Natural Resource Pricing and Rents: An Economic Analysis (Contributions to Economics) 3030767523, 9783030767525

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Contributions to Economics

Andrey Vavilov Georgy Trofimov

Natural Resource Pricing and Rents An Economic Analysis

Contributions to Economics

The series Contributions to Economics provides an outlet for innovative research in all areas of economics. Books published in the series are primarily monographs and multiple author works that present new research results on a clearly defined topic, but contributed volumes and conference proceedings are also considered. All books are published in print and ebook and disseminated and promoted globally. The series and the volumes published in it are indexed by Scopus and ISI (selected volumes).

More information about this series at http://www.springer.com/series/1262

Andrey Vavilov • Georgy Trofimov

Natural Resource Pricing and Rents An Economic Analysis

Andrey Vavilov Institute for Financial Studies Moscow Region, Russia

Georgy Trofimov Institute for Financial Studies Moscow Region, Russia

ISSN 1431-1933 ISSN 2197-7178 (electronic) Contributions to Economics ISBN 978-3-030-76752-5 ISBN 978-3-030-76753-2 (eBook) https://doi.org/10.1007/978-3-030-76753-2 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To my mother Zinaida Pavlovna Vavilova Andrey Vavilov To my mother Antonina Ilyinichna Trofimova, and to my wife Karolina Yakubovich Georgy Trofimov

Preface

The theory of resource pricing is the core of natural resource economics and the economics of resource markets. In this book, we consider theoretical models that highlight fundamental determinants of resource prices and the economic nature of rents for nonrenewable and renewable resources. The focus of resource economics is the problem of optimal resource allocation over time, and the price mechanism plays a key role in making a rational choice between using and conserving a unit of natural resource. These issues are important for understanding the long-term tendencies of price dynamics, the processes of mineral resource depletion, the development of underground mineral reserves, and the rational use of environmental resources. The starting point of this book is the seminal article “The Economics of Exhaustible Resources” written by Harold Hotelling 90 years ago. This article explained the economic nature and pricing of exhaustible resource scarcity and laid the ground for natural resource economics as a research field. The large number of academic papers that have appeared since the 1970s have revived and developed this field, but we do not intend to trace back this development in our monograph. Instead, we have selected several important contributions to resource economics that, in our view, reflect different conceptual visions in this area. We present Hotelling’s model and its extensions and consider the alternative approaches that revised, in essential aspects, the concept of resource scarcity. Although this book is neither a survey of literature on resource economics nor a textbook in this discipline, it has the features of both. Several chapters include analysis of selected papers, while others contain the authors’ original contributions. The style of all chapters is close to the style of textbooks for graduate students. We gave priority to simplicity and clarity in the exposition of models and preferred to derive, when possible, explicit analytical solutions. Although exposition in this book presumes a grounding in the principles of dynamic economic analysis, it intends to be accessible to nonspecialists. We use numerical examples illustrating model solutions. Many graphs and tables present numerical results for combinations of model parameters selected for specific examples of resource markets. Most of these examples relate to oil production, and there are two reasons for this. Firstly, oil still plays the primary role as an energy resource in the global economy. vii

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Preface

The economics of the world oil market is an important subject for many analysts in oil-importing and oil-exporting countries. Secondly, one of the authors of this book had the experience of managing an oil-extracting company that made him familiar with the practical issues of oil production and trade. This experience, to some extent, justifies our interest in theoretical research in the economics of oil. We concern evidence about the world oil market in parts of the book devoted to commodity storage behavior and to the effects of the resource cartel, by which we mean the international petroleum cartel. The idea of this book originated from our collaboration with colleagues from the Economics Department of Pennsylvania State University during the period 2007– 2014. We are grateful to Barry Ickes for very interesting discussions on the history of oil and useful references to related literature, to Alexey Pomansky for thoroughly reading all the chapters of the monograph and valuable comments that stimulated improvements, and to the anonymous referee for valuable comments and recommendations that facilitated our work at the final stage. Moscow Region, Russia

Andrey Vavilov Georgy Trofimov

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Differential Rent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Resource Rents and Opportunity Costs . . . . . . . . . . . . . . . . . . . 1.3 Commodity Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Resource Cartel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I 2

1 5 7 10 11 13

Resource Rents and Opportunity Costs

The Economics of Exhaustible Resources . . . . . . . . . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Hotelling’s Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Hotelling Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Decentralized Solution . . . . . . . . . . . . . . . . . . . . . 2.2.3 Remarks on the Discount Rate and the Real Interest Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Equilibrium Extraction Paths . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Constant-Elasticity Demand . . . . . . . . . . . . . . . . . . . . 2.4 Resource Exhaustion and the Backstop Technology . . . . . . . . . 2.4.1 Solution for Constant-Elasticity Demand . . . . . . . . . . . 2.5 The Herfindahl Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Solution for Two Sites . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 A Modified Hotelling Rule . . . . . . . . . . . . . . . . . . . . . 2.6 The Principle of Comparative Advantage . . . . . . . . . . . . . . . . . 2.6.1 Two Cities, Two Sites . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 20 20 22 23 24 25 26 27 30 30 32 35 36 38 40 41 43

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Contents

3

Prices and Rents of Economically Recoverable Resources . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 The Model of Economically Recoverable Resources . . . . . . . . . 3.2.1 The Equilibrium Extraction Path . . . . . . . . . . . . . . . . . 3.2.2 Dynamics of Marginal Rents . . . . . . . . . . . . . . . . . . . . 3.2.3 Solution for the Linear Case . . . . . . . . . . . . . . . . . . . . 3.2.4 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Incomplete Resource Exhaustion in Hotelling’s Model . . 3.3 The Reservoir Model and the Structure of Oil Rents . . . . . . . . . 3.3.1 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2 Decision Rule for Reserves Addition . . . . . . . . . . . . . . 3.3.3 Equilibrium Dynamic in the Linear Case . . . . . . . . . . . 3.3.4 The Depletion Rate and the Initial Reserve Stock . . . . . 3.3.5 The Structure of Rents . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Resource Development and Extraction Without Depletion . . . . . 3.4.1 The Stationary Equilibrium . . . . . . . . . . . . . . . . . . . . . 3.4.2 The Linear-quadratic Case . . . . . . . . . . . . . . . . . . . . . 3.5 Measuring Resource Scarcity . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 45 48 49 51 54 55 56 56 59 60 61 63 64 66 67 68 69 72 73 75

4

Pricing Energy Resources Under Transition to Alternative Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Energy Consumption . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Resource Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Alternative Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Equilibrium Conditions . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Valuation of Energy Resource and Energy Capital . . . . . . . . . . 4.3.1 Valuation of Conventional Energy Resource . . . . . . . . 4.3.2 Valuation of Alternative Energy Capital . . . . . . . . . . . . 4.4 The Equilibrium Transition Path . . . . . . . . . . . . . . . . . . . . . . . 4.4.1 The Structure of Demand . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Dynamic of the Energy Price Index . . . . . . . . . . . . . . . 4.4.3 Dynamic of Energy Capital . . . . . . . . . . . . . . . . . . . . . 4.4.4 Dynamic of Resource Extraction . . . . . . . . . . . . . . . . . 4.5 The Resource Price Determination . . . . . . . . . . . . . . . . . . . . . . 4.5.1 The Equilibrium Utility Level . . . . . . . . . . . . . . . . . . . 4.5.2 An Example: σ ¼ 2 . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6 Transition Paths of Energy Capital and Extraction . . . . . . . . . . . 4.6.1 Variation of the Elasticity of Substitution . . . . . . . . . . . 4.6.2 Variation of the Preference Weight . . . . . . . . . . . . . . .

77 77 81 81 82 83 83 84 84 85 86 87 88 89 91 92 93 94 95 96 97

Contents

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4.7 The Green Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106 5

Global Carbon Budgeting and the Social Cost of Carbon . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Modelling Climate-Economy Interactions . . . . . . . . . . . . . . . . . 5.3 The Overlapping-Generations Model . . . . . . . . . . . . . . . . . . . . 5.3.1 The Damage Function and the Greenhouse Effect . . . . . 5.4 The Laissez-Faire Dynamic Equilibrium . . . . . . . . . . . . . . . . . . 5.4.1 The Long-Term Equilibrium . . . . . . . . . . . . . . . . . . . . 5.4.2 Equilibrium Transition Paths . . . . . . . . . . . . . . . . . . . . 5.5 The Social Planner’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.1 The Long-Term Optimum . . . . . . . . . . . . . . . . . . . . . . 5.6 Numerical Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 The Social Cost of Carbon . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 The Long-Term Carbon Tax . . . . . . . . . . . . . . . . . . . . 5.7.2 The Social Cost of Carbon and Excess Returns . . . . . . 5.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part II 6

107 107 111 114 115 117 118 119 121 123 125 129 130 131 132 134 138

Commodity Prices

Commodity Prices, Convenience Yield and Inventory Behaviour . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Consumption-Based Model of Commodity Storage . . . . . . . . . . 6.3 Convenience Yield . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Deterministic Dynamic of Commodity Price and Storage . . . . . . 6.5 Stochastic Dynamic of Commodity Market . . . . . . . . . . . . . . . . 6.5.1 Current Availability of Commodity . . . . . . . . . . . . . . . 6.5.2 Inventory Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Solution for the Convenience Yield and the Commodity Price . . . 6.6.1 Two Limit Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.2 The First-Order Autocorrelation for Commodity Price . . . 6.6.3 The Vector Autoregression . . . . . . . . . . . . . . . . . . . . . 6.7 Price-Stabilizing and -Destabilizing Inventory Behaviour . . . . . . 6.7.1 The Adjustment of Storage . . . . . . . . . . . . . . . . . . . . . 6.7.2 The Shock Effects on the Convenience Yield . . . . . . . . 6.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 147 148 150 151 152 153 155 156 157 159 160 160 162 164 166 171

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7

8

Contents

Commodity Prices and Competitive Storage . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 The Base Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Two Regimes of Inventory Management . . . . . . . . . . . . . . . . . 7.4 Stationary Rational Expectations Equilibrium . . . . . . . . . . . . . . 7.5 Autocorrelation of Supply . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 The Equilibrium Price Function . . . . . . . . . . . . . . . . . . 7.5.2 Contribution of Storage to Autocorrelation of Prices . . . 7.6 Stochastic Demand: Persistence of Shocks and Volatility of the Oil Price . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 The Trader’s Problem . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.2 Equilibrium Prices . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.3 Income Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.4 Price-Stabilizing and -Destabilizing Inventory Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.5 The Three Epochs of Oil . . . . . . . . . . . . . . . . . . . . . . . 7.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commodity Trade in Continuous Time, Long-Term Availability and Storage Capacity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 The Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Commodity Market . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Traders and Storage . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3 The Value Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4 Stationary Rational Expectations Equilibrium . . . . . . . . . . . . . . 8.4.1 The Equilibrium Price Function . . . . . . . . . . . . . . . . . . 8.4.2 Free-Boundary Conditions . . . . . . . . . . . . . . . . . . . . . 8.5 The Saddle-Path Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Existence and Uniqueness of Equilibrium . . . . . . . . . . 8.6 The Storage Capacity Constraint . . . . . . . . . . . . . . . . . . . . . . . 8.6.1 Storage Capacity and Trading Zones . . . . . . . . . . . . . . 8.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

173 173 175 177 178 182 183 187 189 189 190 191 191 193 194 195 197 197 200 200 202 202 204 205 207 208 210 211 215 216 218 221

Contents

Part III

xiii

Resource Cartel

9

Cartel Behaviour in an Exhaustible Resource Industry . . . . . . . . . . 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.2 The Base Competitive Model . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3 The Pure Monopoly Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.3.1 Dynamic of Marginal Revenue . . . . . . . . . . . . . . . . . . 9.3.2 Solution for Linear Demand . . . . . . . . . . . . . . . . . . . . 9.3.3 Conservationism of Resource Monopoly . . . . . . . . . . . 9.3.4 The Case of Low-Elasticity Demand . . . . . . . . . . . . . . 9.4 Perfect Competition of Two Regions . . . . . . . . . . . . . . . . . . . . 9.5 The Cartel-Fringe Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.1 Dynamics of Outputs . . . . . . . . . . . . . . . . . . . . . . . . . 9.5.2 The Bounds of the Cost Differential . . . . . . . . . . . . . . 9.6 Cartel and Fringe in the Presence of a Backstop Technology . . . 9.6.1 The Cartel’s Strategic Choice . . . . . . . . . . . . . . . . . . . 9.6.2 Cournot-Nash Equilibrium . . . . . . . . . . . . . . . . . . . . . 9.6.3 The Equilibrium Resource Allocation . . . . . . . . . . . . . 9.6.4 The Cartel’s Distorting Effects . . . . . . . . . . . . . . . . . . 9.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

225 225 229 230 231 231 234 235 235 237 238 241 242 244 245 247 248 249 251 253

10

The Oil Cartel and Misallocation of Production . . . . . . . . . . . . . . . 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 A Model of the Competitive Global Oil Industry . . . . . . . . . . . . 10.3 The Competitive Oil Price and the Allocation of Production . . . 10.3.1 The Stationary Competitive Equilibrium . . . . . . . . . . . 10.3.2 Competitive Production . . . . . . . . . . . . . . . . . . . . . . . 10.3.3 The Firm’s Value and the Social Benefit of Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4 Equilibrium in the Global Oil Industry with a Cartel . . . . . . . . . 10.4.1 Resource Reallocation . . . . . . . . . . . . . . . . . . . . . . . . 10.4.2 The Pure Monopoly Case . . . . . . . . . . . . . . . . . . . . . . 10.4.3 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 10.5 Production Shifting and Welfare Losses . . . . . . . . . . . . . . . . . . 10.5.1 Cartel’s Value Gain . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.2 Welfare Losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5.3 The Graphical Illustration . . . . . . . . . . . . . . . . . . . . . . 10.5.4 Continuation of the Numerical Example . . . . . . . . . . . . 10.6 The Effects of Technological Change . . . . . . . . . . . . . . . . . . . . 10.7 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

255 255 258 259 261 262 262 265 267 269 270 271 272 273 274 275 276 280 282 287

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Contents

Anti-Conservationist Effects of the Conservationist Oil Cartel . . . . 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 The Base Model of a Competitive Market . . . . . . . . . . . . . . . . . 11.3 The Pure Monopoly Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3.1 A Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . 11.3.2 Comparison of Transition Paths . . . . . . . . . . . . . . . . . . 11.4 The Two-Region Model with Competitive Market . . . . . . . . . . . 11.4.1 The Marginal Costs Equalization . . . . . . . . . . . . . . . . . 11.4.2 The Equilibrium Extraction Path . . . . . . . . . . . . . . . . . 11.5 The Two-Region Model with a Cartel . . . . . . . . . . . . . . . . . . . 11.5.1 The Dynamic System . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.2 The Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.3 Time Profiles of Extraction Rates and the Cartel’s Market Share . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.5.4 Production Shifting Over Time . . . . . . . . . . . . . . . . . . 11.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

289 289 292 295 297 297 298 299 300 301 302 303 304 306 307 308 313

Chapter 1

Introduction

This book can be viewed as an introduction to the theory of natural resource pricing, which is grounded on the premise of rational use of resources by economic agents. Their decisions result from resource evaluation based on dynamic optimization: maximization of some economic value with respect to resource constraints that can be given in various forms. Theoretical models of rational use of natural resources can be applied for the cases of non-renewable resources, such as underground stocks of fossil fuels, minerals, ores, etc., and for the cases of renewable resources of biosphere, water and atmosphere. The case of global atmospheric resources, among others considered in this book, is especially important in the context of the climate change problem. There are two important features in modelling resource pricing. First, any valuable stock of a non-renewable or renewable resource should be regarded as a real economic asset of dual nature. On the one hand, this asset serves a store of wealth bringing capital returns to a resource owner. On the other hand, a stock of natural resource has to be used for consumption during a period of time. Rational use of a resource thus means the optimal allocation of resource consumption over time. The price of resource stock as an economic asset is determined jointly with the programme for such allocation and with the price path for the flow of the extracted resource. To illustrate this point, one can compare models of resource pricing with models of equity pricing. The equilibrium equity price is equal to the present value of future dividend flow, which can be given exogenously. In resource models, the flow of dividends is represented by resource rents. The value of resource stock equals the present value of the flow of rents that have to be determined from the solution for optimal resource allocation over time. Second, the volume of underground mineral resources is not known with certainty. For example, an oil stock in the ground can be measured by proven reserves that are currently economically recoverable, but there are no exact data on the amount of oil to be extracted ultimately from a deposit. Nevertheless, the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_1

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1 Introduction

underground stocks of oil and other resources are priced without exact knowledge of their volumes. In traditional resource economics, it was assumed that an exhaustible resource stock is given exogenously and constrains the time path of resource consumption. This assumption was the key one in the pioneering article by Harold Hotelling (1931), who formulated the problem of exhaustible resource extraction as the model of dynamic optimization. The solution of this problem gives the optimal allocation over time of the initial resource stock and the optimal time path of resource price satisfying conditions of asset market equilibrium. Hotelling’s model, however, rests on the stylized description of a resource in the ground as a perfectly known quantity to be used for consumption. An alternative assumption, which is relevant to renewables and non-renewables, is that resource stocks are not observable in advance and have to be determined in the process of resource exploitation. The volume of a depletable mineral resource recovered ultimately from a field or a mine can be determined in the process of extraction from the relationship between the resource price and the extraction costs. In the resource models with no explicit resource constraints that will be considered in this book, an endogenous variable of choice is the resource availability, which is limited and determined along with the time paths of extraction and resource price. For example, there is no explicit resource constraint for an oil deposit, but the volume of resource extracted from the deposit is determined implicitly for the breakeven point, when a further increase of extraction costs makes extraction unprofitable. For a region of oil, a depletion of the low-cost deposits provides incentives to develop the high-cost ones. On the global level, the use of fossil fuels leads to the accumulation of greenhouse gases in the atmosphere, which severely limits the use of fossil fuels in the future. In all these cases, the limited resource availability influences the dynamic choices of rational economic agents. Under an explicit or implicit resource constraint, resource extraction at present implies losses in the future defined as the opportunity costs of resource extraction. In the case of Hotelling’s resource stock, which can be exemplified by a reservoir of oil, the opportunity cost of present resource consumption is the foregone future consumption simply because there is less oil to extract from the reservoir. In the case of the oil region, the opportunity cost may reflect a decreasing quality of oil deposits that results in increasing extraction costs in the future. The concept of limited resource availability is relevant to cases when both non-renewable and renewable resources are used for energy supply. The energy supply includes fossil fuels and renewable energy that are substitutes in energy consumption. The pricing of non-renewable resources of fossil energy depends on the development of renewable energy production underlying the process of transition to alternative energy. Another case considered in this book is the atmospheric resource of the planet, which complements the world resources of fossil fuels. A unit of oil extracted and burned causes carbon dioxide emission, which means consumption of a unit of atmospheric resource. This is a renewable resource with limited absorbing capacity that should be priced according to the social opportunity costs of greenhouse gas emission.

1 Introduction

3

Fig. 1.1 The real price of crude oil, 1946–2019. Source: BP Statistical Review of World Energy (2020), FRED Economic Data (2020)

Evaluation of opportunity costs depends on the level and the time horizon of decision-making. An oil company with a time horizon of several decades takes into account the opportunity costs of exploiting fields under its control, but it usually neglects the social costs of this activity. A socially responsible national government of an oil-rich country determines a trade-off between long-term welfare gains and losses from oil extraction for present and future generations. A group of national governments involved in international cooperation to prevent climate change takes care when it comes to risks and damages for far-distant future generations from the present consumption of fossil fuels. For extractive industries, the opportunity costs are determined via market prices of resources. A resource price net of the marginal costs of resource extraction, processing and transportation is the marginal resource rent. Under conditions of competitive equilibrium, this rent covers the user cost of a mining company, direct and indirect, incurred to obtain a unit of a resource. The user cost, also called “in situ value”, indicates the opportunity cost of resource extraction caused by decreasing the future availability of the resource and a reduction at the margin of resource value. The resource rent thus governs the intertemporal choice of producers and the selection of extraction paths insofar as the opportunity cost indicates the impact of the resource constraint or the limited resource availability. The question is to what extent the movements of actual resource prices reflect such an impact. Figure 1.1 demonstrates the dynamic of the world oil price in real terms in the post-World War II period. The real oil price features significant fluctuations around the exponential trend of long-term growth drawn with a dashed line in the figure. The growth rate of this trend is 2.4% per annum. The most significant oil price spikes occurred in the 1970–1980s and were caused by two oil crises. The significant oil price increase in the 2000s resulted from the upsurge of Asian economies that gave rise to the rapid growth of Asian energy demand.

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1 Introduction

The rising trend of oil prices reflected market perceptions of the increasing scarcity of oil resources that prevailed in the 1970–1980s due to the energy crisis. The similar price dynamic in the 2000s was caused to a large extent by the long-term tendency of resource supply growth to lag behind the rapidly growing energy demand of the new industrial economies. At the same time, the oil price dynamic was influenced by the behaviour of various market participants, including major oil-producing companies, oil traders and financial players, and, most importantly, by the activity of the international petroleum cartel OPEC (Organization of Petroleum Exporting Countries). The latter has played a prominent role in the world oil market since it imposed an oil embargo against Western countries in 1973. For these reasons, there cannot be a clear-cut causal link between any measure of resource availability and resource prices. Price fluctuations around the trend curve shown in Fig. 1.1 were caused by demand and supply shocks that were amplified by the influence of market participants that base their trade on future price expectations. A high price volatility provides incentives to hold physical inventories of resources and to participate in intertemporal trade. The inventory behaviour of oil producers and traders might in some cases have a destabilizing impact on the oil price dynamic. In general, intertemporal trade and market imperfections are a feature of the mineral commodity markets (for resources of energy, metals and non-metals), which are similar in essential aspects to the world oil market. As already mentioned, in this book we will consider theoretical models of resource pricing based on the premise of rational decisions by economic agents. On the one hand, we examine fundamental determinants of resource prices for various concepts and types of natural resources, including non-renewables, renewables and environmental resources. On the other hand, we consider the influence on the resource price level and dynamic of the intertemporal trade in commodity markets and of the resource cartel’s activity, which have distorting effects on resource allocation over time and in space. To embrace these issues, we will analyse the models of resource pricing from two perspectives: natural resource economics and the economics of resource markets. Natural resource economics is discussed in Chaps. 2, 3, 4, and 5, where we consider the following concepts and models of natural resources: • A homogeneous exhaustible resource stock within the framework of Hotelling’s model and its extensions to cases of heterogeneous resources and consumers; • A limited resource availability in models of economically recoverable resources and models of reserves development; • Fossil energy resources in the model of transition to alternative energy based on renewables as backstop resources; • A limited atmospheric resource determined by an implicit constraint for fossil fuel consumption in the context of climate change. The economics of resource markets is described in Chaps. 6, 7, 8, 9, 10, and 11, which deal with the important features of resource markets that influence the determination and dynamics of resource prices:

1.1 Differential Rent

5

• Inventory behaviour of commodity market traders and the effects of intertemporal commodity trade on resource prices; • The petroleum cartel’s activity and the effects of cartel pricing on the spatial and intertemporal allocation of oil production and global oil resources. Methodology We consider simple theoretical models that require the base knowledge of calculus, differential equations, dynamic optimization and elements of stochastic calculus. Analysis of the models’ special cases allows for explicit solutions illustrated by numerical examples that are solved with a pocket calculator, Excel or Scientific WorkPlace. The models are deterministic or stochastic. The latter are applied in part II of the book devoted to commodity markets and storage behaviour. In part I on resource economics and in part III on the resource cartel, we use deterministic models based on the assumption of explicit resource constraints or on the concept of limited resource availability. Each chapter contains a short informal introduction to the subject to provide a better understanding of the formal problems and to help economic intuition. In what follows in the introduction we consider the Ricardian theory of differential rent formalized as a static model of partial equilibrium with resource constraints and then we provide a summary of the parts and chapters of this book.

1.1

Differential Rent

Traditionally, economists considered rental incomes in connection with economic policy issues. Land rent turned out to be in the focus of economic policy in the United Kingdom in the early nineteenth century, when the primary issue was food supply to the increasing population. The main concern of economists was about the effects of the Corn Laws adopted in 1814 by the British parliament to protect domestic agricultural producers from foreign imports of cheap corn. At that time, the challenge for economic theory was to explain the nature of rents paid to landowners and the dynamics of food prices and land rents in the light of economic progress. The answer was provided by the classical theory of differential rent. This was formulated by David Ricardo, who defined land rent as “that portion of the produce of the earth that is paid to the landlord for the use of the original and indestructible powers of soil” (1817, p. 39). Ricardo argued that land rent exists for two reasons: pieces of land differ in the fertility of soil and pieces with superior fertility are scarce. There would be no rent if there was an abundance of fertile lands or the quality of soil was the same everywhere. Plots of land of higher quality ensure lower production costs and therefore yield production surpluses that landowners receive from tenants in the form of rent. The price of an agricultural commodity is defined from the condition of demand and supply equilibrium and is equal to the production cost of the marginal land that brings no rent. Thus, the equilibrium commodity price does

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1 Introduction

Fig. 1.2 The Ricardian model of land rent

not depend on the amount of rent paid to landowners, whereas the rent increases with the growth of this price. For expositional purposes, it is worthwhile formalizing here the arguments of Ricardo by using a simple static model of partial equilibrium. Consider an industry producing an agricultural commodity with competitive producers owning pieces of land that differ in terms of fertility and area. Suppose that grades of land are ranked according to the increasing unit costs of production, c1 < c2 < . . . . Let the production capacity of grade j be defined by its area and equal to qj. Producers within each grade are homogeneous and choose outputs xj to maximize the land rent ( p
cj)xj subject to the capacity constraint: xj  qj

ð1:1Þ



 The Lagrangian for this problem is L j ¼ p
c j x j þ μ j q j
x j , where μj is the shadow value of land of grade j. The first-order condition of optimality is p ¼ cj þ μj

ð1:2Þ

implying that the rent per unit of commodity is equal to the shadow value of land, p
cj ¼ μj. For grades with a positive rent, μj > 0, the capacity constraint is binding, xj ¼ qj. The rent is zero for the marginal grade m, which does not produce at full capacity, xm < qm. Hence, the equilibrium commodity price is Pequal to the unit cost of the marginal grade, p ¼ cm, and the total supply x ¼ mj¼1 x j matches the demand:   P p ¼ P xm þ m
1 q j¼1 j , where P ðxÞ is the inverse demand function. Figure 1.2

1.2 Resource Rents and Opportunity Costs

7

illustrates the equilibrium solution. The horizontal axis denotes the commodity supply, while the vertical axis P relates to price and unit costs. The total amount of differential rent is equal to m
1 j¼1 μ j q j and depicted by the shaded area in this figure. This model captures the basic arguments of Ricardo. First, the grades of land differ in the unit production costs, and second, the superior and intra-marginal grades are scarce in the sense that the capacity constraint (1.1) is binding in equilibrium. One can interpret this inequality as a resource constraint in view of the fact that land is the main resource for agricultural production. In the model of differential rent, the production cost of the marginal piece of land defines the price, while a land of higher grade yields a rent per unit of output, which equals the cost differential with the marginal land. Although the Ricardian model is static, the policy implications of the classical theory concerned the time paths of land rents. Ricardo pointed out a possible scenario of economic development outlined earlier by Thomas Malthus (1815): accumulation of national wealth “would lead to an increased demand for labour, to higher wages, to an increased population, to a further demand for raw produce, and to an increased cultivation” (Ricardo 1817, p. 47). As a result, food prices and rents tend to rise as more land with inferior fertility and higher production costs is involved in cultivation. Landowners in such a scenario receive an increasing share of national product at the expense of manufacturers and workers, implying the need to promote competition in food supply and remove trade barriers to corn imports.

1.2

Resource Rents and Opportunity Costs

Differential rent in the static model is defined as the shadow value of resource constraint (1.1), which measures the contribution of land as the production factor to output. The “indestructible power of soil” is rewarded by the rental income, as follows from the first-order condition (1.2). Although the static model explains the nature of differential rent, it disregards the depletion of soil that requires investment to ameliorate the quality of land. Investment in land as a production asset means intertemporal choice by landowners that can be considered only within a dynamic framework. Economists of the nineteenth century did not pay much attention to the problem of natural resource exhaustibility. A notable exception was the “Coal Question” raised in 1865 by William Jevons in the book of the same title (Jevons 1865). Jevons predicted that British coal reserves would be exhausted within a hundred years. In 1914, Lewis Gray offered informal arguments to revise the Ricardian theory of rent with accounts of natural resources exhaustibility (Gray 1914). In 1931, Hotelling suggested the dynamic model of exhaustible resources mentioned above, which became the basis of resource economics 40 years after publication. We will expound Hotelling’s theory and its extensions in Chap. 2, “The economics of exhaustible resources”. In the base model of homogeneous exhaustible

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1 Introduction

Fig. 1.3 Resource rent for identical fields

resources, a producer faces a resource constraint, and the marginal resource rent is equal to the shadow value of this constraint. The resource rent is positive under a binding constraint, even though all fields are identical and have the same extraction costs per unit of output c, as shown in Fig. 1.3. The marginal resource rent in this figure is the difference between resource price and unit extraction cost. The volume of production by field j is xj. The total resource rent of all producing fields j ¼ 1, . . ., m is depicted by the shaded rectangle. Unlike the Ricardian model, where land rent does not influence the price of agricultural commodities, the resource rent in Hotelling’s model determines the resource price. The marginal rent is the return on the resource stock as an asset and, from the no-arbitrage condition, the rate of return equals the real rate of interest. The marginal resource rent grows at this rate, and the equilibrium resource price meets this condition, known as “Hotelling’s rule”. This rule modifies for the extensions of Hotelling’s model to heterogeneous resources: the marginal rent falls if extraction switches from low-cost to high-cost resource stocks. The marginal rent in this model framework indicates the opportunity cost of extraction and the increasing resource scarcity, because it grows as the stock of an exhaustible resource decreases in size. However, the assumption of natural resource exhaustibility is not necessary for understanding the nature of resource prices and rents. As an alternative, one can consider a factor of resource depletion materialized in higher extraction costs for resource grades of lower quality. If a resource in the ground cannot be exhausted completely due to the depletion effect, the assumption of exogenous resource constraint does not make much sense.

1.2 Resource Rents and Opportunity Costs

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In Chap. 3, “Prices and rents of economically recoverable resources”, we consider a model with limited resource availability. A potential resource in the ground is supposed to be inexhaustible but depleting in the sense that the marginal extraction cost increases with cumulative production. In the case of oil, depletion of conventional resources leads to the development of higher-cost deposits and unconventional resources such as tight and deep-water oil. The economically recoverable resource is determined for the long-term equilibrium of the model as the ultimate cumulative extraction. The opportunity cost of extraction is measured as the present value of future excess extraction costs caused by current extraction. Under a stationary resource demand, the marginal resource rent compensating for this opportunity cost does not necessarily increase over time and does not indicate an increasing resource scarcity. Resource pricing has specific features in the presence of alternative technologies providing substitutes for conventional non-renewable resources. The sources of alternative energy for fossil fuels include wind, solar and biofuel energy. In Chap. 4, “Pricing energy resources under transition to alternative energy”, we consider a model of gradual energy transition for an energy-supplying industry. Exhaustion of a conventional non-renewable resource in the model causes growth of the opportunity cost of extraction and a decline of the relative price of alternative energy. The market share of this energy in the energy mix of consumers increases as a result of gradual substitution of renewables for conventional resources. An important property demonstrated for the model of Chap. 4 is the Green Paradox: a higher consumer preference for alternative energy implies a higher intensity of the conventional resource extraction in the near term. On the global level, the limited availability of natural resources relates primarily to the influence of fossil energy use on climate change, rather than to the exhaustion of fossil energy resources. It is widely recognized that global warming has been caused by the build-up of carbon dioxide in the atmosphere due to the burning of fossil fuels. In Chap. 5, “Global carbon budgeting and the social cost of carbon”, we consider a simplified version of the Integrated Assessment model proposed by William Nordhaus in 1994 to capture the long-term interactions between the global economy and climate (Nordhaus 1994). This version is a model of overlapping generations with fossil energy as an input in production. In a laissez-faire equilibrium, individuals do not account for the negative external effects of carbon emission. The social planner’s solution can be viewed as a form of carbon budgeting, which is similar to the concept of economically recoverable resources in Chap. 3. The planner takes into account the opportunity costs of fossil fuel consumption by the present generation, which are given by welfare losses of future generations.

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1.3

1 Introduction

Commodity Prices

Allocation of resources over time is based on the property of resource storability, which is, in a sense, similar to the “indestructible power of soil” accentuated by Ricardo. Storability is inherent both for resource stocks under the ground and for resource stocks above the ground, such as commercial or strategic inventories of oil. Physical inventories facilitate intertemporal exchange on the organized commodity markets, where traders decide, similarly to extracting firms, whether to sell a resource now or to store it for the future. Therefore, commodity pricing depends, besides current supply and demand, on traders’ demand for inventories based on their expectations of future price movements. Commodity markets incorporate trade with physical delivery of resources to buyers and trade with futures contracts that mainly redistribute risks related to commodity price fluctuations. Markets for commodity futures specify a particular asset class used by financial investors, speculators and hedgers. In the case of crude oil, the daily turnover of futures contracts exceeds the volume of spot trade, but these contracts virtually do not require physical delivery and do not influence directly the oil price (Smith 2009). The influence can be indirect, through the formation of market expectations that determine the inventory behaviour of forward-looking traders that affect the commodity price dynamic. We focus on this behaviour, because resources withdrawn from the earth are traded in the markets for mineral commodities. In consideration of these markets, we treat commodity traders as the main players instead of resource-extracting firms and distinguish between two types of inventory behaviour. The first type is represented by commercial traders acting on the supply or demand side and extracting tangible benefits from holding commodity storage. The utility benefit of storage is given by the convenience yield from preventing disruptions of resource supply, economizing on transaction costs and other advantages of current resource availability. The second type of inventory behaviour is speculative trade with no direct utility benefits from storage. Speculators seek to gain from expected price changes and select positions in physical inventories that result in expectation-driven dynamics of commodity prices. In Chap. 6, “Commodity prices, convenience yield and inventory behaviour”, we examine a model with the first type of traders extracting utility from holding storage and acting on the demand side. In the crude oil market, these kinds of participants are represented by commodity traders, pipeline companies, midstream and downstream producers, including refineries, petrochemical companies, utilities, and final consumers. The equilibrium commodity price for this model satisfies a modified Hotelling’s rule under stochastic supply shocks: the expected rate of price growth equals the interest rate less a dividend-like term corresponding to the convenience yield. Analysis of the model reveals cases of price-stabilizing and -destabilizing inventory behaviour. In the former case, inventories are adjusted to smooth price fluctuations. In the latter case, the demand for inventories reinforces a price change

1.4 Resource Cartel

11

caused by a supply shock. As will be shown, the price-destabilizing behaviour can take place in some periods of time, if the supply shock is significant in absolute size. The second type of inventory behaviour is discussed in Chap. 7, “Commodity prices and competitive storage”, with the use of a canonical model of rational speculative activity in commodity markets. The salient feature of this model is the non-negativity constraint on storage held by competitive traders. Unlike the futures markets permitting short sales, traders in markets with physical delivery cannot “borrow” commodities from the future. Under random fluctuations on the supply side, competitive traders are supposed to select inventory holdings based on the expectations of future price changes. The equilibrium price for this model is determined as a decreasing function of the current availability defined as the sum of storage and commodity supply per period. Under stochastic supply shocks, the model-generated autocorrelation of prices proved to be not high enough to match the observed time series of commodity prices. In contrast, the model predictions are relevant to persistent demand shocks caused by fluctuations of economic activity and income growth. The canonical model of competitive storage is formulated in discrete time, and this brings about a high non-linearity and computational complexity of equilibrium equations. In Chap. 8, “Commodity trade in continuous time, long-term availability and storage capacity”, we offer a continuous-time commodity market model utilizing the advantages of stochastic calculus that make this model a more convenient analysis tool. Two features distinguish it from the original discrete-time version. First, a stochastic process is defined for net demand and embraces demand and supply fluctuations. Second, we show that the equilibrium price function argument is the long-term availability of a commodity instead of the current availability, which does not make sense for continuous time. We consider the stationary rational expectations equilibrium, and analyse the equilibrium price function for the cases of unlimited and limited storage capacity and demonstrate the existence and uniqueness of this function.

1.4

Resource Cartel

In the case of oil price, the changes of tendencies shown in Fig. 1.1 can be attributed to shifts in market beliefs about two subjects on the supply side. First, political and geopolitical events concerning oil-exporting countries and entailing risks of significant supply disruptions. Second, the activity of OPEC that caused the oil price shocks and contributed to the increase of oil price volatility as compared to the pre-cartel era. The market power of the oil cartel rests on its tremendous resource base and on the competitive advantages in terms of extraction costs featured by most of the cartel members. In addition, the cartel’s power has been supported, since the beginning of the 2000s, by the significant growth in the global oil demand. The resource cartel’s market activity affects consumers and competitive producers outside the cartel differently. On the one hand, the cartel reduces its output

12

1 Introduction

to obtain a monopoly gain, in addition to the competitive resource rent, at the expense of consumer surplus. On the other hand, the reduction of the cartel’s output causes an effect of production shifting to the less advanced oil regions that are competing with the cartel. Likewise, a contraction of the cartel’s investment in the development of new oil reserves results in an increase of such investments outside the cartel. Formal treatment of these structural effects depends on whether the resources of oil in the ground are supposed to be exhaustible or recoverable under limited availability. In the extensions of Hotelling’s model to imperfect competition, which are relevant to the world oil market, the supply side includes a resource cartel and a fringe of competitive producers. We consider these extensions in Chap. 9, “Cartel behaviour in an exhaustible resource industry”. The cartel is supposed to be formed in the advantageous oil region with lower unit extraction costs compared to the region of competitive fringe. In equilibrium, the resource price is equal to the full marginal cost of the highest-cost competitive producer, including its opportunity cost of extraction. The cartel’s excess profit is, therefore, provided by the cost differential with the highest-cost producer and in this respect it is similar to the differential rent in the Ricardian model. The cartel’s activity violates the Herfindahl principle held under a competitive market and implying that the advantageous region depletes its resource stock before the disadvantageous region begins extraction. In the cartelized market, the time sequencing of production dramatically changes and permits the empirically relevant case where the low-cost cartel and the high-cost fringe produce simultaneously. Production shifting from the low-cost cartel to the high-cost competitive fringe implies distortions of the industry structure resulting from the misallocation of production. In the long term, this effect brings about a misallocation of global oil resources. In Chap. 10, “The oil cartel and misallocation of production”, we consider a two-region model of resource extraction with investment in reserves development, in which the global allocation of oil resources is determined endogenously. We show that if the cartel’s advantage in production costs over the competitive fringe is considerable, the distorting effects of the cartel are substantial and the efficiency losses on the industry level may be significant. The cartel captures a part of consumer benefits at the cost of significant losses from resource misallocation and deadweight losses. However, these losses can be reduced due to technological changes improving the competitiveness of the fringe. Hotelling suggested in his paper (1931) that a resource monopoly behaves as a “conservationist” that extracts less today and keeps more resource under the ground than the competitive market does. However, in the presence of a high-cost competitive fringe, the cartel’s “conservationism” turns into a too early and too intensive resource extraction by the fringe. We examine this issue in Chap. 11, “Anti-conservationist effects of the conservationist oil cartel”, for a model with economically recoverable resources. The “anti-conservationism” of the equilibrium extraction path selected by the fringe proves to be a robust property of the model. As a result, the cartel acts as if it was a strategic player planning its long-term moves to accelerate

References

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the depletion of competitors’ resources and to aggravate their competitive disadvantages.

References BP Statistical Review of World Energy, 2020 edition. https://nangs.org/analytics/bp-statisticalreview-of-world-energy. Accessed 12 Sep 2020 FRED Economic Data (2020) Economic research, Federal Reserve Bank of St. Louis, https://fred. stlouisfed.org/tags/series?t¼inflation. Accessed 12 Sep 2020 Gray L (1914) Rent under the assumption of exhaustibility. Q J Econ 28(3):466–489 Hotelling H (1931) The economics of exhaustible resources. J Polit Econ 39:137–175 Jevons WS (1865) The coal question. Macmillan, London, p 213 Malthus T (1815) The nature of rent, the Avalon project, documents in law, history and diplomacy, Yale Law School, Lillian Goldman Law Library. https://avalon.law.yale.edu/19th_century/rent. asp. Accessed 14 Sep 2020 Nordhaus W (1994) Managing the global commons: the economics of climate change. MIT Press, Cambridge, MA Ricardo D (1817) On the principles of political economy and taxation, vol 2001. Batoche Books, Kitchener Smith J (2009) World oil: market or mayhem. J Econ Perspect 23:145–164

Part I

Resource Rents and Opportunity Costs

Chapter 2

The Economics of Exhaustible Resources

Abstract We consider in this chapter Hotelling’s classic model of the economics of exhaustible resources and its extensions. The social planner’s problem in this model is to maximize the discounted social benefit from consumption of a homogeneous resource under the constraint that cumulative consumption is no greater than the initial resource stock. The marginal resource rent for the optimal extraction path grows at the real rate of interest, and the equilibrium resource price meets this condition, known as “Hotelling’s rule”. This rule modifies under the model extensions to heterogeneous resources: the marginal rent falls if extraction switches from low-cost to high-cost resource stocks. We consider the models of resource pricing in the presence of a backstop technology, the Herfindahl principle that a lower-cost resource stock depletes before extraction switches to a higher-cost stock, and the principle of comparative advantage for a more general case of heterogeneous resources and consumers.

2.1

Introduction

The theoretical foundation of natural resource economics was laid by Harold Hotelling in his seminal paper published in 1931, “The Economics of Exhaustible Resources”. On the occasion of the 50th anniversary of this publication, Devarajan and Fisher (1981, p. 65) wrote: “There are only a few fields in economics whose antecedents can be traced to a single, seminal article. One such field is natural resource economics, which is currently experiencing an explosive revival of interest; its origin is widely recognized as Harold Hotelling’s 1931 paper, ‘The Economics of Exhaustible Resources’.” That paper pioneered rigorous analysis of a non-renewable resource consumption in the long term. Hotelling formulated the dynamic model of optimal extraction for a finite and perfectly known stock of a resource. The central planner’s problem is to maximize the discounted social benefit from resource consumption under the

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_2

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2 The Economics of Exhaustible Resources

constraint that cumulative extraction is no greater than the initial resource stock. Hotelling emphasized the normative long-term aspect of this problem as a need to prevent wasteful overexploitation of non-renewable resources by current generations in order to ensure their availability to future generations. The key idea of Hotelling’s theory is that any unit of a non-renewable resource extracted today reduces the amount of this resource that may be used at some future date. The choice between consuming the unit of resource now or delaying consumption to a later time introduces the intertemporal dimension as the most essential point of exhaustible resource economics. At the margin, the resource rent, defined as the resource price less the marginal extraction cost, should be equal to the marginal value of conserving a resource for the future, which represents the opportunity cost of resource extraction. For a resource industry operating under constant marginal extraction costs, the assumption in Hotelling’s model, the marginal rent grows exponentially over time at a rate equal to the real interest rate. This theoretical inference is the Hotelling rule, which Robert Solow (1974, p. 3) called the “fundamental principle of the economics of exhaustible resources”. This rule holds for the whole optimal extraction path, regardless of how much of the resource stock is extracted and consumed at any time. The Hotelling rule is based on the assumption that a resource in the ground is a capital asset that generates returns over time. In the world of certainty, the riskless rate of return is equal to the real interest rate. Under a constant marginal cost of extraction, a decrease of resource stock has no marginal effect on the resource rent. The only way that this stock can produce a current rate of return is if its value appreciates. Therefore, the marginal resource value, which is equal to the marginal resource rent, must be growing at a rate equal to the interest rate. This rule is fulfilled both for the optimal extraction path and for a competitive resource industry equilibrium, provided that the social and private discount rates coincide and any externalities are absent. Under the Hotelling rule, the present value of any unit of resource is time-constant and there is no gain for resource suppliers from shifting the extraction programme over time. Therefore, at any point in time, supply meets demand, which is decreasing in price. Under a time-stationary demand function, optimal extraction declines monotonically to zero because the price grows monotonically over time. The extraction programme terminates when the resource stock is fully exhausted. The time of termination is a free variable determined as the solution of the problem. We will consider in this chapter several extensions of Hotelling’s model. After the time of termination, the economy can switch to a more abundant substitute for the exhaustible resource. William Nordhaus (1973, p. 532) introduced the notion of backstop technology in connection with the problem of energy resource availability: “Ultimately, if and when the transition is completed to an economy based on plentiful nuclear resources (either through breeder or fusion reactors), the economic importance of scarce resources will disappear, and capital and labour costs alone will determine prices. The ultimate technology—resting on a very abundant resource base—is the ‘backstop technology’ and is crucial to the allocation of scarce energy resources.”

