example, EU is willing to pay $40 for reducing an additional ton of carbon and accepting that EEC only pays $4 for reducing an additional ton of carbon in the year 2050 (see ﬁgure 4.32). In the absence of a permit trading or side-payment scheme, this Lindahl equilibrium outcome is efﬁcient and incentive compatible. However, in the reality that RICE attempts to capture, various schemes of technological transfers and side payments are feasible, though they might not be necessarily built into each IA model. If side payments are allowed in RICE, we can clearly see possible Pareto improvements from the Lindahl equilibrium outcome in ﬁgure 4.32. For example, there would be about $36 in net gains by shifting a ton of carbon emission reduction from EU to EEC in the year 2050. These two regions can divide $36 among themselves through bargaining. If all regions participate in such side-payment negotiations until no additional opportunities can be exploited, the marginal GHG mitigation costs across regions will be equalized and global efﬁciency will be achieved. For instance, the Nash bargaining rule can be applied here to achieve mutually acceptable outcomes.

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101

Figure 4.33 Marginal GHG mitigation costs (utilitarian)

We do not calculate the welfare gains from potential side payments in RICE. Given the volumes of regional GHG emissions and marginal costs of mitigations, the total amounts of side payments or transfers should not exceed the billion-dollar scale in the Lindahl equilibrium outcome. The room for welfare improvements is not large, compared with the scales of global economy and magnitude of cooperative GHG emission mitigations. Such estimation indicates another desirable property of the cooperative game solutions: using the cooperative game solution as the initial emission quota allowance, further trading these quotas invokes very low transaction costs and minimal shocks to the economy. In the Lindahl equilibrium outcome, differences among marginal mitigation costs are moderate. In contrast, I plot the marginal GHG mitigation costs under the utilitarian weights in ﬁgure 4.33. Here, high and low marginal mitigation costs among regions are hundreds of dollars apart. Starting from this initial quota arrangement, equalization of marginal mitigation costs requires huge amounts of side payments.6 Regions can beneﬁt from the Pareto improvements from carbon permit trading. The question is: Should a rational decision maker place himself in a position like OHI or USA in ﬁgure 4.33? The answer is deﬁnitely not. Because the initial GHG emission quotas given to OHI and USA are too low, this starting point for side payments is not incentive compatible. It is a nonstarter.

102

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In summary, we can deﬁnitely (except for the Negishi outcome) ﬁnd the Pareto-improving side payments in RICE. The magnitude of transfers from cooperative game solutions is not large. Others are not necessarily so. More detailed discussion of magnitudes and directions of technological transfers in the RICE framework can be found in Yang and Nordhaus 2006. I will revisit the transfer issues in section 6.3.

5

5.1

Analysis of Game-Theoretic Solutions in RICE

Stability of the Grand Coalition

In the last chapter, I identiﬁed a set of game-theoretic solutions in the RICE model. Subsequent assessments of these solutions demonstrate their superior properties over conventional solutions. All solutions are based on selections of social welfare weights in an otherwise identical RICE model. Although the game-theoretic solutions in RICE are desirable, whether they can withstand various shocks remains to be seen. In other words, we want to know how stable these game-theoretic solutions are. In dynamic economies, the stability of solutions often hinges on the rate of pure time preference, or discount factor, of the system. If an agent, or a society, discounts the future deeply, shortsighted selfinterests outweigh long run beneﬁts. The agents would invest less in the future. Pollution control is a form of public investment. Therefore, discounting plays an important role here too. In examining global environmental issues with long-lasting impacts, such as climate change, discounting has been one of the central issues in the research (see Portney and Weyant 1999). Because discounting affects trade-offs between current and future costs vs. beneﬁts, it inﬂuences the formation of the grand coalition. In addition, it also affects the stability of the coalition formed under a particular rate of time preference. From a strategic perspective, discounting also plays an important role in repeated games and all the games in a dynamic setting. A series of results, labeled as the ‘‘Folk Theorems,’’ connects agents’ strategic behavior with discount factors. In an open-loop strategic setting, although the game is ‘‘one-shot’’ and decisions are made at the beginning, discounting directly relates to the severity of penalty rules of the game. In the cooperative game implemented in RICE, whether a region

104

Chapter 5

has the incentive to defect GHG mitigation obligation speciﬁed by fji g depends on discounting. Testing the stability of the grand coalition in RICE consists of two aspects. First, we want to know whether the Lindahl equilibrium outcome in chapter 4, calculated with a 3 percent rate of pure time preference ðrÞ, holds for alternative discount factors. Second, we want to know whether regions have an incentive to defect from the grand coalition, facing the penalty rule speciﬁed in chapter 2, under r ¼ 3% and alternative discount rates. 5.1.1 Stability of the Lindahl Equilibrium When the rate of pure time preference changes, the paths of regions’ control and state variables adjust accordingly. Parameter sensitivity analyses conducted in Nordhaus 1994 and Nordhaus and Yang 1996 all show that the rate of pure time preference is the most sensitive among all parameters in the DICE/RICE model. In particular, if the rate of pure time preference decreases, namely, the future becomes more important to all regions, regional GHG control rates increase because of the cumulative nature of the greenhouse effect. In ﬁgure 5.1, an example is provided. It shows the paths of mðtÞ of USA under different rates of pure time preferences. Further studies (Yang 2001, 2003a) show that the ‘‘distance’’ or ‘‘gap’’ between cooperative and noncooperative outcomes (solutions of control problem (A-1) and differential game (A-14)) tends to converge as the rate of pure time preference goes to zero. It is also due to cumulative effects. In this section, I investigate the invariance properties of incentives and stability of the bargaining outcome with respect to discounting, not paths of state and control variables. The testing procedure for this task is simple. Under the original Lindahl social welfare weights, we alter the rate of pure time preference to see when and if incentive criteria no longer hold. That is, we increase or decrease the rate of pure time preference incrementally, then run the model and inspect the signs of incentive checking tables as shown in tables 3.3 through 3.5. If all signs in tables 3.3 and 3.5 are positive, and all signs in table 3.4 are negative, under an alternative discount factor, then the original Lindahl equilibrium outcome (with r ¼ 3%) is a Lindahl equilibrium for the model under this alternative discount factor. We tested a wide range of the rate of pure time preference, from r ¼ 0:1% to r ¼ 8%. With alternative r and the original j L , the optimal

Analysis of Game-Theoretic Solutions in RICE

105

Figure 5.1 GHG control rates under different discount rates (USA)

solutions of RICE maintain the homogeneous signs in all three incentive checking tables. That is, the strong incentive properties of the Lindahl equilibrium obtained under r ¼ 3% are preserved throughout the entire reasonable spectrum of r. Since r is the most sensitive parameter in RICE, this testing result is signiﬁcant. Despite responses in solution paths (ﬁgure 5.1 provides a glimpse), the negotiated share of GHG mitigation is invariant with respect to changes of r. If r increases (decreases) from the default, all regions discount the future more (less). Such attitude changes do not warrant the shifts of relative GHG obligations fji g from one region to another. The fact shows that the incentive properties of the Lindahl equilibrium are stable. The result of the ﬁrst stage of the cooperative game deﬁned in section 2.3.3 holds for wide range of r; only the execution stage (solving the optimal control problem) is affected by choice of r. The invariance of GHG obligation share is in j L , not the actual GHG control rates and GHG emissions. Such invariance of incentives and continuous adjustments in control variables should hold for all reasonable changes in common (not regional speciﬁc) parameters.

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5.1.2 Stability of the Cooperative Game As I have indicated repeatedly, the cooperative game I am considering here is an ‘‘open-loop,’’ or ‘‘one-shot’’ game in a dynamic setting. The collective bargaining decision on fji g is binding for the entire time horizon. In the RICE model, decisions are made at the beginning ðt ¼ 1Þ for all junctures in the future. From a strategic perspective, regions can still reassess their own commitment made at the beginning to see whether it is in their best interests to honor the agreement. If the penalty rule speciﬁed in section 2.3.3 cannot stop one or more regions from defecting from the grand coalition, the game outcome is not stable. In fact, the higher the discount rate, the more likely a region wants to defect. The testing steps here are to check what could happen to the grand coalition under the default r ¼ 3% and what happens if r increases from the default. Even without any penalty rule being speciﬁed, there is not much room for gains from defection in the Lindahl equilibrium. The analysis in section 4.3.1 shows that a free-rider can gain only if (a) all other regions do not retaliate, and (b) no more than two defectors act simultaneously (see table 4.5). In section 4.3.1, defection is allowed (or assumed) from the ﬁrst period through the entire time horizon. Costbeneﬁt trade-offs of defection are calculated as one-shot deals. Here, we incorporate the penalty rule and consider whether a region has an incentive to defect at t ¼ 2 facing others’ retaliation in subsequent decision junctures. A defection scenario is played out as follows. When a region defects at t ¼ 2, other regions will retaliate by playing the noncooperative Nash game at t ¼ 3 and to the end of the time horizon. Such a scenario is exactly the penalty rule of the cooperative game prescribed in chapter 2: if one player is discovered cheating, all other players will forgo the coalition and play the Nash game forever. The payoff of the defecting region is the following: Vi ¼ fUð1Þ j cooperating with other regionsg þ fUð2Þ j playing Nash game with 5-region coalitiong nX o þ UðtÞ ð3 a t a 40Þ j all regions playing Nash game . If the above payoff is higher than the payoff in the grand coalition, the region is likely to defect and the game is unstable; if lower, the region will stay in the grand coalition and the game is stable. I conducted simulations at r ¼ 3%; 5%; 7%, and 9% with the Lindahl social welfare

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107

Table 5.1 Welfare losses from defection r ¼ 3% USA

Cooperation 0.14059063

Defection 0.14059349

r ¼ 5% C-D 2.86E-06

Cooperation 0.09023918

Defection 0.09023943

C-D 2.56E-07

OHI

0.07413825

0.07414069

2.44E-06

0.05046940

0.05046972

3.22E-07

EU

0.18381218

0.18381681

4.63E-06

0.12437746

0.12437806

5.98E-07

CHN

1.38696245

1.38698554

2.31E-05

0.87184114

0.87184451

3.37E-06

EEC

0.25592638

0.25592986

3.49E-06

0.16732647

0.16732693

4.51E-07

ROW

3.48657099

3.48660624

3.52E-05

2.20006739

2.20007329

5.90E-06

r ¼ 7% USA

r ¼ 9%

Cooperation

Defection

C-D

Cooperation

Defection

C-D

0.06774593

0.06774597

4.12E-08

0.05510110

0.05510111

1.07E-08

OHI

0.03915027

0.03915034

7.15E-08

0.03263889

0.03263891

2.14E-08

EU

0.09645421

0.09645434

1.32E-07

0.08023279

0.08023283

4.09E-08

CHN

0.64685428

0.64685508

8.00E-07

0.52242870

0.52242896

2.52E-07

EEC ROW

0.12759149 1.63740937

0.12759159 1.63741090

1.01E-07 1.53E-06

0.10521641 1.32483011

0.10521644 1.32483065

3.04E-08 5.38E-07

weights. The results are shown in table 5.1 (Note: The negative values in the table are due to metrics of utility functions.) The simulation outcomes are clear-cut: no region can gain by defecting from the grand coalition under the given penalty rule. All entries in the Cooperation-Defection (C-D) columns are positive, indicating that payoffs of all regions in defection scenarios are constantly lower than the payoffs in the grand coalition (full cooperation). Although the losses from one-shot defection decrease as r increases, the losses never turn to gains within the ranges of r I tested. In a dynamic system like RICE, the explanations of the outcomes from defection simulations are more complicated than the ‘‘folk theorems.’’ As in any other public good or externality schemes, here a region might gain by ‘‘free-riding’’ on a prearranged efﬁcient solution. However, such a conclusion is often based on the assumption that other regions still stick to the original solution paths and do not retaliate against free-riding behavior. Such gaining potentials from freeriding in RICE are depicted in table 4.5. The plus signs in table 4.5 indicate welfare gains of outsiders of certain coalitions while others are cooperating.1 In the RICE model, both mitigation costs and climate damage are very moderate in early periods. Therefore, the gain from one-period

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defection (if any) is not sufﬁcient to recoup the losses from inefﬁcient Nash equilibrium outcomes for later periods when climate damage is signiﬁcant as the result of the cumulative effect. This feature renders quite different implications between stock externality provisions and ‘‘regular’’ repeated game scenarios. Moderately high discounting does not outweigh cumulative effects or a changing state variable space in the future. I do not conduct simulations with discount rates higher than 9 percent because it is unrealistic for IA models and causes numerical irregularities for a model spanning 250 years. The penalty rule here is indeed very severe. Nevertheless, such a penalty rule has been adopted in many game-theoretic models. The penalty is so severe that it seems unrealistic to impose such a penalty in negotiations of international environmental agreements. Yet, the global cooperation that is consistent with regions’ incentives should be stable from examination of dynamic interactions under such a penalty rule. Here, the future is not a reoccurrence of today, but a world with increasing damage potentials from climate change. The increasing risks in the future would have very strong deterrence effects on rational agents from defecting, even if they have a relatively high discount rate. In summary, the Lindahl equilibrium resulting from the cooperative game deﬁned in this research is stable. Due to limited space, I will not check the stabilities of other possible outcomes from the cooperative game, such as the Shapley value. Our economic intuitions point to similar results for all cooperative game solutions. 5.2

Sensitivity Analysis from a Strategic Perspective

In chapter 4, I identiﬁed various numerical solutions related to the cooperative game, other efﬁcient solutions, and relevant benchmark solutions in RICE. As I have indicated, all calculations are carried out under a single set of parametric assumptions. The only alternations are the social welfare weights in (A-1), which are decision variables in the bargaining process. Usually, sensitivity analysis is a crucial part of any comprehensive modeling undertaking. In the process of DICE/RICE model development, exhaustive sensitivity analysis of parametric disturbances have been conducted in Nordhaus 1994, Nordhaus and Yang 1996, as well as Nordhaus and Boyer 2000. In the ongoing development of RICE2007 by Nordhaus and this author, sensitivity analysis is also part of the process. However, detailed sensitivity analysis regarding parameters is not the focus of this study.