2.1 Introduction

19

The existence of a backstop technology in the extended Hotelling’s model matters for a current extraction programme and for non-renewable resource pricing. We will assume, following Nordhaus (1973) and other authors, that the backstop technology produces a perfect substitute for the conventional non-renewable resource. This assumption makes sense, for example, with regard to energy consumption. The backstop technology may have an unlimited resource base, but it is more expensive initially than the conventional resource. An optimal extraction programme implies that the natural resource is fully exhausted by the time of the switch to the backstop technology, when the resource price coincides with the backstop price. It is important that during the phase of transition to this technology, the resource price depends on the expected backstop price. Another extension of Hotelling’s model relates to heterogeneous resources. In the basic Hotelling model, the resource is homogeneous in the sense that the quality of output and the marginal extraction costs are the same for all producers. Usually, several grades of the same resource exist: mineral ores of higher and lower quality or fossil fuels with lower and higher costs of extraction. For example, the quality and production costs of oil substantially differ for the Middle East reserves and for the Canadian oil sands or deep-water oil resources of the North Sea. In many cases, the lower-cost resources, including conventional oil and gas, are relatively scarce, while the higher-cost resources, such as tight oil or gas hydrates, are abundant. Within the Hotelling framework, resource sites with different extraction costs cannot operate simultaneously. The Herfindahl principle states that a lower-cost resource is exploited until full exhaustion and before extraction proceeds to a highercost resource. This principle determines a link between extraction programmes of heterogeneous resource sites. A lower-cost site is valued higher than a higher-cost one; therefore, the marginal resource rent decreases as production switches from lower-cost to higher-cost resources. The Hotelling rule holds for any individual resource site, but it does not hold for the whole extraction path, because of the tendency of the marginal resource rent to fall with the transition to higher-cost and more abundant resources. The model with heterogeneous resource suppliers is further extended to incorporate heterogeneous resource users. This is a more realistic description of resource markets reflecting geographical location and transportation costs and also the costs of resource conversion that usually differ across users. For this extension of the model, it is assumed that marginal production costs vary between sites and users and are arrayed in a matrix of unit costs. The solution to the social planner’s problem is the optimal allocation of production in time and space that assigns time paths of output flows from each site to each user. The guideline for such allocation is the principle of comparative advantage based on selecting the most efficient user for each supplier at any instant in time. This principle explains, in particular, why the switch to a higher-cost resource can occur before the lower-cost resource is fully exhausted.

20

2.2

2 The Economics of Exhaustible Resources

Hotelling’s Model

Hotelling’s model describes the choice by a central planner of the optimal extraction path for a finite, perfectly known and homogeneous resource stock. The initial resource stock S0 ¼ S(0) is available for exploitation at the initial time t ¼ 0 and the extraction technology is specified by a time-constant marginal cost of extraction c. The planner’s objective function is given by the discounted social welfare derived from resource exploitation. Instantaneous welfare gain is the net benefit for society defined as pecuniary utility of consumption less extraction cost, U(q(t))  cq(t), where q(t)  0 is the amount of resource extracted and consumed at time moment t, U(q(t)) is the instantaneous pecuniary utility of resource consumption, which is a strictly increasing, differentiable and strictly concave function with U0(0) > c. The latter condition ensures positive net benefits from resource extraction. It is assumed that social and individual utilities coincide due to the absence of external effects like environmental impacts. Thus, the planner’s problem is to maximize the discounted welfare function, ZT max

qðt Þ, T

ert ½U ðqðt ÞÞ  cqðt Þdt,

ð2:1Þ

0

subject to the resource constraint on cumulative extraction, ZT qðt Þdt  S0 ,

ð2:2Þ

0

where r is the discount rate, and T  1 is the unknown time of extraction termination.

2.2.1

Hotelling Rule

The Lagrangian for the planner’s problem (2.1)–(2.2) is written as ZT L¼ 0

0 ert ½U ðqðt ÞÞ  cqðt Þdt þ v0 @S0 

ZT 0

1 qðt Þdt A,

ð2:3Þ

2.2 Hotelling’s Model

21

where v0 is the dual variable related to the resource constraint (2.2) or the marginal resource value at the initial time. The first-order condition for positive q(t) is ∂L=∂q ¼ 0: U 0 ðqðt ÞÞ  c  vðt Þ ¼ 0,

ð2:4Þ

vðt Þ ¼ ert v0

ð2:5Þ

where

is the marginal resource value at time t. This value measures the opportunity cost of depleting an extra unit of resource in terms of the net benefit of resource consumption foregone in the future. Condition (2.4) equates the marginal utility of resource consumption and the full marginal cost of resource extraction, which is the sum of direct extraction cost c and the opportunity cost v(t). Define the resource price as the marginal utility of resource consumption, p (t) ¼ U0(q(t)). Then condition (2.4) also means that the marginal resource value v (t) is equal to the marginal resource rent or profit per unit of output, p(t)  c. Condition (2.5) implies that v_ ðt Þ ¼ rvðt Þ, which is represented as Hotelling’s rule for the resource price: p_ ðt Þ ¼ r ðpðt Þ  cÞ,

ð2:6Þ

where a dot over a variable denotes a time derivative. Thus, the marginal resource rent must grow at a rate equal to the discount rate. The marginal resource rent p (t)  c indicates the resource scarcity, since it increases as the resource stock is depleted. Define the choke price as the zero-consumption resource price, α ¼ U0(0). For a finite choke price, the first-order condition for termination time results from differentiating the Lagrangian (2.3) with respect to T: U ðqðT ÞÞ  ðc þ vðT ÞÞqðT Þ ¼ 0:

ð2:7Þ

The welfare gain net of the full cost of resource extraction is zero at the termination date. This terminal condition is fulfilled if qð T Þ ¼ 0

ð2:8Þ

and U(0) ¼ 0. By the time of termination, the resource stock must be fully exhausted, S(T ) ¼ 0. Hence, extraction is choked off at the time when the resource stock runs out. Otherwise, the extraction programme could be improved by delaying somewhat the termination date.

22

2.2.2

2 The Economics of Exhaustible Resources

The Decentralized Solution

The central-planning solution can be decentralized as a competitive equilibrium. Suppose that the resource is supplied by a competitive industry consisting of homogeneous production firms that control the resource stock depletion. Each firm takes the path of resource price p(t) as given and chooses the extraction path q(t) maximizing the present value of future rents: ZT V 0 ¼ max

qðt Þ, T

ert ðpðt Þ  cÞqðt Þdt,

ð2:9Þ

0

subject to the resource constraint the same as (2.2). The discount rate in (2.9) is the same as in the planner’s objective function (2.1). At any instant, the market price clears the market qðt Þ ¼ yðt Þ,

ð2:10Þ

where y(t) is consumer demand. Let P ðyðt ÞÞ  U 0 ðyðt ÞÞ denote inverse demand. The market-clearing condition (2.10) can be written as: pðt Þ ¼ P ðqðt ÞÞ. Hotelling’s rule (2.6) then fulfils for the competitive resource price p(t). Under this rule, the present value of a unit extracted is the same for all future periods when extraction occurs. There is no gain from shifting production over time, and therefore at any instant supply matches demand. Hotelling’s resource pricing rule (2.6) implies the principle of resource valuation: ZT V 0 ¼ ð p0  c Þ

qðt Þdt ¼ ðp0  cÞS0 0

with p0 ¼ p(0), which is called “Hotelling’s valuation principle”. The initial resource stock is valued through the marginal resource rent at the initial time. One can show, similarly, that at any moment t the present value of future rents is equal to: V(t) ¼ ( p RT (t)  c)S(t), where Sðt Þ ¼ qðτÞdτ is the remaining resource stock at time t. t

Hotelling’s rule thus gives a no-arbitrage condition for investment in resource deposits. The yield on this investment equals the capital gain from growing marginal resource rent. In a world of certainty, the rate of marginal rent growth coincides with the return on the alternative riskless assets given by the real interest rate.

2.2 Hotelling’s Model

2.2.3

23

Remarks on the Discount Rate and the Real Interest Rate

In Hotelling’s model, the discount rate in the planner’s objective function (2.1) with the pecuniary utility of consumption is assumed to be equal to the real interest rate indicating the opportunity cost of investment in resource deposits. This rate is given exogenously for the partial equilibrium model of a resource-extracting industry. In a general equilibrium framework, the economy-wide discount rate would result from an allocation of investment in production capital in the economy. One can assume, for example, that capital is allocated through the market for credit that determines the real interest rate. Under provisions of perfect competition, the borrowing rate adjusted for inflation equals the marginal product of capital, which is equalized across sectors. This interest rate can define the opportunity cost of resource extraction and represent a discount rate for the resource-extracting industry. The discount rate in Hotelling’s model is a constant parameter, which is usually assumed to be positive. However, the real interest rate in the real world is timevarying and exhibits a substantial decline over time in the long-term retrospective. Figure 2.1 shows dynamics of the annual yield on ten-year U.S. Treasury inflationindexed security, which was close to zero and even negative in some periods. The phenomenon of negative long-term inflation-adjusted interest rates can be viewed through the lenses of the revived concept of secular stagnation as suggested by Rachel and Summers (2019). They showed that a declining real interest rate for the industrial economies as a whole results from the investment dearth and the savings glut, which are explained by the demographic shifts, the rise in income inequality and the long-term tendency of slowing GDP growth.

Fig. 2.1 Ten-year U.S. Treasury inflation-indexed security yield in 2003–2020, percent. Source: FRED Economic Data (2020)

24

2 The Economics of Exhaustible Resources

From a mathematical point of view, the welfare function (2.1) and the value function (2.9) can be not defined for a non-positive discount rate, r  0, if the time of extraction termination T is infinity. Solutions with infinite T are possible for Hotelling’s model, as we will see below. Nevertheless, to simplify matters in such cases, we will assume that the discount rate is positive, r > 0. Note that using the real interest rate for welfare discounting may be inappropriate for the problems of intergenerational resource allocation. For a social planner model with overlapping generations, a discount rate in the objective function defines the welfare weights of future generations and it may differ from the long-term real interest rate, as we will see for a climate change model in Chap. 5.

2.3

Equilibrium Extraction Paths

The solution for Hotelling’s rule (2.6) is pðt Þ ¼ c þ v0 ert :

ð2:11Þ

For a finite choke price α we have it that at the termination date pðT Þ ¼ P ðqðT ÞÞ ¼ P ð0Þ ¼ α and the initial marginal rent is given by v0 ¼ ðα  cÞerT :

ð2:12Þ

To calculate the equilibrium extraction path, consider the case of linear inverse demand: P ðqÞ ¼ α  βq

ð2:13Þ

with β denoting the slope coefficient. The market-clearing price is pðt Þ ¼ P ðqðt ÞÞ. Combining (2.11), (2.12) and (2.13) implies the solution for equilibrium intensity of extraction: qð t Þ ¼

  αc 1  erðtT Þ : β

ð2:14Þ

Inserting q(t) into the resource constraint (2.2), which is binding and holds with equality in equilibrium, and integrating it implies the equation for the time of termination T:  α  c rT þ erT  1 ¼ S0 : βr

ð2:15Þ

The left-hand side of this equation is increasing in T, hence the time of termination can be considered as the function of initial resource stock:

2.3 Equilibrium Extraction Paths

25

Fig. 2.2 The time of termination

T ¼ T ðS0 Þ, Figure 2.2 depicts this function, which is increasing, concave and has the linear asymptote for arbitrarily large S0: T¼

βS0 þ r 1 : αc

An increase of the initial resource stock influences the resource supply at the extensive and intensive margins. At the extensive margin, such an increase leads to a lengthening of the time period of extraction T(S0). At the intensive margin, an increase of resource stock results in raising equilibrium extraction q(t) at any instant. The marginal resource rent (2.12) is decreasing with T(S0), indicating a lower degree of resource scarcity for a larger resource stock.

2.3.1

A Numerical Example

We can illustrate the solution for linear inverse demand with a numerical example. Suppose that the resource is crude oil, the choke price is α ¼ $300 per barrel, the slope of inverse demand is β ¼ 5, the marginal extraction cost is c ¼ $20 per barrel, the annual real interest rate is r ¼ 0.05 and the initial resource stock is S0 ¼ 3000 million barrels. Then the termination time is the solution of eq. (2.15): T ¼ 73 years. The initial marginal rent is v0 ¼ (α  c)erT ¼ $7.3 per barrel and the initial resource price is p0 ¼ c + v0 ¼ $27.3 per barrel. Figures 2.3 and 2.4 demonstrate the time paths of equilibrium resource price (2.11) and annual extraction (2.14), respectively, simulated for this numerical example. The solid curves show the case with the interest rate r ¼ 0.05, and the dashed curves the case with r ¼ 0.02, other parameters being the same.

26

2 The Economics of Exhaustible Resources

Fig. 2.3 Equilibrium price paths

Fig. 2.4 Equilibrium extraction paths

These figures demonstrate that a higher interest rate encourages a more intensive extraction at the early stage and a more rapid resource depletion. The time period of extraction is longer for the interest rate r ¼ 0.02: T ¼ 96 years. One can see from Fig. 2.3 that for a higher interest rate the resource price is lower at the early stage, but it grows more rapidly at the later stage.

2.3.2

Constant-Elasticity Demand

Consider the case of isoelastic inverse demand:

2.4 Resource Exhaustion and the Backstop Technology

27

P ðyÞ ¼ y1=σ , with σ > 0 denoting the constant price elasticity of demand. In this case, the inverse demand tends toward infinity for arbitrarily small y. The period of extraction T is therefore infinite and the terminal condition (2.8) holds asymptotically as lim qðT Þ ¼ 0:

T!1

The solution for the Hotelling rule is (2.11) and the market-clearing supply is equal to σ

qðt Þ ¼ ðc þ v0 ert Þ : The resource constraint (2.2) is given by Z1

σ

ðc þ v0 ert Þ dt ¼ S0 :

ð2:16Þ

0

The solution of this equation is the initial marginal resource value v0(S0), which is the decreasing function of the initial resource stock. Let c ¼ 0. Then the integral on the left-hand side of (2.16) equals vσ 0 =rσ and the initial marginal resource value is v0 ¼ (rσS0)1/σ . From (2.11), the equilibrium price is pðt Þ ¼ ðrσS0 Þ1=σ ert

ð2:17Þ

and the equilibrium intensity of extraction is qðt Þ ¼ ðrσS0 Þerσt ¼ rσSðt Þ,

ð2:18Þ

because S_ ðt Þ ¼ qðt Þ, hence S(t) ¼ S0erσt. The equilibrium extraction is thus the infinitesimally small share rσdt of the remaining resource stock S(t).

2.4

Resource Exhaustion and the Backstop Technology

Suppose that there exists a backstop technology for the production of a perfect substitute for the exhaustible natural resource. This technology does not use non-renewable resources and has an unlimited resource base. The substitute can be supplied competitively with the marginal cost F ¼ const. The backstop production is

28

2 The Economics of Exhaustible Resources

available initially, but it is more expensive than resource extraction in the sense that F > c. As above, the initial stock of depletable resource is S0. The planner’s problem is to choose the paths of resource extraction q(t) and backstop production b(t) to maximize the discounted welfare function: Z1 max

qðt Þ, bðt Þ

ert ½U ðqðt Þ þ bðt ÞÞ  cqðt Þ  Fbðt Þdt,

ð2:19Þ

0

subject to the resource constraint: Z1 qðt Þdt  S0 :

ð2:20Þ

0

The Lagrangian for this problem is: 0

Z1 LB ¼

e

rt

½U ðqðt Þ þ bðt ÞÞ  cqðt Þ  Fbðt Þdt þ v0 @S0 

0

Z1

1 qðt Þdt A, ð2:21Þ

0

where v0 is the marginal resource value at the initial date. As above, the marginal resource value v(t) rises exponentially at a rate equal to the discount rate: vðt Þ ¼ ert v0 :

ð2:22Þ

The first-order conditions for q(t) and b(t) are U 0 ð qð t Þ þ bð t Þ Þ  c  v ð t Þ  0

ð2:23Þ

½U 0 ðqðt Þ þ bðt ÞÞ  c  vðt Þqðt Þ ¼ 0

ð2:24Þ

0

U ð qð t Þ þ bð t Þ Þ  F  0

ð2:25Þ

½U 0 ðqðt Þ þ bðt ÞÞ  F bðt Þ ¼ 0:

ð2:26Þ

Conditions (2.24) and (2.26) hold with complementary slackness, and q(t) or b(t) equals zero if the net marginal benefit in (2.23) or (2.25), respectively, is negative. The extraction is positive and the backstop production is zero at time t if the full marginal cost of extraction is below the backstop price, c + v(t) < F. In the opposite case, c + v(t) > F, the extraction is zero and the backstop production is positive. These alternative technologies of production cannot be used simultaneously for c + v (t) 6¼ F due to the linearity of production costs in the objective function (2.19).

2.4 Resource Exhaustion and the Backstop Technology

29

Fig. 2.5 The resource price and the backstop price

Fig. 2.6 Resource extraction and the backstop production

There are two phases of production shown in Figs. 2.5 and 2.6. Let T be the time of switching between these phases. At this time, the marginal costs coincide: c þ vðT Þ ¼ F:

ð2:27Þ

During the first phase, resource extraction occurs without backstop production until full exhaustion of the resource at time T. At any moment t < T the resource price is the unit extraction cost plus the net backstop price discounted back to the present, p(t) ¼ c + er(T  t)(F  c), as follows from (2.22), (2.27). During the second phase that continues indefinitely, only the backstop technology is used and the price is p(t) ¼ F. The time of switching T satisfies the first-order condition, ∂L B =∂T ¼ 0: U ðqðT ÞÞ  ðc þ vðT ÞÞqðT Þ ¼ U ðbðT ÞÞ  FbðT Þ:

ð2:28Þ

Due to the equalization of marginal costs (2.27), this condition is fulfilled if the terminal extraction is equal to the backstop production:

30

2 The Economics of Exhaustible Resources

qðT Þ ¼ bðT Þ: For the constant-elasticity inverse demand, P ðyÞ ¼ y1=σ , the terminal extraction is q(T) ¼ Fσ , as shown in Fig. 2.6.

2.4.1

Solution for Constant-Elasticity Demand

Let, as above, c ¼ 0. Then p0 ¼ v0, p(t) ¼ p0ert, q(t) ¼ ( p0ert)σ and the resource constraint at any time t < T is ZT

ðp0 erτ Þσ dτ ¼ Sðt Þ:

t

Taking the integral on the left-hand side of this equation implies: ðp0 ert Þσ  ðp0 erT Þ rσ

σ

¼ Sðt Þ:

The optimal resource price is: pðt Þ ¼ ðF σ þ rσSðt ÞÞ1=σ , because F ¼ p0erT. This formula was derived by Dasgupta and Stiglitz (1975). The optimal price is the CES (constant elasticity of substitution) combination of the backstop price F and the resource price (rσS(t))1/σ . The latter corresponds to the extraction rule (2.18) that prevails in the absence of a backstop technology.

2.5

The Herfindahl Principle

Now we extend the model of Sect. 2.2 to the case of multiple heterogeneous resource sites producing homogeneous outputs with different extraction costs. This extension was suggested by Orris Herfinfahl (1967) and examined subsequently, for example, by Solow and Wan (1976), Alistair Ulph (1978), and Gaudet and Salant (2014). There are n sites that are productive in the sense that the net marginal benefits are positive under the choke price, U0(0) > ci, i ¼ 1, . . ., n. The choke price is assumed to be finite, U0(0) < 1. The sites are indexed so that a higher-cost site has a higher index:

2.5 The Herfindahl Principle

31

c1 < c2 < . . . < cn : Denote the total supply by all sites at time t as qð t Þ ¼

n X

qi ðt Þ:

i¼1

The planner chooses the paths of resource extraction qi(t) maximizing the welfare function: Z1 max

fqi ðt Þg

h i Xn ert U ðqðt ÞÞ  c q ð t Þ dt, i¼1 i i

ð2:30Þ

0

subject to n resource constraints: Z1 qi ðt Þdt  Si0 ,

ð2:31Þ

0

where Si0 is the initial resource stock of site i. The marginal resource value of site i is vi(t). For any i ¼ 1, . . ., n, this marginal value grows exponentially at rate r: vi ðt Þ ¼ vi0 ert

ð2:32Þ

and the first-order conditions are similar to those from the previous section: U 0 ð qð t Þ Þ  c i  v i ð t Þ  0

ð2:33Þ

½U 0 ðqðt ÞÞ  ci  vi ðt Þqi ðt Þ ¼ 0:

ð2:34Þ

The full marginal cost of site i is ci + vi(t). The extraction costs are linear, and therefore at any instant t the planner should choose the site with minimal full marginal cost. The resource price pðt Þ ¼ U 0 ðqðt ÞÞ is set equal to this cost: pðt Þ ¼ ck þ vk ðt Þ,

ð2:35Þ

if qk(t) > 0, while p(t) < ci + vi(t) for all other sites i 6¼ k, for which qi(t) ¼ 0. Since each site is productive, it must be exploited. This means that the full marginal costs of any site must be minimal during some period of time. The initial marginal values vi0 are assigned to ensure that this condition is fulfilled. First, the initial marginal values must be selected so that the value of a lower-cost site is higher:

32

2 The Economics of Exhaustible Resources

v10 > v20 > . . . > vn0 :

ð2:36Þ

Otherwise, the higher-cost sites would never be used. For example, if ck < ci and vk0 < vi0, then the full marginal cost of the higher-cost site i would be higher at any time: ci + vi0ert > ck + vk0ert. Second, the full marginal costs must be assigned so that the sites could be used in ascending order of extraction costs: c1 þ v10 < c2 þ v20 < . . . < cn þ vn0 :

ð2:37Þ

Conditions (2.36) and (2.37) imply the principle of intertemporal cost minimization offered by Herfindahl (1967). Simultaneous extraction of resources with different extraction costs should never occur and a lower-cost resource is exhausted completely before a higher-cost resource is exploited. The time of switching from lower-cost to higher-cost sites is determined similarly to the time of switching to the backstop technology described in the previous section. Let Ti denote the moment of switching between sites i and i + 1 such that the full marginal costs are equal: ci þ vi ðT i Þ ¼ ciþ1 þ viþ1 ðT i Þ:

ð2:38Þ

For this moment, the first-order condition, similarly to (2.28), must fulfil: U ðqðT i ÞÞ  ðci þ vi ðT i ÞÞqi ðT i Þ ¼ U ðqðT i ÞÞ  ðciþ1 þ viþ1 ðT i ÞÞqiþ1 ðT i Þ: This and (2.38) imply that the supplies of adjacent sites are matched at the point of switching: qi ðT i Þ ¼ qiþ1 ðT i Þ

ð2:39Þ

for 1  i  n  1. These conditions link extraction paths for all sites. The time of termination for the highest-cost site Tn is the time of extraction completion for the planner’s problem, hence qn(Tn) ¼ 0. In the next subsection, we will demonstrate the implications of the Herfindahl principle for the case of two sites.

2.5.1

Solution for Two Sites

Suppose that the resource industry consists of two sites. Site 1 is low-cost and site 2 is high-cost, c1 < c2. Consider the competitive market solution for the planner’s problem. Let the inverse demand be linear:

2.5 The Herfindahl Principle

33

Fig. 2.7 The price path in the two-site model

P ðyÞ ¼ α  βy and at any instant the price clears the market, pðt Þ ¼ P ðq1 ðt Þ þ q2 ðt ÞÞ: According to the Herfindahl principle, the low-cost site 1 depletes its resources first, and the high-cost site 2 thereafter. There are two periods of production such that only site 1 produces in the first period, q1(t) > 0, q2(t) ¼ 0, and only site 2 in the second period, q1(t) ¼ 0, q2(t) > 0. The first period is 0  t  T1 and the second one is T1  t  T2, where T1 is the time of switching between the sites and T2 is the time of extraction termination. The equilibrium price path p(t) is shown in Fig. 2.7. It consists of two pieces, p1(t) and p2(t), given by the solution for price (2.35): p1 ðt Þ ¼ c1 þ v10 ert , for 0  t  T 1 :

ð2:40aÞ

p2 ðt Þ ¼ c2 þ v20 e , for T 1  t  T 2 :

ð2:40bÞ

rt

At the initial time the full marginal costs of the sites are c1 + v10 < c2 + v20. For any t > 0 the price path is the lower envelope of these marginal costs. At the switching point T1 the price path shown in Fig. 2.7 is kinked, but it must be continuous to satisfy the condition of supply matching (2.39): q1 ðT 1 Þ ¼ q2 ðT 1 Þ:

ð2:41Þ

This implies the condition of price matching, p1(T1) ¼ p2(T1), that links the differential of initial marginal rents with the extraction-cost differential:

34

2 The Economics of Exhaustible Resources

Fig. 2.8 The extraction path in the two-site model

v10  v20 ¼ ðc2  c1 ÞerT 1

ð2:42Þ

due to price eqs. (2.40a) and (2.40b). The differential of marginal rents, v10  v20, can be viewed as the measure of relative scarcity of the low-cost resource. The extraction paths q1(t) and q2(t) are shown in Fig. 2.8. For site 2, the termination price is the choke price, p2(T2) ¼ α, and hence the extraction path is given by formula (2.14):   q2 ðt Þ ¼ ðα  c2 Þ 1  erðtT 2 Þ =β:

ð2:43Þ

The extraction path of site 1 can be represented as the sum: q1 ðt Þ ¼ q01 ðt Þ þ q001 ðt Þ,

ð2:44Þ

  where q01 ðt Þ ¼ ðα  c1 Þ 1  erðtT 1 Þ =β and q001 ðt Þ ¼ q2 ðT 1 ÞerðtT 1 Þ , as we show in Appendix A.1. The term q01 ðt Þ does not depend explicitly on site 2’s activity. The term q001 ðt Þ results from the supply-matching condition (2.41). As one can see in Fig. 2.8, the closer time t is to switching time T1, the less significant is the term q01 ðt Þ and the more significant is the term q001 ðt Þ. At time T1, q01 ðT 1 Þ ¼ 0 and q001 ðT 1 Þ ¼ q2 ðT 1 Þ. The dates of switch and termination, T1 and T2, are determined from resource constraints (2.31). For site 2, eqs. (2.43) and (2.15) imply that the period of extraction depends on the resource stock of this site, T2  T1 ¼ T(S20). For site 1 the solution is the function of resource stocks in both sites, T1 ¼ T1(S10, S20). We show in Appendix A.2 that ∂T 1 =∂S10 > 0, ∂T 1 =∂S20 < 0:

ð2:45Þ

The time of switching to the high-cost site T1 is increasing with S10, but decreasing with S20. The differential of initial marginal rents (2.42) is decreasing with T1.

2.5 The Herfindahl Principle

35

Consequently, this differential, as the measure of relative scarcity of a low-cost resource, is high if the low-cost site is small or if the high-cost resource is abundant.

2.5.2

A Modified Hotelling Rule

Hotelling’s rule is modified for the case of heterogeneous resource sites. This rule is fulfilled for each site as (2.32), but it should be adjusted for the whole extraction path due to the switching from lower-cost to higher-cost sites. Figure 2.9 demonstrates the time path of marginal resource values for the two-site model: v1(t) ¼ v10ert and v2(t) ¼ v20ert. This time path is drawn with solid curves. The marginal values grow with the same rate r, but for the low-cost site the marginal value is higher at any moment in time, because v10 > v20. The time path of the marginal value is therefore discontinuous at the point of switching T1. Curve v10M in the figure shows the time-average path of the marginal resource value that begins at the initial point v10 on the vertical axis and continues until the point of termination M. This path is given by the exponential function v10er 0 t with r0 denoting the average growth rate of the marginal resource value, which is below the interest rate. For the point of termination M we have it that v10 er0T 2 ¼ v20 erT 2 , implying that r  r 0 ¼ ð ln v10  ln v20 Þ=T 2 :

ð2:46Þ

The differential of growth rates is proportional to the log differential of initial marginal values that indicates the relative scarcity of the low-cost resource. As we Fig. 2.9 Modified Hotelling rule

36

2 The Economics of Exhaustible Resources

have shown, the relative scarcity depends on the extraction-cost differential (2.42) and on the difference in size of resource sites. Thus, as follows from (2.46), the deviation of the marginal rent growth rate from the Hotelling rule is significant if the sites vary considerably in extraction costs and size (the low-cost site is small and the high-cost one is big).

2.6

The Principle of Comparative Advantage

The model with n resource sites and single consumer is further extended to include m heterogeneous final users that differ in consumer utilities and, what is more important, in per-unit costs of resource transportation and conversion. This extension was analysed by Chakravorty and Krulce (1994), Gaudet et al. (2001), Chakravorty et al. (2005), and Gaudet and Salant (2014) as the problem of production allocation in space and time. In this section, we consider this problem. We will call users “cities”, following the paper by Gaudet et al. (2001). Let qij(t) denote the amount of resource obtained from site i and delivered to city j at time t. The supply by site i ¼ 1, . . ., n is qi ð t Þ ¼

m X

qij ðt Þ

j¼1

and the amount consumed by city j ¼ 1, . . ., m is q j ðt Þ ¼

n X

qij ðt Þ:

i¼1

The utility of city j, Uj(q j(t)), has the same properties as the single-user utility U(q (t)). The per-unit cost of supplying from site i to city j is cij. Thus, the industry-wide technology of resource supply is given by the m  n matrix of unit costs. The planner’s problem is to choose the flows of resource production and delivery to final consumers qij(t) maximizing the welfare function: "

Z1 max fqij ðtÞg

e

rt

# m X n X  1  n U 1 q ðt Þ þ . . . þ U n ðq ðt ÞÞ  cij qij ðt Þ dt,

0

subject to n resource constraints:

j¼1 i¼1

2.6 The Principle of Comparative Advantage

37

Z1 qi ðt Þdt  Si0 : 0

The marginal resource value of site i is vi(t) ¼ vi0ert, and the full marginal cost of supplying from site i to city j is cij + vi(t). The first-order conditions relate to each flow qij(t):   U 0j q j ðt Þ  cij  vi ðt Þ  0 h  i  U 0j q j ðt Þ  cij  vi ðt Þ qij ðt Þ ¼ 0: Any city j should choose at any instant t the site with the minimal full marginal cost. On aggregate, no statement can be made about the sequence of extraction for an arbitrary unit cost matrix (cij), because, generally, the users rank sites differently. A site with the lowest cost for one user need not be the lowest-cost site for another one. However, as Gaudet et al. (2001) showed, the generalized Herfindahl principle is fulfilled for each user ranking sites in its order of increasing marginal extraction costs. To clarify this principle, consider such a ranking by city j and a pair of adjacent sites with indices i and k, cij < ckj. Let Tij denote the time of switching from site i for city j. At this time, the full marginal costs are equalized:     cij þ vi T ij ¼ ckj þ vk T ij

ð2:47Þ

and supplies are matched, qij(Tij) ¼ qkj(Tij). Prior to date Tij, the left-hand side of (2.47) is smaller than the right-hand side, and just after this date it is larger. The resource price path for city j, p j ðt Þ ¼ U 0j ðq j ðt ÞÞ, is the lower envelope of full marginal costs of all sites selected by this city. The determination of optimal times of switching Tij for any i ¼ 1, . . ., n and j ¼ 1, . . .m is based on the principle of comparative advantage. Let two cities j and j0 use the same site i at time t and decide about the timing of switching to the higher-cost site k. The principle of comparative advantage determines which city will be the first one. Suppose that ckj  cij < ckj0  cij0

ð2:48Þ

so that the marginal cost increases less if city j rather than j0 switches to site k. Then site k has a comparative advantage in using city j, whether or not this site has an absolute advantage with regard to city j0 (whether or not the unit cost of using site k is lower or higher in city j than in city j0: ckj < ckj0 or ckj > ckj0 ). The efficient outcome dictated by the comparative advantage is that city j switches to site k first, while city j0 continues using site i. Indeed, if eq. (2.47) is fulfilled for city j at the switching point Tij:

38

2 The Economics of Exhaustible Resources

cij þ vi0 erT ij ¼ ckj þ vk0 erT ij ,

ð2:49Þ

then cij0 þ vi0 erT ij < ckj0 þ vk0 erT ij for city j0 due to (2.48). Thus, city j switches to the more costly site k, even though the resource of the less costly site i is still available and utilized by city j0.

2.6.1

Two Cities, Two Sites

Let us illustrate the principle of comparative advantage for the case of two sites and two cities. Suppose that cities 1 and 2 cover their energy needs by using different resources extracted from two sites. These resources are perfect substitutes for final energy users but differ in the marginal costs of production. Let site 1 produce natural gas and site 2 coal, and gas is the dominant resource in the sense that it has an absolute advantage over coal for both cities. The extraction and transportation costs are assumed to be higher for coal. Moreover, the social costs of carbon emission that should be accounted for by the social planner are higher for coal as well. For the sake of convenience, we introduce the subscript g for natural gas and c for coal. Then the unit-cost matrix is 

cg1

cg2

cc1

cc2

 :

Because gas is the dominant resource, the order of ranking is the same for both cities: cg1 < cc1 and cg2 < cc2. Both resources are utilized eventually by both cities if conditions (2.36) and (2.37) are fulfilled: the initial marginal rents are assigned in the inverse order, vg0 > vc0, while the initial full marginal costs are higher for coal: cgj þ vg0 < ccj þ vc0 for j ¼ 1, 2. The generalized Herfindahl principle implies that gas is used first in both cities, and the switch to coal occurs when the gas resource is partially or fully exhausted. Suppose that coal has a comparative advantage in serving city 1: cc1  cg1 < cc2  cg2 :

ð2:50Þ

Switching to coal is more expensive for city 2, hence this city delays switching and continues using gas, whereas city 1 has already switched to coal. Figure 2.10 illustrates this outcome. Curves G1 and G2 are the time paths of full marginal costs for gas in cities 1 and 2, respectively: cgj + vg0ert, j ¼ 1, 2. Curves C1 and C2 depict similar paths for coal: ccj + vc0ert, j ¼ 1, 2. The curves for gas are steeper because vg0 > vc0. The paths of resource prices for each city, p1(t) and p2(t),

2.6 The Principle of Comparative Advantage

39

Fig. 2.10 The principle of comparative advantage

are the lower envelopes of the marginal cost curves. The condition of comparative advantage for coal (2.50) is fulfilled for these curves, as one can see by comparing the lengths of the brackets on the vertical axis in the figure. The point of switching from gas to coal by city 1 is Tg1. In Fig. 2.10, this time corresponds to the point of intersection of curves G1 and C1. The time of switching to coal by city 2, Tg2, comes as curves G2 and C2 intersect. The figure demonstrates that Tg1 < Tg2 and city 1 begins using coal earlier than city 2. Thus, gas and coal are produced simultaneously during the period between the switching points. Although coal is more expensive than gas for both cities, using coal starts when the gas resource is not fully exhausted. Hence, the original Herfindahl principle is violated in the model with multiple users. The length of the period of simultaneous production, Tg2  Tg1, depends on the degree of comparative advantage of gas in serving city 1. We show in Appendix A.3 that the condition of switching (2.49) implies for the case of two cities and two sites: T g2  T g1 ¼ r 1 ln

cc2  cg2 : cc1  cg1

ð2:51Þ

The fraction on the right-hand side indicates the degree of comparative advantage of using coal by city 1. The higher this ratio is, the longer is the period of simultaneous extraction of gas and coal.

40

2 The Economics of Exhaustible Resources

As a result, although coal has the absolute disadvantage in serving both cities, it always has a comparative advantage in serving one of them. Each city uses resources in the order of marginal costs, and it is socially efficient for the city with comparative advantage in coal to abandon using gas, even though the other city continues to use it.

2.7

Concluding Remarks

Hotelling was motivated in his work on the economics of exhaustible resources by concerns that natural resources “are now too cheap for the good of future generations, that they are being selfishly exploited at too rapid a rate” (Hotelling 1931, p. 137). Debates about the inadequacy of the natural resource base for industrial growth were popular at the beginning of the last century. However, Hotelling published his paper after discoveries of big oil reserves in Texas, the Middle East and other regions. During the long period of abundant oil supply, the issue of resource scarcity was not of paramount importance for economists. The oil price shocks of the 1970s revived the interest in this issue, which was strengthened by the popular concerns of that period about limits to economic growth. More recently, discussions of resource exhaustibility also focused on the environmental impacts of fossil fuel consumption. The contribution of Hotelling to natural resource economics is seminal regardless of changing policy trends. He developed formal analysis based on dynamic optimization because, in his words, “the static-equilibrium type of economic theory is plainly inadequate for an industry in which the indefinite maintenance of a steady rate of production is a physical impossibility” (Hotelling 1931, p. 139). The basic model of exhaustible resources provided clear-cut implications for resource price dynamics and intertemporal resource allocation. One advantage of this model is its analytical tractability resting on the simplifying assumptions regarding extraction technology and resource industry structure. It is, therefore, not surprising that the basic model of exhaustible resources could not fit to empirical evidence. In this regard, Jeffrey Krautkraemer (1998, p. 2066) wrote: “For the most part, the implications of the basic Hotelling model have not been consistent with empirical studies of non-renewable resource prices and in situ values. There has not been a persistent increase in non-renewable resource prices over the last 125 years, but rather fluctuations around time trends whose direction depends upon the time period selected as a vantage point.” The basic model of exhaustible resource does not account for important real-world processes and features such as the development of resource bases, uncertainties about demand and supply sides, imperfect competition, technological progress etc. The assumption of constant marginal costs is the most restrictive one and it has been relaxed in some modifications of Hotelling’s model, where the extraction costs are non-linear and may depend on the cumulative extraction or the remaining resource stock. The case of constant marginal costs is, nevertheless, useful

Appendices

41

because it allows for the tractable model extensions envisaged in this chapter and concerning the role of heterogeneity of resource suppliers and users. The main inferences of the models presented in this chapter can be summarized as follows. Under constant marginal extraction cost, the growth rate of marginal resource rent is equal to the real rate of interest and the resource stock is evaluated through the marginal resource rent. In this model, optimal extraction decreases over time until full exhaustion at the point of switching to a backstop technology. Homogeneous resource consumers should not use resources with different extraction costs simultaneously, and a lower-cost resource is exhausted completely before a higher-cost resource is exploited. The marginal resource rent decreases as extraction switches from lower-cost to higher-cost resources. Heterogeneous consumers with a comparative advantage in the higher-cost resource should abandon using the lowercost resource, even though this resource is not exhausted and still consumed by other users. Hotelling’s theory emphasizes the role of resource scarcity, although this factor is not “the ultimate cause” of natural resource economics. A more fundamental principle is the opportunity cost of extraction, which serves as a guideline for rational intertemporal choice. In the next chapter, we will consider models of natural resource depletion based on this principle with no exogenous resource constraints. We have not looked in this chapter at the imperfect competition of producers, although Hotelling introduced a monopoly supplier into his model to demonstrate the effects of monopoly power on intertemporal resource allocation. This issue will be examined in part III of this book devoted to resource cartel behaviour.

Appendices A.1 Equation (2.44) Rewrite condition (2.42) as: v10 ¼ v20 þ ðc2  c1 ÞerT 1 :

ð2:A1Þ

We have it that v2 ðT 1 Þ ¼ ðα  c2 ÞerðT 2 T 1 Þ due to (2.12) and because p2(T2) ¼ α. Consequently, v20 ¼ ðα  c2 ÞerT 2 . The market-clearing price for 0  t  T1 is: p1(t) ¼ α  βq1(t), hence, from (2.40a), (2.A1), (2.43): βq1 ðt Þ ¼ α  p1 ðt Þ ¼ α  c1  v10 ert   ¼ α  c1  ðα  c2 ÞerT 2 þ ðc2  c1 ÞerT 1 ert   ¼ α  c1  ðα  c2 ÞerðT 2 T 1 Þ þ c2  c1 erðtT 1 Þ ¼

42

2 The Economics of Exhaustible Resources

    ðα  c1 Þ 1  erðtT 1 Þ  c2  α þ ðα  c2 ÞerðT 2 T 1 Þ erðtT 1 Þ ¼   βq01 ðt Þ þ ðα  c2 Þ 1  erðT 2 T 1 Þ erðtT 1 Þ ¼ βq01 ðt Þ þ βq2 ðT 1 ÞerðtT 1 Þ ¼ βq01 ðt Þ þ βq001 ðt Þ:

A.2 Inequalities (2.45) Due to (2.15) and (2.44), the resource constraint for the low-cost site is written as:  α  c1  rT 1 þ 1  erT 1 þ βr

ZT 1

q001 ðt Þdt ¼ S10 :

0

Insert here q001 ðt Þ ¼ q2 ðT 1 ÞerðtT 1 Þ and rearrange, using (2.43), the left-hand side of this equation:  ZT 1  α  c2  α  c1  rT 1 rðT 2 T 1 Þ þ erðtT 1 Þ dt ¼ rT 1 þ 1  e 1e βr β 0

  α  c2   α  c1  rT 1 þ 1  erT 1 þ 1  erT ðS20 Þ 1  erT 1 ¼ S10 : βr βr From this equation, we have it that ∂T1/∂S10 > 0 and by the implicit function theorem: ðα  c2 ÞerT ðS20 Þ T 0 ðS20 Þð1  erT 1 Þ ∂T 1 ¼ < 0, ∂S20 ðα  c1 Þð1 þ erT 1 Þ þ ðα  c2 Þð1  erT ðS20 Þ ÞerT 1 since T 0(S20) > 0.

A.3 Equation (2.51) Equation (2.49) is fulfilled for the times of switching Tg1 and Tg2: cg1 þ vg0 erT g1 ¼ cc1 þ vc0 erT g1

References

43

cg2 þ vg0 erT g2 ¼ cc2 þ vc0 erT g2 : This is rewritten as   cc1  cg1 ¼ vg0  vc0 erT g1   cc2  cg2 ¼ vg0  vc0 erT g2 and yields cc2  cg2 ¼ erðT g2 T g1 Þ : cc1  cg1

References Chakravorty U, Krulce D (1994) Heterogeneous demand and order of resource extraction. Econometrica 62(6):1445–1452 Chakravorty U, Krulce D, Roumasset J (2005) Specialization and non-renewable resources: Ricardo meets Ricardo. J Econ Dyn Control 29:1517–1545 Dasgupta P, Stiglitz J (1975) Uncertainty and the rate of extraction under alternative arrangements. Inst. Mathemat. Stud. In the Soc. Sci. (IMSSS), tech. rep. no. 179, Sept. 1975 Devarajan S, Fisher A (1981) Hotelling’s ‘economics of exhaustible resources’: fifty years later. J Econ Lit 19(1):65–73 FRED Economic Data (2020) Economic Research, Federal Reserve Bank of St. Louis. https://fred. stlouisfed.org/series/DFII10. Accessed 23 Sep 2020 Gaudet G, Salant S (2014) The hotelling model with multiple demands. Resources for the future discussion paper 1421, Washington DC, p 24 Gaudet G, Moreaux M, Salant S (2001) Intertemporal depletion of resource sites by spatially distributed users. Am Econ Rev 91(4):1149–1159 Herfindahl O (1967) Depletion and economic theory. In: Gaffney M (ed) Extractive resources and taxation. University of Wisconsin Press, Madison, pp 63–90 Hotelling H (1931) The economics of exhaustible resources. J Polit Econ 39:137–175 Krautkraemer J (1998) Non-renewable resource scarcity. J Econ Lit 36(4):2065–2107 Nordhaus W (1973) The allocation of energy resources. Brook Pap Econ Act 3:529–576 Rachel L, Summers L (2019) On secular stagnation in the industrialized world. NBER working paper 26198. National Bureau of Economic Research, Massachusetts Solow R (1974) The economics of resources or the resources of economics. Am Econ Rev 64 (2):1–14 Solow R, Wan F (1976) Extraction costs in the theory of exhaustible resources. Bell J Econ 7 (2):359–370 Ulph AM (1978) A model of resource depletion with multiple resources. Econ Rec 54:334–345

Chapter 3

Prices and Rents of Economically Recoverable Resources

Abstract Models in this chapter are based on the premise of limited resource availability with no explicit resource constraints. A potential resource in the ground is supposed to be inexhaustible but depleting in the sense that the marginal extraction cost increases with cumulative extraction. An economically recoverable resource is determined in the base model for long-term equilibrium as the ultimate cumulative extraction. In the reservoir model, which is relevant to the oil industry, producing reserves are developed through drilling new wells and play a dual role: as reservoirs containing oil inventory and as a factor of production. Finally, we analyse a model with extraction and investment in addition of producing reserves.

3.1

Introduction

In the models of exhaustible resources in the previous chapter, a finite resource stock constrains resource allocation over time. In the real world, no such resource constraints exist. Nobody knows exactly how much oil, gas, coal or any other natural resource is contained under the earth’s surface. Still, different measures of resource availability are applied based on geological, engineering and economic data. William Nordhaus compared three measures: proved reserves, ultimate recoverable resource and crustal abundance (Nordhaus 1974). Table 3.1 taken from his paper presents these measures for several mineral resources as the ratio of resource stock to annual consumption. This ratio, measured in years, is quite low for proved reserves: it varies from several decades to hundreds of years. At the opposite extreme is crustal abundance, which is of the order of hundreds of millions to billions of years provided that everything can be recovered. The data on ultimate recoverable resources in Table 3.1 indicate, albeit with a very high degree of uncertainty, the expected cumulative extraction in the long term at prices that consumers will be able to afford to pay and by using technologies that may become available in the future. Proved reserves are determined with a much higher degree of certainty as resources that are commercially recoverable under existing economic and operating conditions.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_3

45

46

3 Prices and Rents of Economically Recoverable Resources

Table 3.1 Resource availability for important minerals by three measures, years

Copper Iron Phosphorus Molybdenum Uranium Aluminium

Proved reserves/ Annul consumption 45 117 481 65 50 23

Ultimate recoverable resource/ Annual consumption 340 2657 1601 630 6897 8455

Crustal abundance/ Annual consumption 242,000,000 1,815,000,000 870,000,000 422,000,000 1,855,000,000 38,500,000,000

Source: Nordhaus (1974), p. 23 Table 3.2 Measures of resource availability for crude oil and natural gas in the U.S.A., as of January 1, 2018

Crude oil, billion barrels Natural gas, trillion cubic feet

Proved reserves 42 438.5

Unproved resources 302.1 2390.3

Technically recoverable resources 344.1 2828.8

Source: U.S. Energy Information Administration (2020)

A common measure of long-term availability of crude oil and natural gas resources in the United States is the remaining technically available resource, which consists of proved reserves and unproved resources. The latter include resources confirmed by exploratory drilling, resources from undiscovered pools within confirmed fields and undiscovered resources located outside fields where the presence of resource has been confirmed by exploratory drilling (U.S. Energy Information Administration 2020, p. 21). Table 3.2 shows these measures of domestic resource availability for crude oil and natural gas for the U.S.A. Proved reserves tend to increase over time as additional geological and engineering data become available and market conditions change. Exploration activity and the discovery of new deposits also result in the addition of substantial reserves. For example, in the case of the U.S. oil industry, proved reserves amounted to 13 billion barrels in 1930 and about 20 billion in 1988. In the interim, 120 billion were added and used (Adelman 1990, p. 3). The exhaustion of existing world oil reserves occurred more slowly than the addition of reserves, as one can see from Fig. 3.1. The fact that the development of reserves has been outpacing the growth of extraction is at odds with the premise of a fixed resource constraint. In this chapter, we consider models relevant to the concepts of recoverable resources and proved reserves. We assume that a resource in the ground is potentially inexhaustible but depletable in the sense that the marginal production costs increase with the cumulative amount of resource already extracted. The cumulative depletion effect on the marginal extraction costs characterizes, in an abstract manner, the geology and technology of resource development and extraction. Due to this

3.1 Introduction

47

Fig. 3.1 World proved oil reserves and crude oil production in 1980–2019. Source: BP Statistical Review of World Energy (2020)

effect, the volume of recoverable resource can be determined as the model solution resulting from conditions of long-term equilibrium. In the model examined by Yeganeh Farzin (1992), there is no resource constraint and the marginal extraction cost increases with cumulative extraction. As will be shown, the equilibrium solution for this model is given by a saddle path converging to the steady state. The equilibrium extraction path determines the ultimate or economically recoverable resource, which has a finite size in the long term. The amount of this resource is equal to the steady-state cumulative extraction. Although a resource in the ground in this model is potentially unlimited, it has a positive marginal value determined by the opportunity cost of using a unit of resource now rather than conserving it for the future. Producers are supposed to internalize the cumulative depletion effect in their production plans. The opportunity cost per unit of extracted resource indicates future excess extraction costs caused by this effect. The opportunity cost tends toward zero in the long term, because the marginal resource value tends toward zero under the condition of optimality that all valuable resources should be extracted. The marginal resource rent in this model can be decreasing or non-monotonic over time, unlike the scarcity rent in Hotelling’s model (Harold Hotelling 1931), which monotonically increases with the exhaustion of resource stocks. In resource models of another kind, originated by Robert Pindyck (1978), producers invest in the development of new deposits. This activity results in the replenishment and accumulation of reserve stocks and resembles inventory management. Morris Adelman characterized reserve additions in the oil industry as “flows from unknown resources into a reserve inventory” (Adelman 1990, p. 2), which include, first, finding new reservoirs, and second, development drilling in known reservoirs. In practice, most money is spent on the second stage, in the development of known deposits (op. cit., p. 3). Therefore, the exploration and development of

48

3 Prices and Rents of Economically Recoverable Resources

reserve stocks are treated below as a unique process of reserves addition, which we also term “resource development”. To illustrate these points, we will consider a simple reservoir model, which is relevant to the oil industry. In this model, underground oil reserves serve as reservoirs for oil inventory and as a production factor. The contribution of this factor to output is defined by the effect of pressure in reservoirs on the intensity of the flow of oil to the surface. Pressure is the scarce production factor, because it falls with the exhaustion of operating reservoirs. A falling pressure in operating wells encourages the development of new reservoirs. The marginal cost of adding reservoirs increases over time because of the cumulative depletion effect. In equilibrium, there is a positive link between this marginal cost and the marginal value of pressure. This link defines the structure of the marginal resource rent, which is the sum of the scarcity premium for pressure and the marginal value of a resource in the ground. In this model, the initial reserve stock and the economically recoverable resource are determined as the equilibrium solution. In the model of resource development and extraction by Pindyck (1978), producers economize on extraction costs by adding new reserves. We consider a special case of this model with no depletion in the addition of reserves. In this case, the marginal cost of adding reserves is constant and the resource exploitation occurs in a stationary regime. Such a regime is possible if, for example, a depletion in resource development is offset by technical progress. As will be shown, in the absence of a depletion effect, a stationary equilibrium path is given by the intensity of resource extraction and the size of the reserve stock, which are constant over time. For this solution, we consider a distribution of long-term benefits from resource exploitation between consumers and producers.

3.2

The Model of Economically Recoverable Resources

Consider a model of a resource-supplying competitive industry with a depletable resource in the ground. Homogeneous producers in this industry exploit deposits of the same size and quality. Resource depletion leads to an increase of marginal extraction costs with cumulative extraction Q(t). The marginal extraction cost is assumed to be a function of cumulative extraction, c(Q(t)), defined for all Q(t)
0, strictly increasing, twice continuously differentiable and unbounded: lim cðQÞ ¼ Q!1

1. This is a special case of the original Farzin’s model (Farzin 1992), where the extraction cost is a more general function of three variables: cumulative extraction, current intensity of extraction and a variable of technological progress. Firms select extraction paths to maximize the present value of future rents:

3.2 The Model of Economically Recoverable Resources

ZT max

Qðt Þ, qðt Þ, T

ert ½pðt Þ  cðQðt ÞÞqðt Þdt,

49

ð3:1Þ

0

subject to the cumulative extraction equation: Q_ ðt Þ ¼ qðt Þ,

ð3:2Þ

where p(t) is the resource price, r > 0 is the discount rate, q(t) is the intensity of extraction at instant t and T  1 is the time of extraction termination. The initial cumulative extraction is Q(0) ¼ 0. Firms take into account the effect of cumulative extraction on the marginal cost c(Q(t)). The market is cleared at any instant: qðt Þ ¼ yðt Þ,

ð3:3Þ

where y(t) is instantaneous demand. Demand function is y ¼ y( p) with y0( p) < 0 and the inverse demand is p ¼ P ðyÞ with a finite choke price, P ð0Þ ¼ α < 1.

3.2.1

The Equilibrium Extraction Path

The present-value Hamiltonian for this problem is H ¼ ert ½ðp  cðQÞÞq þ μq, where μ  0 is the dual variable related to (3.2). The time argument t is omitted here and henceforth, if possible. Maximizing with respect to q and Q gives the first-order necessary conditions for the interior solution: p  cðQÞ ¼ w

ð3:4Þ

w_ ¼ rw  c0 ðQÞq,

ð3:5Þ

where w ¼  μ is the marginal value of a resource in the ground (the user cost of extraction). It is equal in equilibrium to the marginal resource rent p  c(Q) and evolves according to the costate eq. (3.5). The latter results from the costate equation for μ: μ_ ¼ rμ  ∂H c =∂Q ¼ rμ þ c0 ðQÞq , where H c  ert H is the current-value Hamiltonian. An equilibrium path must satisfy the transversality condition that the marginal value of a resource in the ground is zero at the time of termination:

50

3 Prices and Rents of Economically Recoverable Resources

wðT Þ ¼ 0:

ð3:6Þ

Otherwise, if w(T) > 0, the resource remaining in the ground would be valuable and continuing extraction would benefit producers. We will show below that the time of termination for this model is infinity, hence the transversality condition is fulfilled as lim wðt Þ ¼ 0:

t!1

ð3:60 Þ

Note that conditions (3.4)–(3.6) are sufficient for optimality, whether or not the marginal cost function c(Q) is convex, because the current-value Hamiltonian H c is linear in control variable q (the value of H c maximized with respect to q is identically zero, hence the maximized H c is concave in Q and Arrow’s sufficiency theorem is fulfilled (Seierstad and Sydsaeter 1987, p. 236)). The state variables p and Q satisfy equations of equilibrium dynamics: p_ ¼ r ðp  cðQÞÞ

ð3:7Þ

Q_ ¼ yðpÞ:

ð3:8Þ

Equation (3.7) results from inserting (3.4) into (3.5) and because w_ ¼ p_  c0 ðQÞq. This equation coincides with the Hotelling rule for price growth, p_ ¼ r ðp  cÞ, under constant unit cost of extraction c. Eq. (3.8) is the condition of market clearing. The steady state of this dynamic system is p ¼ α

ð3:9Þ

Q ¼ c1 ðαÞ:

ð3:10Þ

The steady-state price is the choke price and the steady-state cumulative extraction is the increasing function of the choke price. The steady state exists and is unique, because the unit cost function c(Q) is strictly increasing and unbounded. The characteristic equation for system (3.7)–(3.8) is  r  λ   y 0 ð pÞ

 rc0 ðQÞ  ¼ λ2  rλ þ ry0 ðpÞc0 ðQÞ ¼ 0, λ 

where λ is the characteristic root. The intercept of this equation is negative, hence the negative root is unique. The equilibrium solution is the saddle path originating at the initial price p0 ¼ p(0) and converging to the steady state, as one can see in Fig. 3.2, which shows the phase plane of the dynamic system (3.7), (3.8). The saddle path satisfies the transversality condition (3.60) for the infinite time horizon. The non-saddle trajectories reach in a finite time T either the locus of the choke price p ¼ α or the locus of zero marginal rents p ¼ c(Q), drawn with the dashed curves in the figure. The non-saddle paths converging to the zero-rent locus satisfy the

3.2 The Model of Economically Recoverable Resources

51

Fig. 3.2 The saddle-path solution

transversality condition (3.6). However, these trajectories are not optimal, because each of them provides recovery of a lower amount of valuable resource than the saddle path does: Q(T) < Q, as shown in Fig. 3.2. The saddle path is, therefore, selected as the optimal solution among all candidate trajectories satisfying the necessary conditions (3.6), (3.7), (3.8). As a result, one can conclude that the steady-state cumulative extraction Q is the economically recoverable resource.

3.2.2

Dynamics of Marginal Rents

For the saddle-path solution, the marginal rent converges to zero as time tends toward infinity, because the marginal extraction cost converges to the resource price, as follows from eq. (3.7). Integrating the costate eq. (3.5) over time implies: Z1 wðt Þ ¼

erðτtÞ c0 ðQðτÞÞqðτÞdτ:

ð3:11Þ

t

The marginal resource rent equals the present value of incremental cost increases at any τ
t caused by raising the level of cumulative extraction at time t. An additional unit of extraction at this time exerts a cumulative depletion effect on current and future costs. Thus, the marginal rent w(t) measures the opportunity cost of resource extraction that accounts for this effect. It is important to note that one can regard the marginal resource rent w(t) as a measure of a decreasing resource scarcity. In the long term, the remaining stock of economically recoverable resource Q  Q(t) reduces to zero, but w(t) declines to

52

3 Prices and Rents of Economically Recoverable Resources

Fig. 3.3 Marginal extraction cost functions

zero as well. The marginal resource rent indicates a falling economic scarcity of this resource in the long term, despite the fact that its volume shrinks over time. The transition dynamics of the marginal resource rent depends on the shape of the unit cost function c(Q). Consider the following class of these functions: cðQÞ ¼ c0 þ c1 Qσ

ð3:12Þ

with parameters c0
0, c1, σ > 0. The minimal unit cost is assumed to be below the choke price, c0 < α. The function (3.12) is convex if σ
1 and concave if σ  1. These two types are shown in Fig. 3.3. In the case of a concave function, with σ quite low, the effect of cumulative depletion is strong at the early stage of resource extraction when Q is small. In the case of a convex function with σ quite high, this effect is more pronounced at the later stage, when Q is large. For the class of extraction cost functions specified by (3.12), consider dynamic equations for the marginal rent (3.5) and cumulative extraction (3.8), which are rewritten as: w_ ¼ rw  c1 σQσ1 Q_

ð3:13Þ

Q_ ¼ yðw þ cðQÞÞ,

ð3:14Þ

because c0(Q) ¼ c1σQσ  1. The saddle-path equilibrium trajectories for this system are drawn in Fig. 3.4. The marginal resource rent is monotonically declining along the equilibrium path for σ < 1. It is shown in Appendix A.1 to this chapter that for a concave unit cost function c(Q), the marginal resource rent w(t) decreases over time for any t
0. The equilibrium trajectory is non-monotonic in the case where σ > 1, because w_ ð0Þ > 0 for Q(0) ¼ 0. As one can see from eq. (3.13), in this case the marginal rent is increasing at the early stage of extraction, and has a peak, because it must be decreasing at the late stage. Both saddle paths drawn in Fig. 3.4 converge to the steady states with marginal rent w ¼ 0 and economically recoverable resource

3.2 The Model of Economically Recoverable Resources

53

Fig. 3.4 The two types of saddle-path trajectories

Q ¼

 1=σ α  c0 c1

as follows from the steady-state solution (5.9), (5.10). In Fig. 3.4, the recoverable resource is Q1 for the case where σ < 1 and Q2 for the case where σ > 1, hence Q1 > Q2 (provided that α  c0 > c1). One can attribute the concavity of the marginal cost function c(Q) to the premise that low-cost resources are relatively scarce. Suppose that this function represents a continuum of heterogeneous finite-stock deposits ranked in the order of increasing unit extraction costs. Then the cumulative depletion effect results from a continuous process of exhaustion of the lower-cost deposits and switching to the higher-cost ones. For a concave function c(Q), the lower-cost deposits are small in size and depleting rapidly, hence the marginal resource rent declines rapidly at the beginning of extraction (for σ < 1 eq. (3.13) implies that w_ ð0Þ ¼ 1 ). In Chap. 2, we demonstrated for the finite-stock model with two resource sites a somewhat similar property that switching to the abundant higher-cost site brings about a drop of the marginal resource rent. In contrast, one can interpret the convex unit cost function c(Q) as representing an industry with abundant low-cost resources. If σ is high enough, the marginal extraction cost is approximately constant at the early stage, as depicted in Fig. 3.3. This case is representative of extraction patterns for resource-rich world oil regions such as the Middle East. The behaviour of marginal resource rent w(t) at the early stage of resource extraction for such cases resembles the behaviour of the scarcity rent in Hotelling’s model with a constant marginal extraction cost. That is why w(t) is increasing for σ > 1 at the early stage, as is shown in Fig. 3.4.

54

3 Prices and Rents of Economically Recoverable Resources

3.2.3

Solution for the Linear Case

Suppose that the marginal cost is the linear function of cumulative extraction: cðQÞ ¼ c0 þ c1 Q, where c0
0 is the minimal unit cost and c1 > 0 is the slope coefficient. This is the intermediate case, σ ¼ 1, for the class of marginal cost functions defined by (3.12). Also, let the inverse demand function be linear, P ðyÞ ¼ α  βy with α denoting the choke price and β the slope of inverse demand. The dynamic system (3.7), (3.8) for the linear case is given by equations: p_ ¼ r ðp  c0  c1 QÞ αp Q_ ¼ : β

ð3:70 Þ ð3:80 Þ

The steady state of this system is: p ¼ α α  c0 Q ¼ c1 and the characteristic equation is  rλ   1=β

 rc1  ¼ λ2  rλ  rc1 =β ¼ 0: λ 

ð3:15Þ

We show in Appendix A.2 that the saddle-path solution is pðt Þ ¼ α  βθQ eθt   Qðt Þ ¼ Q 1  eθt ,

ð3:16Þ ð3:17Þ

where θ ¼ λ ¼

 ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ 4rc1 =β  r 2

ð3:18Þ

is the equilibrium extraction rate. At any instant, extraction is the constant fraction θ of the remaining recoverable resource:

3.2 The Model of Economically Recoverable Resources

55

qðt Þ ¼ Q_ ðt Þ ¼ θðQ  Qðt ÞÞ: The saddle-path solution (3.16), (3.17) converges to the steady state p, Q and satisfies the initial condition Q(0) ¼ 0. The initial resource price is:   βθ βθ p0 ¼ α  βθQ ¼ 1  α þ c0 : c1 c1 

ð3:19Þ

This is the linear combination of the choke price α and the initial unit cost c0 with weight βθ/c1 indicating a closeness of the initial price to the initial unit cost. We show in Appendix A.3 that βθ < c1, and this weight is below unity.

3.2.4

A Numerical Example

Consider a numerical example for the linear case. Let the discount rate be r ¼ 0.05, the choke price be α ¼ $250 per barrel and the slope of inverse demand be β ¼ 10. Let the parameters of the unit cost function be the following ones: c0 ¼ $5 per barrel, c1 ¼ 0.1. We can utilize a benefit of the linearity and introduce arbitrarily (for a numerical example) a scale parameter N for a unit of measurement of resource so that the inverse demand is P ðyÞ ¼ N ðα  βyÞ (the scale parameter is normalized to one in the model). Let N ¼ 106. Then the economically recoverable resource is Q ¼ (α  c0)N/c1 ¼ 245 ∙ 106/0.1 ¼ 2,450 ∙ 106 barrels. The equilibrium extraction rate is calculated from (3.18): θ ¼ 0.0085. The initial price of a barrel of oil is found from (3.19): p0 ¼ α  βθQ ¼ 250  10 ∙ 0.0085 ∙ 2,450 ¼ $40.7 per barrel. Figure 3.5 demonstrates the time profiles of equilibrium extraction q(t), resource price p(t) and marginal extraction cost c(Q(t)) for this numerical example. The

Fig. 3.5 The time profiles of equilibrium price, marginal cost and extraction

56

3 Prices and Rents of Economically Recoverable Resources

marginal resource rent is the difference between price and marginal cost. The figure shows that this rent is decreasing in time, unlike the scarcity rent in Hotelling’s model.

3.2.5

Incomplete Resource Exhaustion in Hotelling’s Model

For Hotelling’s model presented in the previous chapter, full exhaustion of the initial resource stock was guaranteed by the assumption of constant marginal cost of extraction. If this cost was increasing with cumulative extraction, the exhaustion could be incomplete and the recovered resource would be determined as the model solution. Suppose that cumulative extraction at termination time is constrained by the finite resource stock Q: QðT Þ  Q:

ð3:20Þ

Adding this resource constraint to the producer problem (3.1)–(3.2) gives Hotelling’s model with cumulative depletion effect. This model was examined by Levhari and Liviatan (1977), who showed that the marginal resource rent is the sum of two terms (op. cit., eq. 8, p. 181):

p ð t Þ  c ð Qð t Þ Þ ¼ e

rðTt Þ

ZT ð pð T Þ  c ð Q ð T Þ Þ Þ þ

erðτtÞ c0 ðQðτÞÞqðτÞdτ:

t

The first term on the right-hand side is Hotelling’s scarcity rent and the second one is the present value of excess costs caused by the depletion effect, similar to (3.11). The scarcity rent is zero for the non-binding resource constraint (3.20) and positive otherwise. Figure 3.6 (a, b) illustrates these cases. In the first case the resource constraint does not matter, and the outcome is the same as in the model in this section. This constraint matters in the second case, but the question arises about its meaning from geological and engineering points of view. Which limiting factors can terminate further recovery of a valuable resource and bring the increasing scarcity rent? We will consider this factor and its role in oil extraction in the next section.

3.3

The Reservoir Model and the Structure of Oil Rents

This factor refers to the tremendous natural pressure compressing an underground petroleum reservoir, which is an area of porous rock saturated with oil (Livernois 1987). A well drilled into the reservoir forces oil to flow into and up the well. The

3.3 The Reservoir Model and the Structure of Oil Rents

57

Fig. 3.6 Incomplete and complete exhaustion in Hotelling’s model

natural reserve pressure decreases as the contents of the reservoir are extracted, and secondary recovery techniques (most commonly injection of water into reservoir rock) are used to artificially maintain the pressure in the reservoir. This feature of conventional oil extraction is captured by the reservoir models of oil production that were developed by Kuller and Cummings (1974), John Livernois (1987), Morris Adelman (1990), and Cairns and Davis (2001). The model in this section is based on the approach of these authors. Suppose that a resource in the ground is unlimited, and the technology of resource extraction involves two stages. In the first stage, a producing reserve stock is prepared for extraction, and in the second stage, the resource is extracted from this stock. We will analyse in this section a reservoir model, and the key assumption of this model is that the stock of oil reserve under the ground plays a dual role—as a reservoir of oil inventory and as a factor of production. Consider an industry consisting of homogeneous competitive producers who own an underground resource and control reserves. They develop known reservoirs by drilling new wells and bringing them to a condition where extraction can take place. The initial reserve stock of each new reservoir has a finite size normalized to unity. A production firm chooses at any instant the amount of reserve addition, which coincides with the number of new oil reservoirs. A cohort of reservoirs opened at time j
0 is characterized at time t
j by the volume of reservoir sj(t) and the intensity of extraction qj(t). The reserve stock of cohort j changes as: s_ j ðt Þ ¼ q j ðt Þ:

ð3:21Þ

We assume that extraction by any cohort is constrained by the stock in the reservoir:

58

3 Prices and Rents of Economically Recoverable Resources

q j ðt Þ  γs j ðt Þ

ð3:22Þ

for j 2 [0, t] where γ ¼ const is the physically feasible extraction rate. This constraint captures the dependence of the intensity of the oil flow from the reservoir to the surface on the pressure within the reservoir. Under a regime of free flow of oil, the pressure falls with the reduction of the reservoir and, according to (3.22), the feasible intensity of extraction falls in constant proportion γ to the remaining stock in the reservoir. Cairns and Davis (2001) used a constraint similar to (3.22) in the dynamic optimization model of oil extraction. They considered pressure as the model variable and assumed a negative linear dependence of pressure on the cumulative extraction from the reservoir. Let us assume, for simplicity, that secondary methods of oil recovery are not used and the extraction cost is zero for all wells producing in the flowing regime. Then the instantaneous return on cohort j at time t Ris p(t)qj(t) and the firm’s operating profit is Rt t p ð t Þq ð t Þdj ¼ pðt Þqðt Þ, where qðt Þ ¼ 0 q j ðt Þdj denotes the output from all proj 0 ducing reservoirs. The constraint on output facing the firm is obtained by summing up (3.22) over all installed wells: q(t)  γS(t), where Zt Sð t Þ ¼

s j ðt Þdj

ð3:23Þ

0

is the total stock of producing reserves. The reserves addition x ¼ x(t) is equal to the volume of new reservoirs opened at time t: x(t) ¼ st(t). Let the cost of reserves addition be given by the function z(X)x, where z(X) is the marginal cost function, which is increasing with the cumulative reserves addition X ¼ X(t). This function characterizes the effect of resource depletion on the addition of new reserves. Thus, the firm’s control variables are the intensities of extraction q(t)
0 and the reserves addition x(t)
0. The state variables are the current reserve stock S(t) and the cumulative reserves addition X(t). The initial reserve stock is equal to the initial reserve creation, S(0) ¼ X(0). The firm’s problem consists of two stages. At the first stage, the initial reserve stock S(0) is determined. At the second stage, the firm selects the time paths of variables S(t) and X(t) to maximize the present value of future rents: Z1

ert ½pðt Þqðt Þ  zðX ðt ÞÞxðt Þdt,

0

subject to the balance of reserves stock:

ð3:24Þ

3.3 The Reservoir Model and the Structure of Oil Rents

59

S_ ðt Þ ¼ xðt Þ  qðt Þ,

ð3:25Þ

qðt Þ  γSðt Þ,

ð3:26Þ

the feasibility constraint:

and the equation for cumulative reserves addition: X_ ðt Þ ¼ xðt Þ:

ð3:27Þ

The change of reserves (3.25) is the difference between addition to the stock and extraction. This equation follows fromRtaking the time derivative of S(t) in (3.23) and t accounting for (3.21): S_ ðt Þ ¼ st ðt Þ þ 0 s_ j ðt Þdj ¼ xðt Þ  qðt Þ. The reserve stock S(t) fulfils the role of production factor, only if the feasibility constraint (3.26) is binding. The market clears at any instant, hence pðt Þ ¼ P ðqðt ÞÞ: As above, the choke price is finite, P ð0Þ ¼ α < 1.

3.3.1

Equilibrium Conditions

The present-value Hamiltonian for the second-stage problem (3.24)–(3.27) is H ¼ ert ½pq  zðX Þx þ vðx  qÞ þ ξðγS  qÞ þ ηx,

ð3:28Þ

where v
0 is the shadow value of the reserve stock as inventory of oil (3.25), ξ
0 is the shadow value of the reserve stock as production factor (3.26) and η  0 is the shadow value of cumulative reserve addition (3.27). One can interpret ξ as the marginal value of aggregate pressure in producing reservoirs. The first-order conditions for the positive control variables are: p¼vþξ

ð3:29Þ

v  zðX Þ ¼ w

ð3:30Þ

∂H c ¼ rv  γξ ∂S

ð3:31Þ

∂H c ¼ rw  z0 ðX Þx, ∂X

ð3:32Þ

v_ ¼ rv  w_ ¼ rw þ

where, as above, w ¼  η is the marginal value of a resource in the ground and H c is the current-value Hamiltonian. The resource price is the sum of the marginal value of

60

3 Prices and Rents of Economically Recoverable Resources

reserve stock v and the marginal value of pressure ξ. Eqs. (3.29) and (3.30) imply that the marginal resource value is equal to w ¼ p  ξ  zðX Þ:

ð3:33Þ

Equations (3.31) and (3.32) are costate for the reserve stock and the cumulative reserve addition, respectively. In addition to the first-order eqs. (3.29)–(3.32), the equilibrium path must satisfy the following transversality conditions: lim Sðt Þ ¼ 0, lim wðt Þ ¼ 0:

t!1

t!1

ð3:34Þ

The first condition means that the reserve stock must be fully utilized by the termination of resource exploitation. The second one means that the current marginal resource value tends to zero in the long term. This condition guarantees transformation of all currently valuable resources of oil in the ground into producing reserves. The marginal resource value w indicates the opportunity cost of reserves addition caused by “the reduced long-term development prospects” (Krautkraemer and Toman 2003, p. 10). The first-order condition (3.30) means that the marginal value of reserve stock v is the sum of the marginal cost of new reserves addition z(X) and the marginal resource value w. Integrating the costate eq. (3.32) over time and taking into account the transversality condition (3.34) on w implies: Z1 wðt Þ ¼

erðτtÞ z0 ðX ðτÞÞxðτÞdτ:

t

The marginal resource value equals the present value of incremental increases of development costs in the future caused by reserves additions. Thus, the marginal value of existing reserve stock v covers the marginal cost of current reserves addition and the cumulative depletion effect on the reduction of long-term development prospects.

3.3.2

Decision Rule for Reserves Addition

Consider the first-order conditions (3.29)–(3.32). Inserting (3.29) into the costate equation for reserve stock (3.31) implies the equation for price growth: _ p_ ¼ r ðp  ξÞ  γξ þ ξ,

ð3:35Þ

3.3 The Reservoir Model and the Structure of Oil Rents

61

_ Inserting (3.29) and (3.30) into the costate equation for cumubecause v_ ¼ p_  ξ. lative reserve additions (3.32) yields another equation for price growth: _ p_ ¼ r ðp  ξ  zðX ÞÞ þ ξ,

ð3:36Þ

because w_ ¼ p_  ξ_  z0 ðX Þx due to (3.33). Reserves addition and extraction occur simultaneously if, and only if, the equations for price growth (3.35) and (3.36) coincide. This is the case if ξ¼

rzðX Þ : γ

ð3:37Þ

This condition establishes a link between the marginal value of pressure ξ and the marginal cost of reserve addition z(X). The time derivative of ξ is ξ_ ¼ rz0 ðX Þx=γ . Inserting this into (3.36) yields the following decision rule for reserve addition: x¼γ

P 0 ðqÞq_  rw , rz0 ðX Þ

ð3:38Þ

_ The reserve addition is positive only if the equilibrium extraction since p_ ¼ P 0 ðqÞq. is decreasing in time, q_ < 0. In other words, the development of new producing reserves is aimed at offsetting a decline of extraction.

3.3.3

Equilibrium Dynamic in the Linear Case

Consider the case of linear inverse demand, P ðqÞ ¼ α  βq, and linear marginal cost of reserve addition: zðX Þ ¼ ρX,

ð3:39Þ

where ρ is a positive parameter. The constraint (3.26) is binding in equilibrium, hence q ¼ γS, provided that the resource price is positive, P ðqÞ > 0. The dynamic equilibrium equations for the state variables are:

where

S_ ¼ X_  γS

ð3:40Þ

X_ ¼ AS þ BX  C,

ð3:41Þ

62

3 Prices and Rents of Economically Recoverable Resources

Fig. 3.7 The equilibrium path of reserve stock

A¼

βγ 2 ðr þ γ Þ rρðr þ γ Þ αγr ,B ¼ ,C ¼ : 2 rρ þ βγ 2 rρ þ βγ rρ þ βγ 2

Equation (3.40) is the reserve balance (3.25). Eq. (3.41) is the decision rule for reserve addition (3.38) in the linear case derived in Appendix A.3. Figure 3.7 shows the phase plane for system (3.40) and (3.41). The steady state is S ¼ 0 X ¼

C αγ ¼ : B ρð r þ γ Þ

The steady-state reserve stock is zero. The steady-state cumulative reserve addition is the economically recoverable resource. The steady state satisfies the condition of zero current marginal value of resource, which follows from (3.33), (3.37) and (3.39): w ¼ α  βγS 

rþγ ρX ¼ 0: γ

The line of zero marginal resource value, w ¼ 0, is drawn in Fig. 3.7. The equilibrium trajectory shown in Fig. 3.7 is the stable saddle path: Sðt Þ ¼ S0 eθt

ð3:42Þ

X ðt Þ ¼ X   ðX   X 0 Þeθt ,

ð3:43Þ

where S0 ¼ S(0), X0 ¼ X(0) denote the initial reserve stock and cumulative reserve addition and θ is the negative characteristic root of the system (3.40), (3.41):

3.3 The Reservoir Model and the Structure of Oil Rents

θ¼

63

 ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ 4γB  r , 2

as is demonstrated in Appendix A.3. The saddle path is drawn in Fig. 3.7 with an arrow originating at the initial point (S0, X0). This path converges to the steady state (S, X) and satisfies both transversality conditions (3.34). The non-saddle paths are drawn in this figure with curved arrows. They reach in finite time the zero-marginalvalue line w ¼ 0 or the zero-reserve axis S ¼ 0, implying a lower amount of economically recoverable resource compared to X, which is provided by the saddle-path solution.

3.3.4

The Depletion Rate and the Initial Reserve Stock

The characteristic root θ is the depletion rate indicating the rate of exponential decline of the reserve stock S(t) and the intensity of extraction q(t) ¼ γS(t) ¼ γS0eθt. Adelman (1990, p. 5) adopted the assumption of exponential decline of extraction, because “it is conventional among reservoir engineers”. Note that in this model the depletion rate θ differs from the extraction rate γ, whereas these rates coincide in the linear model of recoverable resource of the previous section of this chapter. The equilibrium path originates at the initial point (S0, X0) located on line S ¼ X shown in Fig. 3.7. The initial reserve stock S0 is determined at the first stage of the firm’s problem and defines the initial supply as q0 ¼ γS0 and the initial equilibrium price as p0 ¼ α  βγS0, which determine the saddle-path solution for resource price. We assume that the volume of investment required to create the initial reserve stock S0 is below the present value of rents given by the maximized objective function (3.24), so that the development and extraction of the resource is profitable. To determine the initial reserve stock for the saddle-path equilibrium path one can insert the solution (3.42), (3.43) into the differential eq. (3.40), implying that θS0 ¼ θðX   X 0 Þ  γS0 : Taking into account that X0 ¼ S0 we obtain: θ S0 ¼ X  : γ

ð3:44Þ

The ratio of initial reserve stock to the economically recoverable resource is equal to the ratio of the depletion rate to the extraction rate. The inverse ratio γ/θ indicates the rate of cumulative extraction growth during the whole period of exploitation. Taking the time derivative in (3.43) yields the equilibrium intensity of the reserve addition, provided that γ > θ:

64

3 Prices and Rents of Economically Recoverable Resources

xðt Þ ¼ θðX   X 0 Þeθt ¼ ðγ  θÞS0 eθt due to eq. (3.44). One can check that the feasibility constraint (3.26) is binding along the equilibrium path, if the following condition on the model parameters is fulfilled: β
0. Furthermore, the marginal extraction cost is decreasing with the reserve stock, 2

∂ C < 0: ∂q∂S

ð3:46Þ

The firm selects the time paths of these variables to maximize the net present value of rents with I0 denoting the initial investment in reserves creation: Z1

ert ½pðt Þqðt Þ  CðSðt Þ, qðt ÞÞ  zxðt Þdt  I 0 ,

0

subject to the reserve stock balance the same as (3.25) and the equation for cumulative reserve addition the same as (3.27). The Hamiltonian for this problem is written as H ¼ ert ½pq  C ðS, qÞ  zx þ vðx  qÞ þ ηx: The first-order necessary conditions are p  Cq ðS, qÞ ¼ v

ð3:47Þ

vz¼w

ð3:48Þ

v_ ¼ rv þ Cs ðS, qÞ

ð3:49Þ

w_ ¼ rw

ð3:50Þ

with subscripts denoting partial derivatives and w ¼  η. The marginal resource rent w satisfying eq. (3.50) must be identically zero to rule out bubble solutions for the infinite time horizon T ¼ 1. For w  0, the marginal value of reserves in (3.48) is equal to the marginal cost of reserve addition:

3.4 Resource Development and Extraction Without Depletion

67

v¼z and is time-constant, v_ ¼ 0. Consequently, eqs. (3.48) and (3.49) transform to the static system for S and q: P ðqÞ  Cq ðS, qÞ ¼ z

ð3:51Þ

C s ðS, qÞ ¼ rz:

ð3:52Þ

Since the equilibrium reserve stock is constant, the intensity of reserve addition coincides with the intensity of extraction at any time moment: x ¼ q: As a result, under constant marginal development cost the equilibrium solution is a time-stationary path.

3.4.1

The Stationary Equilibrium

Each eq. (3.51) or (3.52) defines an extraction curve as an implicit dependence of q on S. The extraction curve q1(S) implied by eq. (3.51) is increasing: CqS dq1 >0 ¼ 0 dS P ðqÞ  C qq due to condition (3.46) and since P 0 ðqÞ < 0, Cqq > 0. The extraction curve q2(S) implied by eq. (3.52) is also increasing: dq2 C ¼  SS > 0: dS C Sq The slope of q1(S) is lower than the slope of q2(S) for any S, because C SS Cqq
C 2qS for the jointly convex cost function. Consequently, the stationary equilibrium (q, S) exists and is unique. This equilibrium is demonstrated in Fig. 3.10. Curve q1(S) in this figure relates to the marginal value of resource extraction, and curve q2(S) to the marginal value of resource addition to the reserve stock. The point of stationary equilibrium (q, S) ensures equalization of these marginal values. Thus, the equilibrium path is S(t) ¼ S, q(t) ¼ x(t) ¼ q for any t > 0. An initial instantaneous investment in reserve stock creation is required to increase it from the initial level S0 ¼ 0 to the stationary-equilibrium level S. The initial investment I0 ¼ zS thus places the equilibrium trajectory onto the stationary path without a saddle-path transitory regime.

68

3 Prices and Rents of Economically Recoverable Resources

Fig. 3.10 The stationary equilibrium

3.4.2

The Linear-quadratic Case

This case allows for an explicit solution for the stationary equilibrium. Consider the following extraction cost function: C ðS, qÞ ¼ q2 =2δS,

ð3:53Þ

where δ is the parameter of extraction productivity. This function is quadratic in argument q and linearly homogenous in both arguments. Let the inverse demand be linear with unit slope, P ðyÞ ¼ α  y. We have: CS ¼  q2/2δS2 , Cq ¼ q/δS and the system (3.51), (3.52) represented as α  q  q=δS ¼ z

ð3:510 Þ

ðq=SÞ2 ¼ 2δrz:

ð3:520 Þ

Inserting q/S from (3.510 ) into (3.520 ) and rearranging terms yields the stationary solution: q ¼ α  z  ð2rz=δÞ1=2 αz S ¼  δ1=2 : ð2δrzÞ1=2 The intensity of extraction and the reserve stock in stationary equilibrium increase with the productivity of extraction δ and decrease with the marginal development cost z and the discount rate r. By contrast, in Hotelling’s model a higher discount rate encourages a more intensive extraction at the early phase of resource exploitation. The stationary resource price is

3.5 Measuring Resource Scarcity

69

p ¼ α  q ¼ z þ ð2rz=δÞ1=2 : It is equal to the marginal development cost plus the marginal extraction cost Cq ¼ (2rz/δ)1/2. The total extraction cost is C(S, q) ¼ Cqq/2 ¼ (rz/2δ)1/2q. Since x ¼ q, the instantaneous profit is pq  zq  C(S, q) ¼ (2rz/δ)1/2q  (rz/2δ)1/ 2  q ¼ (rz/2δ)1/2q. The net present value of profits is V 0 ¼ r 1 ðp q  zq  CðS , q ÞÞ  I 0 : The initial investment is I0 ¼ zS. Due to (3.520 ), the net present value is equal to zero:  rV 0 ¼ ðrz=2δÞ1=2 q  rzS ¼ ðrz=2δÞ1=2 q  rzð2δrzÞ1=2 q ¼ ð2δrzÞ1=2 ðrz  rzÞq ¼ 0: Thus, the present value of future profits exactly covers the initial investment in reserve stock creation. The net present value is zero—first, due to the absence of depletion in resource development implying that the marginal resource rent w is zero at any time, and second, because the extraction technology given by the extraction cost function (3.53) is linearly homogeneous in production factors. As a result, the net long-term gain of producers from resource exploitation is zero, while consumers get all social benefit through consumer surplus.