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109

In the RICE model and other IA models, the relationships between parametric assumptions and optimal paths of state and control variables are straightforward. Marginal or incremental changes in parameters lead to marginal or incremental shifts in optimal paths of state and control variables (assuming the model is well behaved). For example, the lower the rate of pure time preference, the higher the private investment level, for agents care about the future more; the higher the capital depreciation rate, the faster the capital formation to replenish capital stocks. Nonetheless, I do not examine those types of parameter and outcome relationships here. In this research, I want to investigate the scope of incentive preservation of the model as well as the impact of model structure (including parameter alternation) on cooperative game solutions. Such examination can be characterized as coalitional sensitivity analysis. 5.2.1 The ‘‘Size’’ of the Core and the Lindahl Neighborhood Two important cooperative game solutions identiﬁed in chapter 4 are the Lindahl equilibrium and the Shapley value. Numerical testing shows that both solutions are inside the core of the cooperative game deﬁned in RICE. Algorithmically, these two cooperative game solution concepts (j L and j S ) are solved with certain numerical precision criteria. Those two speciﬁc solution outcomes are approximations. The definitions of the Lindahl equilibrium and the Shapley value would hold numerically in a close neighborhood of the obtained solutions. Here, I want to probe the boundary inside which the Lindahl properties hold. On the other hand, the core allocation is a broader solution concept. I am also interested in the scope (size) of the core of the RICE model. The criteria of the Lindahl equilibrium property are the homogeneous signs in all three incentive checking tables (as in tables 3.3 through 3.5). The criteria of a core allocation are as follows: the ﬁrst row of table 3.3 has all ‘‘þ’’ signs, all entries in table 3.4 have ‘‘’’ signs (no subcoalition can block the grand coalition), and there are no requirements for signs in table 3.5. Evidently, the former requires much stronger incentives than the latter. Using set theory language, the latter encloses the former. Measuring the boundary of the Lindahl property or the size of the core is to determine the area on simplex P S ¼ fji j ji ¼ 6g that maps to the solutions with the above required signs in the relevant tables. P Simplex S ¼ fji j ji ¼ 6g is a subspace in R 5 . With simple numerical methods here, it is very difﬁcult to characterize the shape of the

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core of this complicated system. We assume that, without proof, the core corresponds to a connected set in the simplex. Based on the above assumption, we come up with the following crude approach to measure the scope of the core as well as the boundary of the Lindahl property. We consider the Lindahl weights obtained in chapter 4 as an imprecise ‘‘central’’ point of the set with the Lindahl properties and the core. Starting from this default point, we move a single region’s weight up or down while holding other regions’ weights at the Lindahl weight levels and execute the entire hybrid-Nash equilibrium procedure under the new weights. When a region’s weight increases while other regions’ weights remain the same as in j L , other regions’ welfares are negatively affected. And when a region’s weight decreases while other regions’ weights remain the same as in j L , this region’s welfare is hurt. After getting all coalitional solutions under the new weights, we test the signs in the three tables. We marked down the point where the sign requirements in the three tables begin to sway. For example, consider as a baseline j USA ¼ 1:574 in j L . Holding other regions’ weights as in j L , all three tables maintain homogeneous signs until USA’s weight is above j USA ¼ 2:065 or below j USA ¼ 1:320. We probe the upper and lower bounds that maintain the strongest incentives (homogeneous signs in all three tables) for each region. The results are presented as in U1 and D1 rows in table 5.2. Probing the boundary of the core properties is slightly more subtle. If the ﬁrst row of table 3.3 is all ‘‘þ’’ and all entries in table 3.4 are ‘‘,’’ the allocation is deﬁnitely in the core (a sufﬁcient condition). We test the upper and lower bounds for which such consistencies of signs are preserved. For example, when USA’s weight is above j USA ¼ 2:202 or below j USA ¼ 1:140, a ‘‘þ’’ sign begins to show up in table 3.4. We recTable 5.2 Calculation results of the Lindahl and the core boundaries USA

OHI

EU

CHN

EEC

ROW

U3

2.578

4.051

2.325

0.355

2.802

0.201

U2

2.202

2.189

1.391

0.287

1.465

0.177

U1

2.065

2.061

1.265

0.256

1.377

0.123

Default

1.574

1.805

1.094

0.202

1.229

0.096

D1

1.320

1.397

0.819

0.159

0.957

0.089

D2

1.140

0.839

0.573

0.130

0.560

0.086

D3

1.091

0.837

0.565

0.125

0.559

0.082

Analysis of Game-Theoretic Solutions in RICE

111

ord such upper and lower bounds for each region in U2 and D2 rows in table 5.2. If there are mixed signs in a row in table 3.4, the grand coalition may or may not be blocked by a subcoalition (all ‘‘þ’’ in a row is a block). Whether the ‘‘þ’’ region(s) can compensate for the ‘‘’’ region(s) in a subcoalition to block the grand coalition depends on actual gains and losses among the regions. Because we do not assess such potential compensations, a rule of thumb is employed to measure the ‘‘perimeter’’ (used as a looser term for boundary) of the core. The rule is that there are no more than one ‘‘þ’’ signs in a row and there are no ‘‘þ’’ signs in any two-member subcoalitions (no single winner and no single loser in a two-member coalition) in table 3.4. Of course, the ﬁrst row in table 3.3 has to be all ‘‘þ.’’ We test the upper and lower limits of such sign preservation for each region. For example, when USA’s weight is above j USA ¼ 2:702 or below j USA ¼ 1:091, this rule of thumb no longer holds. The testing results are shown in rows U3 and D3 of table 5.2. After locating these upper and lower bounds region by region, we normalize the new welfare weight shares onto the simplex. Once all six regions’ upper and lower bounds are determined, we have an approximate ‘‘boxed’’ dimension of the Lindahl properties and the core. B1 (the ‘‘box’’ with the Lindahl properties) is smaller than B2 (the ‘‘box’’ deﬁnitely with the core properties), which in turn is smaller than B3 (the ‘‘box’’ that likely has the core properties). In addition, B1 H B2 H B3. Each box is a convex hull with twelve points as vertexes. The numerical results of rescaled social welfare weights are summarized in tables 5.3 through 5.5. In table 5.3, the searching results for P the Lindahl properties are rescaled onto simplex S ¼ fji j ji ¼ 6g. Tables 5.4 and 5.5 contain the same results for the core allocation. Interpretations of these tables are as follows. Taking USA as an example, the ‘‘willingness to pay’’ property of the Lindahl equilibrium would be preserved between ½D1; U1 ¼ ½1:320; 2:065 for the weight of USA while other regions’ weights are at the default level; the core property deﬁnitely holds between ½D2; U2 ¼ ½1:140; 2:202 for the weight of USA while other regions’ weights are at the default level; and the core property likely holds between ½D3; U3 ¼ ½1:091; 2:202 for the weight of USA while other regions’ weights are at the default level in table 5.2. The new weights of USA along with default weights of other regions are shown in tables 5.3 and 5.4 after scaling to the

112

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Table 5.3 Rescaled new weights mapping to the Lindahl equilibrium USA

OHI

EU

CHN

EEC

ROW

Default USA_U1

1.574 1.9088

1.805 1.6685

1.094 1.0112

0.202 0.1867

1.229 1.1360

0.096 0.0887

USA_D1

1.3784

1.8848

1.1424

0.2109

1.2833

0.1002

OHI_U1

1.5096

1.9767

1.0492

0.1937

1.1787

0.0921

OHI_D1

1.6888

1.4989

1.1738

0.2167

1.3187

0.1030

EU_U1

1.5304

1.7550

1.2299

0.1964

1.1949

0.0933

EU_D1

1.6496

1.8917

0.8583

0.2117

1.2880

0.1006

CHN_U1

1.5600

1.7889

1.0842

0.2537

1.2180

0.0951

CHN_D1 EEC_U1

1.5854 1.5361

1.8180 1.7615

1.1019 1.0677

0.1601 0.1971

1.2379 1.3439

0.0967 0.0937

EEC_D1

1.6487

1.8907

1.1459

0.2116

1.0024

0.1006

ROW_U1

1.5669

1.7969

1.0891

0.2011

1.2235

0.1224

ROW_D1

1.5758

1.8071

1.0953

0.2022

1.2304

0.0891

Table 5.4 Rescaled new weights mapping inside the core USA

OHI

EU

CHN

EEC

ROW

Default

1.574

1.805

1.094

0.202

1.229

0.096

USA_U2

1.9934

1.6340

0.9903

0.1829

1.1126

0.0869

USA_D2

1.2289

1.9457

1.1793

0.2178

1.3248

0.1035

OHI_U2

1.4793

2.0573

1.0282

0.1898

1.1551

0.0902

OHI_D2

1.8760

1.0000

1.3039

0.2408

1.4648

0.1144

EU_U2

1.4998

1.7199

1.3254

0.1925

1.1710

0.0915

EU_D2 CHN_U2

1.7237 1.5520

1.9766 1.7798

0.6275 1.0787

0.2212 0.2830

1.3459 1.2118

0.1051 0.0947

CHN_D2

1.5931

1.8269

1.1073

0.1316

1.2439

0.0972

EEC_U2

1.5144

1.7367

1.0526

0.1944

1.4096

0.0924

EEC_D2

1.7715

2.0315

1.2313

0.2273

0.6303

0.1080

ROW_U2

1.5530

1.7810

1.0794

0.1993

1.2126

0.1746

ROW_D2

1.5766

1.8080

1.0958

0.2023

1.2311

0.0861

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Table 5.5 Rescaled new weights likely mapping to the core USA

OHI

EU

CHN

EEC

ROW

1.574 2.2085

1.805 1.5463

1.094 0.9372

0.202 0.1730

1.229 1.0528

0.096 0.0822

USA_D3

1.1865

1.9630

1.1898

0.2197

1.3366

0.1044

OHI_U3

1.1453

2.9476

0.7960

0.1470

0.8943

0.0699

OHI_D3

1.8768

0.9980

1.3045

0.2409

1.4654

0.1145

EU_U3

1.3060

1.4977

1.9292

0.1676

1.0198

0.0797

EU_D3

1.7262

1.9795

0.6196

0.2215

1.3478

0.1053

CHN_U3

1.5349

1.7601

1.0668

0.3462

1.1984

0.0936

CHN_D3 EEC_U3

1.5945 1.2471

1.8285 1.4301

1.1082 0.8668

0.1266 0.1600

1.2450 2.2200

0.0972 0.0761

EEC_D3

1.7719

2.0319

1.2315

0.2274

0.6293

0.1081

ROW_U3

1.5469

1.7740

1.0752

0.1985

1.2079

0.1975

ROW_D3

1.5777

1.8092

1.0966

0.2025

1.2319

0.0822

Default USA_U3

simplex (rows ‘‘USA_U1’’ and ‘‘USA_D1’’ in table 5.3, ‘‘USA_U2’’ and ‘‘USA_D2’’ in table 5.4, as well as ‘‘USA_U3’’ and ‘‘USA_D3’’ in table 5.5). If USA’s weight increases to 2.203, at least one region’s (not USA’s) incentive requirement for the core property is violated. On the other hand, USA might be willing to reduce its welfare weight down to 1.140, while other regions stay at the default. If its weight decreases to 1.139, USA would rather not be in this grand coalition. Other entries in these tables can be explained exactly the same way. The sizes or scopes of these three ‘‘boxes’’ are estimated by the following method. We construct three convex hulls using the twelve points in tables 5.3, 5.4, and 5.5, respectively. Then we calculate the volumes of the three convex hulls as the percentage of the volume P of simplex S ¼ fji j ji ¼ 6g. To do so, we used a shareware called Qhull, provided by Barber, Dobkin, and Huhdanpaa (1996). Although this is not an accurate sizing of the Lindahl equilibrium allocation or the core, the calculations provide approximate geometric ratios of the entire efﬁciency set and the set with desirable properties. The results are as follows: P The volume of simplex S ¼ fji j ji ¼ 6g: 64.8; The volume of convex hull B1: The volume of convex hull B2: The volume of convex hull B3: B1/S ¼ 0.3932%; B2/S ¼ 0.6432%;

0.25477044; 0.41679085; 0.74225675. B3/S ¼ 1.1455%.

114

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The ratio B1/S indicates that roughly 0.39 percent of all efﬁcient solutions possess the strongest incentive properties from all regions. That is, the bargaining outcome with any fji g inside B1 has homogeneous signs in all three tables and all regions are ‘‘willing to pay’’ for P such results. B1 is a small portion of S ¼ fji j ji ¼ 6g. This fact implies that the Lindahl equilibrium without endowment transfers is achievable through bargaining and cooperation. One point inside B1 maps to the Lindahl equilibrium. Numerically, we treat the ‘‘center point’’ j L as the Lindahl equilibrium while admitting that our heuristic algorithm for searching the Lindahl equilibrium is not highly accurate. Since the core of the cooperative game in RICE contains all the possible bargaining outcomes among efﬁcient allocations, we are more interested in B2 and B3. The volume of B3 is closer to the actual size of the core. The ratio B3/S ¼ 1.146 shows that the size of the core is moderate. This calculation result is a good indication for a ‘‘shrinking’’ core: the opportunity for not being blocked by any coalition is small in this six-region model. One can infer that more regions interacting with one another would make this ratio even smaller. Most important, the popular utilitarian and Negishi weights are not inside the core. The scope of this imprecise ‘‘boundary’’ of the core can be seen more clearly from table 5.6. In the table, the scaled upper and lower bounds of each region are listed along with the default weights. In addition, one-dimensional width ¼ ðupper-lowerÞ and scaled width ¼ ½ðupperlowerÞ/default of the boundaries are also reported. It is interesting to note that the scaled ‘‘bandwidth’’ of the six regions is narrower for heavy emitters (USA, CHN, and ROW) and wider for light emitters. It is consistent with our intuition that other regions are less sensitive to the GHG mitigation obligation changes of light emitters. Despite multiangle deciphering of these numerical results on ‘‘boundaries,’’ it is very difﬁcult to measure the real size and shape of Table 5.6 Further results from core boundary calculation USA

OHI

EU

CHN

EEC

ROW

U3

2.578

4.051

2.325

0.355

2.802

0.201

Default D3

1.574 1.091

1.805 0.837

1.094 0.565

0.202 0.125

1.229 0.559

0.096 0.082

Width

1.487

3.214

1.76

0.23

2.243

0.119

Scaled width

1.3630

3.8399

3.1150

1.8400

4.0125

1.4512

Analysis of Game-Theoretic Solutions in RICE

115

the core domain on the simplex. Nevertheless, the twelve points in table 5.5 (vertices of box B3) map onto the ‘‘border’’ or ‘‘near border’’ of the core. They also represent the maximal tolerance of the grand coalition to changes of the initial ‘‘quota’’ of a single region. It would be informative to inspect the optimal paths of state, control, and costate variables under these social welfare weights. Presumably, they are all potential outcomes of the quota bargaining process, or the cooperative game deﬁned in chapter 2. First, the global GHG emission paths of these twelve solutions are plotted along with the default Lindahl equilibrium in ﬁgure 5.2. The lines are almost indistinguishable, though not identical, in the graph. Since the twelve solution paths are supposed to be on the boundaries of the core in different directions, we can tentatively propose the following hypothesis: The core solutions are numerically invariant with respect to aggregate ﬂows of externalities (bðtÞ in chapter 2); thus the invariance propagates to stock of externalities too (BðtÞ in chapter 2), namely, global GHG emissions and concentrations. As demonstrated in ﬁgure 4.18, such invariance does not hold for all efﬁcient solutions. The utilitarian weights result in signiﬁcantly different paths of ﬂow and stock

Figure 5.2 Global GHG emissions on the core boundaries

116

Chapter 5

Figure 5.3 GHG control rates on the core boundaries (USA)

of externalities. This apparent invariance phenomenon implies that various bargaining outcomes that meet IR and IC criteria lead to the same aggregate optimal paths of stock externalities. If a prescribed path of externality does not meet these incentive criteria, the path, even being efﬁcient, might need to be changed. On the other hand, regional GHG control rates (control variables) and carbon taxes (costate variables) respond to the welfare weight changes. They do not show invariance. In ﬁgures 5.3 to 5.14, each region’s GHG control rates and carbon taxes in the twelve scenarios are plotted along with the default Lindahl equilibrium separately. Magnitudes of deviations from the default vary from region to region. However, some common patterns across regions are worth noting. For each region, the highest and the lowest GHG control rates and carbon taxes are in the cases where its own welfare weight is altered. In ﬁgure 5.3, for example, when USA’s weight increases from 1.574 to 2.578, its GHG control rate, labeled as ‘‘USA_U,’’ is the lowest of all; when its weight decreases from 1.574 to 1.091, its GHG control rate, labeled as ‘‘USA_D,’’ is the highest. These two paths clearly separate

Analysis of Game-Theoretic Solutions in RICE

117

themselves from others. The same pattern holds for all six regions except for ROW_D in ﬁgure 5.13. These two most drastic adjustments are with respect to the region’s own share changes (in terms of other regions’ default relative shares): increased share leads to lowered mitigation obligations, and vice versa. Such bands in GHG control rates and carbon taxes reﬂect the ‘‘size’’ of the bargaining ‘‘room.’’ The upper bands are the maximum amounts that the regions are willing to pay without giving up on the grand coalition. From ﬁgures 5.6 and 5.8 we can see that OHI and EU may be willing to pay up to $600 per ton of carbon in late periods at their highest yet acceptable social welfare weights, providing other regions abide by their GHG mitigation obligations. On the other hand, the lower band is a region’s lightest possible GHG mitigation obligations that are tolerable to others. In any incentive compatible grand coalition, no region’s GHG control rates or carbon taxes can go below such bands. Regions also react to other regions’ welfare weight changes. In most cases, such reactions are marginal to moderate. They center on the default. For all ﬁve regions except ROW, the strongest reactions are with respect to ROW_U. That is, when ROW’s weight increases from 0.096 to 0.201, other regions’ GHG control rates and carbon taxes move up signiﬁcantly (see ‘‘ROW_U’’ curves in ﬁgures 5.3 to 5.12). Because ROW is the dominant GHG emitter in the long run, other regions have to compensate much more if ROW’s mitigation obligation reduces from the default. As for ROW (see ﬁgures 5.13 and 5.14), all paths bundle together except for ‘‘ROW_U,’’ where both GHG control rates and carbon taxes are substantially lower. The explanation for such a pattern is that there is not much room for ROW to lower its weight (from 0.096 to 0.082). A moderately decreased control rate by ROW results in a large increase of GHG emission. Reduced mitigation obligations by the dominant emitter have to be shouldered by other regions. Other regions can be better off by blocking ROW and forming subcoalitions. In such a case, the core property of the grand coalition no longer holds. Another interesting observation is that the highest carbon tax (corresponding to the region’s lower bound of weight) is from more than doubling to more than six times the lowest carbon tax (corresponding to the region’s upper bound of weight) in all six regions. This phenomenon tells us that there is sufﬁciently large room for negotiation on mutually agreeable terms of collective mitigation actions. Although a region deﬁnitely prefers paying less for GHG mitigation, it is willing