3.5

Measuring Resource Scarcity

The marginal resource value in the above models is an indicator of economic scarcity of a natural resource. As was shown in the previous chapter for the base Hotelling’s model, the marginal resource value is increasing in time, but it is decreasing for the extension of this model to heterogeneous resource fields as extraction switches from higher- to lower-quality grades. As was demonstrated in this chapter, under a limited resource availability the marginal resource value can be increasing or decreasing with cumulative extraction. Dynamics of a resource scarcity measure indicate whether a resource under consideration is getting more scarce or abundant from the economic point of view. The tendencies of non-renewable resource scarcity in the real world depend on a number of factors on demand and supply sides that are not captured by the simple resource models. These factors include growth of income and population, technological innovations, discoveries of non-conventional resources, substitution of physical capital for natural resources, etc. Therefore, empirical tests based on the first-order conditions implied by these models do not account for the variety of

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3 Prices and Rents of Economically Recoverable Resources

factors that can complicate the optimal paths of resource price. Another obstacle to empirical testing of resource scarcity is that it is inherently difficult to measure the marginal resource value or the marginal rent of resource owner. The resource rent is not directly observable because data on extraction and development costs are usually proprietary information. For vertically integrated resource-extracting companies, it may be difficult to isolate these costs from other production expenses such as transportation and processing costs. However, it is possible to estimate an industry-level extraction cost function using observable data on extraction and capital-labour inputs in order to calculate time series of marginal rents for such cost function. For example, Halvorsen and Smith (1991) estimated a cost function implied by a constrained cost minimization and based on aggregate data for the Canadian metal mining industry for the period 1954 to1978. They provided empirical tests of resource scarcity for the dynamic first-order conditions, which are similar to eq. (3.5) or (3.13) in the model of recoverable resource in this chapter. These tests rejected the theory of exhaustible resources for annual discount rates ranging in value from 2 to 20%. The estimated average annual growth rate of marginal resource rent was 0.57%, but proved to be statistically insignificant and did not provide a conclusion as to whether the resource scarcity in this case was increasing or decreasing. A different approach to measuring scarcity rents on producing reserves is based on the model of resource exploration by Devarajan and Fisher (1982). Their idea was that these rents are linked to expenditures of firms on exploration that make the resource available for extraction. Pierre Larraine (1985) extended this model to derive a measure of resource rent as the full marginal costs of exploration, which is defined as the sum of the marginal discovery cost and the scarcity rent on limited exploratory prospects. This relationship is similar to the equations of this chapter: (3.30) in the reservoir model and (3.48) in the resource development model stating that the marginal rent on reserves v is the sum of the marginal development cost z and the marginal resource value w. Both Devarajan and Fisher (1982) and Larraine (1985) demonstrated the empirical evidence that costs of oil discovery were rising in the post-World War II period in the United States and Canada indicating an increasing scarcity of oil-producing reserves. However, due to technological inventions, discovery and development costs in the U.S. oil industry at the beginning of the 2000s were one-third of what they had been twenty years before (Krautkraemer 2005, p. 25). An alternative measure of resource scarcity indicating decreasing or increasing resource availability is the resource price. Unlike resource rents, it is observable and incorporates direct and indirect sacrifices made to obtain a unit of resource and measured, respectively, by the current extraction cost and the opportunity cost of extraction. However, the price can be a misleading indicator of a future resource availability, if it decreases despite of the increasing scarcity rent. This can occur due to technological innovations in extraction that bring about a rapid decline of extraction costs (Krautkraemer 1998, p. 2089). Improvements in extraction technology resulting in economizing on inputs of production capital and labour lead to reductions of the resource price, but do not necessarily increase resource availability. For

3.5 Measuring Resource Scarcity

71

Fig. 3.11 Index of real prices of metals and minerals, January 2010 ¼ 100. Sources: World Bank (2020), FRED Economic Data (2020)

example, technical progress steadily reduced unit extraction costs in the U.S. lumber industry in the nineteenth century, but forest resources were disappearing at a rapid rate (Eric Neumayer 2000, p. 21). Technical progress and discovery of new resource fields were the main factors behind the decline of real unit costs and prices of the majority of mineral commodities in the twentieth century. The first empirical study that documented this tendency was the work by Barnett and Morse (1963), who revealed a general downward trend for the index of mineral prices relative to the price index of non-extractive commodities over the period 1870–1957. Margaret Slade (1982) showed that the long-term trend of the mineral aggregate price index proved to be U-shaped because of the increase in mineral prices in 1945–1980. This change of trend was caused by the growth of marginal extraction costs and marginal resource values in the post-World War II period. In particular, a significant increase of the fossil fuels prices in the 1970s resulted, to a large extent, from the market power exercise by the petroleum cartel OPEC. However, the upward trend of resource prices did not continue after the 1970s, as can be seen from Fig. 3.11 showing the index of non-fuel mineral prices deflated by the U.S. consumer price index and Fig. 3.12 demonstrating the index of real prices of fossil fuels. Nevertheless, a significant growth of these price indices in the 2000s indicates the effect of increasing resource scarcity caused by the rise of Asian resource demand.

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3 Prices and Rents of Economically Recoverable Resources

Fig. 3.12 Index of real energy commodity prices, January 2010 ¼ 100. Sources: World Bank (2020), FRED Economic Data (2020)

3.6

Concluding Remarks

We have considered in this chapter the simple theoretical models with limited resource availability represented by economically recoverable resource and producing reserves. These concepts can be viewed as alternatives to the fixed-stock assumption that are relevant to the practical issues of resource exploration, development and extraction. Adelman (1990, p. 1) noted in connection with these issues: “If expected finding-development costs exceed the expected net revenues, investment dries up, and the industry disappears. Whatever is left in the ground is unknown, probably unknowable, but surely unimportant; a geological fact of no economic interest.” If the costs and prices determine the volume of resource to be recovered, the fixed-stock assumption proves to be superfluous, according to Adelman’s view (op. cit., p. 2). For the recoverable resource model, it was assumed that the extraction cost increases with cumulative extraction. The gains from current extraction are equalized with the expected losses from increasing extraction costs in the future. The resource rent thus compensates for losses caused by the cumulative depletion effect. In the long term, the marginal rent decreases over time, because this effect abates with the decline of resource extraction. For the reservoir model, the marginal resource rent also decreases over time, whereas the marginal value of pressure, as a scarce resource, increases. In the special case of Pindyck’s model with no depletion effects, the equilibrium extraction path is time-stationary, and the marginal resource rent is zero. The volume of recoverable resource depends, apart from the cumulative depletion effect, on the willingness of consumers to pay for the resource. In the models considered in this chapter, consumer demand is stationary and the choke price is

Appendices

73

finite and time-constant. Taking into account the tendency of resource demand to grow over time would imply increasing volumes of recoverable resource. The models of resource development and extraction include, as a state variable, the reserve stock, which plays a dual role—as an inventory of resource and as a factor of production. We did not use the term “proved reserves” for this factor, because the influence of the size of proved reserves on the productivity of extraction is questionable. This link exists for operating oil wells due to the fall of pressure with cumulative extraction, but it is hardly possible to corroborate such a link for proved reserves. We therefore used a narrower notion of “producing reserves” measured as resources recoverable from operating wells. We have focused on the deterministic processes of resource development without considering exploration activity, because the latter is intrinsically uncertain. Exploration uncertainty relates to results of geological search for new deposits of random quantity and quality. Therefore, exploratory outcome may follow a stochastic process linking exploratory inputs with outputs at any instant of time. The key motivation for exploration examined, for example, by Devarajan and Fisher (1982) is to enhance reserves to mitigate the effect of resource depletion on the future productivity of extraction, which increases with the addition of reserves. The models of recoverable resource examined in this chapter can be extended to introduce stochastic shocks to inverse demand or marginal extraction costs. For example, the marginal extraction cost c(Q) can be represented as a stochastic process with drift depending on cumulative extraction. In such cases, the expected economically recoverable resource becomes a stochastic variable adapting to current information about demand and supply conditions. The models of this chapter will be used further in this book: in Chap. 10 for the analysis of a resource cartel’s effects on the global production structure, and in Chap. 11 for the analysis of the consequences of the oil cartel’s activity for time patterns of extraction by different oil regions.

Appendices A.1 Decreasing Marginal Resource Rent w(t) for σ < 1 This statement is a special case of Farzin’s (1992, p. 817) more general result. Integrating (3.11) by parts yields, after some manipulation,

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3 Prices and Rents of Economically Recoverable Resources

Z1 wðt Þ ¼

erðτtÞ c0 ðQðτÞÞqðτÞdτ

t 1 0

¼ r c ðQðt ÞÞqðt Þ þ r

1

Z1

erðτtÞ

d ½c0 ðQðτÞÞqðτÞ dτ: dτ

t

Substituting rw(t) into (3.5) implies Z1 w_ ðt Þ ¼

erðτtÞ

d ½c0 ðQðτÞÞqðτÞ dτ: dτ

t

_ þ c0 ðQÞq_ ¼ c00 ðQÞq2 þ c0 ðQÞq_  0 We have it that dðc0 ðQÞqÞ=dτ ¼ c00 ðQÞQq 00 because c (Q)  0 for the concave function and q_ < 0 for the equilibrium extraction path (we have it for system (3.7)–(3.8) that p_ ðt Þ > 0 for all t
0). Consequently, w_ ðt Þ  0.

A.2 The Solution (3.16), (3.17) On the one hand, inserting (3.16), (3.17) into (3.70) we have:   p_ ¼ r ðp  c0  c1 QÞ ¼ r α  βθQ eθt  c0  c1 Q þ c1 Q eθt   ¼ r c1 Q eθt  βθQ eθt ¼ r ðc1  βθÞQ eθt , because Q ¼ (α  c0)/c1. On the other hand, differentiating (3.16) with respect to time yields p_ ¼ βθ2 Q eθt : These two equations on price growth coincide if βθ2 ¼ r ðc1  βθÞ, which is fulfilled, because θ ¼  λ satisfies the characteristic eq. (3.15). From (3.18), condition θβ/c1 < 1 is equivalent to: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2c1 þ r: r 2 þ 4rc1 =β < β Squaring both sides implies

References

75

r 2 þ 4rc1 =β < r 2 þ 4rc1 =β þ 4ðc1 =βÞ2 :

A.3 System (3.40)–(3.41) In the linear case, eq. (3.38) transforms to rρX_ ¼ γβq_  γr ðα  βq  ξ  ρX Þ: Inserting here q ¼ γS, ξ ¼ rρX/γ from (3.37) and S_ ¼ X_  q ¼ X_  γS and rearranging terms implies 

 rρ þ γ 2 β X_ ¼ γ 2 βðγ þ r ÞS þ rρðγ þ r ÞX  αγr,

which yields (3.41). The characteristic equation for system (3.40), (3.41) is  A  γ  λ   A

   ¼ λ2  ðA þ B  γ Þλ  γB ¼ λ2  rλ  γB ¼ 0, B  λ B

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi since A + B ¼ r + γ. The negative root is λ ¼ 12 r  r 2 þ 4γB ¼ θ.

References Adelman M (1990) Mineral depletion with special reference to petroleum. Rev Econ Stat 72 (1):1–10 Barnett H, Morse C (1963) Scarcity and growth: the economics of natural resource availability. John Hopkins U. Press (for Resources for the Future), Baltimore BP Statistical Review of World Energy (2020), (2018) Cairns R, Davis G (2001) Adelman’s Rule and the petroleum firm. Energy J 22(3):31–54 Devarajan S, Fisher A (1982) Exploration and scarcity. J Polit Econ 90(6):1279–1290 Farzin Y (1992) The time path of scarcity rent in the theory of exhaustible resources. Econ J 102 (413):813–830 FRED Economic Data (2020) Economic research, Federal Reserve Bank of St. Louis https://fred. stlouisfed.org/series/CPIAUCSL. Accessed 30 Sep 2020 Halvorsen R, Smith T (1991) A test on the theory of exhaustible resources. Q J Econ 106 (1):123–140 Hotelling H (1931) The economics of exhaustible resources. J Polit Econ 39:137–175 Krautkraemer J (1998) Non-renewable resource scarcity. J Econ Lit 36(4):2065–2107 Krautkraemer J (2005) Economics of natural resource scarcity: the state of the debate. Resources for the Future. Washington D.C. Discussion Paper 05–14, p 45

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Krautkraemer J, Toman M (2003) Fundamental economics of depletable energy supply. Resources for the Future. Washington D.C. Discussion Paper 03–01, p 30 Kuller R, Cummings R (1974) An economic model of production and investment for petroleum reservoirs. Am Econ Rev 64(1):66–79 Larraine P (1985) Discovery cost as a measure of rent. Can J Econ 18(3):474–483 Levhari D, Liviatan N (1977) Notes on Hotelling’s economics of exhaustible resources. Can J Econ 10(2):177–192 Liu P, Sutinen J (1982) On the behaviour of optimal exploration and extraction rates for non-renewable resource stocks. Resour Energy 4:145–162 Livernois J (1987) Empirical evidence on the characteristics of extractive technologies: the case of oil. J Environ Econ Manag 14:72–86 Neumayer E (2000) Scarce or abundant?: the economics of natural resource availability. LSE Research Online. The London School of Economics and Political Science, p 68 http://eprints. lse.ac.uk/18905/. Accessed 1 Oct 2020 Nordhaus W (1974) Resource as a constraint on growth. Am Econ Rev 64(2):22–26 Pindyck R (1978) The optimal exploration and production of non-renewable resources. J Polit Econ 86:841–861 Seierstad A, Sydsaeter K (1987) Optimal control theory with economic applications. Elsevier, Amsterdam, p 463 Slade M (1982) Trends in natural resource commodity prices: an analysis of the time domain. J Environ Econ Manag 9:122–137 U.S. Energy Information Administration (2020) Assumptions to the Annual Energy Outlook 2020, oil and gas supply module, p 22 https://www.eia.gov/outlooks/aeo/assumptions/pdf/oilgas.pdf. Accessed 3 Oct 2020 World Bank (2020) World Bank commodity price data (The Pink Sheet) http://pubdocs.worldbank. org/en/561011486076393416/CMO-Historical-Data-Monthly.xlsx. Accessed 30 Sep 2020

Chapter 4

Pricing Energy Resources Under Transition to Alternative Energy

Abstract Resource pricing has specific features in the presence of alternative technologies providing substitutes for conventional non-renewable resources. The sources of alternative energy for fossil fuels include wind, solar and biofuel energy. In this chapter, we consider a model of gradual energy transition for an energysupplying industry. Exhaustion of a conventional non-renewable resource in this model causes a decline of the relative price of alternative energy. The market share of this energy in the energy mix of consumers is increasing as a result of gradual substitution of renewables for conventional resources. An important property demonstrated below is the Green Paradox: a higher consumer preference for alternative energy implies a higher intensity of the conventional resource extraction in the near term.

4.1

Introduction

In Chap. 2 we considered models of exhaustible resources in the presence of a backstop technology. It was shown that a limited natural resource should be fully exhausted by the time of switching to the backstop technology, when the resource price becomes equal to the backstop price. This inference resulted from the assumption that the backstop technology provides a perfect substitute for the conventional non-renewable resource. In this chapter, we will examine prices and extraction paths of non-renewable resources in the context of their gradual substitution by alternative renewable resources. This approach is relevant to energy production. Conventional energy resources are represented by proven reserves of fossil fuels including oil, natural gas and coal. Alternative energy embraces non-fossil resources that are unlimited or continually replenished through natural processes: hydropower, wind, solar, geothermal, ocean and biofuel energy. One can extend the list of renewables by including nuclear energy, classified sometimes as quasi-renewable. This addition is justified by the potential opportunities of advanced nuclear technologies such as fast breeder reactors or hybrid fusion-fission reactors with very high fuel efficiency using the isotope of uranium-238 that is plentiful in nature. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_4

77

78

4 Pricing Energy Resources Under Transition to Alternative Energy

Table 4.1 Estimated cost structure and unit cost of electricity for expected lifetime of new power sources entering service in 2022 (2019 dollars per megawatt-hour) Plant Type Share of capital cost, % Share of variable cost (incl. Fuel), % Unit cost, $/MWh

Combined cycle 22.2

Combustion turbines 25.7

Wind onshore 74.0

Solar photovoltaic 75.6

Hydroelectrica 70.6

70.3

64.8

0.0

0.0

5.8

11.1

33.5

64.2

36.7

37.4

52.8

81.7

Advanced nucleara 68.7

Source: Authors’ calculation based on U.S. Energy Information Administration (2020, p. 7, 15) a Power sources entering service in 2025

The most essential feature of alternative energy is that it is derived from various natural processes like air and water flows, sunlight or nuclear fission by means of production capital—wind turbines, hydropower plants, solar cells or nuclear reactors. Physical capital plays a dominant role in the production of non-fossil energy, and this role is indicated by a high share of capital costs per unit of energy output. Table 4.1 presents data on the cost structure of electric energy generation and on the averaged production costs in dollars per megawatt-hour ($/MWh) for some energy sources. As one can see from Table 4.1, the share of capital in production costs of renewable energy varies between 68 and 76%. This share is around a quarter for non-renewable energy generation based on natural gas. The share of variable costs including fuels is zero for wind and solar energy, only 6–11% for hydropower and nuclear energy but roughly two thirds for non-renewable energy. Thus, the cost structures of alternative and conventional energy are inversely related to each other: the capital costs dominate for the former and the fuel expenditure for the latter. Under high capital intensity, technological changes in alternative energy production have been materialized in new capital equipment and have caused a significant fall in levelized costs of electricity for wind and solar energy. Levelized cost represents “the average revenue per unit of electricity generated that would be required to recover the costs of building and operating a generating plant during an assumed financial life” (U.S. Energy Information Administration 2020, p. 1). The levelized costs reduced during the decade 2009–2019 by 70% for wind turbines and by 89% for utility-scale solar plants (Lazard 2020). Cost reductions were driven by improvements in efficiency and declines in the prices of system components. Besides that, government subsidies to support investment in alternative energy in many countries have contributed to expansion of its market share. Figure 4.1 demonstrates dynamics of the renewables shares of power generations around the world including wind, solar, geothermal, biofuels, biomass and other. The total share of these sources of renewable energy in global energy production has increased from 1.4% in 2000 to 10.4% in 2019.

4.1 Introduction

79

Fig. 4.1 Renewables shares of the world power generation. Source: Author’s calculation based on BP Statistical Review of World Energy (2020)

These tendencies indicate growth of the competitiveness of alternative energy with regard to conventional energy. Renewables are still relatively expensive in many countries and need subsidizing, but in the long term, their growth will be supported, aside from technical innovations, by the exhaustion of traditional resources. In this chapter, we will consider a model of gradual transition from conventional to alternative energy from the perspective of natural resource economics. The exhaustion of conventional resources in this model causes growth of their relative price and of the market share of alternative energy. The extension of this share leads to the accumulation of production capital of renewables that replaces the extracted conventional resources and thus brings about a gradual transition of the economy to alternative energy. Such a transition is possible if conventional and alternative energies are imperfect substitutes in consumption. We assume that these types of energy are differentiated goods to capture the specific features of energy transformation, transmission and distribution. Energy is consumed in various forms, such as lighting, heating and transportation, which are per se imperfect substitutes or even complementary goods. The consumption of some kinds of renewables like hydropower and geothermal energy is constrained by geographic location. Other kinds, like wind and solar energy, are generated through intermittent natural processes and should usually be supported by conventional energy. The provision of energy security at the national level requires a diversification of energy sources, regardless of production and transportation costs, to ensure energy availability at any time. The variation of energy generation costs, shown, for example, in the last row of Table 4.1, is indicative of the fact that various types of energy are imperfect substitutes.

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4 Pricing Energy Resources Under Transition to Alternative Energy

We also assume that consumer preferences are asymmetric with regard to conventional and alternative energy. One can point out the merits and drawbacks of both types of energy. On the one hand, alternative energy is inexhaustible and clean: it does not release carbon dioxide emissions, unlike conventional energy production. On the other hand, alternative energy incurs environmental and social costs too. Nuclear power plants bear the risks of large-scale catastrophes; building dams for hydropower plants destroys local ecosystems and poses the risk of dams breaking and flooding surrounding territories. Wind turbines are noisy and have an adverse visual impact; creating large-scale solar plants requires wide areas of land. Because of this ambiguity of merits and drawbacks, it is reasonable to assume that alternative energy can be more or less preferable for consumers. In the model presented below, there are two types of energy producers supplying two types of energy—non-renewables and renewables—to consumers. Initially, the suppliers are endowed with two factors of energy production: the resource stock for extraction of non-renewable energy resources and the physical capital for production of renewable energy, which we will call the energy capital. Transition to alternative energy means that the resource stock is being exhausted, while the energy capital is being accumulated. The question is how much capital is created in the long term to replace the initial resource stock, and how rapidly the transition to alternative energy occurs. It will be shown that the transition path depends on the initial factor endowments that determine the initial resource price and the initial market share of alternative energy. We analyse a partial equilibrium, unlike macroeconomic endogenous growth models of energy transition, for example the model of directed technical change by Daron Acemoglu et al. (2012). In that model, the final good is produced from “clean” and “dirty” inputs, which are imperfect substitutes, and dirty production uses exhaustible resources. As was shown by these authors, sustainable economic growth with innovations directed toward clean energy can be achieved with various environmental policy tools if the degree of substitution is sufficiently high. Another example is the paper by Baris Vardar (2013), who examined the optimal growth path and resource use of an economy under energy transition and imperfect substitutability of renewable and non-renewable energy resources. To simplify our analysis, we focus on a stationary equilibrium and understand stationarity in the following sense. Energy demand is assumed to be of Hicksian type: it is a function of prices for conventional and alternative energy and of energy consumption utility, whereas the more common Marshallian demand is a function of prices and household income. We assume that the utility level, as an argument of Hicksian demand function, is constant in time, similarly to the case of Marshallian demand with constant income. The assumption of time-invariant utility is justified since the real interest rate in the model is constant and equal to the discount rate. The utility level is an endogenous variable determined in equilibrium and depending on the initial size of production factors. For example, as will be shown below, the equilibrium utility of energy consumption is high if the economy has sufficiently large initial stocks of energy resource or energy capital or both.

4.2 The Model

81

Thus, we will consider a process of pure substitution of alternative energy for the conventional one. “Pure” means that we focus on the stationary regimes of energy substitution that maintain the equilibrium energy mix on the initial isoquant of consumption utility. In other words, we examine a welfare-invariant energy transition and are not concerned with the non-stationary processes of transition that induce welfare growth or decline.

4.2

The Model

The energy industry is represented by competitive producers that own factors of production and supply two types of energy to consumers. The first type is conventional energy generated from fossil fuel, which is extracted from a finite resource stock. The second type is alternative energy generated with zero marginal cost with physical capital, which is the sole factor of this energy production. The resource stock is exhaustible, while the energy capital is accumulated over time. Consumer preferences play the central role in the model. We assume that conventional and alternative sources of energy are imperfect substitutes. At any moment in time, consumers choose a combination of energy based on the consumption utility function with a constant elasticity of substitution. Preferences of energy types are asymmetric: consumption of a marginal unit of alternative energy may be more or less preferable to a marginal unit of conventional energy. This asymmetry is indicated by a weight of alternative energy in the utility function. Energy producers of each type are homogeneous and their number is normalized to one. There is no production chain with midstream and downstream technological processes of energy supply to final consumers, and we focus only on upstream production. The extraction costs for the conventional resource are assumed to be zero, and the price of conventional energy is equal to the resource rent. The model is highly stylized, but it captures in a simple way the evolution of industry structure resulting from the process of conventional resource exhaustion.

4.2.1

Energy Consumption

Consumer preferences regarding energy mix are given by the utility function:
θ 1=θ uðy1 , y2 Þ ¼ y1 þ ðλy2 Þ1=θ ,

ð4:1Þ

where y1 and y2 are consumption of energy from exhaustible and alternative sources, respectively, and λ > 0 is the preference weight of alternative energy, θ ¼ σ/(σ  1), where σ is the elasticity of substitution between conventional and alternative energy, and σ ¼ θ/(θ  1). If the preference weight λ is above 1, the alternative energy is

82

4 Pricing Energy Resources Under Transition to Alternative Energy

preferable to the conventional one and vice versa. Energy sources are imperfect substitutes implying that 1 < σ < 1 or 1 < θ < 1. For the sake of notational convenience, we will use both parameters θ and σ interchangeably. At any instant, consumers choose the energy mix y1, y2 minimizing consumer expenditures subject to the constraint on the utility level: min p1 y1 þ p2 y2

ð4:2Þ

uðy1 , y2 Þ  U,

ð4:3Þ

y1 , y2

where p1 and p2 are the prices of conventional and alternative energy, and U is a lower-bound utility level.

4.2.2

Resource Extraction

Resources for conventional energy are extracted by firms maximizing the present value of future rents: ZT V 0 ¼ max x1 ðt Þ

ert p1 ðt Þx1 ðt Þdt,

ð4:4Þ

0

subject to the resource stock balance: s_ ðt Þ ¼ x1 ðt Þ,

ð4:5Þ

where t is the time variable, r is the discount rate, x1(t) is resource extraction and T  1 is the time of extraction termination. The initial resource stock is s(0) ¼ s0 and the resource constraint is given by the equation: ZT x1 ðt Þdt ¼ s0 :

ð4:6Þ

0

Under zero extraction cost, cumulative extraction by the date of termination must be equal to the initial resource stock.

4.2 The Model

4.2.3

83

Alternative Energy

Alternative energy is supplied by companies using a one-factor linear technology of production: x2 ðt Þ ¼ ak ðt Þ,

ð4:7Þ

where x2(t) denotes the alternative energy production, k(t) the energy capital and a the constant productivity of capital. Capital is accumulated over an infinite time horizon as: k_ ðt Þ ¼ qðt Þ  δkðt Þ,

ð4:8Þ

where q(t) is the capital addition and δ is the depreciation rate. The initial capital stock is k(0) ¼ k0. The cost of capital creation is linear and equal to: cq(t), where c is the constant marginal cost. Instantaneous profit is p2(t)x2(t)  cq(t). The energy-producing company chooses the paths of energy generation and capital addition to maximize the present value of future profits: Z1 W 0 ¼ max

x2 ðt Þ, qðt Þ

ert ðp2 ðt Þx2 ðt Þ  cqðt ÞÞdt,

ð4:9Þ

0

subject to the production function (4.7) and the equation of capital accumulation (4.8).

4.2.4

Equilibrium Conditions

The market-clearing conditions: y1 ðt Þ ¼ x1 ðt Þ, y2 ðt Þ ¼ x2 ðt Þ are fulfilled at any instant under equilibrium energy prices p1(t) and p2(t).

ð4:10Þ

84

4.3

4 Pricing Energy Resources Under Transition to Alternative Energy

Valuation of Energy Resource and Energy Capital

Consider the models of energy producers (4.4)–(4.6) and (4.7)–(4.9). Production assets are given by the resource stock and the energy capital. In this section, we analyse the value functions for these assets and derive the equations for energy prices.

4.3.1

Valuation of Conventional Energy Resource

Consider the model of resource extraction (4.4)–(4.6). Suppose that the time of extraction termination is infinite, T ¼ 1 (we will show below that this is the case for the equilibrium transition path). Then, at any instant t  0, the present value of rents is a function of two state variables—the remaining resource stock s(t) and the resource price p1(t). The value function V ¼ V(s, p1) satisfies the continuous-time Bellman equation: rVdt ¼ max ðp1 x1 dt þ dV Þ, x1 , s

ð4:11Þ

where dV is the value differential. We omit the time argument here and henceforth, if it is not necessary. At any instant, the firm chooses the volume of extraction to maximize the sum of current rent, p1x1dt, and the expected value gain, dV, over a small interval of time. The rate of return for resource value in (4.11) is equal to the real interest rate. The value differential is approximately equal to the sum of the effects of stock reduction and the resource price change: dV ¼ V s ds þ V p1 dp1 with subscripts denoting partial derivatives. Inserting this and the stock balance (4.5) into Eq. (4.11) yields:   p1 x1 dt  V s x1 dt þ V p1 dp1 : rVdt ¼ max x 1

ð4:12Þ

Taking the maximum with respect to x1 implies that the marginal resource value must be equal to the price: V s ¼ p1 :

ð4:13Þ

Under this condition, a firm is indifferent as to how much to extract and produces to meet the condition of market clearing (4.10) for x1. Hence, due to (4.13), the Bellman Eq. (4.12) is reduced to:

4.3 Valuation of Energy Resource and Energy Capital

rVdt ¼ V p1 dp1 :

85

ð4:14Þ

We can apply Hotelling’s valuation principle that we talked about in Chap. 2 for the resource stock: V ¼ p1 s: For this function, we have it that V 0p1 dp1 ¼ sdp1 and Eq. (4.14) transforms to Hotelling’s rule for resource price growth: p_ 1 ¼ rp1 :

ð4:15Þ

The initial price and the extraction path are determined from the resource constraint (4.6) that we will examine further below.

4.3.2

Valuation of Alternative Energy Capital

Now consider the model of alternative energy production (4.7)–(4.9).The present value of profits is a function of the state variables—energy capital and alternative energy price. At any instant, this value function, W ¼ W(k, p2), satisfies the Bellman equation: rWdt ¼ max ½ðp2 x2  cqÞdt þ dW , x2 , q

ð4:16Þ

where dW is the value differential, which is equal to: dW ¼ W k dk þ W p2 dp2 :

ð4:17Þ

The value differential is the sum of the effects of capital creation and price change. Inserting (4.17), the production function (4.7) and the capital accumulation Eq. (4.8) into (4.16) yields:   ðp2 ak  cqÞdt þ W k ðq  δkÞdt þ W p2 dp2 : rWdt ¼ max q

ð4:18Þ

Differentiating the right-hand side of (4.18) with respect to q implies that the marginal value of capital is equal to the marginal cost of capital addition: W k ¼ c:

ð4:19Þ

86

4 Pricing Energy Resources Under Transition to Alternative Energy

Under this condition, a firm is indifferent as to how much to invest in energy capital and produces to meet the condition of market clearing (4.10) for x2. With (4.19), we rewrite the Bellman Eq. (4.18) as rWdt ¼ ðp2 a  cδÞkdt þ W p2 dp2 :

ð4:20Þ

We make a guess from equality (4.19) that the value function is linear in capital: W ¼ ck,

ð4:21Þ

which is similar to Hotelling’s valuation principle. Then the alternative energy price is time-constant and equal to: p2 ¼

rþδ c, a

ð4:22Þ

because the second term on the right-hand side of (4.20) is zero and inserting (4.21) into (4.20) implies that rc ¼ p2a  cδ. As a result, the alternative energy price (4.22) is equal to the marginal cost of capital creation, c, multiplied by the cost-to-productivity ratio, (r + δ)/a. One can interpret this ratio as a price mark-up over the marginal cost that ensures equality between capital return, ap2, and the full unit cost of energy production, (r + δ)c. At any instant, the supply of alternative energy x2(t) ¼ ak(t) results from the decision about capital addition q(t), which is determined from the market-clearing conditions that we will consider further below.

4.4

The Equilibrium Transition Path

The transition to alternative energy is viewed here as a gradual process of substitution of this energy for a conventional one. At any instant, the energy supply in the model adjusts to changes of demand driven by a decline of the relative price of alternative energy. We consider an equilibrium transition path with a constant utility level U. The utility level is an endogenous variable determined from conditions of equilibrium for the initial resource stock s0 and the initial capital stock k0. The initial stocks s0 and k0 are given. Figure 4.2 shows the plane of indifference curves for the utility function u(y1, y2) with the equilibrium transition path on the utility isoquant U. The transition path begins at the initial point (y1(0), y2(0)) and moves to the north-west. The initial energy capital determines the initial consumption of alternative energy as y2(0) ¼ ak0. As we have shown, the alternative energy price is constant and equal to p2 . The initial consumption of conventional energy y1(0) and the initial price p1(0) are unknown. The initial relative price p1 ð0Þ=p2 is depicted in Fig. 4.2 as line S, the slope of isoquant U at the initial point (y1(0), y2(0)).

4.4 The Equilibrium Transition Path

87

Fig. 4.2 The path of transition to alternative energy

The transition path in Fig. 4.2 converges to the terminal point ð0, y2 Þ, where the resource stock is fully exhausted and the terminal resource consumption is zero. The terminal consumption of alternative energy y2 is finite for any isoquant of utility function (4.1) under imperfect substitution. The market-clearing condition (4.10) implies that y2 ¼ ak, where k is the finite long-term volume of capital, which is an endogenous variable of the model, as well as the utility level U. The choice of the equilibrium transition path includes the choices of: (a) The initial consumption of conventional energy y1(0); (b) The initial price of conventional energy p1(0); (c) The equilibrium utility isoquant U and the long-term volume of capital k. As one can see from Fig. 4.2, the choice of any of these variables predetermines the choices of the other ones from this list.

4.4.1

The Structure of Demand

The solution for the consumer problem (4.1)–(4.3) is given by Hicksian demand functions: y1 ¼ ðP=p1 Þσ U, where

y2 ¼ λσ1 ðP=p2 Þσ U,

ð4:23Þ

88

4 Pricing Energy Resources Under Transition to Alternative Energy 1
1σ P ¼ p1σ þ ðp2 =λÞ1σ 1

ð4:24Þ

is the price index for the utility function (4.1). Demand functions (4.23) are derived in Appendix A.1. The demand for each type of energy is proportional to the utility level U. The price elasticity of demand coincides with the elasticity of substitution σ. The price index P is equal to the energy expenditures per unit of utility: P ¼ ( p1y1 + p2y2)/U. Denote the initial resource price as p1 ð0Þ ¼ p01 and the initial market share of alternative energy as   1σ p =λ p2 y2 ð0Þ ψ¼ 0 ¼  1σ 2  1σ  p1 y1 ð0Þ þ p2 y2 ð0Þ p0 þ p =λ 1

ð4:25Þ

2

as implied by demand functions (4.23). In what follows in this section, we will characterize dynamics of the energy price index, the energy capital and the resource extraction.

4.4.2

Dynamic of the Energy Price Index

From (4.23), the demand for conventional energy y1 is positive at any time, and therefore the period of extraction is infinite, T ¼ 1. Hotelling’s rule (4.15) implies that the conventional resource price grows exponentially: p1 ðt Þ ¼ p01 ert :

ð4:26Þ

Inserting (4.26) into (4.24), we represent the price index as the generalized logistic function of time: Pðt Þ ¼

1
 1σ  1σ 1σ p01 erð1σ Þt þ p2 =λ :

ð4:27Þ

We show in Appendix A.2 that this function is the solution of the so-called “Richards growth equation”: h i  1σ P_ ðt Þ ¼ rPðt Þ 1  p2 =λ Pðt Þσ1 for the initial condition P0 ¼ Pð0Þ ¼

ð4:28Þ

1
   1σ 1σ 1σ p01 þ p2 =λ . The usual logistic

differential equation corresponds to the case σ ¼ 2. Equation (4.28) is a

4.4 The Equilibrium Transition Path

89

Fig. 4.3 Dynamics of the energy price index

generalization of hHotelling’s rule: the growth rate of the energy price index is equal i  1σ to: P_ ðt Þ=Pðt Þ ¼ r 1  p2 =λ Pðt Þσ1 and it is decreasing in time for σ > 1. Equation (4.28) implies that the price index P(t) is increasing and converging to the preference-adjusted price of alternative energy: P ¼ p2 =λ as t tends toward infinity. The time profile of the price index can be concave or convex-concave as demonstrated in Fig. 4.3. We show in Appendix A.2 that the function P(t) is convex-concave if, and only if, 1 ψ : σ

ð4:29Þ

Under this condition, the growth of the price index in the early period of transition resembles the resource price growth in Hotelling’s model. Condition (4.29) holds if ψ is small, that is, the initial relative price of conventional energy p01 =p2 is low. In this case, the conventional energy dominates because the alternative energy is expensive and its market share ψ is small.

4.4.3

Dynamic of Energy Capital

Consider the market-clearing condition for alternative energy, x2(t) ¼ y2(t), which can be rewritten as:  σ akðt Þ ¼ λσ1 Pðt Þ=p2 U due to (4.7) and (4.23). Take the time derivative of capital and use (4.28):

ð4:30Þ

90

4 Pricing Energy Resources Under Transition to Alternative Energy

Fig. 4.4 Dynamics of capital and extraction

h i  1σ dPðt Þ λσ1 U λσ1 U ¼ σPðt Þσ1 rσPðt Þσ 1  p2 =λ Pðt Þσ1 : k_ ðt Þ ¼ σ σ dt ap2 ap2  σ Substituting for Pðt Þ=p2 ¼ ak ðt Þ=λσ1 U (as implied by (4.30)) and rearranging terms yields the differential equation for capital growth: h i 1 k_ ðt Þ ¼ rσkðt Þ 1  ððaλ=U Þk ðt ÞÞθ :

ð4:31Þ

Remember that θ ¼ σ/(σ  1). The capital addition q(t) is determined at any instant from this equation and the equation of capital accumulation (4.8): qðt Þ ¼ k_ ðt Þ þ δkðt Þ: The equilibrium capital path is the solution of Eq. (4.31). As time goes to infinity, capital converges to the steady-state level: k ¼ U=aλ

ð4:32Þ

for any initial capital k0. The steady-state capital is equal to the utility level U adjusted by the capital productivity a and the preference weight λ. We show in Appendix A.3 that the solution for capital growth Eq. (4.31) is the generalized logistic function of time:  θ 1  ψ rðσ1Þt k ðt Þ ¼ k 1 þ e : ψ

ð4:33Þ

The capital path (4.33) depends on the two unknown variables: the long-term capital k and the initial market share ψ. Figure 4.4 depicts the case of convexconcave function k(t). The time profile of capital (4.33) implies a simple link between long-term and initial volumes of capital:

4.4 The Equilibrium Transition Path

91

k ¼ k0 ψ θ :

ð4:34Þ

The long-term growth rate of capital, k=k 0 ¼ ψ θ , is inversely related to the initial market share of alternative energy ψ, which increases with the initial relative price p01 =p2 . Thus, the lower the initial relative price of conventional energy, the higher is the long-term level of energy capital. A lower relative price of conventional energy indicates a more abundant initial stock of natural resource, which is replaced in the long term with a larger volume of energy capital.

4.4.4

Dynamic of Resource Extraction

Extraction is linked to energy capital because, from (4.23), demands for traditional and alternative energy are related as: y1 ¼ y2 ðp2 =p1 Þσ =λσ1 and the market-clearing conditions (4.10) imply that  1σ !θ  θ  σ x2 ðt Þ p2 ak ðt Þ σ p2 =λ ψ σrt ¼ λ e ¼ ak ð t Þλ eσrt : x1 ðt Þ ¼ σ1   1σ 1ψ p01 ert λ λσ1 p01 Inserting here the solution for capital (4.33) and rearranging terms yields:  x1 ðt Þ ¼ aλk 1 þ

ψ erðσ1Þt 1ψ

θ

:

ð4:35Þ

The equilibrium extraction is proportional to the long-term capital and decreases over time. It tends toward zero as time goes to infinity, as shown in Fig. 4.4. The initial extraction is  θ 1ψ x1 ð0Þ ¼ aλkð1  ψ Þ ¼ aλk0 ψ θ

ð4:36Þ

due to (4.34). It is proportional to the initial capital and inversely related to the initial market share of alternative energy.

92

4.5

4 Pricing Energy Resources Under Transition to Alternative Energy

The Resource Price Determination

The time paths of energy capital and resource extraction depend on the initial market share ψ, which is found from the resource constraint (4.6) that we rewrite for T ¼ 1 as: Z1 x1 ðt Þdt ¼ s0 ,

ð4:37Þ

0

where x1 ðt Þ ¼ aλk 0 ψ

θ



ψ erðσ1Þt 1þ 1ψ

θ

from Eqs. (4.34) and (4.35). The resource constraint (4.37) can be rewritten as F ðψ Þ ¼

ψ θ s0 , aλk0

ð4:38Þ

where Z1  1þ F ðψ Þ ¼

ψ erðσ1Þt 1ψ

θ dt:

ð4:39Þ

0

The function F(ψ) is defined for ψ 2 (0, 1) and positive. It is decreasing because the function inside the integral in (4.39) is decreasing in ψ. Moreover, F(0) ¼ 1 and F(1) ¼ 0. The right-hand side of Eq. (4.38) is increasing in ψ. Consequently, this equation has a unique solution ψ , 0 < ψ  < 1, as shown in Fig. 4.5. Curve A in this figure depicts the right-hand side of (4.38). The existence and uniqueness of the equilibrium initial market share ψ  implies that the equilibrium transition path exists and is unique. This market share determines the initial extraction x1(0) and the equilibrium utility isoquant U depicted in Fig. 4.2. The equilibrium transition path shown in that figure is determined by the combination of initial production factors, because the equilibrium market share ψ  depends on the ratio of productivity-adjusted capital to resource, aλk0/s0, as follows from Eq. (4.38). The graph in Fig. 4.5 demonstrates that an increase of this ratio shifts curve A downwards to curve A0 and brings about an increase of the equilibrium market share from ψ  to ψ 0. As a result, the equilibrium market share is an increasing function of the initial capital-resource ratio, ψ  ¼ ψ(K0/s0), where K0 ¼ aλk0 denotes the

4.5 The Resource Price Determination

93

Fig. 4.5 The equilibrium initial market share of alternative energy

productivity-adjusted energy capital. The equilibrium initial resource price p01 is increasing in this ratio too, as it is increasing in ψ :   p01 ¼ p2 =λ



ψ 1  ψ

θ1

ð4:40Þ

as follows from Eq. (4.25). A higher ratio K0/s0 thus implies a higher level of resource price. The significance of this effect depends on the elasticity of substitution. If σ is low (θ is high), an increase of capital-resource ratio causes a change of ψ  that can lead to a substantial increase of the resource price.

4.5.1

The Equilibrium Utility Level

Equations (4.32) and (4.34) imply that the equilibrium utility level is equal to: U ¼ K 0 ψ θ ¼ K 0 ψ ðK 0 =s0 Þθ :

ð4:41Þ

The effect of the initial capital on utility is ambiguous. The first factor in (4.41) corresponds to a positive direct effect, and the second factor to a negative indirect effect of the initial capital-resource ratio on the capital growth rate k=k0 ¼ ψ ðK 0 =s0 Þθ . The indirect effect is strong if the elasticity of substitution is low and θ is high. In this case, the replacement of conventional energy requires the creation of large volumes of energy capital. Thus, the equilibrium utility level is determined by a combination of initial production factors. If the economy is resource-rich, then the capital-resource ratio

94

4 Pricing Energy Resources Under Transition to Alternative Energy

is low and the conventional energy is cheap at the early stage of transition. Although such economy lacks the initial energy capital, the utility of energy consumption is high, because the expected rate of capital growth is significant.

4.5.2

An Example: σ = 2

The function F(ψ) is a binomial integral if σ is a rational number. Even if F(ψ) can be derived explicitly, the equation for market share (4.38) is hard to treat analytically except for some simple cases. As an example, we consider σ ¼ θ ¼ 2 and show in Appendix A.4 that in this case: F ðψ Þ ¼

ψ  lnψ  1 : r

ð4:42Þ

Hence, the resource constraint (4.38) is represented as: ψ  lnψ  1 ¼ rψ 2

s0 : K0

ð4:43Þ

Curve A in Fig. 4.6 depicts the numerical solution for this equation, ψ  ¼ ψ(K0/ s0), for r ¼ 0.03 and the ratio K0/s0 varying from 0.01 to 0.3. Consider an approximation of this solution for ψ close to 1, that is, for the case of a high initial capital-resource ratio K0/s0. We apply the following second-order approximation:

Fig. 4.6 The initial market share ψ(K0/s0)

4.6 Transition Paths of Energy Capital and Extraction

95

lnψ  ψ  1  0:5ð1  ψ Þ2 so that F(ψ)  0.5(1  ψ)2/r, and the resource constraint (4.43) transforms to the equation: ð1  ψ Þ2 ¼ 2rψ 2 s0 =K 0 with the solution: ψ ¼

ðK 0 =s0 Þ1=2 ðK 0 =s0 Þ1=2 þ ð2r Þ1=2

:

ð4:44Þ

Curve B in Fig. 4.6 shows this approximation for the same numerical example. From (4.40), the initial resource price for this approximate solution is equal to: p01

¼



p2 =λ





ψ 1  ψ

¼



p2 =λ





K0 2rs0

1=2 ¼

p2



ak 0 2rλs0

1=2 :

Note that the resource price decreases with the preference weight of alternative energy λ. From (4.41), (4.44), the utility level for the approximate solution is equal to
2 U ¼ K 0 ψ 2 ¼ ð2rs0 Þ1=2 þ ðλak 0 Þ1=2 : Comparing this formula with the utility function (4.1) yields the initial consumption mix: y1(0) ¼ 2rs0 and y2(0) ¼ ak0. Thus, the initial extraction rate is approximately equal to 2r.

4.6

Transition Paths of Energy Capital and Extraction

The approximation of Eq. (4.43) gives the simple analytical solution for σ ¼ 2, which is suitable only if the market share ψ(K/s) is close to one. If this share is small, one has to resort to numerical analysis. In what follows, we examine the effects of changes in parameters of consumer preferences σ and λ on the transition dynamics of energy capital and resource extraction. Suppose that the initial production factors are k0 ¼ 1 trillion dollars and s0 ¼ 10 billion tonnes of oil equivalent and the unit cost of capital creation is normalized to one, c ¼ 1. Parameters of energy production technology are selected as a ¼ 0.1 and δ ¼ 0.07 and the discount rate is r ¼ 0.03. From (4.22), the equilibrium price of alternative energy is p2 ¼ ð0:03 þ 0:07Þ=0:1 ¼ 1.

96

4.6.1

4 Pricing Energy Resources Under Transition to Alternative Energy

Variation of the Elasticity of Substitution

Let λ ¼ 1 and σ ¼ 2, 3 or 5. We solve numerically the resource constraint Eq. (4.38) for ψ and then calculate the initial resource price p01 from (4.40) and the long-term capital k from (4.34). Table 4.2 presents the results of these calculations. The equilibrium market share of alternative energy ψ  decreases with the elasticity of substitution, while the initial resource price p01 increases. Energy capital increases during the transition roughly by a factor of 7–8.5 compared to the initial level k0. Using the data in Table 4.2, we calculate the time paths for capital k(t) given by (4.33) and extraction x1(t) given by (4.35) for σ ¼ 2, 3, 5. The results of the calculations are shown in Figs. 4.7 and 4.8. For a higher elasticity of substitution, capital growth at the early stage of transition is more rapid, as can be seen in Fig. 4.7. On the one hand, the rapid capital growth is spurred by a lower initial market share ψ  for higher σ, as one can see from Table 4.2 (and Eq. (4.34)). On the other hand, the substitution of alternative energy for a more expensive conventional energy is easier at the later stage of transition for greater σ. This substitution dictates a more rapid pace of capital accumulation and a more intensive resource extraction at the early stage shown in Fig. 4.8. The graph in this figure demonstrates a less conservative path of resource extraction under a higher elasticity of substitution. Table 4.2 Equilibrium variables for various σ

σ ψ p01 k

Fig. 4.7 Energy capital paths for various σ

2 0.358 0.558 7.79

3 0.269 0.607 7.16

5 0.180 0.685 8.51

4.6 Transition Paths of Energy Capital and Extraction

97

Fig. 4.8 Extraction paths for various σ Table 4.3 Equilibrium variables for various λ

λ ψ p01 k

0.5 0.202 1.006 11.01

1 0.269 0.486 7.16

2 0.350 0.367 4.84

The key feature of this effect is the exhaustibility of a conventional resource that determines the intertemporal choice between current and future extraction. A more intensive extraction at the early stage for higher σ implies a less intensive extraction at the late stage. A higher capital-resource ratio at the late stage, in turn, requires a more rapid capital growth at the early stage in addition to a more intensive resource extraction at this stage, as if energy capital and resource were complementary factors of production. A somewhat paradoxical feature of this causality is that such an effect of “induced complementarity” at the beginning of transition is more pronounced for a higher degree of substitution between resource and capital.

4.6.2

Variation of the Preference Weight

Suppose that technology parameters are the same as above and let σ ¼ 3. Consider variations of the preference weight of alternative energy: let λ be equal to 0.5, 1 or 2. As in the previous examples, the alternative energy price is equal to one. The results of equilibrium calculations are presented in Table 4.3. The initial market share of alternative energy increases with the preference weight, whereas the initial resource price and the long-term capital decrease. The resource price p01 is linked to the preference-adjusted alternative energy price p2 =λ

98

4 Pricing Energy Resources Under Transition to Alternative Energy

Fig. 4.9 Energy capital paths for various λ and σ ¼ 3

Fig. 4.10 Extraction paths for various λ and σ ¼ 3

through Eq. (4.40), and due to this link a higher preference weight implies a lower resource price. The long-term capital k is decreasing in λ, because, from (4.34), it is decreasing in ψ , which is increasing in λ (because σ > 1, as follows from (4.25)). The effects of the preference weight variations on the transition paths of energy capital and extraction are presented graphically in Figs. 4.9 and 4.10. Thus, a higher preference for alternative energy leads to a lower initial resource price that enhances consumer demand for conventional energy and slows down the demand for alternative energy. This inference leads to another paradoxical

4.7 The Green Paradox

99

conclusion that a higher preference weight of alternative energy results in a slower transition to this energy and a more intensive extraction of conventional resources at the early stage. This effect is also explained by the influence of the intertemporal resource constraint that brings about a higher initial market share of alternative energy and a lower growth rate of energy capital, even though this energy is preferable to consumers.

4.7

The Green Paradox

The paradoxes of energy transition that we have demonstrated for the numerical examples are relevant to the practical issue of the effectiveness of environmental regulation and climate change policy. In practice, the green policies adopted by many countries have not led to a significant reduction of greenhouse gas releases on a global scale. The Green Policy Paradox was first coined in 2008 by Hans-Werner Sinn, who argued that imposing a carbon tax to curb the demand for fossil fuels may be counterproductive because of the intertemporal nature of natural resource exhaustion (Sinn 2008). Sinn showed that if the carbon tax rises over time, the anticipation of this policy encourages the suppliers of exhaustible energy resources to bring forward extraction. Taxation of carbon emission can thus reinforce this emission, contrary to the intention of green policymakers. Similar problems arise with government policies to support the switch to alternative energy through subsidizing R&D and new capital investments in renewables. This subsidizing may have ambiguous effects on the production of fossil fuels. For example, Michael Hoel (2008) showed that when the supply side of fossil fuels is taken into consideration, fossil energy prices may decline as a consequence of technological improvements in renewable energy production. The decline of conventional energy prices encourages demand for exhaustible resources and reduces the effectiveness of policy against carbon emissions. We do not consider the climate policy in this chapter, but we can illustrate the nature of the Green Paradox in a simple way, by using our calculations from the previous section. The positive link between the prices of conventional and alternative energy is given in our model by Eq. (4.40), which we rewrite here for the sake of convenience:   p01 ¼ p2 =λ



ψ 1  ψ

θ1

:

ð4:45Þ

The initial resource price p01 is proportional to the preference-adjusted price of alternative energy p2 =λ. In our numerical examples, an increase of the alternative energy preference weight λ leads to a decrease of the resource price, as can be seen in Table 4.3. The direct effect of decreasing factor p2 =λ in (4.45) overweighs the indirect effect of increasing market share ψ .

100

4 Pricing Energy Resources Under Transition to Alternative Energy

Fig. 4.11 (a) Extraction paths for σ ¼ 5 (Billion tonnes of oil equivalent). (b) Extraction paths for σ¼2

Suppose now that there is a social planner making an optimal climate policy and taking into account the external effects of conventional energy production like carbon emission that we do not model explicitly. Suppose that individual energy consumers do not internalize these externalities in their demand decisions. Let the social utility function be given by (4.1), but differ from the individual utility in the preference weight of alternative energy λ, and this weight is higher for social utility. Let λ ¼ 1 for individuals and λ ¼ 2 for the social planner who subsidizes the production of clean energy. Subsidizing reduces the marginal cost of energy capital creation from c to c/2, so that the optimal price of alternative energy is set at the level p2 =2 , corresponding to the social weight of clean energy λ ¼ 2. Then, from Table 4.3, the initial resource price is p01 ¼ 0:486 without subsidizing (for λ ¼ 1) and p01 ¼ 0:367 with subsidizing (for λ ¼ 2). As a result, the green energy policy encourages consumption of a cheaper conventional energy and implies a more rapid resource extraction at the early stage of transition, as demonstrated in Fig. 4.10. This effect is robust to variations in the elasticity of substitution. Figure 4.11(a, b) shows the extraction paths for the above numerical example with λ ¼ 0.5, 1, 2 and σ ¼ 2 and 5. Extraction at the early stage is more intensive for a higher weight of alternative energy. From (4.36) and (4.40), the effect of λ on the initial extraction can be represented as the product of the wealth effect of initial energy capital and the σ terms-of-trade effect: x1 ð0Þ ¼ aλk0 ðp2 =λp01 Þ . The former overweighs the latter for all our numerical examples. The Green Paradox is also manifested in our model as the negative effect of clean energy subsidizing on the accumulation of energy capital. One can see from Table 4.3 that the initial market share of alternative energy increases, because of the subsidizing, from ψ  ¼ 0.269 for λ ¼ 1 to ψ  ¼ 0.350 for λ ¼ 2 (for the case σ ¼ 3). Equation (4.34) implies that the long-term growth rate of energy capital relates to this market share as k=k0 ¼ ðψ  Þθ ¼ 7:17 in the former case and falls to k=k 0 ¼ 4:83 in the latter case. A higher preference weight λ implies a higher market

4.8 Concluding Remarks

101

share of alternative energy ψ  and a lower initial resource price p01 indicating a lower value of the initial resource stock s0, which has to be replaced in the long term with a lower volume of energy capital k. As a result, the clean energy policy, modelled as the switch of preference weight from λ ¼ 1 to λ ¼ 2, has a detrimental effect on clean energy development in the long term. Finally, it is possible to interpret the effects of increasing the elasticity of substitution between clean energy and fossil fuels shown in Figs. 4.7 and 4.8 as a kind of Green Paradox outcome too. This interpretation was suggested in the paper by Ngo Van Long (2014), where an increase of σ is associated with a technological change making the alternative energy a closer substitute for fossil fuels. Long demonstrated that such a change can result in the increase of cumulative extraction. In our model, a similar outcome of an increased substitutability is the effect of “induced complementarity” with a more rapid extraction supplementing a more rapid capital growth at the early stage of transition.

4.8

Concluding Remarks

The exhaustion of conventional energy resources does not mean that energy consumption will decline in the future, because fossil fuels are being gradually replaced with alternative sources. In the model in this chapter, an economy simultaneously uses conventional and alternative energy and gradually increases the share of renewables in the energy mix of consumers. In reality, the transition of economies to renewable energy is driven by the relative price changes resulting from the influence of two factors—technical progress making the alternative energy cheaper and the depletion of fossil resources making the conventional energy more expensive. We assumed that the marginal cost of energy capital c is constant and focused on the exhaustion of conventional resources, but qualitatively these two factors exert a similar influence on the energy transition. Differentiated energy resources are used simultaneously in our model because they are imperfect substitutes. The difference in energy sources does not matter for power consumption by households, if they are indifferent to environmental issues. Nonetheless, the imperfect substitution is a reasonable hypothesis, if one considers final energy consumption for regional or national systems of energy production, transportation and distribution. Under the system approach, the technological and geographical constraints matter for managerial decisions, and different types of energy prove to be suitable for different purposes. Due to the imperfect substitution, prices of different energy sources may vary and diverge in time, implying that their market shares also diverge depending on the degree of substitution. We have shown in this chapter that the price of an exhaustible energy resource satisfies Hotelling’s rule, whereas the energy price index is a generalized logistic function of time. An equilibrium transition path exists and is unique, and the initial equilibrium market share of alternative energy increases with the initial

102

4 Pricing Energy Resources Under Transition to Alternative Energy

capital-resource ratio. The long-term growth of alternative energy capital decreases with the initial market share, while the equilibrium utility level increases in the initial production factors—the resource stock and the energy capital. Equilibrium energy transition exhibits the property of the Green Paradox: a higher preference weight of alternative energy implies a more intensive near-term resource extraction. This effect is a consequence of the exhaustibility of a conventional fossil resource that determines the intertemporal choice between current and future extraction. In the narrow sense, the resource constraint means that extraction is limited in the long term by the availability of resource in the ground. In the broad sense, this constraint can be predetermined by the environmental constraints such as a long-term carbon budget, which means a feasible volume of cumulative carbon emission into the atmosphere caused by the fossil resource extraction and use. In the Green Paradox context, we considered the utility weight of alternative energy as the parameter of a social planner’s preference with λ > 1 indicating a policy to support alternative energy, though we have not modelled this policy explicitly. The political priority of clean energy is common for developed countries that use large-scale subsidizing of renewables. Total subsidies for renewable energy amounted to $166 billion around the world in 2017, according to the International Renewable Energy Agency (2020, p. 45). However, as the International Energy Agency estimated, in the same year national governments spent a twofold amount of funds, $335 billion, to subsidize fossil fuels (Gould and Adam 2020). Subsidizing fossil fuel consumption does not necessarily mean that the support of conventional energy is a political goal of some national governments or, in terms of our model, that λ < 1 for the social utility function. Such subsidizing is typical of resource-rich countries, where governments aim to keep domestic energy prices low compared to international markets. The model of this chapter implies that the longterm growth of energy capital is higher under a cheap initial resource price that ensures a lower initial market share of alternative energy and, as a result, a higher rate of energy capital accumulation. From this point of view, the price wedge that exists between world and domestic energy prices due to the protectionist barriers allows resource-rich economies to use their natural advantages and get more longterm benefits from energy transition.

Appendices A.1 Hicksian Demand Functions (4.23) The Lagrangian for the consumer problem (4.1)–(4.3) is

Appendices

103


θ 1=θ 1=θ L ¼ p1 y1 þ p2 y2  μ y1 þ ðλy2 Þ U , where μ is the dual variable for constraint (4.3). The first-order conditions are:
θ1 ∂L 1=θ1 1=θ ¼ p1  μy1 y1 þ ðλy2 Þ1=θ ¼0 ∂y1
θ1 ∂L 1=θ1 1=θ ¼ p2  μλ1=θ y2 y1 þ ðλy2 Þ1=θ ¼0 ∂y2
θ ∂L 1=θ ¼ U  y1 þ ðλy2 Þ1=θ ¼ 0: ∂μ

ð4:46Þ ð4:47Þ ð4:48Þ

From (4.46), (4.47), the relative price is: p2 =p1 ¼ λ1=θ ðy2 =y1 Þ1=θ1 ¼ λ1=θ ðy2 =y1 Þ1=σ , implying that
σ y2 ¼ y1 λ1=θ p1 =p2 ¼ y1 λσ1 ðp1 =p2 Þσ :

ð4:49Þ

Insert this into (4.48) and rearrange terms
θ  1=θ θ
1=θ 1=θ U ¼ y1 þ λ1=θ y1 λσ1 ðp1 =p2 Þσ ¼ y1 þ y1 1=θ λσ=θ ðp1 =p2 Þσ=θ
θ
θ ¼ y1 1 þ ðλp1 =p2 Þσ1 ¼ y1 1 þ ððp2 =λÞ=p1 Þ1σ !θ p1σ þ ðp2 =λÞ1σ 1 ¼ y1 : p1σ 1 Consequently, y1 ¼




p1σ þ ðp2 =λÞ1σ 1

1 1σ

=p1

σ

due to (4.24). Inserting this into (4.49) yields: y2 ¼ λσ1 ðP=p2 Þσ U:

U ¼ ðP=p1 Þσ U

104

4 Pricing Energy Resources Under Transition to Alternative Energy

A.2 Time Derivatives of the Energy Price Index Differentiate (4.27): 1 1  1σ rð1σ Þt 1
 0 1σ rð1σÞt   1σ 1σ p1 e e þ p2 =λ P_ ðt Þ ¼ r ð1  σ Þ p01 1σ  0 1σ rð1σ Þt p e ¼ rPðt Þ  1σ 1  1σ p01 erð1σÞt þ p2 =λ !   1σ h i  1σ p2 =λ ¼ rPðt Þ 1   1σ Pðt Þσ1 : ¼ rPðt Þ 1  p2 =λ   1σ p01 erð1σÞt þ p2 =λ

The second-order time derivative of the price index is
   € ¼ r P_ 1  σ p2 =λ 1σ Pðt Þσ1 : P The inflection point of P(t) is   1 e ¼ p2 =λ σ 1σ : P The initial price index is 1 !σ1   1σ 1
    1σ 1σ    p2 =λ 0 1σ þ p2 =λ ¼ p2 =λ  1σ  P0 ¼ p1 1σ p01 þ p2 =λ   1 ¼ p2 =λ ψ σ1 :

e  P0 or Consequently, P(t) is convex-concave if, and only if, P 1 ψ : σ

A.3 The Time Path of Capital (4.33) From (4.30), (4.32):

Appendices

105

!σ  σ  σ Pðt Þ λσ1 U Pðt Þ σ P ðt Þ k ðt Þ ¼ ¼ kλ ¼k    : p2 p2 a p2 =λ

ð4:50Þ

From (4.25), (4.27): Pðt Þ ¼

 1 1
   1σ 1σ   1σ rð1σ Þt 1  ψ rðσ1Þt 1σ p01 e e þ p2 =λ ¼ p2 =λ 1 þ : ψ

Inserting this into (4.50) yields  θ 1  ψ rðσ1Þt e k ðt Þ ¼ k 1 þ : ψ

A.4 The Case σ = 2 Consider function F(ψ): Z1  F ðψ Þ ¼ 1þ

ψ erðσ1Þt 1ψ

θ dt:

0

Denote γ ¼ er(σ

 1)t

. Then t¼

lnγ dγ , dt ¼ r ð σ  1Þ γr ðσ  1Þ

and 1 F ðψ Þ ¼ r ðσ  1Þ

Z1 1

For σ ¼ θ ¼ 2, we have:

 γ 1þ 1

ψ γ 1ψ

θ dγ:

106

4 Pricing Energy Resources Under Transition to Alternative Energy

Z1 rF ðψ Þ ¼

 γ 1þ 1

1

Z1 ¼ 1

ψ γ 1ψ

 ln

dγ

! ψ ð1  ψ Þ 1 ψ   dγ γ 1  ψ þ ψγ ð1  ψ þ ψγ Þ2

  ¼ lnγ  ln 1 þ ¼

2

1

ψ 1ψ

γ þ 1ψ 1  ψ þ ψγ 1

1

γ 1ψ

¼ lnψ  ð1  ψ Þ þ ln ð1  ψ Þ þ 1  ψ þ ψγ 1  ψ þ ψγ 1

¼ ψ  lnψ  1: implying (4.42).

References Acemoglu D, Aghion P, Bursztyn L, Hemous D (2012) The environment and directed technical change. Am Econ Rev 102(1):131–166 BP Statistical Review of World Energy (2020). https://www.bp.com/en/global/corporate/energyeconomics/statistical-review-of-world-energy.html. Accessed 10 Oct 2020. Gould T, Adam Z (2020) Low fuel prices provide a historic opportunity to phase out fossil fuels consumption subsidies. The International Energy Agency article. https://www.iea.org/articles/ low-fuel-prices-provide-a-historic-opportunity-to-phase-out-fossil-fuel-consumption-subsidies. Accessed 12 Oct 2020. Hoel M (2008) Bush meets Hotelling: Effects of improved renewable energy technology on greenhouse gas emissions. Memorandum 29, Department of Economics, University of Oslo, p 30 International Renewable Energy Agency (2020) Renewable power generation costs in 2019. https:// www.irena.org/-/media/Files/IRENA/Agency/Publication/2020/Jun/IRENA_Power_Genera tion_Costs_2019.pdf. Accessed 12 Oct 2020 Lazard (2020) Lazard’s levelized cost of energy analysis – Version 13.0. https://www.lazard.com/ media/451086/lazards-levelized-cost-of-energy-version-130-vf.pdf. Accessed 15 Oct 2020 Long N V (2014) The green paradox in open economies. Econstor, CESifo working paper no. 4639, p 30 Sinn H-W (2008) Public policies against global warming. CESifo Working Paper No. 2087, p 42 U.S. Energy Information Administration (2020) Levelized cost and levelized avoided cost of new generation resources in the 2020 https://www.eia.gov/outlooks/aeo/pdf/electricity_generation. pdf. Accessed 13 Oct 2020. Vardar B (2013) Imperfect resource substitution and optimal transition to clean technologies. CORE Discussion paper, CORE-UCL, Louvain-la-Neuve, Belgium, p 19

Chapter 5

Global Carbon Budgeting and the Social Cost of Carbon

Abstract The limited availability of natural resources on the global level relates primarily to the influence of fossil energy use on climate change, rather than to the exhaustion of fossil energy resources. In this chapter we consider a version of the Integrated Assessment climate-economy model with overlapping generations and fossil energy as an input in production. In a laissez-faire equilibrium, individuals do not account in their production plans for the negative external effects of carbon dioxide emission. These effects are internalized by a “social planner” solving the problem of optimal accumulation of global wealth and carbon dioxide concentration in the atmosphere. The carbon budget is defined for this problem as the optimal concentration of carbon dioxide in the long term. The social opportunity cost of fossil fuel consumption supporting this budget is given by the present value of marginal welfare losses of future generations. We examine the effects of welfare discounting on the long-term outcome and, in particular, on the welfare of distant future generations.

5.1

Introduction

The ultimate aim of transition to alternative energy considered in Chap. 4 is to preserve environmental and natural resources for future generations. The burning of fossil fuels as the main energy source and other human activities have led to emission into the atmosphere of greenhouse gases—carbon dioxide, methane and nitrous oxygen (see Box 5.1). The human-induced excess concentration of these gases in the atmosphere causes a greenhouse effect by absorbing the excess heat energy of infrared radiating reflected from the earth’s surface. There is a wide consensus among climate scientists that the greenhouse effect leads to global warming and climate change. It manifests itself in increases of global average air and ocean temperatures, increases in the frequency of extreme climate events (such as droughts, floods and hurricanes), widespread melting of sea ice and glaciers that raised global average sea levels, ocean acidification and other processes. Figure 5.1 shows the correlation between the growth of carbon dioxide concentration in the atmosphere since 1900 and the tendency of the mean global © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_5

107

108

5 Global Carbon Budgeting and the Social Cost of Carbon

Fig. 5.1 Carbon dioxide concentration in atmosphere, ppm and mean global temperature, T
C. Sources: SeaLevel.info (2020); Datahub (2020)

temperature to increase. A strong link between the temperature trend and the growth of concentration is obvious for the post-World War II period. The concentration of carbon dioxide in the atmosphere is measured in parts per million (ppm) and each part represents approximately 7.8 billion metric tons of CO2 in the atmosphere. The CO2 concentration has increased since 1900 by 39.3%, from 295 to 411 ppm in 2019, while the mean global temperature has raised by nearly 1
C. These tendencies prompted international efforts to curb global warming. The Paris Agreement on Climate Change was signed in December 2015 and 195 participating countries adopted commitments to strengthen the global response to climate change by “holding the increase in the global average temperature well below 2
C above pre-industrial levels and pursuing efforts to limit the temperature increase to 1.5 degrees” (Paris Agreement 2015, Article 2). These goals are ambitious, given the human-induced global warming of 1
C already reached and climate inertia. It is widely acknowledged that fulfilment of these goals requires immediate and considerable actions on global and national levels to ensure transformations of the world energy system aimed at reducing substantially the fossil energy production. Box 5.1 Emission of Human-Induced Greenhouse Gases Fossil fuels are responsible for 85% of the world’s carbon dioxide (CO2) emission and 64% of all greenhouse gas emissions. Carbon dioxide from fossil fuel combustion provides over 90% of energy-related emissions of greenhouse gases. Methane (CH4) accounts for 10% of energy sector emissions and originates mainly from oil and gas extraction. Since 2000, the energy share (continued)

5.1 Introduction

109

Box 5.1 (continued) of the total emission of greenhouse gases has steadily fluctuated around two-thirds. The other sources of carbon dioxide and nitrous oxide (N2O) emissions originate from energy transformation, transport, industry such as steel and cement production, buildings and land use. The latter include CO2 accumulation in the atmosphere caused by deforestation and other human activities on the land (International Energy Agency 2015). The ambitious climate policy targets impose constraints on future emission of greenhouse gases. By the end of 2018, the volume of carbon dioxide emitted since 1870 was about 2200 billion tons. A carbon budget gives an estimate of how much cumulative emission of CO2 and its equivalents is left for the future to fulfil the temperature targets with some probability. Figure 5.2 shows carbon budgets corresponding to targets between 3 and 1.5 degrees Celsius to be fulfilled with a 66% chance. The carbon budget uncertainty arises from uncertainties in future emissions of greenhouse gases and climate responses to radiative forcing. Figure 5.2 also shows the amount of carbon dioxide contained potentially in the world’s proved reserves of fossil fuels—about 2900 billion tons. This is several times above the carbon budgets for 2
C and 1.5
C temperature targets, which are equal to 800 and 280 billion tons, respectively. It means that a major part of world reserves cannot be extracted without exceeding these targets. Therefore, the carbon budgets for tough temperature targets “well below 2
C and close to 1.5
C” can be viewed as the binding resource constraints on cumulative emission to be released in

Fig. 5.2 Carbon budgets, fossil fuel reserves and lock-in emission in 2018. Source: The authors’ calculation based on the data from International Energy Agency and International Panel on Climate Change

110

5 Global Carbon Budgeting and the Social Cost of Carbon

the future. Under these constraints, proved fossil fuel reserves cannot be exhausted and must largely remain in the ground forever (as was illustrated by Bill McKibben (2012) with his “new mathematics” of global warming). Moreover, the ambitious temperature targets are incompatible with the upfront capital invested in fossil fuel extraction and energy supply. The “lock-in” global capital is worth 25 trillion dollars (Carbon Tracker Initiative 2018, p. 9) and includes equipment for extraction of fossil resources, vehicles and industrial facilities, infrastructure for energy generation, transportation and consumption with the usual lifetime of several decades. As shown in Fig. 5.2, the total amount of carbon dioxide emissions “locked in” existing fossil fuel production capacities and infrastructure around the world equals 550 billion tons. This is far below the carbon budgets of 2200 and 1700 billion tons for the temperature targets of 3
C and 2.5
C, respectively, but is nearly twice as high as the remaining carbon budget for the target of 1.5
C. Under these circumstances, the implementation of the Paris Agreement is possible only if two conditions are fulfilled. Firstly, national governments challenge fossil fuel dependence and actively influence investment in energy development. Private investors perceive, quite correctly, that their individual actions do not affect the atmospheric concentration of CO2, and governments have to internalize as much as possible these external effects. Secondly, the ambitious goals of global climate policy require breakthroughs in technologies of negative emission and carbon capture and storage, in addition to further progress for renewables and energy efficiency. Otherwise, fossil fuels will continue to dominate energy use in the long term, as the International Energy Agency predicted in its “New Policies Scenario” with the estimate of global warming above 2.7 degrees, which does not align with the goals of the Paris Agreement (International Energy Agency 2017). In this chapter, we will consider the interdependence of global warming, carbon budgets and investment decisions of economic agents. We will use the approach in Chap. 3, where we modelled economically recoverable natural resources. Remember that the ultimate recoverable resource was defined as the expected cumulative extraction in the long term at prices that consumers will be able to afford to pay. Likewise, an effective carbon budget can be defined as the expected cumulative emission of greenhouse gases available in the long term with regard to climate response to emissions and economic response to climate change. Here “effective” means that cumulative emission refers to carbon dioxide and its equivalents emitted and accumulated in the atmosphere and affecting global warming. The temperature target carbon budgets in Fig. 5.2 are derived from simulation runs of the sophisticated climate science models. As we said, any such budget defines an exogenous resource constraint for the global economy in terms of a finite environmental resource available under a specified temperature target. In a simple economy-climate model in this chapter, the carbon budget is determined endogenously as a result of economic activity and government policy aimed at mitigating climate change. We assume that the policy goal is to maximize the social welfare of current and future generations so that the external effects of using fossil energy are internalized on the global level. The optimal carbon budget is calculated as the

5.2 Modelling Climate-Economy Interactions

111

solution of the problem of atmospheric resource allocation over time with regard to the opportunity costs of CO2 emission.

5.2

Modelling Climate-Economy Interactions

The model in this chapter is based on the methodology of Integrated Assessment Models such as the Dynamic Integrated Model of Climate and Economy (DICE) pioneered by William Nordhaus (1994, 2014, 2017). These models formalize the causal links between carbon dioxide emission by the world economy, atmospheric concentration of CO2 and global warming that has a negative impact on the economy. This causality is shown in the scheme of main DICE modules in Fig. 5.3. The core of DICE is the neoclassical model of economic growth with an infinitely living representative agent choosing a consumption-saving path. The economy module in Fig. 5.3 includes competitive producers using fossil energy that causes emission of carbon dioxide into the atmosphere. The carbon cycle module describes CO2 circulation between three environmental layers: ocean, biosphere and atmosphere. An increase of atmospheric concentration of CO2 leads to a growth in temperature owing to the greenhouse effect and the exchange of energy between layers. Climate change, described by the climate module in Fig. 5.3, results in damage to the global economy measured as a share of total factor productivity in aggregate output. The share of damage in DICE models is assumed to be a function of mean global temperature. This function describes how much the economy loses from global warming in any time period. The estimates of aggregate damage function incorporate a variety of monetized factors and social damages that can be evaluated through equivalent output losses. Nordhaus and his colleagues used a large number of microeconomic studies on various consequences of climate change, including damages to agriculture, coastal regions, amenity values and human health. Their estimates of damage from global warming omit, however, several important factors,

Fig. 5.3 The structure of the DICE model

112

5 Global Carbon Budgeting and the Social Cost of Carbon

such as losses from changes in biodiversity, sea-level rises, ocean acidification and changes in ocean circulation (Nordhaus and Sztorc 2013, p. 11). As tools for policy assessment, the DICE models (and other economy-climate models) can simulate different scenarios that represent different assumptions about climate policy to be undertaken in the future. The last version, DICE-2016 (Nordhaus 2016), includes four such scenarios: (a) baseline with no climate policy above the policies already adopted in 2015; (b) welfare-maximizing policy over an indefinite future; (c) temperature-limited optimal policy with the constraint that global temperature does not exceed 2.5
C; and (d) scenario with an extremely low rate of social welfare discounting 0.1% per year offered by Nicolas Stern (2007). The last scenario provided new guidelines on climate policy based on intergenerational fairness that dictates no welfare discounting for future generations. Among the central issues of practical applications of economy-climate models is calculation of the social cost of carbon. This is the external cost to society of a unit of carbon emission, which is ignored by economic agents in the absence of climate policy. Since carbon emission has long-lasting cumulative effects, the social cost of carbon is equal to the present value of the future damage stream resulting from a marginal unit of carbon emission. This cost should be covered by the Pigouvian tax paid by emitters. The economy-climate models are employed to analyse how the time path of carbon tax depends on a selected policy scenario (Nordhaus 2014). It is important to emphasize the link between the social cost of carbon and the opportunity cost of resource extraction that played the key role in the previous chapters of this book. As was shown in Chap. 3, the opportunity cost of extraction is defined for the optimal extraction problem as the shadow value of cumulative extraction, and is calculated as the present value of future incremental cost increases caused by resource depletion. The opportunity cost of extraction is taken into account by extracting companies internalizing the effect of resource depletion in their production plans. This opportunity cost may be paid as resource rent to land owners to compensate for their losses caused by this effect. Similarly, the social cost of carbon can be defined via the shadow value of effective cumulative emission for a problem of social welfare maximization. Carbon emitters paying the optimal Pigouvian carbon tax compensate for social losses caused by the external damage from emission. In the model presented below, we consider overlapping generations and the intergenerational allocation of carbon budgets. Emissions of greenhouse gases can cause damages very far in the future, hence climate changes concern the well-being of future generations over a very long period of time. The economic dynamic in our model is described by two state variables: carbon dioxide concentration in the atmosphere and wealth per capita. The economic response to climate change results from the allocation of wealth between two factors of production—non-fossil physical capital and fossil energy. For the sake of simplicity, we assume that renewables, as a fossil energy substitute, are incorporated into the non-fossil capital factor. We will analyse the model economy evolving under laissez-faire with no climate policy and under optimal policy conducted by a “social planner”. The former case can be called “business as usual”, because private agents ignore the effects of their

5.2 Modelling Climate-Economy Interactions

113

own fossil use on current and future outcomes. They have, therefore, no incentives to replace fossil fuels with renewables, and the factor share of fossil energy in the final good production is constant over time for the Cobb-Douglas production function. In contrast, in the case of optimal policy, capital gradually substitutes for fossil energy, because the social planner accounts for climate externalities modelled by the damage function. It is assumed that the planner maximizes a Bergson-Samuelson social welfare function specified as the weighted sum of consumption utilities of all members of current and future generations. The weights (discount factors) represent policy preferences with respect to the welfare of future generations. The rate of welfare discounting fulfils the role of a policy parameter that can notably differ from the usual time preferences of a representative agent. We consider two cases of optimal policy: with a discount rate typical of households’ investment decisions and with Stern discounting based on the principle of intergenerational fairness. The optimal effective carbon budget is found in each case as the steady-state solution for the long-term optimum. The stationarity of the carbon dioxide concentration in the atmosphere means net carbon neutrality, when carbon dioxide emission into and removal from the atmosphere (due to natural sinks such as ocean and terrestrial biosphere) coincide. In broad sense, net greenhouse gas neutrality is defined as “the balance between anthropogenic emissions of greenhouse gases from sources and the removal of such gases by sinks” (e.g. Appunn et al. 2020). The steady-state concentration plays the key role in our model, because it determines all other steady-state variables, including long-term wealth, consumption and factor returns. We use numerical analysis for comparison of outcomes under different rates of social time preferences. To ensure the plausibility of numerical results, parameters of the damage function of our model are calibrated to match the values of the damage function in the last version of the DICE model. A surprising result of this numerical exercise is that the optimal policy with Stern discounting has an insignificant effect on the welfare of distant-future generations as compared to the policy with the usual time preference. In other words, the policy choice based on intergenerational fairness essentially does not matter for the wellbeing of those generations who are supposed to be the main beneficiaries of such a policy. A higher policy preference in favour of future generations implies a more stringent policy to abate emissions incurring higher abatement costs for the economy. Therefore, a potential welfare gain of distant-future generations from intergenerational equity is partially offset by a slowdown of capital accumulation that results from a more stringent climate policy. Optimal climate policy can be implemented via the tax on CO2 emission that represents the social cost of carbon measured in units of real wealth. The optimal rate of carbon tax is, therefore, proportional to the ratio of the shadow price for CO2 concentration (or, equivalently, for the current-period effective carbon budget) to the shadow value of current wealth. This means that the dynamic of carbon tax depends on two factors: cumulative net emission and accumulation of wealth. All other things being equal, a wealthier society, characterized by a lower shadow value of wealth, should pay a higher carbon tax. It is also shown that under climate policy aimed at

114

5 Global Carbon Budgeting and the Social Cost of Carbon

restricting the supply of fossil fuels, a higher carbon tax implies higher excess returns on investment in fossil energy production.

5.3

The Overlapping-Generations Model

Consider a general equilibrium model of the world economy with a simple structure of overlapping generations, each with a lifespan of two periods. Members of each generation are homogeneous, and the population size of each generation is timeconstant and normalized to one. Each individual works in youth and consumes in old age. An individual born in period t  1 supplies a unity of labour to production sectors, receives a wage and accumulates wealth Wt  1 in the first period of life. Wealth is invested into production factors to be used in the next period t. In the second period of life, the individual receives a yield on investment and consumes Ct units of final output of the economy. The utility of an old individual born in period t  1 is the increasing and concave function of consumption, u(Ct). In any period t, global output Yt is distributed between consumption by the old generation and capital accumulation by the young generation: Y t ¼ Ct þ W t :

ð5:1Þ

Production begins in period 1 and the wealth of the initial old generation living in this period is W0. There are two sectors in the economy producing final goods and fossil energy. The final goods sector uses Cobb-Douglas technology with three production factors: labour, capital and energy: Y t ¼ D ðQt Þkαft eδt ,

ð5:2Þ

where labour input is unity, kft is capital employed by the final goods sector, et is fossil energy input, α and δ are the factor elasticities of capital and energy, respectively. The variable Qt denotes CO2 concentration in the atmosphere above a base level, for which the anthropogenic impact on the environment is negligible. The function D ðQt Þ, 0  D ðQt Þ  1, indicates damage to the economy from the excess CO2 concentration. The damage function D ðQt Þ is defined for Qt  0, decreasing, concave and satisfying condition D ð0Þ ¼ 1. Concavity means an increasing marginal effect on the economy of a higher CO2 level. The labour share in the final output is 1  α  δ and the end-of-period wealth of the young generation born in period t is W t ¼ ð1  α  δ ÞY t :

ð5:3Þ

5.3 The Overlapping-Generations Model

115

All labour income is invested in the production capital of the economy, hence the share of investment in final output coincides with the labour share in production. The energy sector uses capital ket as the only factor of production: et ¼ k et :

ð5:4Þ

We treat fossil energy resources as production capital of the energy sector and assume the absence of depletion effects in the development of a resource base for energy production. The total production capital of the economy in period t is created from the wealth of the young generation born in period t  1 and is allocated between the sectors as W t1 ¼ kft þ k et :

ð5:5Þ

Production capital fully depreciates during one period. Consumption by the old generation in period t is Ct ¼ (α + δ )Yt, as follows from Eqs. (5.1) and (5.3). A carbon dioxide emission into the atmosphere is assumed to be proportional to fossil energy production, θet, where θ > 0 is the emission rate. The concentration of CO2 evolves over time as Qt ¼ ð1  μÞQt1 þ θet ,

ð5:6Þ

where μ 2 (0, 1) is the outflow rate—the rate of CO2 net removal from the atmosphere and absorption by ocean and terrestrial biosphere. The change of concentration in period t is equal to emission less the net outflow from the atmosphere. The initial CO2 concentration is Q0  0. Figure 5.4 illustrates Eq. (5.6). It shows the evolution of the balance of carbon dioxide sources and sinks since 1900 presented by the Global Carbon Project. The term θet in the equation corresponds to sources of emission (fossil fuels and land use), the term μQt  1 to sinks (ocean and land) and the change of concentration Qt  Qt  1 to accumulation of CO2 in the atmosphere.

5.3.1

The Damage Function and the Greenhouse Effect

In the Nordhaus model (e.g. Nordhaus 1994, 2014) and other climate-economy models, damage to the global economy is caused by an increase of the mean global temperature in the atmosphere. The temperature increase results from the emission of carbon dioxide that circulates in the environment and causes the greenhouse effect. This effect can be specified for our model with the example of the concentrationdefined damage function that will be used in this chapter:

116

5 Global Carbon Budgeting and the Social Cost of Carbon

Fig. 5.4 Annual balance of carbon dioxide sources and sinks, 1900–2018. Sources: Global Carbon Budget (2019); Fossil fuels and industries: Gilfillan et al. (2019); Land-use change emissions: Houghton and Nassikas (2017), Hansis et al. (2015); Atmospheric CO2 growth: Joos and Spahni (2008), The ocean CO2 sink: DeVries (2014), Khatiwala et al. (2013). The land sink is the average of several dynamic global vegetation models that reproduce the observed mean total land sink of the 1990s

D ðQt Þ ¼ 1  að ln ð1 þ Qt ÞÞρ ,

ð5:7Þ

which is decreasing and concave for a > 0 and ρ  2. The damage function (5.7) can be derived from three relationships describing in a simplified way the causal links between CO2 emission, temperature increase and damage to the economy, and corresponding to three modules of the DICE model depicted in Fig. 5.3. First, damage to the economy can be defined as the function of the mean atmospheric temperature increase ΔTt above the level of a base year, for which damage was absent: ΩðΔT t Þ ¼ 1  b1 ΔT t ρ :

ð5:7aÞ

The temperature-defined damage function Ω(ΔTt) is decreasing, concave and satisfying condition Ω(0) ¼ 1. Similar specifications of damage functions are used in the studies of temperature impact on global output surveyed by Nordhaus and Moffat (2017). Second, the temperature increase is supposed to be proportional to the energy surplus ΔEt caused by the increased radiative forcing,

5.4 The Laissez-Faire Dynamic Equilibrium

ΔT t ¼ b2 ΔEt :

117

ð5:7bÞ

This relationship results from the linear approximation of the dependence between atmospheric temperature increase and anthropogenic radiative forcing (Roe and Baker 2007). We thus abstract away from the diffusive inertia terms that are present in the dynamic equation for atmospheric temperature in the two-level global climate model (Nordhaus and Sztorc 2013, p. 17). Third, the energy surplus is proportional to the logarithm of the ratio of atmospheric CO2 concentration to the base level, ΔEt ¼ b3 ln ð1 þ Qt Þ= ln 2,

ð5:7cÞ

provided that the base-level concentration is normalized to one. We assume that exogenous radiative forcing is absent. The logarithmic dependence between man-made radiative forcing and CO2 concentration is the standard assumption used to describe the greenhouse effect in climate-economy models (e.g. Nordhaus and Sztorc 2013; Nordhaus 2016; Cai et al. 2013; Golosov et al. 2014). Inserting (5.7c) into (5.7b) implies that the product of parameters b2b3 indicates the increase of mean global temperature for CO2 concentration at twice the base level, that is, for Qt ¼ 1. In the long term, such an increase of temperature defines an equilibrium climate sensitivity as the response of the earth’s climate system to a doubling of the concentration of atmospheric carbon dioxide. Combining Eqs. (5.7a)–(5.7c) yields the concentration-defined damage function (5.7) under the following relationship between the model parameters: a ¼ b1(b2b3/ ln 2)ρ. In the general case, the damage function D ðQt Þ results from a representation of the greenhouse effect and global warming similar to the one given by Eqs. (5.7a)– (5.7c).

5.4

The Laissez-Faire Dynamic Equilibrium

In the absence of climate policy, competitive producers do not take into account the damage to the economy from fossil energy use. Let pt denote the real energy price and Rt the real rate of interest. Production firms maximize profits in the final good sector: π tf ¼ Y t  Rt kft  pt et

ð5:8Þ

π et ¼ pt et  Rt k et :

ð5:9Þ

and in the energy sector:

The first-order conditions for profit maximization are

118

5 Global Carbon Budgeting and the Social Cost of Carbon δ αD ðQt Þk α1 ft et ¼ Rt

δD ðQt Þk αft eδ1 ¼ pt t for the final good sector and pt ¼ Rt for the energy sector due to (5.4). Combining these conditions implies the allocation of wealth between the sectors in proportion to factor elasticities: kft/ket ¼ α/δ or, from (5.5): kft ¼

αW t1 δW t1 , ket ¼ et ¼ : αþδ αþδ

ð5:10Þ

The allocation of wealth is determined by the shares of inputs in the final good production and is not influenced by damage from carbon dioxide emission. Inserting (5.10) into (5.2) yields final output as the function of the beginning-of-period wealth: Y t ¼ D ðQt ÞAW αþδ t1 ,

ð5:11Þ

where A¼

αα δδ : ðα þ δÞαþδ

Consequently, we can rewrite the equations for accumulation of wealth (5.3) and CO2 concentration (5.6) as W t ¼ ð1  α  δ ÞD ðQt ÞAW αþδ t1 Qt ¼ ð1  μÞQt1 þ θ

δW t1 : αþδ

ð5:12Þ ð5:13Þ

This system of difference equations determines a dynamic equilibrium path of wealth and CO2 level for the initial conditions W0 and Q0.