118

Figure 5.4 Carbon taxes on the core boundaries (USA)

Figure 5.5 GHG control rates on the core boundaries (OHI)

Chapter 5

Analysis of Game-Theoretic Solutions in RICE

Figure 5.6 Carbon taxes on the core boundaries (OHI)

Figure 5.7 GHG control rates on the core boundaries (EU)

119

120

Figure 5.8 Carbon taxes on the core boundaries (EU)

Figure 5.9 GHG control rates on the core boundaries (CHN)

Chapter 5

Analysis of Game-Theoretic Solutions in RICE

Figure 5.10 Carbon taxes on the core boundaries (CHN)

Figure 5.11 GHG control rates on the core boundaries (EEC)

121

122

Figure 5.12 Carbon taxes on the core boundaries (EEC)

Figure 5.13 GHG control rates on the core boundaries (ROW)

Chapter 5

Analysis of Game-Theoretic Solutions in RICE

123

Figure 5.14 Carbon taxes on the core boundaries (ROW)

to pay such amounts as long as it is better off than playing the nocooperative Nash game or forming subcoalitions. The six graphs of regional carbon taxes (ﬁgures 5.4, 5.6, 5.8, 5.10, 5.12, and 5.14) depict the acceptable ranges for each region in the initial quota bargaining. 5.2.2 False Perceptions of Climate Damage In the RICE model used for this study, only USA and CHN are sovereign countries among the six regions. Other regions are blocs of multiple countries. While the European Union may act in unison on international environmental issues, the other three regions in the model are not necessarily in complete harmony in dealing with climate change. As we have discussed before, ROW is particularly problematic because it is a huge bloc formed by many heterogeneous countries. But as far as efﬁcient solutions are concerned, it does not matter at what hierarchical layer or regional breakdowns we internalize the externality. However, if we want to study strategic interactions and cooperation in providing externalities, aggregation affects the outcomes signiﬁcantly. Based on the model assumption, ROW is a bloc in which climate externalities are already fully internalized. Then, it behaves as a single

124

Chapter 5

region that either cooperates with other regions or interacts strategically with other regions. In reality, either ROW as a whole, or individual countries inside ROW, would be ‘‘willing to pay’’ its mitigation obligations in the Lindahl equilibrium or the Shapley value solutions in chapter 4. Countries in ROW prefer paying much less than those in the cooperative game solutions. To resolve this controversy, we need to reexamine the assumptions made about ROW. Apparently, further disaggregating ROW into more subregions is not an option. The calculation time for solving ‘‘hybrid’’ Nash equilibrium solutions would be prohibitive. Furthermore, as long as the regional breakdown is not at the sovereign-country level (nearly 200 countries by UN recognition), the abovementioned controversy always exists. Keeping ROW as a single entity, a correction method adopted in Nordhaus and Yang 1996 is simple and effective. The method separates real climate damages and the perception of climate damages in ROW. Its rationales are as follows. Many assumptions in the model are based on data and scholars’ estimations, such as predictions of population growth, energy intensity trend, and total factor productivity growth. They have no direct implications for incentives of regions. Revising these assumptions from the default is part of conventional sensitivity analysis. What is the most important factor that inﬂuences ROW’s attitude toward climate change? The answer is its perception of the potential damage of climate change fallen on itself. An individual country in ROW may believe that domestic GHG emissions will not cause much damage to itself but rather to others; it will beneﬁt from other countries’ mitigation efforts. Such an attitude is typical free-riding behavior. Such behavior adds up to all countries in ROW; thus ROW as a whole would behave as if it does not fully perceive the damage from climate change. From a modeling perspective, implementing the above consideration is straightforward. The default calibration of climate damage is based on the best knowledge about climate impacts. It is the ‘‘real’’ damage. If a region does not perceive such ‘‘real’’ climate damage and behaves as if there is less damage, its actions can be generated from a lower damage calibration. Therefore, the issue here is to see what happens if ROW is assumed to have a much less severe damage function. Strategically, the scenario can be treated in the following setting. Countries in ROW do not cooperate with one another to internalize the externality. Instead, they play the Nash game or stay in BaU inside

Analysis of Game-Theoretic Solutions in RICE

125

Table 5.7 Lindahl weights under false damage perception by ROW USA

OHI

EU

CHN

EEC

ROW

Default n¼5

1.574 1.4955

1.805 1.7150

1.094 1.0394

0.202 0.1919

1.229 1.1677

0.096 0.3905

n ¼ 10

1.4454

1.6575

1.0046

0.1855

1.1286

0.5785

n ¼ 15

1.3972

1.6023

0.9711

0.1793

1.0910

0.7590

n ¼ 20

1.3559

1.5549

0.9424

0.1740

1.0587

0.9140

the bloc. The result would be very low GHG control rates and carbon taxes inside the bloc. After they settle from inside, a proxy for them (labeled ROW here) represents their position interacting with other regions, either cooperatively or strategically. This interpretation would render similar results as blindness on damage (not perceiving climate damage correctly). The modeling change made here is very simple. In (A-7), we change the damage function coefﬁcient a1 of ROW to a1 =n, with n ¼ 5; 10; 15, and 20. Evidently, the larger the value of n, the lower the climate damage for ROW. All other regions’ a1; i are kept at the default. The scenarios capture different degrees of false perception of climate damage by ROW, assuming a1 is the true damage coefﬁcient. Under those new speciﬁcations, we search for new Lindahl equilibrium and Shapley value weights, repeating the procedures in chapter 4. Four sets of new Lindahl equilibrium welfare weights, along with the default, are listed in table 5.7. Under the new assumptions on ROW’s damage function, these sets of welfare weights ensure homogeneous signs in all three incentive checking tables (as in tables 3.3 through 3.5). One obvious pattern in the table is that the welfare weight share of ROW increases as n increases. The fact implies that ROW would share lighter burdens in the grand coalition. The outcome is consistent with common sense: a region with lower climate damage contributes less to GHG mitigation voluntarily. The searching procedure reveals another interesting observation. The relative shares of the other ﬁve regions remain the same under the assumption of different degrees of false perception of climate damage by ROW. Actual searching of new Lindahl weights is simple. We just increase ROW’s weights and keep other regions’ weights ﬁxed at the default j L until all incentive requirements are met. Then, we rescale the weights onto the simplex. The observation suggests a hypothesis:

126

Chapter 5

Figure 5.15 GHG control rates under false perceptions (USA)

Coalitional structure is invariant with respect to cost/damage shifts of outside players. Here the invariance refers to relative share ji . Here, the {USA, OHI, EU, CHN, EEC} coalition would have the same fji g under different assumptions of ROW’s damage function. Again, the optimal paths of state and control variables in the coalition will adjust in response to outsiders’ changes. Another phenomenon is ROW’s insensitivity of incentives as n increases. When n increases, incentive criteria in all three tables are met with a wider range of ROW’s welfare weights (U1 and D1 in table 5.2). When n ¼ 20, ½D1; U1 ¼ ½0:794; 1:326 for ROW.2 It is as if ROW can bear a much broader range of mitigation burdens without affecting the incentives of itself or others. The paths of state, control, and costate variables of regions respond to the false perception of climate damage by ROW. Patterns of reactions from the other ﬁve regions are very similar. I just report the GHG control rates and carbon taxes of USA in ﬁgures 5.15 and 5.16. First, the open-loop Nash equilibrium paths of USA under different n almost coincide with one another yet are not identical to each other. In

Analysis of Game-Theoretic Solutions in RICE

127

Figure 5.16 Carbon taxes under false perceptions (USA)

a noncooperative setting, regions react to ROW’s varied GHG emissions and decide their own emission paths. Because ROW’s emission baseline is dominantly large, strategic reactions to small changes of a large portion of GHG emissions by others is very small. Second, the paths of control rates in the Lindahl equilibrium with different damage coefﬁcients of ROW (n ¼ 5; 10; 15, and 20) are very close to one another. They are all lower than the default path. Corresponding carbon taxes in ﬁgure 5.16 are lower too. Here, facing ROW’s reduced GHG mitigation contribution, USA and other regions (not depicted) do not compensate for the lost share. Instead, they reduce their own share in the Lindahl outcome and stay put in the noncooperative Nash equilibrium. Explanation of USA’s reactions has to be incorporated with ROW’s behavior under the false perception of climate damage, which is reported in ﬁgures 5.17 and 5.18. As n increases, the GHG control rates and carbon taxes of ROW decrease monotonically, both in the openloop Nash equilibrium and in the Lindahl equilibrium. A striking observation is that in all four false perception scenarios, ROW’s GHG

128

Chapter 5

Figure 5.17 GHG control rates under false perceptions (ROW)

control rates and carbon taxes in the Lindahl equilibrium are lower than its Nash equilibrium paths in the default. If in the eyes of ROW climate damage is low, its ‘‘willingness to pay’’ level in the Lindahl equilibrium is also low and lower than the Nash equilibrium with correct damage assessment. Once ROW’s GHG control rates become very low under false perception of climate damage (2% to 3% in the Nash equilibrium and 4% to 5% in the Lindahl equilibrium), other regions just treat ROW as a large ‘‘insensitive’’ emitter. Consequently, other regions’ reactions to ROW’s extremely low control rates under different n become marginal. ROW’s behavior under false perceptions of climate damages captures attitudes of developing countries in negotiations of international cooperation in GHG mitigation. They are not a uniﬁed bloc. Nevertheless, if the international community asks very little for their GHG mitigation commitment, they will readily agree. Their proxy, such as ROW in our model setting, would be willing to cooperate with other regions on such a basis. However, if ROW asks for more GHG mitigation contributions from other willing participants in the grand coalition to com-

Analysis of Game-Theoretic Solutions in RICE

129

Figure 5.18 Carbon taxes under false perceptions (ROW)

pensate for its lowered obligation, it would not be incentive compatible with those regions. Our simulations show the point. Calculations of the Shapley values under the false perception of damage by ROW are more complicated. The invariance of shares with respect to the other ﬁve regions in the Lindahl equilibrium does not hold here because ROW’s alternative damages affect every region’s ‘‘equal gain’’ calculation. Therefore, full-length searching is conducted for n ¼ 5; 10; 15; 20. The iteration termination criterion is set at the maximum relative error, which is lower than 3 percent. All absolute errors are at or below 104 level. Table 5.8 contains the calculation results along with the default for comparison. Testing the results using the three incentive checking tables shows that these four Shapley values are in the core. The results clearly show that the Shapley value share of ROW monotonically decreases as n increases. When n ¼ 10, ROW’s Shapley value share (20.92%) is overtaken by CHN (31.6%) and is followed closely by USA (17.3%). When n ¼ 20, the Shapley value share of USA (18.81%) also surpasses CHN (16.23%). The social welfare weight shifts

130

Chapter 5

Table 5.8 Shapley value calculation under false damage perception by ROW USA

OHI

EU

CHN

EEC

ROW

Weight n¼1

2.1264

1.7569

0.9302

0.1431

0.9523

0.0911

n¼5

2.0020

1.8062

0.8800

0.1425

0.8802

0.2871

n ¼ 10

1.9740

1.6710

0.8722

0.1469

0.8539

0.4820

n ¼ 15

1.8705

1.6431

0.8139

0.1426

0.8360

0.6759

n ¼ 20

1.8033

1.5779

0.8330

0.1392

0.8135

0.8331

n¼1

0.005189

0.002567

0.00392

0.007823

0.002806

0.033805

n¼5

0.003808

0.001945

0.002854

0.006664

0.002007

0.007106

n ¼ 10 n ¼ 15

0.003373 0.003118

0.001652 0.001576

0.002515 0.002315

0.00616 0.005326

0.001718 0.001631

0.004078 0.003168

n ¼ 20

0.002959

0.001498

0.00226

0.004908

0.001553

0.002554

n¼1

0.5549

0.2745

0.4192

0.8365

0.3001

3.6148

n¼5

0.9370

0.4786

0.7023

1.6398

0.4938

1.7485

n ¼ 10

1.0381

0.5084

0.7740

1.8958

0.5287

1.2550

n ¼ 15

1.0919

0.5519

0.8107

1.8651

0.5711

1.1094

n ¼ 20

1.1285

0.5713

0.8619

1.8719

0.5923

0.9741

Shapley value

Scaled Shapley value

Shapley value share % n¼1

9.2479

4.5749

6.9863

13.9423

5.0009

60.2477

n¼5 n ¼ 10

15.6168 17.3010

7.9765 8.4735

11.7044 12.9001

27.3294 31.5962

8.2308 8.8121

29.1421 20.9171

n ¼ 15

18.1977

9.1981

13.5111

31.0844

9.5191

18.4896

n ¼ 20

18.8088

9.5220

14.3656

31.1976

9.8716

16.2344 0.0003

R. error % n¼1

3.743

2.863

3.292

0.811

2.679

n¼5

2.78

1.969

0.011

0.913

1.324

0.473

n ¼ 10

0.791

2.867

1.43

0.708

0.861

2.096

n ¼ 15

1.793

2.899

1.188

0.198

1.422

2.255

n ¼ 20

1.755

2.865

2.147

0.272

1.363

0.197

A. error(xE-4) n¼1

1.94

0.74

1.29

0.63

0.75

1.06

0.38

0

0.61

0.27

0.34

0.27

0.47

0.36

0.44

0.15

0.85

n¼5 n ¼ 10

0

n ¼ 15

0.56

0.46

0.27

0.11

0.23

0.71

n ¼ 20

0.52

0.43

0.49

0.13

0.21

0.05

Analysis of Game-Theoretic Solutions in RICE

131

are also consistent with the alternative assumptions on ROW’s damage. As ROW’s climate damage coefﬁcient gets smaller, its Shapley value weights (not the Shapley value share) become larger. Consequently, ROW’s GHG mitigation obligations in the grand coalition are lower as n increases. The simulation results here show that if ROW’s climate damage becomes less and less severe, whether it is real or due to false perception, its Shapley value share goes down steadily. For a large region such as ROW with the most GHG emissions and fast economic growth, climate change is less relevant to the region if it would not cause significant damage. In such a case, ROW’s role is less crucial in the Shapley value solution of the cooperative game. Although other regions can beneﬁt from GHG mitigations in ROW, combating climate change is no longer as important for ROW. The other regions would take whatever mitigation level ROW is willing to offer and renegotiate their own shares. The severity of climate damage of CHN is only second to ROW in the default calibration (see appendix 3). As a result, CHN’s Shapley value share increases more signiﬁcantly than any of the other regions as ROW’s share goes down due to the false perception. When n ¼ 10; 15, and 20, CHN’s Shapley value share is around 31 percent, much higher than any other regions. As I have indicated before, the false perception of climate damage by ROW is mainly due to its inefﬁciency internally. That is, ROW is not a single decision maker. Such an assumption is actually much more realistic for international environmental politics. Once we accept such a premise, CHN and USA are the two most critical players (single decision makers) in the cooperative game of GHG mitigation. This assertion has been proven true by the Shapley value calculations here and by the impasse of post-Kyoto negotiations caused by the USA and China. Finally, the sensitivity analysis from strategic perspectives can be expanded under many alternative assumptions. The study here is by no means an exhaustive one. The exercises in this chapter are good examples of the incorporation of game-theoretic approaches into IA modeling. They offer a new angle of sensitivity analysis in IA modeling of climate change.