5.4.1

The Long-Term Equilibrium

The steady state of system (5.12), (5.13) satisfies conditions: Wt ¼ Wt  1 and Qt ¼ Qt  1. Equation (5.13) implies the steady-state link between wealth and CO2 concentration:

5.4 The Laissez-Faire Dynamic Equilibrium

119

Fig. 5.5 The long-term equilibrium

W¼

μðα þ δÞ Q: θδ

ð5:14Þ

Here and henceforth, we omit the time subscript in the notation of steady-state variables. Inserting (5.14) into the dynamic equation for wealth (5.12) and rearranging terms yields the equation for steady-state concentration: ðBQÞ1αδ ¼ D ðQÞ, ð1  α  δ ÞA

ð5:15Þ

where B¼

μðα þ δÞ : θδ

The left-hand side of Eq. (5.15) is increasing in Q and the right-hand side is decreasing, hence this equation has the unique solution Q shown in Fig. 5.5. One can see from this figure and (5.15) that the long-term concentration Q is increasing with the emission rate θ and decreasing with the outflow rate μ.

5.4.2

Equilibrium Transition Paths

In Appendix A.1 we consider the backward-dynamic system corresponding to (5.12), (5.13) and mapping the state variables in period t into the state variables in period t  1. We show that the characteristic equation for the steady state of the backward-dynamic system is given by the quadratic polynomial:

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5 Global Carbon Budgeting and the Social Cost of Carbon

Fig. 5.6 Characteristic roots for the backward-dynamic system

Fig. 5.7 Transition paths in the plane of state variables Wt and Qt

ð1  μÞðα þ δÞη2  ð1 þ α þ δ  ð1 þ εðQ ÞÞμÞη þ 1 ¼ 0,

ð5:16Þ

where η is the characteristic root and εðQÞ ¼ D 0 ðQÞQ=D ðQÞ > 0 is the elasticity of damage function. The characteristic Eq. (5.16) is portrayed in Fig. 5.6. It is shown in Appendix A.1 that this equation has two real roots η1 and η2 that are positive and above one, if ε(Q) is sufficiently small. Under this condition, the steady state of the backward-dynamic system is an unstable node implying that the same steady state for the forwarddynamic system (5.12), (5.13) is a stable node. Figure 5.7 shows the state plane and equilibrium paths depicted with continuous curved arrows for different initial conditions W0 and Q0. Equilibrium paths converge to the stationary point S ¼ (W, Q), where W ¼ BQ, as follows from (5.14). The

5.5 The Social Planner’s Problem

121

stationary point is the intersection of locus LQ where Qt ¼ Qt  1 with locus LW where Wt ¼ Wt  1. Locus LQ is given by Eq. (5.14) and locus Lw satisfies Eq. (5.15), which can be rewritten as W 1αδ ¼ ð1  α  δ ÞAD ðQÞ: Equilibrium paths in Fig. 5.7 demonstrate various time profiles of CO2 concentration and wealth depending on the initial conditions. Path 1 demonstrates growth of both state variables, because this path is located below loci LQ and LW. Path 2 is characterized by a low initial wealth and a high initial CO2 concentration, which decreases in the early phase of transition (since the outflow of CO2 from the atmosphere exceeds emission) and then increases, whereas wealth increases. Path 3 starts at the initial point with a high level of wealth, which increases during the early phase and then decreases, whereas the CO2 level for this path increases. The long-term equilibrium wealth is positive, because the outflow rate is positive. For μ ¼ 0, the stationary state of dynamic system (5.12), (5.13) would be W ¼ 0, Q ¼ 1. In other words, in the absence of carbon dioxide outflow from the atmosphere, the economy would degenerate in the long term to a state with zero wealth and 100% damage from carbon dioxide emission.

5.5

The Social Planner’s Problem

Suppose that a social planner chooses the optimal path of resource allocation and production by accounting for the external effects of carbon dioxide emission on final output. The optimal structure of production is determined by the allocation of wealth between the sectors of final good and fossil energy, which is similar to (5.10): kft ¼ λt W t1 , k et ¼ et ¼ ð1  λt ÞW t1 :

ð5:17Þ

The weight λt is the share of the beginning-of-period wealth invested in the final good sector. For the laissez-faire equilibrium this share is λt ¼ α/(α + δ). We can represent the final output as a function of the beginning-of-period wealth similar to (5.11): Y t ¼ D ðQt ÞAðλt ÞW αþδ t1 , where Aðλt Þ ¼ λαt ð1  λt Þδ :

ð5:18Þ

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5 Global Carbon Budgeting and the Social Cost of Carbon

The social planner uses the discount factor β < 1 to weight welfare (utility functions) of current and future generations. The planner chooses the sequence of wealth allocation λt to maximize the integral discounted welfare of all generations: max

1 X

λt

βt uðC t Þ,

ð5:19Þ

t¼1

subject to the condition of final output distribution in any period: C t ¼ ðα þ δÞY t

ð5:20Þ

and the dynamic state Eqs. (5.12), (5.13) rewritten as: W t ¼ ð1  α  δ ÞY t

ð5:21Þ

Qt ¼ ð1  μÞQt1 þ θð1  λt ÞW t1 :

ð5:22Þ

Equation (5.20) follows from Eqs. (5.1) and (5.3) and means that the old generation in period t consumes the product of non-labour factors of final output. The state Eq. (5.21) expresses the dynamic of wealth and (5.22) is the equation for carbon dioxide accumulation in the atmosphere. Combining Eqs. (5.20) and (5.21) implies a link between consumption and end-of-period wealth: Ct ¼ ρW t ,

ð5:23Þ

where ρ¼

αþδ 1αδ

is the relative weight of non-labour factors in the final output. Let ψ t  0 denote the shadow value of wealth and vt  0 the shadow value of CO2 removal from the atmosphere indicating the marginal value of the effective, that is concentration-defined, carbon budget remaining in period t. This budget is equal to Qopt  Qt, with Qopt denoting the long-term optimal concentration that will be determined below. We derive in Appendix A.2 the first-order condition for optimal allocation of capital between the final good sector and the energy sector: ψ t ln0 Aðλt ÞW t ¼ vt θW t1

ð5:24Þ

vt1 ¼ ψ t1 W t1 ln 0 D ðQt1 Þ þ βð1  μÞvt

ð5:25Þ

and the costate equations:

5.5 The Social Planner’s Problem

123

ψ t1 ¼ ρu0 ðρW t1 Þ þ βð1  α  δ ÞRt ψ t ,

ð5:26Þ

where, as above, Rt ¼ ∂Yt/∂kft is the rate of return on capital in the final good production, which is equal to the real interest rate. The first-order condition (5.24) establishes a balance between the marginal contribution of energy to wealth accumulation and the marginal external effect of CO2 emission. This condition implies a link in any time period between the marginal value of wealth ψ t and the marginal value of the effective carbon budget vt. The costate equation for CO2 concentration (5.25) implies that the marginal carbon budget value is equal to the marginal welfare gain from reducing CO2 emission plus the discounted next-period marginal carbon budget value. The discount factor β(1  μ) equals the social planner’s discount factor adjusted for the outflow rate μ. The higher this rate, the faster carbon dioxide is removed from the atmosphere and the lower is the marginal effect of current emission on future damage. The costate equation for wealth (5.26) means that the marginal value of wealth is the sum of the marginal welfare gain of the current old generation and the discounted marginal value of wealth of future generations. The time-varying discount factor β(1  α  δ )Rt is the product of the social planner’s discount factor, the labour share of output and the rate of return on capital.

5.5.1

The Long-Term Optimum

The system of Eqs. (5.21), (5.22), (5.24)–(5.26) determines the dynamic of five variables: Wt, Qt, λt, vt and ψ t. Consider the steady-state solution for the planner’s problem. Equation (5.22) implies the steady-state link between wealth and CO2 concentration: W¼

μQ , θ ð1  λ Þ

ð5:27Þ

which is similar to (5.14). For the steady state, the first-order condition (5.24) is written as ψln0 AðλÞ ¼ vθ

ð5:28Þ

and the costate equation for CO2 accumulation (5.25) is represented as ð1  βð1  μÞÞv ¼ ψWln0 D ðQÞ:

ð5:29Þ

Combining Eqs. (5.27)–(5.29) yields the steady-state rule for wealth allocation:

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5 Global Carbon Budgeting and the Social Cost of Carbon

Fig. 5.8 The long-term CO2 concentration

λðQ Þ ¼

α , α þ δ  ð1  βð1  μÞÞ1 μεðQÞ

ð5:30Þ

as shown in Appendix A.3. The steady-state share of non-fossil investment λ(Q) is increasing in Q (since ε0(Q) > 0) and exceeds this share for the laissez-faire equilibrium: λðQ Þ >

α : αþδ

The share of fossil energy investment 1  λ(Q) is decreasing with the social planner’s discount factor β, which means that a higher preference weight of future generations implies a lower input of fossil energy in production. Inserting λ(Q) into the steady-state equation for wealth, as implied by (5.18) and (5.21), and rearranging terms yields the equation for steady-state CO2 concentration: ðBðQÞQÞ1αδ ¼ D ðQÞ, ð1  α  δ ÞAðλðQÞÞ

ð5:31Þ

where BðQÞ ¼

μ : θð1  λðQÞÞ

Equation (5.31) is similar to (5.15) and determines the long-term optimal solution Qopt shown in Fig. 5.8. It is unique because the right-hand side of (5.31) is decreasing in Q and the left-hand side is increasing, because λ0(Q) > 0, B0(Q) > 0 and A0(λ) < 0. The latter inequality follows from the first-order condition (5.24).

5.6 Numerical Analysis

125

As was mentioned in the introduction to this chapter, the economic meaning of carbon budget is similar to the concept of economically recoverable resource examined in Chap. 3 of this book. One can see from Fig. 5.8 that the initial optimal carbon budget Qopt  Q0 is below the initial laissez-faire budget Q  Q0. The optimal steady-state concentration Qopt is increasing with the emission rate θ and decreasing with the discount factor β. The latter follows from the fact that the lefthand side of (5.31) is increasing with λ(Q), which is increasing with β.

5.6

Numerical Analysis

Consider a numerical example to compare the long-term outcomes under laissezfaire equilibrium and the optimal programme under different policy scenarios. We will use the example of damage function (5.7) that we rewrite for the sake of convenience: D ðQÞ ¼ 1  að ln ð1 þ QÞÞρ : The values of the model parameters are the following. The factor elasticities of final output are α ¼ 0.50, δ ¼ 0.05; the rates of CO2 outflow and emission are μ ¼ 0.28, θ ¼ 40; parameters of the damage functions D ðQÞand Ω(ΔT ) ¼ 1  b1ΔTtρ are: a ¼ 0.044, ρ ¼ 2.018 and b1 ¼ 0.0023. The choice of parameters of the production function δ ¼ 0.05, α ¼ 0.50 reflect the standard estimates of fossil energy contribution to final output and the long-term tendency of a declining labour share in global production. We assume that the calendar period corresponding to one period in our model, half of a generational lifetime, amounts to 25 years. The outflow rate μ ¼ 0.28 corresponds to the annual rate 1.3% of CO2 absorption by ocean and biosphere since μ ¼ 1  (1  0.013)25 ¼ 0.28 for the 25-year period. The emission rate θ ¼ 40 is a calibration parameter. Box 5.2 explains the calibration of parameters for the damage functions D ðQÞ and Ω(ΔT ). Box 5.2 Parameters of the Damage Functions Parameters of functions D ðQÞ and Ω(ΔT ) have been selected to match the estimates of the long-term climate sensitivity and the damage function used in DICE models. First, the mean global temperature increase for Q ¼ 1, that is, the doubled base level of CO2 concentration, is supposed to be equal to
3 C. This is the mean of the assessed likely range of equilibrium climate
sensitivity between 1.5 and 4.5 C (Intergovernmental Panel on Climate Change 2014, p. 43). Second, parameters of damage functions D ðQÞ and Ω(ΔT ) should correspond to the core estimates of damage in the 2016 DICE version: 2.1% of (continued)

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5 Global Carbon Budgeting and the Social Cost of Carbon

Box 5.2 (continued)

global income at 3 C global warming and 8.5% of income at 6 C warming (Nordhaus, 2016, p. 13). Hence, a damage of 2.1% is expected for the doubled base level of CO2 concentration. Thus, we have three conditions on the parameters of damage functions D ðQÞ and Ω(ΔT ) that can be written as D ð1Þ ¼ Ωð3Þ ¼ 0:979 Ωð6Þ ¼ 0:915 From (5.7) and (5.7a), this is equivalent to að ln 2Þρ ¼ b1 3ρ ¼ 0:021 b1 6ρ ¼ 0:085 These three equations are fulfilled for a ¼ 0.044, ρ ¼ 2.018 and b1 ¼ 0.0023. For the selected set of parameters, the long-term equilibrium CO2 concentration under laissez-faire is the solution of Eq. (5.15): Q ¼ 1.41. The damage function for this level of concentration equals D ðQ Þ ¼ 0:966 implying that the long-term damage from carbonization of the atmosphere with no climate policy is 3.4% of GDP. The temperature increase for this level of damage, as
implied by the damage function Ω(ΔT ), is 3.8 C. The share of investment in fossil energy in the absence of climate policy is δ/(α + δ) ¼ 9.1%. Consider the optimal problem for two rates of social time preference: (a) 1.5% per year, which was used by Nordhaus in the baseline scenario of optimal climate policy and which is close to the rate of individual welfare discounting by households; and (b) 0.1% per year, as advocated by Stern in his Review (2007). For the 25-year calendar period, the discount factors are: β ¼ 1.01525 ¼ 0.689 for Nordhaus discounting and β ¼ 1.00125 ¼ 0.975 for Stern discounting. The long-term optimal CO2 concentration is the solution of steady-state Eq. (5.31). For the annual discount rate 1.5% we obtain Qopt ¼ 0.98 and the value of damage: D ðQopt Þ ¼ 0:980


The long-term damage is 2.0%, implying a temperature increase of 2.95 C for this level of damage.

5.6 Numerical Analysis Table 5.1 The long-term optimal solution

127 Discount rate 0.015 0.001

Qopt 0.98 0.82

λ(Qopt) 0.938 0.948

ε(Qopt) 0.030 0.024

D ðQopt Þ 0.980 0.984

Table 5.2 Indicators of long-term climate change in various model scenarios

Warming, degrees Celsius Damage, percent

Laissezfaire 3.8

Mean climate sensitivity 3.0

Nordhaus discounting 2.95

Stern discounting 2.57

Temperature target 2.0

3.4

2.1

2.0

1.6

0.9

For the annual discount rate of 0.1% the long-term CO2 level is Qopt ¼ 0.81 and the damage function is D ðQopt Þ ¼ 0:984


The long-term damage is 1.6% and the temperature increase is 2.57 C. It is important to note that the absolute level of pre-industrial CO2 concentration is 285 ppm. For this base level, we can obtain the long-term absolute concentration: 285 ∙ 1.98 ¼ 564 ppm for a discount rate of 1.5% and 285 ∙ 1.81 ¼ 516 ppm for a discount rate of 0.1%. For comparison, the level of concentration 450 ppm or, in relative terms, 1 + Q450 ¼ 450/285 ¼ 1.58, was adopted by the International Energy Agency in 2011 as the long-term goal of energy policy ensuring achievement of the
temperature target of 2.0 C . The IEA’s climate policy scenario specifying the assumptions for this outcome was called “450 Scenario” (International Energy Agency 2011). Table 5.1 summarizes the long-term optimal solution for our model for two discount rates. The share of investment in the energy sector 1  λ(Qopt) is 6.2% under the discount rate 0.015 and 5.2% under Stern discounting. The long-term damage function elasticity decreases with the lowering of the discount rate. As a result, the optimal policy for our numerical example provides a reduction of damage from 3.4 to 2.0% under welfare discounting with an annual rate of 1.5%. Under near-zero discounting the optimal policy gives a further reduction of damage to 1.6%. The long-term CO2 levels Qopt ¼ 0.98 and Qopt ¼ 0.82 are below the benchmark of the doubled base level Q ¼ 1. The long-term indicators of climate change are presented in Table 5.2. It shows the temperature increase and economic damage under laissez-faire, for the mean
level of climate sensitivity of 3.0 C, for the optimal policy with Nordhaus and Stern
discount rates, and for the long-term temperature target of 2.0 C. In the latter case, the estimate of damage is 0.9%, because Ω(2) ¼ 0.991. This coincides, for the selected set of model parameters, with the estimate   of damage for the targeted level of concentration for the “450 Scenario”: D Q450 ¼ D ð0:58Þ ¼ 0:991.
The long-term temperature increase under Stern discounting is 2.57 C for our
model. This is close to the temperature asymptote 2.5 C under Stern discounting

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5 Global Carbon Budgeting and the Social Cost of Carbon


calculated for the DICE-2016R model, which exceeds significantly the 2.0 C target.
As Nordhaus recognizes, “the international goal of 2.0 C is not feasible with current DICE estimates without technologies that allow negative emissions by the midtwenty-first century” (Nordhaus 2017, p. 5). A surprising result of our model is that climate policy with Stern discounting leads to an outcome for distant-future generations that is nearly the same as with Nordhaus discounting. One can compare the levels of long-term consumption, which is proportional to the long-term wealth, as follows from Eq. (5.23). From (5.14), (5.27), the ratio of wealth under optimal policy and laissez-faire is Qopt ð1  λ Þ W opt ,  ¼  W Q ð1  λðQopt ÞÞ where 1  λ ¼ δ/(α + δ) is the share of investment in fossil energy in the absence of climate policy. We have it that Q ¼ 1.41, λ ¼ 0.091 and obtain, by using a finer approximation of Qopt than those in Table 5.1: W opt =W  ¼ 1:021 for a discount rate of 0.015 and W opt =W  ¼ 1:024 for a discount rate of 0.001. The optimal policy with Nordhaus discounting leads to an increase of long-term wealth and consumption by 2.1% as compared to a no-policy scenario, while nearzero discounting results in an increase of only 2.4%. The principle of intergenerational equity underlying Stern discounting implies higher welfare weights for distant-future generations. For example, for time period T ¼ 10 the discount factors are βT ¼ 0.68910 ¼ 0.02 under Nordhaus discounting and βT ¼ 0.97510 ¼ 0.77 under Stern discounting. It is somewhat surprising that such a dramatic difference in welfare weights essentially does not matter for the size of wealth of distant-future generations (their relative gain from the climate policy fairness is 0.3% in our example). The reason for such an outcome is that a potential welfare gain of these generations from intergenerational equity is largely offset by the slowing of capital accumulation by previous generations. This slowing results from a more stringent policy aimed at reducing the fossil energy supply.

5.7 The Social Cost of Carbon

5.7

129

The Social Cost of Carbon

Optimal climate-economy models are used to determine time paths of the social cost of carbon. This cost indicates the marginal welfare loss from carbon emission, which equals the present value of future damage stream caused by emission in the current period. The social cost of carbon coincides with the optimal carbon tax if there is no interaction between climate policy and other market failures. The carbon tax is used in practice as an instrument for climate policy implementation. The rate of this tax is measured in real-term currency units per metric ton of carbon or CO2 or, in terms of our model, in units of final output per unit of emission. To calculate the optimal tax rate, let us return to the equilibrium model of Sect. 5.3. Suppose that in any period t the final good producers pay a carbon tax, which is transferred as a lump sum to currently living generations. At the beginning of this period, a member of the old generation allocates wealth between production sectors to maximize the total profit defined from (5.8) and (5.9) as   π tf þ π et ¼ Y t  Rt kft þ ket  τt θet ,

ð5:32Þ

where τt denotes the rate of tax on CO2 emission in period t. Remember that the energy production function is given by et ¼ ket, the capital allocation is defined in terms of shares of wealth: kft ¼ λt W t1 , k et ¼ ð1  λt ÞW t1 and the final output is Y t ¼ D ðQt ÞAðλt ÞW αþδ t1 . Inserting these into the objective function (5.32) implies profit maximization via the decentralized choice of λt:   max D ðQt ÞAðλt ÞW αþδ t1  Rt W t1  τt θð1  λt ÞW t1 : λt

ð5:33Þ

The first-order condition is DðQt ÞA0 ðλt ÞW αþδ t1 ¼ τt θW t1 : Due to the equation for wealth accumulation (5.3) and the final output Eq. (5.18), it transforms to ln0 Aðλt ÞW t ¼ ð1  α  δ Þτt θW t1 :

ð5:34Þ

The first-order condition for profit maximization coincides with the social planner’s first-order condition (5.24) if the rate of carbon tax is equal to

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5 Global Carbon Budgeting and the Social Cost of Carbon

τt ¼

vt : ψ t ð1  α  δ Þ

ð5:35Þ

The optimal tax rate is proportional to the ratio of shadow values of the social planner’s problem, vt/ψ t. The numerator is the marginal value of the carbon budget, while the denominator is the marginal value of wealth multiplied by the labour share in production. The latter coincides with the ratio of the end-of-period wealth to final output (5.3), hence the carbon tax is measured in units of final output, according to (5.32).

5.7.1

The Long-Term Carbon Tax

It is straightforward to calculate the long-term carbon tax by using the conditions of long-term optimum. The steady-state ratio of shadow values is found from Eq. (5.29): Wln0 D ðQÞ v ¼ : ψ 1  β ð1  μ Þ Substituting (5.27) for W in this formula implies: μεðQÞ v ¼ : ψ θð1  λÞð1  βð1  μÞÞ Hence, the long-term optimal carbon tax rate (5.35) equals τopt ¼

μεðQopt Þ : θð1  λðQ ÞÞð1  βð1  μÞÞð1  α  δ Þ opt

ð5:35aÞ

It should come as no surprise that the optimal tax increases with the outflow rate μ and decreases with the emission rate θ. This is a consequence of the steady-state proportional link between wealth and concentration (5.27). The optimal tax is the inverse of the marginal value of wealth, hence it increases with the steady-state wealth, which increases with the ratio of parameters μ/θ due to (5.27). Using the simple formula (5.35a), we can calculate the steady-state tax rates for the annual rates of social time preference 0.015 and 0.001. From Table 1 we have it that λ(Qopt) ¼ 0.938, ε(Qopt) ¼ 0.03 for the discount rate 0.015. The optimal tax rate (5.35a) in this case is

5.7 The Social Cost of Carbon Table 5.3 Global social cost of carbon in DICE-2013R model, 2005 U.S. dollars per ton

131 Discount rate 0.015 0.001 The ratio of costs

2020 21.2 25.0 1.18

2025 25.0 30.1 1.20

2030 29.3 35.9 1.23

2050 51.5 66.9 1.30

Source: Nordhaus (2014, p. 288)

τopt ð0:015Þ ¼

0:28  0:03 40  ð1  0:938Þ  ð1  0:689  ð1  0:28ÞÞ  ð1  0:50  0:05 Þ

¼ 0:015: Similarly, we calculate the optimal tax rate for the discount rate 0.001 as τopt(0.001) ¼ 0.024. The absolute value of the optimal tax rate depends on the choice of units of measurement for CO2 level, but the ratio of tax rates does not: τopt ð0:001Þ ¼ 1:60: τopt ð0:015Þ It is interesting to compare this ratio with estimates of the global social cost of carbon for the DICE-2013R model presented in Table 5.3. It shows the absolute and relative costs for the welfare discount rates 0.015 and 0.001. In the latter case, a recalibration of the DICE model was used to maintain a baseline real return on capital simulated for the first 30 years (Nordhaus 2014, p. 287). The estimate of the long-term ratio of carbon costs for the discount rates 0.001 and 0.015 in our numerical example is 1.6. It exceeds the same ratios presented in Table 5.3 and varying in the range 1.18–1.30. Nonetheless, the DICE model ratios increase over time and come closer to our estimate.

5.7.2

The Social Cost of Carbon and Excess Returns

The social cost of carbon is internalized in factor returns due to the carbon tax. To demonstrate this effect, consider the profit-maximization problem (5.33) and the first-order-condition (5.34), which is rewritten as ln0 Aðλt ÞY t =W t1 ¼ θτt

ð5:34aÞ

due to the equation of wealth accumulation (5.3). We have it that ln0 Aðλt Þ ¼

δ α  ð1  λt Þ λt

and the left-hand side of (5.34a) equals the difference of returns: Ret  Rt , where Ret ¼ ∂Y t =∂k et ¼ δY t =ðð1  λt ÞW t1 Þ is the return on the fossil energy capital in

132

5 Global Carbon Budgeting and the Social Cost of Carbon

Table 5.4 The long-term real interest rate and the return on fossil energy capital

Discount rate 0.015 0.001

R 1.185 1.176

1+r 1.0068 1.0065

Re 1.786 2.136

1 + re 1.023 1.039

re  r 0.016 0.032

production and Rt ¼ ∂Yt/∂kft ¼ αYt/(λtWt  1) is the return on the final good capital, which is equal to the real interest rate in the model. Condition (5.34) is represented as Ret  Rt ¼ θτt :

ð5:36Þ

As a result, the excess return on fossil energy is proportional to the carbon tax rate. The higher this rate or CO2 emission rate, the more significant is the excess return. One can calculate the long-term rates of return for our numerical example using the results in Table 5.1 and Eq. (5.27) for long-term wealth. The long-term real interest rate is calculated as R ¼ αD ðQÞAðλÞW αþδ1 =λ and the annualized real interest rate is 1 + r ¼ R1/25. The return on fossil energy capital Re is found from Eq. (5.36). The calculation results are given in Table 5.4 for the rates of social time preference 0.015 and 0.001. The last two columns represent the annual return on energy capital 1 + re ¼ (Re)1/25 and the annual excess return re  r. The steady-state real interest rate is 0.65–0.68% per year. The rates of return on fossil energy investment are 2.3 and 3.9% per year. The annual excess return to this investment is 1.6% for a welfare discount rate of 0.015 and 3.2% for a discount rate of 0.001. The excess return on fossil energy capital results from the reduction of this energy supply as compared to the laissez-faire equilibrium due to internalization of the social cost of carbon by market participants paying the carbon tax. In other words, the rate of excess return on fossil energy indicates the cost of climate abatement policy for the economy, similarly to the carbon tax rate. Both rates are increasing with the welfare weight of future generations.

5.8

Concluding Remarks

As was mentioned at the beginning of this chapter, there is an analogy between the notions of economically recoverable resource and carbon budget, on the one hand, and between resource rent and the social cost of carbon, on the other. Rental relations arise if natural resources in the ground are in private or public ownership. One can suggest that the owner of the global atmospheric resource is humanity, including current and future generations. The model’s social planner is a proxy setting a “rent” in the form of a carbon tax paid for wasting the atmospheric resource with emissions of greenhouse gases. The current generation pays such a rent that can be transferred to members of the same generation to subsidize development of alternative energy or it can be redistributed among generations by means of environmental finance.

5.8 Concluding Remarks

133

Consideration of these issues requires analysis of a fully fledged dynamic model of climate policy, which is beyond our consideration. The model of this chapter also does not account for global dynamic factors such as population growth and technical change. It is possible to adjust the model to include these two factors, which have opposite influences on carbon dioxide emission (if technical change leads to a decreasing cost of renewables) and can largely offset each other. The model can be modified to account for the uncertainty of climate change by randomizing parameter a of the damage function (5.7) or parameter μ of the equation for concentration dynamic (5.6). The uncertainty of parameter a reflects incomplete scientific knowledge about climate sensitivity and damage to the economy from increased global temperatures. Deep structural uncertainties about “the unknown unknowns” arise due to the positive climate-carbon feedback with a selfamplification potential of greenhouse warming and the threats of abrupt changes of climate. The issues of climate policy uncertainties include the consequences of fat-tailed probabilities of disastrous changes for the analysis of climate change (e.g. Weitzman 2011) and policy choices under the ambiguity of climate-economy models resulting from the incompleteness of scientific knowledge about climate and socio-economic consequences of climate change (e.g. Heal and Millner 2017). Nevertheless, deterministic models of climate change and climate-economy interplay (such as DICE) are widely applied for climate policy analysis. Parametric uncertainty in these models is taken into account by using probabilistic estimates for ensembles of Monte-Carlo simulation runs. Although we have not considered here transition paths of the optimal problem, one can make important inferences about the goals of climate policy from analysis of the long-term optimum. In our model, this is a steady state of net carbon neutrality defined by the balance between emissions of carbon dioxide from sources and its removal by sinks. Such a state is relevant not only to welfare of distant-future generations in dynamic climate-economy models, but also to ambitious programmes elaborated recently by some countries. For example, the European Green Deal, a set of policy initiatives adopted by the European Commission in December 2019, aims to achieve the state of net greenhouse gas neutrality by 2050. Germany, one of the leading countries in tackling climate change, plans to reach net-zero emissions by that year through expansion of renewable power and development of hydrogen economy. Such programmes are ambitious because of a very short period of transition assumed in the European Green Deal and similar programmes. We have to admit, however, that transition of the global economy to a state of net carbon neutrality will take a longer time. We tried to show that carbon budgets can play a central role in the interaction between economy and climate underlying such transition. Our focus was on the effective, that is, concentration-defined, carbon budgets rather than on the conventional ones defined as available cumulative emissions such as those demonstrated in Fig. 5.2. Effective carbon budgets matter, because it is the concentration of carbon dioxide in the atmosphere, rather than cumulative emissions per se, that causes the greenhouse effect. An increase of concentration in any period results from the

134

5 Global Carbon Budgeting and the Social Cost of Carbon

circulation of CO2 in the atmosphere (corresponding to the carbon cycle module in Fig. 5.3) and is given by emission net of cumulative CO2 uptakes by ocean and biosphere. Calculation of the effective carbon budgets as cumulative net emission dramatically differs from the conventional carbon budgeting based on primary emissions. For example, between 2007 and 2018 the annual average emission of carbon dioxide from burning fossil fuels and land use was 39.7 billion tons, while the emission net of uptakes was only 17.3 billion tons on average, less than half of the primary emission (Global Carbon Budget 2018). This difference is important for quantification of strategic goals of climate policy. The conventional carbon budgeting presented by diagram in Fig. 5.2 implies zeroemission commitments for some dates in the future. For example, the projected level
of zero emission for the temperature target of 1.7 C is reached by 2080 (Global Carbon Budget 2018). The zero-emission commitment for some future date implies negative net emission afterwards because of the carbon uptakes by ocean and biosphere. This, in turn, implies an implicit commitment of zero human-induced atmospheric concentration of carbon dioxide. Such an outcome seems very attractive as the ultimate goal of long-term climate policy, but may be questionable in the context of economy-climate interactions and intergenerational welfare maximization. In the case of our model, the optimal human-induced concentration is positive in the long term, hence any commitment of zero concentration is suboptimal. Varying welfare discounting in climate-economy models allows for comparison of outcomes, including pathways of carbon tax, CO2 concentration, damage to the economy, etc. under different policy scenarios. However, the integral welfare functions corresponding to different intergenerational preferences are incomparable in absolute values because of different welfare weighting of generations. Nevertheless, welfare comparisons in this context make sense with regard to distant-future generations. Such a comparison is based on conditions of the long-term optimum and is of practical interest, because the welfare of these generations is the primary objective of the “fair policy” scenarios with near-zero discount rates. Yet, as it turns out for the numerical analysis of this chapter, it is quite possible that the welfare gain of future generations from fairness is insignificant. The choice of discount rate can exert a significant influence on the carbon tax dynamic (as one can see from Table 5.3), but it does not matter much to those who are supposed to benefit most of all from the intergenerational equity principle.

Appendices A.1 Characteristic Eq. (5.16) The backward-dynamic system corresponding to (5.12), (5.13) is

Appendices

135



1 αþδ Wt ð1  α  δ ÞAD ðQt Þ   θδW t1 ¼ ð1  μÞ1 Qt  : αþδ

W t1 ¼

ð5:37Þ

Qt1

ð5:38Þ

Take the derivatives:

∂W t1 ∂Qt

1  αþδ ∂W t1 W t 1 Wt 1 W ¼ ¼  t1 α þ δ Wt α þ δ ð1  α  δ ÞADðQt Þ ∂W t 1  αþδ 1 DðQt Þαþδ1 0 D 0 ðQt Þ Wt 1  ¼ ¼ D ðQt Þ  W t1 α þ δ DðQt Þ αþδ ð1  α  δ ÞA

∂Qt1 θδ ∂W t1 ¼ ð1  μÞ1  α þ δ ∂W t ∂W t   ∂Qt1 θδ ∂W t1 1  ¼ ð1  μÞ 1 : α þ δ ∂Qt ∂Qt For the steady state, Wt ¼ Wt  1 ¼ W, Qt ¼ Qt  1 ¼ Q and, from (5.14): W ¼

μðα þ δÞ  Q : θδ

Hence, for the steady state we have: ∂W t1 1 ¼ αþδ ∂W t D 0 ðQ Þ ∂W t1 1 μ D 0 ðQ ÞQ μεðQ Þ  ¼ ¼     W ¼ θδ α þ δ DðQ Þ θδ DðQ Þ ∂Qt ∂Qt1 θδ ¼ ∂W t ð1  μÞðα þ δÞ2   μεðQ Þ ∂Qt1 1 : ¼ ð1  μÞ 1 αþδ ∂Qt The characteristic equation for the backward-dynamic system (5.37), (5.38) is

136

5 Global Carbon Budgeting and the Social Cost of Carbon

  1  η  α þ δ   θδ   ð1  μÞðα þ δÞ2 ¼ η2 

  μεðQ Þ   θδ      με ð Q Þ 1  η  ð1  μ Þ 1 αþδ

1  μ þ α þ δ  μεðQ Þ 1 ¼0 ηþ ð1  μÞðα þ δÞ ð1  μÞðα þ δÞ

or ð1  μÞðα þ δÞη2  ð1 þ α þ δ  ð1 þ εðQ ÞÞμÞη þ 1 ¼ 0:

ð5:39Þ

This equation has two positive real roots if ð1  μ þ α þ δ  εðQ ÞμÞ2 > 4ð1  μÞðα þ δÞ, which is the case for small ε(Q). The left-hand side of (5.39) is positive for η ¼ 1, because ð1  μÞðα þ δÞ > α þ δ  ð1 þ εðQ ÞÞμ (since α + δ < 1 + ε(Q)) implying that the minimal real root of (5.39) is above unity. Consequently, both characteristic roots η1 and η2 depicted in Fig. 5.6 are above one, implying that the steady state of the backward-dynamic system (5.37), (5.38) is an unstable node.

A.2 The First-Order Conditions (5.24)–(5.26) The Lagrangian for the social planner’s problem (5.18)–(5.22) is L¼

1 X    βt uðρW t Þ þ ψ t ð1  α  δ ÞD ðQt ÞAðλt ÞW αþδ t1  W t t¼0

þvt ðQt  ð1  μÞQt1  θð1  λt ÞW t1 Þ,

ð5:40Þ

where we have used equations for output (5.18) and consumption (5.23). Differentiating with respect to λt implies ψ t ð1  α  δ ÞDðQt ÞA0 ðλt ÞW αþδ t1 ¼ vt θW t1 : Due to (5.18), (5.21), the left-hand side of this equation can be written as ψ tln0A (λt)Wt, implying (5.24). Differentiating (5.40) with respect to Qt and taking into account (5.8), (5.21) yields the costate equation:

Appendices

137

vt ¼ ψ t W t ln0 D ðQt Þ þ βð1  μÞvtþ1 , which is equivalent to (5.25). Consider the costate Eq. (5.26). Differentiating (5.40) with respect to Wt and taking into account (5.18), (5.21) gives: ρu0 ðρW t Þ  ψ t þ βψ tþ1 ðα þ δ Þ

W tþ1  βθvtþ1 ð1  λtþ1 Þ ¼ 0: Wt

ð5:41Þ

From the first-order condition (5.24), we have: θvtþ1 ¼ ψ tþ1 ln0 Aðλtþ1 Þ

W tþ1 : Wt

Inserting this into (5.41) implies ρu0 ðρW t Þ  ψ t þ βψ tþ1

W tþ1 ðα þ δ þ ð1  λtþ1 Þln0 Aðλtþ1 ÞÞ ¼ 0: Wt

We have it that ln0 Aðλtþ1 Þ ¼

α δ  , λtþ1 ð1  λtþ1 Þ

hence, α þ δ þ ð1  λtþ1 Þln0 Aðλtþ1 Þ ¼ α þ α

1  λtþ1 α ¼ : λtþ1 λtþ1

We can rewrite (5.42) as ρu0 ðρW t Þ  ψ t þ βψ tþ1

W tþ1 α  ¼ 0: W t λtþ1

This is equivalent to ψ t ¼ ρu0 ðρW t Þ þ βð1  α  δ Þψ tþ1 Rtþ1 , because Wt + 1 ¼ (1  α  δ )Yt + 1 from (5.21) and Rtþ1 ¼

∂Y tþ1 αY tþ1 αY tþ1 ¼ ¼ : k ftþ1 λtþ1 W t ∂kftþ1

ð5:42Þ

138

5 Global Carbon Budgeting and the Social Cost of Carbon

A.3 The Steady-State Rule (5.30) Inserting (5.27) and (5.28) into (5.29) and cancelling ψ out of both sides implies ð1  βð1  μÞÞð1  λÞln0 AðλÞ ¼ μQln0 D ðQÞ: Rearrange terms:   1λ ð1  β ð1  μ ÞÞ α  δ ¼ μεðQÞ λ λ α ¼ : 1  λ δ  ð1  βð1  μÞÞ1 μεðQÞ This yields (5.30).

References Appunn K, Eriksen F, Wettengel J (2020) Germany’s greenhouse gas emissions and energy transition targets. Clean Energy Wire, Factsheets. https://www.cleanenergywire.org/factsheets/ germanys-greenhouse-gas-emissions-and-climate-targets. Accessed 18 Oct 2020 Cai Y, Judd K, Lontzek T (2013) The social cost of stochastic and irreversible climate change. NBER working paper, no. 18704, p 38 Carbon Tracker Initiative (2018) Carbon budgets: where we are? Carbon budget explainer. https:// www.carbontracker.org. Accessed 17 Oct 2020 Datahub (2020) Global temperature time series. https://datahub.io/core/global-temp#resourceannual. Accessed 20 Oct 2020 DeVries T (2014) The oceanic anthropogenic CO2 sink: storage, air-sea fluxes, and transports over the industrial era. Glob Biogeochem Cycles 28:631–647 Gilfillan D, Marland G, Boden T, Andres T (2019) Global, regional, and national fossil-fuel CO2 emissions. Available at: https://energy.appstate.edu/CDIAC. Accessed 19 Oct 2020 Global Carbon Budget (2018) Global Carbon Project. Future earth: research, innovation, sustainability. Earth system science data. The data publishing journal. Published on 5 December 2018, p. 81 Global Carbon Budget (2019). https://www.icos-cp.eu/global-carbon-budget-2019. Accessed 21 Oct 2020, Global Carbon Project. (2019). Supplemental data of Global Carbon Budget 2019 (Version 1.0) [Data set]. Global Carbon Project. https://doi.org/10.18160/gcp-2019. Accessed 21 Oct 2020 Golosov M, Hassler J, Krussel P, Tsyvinsky A (2014) Optimal taxes on fossil fuel in general equilibrium. Econometrica 82(1):41–88 Intergovernmental Panel on Climate Change (2014) In: Core Writing Team, Pachauri RK, Meyer LA (eds) Climate change 2014: Synthesis report. Contribution of working groups I, II and III to the fifth assessment report of the intergovernmental panel on climate change. IPCC, Geneva, p 151 Hansis E, Davis S, Pongratz J (2015) Relevance of methodological choices for accounting of land use change carbon fluxes. Glob Biogeochem Cycles 29:1230–1246

References

139

Heal G, Millner A (2017) Uncertainty and ambiguity in environmental economics: conceptual issues. Center for Climate Change and Economics Policy Working Paper no. 314 https://www. lse.ac.uk/granthaminstitute/wp-content/uploads/2017/09/Working-Paper-278-Heal-Millner. pdf. Accessed 20 Oct 2020 Houghton R, Nassikas A (2017) Global and regional fluxes of carbon from land use and land cover change 1850–2015. Glob Biogeochem Cycles 31:456–472 International Energy Agency (2011) World energy outlook 2011. IEA, Paris. https://www.iea.org/ reports/world-energy-outlook-2011. Accessed 22 Oct 2020 International Energy Agency (2015) Energy and climate change. World Energy Outlook  2015 special report, p 200 International Energy Agency (2017) World energy outlook 2017. IEA, Paris. https://www.iea.org/ reports/world-energy-outlook-2017. Accessed 22 Oct 2020 Joos F, Spahni R (2008) Rates of change in natural anthropogenic radiative forcing over the past 20,000 years. Proc Natl Acad Sci 105:1425–1430 Khatiwala S, Tanhua T, Mikaloff Fletcher S, Gerber M, Doney S, Graven H, Gruber N, McKinley G, Murata A, Rios A, Sabine C (2013) Global Ocean storage of anthropogenic carbon. Biogeosciences 10:2169–2191 McKibben B (2012) Global warming’s terrifying new math. Three simple numbers that add up to global catastrophe – and that make clear who the real enemy is, Rolling Stone. July 19, 2012 https://www.rollingstone.com/politics/politics-news/global-warmings-terrifying-new-math188550. Accessed 19 Oct 2020 Nordhaus W (1994) Managing the global commons: The economics of climate change. MIT Press, Cambridge, MA Nordhaus W (2014) Estimates of the social cost of carbon: Concepts and results from the DICE-13R model and alternative approaches. J Assoc Environ Resour Econ 1(1/2):273–312 Nordhaus W (2016) Projections and uncertainties about climate change in an era of minimal climate policies. Cowles foundation discussion paper no. 2057, Yale University, p 44 Nordhaus W (2017) Evolution of modelling of the economics of global warming: Changes in the DICE model, 1992–2017. Cowles Foundation Discussion Paper, No. 2084 Nordhaus W, Sztorc P (2013) DICE 2013R: Introduction and user’s manual. Yale University, second edition, p 101 Nordhaus W, Moffat A (2017) A survey of global impacts of climate change: Replication, survey methods and a statistical analysis. Cowles foundation discussion paper no. 2096, Yale University, p 40 Paris Agreement on Climate Change (2015) United Nations framework convention on climate change Roe G, Baker M (2007) Why is climate sensitivity so unpredictable? Supplementary material. Department of Earth and Space Sciences, University of Washington, Seattle, WA 98195, USA, p. 7 SeaLevel.info (2020) Atmospheric Carbon Dioxide (CO2) levels, 1800 – present https://www. sealevel.info/co2.html. Accessed 18 Oct 2020 Stern N (2007) The economics of climate change: The Stern review. Cambridge University Press, London, p 662 Weitzman M (2011) Fat-tailed uncertainty in the economics of catastrophic climate change. Rev Environ Econ Policy 5(2):275–292

Part II

Commodity Prices

Chapter 6

Commodity Prices, Convenience Yield and Inventory Behaviour

Abstract Markets for mineral commodities serve for delivering resources extracted from the earth to consumers. Price dynamics on these markets depend on the behaviour of commodity traders holding inventories. In this chapter, we examine a commodity market model with economic agents acting on the demand side and extracting utility from holding storage. In this model, commodity price fluctuations are driven by stochastic supply shocks. The market equilibrium dynamic is described by the stochastic difference equations for the commodity price and the convenience yield, which indicates the marginal utility of inventory holding. We show how the inventory behaviour of traders contributes to the autocorrelation of prices. Analysis of the model reveals the cases of price-stabilizing and -destabilizing inventory behaviour. In the former case, inventories are adjusted to smooth price fluctuations. In the latter case, the demand for inventories reinforces a price change caused by supply shock.