6

6.1

Policy Applications of Game-Theoretic Solutions in RICE

Incentive Compatibilities of Unilateral Actions

In the past decades, many proposals of international cooperation on GHG emission mitigations have been put on the table. The Kyoto Protocol, drafted in 1997, is the most inﬂuential one. There have been numerous studies done on the cost-effectiveness and fairness of implementing the Kyoto Protocol.1 There are also extensive discussions on implementing the Kyoto Protocol through the mechanisms, such as joint implementation ( JI), clean development mechanism (CDM), and carbon emission permit trading, outlined in the Protocol. Some of these mechanisms have been implemented at early stages in some European Community countries. However, analyses of the Kyoto Protocol from regional incentive perspectives have not been paid sufﬁcient attention. Despite 175 countries having ratiﬁed or accepted the Kyoto Protocol as of June 2007, implementing the Kyoto Protocol is not smooth sailing. Potential dangers of climate change have been pointed out in unequivocal terms by IPCC assessment reports, yet fully implementing the Kyoto Protocol has stalled. Difﬁculties in implementing the Kyoto Protocol occur not because of inaccuracies and uncertainties in mitigation cost estimations but because of incentive incompatibilities among major regions. The USA’s refusal to ratify the treaty is the most signiﬁcant example. On the other hand, China’s ready approval of the treaty might be a telling ‘‘uneventful’’ event. One key characteristic of the Kyoto Protocol is unilateral action in GHG mitigation. By ‘‘unilateral actions,’’ we mean that there are binding GHG mitigation requirements for some countries and no requirements for others. On the one hand, the Kyoto Protocol requires that the Annex-I regions, which include all industrialized countries and Eastern European countries, meet their targeted GHG mitigation goals

134

Chapter 6

before 2012; on the other hand, most developing countries bear no obligation in GHG emission reduction. Such unilateral action clauses might be justiﬁable for historical, political, and economic reasons. But they are hardly acceptable from incentive perspectives. We should remember that any international environmental agreement can ask no more than voluntary participation from sovereign countries. For a truly global cooperation (a grand coalition), the stamps of approval from all major countries are necessary. Analysis from an incentive perspective is a crucial component in understanding the Kyoto Protocol. The modeling approach adopted in this study is a perfect tool for such analysis. From a modeling perspective, the Kyoto Protocol is an agreement on the initial quota of regional GHG mitigations. Some regions are assigned to a stringent quota allowance; some regions are assigned to the amounts up to the BaU baseline. It is similar to the negotiation of shares of social welfare weights fji g in our framework. Annex-I regions are treated as if assigned to very small ji and non-Annex-I regions that bear no burdens of GHG mitigation are assigned very large ji . Such an arrangement would have no incentive conﬂicts with regions having large ji but would have serious problems with those with very small ji . Actually, such unilateral action situations ﬁt one of the ‘‘hybrid’’ Nash equilibria better than a grand coalition under any social welfare weights. Without probing into modeling details of unilateral actions in the Kyoto Protocol, the analyses in chapters 4 and 5 offer good indications of the incentive properties of such actions.2 A collective GHG mitigation proposal is more appealing to those with light burdens and hefty beneﬁts. Such a tendency or property of stock externality provision can be seen clearly through the series of simulations in previous chapters. For example, under some popular and/or convenient arrangements, such as Negishi and utilitarian outcomes, some regions are not willing to join the global GHG mitigation arrangement. They would rather play the noncooperative Nash game instead. The Kyoto Protocol demands more stringent GHG emission reductions from USA, OHI, and EU than the utilitarian scenario in chapter 4. With the parametric assumptions and calibration of RICE, the Kyoto Protocol would make these regions even worse off than the utilitarian results. Therefore, the incentive property of the Kyoto Protocol in RICE is evident from this inferential connection. The Kyoto Protocol is not incentive compatible for those regions that do not like the utilitarian outcomes. The fact that USA has not ratiﬁed the Kyoto Protocol

Policy Applications of Game-Theoretic Solutions in RICE

135

seems to echo our conclusion here. President George W. Bush of the United States made a ‘‘counterproposal’’ to the Kyoto Protocol in May 2007, almost a decade after the signing of the Protocol. In this proposal, USA commits itself to GHG mitigation obligations but insists on ﬁfteen major GHG emitters acting collectively. China and India are among these ﬁfteen countries. Unilateral actions taken on global externalities are not efﬁcient in general. Thus, the Kyoto Protocol represents a ‘‘second-best’’ scenario. This is one of many reasons why most analyses of the Kyoto Protocol are focused on its ‘‘cost-effectiveness’’ but not on its ‘‘efﬁciency.’’ Some people might argue that the clauses in the Protocol serve merely as a starting point for further efﬁciency-improving policies of GHG mitigations, such as tradable emission permits and joint implementations. However, as I have demonstrated in previous chapters, an international GHG mitigation proposal is a difﬁcult one, if not a nonstarter, with initial ‘‘quota’’ allocations that are not incentive compatible. Although a tradable emissions permit scheme and joint implementation can improve cost-effectiveness within the Kyoto Protocol framework, they are not sufﬁcient to compensate for the losses of USA, OHI, and EU. It is not surprising that all research on the Kyoto Protocol conclude various degrees of welfare losses in these regions. On the other hand, the same research indicates welfare gains by the former Soviet Union by selling so-called hot air through permit trading. By assessing welfare changes in different regions, it is hard to imagine the Kyoto Protocol can be agreed on voluntarily by all regions. The incentive properties of the Kyoto Protocol can also be examined from a different angle. The Protocol offers a plan for global GHG emission constraints or an upper bound. Without arguing the costs and beneﬁts of such constraints, one can still ask the following question: Is the treaty a good and fair starting point or initial allocation for achieving the desired GHG emission target from an incentive viewpoint? The answer is clearly no. The argument is similar to the comparison between the Negishi and the Lindahl outcomes in section 4.5. The two solutions lead to very similar global GHG emission paths but their incentive properties are very different. The sole reason for such dichotomy is the initial quota. The sulfur dioxide (SO2 ) emission permit trading scheme in the United States has been a success story of a market approach to dealing with pollution problems (see Ellerman et al. 2000). SO2 emission permit trading within a country has been a frequently cited example for

136

Chapter 6

promoting a global CO2 emission permit trading scheme. However, if some power plants were given very low SO2 emission quotas and others were given the quota up to their production capacities in the United States’ SO2 permit trading market (assuming these power plants are otherwise comparable), would we still be hailing the success story today? The answer is ‘‘probably not.’’ Fortunately, we need not learn the lessons the hard way due to fair and rational initial quota allocations in the SO2 trading market. The initial quotas are equally important for a global GHG emission trading market, just like the much smaller SO2 permit market. Finally, some scholars might argue that the assumptions in RICE lead to such pessimistic conclusions regarding the Kyoto Protocol. Under alternative assumptions, as we have tested in the sensitivity analysis in chapter 5, the paths of state and control variables will change. The cooperative game solution set will change too. However, in a stock externality provision scheme, only prohibitively high mitigation costs and/or negligible climate damages warrant near-zero (not zero) mitigation obligation in an incentive-compatible coalition. ROW and CHN do not belong to either category. Therefore, alternative assumptions within a reasonable range will not change the basic conclusions here. 6.2

Renegotiation of International Environmental Agreements (IEAs)

The RICE model is one of a few forward-looking dynamic IA models in the ﬁeld. To acquire the desirable properties, the model has to compromise on its dimensionality by limiting regional and sectoral breakdowns. Comparing with other recursive dynamic IA models, the gain from RICE is in its forward-looking dynamics. Yet, to many, such dynamics are not sufﬁcient and too simplistic. The criticisms are particularly legitimate and relevant from a gametheoretic viewpoint. Forward-looking and perfect-foresight dynamics imply an open-loop decision-making structure. As I have explained, open-loop strategies involve ‘‘one-shot’’ decisions made at the beginning of time. They incorporate all foreseeable information, both deterministic and stochastic, during the entire planning horizon. The decision structure is not capable of dealing with new information acquired and/or with unexpected shocks in the future. In climate change, abrupt and catastrophic climate events are drawing more and more attention from scholars. Economic analyses of such events can hardly ﬁt into any one-shot decision structure. Consequently, the

Policy Applications of Game-Theoretic Solutions in RICE

137

open-loop assumption is still too unrealistic, despite its apparent advance from recursive models, for a comprehensive IA framework. The open-loop decision structure can be improved conceptually by adopting closed-loop or feedback structure. The distinction between open-loop and closed-loop strategies can be found in the literatures of optimal control theories and game theory (for example, Holly and Hughes Hallett 1989; Fudenberg and Tirole 1991). While the advantage of closed-loop strategies over open-loop strategies is easy to establish conceptually because they represent broader strategic situations, implementing closed-loop strategies in empirical models is difﬁcult because of computational complexity. In Yang 2003b, a sequential algorithm that decomposes a discrete closed-loop strategy problem into sequences of open-loop strategies is developed. In addition, the algorithm of such decomposition is applied to RICE in the paper to answer some questions related to coalition formation in GHG mitigation. Here, I extend this approach, treat it as a new solution concept in RICE, and use it to answer some policy questions.3 Instead of treating an open-loop solution as a single decision made at year 2000 for the next 250 years, we can envision that the model solution is a sequence of decisions made at the beginning of each period (5 years per period in RICE). At each decision juncture, the outcome from the previous period is the initial condition for the current decision. If the decision maker has perfect foresight and there are no unexpected shocks, the two approaches will render an identical result. If there is new information available at T ¼ t that was unknown at T ¼ t 1, it will be incorporated into the decision made at T ¼ t. In such situations, the two approaches yield different results. In the second approach, piecemeal solutions are extracted from each decision juncture to constitute a closed-loop or feedback strategy proﬁle for the entire planning horizon. The sequential algorithm of closed-loop strategies expands the application areas of cooperative game solution concepts in RICE, at the expense of much longer computation time. One important area to investigate is the sustainability or stability of the original grand coalition from various unexpected shocks. The following is an interesting question that can be answered: Is the Lindahl equilibrium outcome renegotiation-proof under newly acquired knowledge on climate damage and/or mitigation costs, or under new scientiﬁc conclusions in climatology (such as the probabilities of catastrophic climate events increasing from a to b, and b X a)?

138

Chapter 6

In this study, I conduct a few experiments in a closed-loop paradigm that extends investigations in previous chapters. The ﬁrst experiment is related to ROW. As I conceded, trajectories of state and control variables as well as strategic postures of ROW are controversial in RICE. I explained the calibrations and tested alternative assumptions of ROW. The key issue is that ROW is a decentralized bloc put together for modeling convenience and is not a decision entity. Taking these factors into account, the cooperative game outcomes can be quite different (see the simulation results in section 5.2.2.). Here I look at this issue ‘‘dynamically.’’ As climate change becomes more imminent and apparent over time, regions might be more conscious of its presence. When climate change becomes a clear and present danger and regions become richer from economic growth, the tendency for cooperation might be stronger. If ROW reduces their internal inefﬁciencies gradually, what will happen to the initial grand coalition formed as the Lindahl equilibrium outcome? I use the following parametric assumption to mimic such a shift. In the next twenty years, ROW gradually reduces its internal inefﬁciency. In addition, the magnitude of reduction is not known beforehand. I use the same design as in section 5.2.2, where n is used as an adjustment factor in damage function, to reﬂect internal inefﬁciency over time. That is, T ¼ 1 and n ¼ 20; T ¼ 2 and n ¼ 15; . . . T ¼ 5 and n ¼ 1. These changes are not foreseen in the periods before they occur. For example, at T ¼ 1, ROW and other regions have no information that n will reduce to 15 in the next period. As shown in table 5.6, the Lindahl weight is j L ¼ ð1:3559; 1:5549; 0:9424; 0:1740; 1:0587; 0:9140Þ when n ¼ 20. Comparing with the default assumption n ¼ 1, the above weight for ROW is substantially higher and other regions’ weight share reduces proportionally. As I have explained in the previous chapter, setting n ¼ 20 is more realistic. If the initial grand coalition is formed with ROW’s internal inefﬁciency (namely, n ¼ 20), can this grand coalition survive when ROW improves its internal efﬁciency gradually? The simulations from the sequential algorithm show the following patterns. When T ¼ 2, n reduces from 20 to 15. The three incentivechecking tables (tables 3.3, 3.4, and 3.5) begin to show incentive incompatibilities. In table 6.1, we observe that the grand coalition under the original Lindahl weights might not be in the core, if USA can fully compensate the other four regions in the {USA, OHI, EU, CHN, EEC} coalition (row 2). Further incentive checking in table 6.2 reveals that

Policy Applications of Game-Theoretic Solutions in RICE

139

Table 6.1 Welfare change from grand coalition ðT ¼ 2Þ No.

USA

OHI

EU

CHN

EEC ROW

OHI

EU

CHN

EEC

ROW

33

34

35

36

37

38

39

40

41

42

43

44

45

46

47

1 2

þ

3

4

5

6

7

8

9

10

11

12

13

14

15

16

17

USA

48

18

19

20

21

22

23

24

25

26

27

28

29

30

31

32

No.

49 50

51 52

53

54

55

56

57

58

59

60

61 62

63 64

140

Chapter 6

Table 6.2 Welfare changes from Nash equilibrium ðT ¼ 2Þ No.

USA

OHI

EU

CHN

EEC

ROW

No.

1

þ

þ

þ

þ

þ

þ

2

þ

þ

þ

þ

þ

3

þ

þ

þ

þ

4

þ

þ

þ

þ

5

þ

þ

þ

þ

6

þ

þ

þ

þ

7

þ

þ

þ

8

þ

þ

þ

9

þ

þ

þ

þ

10

þ

þ

þ

þ

11

þ

þ

þ

12

þ

þ

þ

13

þ

þ

14

þ

þ

þ

15

þ

16

þ

þ

17

þ

þ

þ

þ

þ

þ

þ

OHI

EU

CHN

EEC

ROW

33

þ

þ

þ

þ

þ

34

þ

þ

þ

þ

35

þ

þ

þ

36

þ

þ

þ

37

þ

þ

þ

38

þ

þ

þ

39

þ

þ

40

þ

þ

41

þ

þ

þ

42

þ

þ

þ

43

þ

þ

44

þ

þ

45

þ

þ

46

þ

þ

47

þ

þ

þ

þ

þ

þ

þ

48 þ

þ

þ þ

18

þ

þ

þ

19

þ

þ

þ

20

þ

þ

þ

21

þ

þ

þ

22

þ

þ

þ

23

þ

þ

24

þ

þ

25

þ

þ

þ

26

þ

þ

þ

27

þ

þ

28

þ

þ

29

þ

30

þ

þ

31

32

þ

USA

þ

49 50

þ

51 52

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

þ

53

þ

þ

54

þ

þ

55

þ

þ

þ

56 þ

þ

57

þ

þ

58

þ

þ

59

þ

þ

þ

60 þ

61 62

þ

63 64

þ

Policy Applications of Game-Theoretic Solutions in RICE

141

USA is worse off in some coalitions than in the Nash equilibrium. Therefore, the IR condition for USA is shaky. The grand coalition under the original Lindahl weights is not in the core. In table 6.3, we also see compromises of incentives, all related to USA. Such incentive incompatibilities continue to deteriorate and propagate to other regions when T ¼ 3 and n ¼ 10; T ¼ 4 and n ¼ 5; and T ¼ 5 and n ¼ 1. Avoiding redundancy, I skip the T ¼ 3 and T ¼ 4 cases and only report what happens when T ¼ 5. The results are shown in tables 6.4 through 6.6. The patterns in these tables indicate that all ﬁve regions, except for ROW, are unhappy with the arrangement at T ¼ 1, based on N ¼ 20. In table 6.4, all subcoalitions without ROW, ﬁfteen in total, block the grand coalition (all ‘‘þ’’ signs in a row). In table 6.5, all ﬁve regions except for ROW prefer the Nash equilibrium to the grand coalition. In addition, all regions are worse off in any subcoalition with ROW than in the Nash equilibrium. In table 6.6, only ROW has consistent ‘‘’’ signs. In sum, all regions rebel against the initial mitigation quota based on n ¼ 20 at T ¼ 1. The initial bargaining outcome needs to be renegotiated. The grand coalition with the original Lindahl weights collapses. While all ﬁve other regions maintain the consistent incentives among themselves (see all subcoalitions without ROW in tables 6.4 and 6.5), they demand that ROW do more. In such a case, renegotiation of the grand coalition will start before T ¼ 5. The storyline behind this set of experiments is the dissipation of false perception of climate change by ROW. Other regions accept such false perception at the beginning and negotiate accordingly. Once ROW realizes its mistakes, all regions should renegotiate the initial quota. In the new bargaining outcome, ROW will shoulder more responsibilities and the welfare weight of ROW will be lower. We can see shadows of the Kyoto Protocol in this story. In the Kyoto Protocol, no GHG mitigation obligations are required for developing countries for various reasons. But if climate change begins to do serious damage to the developing countries twenty years from now, economic development itself will not be an excuse for not participating in GHG mitigation: the impact of climate change hinders economic development directly. I hope that politicians and policymakers have better visions with respect to the future. If so, no region would shirk its responsibility in dealing with climate externalities. Unfortunately, myopia is a prevailing mode of thinking and we need closed-loop strategies to capture the consequences of such inefﬁcient decisions.

142

Chapter 6

Table 6.3 Welfare changes between coalitions ðT ¼ 2Þ No.

USA

OHI

EU

CHN

EEC

ROW

1 2

þ

3

4

5

6

þ

7

8

þ

9

10

þ

11

12

þ

13

14

þ

þ

15

16

þ

17

18

þ

19

20

þ

21

22

þ

23

24

þ

25

26

þ

27

28

þ

29

30

þ

31

32 33

34

35

36

Policy Applications of Game-Theoretic Solutions in RICE

143

Table 6.3 (continued) No.