6.1

Introduction

The choice of optimal extraction programmes in the models of non-renewable resources is based on the intertemporal allocation of a resource in the ground. The classical model of exhaustible resource that was examined in Chap. 2 implies Hotelling’s rule stating that the optimal resource price net of the unit extraction cost should grow over time at a rate equal to the real rate of interest. Under this rule, producers are indifferent between present and future extraction of a resource, which is supplied at any time to meet current demand. In the models with limited resource availability in Chap. 3, the stock of producing reserves was treated as an inventory with outflows of extraction and inflows of reserves addition. In both cases, the dynamics of resource price depend on the intertemporal choices of producers, which is possible for one simple reason: the resource in the ground is storable. In this chapter, we consider resource pricing based on the competitive mechanisms of commodity trade. Mineral resources, being withdrawn from the earth, are priced on the organized commodity markets. The storability of resources above the ground allows for intertemporal commodity trade with physical delivery to final © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_6

143

144

6 Commodity Prices, Convenience Yield and Inventory Behaviour

consumers. Physical storage serves as a device facilitating such trade, similarly to reserve stocks under the ground. On the one hand, commodity traders holding inventories decide whether to sell or to leave a unit of resource in storage, and thereby they can affect commodity supply on the spot market. On the other hand, a commodity market clears in any period through adjustment of inventories to price movements. The inventory behaviour is viewed in this chapter as a mechanism reconciling the intertemporal choice of commodity traders with the condition of market clearing. In Chap. 3, we treated underground reserves as a factor of production that affects the cost and intensity of resource extraction. Commodity storage above the ground can fulfil a somewhat similar task of providing various services to owners of storage operating in commodity markets. First, inventories exert a buffering effect on production and consumption through absorbing price fluctuations. Second, inventory holdings eliminate losses from unexpected disruptions of supply and from waiting for delivery of a commodity. Third, storing a physical commodity allows owners of inventories to economize on marketing and transportation expenses. The benefit from holding a unit of physical storage is called the convenience yield. This notion was introduced by Nicolas Kaldor in 1939 and has often been used to explain the phenomenon of persistent backwardation. Strong backwardation is a situation in a commodity market incorporating trade in futures contracts, when the futures price is below the spot price. Backwardation is weak if the same condition is fulfilled for the discounted futures prices. Backwardation in weak or strong form is present in many commodity markets. For example, in the case of the crude oil market, Robert Litzenberger and Nir Rabinowitz (1995) reported that between February 1984 and April 1992, weak backwardation occurred 94% of the time and strong backwardation 77% of the time. A more extended data series provided by Delphine Lautier (2009, p. 3) for the period 1989–2008 demonstrates the violation of the weak backwardation only in three short episodes in 1993–1994, 1998 and 2005–2006. Persistent positive spreads between discounted futures and spot quotations can be explained by the significant benefits of holding physical storage exceeding the costs of storing. In other words, storage brings an additional utility gain to the inventory holder as compared, for example, to holding a long forward contract for delivery of the same commodity. Although the benefit of storage is not directly observable, it can be calculated from data on spreads between spot and futures prices. Empirical evidence suggests that the convenience yield should be regarded as a stochastic variable that drives the relationship between spot and futures prices. Figure 6.1 shows the WTI spot price for crude oil and the convenience yield calculated for the three-month futures for the period 2010–2019. This figure demonstrates the presence of weak backwardation and a positive correlation between the convenience yield and the spot price. Rajna Gibson and Eduardo Schwartz (1989) refuted the assumption of a perfect correlation between the convenience yield and the spot price of crude oil. Moreover, these authors (1990) and Schwartz (1997) examined the joint dynamic of a spot price and a

6.1 Introduction

145

Fig. 6.1 WTI spot price of crude oil and convenience yield for three-month NYMEX crude oil futures (weekly data) in the period 2010–2019. Source: US Energy Information Administration; authors’ calculation of the convenience yield under the assumptions that the annual riskless interest rate is 2% and the annual cost of storage is 3% of the spot price: Annualized conveniece yield ¼ (Log (spot price)  Log( futures price) + quarterly interest rate + quarterly storage cost)  4  100

convenience yield for some commodities and revealed the mean-reverting behaviour of the convenience yield. We will consider in this chapter a simple dynamic model of commodity trade with inventory management, which is relevant to the empirical findings of Gibson and Schwartz and other empirical studies referred to below. The convenience yield in this model is an endogenous variable, which is determined jointly with the equilibrium commodity price. The model actors are economic agents on the demand side choosing a time path of commodity consumption and inventory holding. For a real-world value chain of oil production, such agents are represented by refiners, wholesalers, retailers and end users holding inventories of crude oil and refined products to ensure the flexibility of production and to prevent disruptions in supply. In the model, economic agents on the demand side that will be called in what follows “consumers” derive utility gains both from consumption and inventory holding that are adjusted in any time period to changes of commodity price. The marginal benefit of storage decreases with the size of storage: if inventories are large, the gains from holding a unit of inventory are small. Conversely, the marginal benefit is high when the stocks are small in size and the costs of immediate delivery are substantial. Commodity supply is assumed to be an exogenous variable of the model, which is consumption-based, unlike the production-based models of inventory behaviour examined earlier, for instance, by Michael Brennan (1958), Jose Scheinkman and Jack Schechtman (1983), Jeffrey Williams (1986) and Robert Pindyck (1994, 2004). As was assumed in those models, inventory decisions are adopted by production

146

6 Commodity Prices, Convenience Yield and Inventory Behaviour

firms acting on the supply side and getting benefits from inventory holding. Scheinkman and Schechtman (1983) proved the existence of equilibrium for a model of storage and production under general conditions and established several comparative static results. Brennan (1958), Williams (1986) and Pindyck (1994, 2004) demonstrated how prices of commodity and storage are determined with production plans and inventories of firms through interactions between competitive markets for commodity and storage. In the consumption-based model of this chapter, stochastic fluctuations of commodity prices are driven by random supply shocks, although one can extend the model to incorporate demand shocks. We assume that the stochastic supply process allows for a positive serial dependence and consider the simplest case of a first-order linear autoregression. In the real world, a resource supply is affected by external events that may have long-lasting consequences, and in many cases a single supply shock is capable of affecting production for several years. Examples of negative supply shocks of this kind are disruptions of the oil supply caused by political disturbances and conflicts in oil-exporting countries or by international sanctions forbidding imports from some of these countries. Positive shocks with long-lasting consequences include discoveries of new vast resource fields operating for many decades or a cancelation of sanctions. The positive serial dependence of the supply process reflects lagged effects of supply shocks of various kinds. Inventory behaviour in our model contributes to the serial correlation of the commodity price. In the absence of storage, all serial correlation of price is attributed to the driving supply process, and the persistence of price fluctuations is fully determined by the autocorrelation of supply shocks. The impact of storage behaviour on the autocorrelation of price results from the properties of inventory management. Consumers allocate an available commodity between consumption and storage. Any unit of stored commodity has delayed effects on the future supply to the market, which are similar to the long-lasting effects of supply shocks. The autocorrelation of commodity price is thus reinforced by the rational choice of stock holdings by consumers. For a stationary supply process, we derive an approximate equilibrium solution for the stochastic commodity price and for the convenience yield defined in the model as the relative price of storage in commodity units. This yield is similar to the price of storage in the above-mentioned production-based models with the interacting markets for commodity and storage. The solution for inventory holding in our model is determined as the weighted sum of contributions of all previous supply shocks to storage accumulation plus the terms-of-trade effect of the relative price of storage. The contributions of supply shocks result from the interaction between “the propensity to store” of consumers and the autocorrelation of supply process. The inventory behaviour in the model influences the volatility of commodity price because it can mitigate or reinforce supply fluctuations. The model embraces the cases of price-stabilizing and -destabilizing inventory adjustments to shocks. For example, storage can partially absorb a negative supply disturbance if it decreases to a greater extent than consumption. As a result, the inventory adjustment dampens the

6.2 Consumption-Based Model of Commodity Storage

147

effect of such a disturbance on the current price. In the opposite case of destabilizing behaviour, this adjustment amplifies the effect of price increase caused by a negative supply shock. We show that the mode of inventory adjustment manifests itself in the convenience yield behaviour. The inventory adjustment is price-stabilizing if the convenience yield co-moves with the current price: for example, this yield increases in response to a negative supply shock. The inventory behaviour is price-destabilizing if the convenience yield decreases with the negative supply shock, because it responds to the change of price expectation to a greater degree than to the movement of current price. As will be demonstrated, the inventory behaviour is price-stabilizing if the autocorrelation of supply process is not very high. However, it can be pricedestabilizing under a significant supply shock with persistent lagged effects. Examples of such behaviour have been examined in narrative studies of oil history and in econometric studies of the world oil market. For instance, Lutz Killian and Daniel Murphy (2014) estimated a structural vector autoregressive model of the global market for crude oil. They showed that price-destabilizing speculative demand played an important role during price shock episodes, including in 1979, 1986, 1990 and 2002. In the case of the Persian Gulf War of 1990–1991, the unexpected supply disruption prompted an increase in speculative demand for oil driven by expectations of subsequent supply disruptions. As Killian and Murphy argue, similar episodes occurred during the Venezuelan Crisis and Iraq War of 2002–2003. In most of those cases, a negative supply shock of significant size caused a decline in oil inventories, while the upsurge of speculative demand caused an increase of inventories. As a result, inventories moved little due to the offsetting effects of supply shock and speculation, whereas both of them contributed to the large spike in oil price. Such situations are captured in the cases of pricedestabilizing inventory adjustment that will be analysed on the base of the commodity storage model presented in the next section.

6.2

Consumption-Based Model of Commodity Storage

Consider the choice of commodity consumption and storage by homogeneous consumers. There is no production in the model, and the commodity supply is assumed to be price-inelastic. Time is discrete and in each time period a consumer chooses a combination of commodity consumption and storage to maximize the expected discounted consumer surplus over an infinite time horizon. The consumer’s utility per period is a function of the consumption-storage combination. The number of consumers is constant and normalized to one. For the sake of simplicity, we assume that storage is costless and the storage capacity is unlimited. The consumer’s problem is maximizing the expected present value of consumer surplus:

148

6 Commodity Prices, Convenience Yield and Inventory Behaviour

U ¼ E0

1 X

βt ðuðyt , st Þ  pt ðyt þ it ÞÞ,

ð6:1Þ

st ¼ st1 þ it ,

ð6:2Þ

t¼1

subject to the storage balance

where E0 is the symbol of conditional mathematical expectation at the initial time 0, β ¼ 1/(1 + r) is the discount factor and the discount rate r > 0 is a riskless real interest rate, yt is consumption of a commodity in period t, st  0 is storage at the end of period t, u(yt, st) is the pecuniary utility of consumption and storage, pt is the commodity spot price and it is the inventory addition per period, which may be positive or negative. Consumer surplus in (6.1) is the difference of utility and consumer’s expenditures per period. The utility function u(yt, st) is increasing, concave, twice continuously differentiable and satisfies the Inada condition: ∂u(yt, 0)/∂st ¼ 1 for any yt. This condition guarantees that stock-outs never occur, since the marginal utility of storage is unbounded as the size of storage tends toward zero. The consumer is endowed with the initial storage s0  0. The commodity supply is given by the exogenous variable xt, which, in general, may be deterministic or stochastic. The market-clearing condition is yt þ it ¼ xt :

ð6:3Þ

In any period, the sum of consumption and inventory addition equals the current supply.

6.3

Convenience Yield

The Lagrangian for the consumption-storage problem (6.1)–(6.2) is: L ¼ E0

1 X

βt ðuðyt , st Þ  pt ðyt þ it Þ þ vt ðst1 þ it  st ÞÞ,

ð6:4Þ

t¼1

where vt is the costate variable related to storage balance (6.2). The internal firstorder conditions for variables yt and it are ∂ut =∂yt ¼ pt

ð6:5Þ

pt ¼ v t ,

ð6:6Þ

6.3 Convenience Yield

149

where ∂ut/∂yt denotes ∂u(yt, st)/∂yt. The commodity price is equal to the marginal utility of consumption and coincides with the shadow value of storage. The firstorder condition for st is vt ¼

∂ut þ βE t vtþ1 : ∂st

ð6:7Þ

The transversality condition rules out inflationary bubbles: lim E 0 βt vt st ¼ 0:

t!1

ð6:8Þ

Substituting pt for vt in (6.7) implies the commodity price equation: pt ¼

∂ut þ βE t ptþ1 : ∂st

ð6:9Þ

The marginal utility ∂ut/∂st defines the marginal benefit of inventory holding in period t. The commodity price in (6.9) is the sum of this benefit and the discounted expected price in the next period. In what follows, we will use the additive isoelastic utility function: uð y t , s t Þ ¼

y1θ þ bs1θ  ð 1 þ bÞ t t , 1θ

ð6:10Þ

where θ > 0 is the inverse demand elasticity, and b  0 is the utility weight of storage. The term (1 + b) in the nominator of (6.10) is used to include the case of logarithmic utility with θ ¼ 1. For the utility function (6.10), we have: ∂ut =∂yt ¼ yθ and, from (6.5), cont sumption is equal to: 1=θ

y t ¼ pt

:

The marginal benefit of inventory holding is decreasing with the size of storage: ∂ut =∂st ¼ bsθ t and it tends toward infinity as st goes to zero. Define the convenience yield as the marginal benefit of storage in commodity units: δt ¼

∂ut =∂st : ∂ut =∂yt

ð6:11Þ

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6 Commodity Prices, Convenience Yield and Inventory Behaviour

For the constant-elasticity utility (6.10), it is the power function of the consumption-storage ratio: δt ¼ bðyt =st Þθ :

ð6:12Þ

Since ∂ut/∂yt ¼ pt, the commodity price Eq. (6.9) can be written as E t ptþ1 ¼ ð1 þ r Þð1  δt Þpt  ð1 þ r  δt Þpt , provided that r and δt are quite small. Thus, the expected rate of value gain from holding commodity during one period equals the interest rate less the convenience yield: Et Δptþ1 ¼ r  δt , pt

ð6:13Þ

The expected value gain and the convenience yield cover the cost of servicing a loan used to make investment in storage. Equation (6.13) is the equilibrium condition for intertemporal allocation of commodity, which is similar to Hotelling’s rule for resource price dynamic.

6.4

Deterministic Dynamic of Commodity Price and Storage

Consider the dynamic of commodity price and storage in the case of no uncertainty in supply, which is assumed to be time-constant and normalized to one, xt  1. Future prices are perfectly foreseen, implying that Etpt + 1 ¼ pt + 1 in any period. From the market-clearing condition (6.3), the inventory addition is it ¼ xt  yt ¼ 1=θ . Equations for price growth (6.13) and storage balance (6.2) are 1  pt represented as the system of difference equations on pt and st: ptþ1 ¼ ð1 þ r Þpt  bsθ t 1θ

stþ1 ¼ st þ 1  ptþ1 :

ð6:14Þ ð6:15Þ

The steady state of this system is p ¼ 1, s ¼ ðb=r Þ1=θ : The price growth Eq. (6.13) implies that the stationary convenience yield equals the interest rate:

6.5 Stochastic Dynamic of Commodity Market

151

Fig. 6.2 Equilibrium paths of commodity price and storage

δ ¼ r: Figure 6.2 depicts paths for the dynamic system (6.14), (6.15) as continuous curved arrows. The steady state is the saddle point. The two equilibrium paths satisfying the transversality condition (6.8) are drawn with thick curved arrows. They converge to the steady state from the “north-west” and the “south-east”. We rule out the bubble solutions with perpetual storage accumulation and the solutions with storage degenerating to zero, which are drawn in the figure with thin curved arrows. The initial equilibrium price p0 is high if the initial storage s0 is small, and the price pt is decreasing along this path despite the accumulation of inventories.

6.5

Stochastic Dynamic of Commodity Market

Suppose that supply is stochastic and driven by the first-order autoregression: ln xt ¼ αlnxt1 þ εt ,

ð6:16Þ

where α is the autoregression coefficient, 0  α < 1, and εt is a serially uncorrelated random variable with zero mean and variance σ 2. The unconditional (long-term) mean for the supply process is x ¼ 1. Consider the commodity price Eq. (6.13) for stochastic supply. Using the linear approximation of logarithm, EtΔ ln pt + 1  EtΔpt + 1/pt, we represent the expected growth rate of price as

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6 Commodity Prices, Convenience Yield and Inventory Behaviour

E t Δ ln ptþ1 ¼ r  δt :

ð6:17Þ

The storage is adjusted in any period to make the price growth Eq. (6.17) compatible with the market-clearing condition (6.3), which can be written as: ln pt ¼ θlnðxt  it Þ,

ð6:18Þ

1=θ

¼ xt  it . The logarithm of market-clearing price is proportional to the since pt negative logarithm of supply less the change of inventory holding. The latter depends on the expected price change, hence we have to consider the inventory decision to derive a solution for price.

6.5.1

Current Availability of Commodity

One can express equilibrium consumption and storage chosen by economic agents as shares of a current availability of commodity at defined as the sum of supply in period t and inventory stock by the end of period t  1, at ¼ xt + st  1. Jose Scheinkman and Jack Schechtman (1983), Jeffrey Williams and Brian Wright (1991, p. 28) and other authors used this variable in the analysis of competitive equilibrium models with storage. In our model, the current availability is a state variable that affects decisions adopted by consumers. In any period, the commodity “on hand” supplied by producers or available from previous storage is allocated between consumption and addition to inventory stock. The proportion of allocation is determined by the convenience yield δt, because, from (6.12), the consumption is linked to storage as yt ¼ st ðδt =bÞ1=θ :

ð6:19Þ

The inventory balance (6.2) and the market-clearing condition (6.3) imply that the sum of consumption and storage by the end of period t must be equal to the current availability, y t þ s t ¼ at :

ð6:20Þ

Combining this equation with (6.19) yields the implicit solution in recursive form:

where

st ¼ μt at

ð6:21Þ

yt ¼ ð1  μt Þat ,

ð6:22Þ

6.5 Stochastic Dynamic of Commodity Market

μt ¼

153

b1=θ 1=θ

b1=θ þ δt

is the share of storage in the current availability. This share increases with the utility weight of storage b and decreases with δt. We will consider a model solution for a stationary stochastic process, an analogue of the stationary state ( p, s) in the deterministic case. The initial time for this process is minus infinity, and the logarithm of current supply (6.16) is represented as the infinite-order moving average: ln xt ¼

1 X

ατ εtτ

ð6:23Þ

τ¼0

In what follows we consider an approximate solution for inventory holding for a small neighbourhood of point ( p, s).

6.5.2

Inventory Behaviour

To derive this solution, let us introduce, in addition to current supply xt, an auxiliary stochastic variable qt driven by a mean-reverting process similar to (6.16): ln qt ¼ μlnqt1 þ εt ,

ð6:24Þ

where μ¼

b1=θ b1=θ þ r 1=θ

is the long-term rate of storage defined as the share of storage in the current availability corresponding to the stationary convenience yield δ ¼ r. The process (6.24) is built on the same sequence of stochastic shocks as the process for supply (6.16), but differs in the autoregressive coefficient μ. We call the variable lnqt a stored excess supply because it is the infinite-order moving average of supply shocks: ln qt ¼

1 X

μτ εtτ :

ð6:25Þ

τ¼0

This moving average is the sum of the current supply shock εt and all increments generated by previous supply shocks εt  τ remaining in storage by period t under the

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6 Commodity Prices, Convenience Yield and Inventory Behaviour

long-term rate of storage μ. The variable lnxt represented as the moving average of these shocks (6.23) defines a cumulative excess supply. We suppose that μ 6¼ α. The following proposition characterizes the dependence of storage on the state variables of stored and cumulative excess supply and on the convenience yield. Proposition 6.1 Inventory holding in period t is st ¼ s þ μ

μlnqt  αlnxt  μ  ðδ  r Þ:  θr t μα

ð6:26Þ

Proof: in Appendix A.1. Deviation of inventory holding from the long-term mean s ¼ (b/r)1/θ is the sum of two terms representing an availability effect of storage accumulation and a termsof-trade effect. The former is proportional to the weighted difference of stored and cumulative excess supply, μlnqt  αlnxt. The latter captures the effect of convenience yield fluctuations on the inventory behaviour. As one can see from (6.26), the higher the deviation of this yield from the long-term mean, δt  r, indicating a higher price of storage, the lower is the volume of inventory holding. To clarify the meaning of the term related to the availability effect in (6.26), it makes sense to utilize some algebra from the proof of Proposition 6.1. We use the linear approximation for the state variables of cumulative and stored excess supply: xt  1 þ

1 X τ¼0

ατ εtτ ,

qt  1 þ

1 X

μτ εtτ

τ¼0

and consider a supply shock in period t  τ and its impact on the storage accumulation by period t inclusive. Let εt  τ > 0. In the absence of the terms-of-trade effect, the amount of εt  τ stored in period t  τ equals μεt  τ, and the amount remaining in storage in period t is μτ + 1εt  τ. In the next period t  τ + 1, the contribution of supply shock εt  τ in the current supply is αεt  τ, of which the amount stored in period t is μταεt  τ. We can proceed similarly and get, finally, the contribution of supply shock εt  τ in the current supply in period t as ατεt  τ, of which the amount stored in this period is ματεt  τ. Summing up these terms over time periods t  τ to t yields the contribution of supply shock εt  τ to storage accumulation by the end of period t:   Δtτ,t ¼ μ μτ þ μτ1 α þ . . . þ ματ1 þ ατ εtτ ¼ μτþ1 ð1 þ ðα=μÞ þ . . . þ ðα=μÞτ Þεtτ ¼ μτþ1

 1  ðα=μÞτþ1 μ  τþ1 μ  ατþ1 εtτ , εtτ ¼ μα 1  α=μ

6.6 Solution for the Convenience Yield and the Commodity Price

155

provided that α < μ. For εt  τ < 0, this formula expresses a negative supply shock contribution to storage reduction. Summing up Δt  τ, t over all τ  0 implies the total contribution of current and past supply shocks to storage in period t: 1 X τ¼0

Δtτ,t

μ ¼ μα

μ

1 X

τ

μ εtτ  α

τ¼0

1 X

! τ

α εtτ

τ¼0

¼

μ ðμlnqt  αlnxt Þ: μα

We have thus derived the availability effect term in storage accumulation (6.26). This term proves to be the same for the case α > μ, as shown in the proof of Proposition 6.1. As a result, the total contribution of current and past supply shocks to storage is proportional to the weighted difference of the stored and cumulative excess supply. One can see from the proof of Proposition 6.1 that the availability effect term equals 1 X μ ðμlnqt  αlnxt Þ ¼ μ μτ xtτ  s : μα τ¼0

The right-hand side of this equation is the deviation of total current and lagged supply accumulated in inventories (net of the terms-of-trade effect) from the steadystate storage. The approximate solution for equilibrium storage (6.26) depends only on the current-period state variables qt and xt, thereby greatly simplifying our further analysis.

6.6

Solution for the Convenience Yield and the Commodity Price

The proof of Proposition 6.1 contains the solution for equilibrium consumption, which is similar to the solution for storage (6.26): y t ¼ 1 þ ð1  μ Þ

μlnqt  αlnxt þ ðμ=θÞe δt , μα

ð6:27Þ

where e δt ¼ ðδt  r Þ=r denotes the relative deviation of convenience yield from the interest rate. The deviation of consumption from the long-term mean y ¼ 1 is the sum of terms related to the availability effect and the terms-of-trade effect. The latter has the opposite sign to the terms-of-trade effect for storage. Inserting (6.27) into (6.18) implies the equation for price

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6 Commodity Prices, Convenience Yield and Inventory Behaviour

ln pt ¼ θð1  μÞ

μlnqt  αlnxt  μe δt : αμ

ð6:28Þ

The equilibrium price must satisfy two conditions: Eq. (6.28) that ensures market clearing and the equation for expected price growth (6.17) resulting from the intertemporal allocation of storage. The compatibility of these two conditions is provided by the equilibrium convenience yield. Proposition 6.2 The equilibrium convenience yield is given by: e δt ¼ λq ln qt þ λx ln xt ,

ð6:29Þ

θ ð1  μ Þ2 μ θð1  μÞð1  αÞα , λx ¼ : ðα  μÞðr þ μð1  μÞÞ ðμ  αÞðr þ μð1  αÞÞ

ð6:30Þ

where λq ¼

Proof: in Appendix A.2 The convenience yield in relative terms (6.29) is the linear combination of stored and cumulative excess supply. The coefficients λq and λx have the opposite sign, which means that the effects of stored and cumulative excess supply on the convenience yield partially offset each other. Combining the solution for e δt with the equation for price (6.28) and rearranging terms (see Appendix A.3) yields the solution for price: ln pt ¼ ηq ln qt þ ηx ln xt ,

ð6:31Þ

where ηq ¼ λq

r , 1μ

ηx ¼ λx

r : 1α

ð6:32Þ

Coefficients of a linear combination for price are proportional to the corresponding coefficients for convenience yield and have the same sign.

6.6.1

Two Limit Cases

Consider the solution for commodity price (6.31) for two limit cases. In the first case the share of storage in the current availability is zero, μ ¼ μt ¼ 0, which takes place if the utility weight of storage is zero, b ¼ 0. Then, from (6.30) and (6.32), we have it that ηq ¼ 0, ηx ¼  θ and the price Eq. (6.31) is represented as the first-order autoregression:

6.6 Solution for the Convenience Yield and the Commodity Price

ln pt ¼ θlnxt ¼ θðαlnxt1 þ εt Þ ¼ αlnpt1  θεt :

157

ð6:33Þ

Thus, in the absence of storage, all serial correlation of price is attributed to the autocorrelation of the supply process, which is given by parameter α. In the second case the supply process is serially uncorrelated, α ¼ 0. Thus we have it that ηx ¼ 0 and ηq ¼ 

θr ð1  μÞ : r þ μð1  μÞ

The price Eq. (6.31) reduces to ln pt ¼ ηq ln qt ¼ ηq ðμlnqt1 þ εt Þ ¼ μlnpt1 þ ηq εt : This case illustrates a contribution of storage to autocorrelation of price and a price-stabilizing effect of storage. The autocorrelation of price is given by the longterm rate of storage μ, even though the driving supply process is uncorrelated. The conditional first-order variance of price is Var t1 ln pt ¼ Var t1 ηq εt ¼ η2q σ 2 : One can see that η2q < θ2 for α ¼ 0. Hence, the conditional variance of price is below this variance in the absence of storage (for μ ¼ 0), which is equal to Vart  1 ln pt ¼ θ2σ 2, as follows from (6.33). The reduction of variance results from the inventory behaviour of consumers.

6.6.2

The First-Order Autocorrelation for Commodity Price

The contribution of storage to autocorrelation of prices generally is indicated by comparison of the autocorrelation of prices with this autocorrelation in the absence of storage. In the latter case, the autocorrelation coefficient is equal to α, as we have shown. In general, the first-order autocorrelation coefficient is defined as the ratio: ρ¼

Covð ln pt , ln pt1 Þ : Var ð ln pt Þ

ð6:34Þ

The unconditional variance and covariance are calculated for the solution for price given by (6.31):

158

6 Commodity Prices, Convenience Yield and Inventory Behaviour

Fig. 6.3 Autocorrelation of price ρ as the function of α for different values of μ

Var ð ln pt Þ ¼ Covð ln pt , ln pt1 Þ ¼

η2q σ 2 2ηq ηx σ 2 η2 σ 2 þ þ x 2 2 1  αμ 1  α 1μ μη2q σ 2 ðα þ μÞηq ηx σ 2 αη2x σ 2 þ : þ 1  αμ 1  μ2 1  α2

In Fig. 6.3 we show the dependence of the autocorrelation coefficient ρ on α calculated with the use of formulas (6.30), (6.32) for different values of the longterm rate of storage μ ¼ 0, 0.2, 0.4, 0.6, 0.8 and r ¼ 0.05. The horizontal axis shows α in the interval [0, 1), and the vertical line corresponds to the coefficient ρ for the same interval. The values of ρ are interpolated for points with α ¼ μ, which is the special case of the model. The diagonal line shows the autocorrelation of prices for μ ¼ 0. Figure 6.3 demonstrates that the contribution of storage to this autocorrelation is significant. It decreases in relative terms with the autocorrelation of supply process and increases with the long-term rate of storage. Note that the functions depicted in this figure do not depend on parameter θ, because it vanishes from the ratio (6.34). These functions vary slightly with r. Parameters θ and r influence μ, which we consider in Fig. 6.3 as the model parameter. We will take into account the dependence of μ on θ and r in a numerical example further below.

6.6 Solution for the Convenience Yield and the Commodity Price

6.6.3

159

The Vector Autoregression

Generally we can represent the solution for convenience yield (6.29) and the commodity price (6.31) in the form of vector autoregression. Proposition 6.3 1. The dynamic equations for commodity price, convenience yield and supply process are represented as:   Δ ln pt ¼ r  δt1 þ ηq þ ηx εt   Δδt ¼ ð1  μÞðr  δt1 Þ þ r ðα  μÞλx ln xt1 þ r λq þ λx εt Δ ln xt ¼ ðα  1Þ ln xt1 þ εt :

ð6:35Þ ð6:36Þ ð6:37Þ

2. The coefficient of supply shock in the price growth Eq. (6.35) is equal to: ηq þ ηx ¼ 

θr ð1  μÞðr þ μÞ < 0: ðr þ μð1  μÞÞðr þ μð1  αÞÞ

ð6:38Þ

Proof: in Appendix A.4. The vector autoregression includes dynamic equations for the price (6.35), for the convenience yield (6.36) and for the supply process (6.37), replicating (6.16). The stochastic term in the equation for price (6.35) affects the price change unambiguously: a negative shock contributes to an increase and a positive one to a decrease of commodity price. The equation for the convenience yield (6.36) defines the change of this yield as the sum of three terms related to the mean-reverting effect, to the pastperiod supply effect and to the effect of current supply shock. The mean-reverting effect is weak under a high long-term rate of storage μ. The effect of previous-period cumulative excess supply lnxt  1 on Δδt in Eq. (6.36) is negative, because, from (6.30): r ðα  μÞλx ¼ 

θr ð1  μÞð1  αÞα < 0: ðr þ μð1  αÞÞ

For example, a high previous-period cumulative excess supply has a downward impact on Δδt. In other words, the excess supply causes storage accumulation that leads, all things being equal, to a reduction of the relative price of storage. The effect of current-period supply shock εt on Δδt in Eq. (6.36) depends on the sign of coefficient r(λq + λx), which is ambiguous:

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6 Commodity Prices, Convenience Yield and Inventory Behaviour



r λq þ λx



  θr ð1  μÞ ð1  μÞμ ð1  αÞα ¼  ⋚0: αμ r þ ð1  μÞμ r þ ð1  αÞμ

For a small coefficient α, we have it that r(λq + λx) < 0 implying that the effect of current supply shock on the convenience yield is similar to the effect of this shock on the price. However, if α is high enough, the sign of r(λq + λx) can be positive, as we shall demonstrate in what follows.

6.7

Price-Stabilizing and -Destabilizing Inventory Behaviour

Inventory behaviour can mitigate or amplify the impact of supply shocks on the commodity price dynamic. This feature closely relates to the ambiguity of the supply shock effect on the convenience yield. Rewrite the equation for expected price growth (6.17) as: δt ¼ r þ ln pt  E t ln ptþ1 :

ð6:39Þ

Suppose that a negative shock εt occurs and results in an increase of the current price and the price expectation. If the effect on current price dominates the effect on price expectation, then from (6.39), the convenience yield increases and co-moves with the current price. If the effect on expectation is dominating, the convenience yield decreases and moves counter to the current price. In what follows, we show that in the former case the inventory behaviour has a stabilizing effect on the price dynamic, and in the latter case the effect is destabilizing.

6.7.1

The Adjustment of Storage

A supply shock translates into changes in the current and future prices through the adjustment of equilibrium consumption and storage as given by Eqs. (6.21)–(6.22). First, the supply shock in period t causes a change of availability at that can be expressed as Δat ¼ ð1  μÞða  at1 Þ þ αlnxt1 þ εt ,

ð6:40Þ

where a ¼ x + s ¼ 1 + (b/r)1/θ is the long-term availability, as we show in Appendix A.5. Second, this shock leads to a shift of the convenience yield Δδt that results from the solution in autoregressive form (6.36) and implies the terms-of-trade effect on inventory behaviour.

6.7 Price-Stabilizing and -Destabilizing Inventory Behaviour

161

Fig. 6.4 Adjustment of storage under a negative shift of availability (a) Stabilizing adjustment (b) Destabilizing adjustment

Figure 6.4(a, b) illustrates the adjustment of the consumption-storage combination under a negative supply shock εt. Point A depicts the equilibrium combination (yt  1, st  1) in period t  1. This point is located on the intersection of line at  1, showing the availability constraint (6.20) in that period, with line OA of consumption-storage combinations (6.19) corresponding to δt  1. Suppose that the supply shock εt < 0 is significant and causes a decrease of current supply, Δxt < 0, that leads to a reduction of availability from at  1 to at. Line at  1 in the figure moves downwards to the parallel line at. The reduction of availability shifts the consumption-storage combination from point A to point B on line OA. Since the effect of supply shock on convenience yield is ambiguous, we consider two cases. In the case shown in Fig. 6.4a, the convenience yield increases from δt  1 to δt. Line OA shifts upward to line OC corresponding to δt. The consumption-storage combination moves to point C, the intersection of line at with line OC. The negative effect of supply shock on consumption is partially offset by the significant reduction of inventories, as seen from the figure. The increase of price is also partially offset, since pt ¼ yθ t . Thus, for Δat < 0 and Δδt > 0 the inventory adjustment is pricestabilizing. Figure 6.4b shows the case when the convenience yield decreases. Line OA shifts downward to line OC, and the equilibrium consumption-storage combination moves from point A through point B to C, the intersection of line at with OC. The change of δt reinforces the effect of supply shock on the fall of consumption and the price increase. Hence, the inventory adjustment with Δat < 0 and Δδt < 0 is pricedestabilizing. Figure 6.5 is similar to Fig. 6.4 and demonstrates the effects of a significant positive supply shock εt > 0 that causes a positive shift of availability, Δat > 0. The adjustment of storage is stabilizing in Fig. 6.5a for Δδt < 0 and destabilizing in Fig. 6.5b for Δδt > 0.

162

6 Commodity Prices, Convenience Yield and Inventory Behaviour

Fig. 6.5 Adjustment of storage under a positive shift of availability. (a) Stabilizing adjustment, (b) Destabilizing adjustment

One can summarize the cases drawn in Figs. 6.4 and 6.5 as follows: the adjustment of storage is price-stabilizing if Δat < 0, Δδt > 0 orΔat > 0, Δδt < 0 and price-destabilizing if Δat < 0, Δδt < 0 orΔat > 0, Δδt > 0:

6.7.2

The Shock Effects on the Convenience Yield

Consider again the case of a negative supply shock εt. Whether or not it has a pricedestabilizing effect depends on the sign of coefficient r(λq + λx) in the dynamic Eq. (6.36) for δt. If λq + λx < 0, the supply shock εt < 0 exerts an upward pressure both on the price and on the convenience yield, and therefore the inventory adjustment is price-stabilizing. In the opposite case, if λq + λx > 0, a negative supply shock causes an upward pressure on the price and a downward pressure on the convenience yield. In this case, storage reduces to a less extent than consumption, as one can see in Fig. 6.4b. If the shock is significant, then Δat < 0 and we have it from Eq. (6.36) that Δδt < 0, and the inventory adjustment is price-destabilizing. To illustrate this inference, consider the following numerical example. Let r ¼ 0.05, b ¼ 0.2 and the inverse demand elasticity takes three values: θ ¼ 0.8, 2, 3. The long-term rate of storage corresponding to these values is μ ¼ 0.85, 0.59,

6.7 Price-Stabilizing and -Destabilizing Inventory Behaviour

163

Fig. 6.6 Coefficient r(λq + λx) as the function of α for different values of θ

0.56, respectively. Figure 6.6 shows the dependence of the coefficient of supply shock r(λq + λx) in Eq. (6.36) on the coefficient of supply autocorrelation α. One can see from this figure that the coefficient r(λq + λx) is positive for high α and negative otherwise. In the former case, a significant supply shock has a destabilizing impact on the commodity price, which occurs because the shock’s effect on the price expectation dominates the effect on the current price (according to Eq. (6.39)). The expectation effect proves to be strong for high α, because a significant negative supply shock in period t induces a sequence of negative supply increments ατεt in subsequent periods t + τ bringing about an upward pressure on the future price dynamic and on the price expectations in period t. Thus, the expectation effect can be dominating under a high autocorrelation of the supply process. This destabilizing effect can be reinforced in situations of mean reversion. Suppose that a significant supply shock εt < 0 occurs after a series of supply shocks with prevailing positive values, so that lnxt  1 > 0. Such a past dynamic exerted a downward pressure on the price and an upward pressure on the convenience yield (this follows from Eqs. (6.35) and (6.36) because ηq + ηx < 0 and r(λq + λx) > 0 under high α). Suppose that by this reason the convenience yield in period t  1 was above the long term average, δt  1 > r. In this situation, the negative effect of supply shock on the convenience yield is reinforced by the negative mean-reverting effect and the negative effect of past-period cumulative excess supply. Indeed, the first and the second terms on the right-hand side of Eq. (6.36) are negative, because δt  1 > r and r(α  μ)λx < 0. These two effects caused by mean reversion contribute to the downward pressure on the convenience yield caused by the negative supply shock in period t.

164

6 Commodity Prices, Convenience Yield and Inventory Behaviour

There is a similarity between situations of price-destabilizing inventory adjustment in the model and the real-world cases of mean reversion in commodity price movements. A downward movement of price can be reverted into the opposite tendency by a significant negative supply shock that causes a significant price increase. The expectations of a further price growth arise due to the market beliefs that the shock may have long-lasting negative consequences for commodity supply or may signal about such consequences. If the preceding downward deviation of price from a long-term fundamental level was significant, the market could expect a significant mean-reverting effect that reinforces the further growth of price. The expectation of price growth can strengthen the accumulation of inventories instead of sales of inventories that a sharp reduction of supply and a price jump dictate for the price-stabilizing inventory adjustment.

6.8

Concluding Remarks

In this chapter, we have considered the model of commodity price based on the consumption-storage choice by utility-maximizing economic agents. The key assumption of the model is that storage is an argument of their utility function. The storage-in-utility hypothesis means that inventory holding is beneficial for a variety of reasons: for example, because of supply-demand fluctuations that may occur within a time period and that are not modelled explicitly. The marginal utility of storage decreases with the storage size and tends toward zero under abundant stocks. The equilibrium convenience yield is always positive in the model, in line with the empirical evidence of persistent weak backwardation in the commodity markets. We considered only trade with physical delivery of commodities to buyers and neglected trade with futures or forward contracts that allow for hedging and trading risks related to commodity price fluctuations. Commodity futures markets provide reallocation of these risks among market participants and reveal information about price expectations. In the case of crude oil, the daily turnover of futures contracts exceeds the volume of spot trade, but these contracts virtually do not require physical delivery and do not influence directly the oil price (Smith 2009). The influence can be indirect, through the formation of market expectations that determine the inventory behaviour of forward-looking traders affecting the commodity price dynamic. By contrast, physical storage exerts a direct influence upon the current demand and supply of commodities and thereby plays an essential role in commodity price formation.

6.8 Concluding Remarks

165

Market participants in the model determine in any period the level of consumption jointly with the choice of inventory holding by comparing the current commodity price with the marginal utility of storage. The relative price of storage, as measured by the convenience yield, determines the equilibrium share of storage in the current availability of commodity. The equilibrium commodity price satisfies a modified Hotelling’s rule for stochastic supply: the expected rate of price growth equals the interest rate less the dividend-like term given by the convenience yield. We have shown that the optimal choice of inventory holding depends on the linear combination of two stochastic state variables: the cumulative and stored excess supply. The latter is unobservable, but the equilibrium dynamic of commodity market can be represented as the vector autoregression for three observables: the price, the convenience yield and the current supply. Equations of this autoregression are somewhat similar to the econometric model of commodity price and convenience yield examined by Gibson and Schwartz (1990). We have demonstrated that commodity trade with storage can increase the volatility of commodity price. First, this trade contributes to the positive autocorrelation of price implying an increase of the price volatility for a multi-period time horizon. Second, in the short term, the price volatility can increase through the adjustment of convenience yield to supply shocks. We have revealed the cases of price-stabilizing and -destabilizing inventory behaviour. In the former case, inventories are adjusted to smooth price fluctuations and, as a result, the convenience yield co-varies with the current price. This agrees with the estimates by Gibson and Schwartz (1990) of a positive correlation between the price and the convenience yield. In the latter case, the demand for inventories reinforces a price change caused by supply shock and the convenience yield moves in the opposite direction to the commodity price. As has been shown in this chapter, the price-destabilizing inventory behaviour can occur in some periods when the supply shock is significant in absolute size. It may have a destabilizing impact on the commodity price, because the shock’s effect on the price expectation dominates the effect on the current price. The expectation effect is strong under a high autocorrelation of the supply process, because a significant supply shock induces significant lagged effects on future supply and price dynamic that are mirrored in the price expectation. The assumption that commodity storage delivers utility to inventory holders provides a simple explanation for the phenomenon of backwardation and eases formal analysis of the consumption-storage model. The Inada condition on the utility function ensures that the marginal benefit of storage is unbounded for small stocks and the non-negativity constraint on storage is non-binding. However, the situations of stock-outs may be empirically relevant, if one considers a pure speculative motive for inventory holding, when storage does not deliver direct utility gains to consumers. We shall examine the implications of this hypothesis in the next chapter.

166

6 Commodity Prices, Convenience Yield and Inventory Behaviour

Appendices A.1 Proposition 6.1 The solution for st, yt is found as a linear approximation of Eqs. (6.21), (6.22) near the stationary point s ¼ (b/r)1/θ, x ¼ 1. Consider the storage-consumption ratio (6.19): st =yt ¼ ðb=δt Þ1=θ : The linear approximation of the right-hand side for δ ¼ r implies: ðb=δt Þ1=θ ¼ s ðr=δt Þ1=θ  s ð1  ðδt  r Þ=θr Þ: Hence, ln ðst =s Þ  ln yt  

ðδt  r Þ , θr

ð6:41Þ

We will show that the solution for st ¼ μtat satisfying this approximation is given by   st ¼ sat þ μ0r ðδt  r Þa ¼ sat þ μ0r ðδt  r Þ 1 þ ðb=r Þ1=θ ,

ð6:42Þ

where a ¼ x + s ¼ 1 + (b/r)1/θ is the stationary availability, 1 X   μτ xtτ sat ¼ μ xt þ sat1 ¼ μ

ð6:43Þ

τ¼0

is the amount of storage defined by the availability effect, b1=θ r 1=θ1 μ0r ¼   2 θ b1=θ þ r 1=θ is the derivative of μ with respect to r. Inserting this derivative into (6.42) implies   b1=θ r 1=θ1 1 þ ðb=r Þ1=θ ðδt  r Þ ¼ sat  ðμ=θr Þðδt  r Þ: st ¼ sat   1=θ 2 1=θ θ b þr The linear approximation for supply process (6.16) is:

ð6:44Þ

Appendices

167

xt ¼ αxt1 þ ð1  αÞ þ εt : Iterating terms yields: xt ¼

1 X

ατ εtτ þ ð1  αÞ

τ¼0

1 X

ατ ¼ 1 þ

τ¼0

1 X

ατ εtτ :

τ¼0

Inserting this into (6.43) implies: sat ¼

1 X

μτþ1 1 þ

1 X

τ¼0

! α j εtτj

¼

j¼0

X X μ þ μτþ1 α j εtτj : 1  μ τ¼0 j¼0 1

1

ð6:45Þ

Suppose that α > μ and rearrange the dual sum: 1 X τ¼0

μτþ1

1 X

α j εtτj ¼ μ

j¼0

1 X j¼0

τ¼0

α j εt1j þ . . .