USA

OHI

EU

CHN

EEC

37

38

39

40

41

42

43

44

45

46

47

ROW

48 49

50

51

52

53

54

55

56 57

58

59

60 61 62 63 64

144

Chapter 6

Table 6.4 Welfare change from grand coalition ðT ¼ 5Þ No.

USA

OHI

EU

CHN

EEC

ROW

1 2

þ

þ

þ

þ

þ

3

þ

4

þ

þ

þ

þ

5

6

þ

þ

þ

þ

7

8

þ

þ

þ

9

þ

10

þ

þ

þ

þ

11

þ

12

þ

þ

þ

13

14

þ

þ

þ

15

þ

16

þ

þ

17

þ

USA

OHI

EU

CHN

EEC

ROW

33

34

þ

þ

þ

þ

35

36

þ

þ

þ

37

38

þ

þ

þ

39

40

þ

þ

41

42

þ

þ

þ

43

44

þ

þ

45

46

þ

þ

47

48

þ

18

þ

þ

þ

19

þ

20

þ

þ

þ

21

22

þ

þ

þ

23

þ

24

þ

þ

25

þ

26

þ

þ

þ

27

þ

28

þ

þ

29

þ

30

þ

þ

31

þ

32

No.

49 50

51 52

þ

þ

þ

þ

þ

53

54

þ

þ

55

56

57

58

þ

þ

59

60

61 62

63 64

Policy Applications of Game-Theoretic Solutions in RICE

145

Table 6.5 Welfare changes from Nash equilibrium ðT ¼ 5Þ No.

USA

OHI

EU

CHN

EEC ROW

1

2

þ

þ

þ

þ

þ

3

4

þ

þ

þ

þ

5

6

þ

þ

þ

þ

7

8

þ

þ

þ

9

þ

þ

þ

þ

12

þ

þ

þ

13

þ

þ

þ

16

þ

þ

17

EEC

ROW

33

þ

34

þ

þ

þ

þ

35

36

þ

þ

þ

37

38

þ

þ

þ

39

40

þ

þ

þ

41

42

þ

þ

þ

þ

43

44

þ

þ

þ

45

46

þ

þ

þ

47

þ

USA

þ

þ

þ

þ

þ

þ

þ

48

þ

18

þ

þ

þ

19

20

þ

þ

þ

21

22

þ

þ

þ

23

24

þ

þ

25

26

þ

þ

þ

27

28

þ

þ

29

30

þ

þ

31

32

CHN

þ

10

14

EU

þ

11

15

OHI

þ

No.

þ

49 50

þ

51 52

þ

þ

þ

þ

þ

þ

þ

þ

þ

53

54

þ

þ

55

þ

þ

56 þ

þ

57

58

þ

þ

59

þ

þ

60 þ

61 62

þ

63 64

þ

146

Chapter 6

Table 6.6 Welfare changes between coalitions ðT ¼ 5Þ No.

USA

OHI

EU

CHN

EEC

ROW

1 2

þ

þ

þ

þ

3

þ

4

5

6

þ

7

þ

8

þ

þ

þ

9

þ

10

þ

þ

þ

11

þ

þ

12

þ

þ

þ

13

þ

14

þ

þ

þ

þ

þ

þ

þ þ

þ

þ

þ

þ

15

þ

þ

16

þ

þ

17

þ

18

þ

þ

þ

19

þ

þ

20

þ

þ

þ

21

þ

22

þ

þ

þ

23

þ

þ

24

þ

þ

25

þ

þ

26

þ

þ

þ

27

þ

þ þ

28

þ

þ

29

þ

þ

30

þ

þ

31

þ

þ þ

þ

32 33

34

þ

þ

þ

35

36

þ

þ

þ

þ

Policy Applications of Game-Theoretic Solutions in RICE

147

Table 6.6 (continued) No.

USA

OHI

EU

CHN

EEC

37

38

þ

þ

þ

39

40

þ

þ

41

42

þ

þ

þ

43

44

þ

þ

45

46

þ

þ

47

ROW

48 49

50

þ

þ

þ

51

52

þ

þ

53

54

þ

þ

55

56 57

58

þ

þ

59

60 61 62 63 64

148

Chapter 6

From a modeling point of view, the outcome of the above set of sequences of decisions shows that the grand coalition is not stable with respect to certain unexpected changes in region-speciﬁc (or private) information. Such changes alter the strategic positions among the regions. Depending on the circumstances, such unexpected shifts may infringe on the renegotiation proofness of the original grand coalition. Any international environmental agreement should be based on the best knowledge concerning environmental issues. Climate change is a gigantic scientiﬁc puzzle with only partial answers known. Predictably, in the next few decades, scientists will know more about climate change both from research and from new observations. Tentative conclusions on climate change today will certainly be revised in the future. Our question here is whether the cooperative game outcome with the strongest incentives (the Lindahl outcome) needs renegotiation when new knowledge about climate change is acquired in the future. In a reduced and simplistic form, revised knowledge can lead to either of two directions: climate change may be more severe than it is predicted today or less severe. RICE incorporates an extremely simple box module of carbon cycle. It cannot address the issue from a scientiﬁc perspective but can simulate, in a simple way, the revision of scientiﬁc knowledge through reparameterization of the carbon module over time. To simulate a new understanding of the severity of climate change, I assume that there are unexpected shifts in the value of e3 in equation (A-11). e3 is the coefﬁcient of radiative forcing. If it increases, a unit of GHG emission causes larger atmospheric temperature increases, and a decrease in it causes lower temperature increases. In the following sets of simulations, I assume there are unexpected increases and decreases of e3 . From its default value of 0.1075, e3 decreases to 0.105, 0.1025, 0.100, and 0.0975 incrementally between T ¼ 2 and T ¼ 5; it increases to 0.110, 0.115, 0.120, and 0.125 in another set of simulations. Again, these changes are not foreseeable from decision makers’ perspectives. We investigate the renegotiation proofness of the original Lindahl equilibrium outcome in facing these common knowledge shocks. When e3 decreases unexpectedly, it implies that climate change will cause less damage than what people originally thought. Regions will adjust their control variable trajectories accordingly. In this case, regions will lower their respective GHG control rate m i ðtÞ in response to lower damages. In contrast to these predictable outcomes, there is an interesting question concerning the incentives of regions: Can the

Policy Applications of Game-Theoretic Solutions in RICE

149

original Lindahl equilibrium outcome survive these shocks? That is, we want to know whether regions’ incentives change under the original Lindahl social welfare weights. Our simulations show that the three incentive checking tables (tables 3.3 through 3.5) in all four decision junctures maintain homogeneous signs under the original Lindahl weights and consecutive shocks of e3 . (The tables are not shown here.) The outcomes concerning regional incentives are not apparent from conventional modeling. These incentive properties are revealed only through meticulous and lengthy simulations. Conceptually, they are consistent with the term ‘‘no regret policy’’ referred to frequently in the IA literature. ‘‘No regret’’ should not mean that there are no adjustments of actual responses to newly acquired knowledge; no regret here implies that no region would regret its signature on the initial agreement on GHG mitigation burden sharing. Facing a lower threat of climate change, regions are content with reducing GHG mitigation proportionally under the same Lindahl weights. Preservation of incentives or renegotiation proofness in response to lowered climate damage is one of many good properties of the Lindahl equilibrium. Finally, to test the robustness of renegotiation proofness of the original Lindahl equilibrium in facing reduced climate damage, we experimented with an extremely low value of e3 . It is e3 ¼ 0:05, at T ¼ 5.4 The signs in the three tables are still homogeneous. That is, the renegotiation proofness of the Lindahl equilibrium is generally true in the direction of lower climate damages. The second set of simulations on increasing climate damage is intentionally ‘‘exaggerated.’’ At T ¼ 5, the value of e3 is threefold higher than the default. Some scientists predict catastrophic consequences of climate change. This set of experiments mimics what happens to the grand coalition if catastrophe or much graver damage materializes unexpectedly. The incentive results from unexpected increases in climate damage are identical to those in decreasing climate damage. Confronted with signiﬁcantly increased climate damage, regions still honor the original Lindahl equilibrium arrangement. All three tables maintain homogeneous signs through the sequence of shocks (tables not shown). As I have stated repeatedly, such invariance or renegotiation proofness does not imply unresponsiveness in control and state variable paths. Much higher GHG control rates are expected from all regions in facing such adverse shocks. The conclusion here states that regions would

150

Chapter 6

follow the initial share of obligations negotiated through the cooperative game. Actions under the Lindahl weights preserve the strongest incentives under such shocks. The simulations here suggest that renegotiation proofness of the Lindahl equilibrium holds when there are common knowledge shocks (or symmetric shocks); it may fail when there are asymmetric shocks, such as changes in mitigation costs or climate damage in one or more regions. (In Yang 2003b, extensive simulations are performed with asymmetric shocks.) Considering the conclusions from the stability analysis of the Lindahl equilibrium in section 5.1.1 and here, we have more comprehensive information on the good properties of the Lindahl equilibrium. The cooperative game and feedback solution algorithm form a powerful analytical tool. A combination of the two can be used in many other strategic situations involving future shocks. I do not intend to visit them exhaustively here. The approach here provides the necessary tools to conduct these analyses when needed. 6.3 Distribution Analysis of GHG Mitigation Policies from Strategic Perspectives In debating GHG mitigation policies, modelers often have trouble separating positive and normative issues. Some scholars have studied the distributional issues of GHG mitigation policies extensively, such as in Rose, Stevens, and Edmonds 1998. In fact, distribution analysis is a component of most IA models. Any speciﬁc GHG mitigation policy has its distributional implications. The most signiﬁcant distributional consequence of GHG mitigation policies involves monetary transfers in carbon permit trading or other bilateral/multilateral cooperation. Conclusions from IA modeling on distributional issues are wideranging. Some require huge amounts of unilateral transfers from industrialized countries to developing countries. The rationale for such transfers is mainly technological. Developing countries might reduce GHG emissions less expensively with advanced technologies from developed countries. If developed countries can take credit for such mitigations, they are willing to pay for them. Others suggest reversal of the transfer direction. Because developing countries are more vulnerable to climate change damage, they should encourage developed countries to mitigate more GHG emissions by providing them with ﬁnancial incentives.

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From a global perspective, transferring wealth from rich countries (those regions with lower marginal utility of income) to poor countries will increase global welfare. However, such transfers are feasible only with the consent of donors. There is no ‘‘federal government’’ to enforce such redistributions. In IA modeling, regional welfare changes are the results of GHG mitigation policies. If the IA model allows free transfers at the ‘‘social planner’s’’ discretion, the monies will ﬂow with or without GHG mitigation policies. An IA model with utilitarian social welfare weights and open channels for transfers has exactly such properties: a portion of transfers is to improve cost-effectiveness of GHG emissions through permit trading; a portion of transfers is to improve global welfare through pure wealth transfers. If a social welfare function is used as the objective function in an IA model, sometimes it is difﬁcult to separate redistribution from efﬁciency. I believe that the huge amounts of transfers recorded in the literature are partially due to model speciﬁcations instead of necessities from GHG mitigation policies. Redistribution of wealth and efﬁciency enhancements of GHG mitigation policies are intertwined in the calculations. In addition, incentive consideration is not a part of those calculations. Though global welfare might be improved, the outcome is not necessarily compatible with an individual country’s incentives at the starting points of such calculations. Transferors are not willing to donate the monies if they end up worse off. Taking incentive compatibility into account, the utilitarian solution should not be treated as a feasible starting point for transfers. The results from the calculations with the utilitarian approach are misleading. In this study, I examine the distributional changes of GHG mitigation policies by separating mitigation policy consequences from general wealth redistributions. This makes it possible to identify the necessary and minimum amounts of welfare changes in achieving the policy target. Concretely, I calculate the necessary amounts of transfers for equalizing global GHG marginal mitigation costs from different benchmarks. The procedures include a series of side calculations. First, I calculate equalized global marginal costs of GHG mitigations that meet the same global emission paths in different benchmarks. I select the Lindahl and utilitarian scenarios in chapter 4 for this analysis. The Negishi outcome, as the MC equalizing outcome without transfers, is also presented for comparison purposes. Second, I calculate regional GHG control rate changes necessary to equalize regional MC from the

152

Chapter 6

Figure 6.1 Equalized marginal GHG mitigation costs

Lindahl and utilitarian outcomes. Third, I calculate regional mitigation cost changes between the benchmarks (Lindahl or utilitarian) and the respective MC equalization scenarios. The difference between total reduced costs and increased costs across regions is the net gain (if it is positive) from the potential transfer scheme that equalizes the marginal mitigation costs across regions. In such an MC equalization scenario, those regions with lowered costs can fully compensate the regions incurring higher costs and still have something left over (surplus) to improve global welfare. The paths of equalized marginal GHG mitigation costs are plotted in ﬁgure 6.1. The paths of the Lindahl and Negishi outcomes are close. The range of marginal mitigation costs is between $5 per ton of carbon in early years to $60 and $80 in late years respectively. As ﬁgure 4.18 has shown, the global GHG emission paths of the Lindahl and Negishi outcomes are very close (they are almost indistinguishable under the resolution of the graph). The slightly lower MC paths of the Lindahl outcome might be due to the efﬁciency improvement of the transfer scheme and to the slightly higher global GHG emission paths (in ﬁgure 4.18, global GHG emissions of the Negishi outcome are slightly lower than that of the Lindahl outcome). The utilitarian outcome demands much higher equalized MC globally. Its range is from the mid $10s to $140 per ton of carbon during the time horizon I plotted. The reason

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Figure 6.2 GHG control rates reduction (Lindahl)

for such high marginal GHG mitigation costs is that the utilitarian outcome results in much more stringent global GHG emissions (again, see ﬁgure 4.18). Collectively, it requires that each region engage in more GHG mitigation efforts to achieve such a target. Without transfers, it asks developed countries to mitigate huge quantities of GHG emissions (see ﬁgures 4.6 and 4.7). Consequently, the path of equalized MC of the utilitarian outcome is through transferring and from an incentive-incompatible initial quota arrangement. Figures 6.2 and 6.3 reveal more details about distributional implications of a transfer scheme in the Lindahl outcome. Figure 6.2 depicts the differences between the regional GHG control rates in the Lindahl equilibrium outcome and the equalized MC outcome. It shows that CHN and EEC need to control more GHG emissions to achieve equalized MC globally. Increased GHG emission control rates in these two regions are moderate. They are around a 4 percent increase for CHN and a 10 percent increase for EEC. As a result of such increases, they are the recipients of transfers. Other regions’ GHG control rates reduce slightly, ranging from 2 percent to 4 percent. They are donors in the transfer scheme for equalization of global marginal costs of GHG mitigations. As for total mitigation costs, extra burdens from increased control rates in CHN and EEC, as well as reduced burdens in other regions,

154

Chapter 6

Figure 6.3 Total GHG mitigation cost reductions (Lindahl)

are extremely moderate. Figure 6.3 conﬁrms the results. In the graph, the differences between regional mitigation costs in the Lindahl benchmark and equalized MC outcome are plotted. A negative reading implies mitigation cost increases in the latter. CHN and EEC have to increase their GHG control rates; thus, their costs increase. Other regions save the GHG mitigation costs. For most regions, they are measured in billions of US dollars in a century-long time span. Only ROW gains more signiﬁcantly than others.5 Small mitigation cost changes here indicate that only small adjustments are needed in regional control efforts to achieve global equalization of marginal mitigation costs. This result is further proof of the most desirable properties of the Lindahl outcome from the cooperative game in the RICE model. Without any transfers, the initial allocation of GHG mitigation obligations under the Lindahl scheme is very close to the efﬁciency frontier. That is, the original Lindahl allocation that satisﬁes efﬁciency condition (2.11) is very close to the allocation that satisﬁes efﬁciency condition (2.12). Coupled with the strongest incentives from regions, the Lindahl outcome is the most cost-effective and enforceable coalition structure to achieve efﬁciency through transfers. The outcome of the utilitarian case tells a dramatically different story (see ﬁgures 6.4 and 6.5). Regions have to change their GHG control rates signiﬁcantly to equalize the marginal mitigation costs globally.

Policy Applications of Game-Theoretic Solutions in RICE

Figure 6.4 GHG control rates reduction (utilitarian)

Figure 6.5 Total GHG mitigation cost reductions (utilitarian)

155

156

Chapter 6

Developed countries (USA, OHI, and EU) see their control rates reduced from 25 percent to 10 percent; developing countries’ control rates increase from 6 percent to 10 percent (ﬁgure 6.4). For such huge changes in mitigation efforts, regions’ GHG mitigation costs also change considerably. The mitigation cost reductions in developed countries are in the range of 100 billion US dollars per ton of carbon (ﬁgure 6.5). Because the utilitarian allocation requires developed countries to control GHG emission at a very high MC (ﬁgure 4.33), the savings from reduced mitigation obligations in these regions are huge. On the other hand, developing countries’ mitigation costs increase moderately because their original MCs are extremely low. Finally, to assess distributional implications of mitigation cost equalization, I calculated differences between reduced and increased mitigation costs across regions. I expect the difference to be positive because MC equalization is a Pareto improvement. The difference is the maximum gains of MC equalization from the respective benchmarks. Figure 6.6 depicts these gains. Serving as a trivial reference, the Negishi scheme has no gains, because there are no transfers that take place. The gains of the Lindahl outcome are from 0.5 to 15 billion US dollars over time. Again, such moderate gains show that the initial Lindahl allocation is in the neighborhood of the efﬁciency boundary. Neverthe-

Figure 6.6 Total gains from MC equalization

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157

less, the gains can make every region better off by equalizing marginal GHG mitigation costs through certain policy measures, such as a tradable emission permit scheme. The utilitarian case shows much higher gains of up to 300 billion US dollars. Such gains are mainly due to disparate marginal mitigation costs across regions in the utilitarian benchmark. Higher gains here do not mean better results at all. As I have analyzed in chapter 4, the initial GHG mitigation obligations under the utilitarian weights cannot be an outcome from coalition negotiation. The welfare improvements by equalizing MC are measured against a distorted and incentiveincompatible benchmark. Direct reading of such gains without referencing its benchmark is misleading. Although potential transfers/side payments from the utilitarian benchmark are signiﬁcant compared with the Lindahl outcome in RICE, they are still much lower than similar calculations in many other studies. The reason is that we rule out pure wealth redistributions and only capture the potential gains (or losses) necessitated by equalization of marginal GHG mitigation costs across regions. In general, the distributional implications of GHG mitigation policies should not be signiﬁcant, as long as the initial ‘‘quotas’’ are negotiated through the cooperative game deﬁned in this research. 6.4

The Second-Best Subcoalitions of GHG Mitigations

I ‘‘condemn’’ some popular solutions of a social planner’s problem as incentive incompatible in the context of the RICE model. My extensive and multiangle analyses support such a conclusion. I also demonstrate the superior properties of incentive-compatible solutions as the outcomes of the cooperative game of bargaining the share of GHG mitigation obligations. However, the cooperative game solution concepts related to the grand coalitions, such as the Lindahl equilibrium and the Shapley value allocation, can also be blamed as unrealistic and unattainable. Observing what has taken place on the international stage since the adoption of the Kyoto Protocol, such criticisms are quite legitimate. Ten years after its adoption, the Kyoto Protocol, which I characterized as a unilateral action pact, and other proposals for global cooperation in GHG mitigation are in limbo. Although most countries recognize that climate change is an imminent global environmental problem, they are still far apart on how to respond to it. As of 2007,

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the United States demands that China and India (a large country in the ROW bloc of RICE) commit to GHG emission reductions; China, on the other hand, states publicly that it will not share any GHG mitigation obligations in the near future. Such impasses prompt people to pay more attention to subcoalitions of GHG mitigations, both in scholarly research and in policy implementation ( JI and CDM are subcoalitions sanctioned by the Kyoto Protocol and in practice). Since climate change is a stock externality phenomenon, only global cooperation (a grand coalition) can achieve efﬁciency. Subcoalitions might well be Pareto improvements over the noncooperative Nash equilibrium, but they deﬁnitely are still inefﬁcient. Therefore, any subcoalition format, such as JI or CDM, is preferred to the noncooperative status quo. Economic intuition tells us that the superadditive property (deﬁned in section 2.3.3) holds for the cooperative game of stock externality provision in general. If so, the ‘‘size’’ and membership of subcoalitions matters. A larger coalition is always a Pareto improvement on a smaller coalition. Efﬁciency arguments with respect to subcoalitions are very clear. If one or more major GHG emitters insist on not joining the grand coalition, those who are willing to join have to conﬁne the bargaining efforts to the willing partners. Such an outcome is a typical secondbest scenario. The issue is whether the conclusions drawn here for the grand coalition are still valid for subcoalitions. The answer is a resounding yes. The bargaining of social welfare weights (initial GHG mitigation quotas) among the willing participants and execution of the social planner’s optimal control problem should be the same as for the grand coalition. That is, the relative share in social welfare weights are the same as in the grand coalition. Here I take the default Lindahl outcome as an example. The Lindahl weight j L ¼ ðj USAL ; jOHI L ; jEU L ; jCHN L ; jEEC L ; jROW L Þ ¼ ð1:574; 1:805; 1:094; 0:202; 1:229; 0:096Þ. If for whatever reason, CHN and ROW do not want to join this grand coalition, the weight j L ¼ ðj USAL ; jOHI L ; jEU L ; jCHN L Þ ¼ ð1:574; 1:805; 1:094; 0:202Þ has the strongest incentives for all four remaining members. The Lindahl outcome for the 4member coalition has this weight. In table 3.5 (welfare changes between coalitions), the Lindahl outcome has homogeneous ‘‘’’ signs. Therefore, no subcoalitions of this 4-member coalition can block it. In addition, in the incentive checking table 3.4 (welfare changes from Nash) under the Lindahl weights, all signs are ‘‘þ.’’ In particular, all

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159

Figure 6.7 Regional GHG control rates in a ‘‘hybrid’’ Nash equilibrium

four entries in row 6 have ‘‘þ’’ signs. The patterns indicate that this 4-member coalition meets the IR and IC conditions on its own. The above conclusion is based on the fact that the 4-member coalition {USA, OHI, EU, EEC} bargains and cooperates inside the coalition while playing the open-loop Nash game with CHN and ROW strategically. China’s openly stated position that it puts economic development before GHG mitigation seems to ﬁt this scenario. Figure 6.7 records a snapshot of this ‘‘hybrid’’ Nash equilibrium or the secondbest outcome for the 4-member subcoalition. It shows the regional GHG control rates. Compared with the grand coalition under the Lindahl equilibrium (ﬁgure 4.12), the members of the 4-member coalition scale back their control efforts moderately, instead of exerting more control efforts to compensate for CHN and ROW’s nonparticipation. On the other hand, CHN and ROW’s GHG control rates are dramatically lower. Based on this second-best outcome, coalition members can engage in emission permit trading to equalize the marginal mitigation costs internally or communicate with CHN and ROW to open channels for JI or CDM. All of the above can improve the cost-effectiveness of GHG mitigation. Finally, CHN and/or ROW may take the position in the BaU scenario ðmðtÞ ¼ 0Þ while the other four regions form a GHG mitigation coalition. In this case, the conclusion is still the same. Now the

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4-member coalition would treat the GHG emission from CHN and ROW as ‘‘natural’’ instead of ‘‘anthropogenic.’’ Such shifts of common knowledge would not change the relative shares of GHG mitigation obligation inside the coalition. In sum, the modeling methodologies regarding the grand coalition elaborated in this study also apply to the analysis of second-best subcoalitions. However, the number of different possible subcoalitions increases combinatorially as the membership of the potential grand coalition increases. Examining all of them might be repetitive and exhausting. Given the proper policy context, our method can be used to investigate any particular subcoalitions.

Epilogue: Further Research Directions

In this study, I incorporate game-theoretic solutions into a well-known IA model. The extensive simulations of game-theoretic solutions in the RICE model have shown that the regional incentive to join the global coalition of GHG mitigation is an important aspect of IA modeling. Ignoring it would lead to biased and even absurd conclusions in policy analysis of GHG mitigations. Ideally, economic modeling would capture economic behavior in real life. Here such a luxury is partially achieved through game-theoretic solutions in IA modeling: major players’ performances in international negotiation of GHG mitigations are largely predicted in our simulations and analyses. The roles of economics in IA frameworks are in the transition process from straightforward cost-beneﬁt assessments of GHG mitigation policies to in-depth behavioral analysis of international coalitions in GHG mitigation, both in theory and in practice. This research heralds and echoes such transformations. Recognizing the importance of incentives in IA modeling, it is natural to ask whether the modeling methodologies explored here can be extended to other IA frameworks. To a certain degree, the answer is simple: as long as the economic model in the IA framework treats climate change as stock externalities, implementations of game-theoretic solutions in them would not be complicated. The crucial element for game-theoretic solutions in IA modeling is that GHG mitigation costs and climate damages (external effects) have to be in a uniﬁed framework of trade-offs. Otherwise, it would be very difﬁcult to frame strategic postures of regions and there would be no ‘‘game.’’ Bridging the IA models and game-theoretic solutions is deﬁnitely feasible in many existing IA frameworks. While the research in this book is elaborate, it is just the beginning of a wide array of potential research topics. Incorporating incentive

162

Epilogue

components in IA frameworks and seeking game-theoretic solutions in IA models meet the new demands from policy and science arenas of climate change. Some of the interesting research issues include: Coalition formation in an uncertain system Uncertainty is an intrinsic feature of an anthropogenic-climatic system. Facing multifaceted, uncertain, and long-term impacts of climate change, decision makers can hardly sustain a ‘‘one-shot’’ strategy. Although I have shown that the grand coalition under the Lindahl weights can withstand several types of shocks, the investigation is far from complete. In a comprehensive uncertainty analysis, we should identify the sources and ranges of system uncertainties through meticulous modeling in the IA framework, and then investigate coalition formation inside this stochastic system. For example, possible abrupt and catastrophic changes in the climate system have caused serious concerns among scientists and policymakers. The mechanism (or ‘‘triggers’’) of such abrupt changes can be built into the carbon cycle modules in an IA model like RICE. It would be interesting to see how such a possibility of abrupt changes in the climate system affects regions’ incentives toward the grand coalition.

•

In-depth analysis of subcoalitions In the real world of GHG mitigations, the coalition formation is ‘‘bottom-up’’ rather than setting up at the ‘‘top’’ directly. That is, we observe local or regional coalitions on GHG mitigations before reaching full-blown global collaboration. Various schemes of JI and CDM are such examples. Although I have shown that subcoalitions dealing with global stock externalities are the second-best scenarios, they deserve closer examination. Studying incentives to form a subcoalition in the context of IA modeling would reveal interesting strategic behaviors of the regions.

•

Structuring the ‘‘closed-loop’’ strategies in an IA modeling/game-theoretic solution framework The long-term impacts and profound uncertainties of climate change make the ‘‘open-loop’’ strategies or ‘‘one-shot’’ games precarious oversimpliﬁcations of the real world. Because of cumulative effects in GHG emissions and climate change, the regions’ decision spaces are changing and evolving over time. Therefore, no simple repeated game structure can ﬁt such background stories. With the help of faster computing technologies, the methodologies utilized in section 6.2 should be the norm, rather than an extension, of a modeling framework. Once the closed-loop strategies are built in, we will have a much better understanding of regions’ incentives in coalition formation for GHG mitigations.

•

Further Research Directions

163

Involvement in policy-related issues The IA models are set up for responding to policy debates. For any policy issue related to GHG mitigation or adaptations to climate impacts, there will be a gametheoretic angle to assess it. As we all recognize, any international environmental policy initiative can only rely on regions’ voluntary participation. The incentive compatibility is the key factor for cooperation. Cost-beneﬁt analysis is important, and ‘‘incentive analysis’’ is probably even more important.

•

In summary, IA modeling as a comprehensive and multidisciplinary approach to studying climate change is being reﬁned and expanded. There are many ways to incorporate game-theoretic solution concepts in an IA framework. As more and more IA models adopt such an approach, we gain a better grasp of human behavior (sometimes represented by the decision makers) in this ‘‘revolution of nature’’ stirred by human beings.

Appendixes

Appendix 1: The Description of the RICE Model 1 The RICE Model as a Social Planner’s Problem (an Optimal Control Problem): Max

fIi ðtÞ; m i ðtÞg 6 X

W¼

ji ðtÞ ¼ 6;

6 X

Ui ¼

i¼1

6 ðT X i¼1

0

ji ðtÞLi ðtÞ LogðCi ðtÞ=Li ðtÞÞed t dt;

(A-1)

0 < d < 1:

i¼1

s:t: Qi ðtÞ ¼ Ai ðtÞKi ðtÞ g Li ðtÞ 1g

(A-2)

Yi ðtÞ ¼ Wi ðtÞQi ðtÞ

(A-3)

Ci ðtÞ ¼ Yi ðtÞ Ii ðtÞ

(A-4)

K_ i ðtÞ ¼ Ii ðtÞ dK Ki ðtÞ;

0 < dK < 1:

Ei ðtÞ ¼ ð1 mi ðtÞÞsi ðtÞQi ðtÞ; Wi ðtÞ ¼

1 b1; i mi ðtÞ b2; i ; 1 þ a1; i T1 ðtÞ a2; i

0 a mi ðtÞ a 1:

ðIn ðA-2Þ to ðA-7Þ; i ¼ 1; . . . ; 6:Þ

_ ðtÞ ¼ b MðtÞ þ b UU ðtÞ þ M 11 13

6 X

Ei ðtÞ

(A-5) (A-6) (A-7)

(A-8)

i¼1

_ L ðtÞ ¼ b ML ðtÞ þ b MU ðtÞ M 22 23 _ U ðtÞ ¼ b MðtÞ þ b ML ðtÞ þ b MU ðtÞ M 31 32 33

(A-9) (A-10)

166

Appendixes

T_ 1 ðtÞ ¼ e11 T1 ðtÞ þ e12 T2 ðtÞ þ e3 FðtÞ

(A-11)

T_ 2 ðtÞ ¼ e22 ðT2 ðtÞ T1 ðtÞÞ

(A-12)

FðtÞ ¼ h1 LogðMðtÞÞ h2 þ OðtÞ

(A-13)

Deﬁnitions of Variables: Ui

present value of intertemporal utility of region i;

Qi (t)

production function of region i;

Yi (t)

adjusted production function of region i;

Ci (t)

consumption function of region i;

Ki (t)

capital stock level of region i;

Ii (t)

investment function of region i (control variable);

Ei (t)

GHG emission of region i;

mi (t)

GHG emission control rate of region i (control variable);

Wi (t)

adjustment function of GDP;

M(t)

GHG concentration (atmospheric);

ML (t)

GHG concentration (deeper ocean);

MU (t)

GHG concentration (upper ocean);

T1 (t)

atmospheric temperature;

T2 (t)

deep ocean temperature;

F(t)

radiative forcing function.

Deﬁnitions of Time-variant Parameters: Li (t)

labor (population) trend of region i;

Ai (t)

total factor productivity trend of region i;

si (t)

exogenous trend of GHG emission/output ratio of region i;

O(t)

exogenous radiative forcing;

ji (t)

social welfare weight of region i.

2

The RICE Model as an Open-Loop Differential Game: Max

fIi ðtÞ; m i ðtÞg

s.t.

ðT

Li ðtÞ LogðCi ðtÞ=Li ðtÞÞedt dt;

0

(A-2) to (A-7) and (A-9) to (A-13)

0 < d < 1;

i ¼ 1; 2; . . . ; 6: (A-14)

Appendixes

_ ðtÞ ¼ b MðtÞ þ b UU ðtÞ þ M 11 13

167

6 X

(A-8 0 )

Ej ðtÞ þ Ei ðtÞ

j0i

3 Parameter Values of the RICE Model: (See GAMS code of the RICE model in appendix 3) Appendix 2: GAMS Code of Example 2.1 *Social planner’s solution. Set N /Siberia, Fiji/; Alias (N,M); Parameters fai(n) Social welfare weights a(n) Utility share of private good /Siberia 0.8 Fiji 0.7/ c(n) Technology coefficients /Siberia 0.7 Fiji 0.3/ d(n) Damage coefficients /Siberia 0.8 Fiji 0.9/ e(n) Benefit coefficients /Siberia 3 Fiji 3/ ; fai("Siberia") = 0.9; fai("Fiji") = 1 - fai("Siberia"); Variables X(n) Private good b(n) Public bad U1 Individual utility U2 Individual utility W Social welfare function Positive variables X, b; Equations WW Social welfare function UU1 Individual utility function UU2 Individual utility function FF(n) Production transformation function

;

168

Appendixes

WW.. W =e= fai("Siberia")*U1 + fai("Fiji")*U2; UU1.. U1 =e= e("Siberia")*X("Siberia")**a("Siberia")(sum(m, b(m)))**d("Siberia") ; UU2.. U2 =e= e("Fiji")*X("Fiji")**a("Fiji")-(sum(m, b(m)))**d("Fiji"); FF(n).. b(n) =e= c(n)*X(n)**2; Model example1 /all/; Option solprint = off; X.lo(n) = 0.01; b.lo(n) = 0.01; Solve example1 maximizing W using NLP; Display X.l, b.l, U1.l, U2.l, W.l; * Nash equilibrium solution. Set iter /1*5/; Parameter be(iter, n); Loop(iter, b.lo(n) = 0.01; b.up(n) = 100; be(iter, n) = b.l(n); b.fx("Siberia") = be(iter, "Siberia"); Solve example1 maximizing U2 using NLP; b.lo(n) = 0.01; b.up(n) = 100; be(iter, n) = b.l(n); b.fx("Fiji") = be(iter, "Fiji"); Solve example1 maximizing U1 using NLP; Display b.l; ); Display X.l, b.l, U1.l, U2.l;

Appendix 3: GAMS Codes of the RICE Model (Core Part) $TITLE THE RICE MODEL: Version: RICE2007–y * THE RICE MODEL * William D. Nordhaus (Yale), Zili Yang (SUNY Binghamton) OPTION NLP= conopt; SETS T Extended horizon /1*50/ TB(T) Base period TT(T) Terminal period

Appendixes

N

169

Regions of the world /USA, OHI, EU, CHN, EEC, ROW/ MD Categories of initial economic-emission data /Y0, YPC0, K0, L0, E_F0, E_L0/ TD Catego ries of trend data /LGR, LGRGR, TFPGR, TFPGRGR, SIG_I, SIG_A, DEL_TREE/ CB Coefficients of cost and damage functions /A1, A2, B1, B2, DAM1, DAM2/; ALIAS (N, J); ALIAS (T, S); TB(T) = YES$(ORD(T) EQ 1); TT(T) = YES$(ORD(T) EQ CARD(T)); * assign one-dimension initial values and coefficients. SCALARS R Rate of social time /.03/ pref per year DELTAM Removal rate carbon /.0833/ per decade DK Depreciation rate of capital /.1/ per year GAMMA Capital elasticity in output /.3/ * Coefficients for climate module. MAT2000 Concentration in atmosphere /787/ 2000 (b.t.c.) MU2000 Concentration in upper strata /900/ 2000 (b.t.c) ML2000 Concentration in lower strata /21000/ 2000 (b.t.c) * Carbon cycle transition coefficients (percent per decade) TRAA Atmosphere to atmosphere /81.2899/ TRUA Upper box to atmosphere /15.95/ TRAU Atmosphere to upper box /18.7102/ TRUU Upper box to upper box /77.2584/ TRLU Deep oceans to upper box /0.1952/ TRUL Upper box to deep oceans /6.7916/ TRLL Deep oceans to deep oceans /99.8048/ C1 Climate-equation coefficient /.226/ for upper level

170

LAM C3 C4 TE0 TL0 SAT CS HLAL HGLA

Appendixes

Climate feedback factor Transfer coeffic. upper to /.440/ lower stratum Transfer coeffic for lower /.02/ level Initial atmospheric temperature (deg C above pre-ind) Initial temperature of deep oceans (deg C above pre-ind) Speed of adjustment parameter for atm. temperature Equilibrium atm temp increase for CO2 doubling (deg C) Coefficient of heat loss from atm to deep oceans Coefficient of heat gain /0.02/ by deep oceans Utility derivative scaling factor

/1.41/

/0.71/ /0.30/ /0.215/ /2.9078/ /0.44/

Q /.010/ Q1 /0.1/ Q2 /10/ ; * The following table contains the data of the economy and emissions in the initial *period: year 2000. * Y0 —GNP in 2000 US$ (trillions) * K0 —Capital stock in 2000 US$ (trillions) * L0 —Population (billions) * E_F0—GHG emissions from fuel uses (gigatons of CO2 equivalent) * E_L0—GHG emissions from land use changes TABLE INI_DATA(MD,N) Initial macroeconomic data USA OHI EU CHN EEC ROW Y0 9.7648 5.9180 7.3326 1.0807 0.7417 4.1070 K0 13.1989 12.892 12.2452 2.321 1.5417 6.8647 L0 0.2822 0.1807 0.3888 1.2625 0.3262 3.4011 E_F0 0.5911 0.5611 0.7828 0.7553 0.7218 1.4269 E_L0 0 0 0 0.04 0 0.56;

Appendixes

171

* The following table contains coefficients in exogenous trends TABLE TREND_DATA(TD,N) Trend data USA OHI EU CHN EEC ROW LGR 8.7 1 1 10.5 1.5 10.00 LGRGR 22.0 30.5 110.0 25.0 14.0 25.0 TFPGR 6.0 9 8.5 13 10 12 TFPGRGR 4 4 4 6 5 6 SIG_I 8 10.5 9 6.5 6 5.5 SIG_A 0.15 0.40 0.35 0.30 0.20 0.20; TABLE CB_DATA(CB, N) Coefficients in cost-benefit relationship USA OHI EU CHN EEC ROW A1 0.01102 0.01174 0.01174 0.01523 0.00857 0.02093 A2 1.5 1.5 1.5 1.5 1.5 1.5 B1 0.07 0.05 0.05 0.15 0.15 0.10 B2 2.887 2.887 2.887 2.887 2.887 2.887; PARAMETERS GL0(N) Initial growth rate coeff of labor TAXRATE(N,T) Carbon tax in period T and region N ETREE(N,T) Emissions from deforestation L(N,T) Population (millions) LCGR(N,T) Cumulative exponential population growth TFP0(N) Initial calibrated total productivity factor coef TFPCGR(N,T) Cumulative exponential productivity growth rate

172

Appendixes

TFP(N,T) Total factor productivity SIG0(N) Initial E\Y ratio coeff SSIG(N) DSIG(N) SIGMA(N,T) Emission-output ratio GSIG(N,T) LU(N,T) Carbon emissions from land-use change (GTC per year); * Population growth trend. LCGR(N,T)=(TREND_DATA("LGR,"N)/TREND_DATA("LGRGR,"N))* (1–EXP(-(ORD(T)-1)*TREND_DATA("LGRGR,"N)/ 100)); L(N,T)=INI_DATA("L0,"N)*EXP(LCGR(N,T)); * Productivity growth trend. TFP0(N) = INI_DATA("Y0,"N)*(1+CB_DATA("A1,"N)* (TE0/ 2.5)** CB_DATA("A2,"N))/(INI_DATA("K0,"N)**GAMMA* INI_DATA("L0,"N)**(1–GAMMA)); TFPCGR(N,T)=(TREND_DATA("TFPGR,"N)/TREND_DATA("TFPGRGR, "N))*(1–EXP(-(ORD(T)-1)*TREND_DATA("TFPGRGR,"N)/100)); TFP(N,T) = TFP0(N)*EXP(TFPCGR(N,T)); * GHG emissions/ output ratio trend. SSIG(N) = LOG(TREND_DATA("SIG_A,"N)); SIG0(N) = (INI_DATA("E_F0,"N)) / INI_DATA("Y0,"N); DSIG(N) = LOG(1 - LOG(1 + TREND_DATA("SIG_I," N) / 100) / SSIG(N)); GSIG(N,T) = SSIG(N) * (1–EXP(-DSIG(N)*(ORD(T)-1))); SIGMA(N,T) = SIG0(N) * EXP(GSIG(N,T)); * Deforestation rates ETREE(N,T) = INI_DATA("E_L0,"N)*(1–0.1)**((ORD(T)-1)/2); * Initial welfare weights. Parameter lb(t,n); lb(t,n) = 1; PARAMETERS RR(T) Discount factor FORCOTH(T) Exogenous force NN(N) NN0(N) DELTA(N) Delta function of coalition; RR(T) = (1+R)**(5*(1–ORD(T))); NN0(N) = 1;

Appendixes

NN(N) = NN0(N); DELTA(N) = 1; FORCOTH(T) = (-0.1965+(ORD(T)-1)*0.13465/2)$(ORD(T) LE 22) + 1.15$(ORD(T) GT 22); VARIABLES * Variables in economic equations. C(N,T) Consumption Y(N,T) Output YS(T) World aggregate output K(N,T) Capital stock I(N,T) Investment RI(N,T) Interest rate per annum MIU(N,T) Emission control rates E(N,T) CO2 emissions ES(T) World aggregate CO2 emissions * Variables in climate equations MAT(T) Carbon concentration in atmosphere (b.t.c.) MU(T) Carbon concentration in shallow oceans (b.t.c.) ML(T) Carbon concentration in lower oceans (b.t.c.) FORC(T) Radiative forcing TE(T) Atmospheric temperature (deg C from historical) TL(T) Ocean temperature (deg C from historical) * Variables in LHS of objective functions UT1 UT2 UT3 UT4 UT5 UT6 UTILITY1 UTILITY2 UTILITY UTILITA1 UTILITA2 UTILITA the maximand;

173

174

Appendixes

POSITIVE VARIABLES C, Y, K, I, MIU, TE, TL, MAT, MU, ML, E; EQUATIONS * Economy equations KK(N,T) Capital stock balance INVEQ(N,T) Definition of investment KB(N,T) Terminal condition of K YY(N,T) Production function YYS(T) Aggregate production KK0(N,T) Initial capital stock EE(N,T) Emission process EES(T) Aggregate emissions * Climate equations MMAT0(T) Starting atmospheric concentration MMAT(T) Atmospheric concentration equation MMU0(T) Initial shallow ocean concentration MMU(T) Shallow ocean concentration MML0(T) Initial lower ocean concentration MML(T) Lower ocean concentration FORCE(T) Radiative forcing TTE0(T) Initial temperature TEE(T) Temperature equation TTL0(T) Initial lower ocean temperature TLE(T) Lower ocean temperature equation * Objective functions UTT1 UTT2 UTT3 UTT4 UTT5 UTT6 UTILI UTILI1 UTILI2 UTILA UTILA1 UTILA2 Definition of maximand; * Economic module YY(N,T).. Y(N,T) =E= TFP(N,T)*K(N,T)**GAMMA*L(N,T)**(1– GAMMA) *(1–B3(T)*CB_DATA("B1,"N)*(MIU(N,T)**CB_DATA("B2,"N)))

Appendixes

175

/(1+(CB_DATA("A1,"N)/NN(N))*((TE(T)/2.5)**CB_DATA("A2, "N))); YYS(T).. YS(T) =E= SUM(N,Y(N,T)); INVEQ(N,T).. I(N,T) =E= Y(N,T) - C(N,T); KK0(N,TB).. K(N,TB) =E= INI_DATA("K0," N) ; KB(N,T).. R*K(N,T) =L= I(N,T); KK(N,T+1).. K(N,T+1) =E= 5*I(N,T)+(1–DK)**5*K(N,T); EE(N,T).. E(N,T) =E= (SIGMA(N,T)*(1–MIU(N,T))* TFP(N,T)*K(N,T)**GAMMA*L(N,T)**(1–GAMMA) +ETREE(N,T)); EES(T).. ES(T) =E= SUM(N,E(N,T)); * Carbon cycle module. MMAT0(TB).. MAT(TB) =E= MAT2000; MMU0(TB).. MU(TB) =E= MU2000; MML0(TB).. ML(TB) =E= ML2000; MMAT(T+1).. MAT(T+1) =E= MAT(T)*(TRAA/100) +5*ES(T)+ MU(T)*(TRUA/100); MML(T+1).. ML(T+1) =E= ML(T)*(TRLL/100)+(TRUL/100)* MU(T); MMU(T+1).. MU(T+1) =E= MAT(T)*(TRAU/100)+MU(T)*(TRUU/100) +ML(T)*(TRLU/100); FORCE(T).. FORC(T) =E= 4.1*LOG(MAT(T)/596.4)/LOG(2) +FORCOTH(T); TTE0(TB).. TE(TB) =E= TE0; TTL0(TB).. TL(TB) =E= TL0; TEE(T+1).. TE(T+1) =E= TE(T)+(SAT/2)*(FORC(T)-(4.1/ CS)*TE(T)-HLAL*(TE(T)-TL(T))); TLE(T+1).. TL(T+1)=E=TL(T)+(HGLA/2)*(TE(T)-TL(T)); * Objective functions for different solution procedures. UTT1.. UT1 =E= SUM(T,RR(T)*L("USA,"T)*LOG(C("USA,"T)/L("USA,"T))); UTT2.. UT2 =E= SUM(T,RR(T)*L("OHI,"T)*LOG(C("OHI,"T)/L("OHI,"T))); UTT3.. UT3 =E= SUM(T,RR(T)*L("EU,"T)*LOG(C("EU,"T)/ L("EU,"T))); UTT4.. UT4 =E=SUM(T,RR(T)*L("CHN,"T)*LOG(C("CHN,"T)/L("CHN,"T))); UTT5.. UT5 =E= SUM(T,RR(T)*L("EEC,"T)*LOG(C("EEC,"T)/L("EEC,"T)));

176

UTT6..

Appendixes

UT6=E=SUM(T,RR(T)*L("ROW,"T)*LOG(C("ROW,"T)/ L("ROW,"T))); UTILI.. UTILITY =E= R*SUM((T,N), DELTA(N)*LB(T,N)*RR(T)*L(N,T)* LOG(C(N,T)/L(N,T))); UTILI1.. UTILITY1 =E= SUM((T,N),DELTA(N)*LB(T,N)* RR(T)*L(N,T)*LOG(C(N,T)/(L(N,T)*100))); UTILI2.. UTILITY2 =E= 0*SUM((T,N),DELTA(N)*10*LB(T,N)*RR(T)*L(N,T)* LOG(C(N,T)/(L(N,T)*1000)))/.55; UTILA.. UTILITA =E= R*SUM((T,N), (1– DELTA(N))*LB(T,N)*RR(T)*L(N,T)* LOG(C(N,T)/L(N,T))); UTILA1.. UTILITA1 =E= SUM((T,N), (1– DELTA(N))*LB(T,N)*RR(T)*L(N,T)* LOG(C(N,T)/(L(N,T)*100))); UTILA2.. UTILITA2 =E= 10*SUM((T,N), (1–DELTA(N))* 10*LB(T,N) *RR(T)* L(N,T) *LOG(C(N,T)/(L(N,T)*1000)))/.55; *Upper and lower bounds Y.UP("USA,"T) = 20*INI_DATA("Y0,""USA"); Y.UP("OHI,"T) = 20*INI_DATA("Y0,""OHI"); Y.UP("EU,"T) = 20*INI_DATA("Y0,""EU"); Y.UP("CHN,"T) = 150*INI_DATA("Y0,""CHN"); Y.UP("EEC,"T) = 50*INI_DATA("Y0,""EEC"); Y.UP("ROW,"T) = 150*INI_DATA("Y0,""ROW"); I.LO(N,T) = 0.001; MIU.FX(N,T) = 0; MIU.FX(N,"1") = 0; MAT.LO(T)=0.8*MAT2000; ML.LO(T)=0.8*ML2000; MU.LO(T)=0.8*MU2000; TE.LO(T) = 0.9*TE0; TE.UP(T) = 8; TL.LO(T) = TL0; TL.UP(T) = 2.5; E.LO(N,T) = 0.1; OPTION ITERLIM = 99999; OPTION LIMROW = 0; OPTION LIMCOL = 0;

Appendixes

177

OPTION SOLPRINT = OFF; OPTION RESLIM = 99999; MODEL RICE /ALL/; RICE.OPTFILE = 1; * Negishi welfare weights searching. $include weigh04y.gms * BaU solution. $include mark04y.gms * Cooperative solution. $include coop04y.gms * Restart for full decentralized coalition equilibrium solutions. $include fulco04y-02.gms * Restart for subcoalition equilibrium solutions. $include fulco_sub_04y.gms * Restart for a particular coalition equilibrium solution. $include coal99y.gms * Nash solution. $include nash04y.gms Note: Each *.gms file fulfills certain programming or modeling functions. They are not presented here but are available on request.

Notes

Chapter 1 1. Each of these IA modeling undertakings produced a large number of publications, including model documentations. Here, I refer to their Web sites as the authoritative references.

Chapter 2 1. The similar solution concept was proposed in Chander and Tulkens 1995, 1997, where it is called ‘‘partial Nash equilibrium.’’ 2. See Myerson 1991 for a comprehensive discussion. 3. I rule out the situation of the unlikely counterexample offered by Muench (1972). 4. The point has been argued by Uzawa (2003) in a static setting. 5. This can be proved through a simple algebraic property of (2.11).

Chapter 3 1. Appendix 1 also contains deﬁnitions of state and control variables, as well as parameters in the model. 2. The twelve regions in RICE2007 include USA, European Union (EU), Japan ( JPN), Other High Income Countries (OHI), Eastern European Countries (EEC), Russia, China, India, Middle East countries, Sub-Saharan Africa, Latin America, and Other Asian Countries. 3. The rationale for such calibration and other parameterizations of RICE can be found in Nordhaus 1994. P ji ¼ 6g does not 4. In all scenarios except for the Negishi outcome, simplex S ¼ fji j have a time argument. That is, it is time-invariant. 5. The core GAMS codes of the RICE model used in this study are in appendix 3. The complete model codes are available from the author upon request.

180

Notes

6. Note that when m ¼ 5, there is only one region in the coalition. It is equivalent to the noncooperative Nash equilibrium (A-14). 7. The contents of this table are actually part of the studies and will be analyzed in chapter 4 in more detail. To save space, we do not use hypothetical examples in explaining methodologies. 8. Note that table 3.4 is not a ‘‘mirror reﬂection’’ of table 3.3. For example, in row #46, CHN has a ‘‘þ’’ in both tables. 9. More detailed analysis of this aspect will be conducted in chapter 5. 10. IC conditions are also tested for second and third categories of coalitions. To save space, I do not present them here. 11. In this study, the numerical precision of the Shapley value search procedure is controlled at 3 percent relative error and E-4 absolute error across regions.

Chapter 4 1. All ﬁgures of time paths are plotted for thirty periods in this research, unless otherwise noted. 2. In Nordhaus and Yang 1996, this phenomenon is dealt with by assuming different damage functions for ROW in the open-loop Nash equilibrium. 3. The majority of IA models do not model climate change as stock externalities. These models are used for cost-effective analysis of GHG mitigation under various assumptions. Such models are not subject to the criticism here. 4. In chapter 5, I analyze a case in which ROW has a much smaller Shapley value share. 5. Better numerical accuracy is achieved in ﬁgure 4.10 6. Some IA models that adopt utilitarian weights arrive at this conclusion.

Chapter 5 1. In RICE, members of the coalition do react to outsiders’ decisions by strategically following the deﬁnition of the ‘‘hybrid’’ Nash equilibrium. 2. Values in table 5.6 are the middle points of [D1, U1] of ROW.

Chapter 6 1. For example, a special issue of The Energy Journal, edited by Weyant and Hill (1999), is devoted to the cost-effectiveness of implementing the Kyoto Protocol. 2. Simulating the Kyoto Protocol scenarios in long-run forward-looking models requires additional assumptions on regional GHG emissions after 2012 (so-called ‘‘post-Kyoto’’). An example of the post-Kyoto policy is ‘‘Kyoto forever,’’ in which Annex-I regions sustain the same GHG mitigation obligations after 2012 and non-Annex-I regions may or

Notes

181

may not commit to their own GHG mitigations. In general, those policies are more stringent with respect to Annex-I regions than any solutions I have obtained in this research. 3. Refer to Yang 2003b for technical details. 4. This set experiment is purely numerical. No climatologist would accept such a value. 5. Similar to what I have explained about ROW in section 5.2.2, this outlier is due to the assumptions of ROW in RICE.

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Index

AIM, 2 Asia-Paciﬁc Integrated Model (AIM), 2 Atkinson, T., 24 Australia, 32 Austria, 2 Bailey, E. M., 135 Barber, C. B., 113 Barrett, S., 5 Benthamite weight, 34. See also Utilitarian solution Boyer, J., 27, 30, 108 Box model of carbon cycle, 30, 148 Brooke, A., 27 Bush, George W., 135 Business-as-usual solution, 33, 36, 134 simulation results, 53–55, 58, 85–98 Canada, 32 Carbon dioxide (CO2 ), 1, 33, 136 Carbon intensity, 17 Carbon taxes, 35–38 core boundary and, 115–123 RICE model and, 55–61, 65, 70–73, 78–80, 89–98 sensitivity analysis and, 126 Carnegie-Mellon, 2 Carraro, C., 5, 23, 28 CDM, 64, 133, 158 CETA model, 2 CGE models, 3–4 Chander, P., 5, 179 Characteristic function, 20 China, 29, 32, 53, 94, 131, 133, 135, 158, 159 Clean development mechanism (CDM), 64, 133, 158 Clarke, E., 5

Climate damage, 28–32, 56, 85, 88, 90, 96– 97, 107–108, 136–137, 148–150 coefﬁcients, 124–126, 131 false perception of, 123–131 Closed-loop strategy, 137–141 CO2 , 1, 33, 136 Cobb-Douglass production function, 29 Computable general equilibrium (CGE) models, 3–4 Convex hull, 113–114 Cooperative game, 12, 18, 36, 57, 63, 70, 76, 78. See also Lindahl equilibrium; RICE model; Shapley value core of, 21, 23–25, 35, 109–117 penalty rule of, 20, 73–74, 103–108 solution concepts of, 5, 7, 16, 23–25, 34– 36, 86, 136, 157 solutions, 41–48, 87–90, 95, 123, 131, 137 stability of, 106–108 stages of, 18–19, 105 Core allocation, 38, 43, 76, 109 Design matrix, 39–41, 74 DICE (Dynamic Integrated model of Climate and the Economy) model, 27– 28, 104, 108 Discount rate. See Rate of pure time preference Distribution analysis, 150–157 Dobkin, D. P., 113 Dynamic optimization models, 3–4 Edmonds, J., 150 Ellerman, A. D., 135 EPPA model, 3 European Union, 29, 32, 117 Eyckmans, J., 5, 23, 28

188

FEEM, 6 Finus, M., 5, 23, 28 Fisher-Vanden, K., 4 Foley, D., 11, 24 Folk theorems, 103, 107 Folmer, H., 5 Fondazione Eni Enrico Mattei (FEEM), 6 Former Soviet Union, 29, 56, 135 Fragnelli, V., 5 Free-riding, 5, 22, 33, 73–76, 106–107, 124 Fudenberg, D., 137 FUND model, 2, 5 Game theoretic solutions. See Lindahl equilibrium; Open-loop Nash equilibrium; Shapley value GAMS 8, 11, 27–28, 31, 36, 41, 85, 99, 167– 177, 179 G-Cubed model, 2 GDP, 29–31, 38, 53–54, 96–98 Greenhouse gases (GHGs). See speciﬁc types Greenhouse gas (GHG) control rates core boundary and, 115–123 RICE model and, 29–30, 33, 55–62, 71–73, 78–80, 88–98, 148–156, 159 sensitivity analysis and, 104–105, 126 Greenhouse gas (GHG) emissions 3, 14, 26 core boundary and, 114–117 quotas, 33, 52, 98–102, 114, 135 (see also Permits; Side payments; Transfers) RICE model and, 28, 30–31, 33, 53–62, 71–73, 78–80, 85–98, 152–160 sensitivity analysis and, 105, 124–126, 131 Greenhouse gas (GHG) mitigation, 6, 9, 85, 98. See also Marginal mitigation cost burden of, 48, 60–65, 86–98, 125–126, 134, 149, 153 cost of, 9, 18, 26, 30–32, 48–58, 70, 76, 96– 101, 107, 133, 136–137, 150–160 distribution analysis of policies for, 150– 157 initial quota, 26, 101, 117, 134–136, 141, 149, 153, 157–158 international cooperation on, 6–7, 57, 63– 65, 70, 89, 92, 98, 128, 133–134 RICE model and, 30, 33, 51, 56–58, 60–65, 70, 78, 86–89, 94–96 sensitivity analysis and, 105–107, 114– 117, 123–128, 131 strategies for, 3–4, 28

Index

Grand coalition, 20–21, 39–41, 72, 78, 85, 92 blocking by subcoalitions, 38, 43–48, 63– 65, 80, 111, 117 stability of, 22, 74, 88, 103–108, 114–117, 128, 137, 148–149 Groves, T., 5 Hanley, N., 5 Hill, J., 180 Hoel, M., 2 Holly, S., 14, 137 Hughes Hallett, A., 14, 137 Huhdanpaa, H. T., 113 Hybrid coalitional equilibrium. See Hybrid Nash equilibrium Hybrid Nash equilibrium, 22, 37, 39–41, 48, 50, 110, 123, 134, 159 IC, 21, 26, 63, 65, 70, 80, 85, 88, 95–101, 133–138, 153, 157, 159 IEAs, 5, 9, 32, 73, 108, 133–150 IIASA, 2 IMAGE model, 3 IR, 21, 41, 63, 70, 74, 80, 95, 159 Incentive checking, 41–50, 63–70, 73–76, 80–86, 89, 92, 95–97, 138–150, 158– 160 sensitivity analysis and, 104, 109–117, 125–129 Incentive compatibility (IC), 21, 26, 63, 65, 70, 80, 85, 88, 95–101, 133–138, 153, 157, 159 India, 96, 135, 158 Individual rationality (IR), 21, 41, 63, 70, 74, 80, 95, 159 Integrated assessment models, 1–9. See also RICE model Intergovernmental Panel on Climate Change (IPCC), 1–2, 4, 27, 31, 53, 133 International environmental agreements (IEAs), 5, 9, 32, 73, 108, 133–150 International Institute for Applied Systems Analysis (IIASA), 2 Intertemporal optimization models, 3–4 IPCC, 1–2, 4, 27, 31, 53, 133 Italy, 6 Japan, 2, 32 JI, 64, 133, 135, 158, 159 Joint implementation ( JI), 64, 133, 135, 158, 159

Index

Joskow, P. L., 135 Judson, R. A., 30 Kendrick, D., 27 Kolstad, C. D., 4, 85 Kyoto Protocol, 6, 33, 58, 72, 78, 92, 131, 133–135, 141, 157–158 Lindahl, E., 11 Lindahl equilibrium, 11, 35–36, 64–65, 114, 127, 138, 149, 151–159. See also Cooperative game algorithm, 43–50 core properties of, 24, 76, 109–113, 138 deﬁned, 24–25 incentive checking, 73–76 renegotiation proofness of, 137 simulation results, 70–73, 86–98, 100 stability of, 104–105, 148–149 Lindahl weight, 45–48, 71, 74, 106–107. See also Lindahl equilibrium McKibbin, W., 2 Manne, A. S., 2, 35 Marginal mitigation cost, 18, 26, 52, 64, 76, 98–102, 151–157, 159 Mechanism design models, 9 Meeraus, A., 27 MERGE model, 2–3 MIT, 2–3 Montero, J. P., 135 Moulin, H., 51 Muench, T., 179 Myerson, R. B., 25, 179 Nash equilibrium, 39, 63, 65, 70, 108, 158. See also Hybrid Nash equilibrium; Noncooperative game; Open-loop Nash equilibrium formulated as a differential game, 16 payoffs, 41–43 relationship between efﬁciency and, 11– 13, 19 National Institute for Public Health (RIVM), 2–3 Negishi solution, 34, 38, 57, 134–135, 156. See also Walrasian equilibrium algorithm, 37–38 incentive checking, 65–70, 114 simulation results, 60–63, 86–99 Negishi, T., 35 Negishi weight. See Negishi solution

189

Netherlands, 2 Noncooperative game, 6, 12, 20, 31, 33, 57, 106, 117, 127, 134. See also Nash equilibrium Nontransferable utility, 23, 25, 34–36, 70 Nordhaus, W. D., 2, 5, 9, 27, 30, 35, 37, 102, 104, 108, 124, 179–180 OPEC, 85, 96 Open-loop differential game, 29–31, 33, 41, 166–167 Open-loop Nash equilibrium, 16, 20, 25, 31, 33, 39–41, 127, 159. See also Nash equilibrium algorithm, 36–37 simulation results, 55–58, 86–98 Open-loop strategy, 14, 103, 136–137 Paciﬁc Northwest National Laboratory, 2– 3 Pareto improvement, 26, 51, 72, 98–102, 156–157 Parson, E. A., 4 Payoff function, 19, 106 Peck, S. C., 2 Penalty rule. See Cooperative game Permits, 26, 58, 98–102, 133, 136, 150, 157, 159. See also Side payments; Transfers Portney, P. R., 103 Qhull, 113 Quotas. See Greenhouse gas emissions; Side payments; Transfers Raman, R., 27 Rate of pure time preference RICE model sensitivity analysis, 103– 108 in stylized model of externality provision, 15 Renegotiation proofness, 5, 137, 148–150 RICE (Regional dynamic Integrated model of Climate and the Economy) model. See also RICE-99; RICE2007 carbon taxes, 55–61, 65, 70–73, 78–80, 89– 98 climate damage, 56, 85, 88, 90, 96–97, 107–108, 136–137, 148–150 control rates in, 30, 33, 55–62, 71–73, 78– 80, 88–98, 148–156, 159 development of, 5–7, 27–29, 108–109 equations of, 29–31, 165–167

190

RICE (Regional dynamic Integrated model of Climate and the Economy) model (cont.) GHG emissions in, 28, 30–31, 33, 53–62, 71–73, 78–80, 85–98 mitigation in, 30, 33, 51, 56–58, 60–65, 70, 78, 86–89, 94–96 regional breakdown of, 29 sensitivity analysis of, 103–131 solution algorithms in, 36–52 solution concepts in, 33–36 (see also speciﬁc types) RICE-99 model, 27–28. See also RICE model RICE2007 model, 28–29, 108. See also RICE model Richels, R. G., 2, 35 RIVM, 2–3 Rose, A. Z., 150 Samuelson Rule, 10, 17, 24 Samuelson, P., 10, 17 Scenario simulation models, 3 Schmalensee, R., 30, 135 Schneider, S. H., 30 Second-best solutions, 33–34, 135, 157– 160 Second theorem of welfare economics, 24, 65 SGM model, 3 Shapley, L. S., 85 Shapley value, 35–36, 51, 108. See also Cooperative game algorithm, 50–51 core and, 51, 80, 125–131 deﬁned, 25 incentive checking, 80–86 simulation results, 76–80, 86–98 Shubik, M., 85 Side payments, 51–52, 98–102, 157. See also Permits; Transfers Simplex, 15, 18, 23–24, 34–35, 38, 43, 48, 50, 76, 86, 109–111, 113–114, 125 Siniscalco, D., 5 SO2 , 135–136 Social planner’s problem as an outcome of a cooperative game, 38, 63, 70 RICE model and, 29–30, 33, 38, 43, 55–57, 65, 72 in stylized model of externality provision, 15

Index

Social welfare weight, 11–12, 134. See also speciﬁc types RICE model and, 35, 50, 55, 70, 86, 103, 108, 117 in stylized model of externality provision, 15, 20 Stern, N., 4 Stevens, B., 150 Stiglitz, J., 24 Stock externality provision, 7, 85, 108, 134 formulated as a game, 13–23 incentive property, 34 and internalization, 11–13, 63, 87–88, 123–124 (see also Social planner’s problem) private price of, 43 RICE model and, 28, 57, 73, 87–88 stylized model of, 15–18 Stoker, T. M., 30 Sulfur dioxide (SO2 ), 135–136 Superadditivity, 22, 158 Tatonnement procedure, 38, 50 Teisberg, T. J., 2 Temperature, 1, 14, 29–31, 53, 85–87, 148 Thompson, S. L., 30 Tirole, J., 137 Tol, R., 2, 5 Total factor productivity, 29, 53, 124 Tradable permits. See Permits Transferable utility, 23, 34, 70 Transfers, 17, 23, 51–52, 64–65, 76, 98–102, 114, 150–154, 157. See also Permits; Side payments Transformation function, 15, 17 Tulkens, H., 5, 23, 28, 179 Unilateral actions, 8, 133–135 United States, 32, 53, 131, 133, 135, 158 Utilitarian solution, 34–36, 134, 151–157 incentive checking, 41–50, 63–65, 76, 89, 114 simulation results, 57–60, 86–98, 101 Utilitarian weight. See Utilitarian solution Uzawa, H., 179 Walrasian equilibrium, 11, 24, 35, 38, 60, 63, 65, 86. See also Negishi solution Weyant, J. P., 103, 180 Wilcoxen, P., 2 Willingness-to-pay, 24, 35, 43, 48, 56, 70– 71, 78, 86. See also Lindahl equilibrium

Index

sensitivity analysis and, 111, 114–116, 123, 127 Wolak, F. A., 85 Xeon, 50 Yang, Z., 2, 5, 9, 17–18, 27–28, 35, 37, 102, 104, 108, 124, 137, 150, 180, 181 Yohe, G., 2

191