   ¼ μ εt þ ðμ þ αÞεt1 þ μ2 þ μα þ α2 εt2 . . .     ¼ μ εt þ ððμ=αÞ þ 1Þαεt1 þ ðμ=αÞ2 þ ðμ=αÞ þ 1 α2 εt2 . . . 1 X τ¼0

1 X

1 X j¼0



¼μ ¼μ

α j εtj þ μ2

ατ

ατ εtτ

τ X

ðμ=αÞ j

j¼0

1   1  ðμ=αÞτþ1 μ X τþ1 α 1  ðμ=αÞτþ1 εtτ εtτ ¼ α  μ τ¼0 1  μ=α

μ ¼ αμ ¼

α

1 X τ¼0

τ

α εtτ  μ

μ ðαlnxt  μlnqt Þ αμ

1 X

!

τ

μ εtτ

τ¼0

due to (6.23) and (6.25). Inserting this into (6.45) and then into (6.44) implies the solution for st: st ¼ sat 

  μlnqt  αlnxt  μ  μ ðδt  r Þ ¼ s þ μ ðδ  r Þ,  θr θr t μα

since μ/(1  μ) ¼ (b/r)1/θ ¼ s. One can show, similarly, that this formula holds for α < μ. The solution for yt is derived in the same way:

168

6 Commodity Prices, Convenience Yield and Inventory Behaviour

yt ¼ 1 þ ð 1  μ Þ

μlnqt  αlnxt þ ðμ=θr Þðδt  r Þ: μα

The obtained solution for st and yt satisfies condition (6.41), because the left-hand side of (6.41) equals (st/s)  ln yt  st/s  yt ¼  ((1  μ)/θr)(δt  r)  (μ/θr) (δt  r) ¼  (δt  r)/θr.

A.2 Proposition 6.2 Consider the equation for expected price growth (6.17), written as δt : E t Δ ln ptþ1 ¼ re

ð6:46Þ

Using the equation for price (6.28) we have: μE t Δ ln qtþ1  αE t Δ ln xtþ1  μEt Δe δtþ1 : αμ

ð6:47Þ

E t Δ ln qtþ1 ¼ ðμ  1Þ ln qt , E t Δ ln xtþ1 ¼ ðα  1Þ ln xt :

ð6:48Þ

Et Δ ln ptþ1 ¼ θð1  μÞ From (6.16) and (6.24):

For the linear solution e δt ¼ λq ln qt þ λx ln xt we obtain: Et Δe δtþ1 ¼ λq Et Δ ln qtþ1 þ λx E t Δ ln xtþ1 ¼ λq ðμ  1Þ ln qt þ λx ðα  1Þ ln xt :

ð6:49Þ

Inserting (6.47), (6.48), (6.49) and (6.29) into (6.46) implies: μðμ  1Þ ln qt  αðα  1Þ ln xt E t Δ ln ptþ1 ¼ θð1  μÞ αμ    μ λq ðμ  1Þ ln qt þ  λx ðα  1Þ ln xt ¼ r λq ln qt þ λx ln xt : Gathering the terms with lnqt and lnxt in this equality implies equations for λq and λx, respectively: θð1  μÞ

μðμ  1Þ ln qt  μλq ðμ  1Þ ln qt ¼ rλq ln qt αμ

Appendices

169

θð1  μÞ

αðα  1Þ ln xt  μλx ðα  1Þ ln xt ¼ rλx ln xt , μα

which yield: λq ¼

θ ð1  μ Þ2 μ θð1  μÞð1  αÞα , λx ¼ : ðα  μÞðr þ μð1  μÞÞ ðμ  αÞðr þ μð1  αÞÞ

A.3 Equation (6.31) Inserting e δt ¼ λq ln qt þ λx ln xt into the equation for price (6.28):   μlnqt  αlnxt ln pt ¼ θð1  μÞ  μ λq ln qt þ λx ln xt αμ     θð1  μÞμ θð1  μÞα ¼  μλq ln qt þ  μλx ln xt αμ μα we have it that   θð1  μÞμ θð1  μÞμ ð1  μÞμ rθð1  μÞμ  μλq ¼ 1 ¼ αμ αμ r þ μð1  μÞ ðα  μÞðr þ μð1  μÞÞ r , ¼ λq 1μ   θð1  μÞα θð1  μÞα ð1  αÞμ rθð1  μÞα  μλx ¼ 1 ηx ¼ ¼ μα μα r þ μð1  αÞ ðμ  αÞðr þ μð1  αÞÞ r : ¼ λx 1α

ηq ¼

A.4 Proposition 6.3 1. Equations (6.31), (6.47) and (6.17) imply:   Δ ln pt ¼ ηq Δ ln qt þ ηx Δ ln xt ¼ ηq ðμ  1Þ ln qt þ ηx ðα  1Þ ln xt þ ηq þ ηx εt     ¼ E t1 Δ ln pt þ ηq þ ηx εt ¼ r  δt1 þ ηq þ ηx εt : From (6.29), (6.16), (6.24):

170

6 Commodity Prices, Convenience Yield and Inventory Behaviour

  e δt ¼ λq ln qt þ λx ln xt ¼ λq μlnq  t1 þ λx αlnxt1 þ λq þ λx εt ¼ μ λq ln qt1 þ λx ln xt1 þ ðα  μÞλx ln xt1 þ λq þ λx εt   ¼ μe δt1 þ ðα  μÞλx ln xt1 þ λq þ λx εt : Consequently,   Δδt ¼ r ð1  μÞ þ ðμ  1Þδt1 þ r ðα  μÞλx ln xt1  þ r λqþ λx εt ¼ ð1  μÞðr  δt1 Þ þ r ðα  μÞλx ln xt1 þ r λq þ λx εt : 2. ηq + ηx¼ ¼ λq

rθð1  μÞμ rθð1  μÞα r r þ λx ¼ þ 1μ 1  α ðα  μÞðr þ μð1  μÞÞ ðμ  αÞðr þ μð1  αÞÞ   rθð1  μÞ μ α  ¼ ðα  μÞ r þ μð1  μÞ r þ μð1  αÞ ¼

rθð1  μÞ rðμ  αÞ þ μðμð1  αÞ  αð1  μÞÞ ðα  μÞ ðr þ μð1  μÞÞðr þ μð1  αÞÞ

¼

rθð1  μÞ rðμ  αÞ þ μðμ  αÞ ðα  μÞ ðr þ μð1  μÞÞðr þ μð1  αÞÞ

¼

rθð1  μÞðr þ μÞ < 0: ðr þ μð1  μÞÞðr þ μð1  αÞÞ

A.5 Equation (6.40) From (6.26), (6.27), the current availability is at ¼ s t þ y t ¼ a þ

μlnqt  αlnxt , μα

because a ¼ s + 1. Hence, due to (6.16), (6.24): Δat ¼

μlnqt  αlnxt μlnqt1  αlnxt1 ð1  μÞμlnqt1  ð1  αÞαlnxt1  ¼ εt  μα μα μα

References

¼ εt 

171

ð1  μÞμlnqt1  ð1  μÞαlnxt1 ð1  μÞα  ð1  αÞα ln xt1  μα μα

¼ εt þ ð1  μÞða  at1 Þ þ αlnxt1 ¼ ð1  μÞða  at1 Þ þ ln xt :

References Brennan M (1958) The supply of storage. Am Econ Rev 48(1):50–72 Gibson R, Schwartz E (1989) Valuation of long-term oil-linked assets, Anderson graduate School of Management, UCLA, working paper, #6-89 Gibson R, Schwartz E (1990) Stochastic convenience yield and the pricing of oil contingent claims. J Financ 45(3):959–976 Kaldor N (1939) Speculation and economic stability. Rev Econ Stud 7:1–27 Killian L, Murphy D (2014) The role of inventories and speculative trading in the global market for crude oil. J Appl Econ 29:454–478 Lautier D (2009) Convenience yield and commodity markets. Les Cahiers de la Chaire 22, Chaire Finance & Development Durable, p 16 Litzenberger R, Rabinowitz N (1995) Backwardation in oil futures markets: Theory and empirical evidence. J Financ 50(5):1517–1545 Pindyck R (1994) Inventories and the short-run dynamic of commodity prices. RAND J Econ 25 (1):141–159 Pindyck R (2004) Volatility and commodity price dynamics. J Futur Mark 24(11):1029–1047 Scheinkman J, Schechtman J (1983) A simple competitive model with production and storage. Rev Econ Stud 50(3):427–411 Schwartz E (1997) The stochastic behaviour of commodity prices: implications for valuation and hedging. J Financ 52(3):923973. Papers and proceedings fifty-seventh annual meeting, American finance association Smith J (2009) World oil: market or mayhem. J Econ Perspect 23:145–164 Williams J (1986) The economic function of futures markets. Cambridge University Press, New York, p 258 Williams J, Wright B (1991) Storage and commodity markets. Cambridge University Press, Cambridge, p 502

Chapter 7

Commodity Prices and Competitive Storage

Abstract We consider in this chapter a model of rational speculative activity in commodity markets. The salient feature of this model is the non-negativity constraint on storage held by competitive traders. Under random fluctuations on the supply or demand side, traders are supposed to select inventory holdings based on the expectations of future price changes. The equilibrium price for this model is determined as a decreasing function of the current availability defined as the sum of storage and commodity supply per period. Under stochastic supply shocks, the model-generated autocorrelation of prices proves to be not high enough to match the observed time series of commodity prices. In contrast, the model predictions are relevant to persistent demand shocks caused by fluctuations of economic activity and income growth.

7.1

Introduction

The concept of convenience yield is grounded on the premise that storage is valuable for consumers, producers or intermediaries handling the infrastructure for commodity transportation and distribution. This assumption is the key one in the model of the previous chapter, where the marginal value of storage for consumers decreases with storage size. The merit of this model is its analytical tractability, but it disregards the role of a pure speculation motive in stock holding and the commodity price dynamic. As an alternative, one can consider theoretical models of commodity prices with risk-neutral traders holding inventories and maximizing speculative profits. These agents do not consume commodity, nor do they get direct utility gains from inventory holding. The main question in this theory is about the impact of speculative trade and storage behaviour on the dynamics of commodity prices. Does such trade dampen or magnify the price volatility or are both outcomes possible? Another important issue is the extent to which speculation contributes to the autocorrelation of commodity prices. In this chapter, we will examine a canonical competitive storage model with rational traders acting in a commodity spot market with no usage of futures contracts. This activity requires physical delivery of commodities to points of destination and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_7

173

174

7 Commodity Prices and Competitive Storage

physical holdings of stocks, which cannot be negative. Unlike the financial markets permitting short positions in asset trade, physical short sales of commodities are impossible. Angus Deaton and Guy Laroque (1995) compared “permitting the market to borrow the commodity from the future” with consumption of bread from grain that has not yet been harvested. In the convenience yield model of the previous chapter, stock-outs are impossible because the marginal utility of storage is unbounded for stocks close to zero. By contrast, the non-negativity constraint on storage can be binding for profit-maximizing traders that do not get utility gains from physical holding of commodities. Therefore, in the competitive storage model with speculative trade, inventories can be reduced to zero, and stock-outs can be a corner solution. However, empirical evidence suggests that the aggregate measures of commercial commodity inventories never fall to zero. It is, therefore, important to distinguish between so-called “discretionary inventories” defined as the stocks in excess of those committed to production and consumption of commodities and non-discretionary inventories that deliver the benefit of reducing transportation costs, minimizing the risk of disruptions, etc. (e.g. Routledge et al. 2000, p. 1300). One can assume that the non-discretionary inventories utilized in production and consumption are always above zero, while discretionary inventories used by competitive traders can be equal to zero. We will consider below only discretionary inventories for a model of competitive storage and commodity prices examined by Deaton and Laroque (1992). In this model, commodity trade occurs under random supply shocks interpreted by the authors as harvests for agricultural commodities. Such shocks are serially independent and identically distributed over time. The non-negativity constraint implies the essential non-linearity of the model, which is inappropriate for linearization. The equilibrium price is the solution for a stationary rational expectations equilibrium with two regimes. In the trading regime, the expected rate of price growth is equal to the interest rate adjusted for the rate of storage deterioration indicating the physical costs of inventory carryover. In the stock-out regime, the intertemporal link is absent, and price fluctuations are driven only by random supply shocks. One can interpret stock-outs as situations in commodity markets when speculative trade is absent or does not play any significant role in price dynamics. Switching of the equilibrium price between these regimes is determined by a threshold price, which is time-constant in the stationary equilibrium. The equilibrium price is defined as a function of current availability of commodity. It exists under quite general conditions, but cannot be derived analytically because of the non-linearity of equilibrium equations. High non-linearity is a common feature of the commodity storage models with profit-maximizing traders pioneered by Robert Gustafson in his original contribution (1958). An essential non-linearity introduced in these models carries through into non-linearity of the predicted commodity price series. Deaton and Laroque (1992) sought to demonstrate that the standard competitive storage model with random supply is capable of explaining a number of stylized facts about commodity prices, including a high degree of autocorrelation and

7.2 The Base Model

175

skewness of distribution, coupled with the existence of rare but powerful “explosions of prices”. As it turned out, the base competitive storage model with serially uncorrelated supply does not explain satisfactorily the high level of autocorrelation in most of the actual commodity price series. Deaton and Laroque (1995, 1996) and Marcus Chambers and Roy Bailey (1994, 1996) extended the base model to include autocorrelation in stochastic supply process. The question posed for the extended model is about the contribution of speculation to the autocorrelation of prices. Theory predicts that intertemporal commodity trade generates serial dependence in prices. However, estimations and numerical simulations of the extended model with supply shocks show that commodity trade cannot increase the serial dependence in commodity prices to the extent observed in the actual price series. Storage fulfils its customary role of “leaning against the wind”, but in the case of supply shocks its contribution to autocorrelation of commodity prices proved to be insignificant. This conclusion is not valid for the case of persistent shocks of consumer demand driven by fluctuations of the income growth rate. Eyal Dvir and Kenneth Rogoff (2009) extended the commodity storage model of Deaton and Laroque to incorporate demand shocks. Analysis of structural breaks in the long-term series of oil price revealed significant changes in the persistence of income shocks and volatility of the oil price since the beginning of oil production. Moreover, historical narratives reveal a link between the periods of high oil price volatility with the epochs of industrialization in the United States in the second half of the nineteenth century and in East Asia since the 1970s. In both cases, industrialization caused positive shocks of income growth rate that brought about growth of the oil demand that coincided with the restrictions of access to supplies. The inelastic supply in Dvir-Rogoff’s model cannot accommodate persistent demand shocks and, as a result, commodity trade can amplify the oil price volatility and contribute to the persistence of oil price fluctuations.

7.2

The Base Model

There are two groups of competitive actors in the commodity market: commodity traders and final consumers. The number of each group is constant and normalized to one. Neither traders nor consumers have informational advantages. A risk-neutral trader is engaged in inventory management by purchasing or selling qt units of commodity in any period of discrete time t. The trader’s problem is to maximize the expected discounted cash flow from trade over an infinite time horizon: E0

1 X t¼1

β t pt qt ,

ð7:1Þ

176

7 Commodity Prices and Competitive Storage

where β ¼ 1/(1 + r) is the discount factor, r is the riskless real interest rate and pt is the commodity price in period t. The trader is a buyer of commodity if qt  0, and a seller if qt  0. The trader can borrow money or deposit profits in the bank at rate r. We follow Deaton and Laroque (1992) in assuming a simple storage technology with constant returns such that one unit of commodity stored in period t  1 yields 1  δ units of storage in period t. Parameter δ, 0  δ < 1, is the rate of storage deterioration indicating the “shrinkage” charges of carrying inventories over one period. Trading operations result in changes of storage: st ¼ ð1  δÞst1  qt :

ð7:2Þ

Storage at the end of period t is equal to storage transferred from period t  1 minus the volume of sales by traders in period t. The initial storage is s0. In any period, the storage must be non-negative: st  0:

ð7:3Þ

Sales are thus constrained by storage available at the beginning of period t, qt  (1  δ)st  1. Consumer demand is a continuous and strictly decreasing function of price, D( pt) defined on an infinite or finite interval with lower bound p0 and upper bound p1 and tending to infinity as p tends to p0. The market-clearing condition implies that the demand by consumers and traders is equal to supply xt: Dðpt Þ  qt ¼ xt :

ð7:4Þ

Supply is inelastic and given by a sequence of serially independent positive random variables xt with an identical cumulative probability distribution F(xt) having a compact measure support defined as a finite interval with lower bound x and upper bound x. As Deaton and Laroque (1992, p. 5) note, it is also possible to allow demand to be stochastic in which case xt is interpreted as the difference between the supply and the stochastic part of demand. Moreover, an instantaneous supply response can be accounted for by interpreting D( pt) as a non-stochastic excess demand function. In the case where there are no inventory holders, equilibrium price is given by the equation D( pt) ¼ xt. It is assumed that the inverse demand function P(xt) satisfies condition: PðxÞ < 1. It means that when supply takes its lowest possible value, price is finite. In other words, no matter how low the supply, there is always a finite price that clears the market. Figure 7.1 demonstrates an example of the inverse demand function P(xt) for the case: p1 ¼ + 1, p0 ¼ 0. The marked interval on the horizontal axis is the measure support for xt.

7.3 Two Regimes of Inventory Management

177

Fig. 7.1 The inverse demand function and the measure support

7.3

Two Regimes of Inventory Management

By substituting qt ¼ (1  δ)st  1  st from (7.2) into the objective function (7.1) and rearranging terms in this function, we represent the trader’s problem as: max E 0 st 0

1 X

  βt1 βð1  δÞptþ1  pt st ,

ð7:5Þ

t¼1

In this formulation of the problem, traders maximize the expected discounted returns on inventory holdings. In any period, the choice of storage st is based on the expectation of next-period price conditional on information from period t, Etpt + 1. The solution for problem (7.5) satisfies the zero-profit conditions: st  0, if βð1  δÞE t ptþ1 ¼ pt ,

ð7:6Þ

st ¼ 0, if βð1  δÞE t ptþ1 < pt ,

ð7:7Þ

In the trading regime (7.6), the storage is positive only if the expected one-period return on inventory holding is zero. In the stock-out regime (7.7), all inventories are sold out if the expected return is negative. If the expected return was positive, β(1  δ)Etpt + 1 > pt, the risk-neutral traders would have driven the current price up until it equalled the expected next-period price net of the carrying costs. In both regimes, the price meets the market-clearing condition (7.4) rewritten as pt ¼ Pðxt þ qt Þ:

ð7:8Þ

Define a current availability of commodity in period t as the sum of current supply and inventory stock by the beginning of this period:

178

7 Commodity Prices and Competitive Storage

at ¼ xt þ ð1  δÞst1 :

ð7:9Þ

Since st  1 is non-negative, at lies in the infinite interval ½x, 1Þ. For the stock-out regime st ¼ 0, and one can express the market-clearing price (7.8) as the function of current availability: pt ¼ Pðat Þ,

ð7:10Þ

because the following two cases are possible if a stock-out occurs: (a) traders sell all the storage they have transferred from the previous period: if st  1 > 0, st ¼ 0, then qt ¼ (1  δ)st  1 and xt + qt ¼ at; (b) traders have nothing left to sell: st  1 ¼ st ¼ qt ¼ 0, implying that xt + qt ¼ xt ¼ at. In both these cases (7.10) holds, while in the latter case consumer demand equals exogenous supply, D( pt) ¼ xt. The zero-profit conditions (7.6), (7.7) and the price function under stock-out (7.10) imply the following recursive equation for equilibrium price:   pt ¼ max βð1  δÞE t ptþ1 , Pðat Þ :

ð7:11Þ

The equilibrium price is the maximum of prices under trading and stock-out. The current-period price depends on the expectation of next-period price, which must satisfy the similar maximum condition in the next period and so forth. Therefore, a solution for equation (7.11) is a sequence of prices satisfying conditions of rational expectations equilibrium for any period.

7.4

Stationary Rational Expectations Equilibrium

When the uncertainty of period t is resolved, traders observe current supply xt, so they know current availability at. The latter is the only state variable for the trader’s problem, because commodity carried over from the previous period, (1  δ)st  1, and commodity supplied in the current period, xt, are perfect substitutes. Moreover, serially uncorrelated shocks xt provide no information about changes of future supply and movements of future prices. Information about current availability is therefore sufficient for the formation of price expectations and determination of equilibrium price. In a rational expectation equilibrium, the price is a function of the state variable, which is the same on the individual and aggregate levels: pt ¼ pðat Þ:

ð7:12Þ

Unlike the equilibrium price function under stock-out P(at), the function p(at) is unknown and has to be found as the model solution.

7.4 Stationary Rational Expectations Equilibrium

179

Traders account for the market-clearing condition (7.4) and decide about storage on the basis of the observed state variable at, because, from (7.2), (7.4), (7.9), (7.12), st ¼ (1  δ)st  1 + xt  D( pt) ¼ at  D( pt), and storage is represented as the function of current availability: st ¼ sðat Þ ¼ at  Dðpðat ÞÞ,

ð7:13Þ

provided that s(at)  0. Hence, one can rewrite the equilibrium price equation (7.11) as pðat Þ ¼ max ½βð1  δÞE t pðxtþ1 þ ð1  δÞsðat ÞÞ, Pðat Þ,

ð7:14Þ

since the next-period equilibrium price is pt + 1 ¼ p(at + 1) ¼ p(xt + 1 + (1  δ)s(at)). Due to the stationarity of random supply, one can omit time indices in (7.13), (7.14) and define a stationary rational expectations equilibrium as the price function p(a) satisfying the functional equation for all a 2 ½x, 1Þ: pðaÞ ¼ max ½βð1  δÞEpfx þ ð1  δÞða  DfpðaÞgÞg, PðaÞ:

ð7:15Þ

The expectation on the right-hand side of (7.15) is taken with respect to the random variable x 2 ½x, x . The equilibrium price function p(a) provides the intertemporal price link given by (7.15) and ensures that storage s(a) clears the market, as follows from (7.13). A high non-linearity of equation (7.15) results from the maximum condition and the composite functional form of the price expectation in (7.15) that captures the feedback effect of current consumer demand on the dynamic choice of traders. Although it is impossible to solve this functional equation analytically, Deaton and Laroque (1992) proved the existence of equilibrium price function p(a) by using a contraction mapping for (7.15). They showed that the equilibrium price function exists and is unique in the class of continuous non-increasing functions. The structure of p(a) is specified by a threshold price p that determines the regime choice by traders for any realization of state variable a: pðaÞ > PðaÞ for PðaÞ < p ,

ð7:16Þ

pðaÞ ¼ PðaÞ for PðaÞ  p ,

ð7:17Þ

p ¼ βð1  δÞEpðxÞ:

ð7:18Þ

where p is equal to:

The equilibrium price function p(a) is strictly decreasing whenever it is positive. The equilibrium level of storage, s(a) ¼ a  D( p(a)), is strictly increasing whenever P(a) < p.

180

7 Commodity Prices and Competitive Storage

Fig. 7.2 The equilibrium price function

Fig. 7.3 The equilibrium storage function

The cut-off price p equals the discounted expected equilibrium price in the absence of inventories (7.18). At this price, with no inventory demand, a unit held into the next period would make zero expected profit. The choice of regime depends on the relationship between the stock-out price P(a) and the cut-off price p. If P(a) is below p, commodity traders choose trading (7.16) and p(a) exceeds P(a), because total demand is more than the demand for current consumption. If P(a) is above p, traders choose stock-out (7.17) and p(a) coincides with P(a), because demand is equated to current supply (including leftover inventories) and there is no inventory demand. Figure 7.2 shows the equilibrium price function p(a) drawn with the solid kinked curve. The kink point corresponds to the cut-off price p. The dashed curve is the off-equilibrium continuation of the inverse demand function P(a) for P(a) < p. The stock-out takes place if supply x is below the threshold level corresponding to the cut-off price, x ¼ D( p). Figure 7.3 depicts the equilibrium storage function (7.13), which is increasing for x  x, and is identically zero for x < x since in the latter case a ¼ x and D( p(a)) ¼ D(P(x))  x.

7.4 Stationary Rational Expectations Equilibrium

181

Fig. 7.4 The piece-linear autoregression function

The simple rules of inventory management defined by conditions (7.16), (7.17) imply a positive autocorrelation for equilibrium prices. For the trading regime, one can consider an autoregression of equilibrium price given by conditional expectation E( pt + 1| pt) ¼ pt/[β(1  δ)], as implied from (7.15), with innovation term ηt + 1 ¼ p (at + 1)  pt/[β(1  δ)], which is uncorrelated with its previous values and with prices dated t and earlier. For the stock-out regime, equilibrium prices are serially uncorrelated, pt ¼ P(xt), and the autoregression is independent of pt due to (7.18): E( pt + 1| pt) ¼ p/[β(1  δ)] with innovation term ηt + 1 ¼ p(at + 1)  p/[β(1  δ)]. The combined autoregression function for both regimes is piecewise linear:   min ðpt , p Þ E ptþ1 jpt ¼ , β ð1  δ Þ

ð7:19Þ

because p(at) < p under trading and p < p(at) under stock-out, as one can see from Fig. 7.3. The piece-linear autoregression function (7.19) is depicted in Fig. 7.4. It has slope 1/[β(1  δ)] for pt < p and zero for pt > p. In the former case, the expected rate of price growth is approximately equal to the interest rate plus the rate of storage deterioration, since 1/[β(1  δ)]  1 + r + δ for small r and δ, while in the latter case this rate is equal to zero. The conditional variance of prices Var( pt + 1| pt) has a similar structure as the conditional expectation E( pt + 1| pt). Deaton and Laroque (1992) proved that, given the convexity of inverse demand function P(xt), the conditional variance is non-decreasing with the current price:   ∂Var ptþ1 jpt =∂pt  0:

ð7:20Þ

182

7 Commodity Prices and Competitive Storage

Like the autoregression (7.19), the conditional variance increases with pt until pt ¼ p, after which it is constant, because there are no inventories. As follows from (7.20), the next-period price is more volatile at a higher level of the current-period price, because there are fewer inventories under a higher price. The fact that conditional variance is lower under trading than under stock-out indicates a pricestabilizing effect of inventory management. Deaton and Laroque (1992) examined the long-run dynamics of prices and showed that under a positive rate of storage deterioration, 0 < δ < 1, the invariant distribution of storage is bounded and defined for s 2 ½0, ðx  xÞ=δ . Inventories cannot be zero for all t under condition that the lowest price in the absence of inventories is below the threshold level, PðxÞ < βð1  δÞEPðxÞ, but stock-out occurs in finite time with a probability of one so that the price follows a renewal process. The deterioration of inventories at positive rate δ ensures that the equilibrium price in the trading regime, pt ¼ p(at), becomes in finite time above the threshold level p and the process switches to the stock-out regime.

7.5

Autocorrelation of Supply

Autoregression (7.19) implies a serial dependence in prices, even though there is no serial dependence in the underlying supply process. The actual series of commodity prices display high levels of positive autocorrelation that could be explained, at least in part, by intertemporal price smoothing caused by the activity of speculators. If, for example, the current price is low, inventory demand builds up, and the release of inventories in subsequent periods moderates price hikes. However, the base model of competitive storage with serially independent supply predicts positive autocorrelation only for levels of price below the threshold p and zero autocorrelation whenever the price is above the threshold. This is a consequence of the piece-linear autoregression for prices (7.19). If stocks are not being held, the next period’s price is unaffected by the current price. Actual price series demonstrate high autocorrelation both for low and high prices, and the base model does not account for this evidence. Figure 7.5 shows the actual price series for cotton together with the one-step-ahead predictions from the model simulations by Deaton and Laroque (1996). The base model does not track the commodity price data, because the upper limits of simulated price expectations contradict to hikes observed for actual series. One should note that high autocorrelation of commodity prices is captured by a simple first-order linear autoregression (the AR1 model). This autoregression results from the linear inverse demand function, P(xt) ¼ A  Bxt, and the first-order linear autoregression for supply process xt. Figure 7.5 shows simulated series for the AR1 model that lags behind the reversions of peaks, but predicts price movements better than the base storage model. However, the AR1 model attributes all autocorrelation of prices to the supply process and neglects the role of speculation and storage in the commodity price dynamic.

7.5 Autocorrelation of Supply

183

Fig. 7.5 Actual and one-period-ahead expectations of commodity prices. Source: Deaton and Laroque (1996, p. 914)

One can extend the base storage model by assuming positive autocorrelation for the supply process. This assumption is relevant for oil, because discoveries of new fields or inventions of new extraction technologies may have a long-lasting influence on the subsequent growth of oil supply. For agricultural commodities, positive autocorrelation of supply takes place in the case of tree crop, when damage to trees by bad weather, pests or diseases reduces crops for subsequent periods, or even in the case of annuals, if profits obtained from good harvests are invested to improve land and technologies. As Deaton and Laroque (1996, p. 915) point out, “most accounts of price fluctuations by market commentators identify supply shocks as the main source of instability”. The question of interest is whether the activity of traders can enhance the serial dependence that emerges in the supply process.

7.5.1

The Equilibrium Price Function

The extended model of competitive storage was examined by Chambers and Bailey (1994, 1996), and we will follow the exposition of their papers. The stochastic process for inelastic supply is supposed to be first-order Markov, with transition probabilities characterized by the conditional cumulative probability distribution F (x, x0):

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7 Commodity Prices and Competitive Storage

F ðx, x0 Þ ¼ Pr ðxtþ1  x0 jxt ¼ xÞ:

ð7:21Þ

This is the probability that supply in period t + 1 is less than or equal to x0 when supply in period t is x. The conditional probability distribution is defined for a compact measure support X for the next-period supply x0: X ¼ fx0 2 Rjx  x0  xg. The problem of expected profit maximization (7.5) yields the same zero-profit conditions as (7.6), (7.7). These conditions imply the existence of two inventory management regimes and the same maximum condition for the equilibrium price as (7.11). In the model with serially independent supply, the only state variable was the current availability at ¼ (1  δ)st  1 + xt. In the extended model with time dependence of supply, there are two state variables: storage transferred from the previous period yt ¼ (1  δ)st  1 and current supply xt. The latter is the state variable relevant to traders’ decisions because it carries information about the next-period probability distribution of supply (7.21). Consequently, the equilibrium price is determined as the function of two state variables: pt ¼ pðyt , xt Þ: As above, the equilibrium price in the stock-out regime is given by the inverse demand function: pt ¼ Pðat Þ ¼ Pðyt þ xt Þ: In the stationary rational expectations equilibrium, the equilibrium price function p(y, x) satisfies a functional equation similar to (7.15): 

 Z 0 0 0 pðy, xÞ ¼ max βð1  δÞ pðy , x ÞF ðx, dx Þ, Pðy þ xÞ ,

ð7:22Þ

X

where y0 ¼ ð1  δÞ½y þ x  Dðpðy, xÞÞ is the inventory carryover to the next period, as follows from the conditions of storage balance (7.2) and market clearing (7.4). The Lebesgue-Stieltjes integral in (7.22) is the expectation of the one-period-ahead equilibrium price p(y0, x0) conditional on the current-period observation of supply x. Chambers and Bailey (1994) proved that the solution for (7.22) exists, unique and non-increasing in the transferred storage, ∂p/∂y  0. A larger inventory stock transferred from the previous period implies a larger availability of commodity in the market and a lower equilibrium price. Whether or not the equilibrium price function p(y, x) is non-increasing in its second argument depends on the properties of transition probability (7.21).

7.5 Autocorrelation of Supply

185

The transition probability exhibits positive autocorrelation of supply in the sense of first-order stochastic dominance if:   F xð1Þ , x0  F xð2Þ , x0

ð7:23Þ

for any x(1)  x(2) and any x0 2 X. This notion of positive autocorrelation means that the probability of next-period supply at least at level x0 is higher for a larger currentperiod supply x(1). An abundant output in the current period x provides information that with a higher probability output in the next period x0 will also be abundant. This information transfers probability weights upwards and shifts downwards the conditional probability distribution function F(x, x0). Chambers and Bailey (1994) showed that under positively autocorrelated disturbances of supply in the sense of condition (7.23), the equilibrium price function is non-increasing in current supply, ∂p/∂x  0. A large supply in the current period signals a large supply and a low price in the next period, thereby making storage less profitable. A decreasing inventory demand leads to a lower price in the current period despite the increasing consumer demand. The structure of equilibrium price function is similar to the one in the case of serially independent supply process: pðy, xÞ > Pðy þ xÞ for Pðy þ xÞ < p ðxÞ

ð7:24Þ

pðy, xÞ ¼ Pðy þ xÞ for Pðy þ xÞ  p ðxÞ,

ð7:25Þ

where p ðxÞ ¼ βð1  δÞ

Z

pð0, x0 ÞF ðx, dx0 Þ:

ð7:26Þ

X

The equilibrium price function p(y, x) is above the inverse demand function P (y + x) under trading (7.24) and coincides with P(y + x) under stock-out (7.25) for prices exceeding the threshold price p(x). As in the case of serially independent supply process, the threshold price (7.26) equals the discounted expected equilibrium price in the absence of inventories. It is important that p(x) varies with information about current supply, unlike the constant cut-off price p under serially uncorrelated supply. Figure 7.6 generalizes Fig. 7.2 to the case of a positively autocorrelated supply process. It demonstrates the equilibrium price function and switching between the inventory management regimes for large and small current supply, x(1) > x(2). The equilibrium price as the function of transferred storage y is lower for large supply, p (y, x(1)) < p(y, x(2)), for all y. The inverse demand function is lower for large supply too, P(y + x(1)) < P(y + x(2)), and the threshold price is also lower, p(x(1)) < p(x(2)), because from (7.26) and (7.23):

186

7 Commodity Prices and Competitive Storage

Fig. 7.6 The equilibrium price function for positive autocorrelation of supply

dp ðxÞ ¼ βð1  δÞ dx

Z

pð0, x0 Þ X

∂F ðx, dx0 Þ  0: ∂x

The threshold price p(xt) varies with current supply, and the non-linear autoregression for equilibrium price is   min ½pt , p ðxt Þ E ptþ1 jpt , xt ¼ , βð1  δÞ

ð7:27Þ

The expected next-period price is conditional on the current price and supply and is non-increasing with current supply due to the positive autocorrelation (7.23). The autoregression function (7.27) is shown in Fig. 7.7 for high and low levels of supply, x(1) > x(2). The variable of current supply xt in the non-linear price autoregression (7.27) is observable in theory, but in practice, the information on commodity supply is not available at the same quality level as the price series data. Because of the lack of quantity data, equation (7.27) cannot be estimated in a straightforward way. The econometric method elaborated by Deaton and Laroque (1995) includes evaluation of the first and second moments for current supply xt and availability at conditional on current prices. This method requires computation of the theoretical equilibrium price function (7.22) and of the invariant probability distribution for xt and at that depend on the model parameters to be estimated from price data.

7.5 Autocorrelation of Supply

187

Fig. 7.7 The autoregression function for positive autocorrelation of supply

7.5.2

Contribution of Storage to Autocorrelation of Prices

The functional equation (7.22) does not permit an analytical solution and is solved numerically as part of the estimation procedure. Deaton and Laroque (1995) examined numerical solutions for the linear inverse demand function P(xt) ¼ A  Bxt and for supply process driven by the linear autoregression: xtþ1 ¼ ρxt þ ξtþ1 ,

ð7:28Þ

where ρ is the autocorrelation coefficient, 1 < ρ < 1, and ξ’s are independent identically distributed shocks with a positive mean. For these specifications of fundamental factors, they used a cubic spline approximation for iterative calculations of the equilibrium price function as a solution of the functional equation (7.22) and estimated the model parameters A, B, ρ, δ while assuming that the annual interest rate is r ¼ 0.05. The accuracy of the cubic spline approximation was determined by the frequency of the number of grid points between which splines interpolate. The competitive storage model performs much better under serially dependent supply, as Deaton and Laroque (1995) showed. However, the storage model with a serially correlated supply process attributes almost all autocorrelation of prices to supply rather than to the speculative activity. The estimated (through price data) autocorrelation coefficients proved to be quite high for the supply process and even exceeding the autocorrelation coefficients for actual prices for nine of the twelve commodities under consideration (Op. cit., p. S38). Simulations also showed that the effect of storage on autocorrelation of prices is significant for low levels of autocorrelation of supply and insignificant for high

0.7 0.8 0.9 1.0

7 Commodity Prices and Competitive Storage

δ = 0.025 δ = 0.05

0.4 0.5 0.6

δ = 0.1 δ = 0.2 δ = 0.3

0.1 0.2 0.3 0.0

Autocorrelation of price

188

δ = 0.5 δ = 0.8 δ=1 0.0

0.1

0.2

0.3 0.4 0.5 0.6 0.7 Autocorrelation of the harvest

0.8

0.9

1.0

Fig. 7.8 Autocorrelations of price and supply in relation to the rate of storage deterioration. Source: Deaton and Laroque (1996, p. 915)

levels. Deaton and Laroque (1996) demonstrated this effect for a discrete Markov chain for supply processes reproducing a range of autocorrelation coefficients ρ from zero to one. Figure 7.8 shows the dependence between the model-generated autocorrelation coefficients for prices and coefficients ρ for supply processes. In these simulations, the linear demand parameters are A ¼ 0.2, B ¼ 0.5, the interest rate is r ¼ 0.05, and the rate of storage deterioration δ varies between 0.025 and 1. In the extreme case of δ ¼ 1 corresponding to the 45-degree line in Fig. 7.8, storage is zero and the autocorrelation of prices coincides with the autocorrelation of supply. One can see that when the autocorrelation of supply is low, the activity of traders leads to substantial increases in the autocorrelation of prices if the rate of storage deterioration is small. As the autocorrelation of supply increases, the contribution of storage to the autocorrelation of prices decreases and becomes almost negligible for values of ρ close to one. Note that for the similar dependence shown in Fig. 6.3 in Chapter 6, the contribution of storage to the autocorrelation of prices is also significant if the autocorrelation of supply is not high. Deaton and Laroque (1996, p. 899) offered the following intuitive explanation for this result. If prices are highly autocorrelated in the absence of storage, there is little scope to smooth prices over time. In such cases, price swings are large and longlived, hence price smoothing with costly storage requires large stocks to be held for long periods and this activity is “typically neither profitable nor socially desirable”.

7.6 Stochastic Demand: Persistence of Shocks and Volatility of the Oil Price

189

As Deaton and Laroque (1995, 1996) concluded, storage seems to play a small role in generating the autocorrelation in commodity prices. On the one hand, the actions of profit-maximizing traders are not sufficient to generate high autocorrelation coefficients for commodity prices. On the other hand, the estimated autocorrelation of the supply process is too high to infer that supply is the main source of price autocorrelation. For agricultural commodities, weather fluctuations can influence the harvests for several years, but not to an extent consistent with high autocorrelation of actual prices. In contrast, inferences based on the competitive storage model prove to be relevant to cases of persistent demand shocks.

7.6

Stochastic Demand: Persistence of Shocks and Volatility of the Oil Price

Dvir and Rogoff (2009) extended Deaton and Laroque’s model (1992, 1996) to introduce consumer income as a factor of demand for the crude oil market. In the modified competitive storage model, consumer demand is assumed to be a function of oil price and consumers’ income. On the one hand, the income elasticity of oil demand varies between 0.5 and 0.8, according to the International Energy Agency (2007) estimates, implying that income is a significant factor of demand variations. On the other hand, the oil demand depends on a variety of economic factors, including GDP, industrial output, technical change and population growth. These factors may feature a high degree of volatility and commodity trade can contribute to their impact on the oil price. Dvir and Rogoff (2009) examined the very long-run series of the real oil price since 1861 and found strong evidence of changes in the persistence of demand shocks and the volatility of this price. These changes can be attributed to shifts in demand factors that occurred under different stages of economic development and for different regimes of access to oil supplies. The coincidence of rapid growth of oil demand under industrialization with restrictions on oil supply preconditioned a role for commodity trade as an amplifier of oil price fluctuations. We present below Dvir and Rogoff’s model and discuss its empirical implications.

7.6.1

The Trader’s Problem

Dvir and Rogoff’s model (2009) also differs from Deaton and Laroque’s model in the description of storage technology. There is no storage deterioration, δ ¼ 0, but storing is costly and incurs constant unit cost c per period. Traders are supposed to maximize the expected discounted cash flow net of the costs of storing:

190

7 Commodity Prices and Competitive Storage

E0

1 X

βt ðpt qt  cst Þ,

ð7:29Þ

t¼1

subject to the storage balance: st ¼ st1  qt

ð7:30Þ

and the non-negativity constraint st  0. As above, this problem can be represented as: max E0 st 0

1 X

  βt1 βptþ1  pt  c st ,

ð7:31Þ

t¼1

implying zero-profit conditions similar to (7.6), (7.7): st  0 if βEt ptþ1 ¼ pt þ c

ð7:32Þ

st ¼ 0 if βE t ptþ1 < pt þ c:

ð7:33Þ

In any period, positive inventories are held only if the expected discounted future price covers the current price and the cost of storage.

7.6.2

Equilibrium Prices

The oil demand in Dvir and Rogoff’s model is the function of price and income, Dt ¼ D( pt, It), which is decreasing in the first argument and increasing in the second one. A specification of inverse demand used in their model is the isoelastic function of the consumption-to-income ratio: pt ¼ PðDt =I t Þ ¼ ðDt =I t Þγ ,

ð7:34Þ

where γ > 1 is the inverse price elasticity of demand (the income elasticity of demand is unity). Let e at ¼ at =I t and est ¼ st =I t denote “effective” availability and storage, respectively. Then, from the market-clearing condition, Dt =I t ¼ e at  est, and the oil price is given by: at  est Þγ : pt ¼ ðe

ð7:35Þ

The zero-profit conditions (7.32), (7.33) and the price function (7.35) imply the equilibrium price equation:

7.6 Stochastic Demand: Persistence of Shocks and Volatility of the Oil Price

  pt ¼ max βEt ptþ1  c, e at γ ,

191

ð7:36Þ

at γ is price in the stock-out regime. In the trading regime, the effective where pt ¼ e storage est ensures the compatibility of price equations (7.35) and (7.36): ðe at  est Þγ ¼ βE t ptþ1  c:

ð7:37Þ

The equilibrium price solving functional equation (7.36) is a function of two or more state variables—the effective availability of oil and the exogenous variables specifying income dynamics.

7.6.3

Income Processes

There are two alternative stochastic processes for income: one where income moves around a deterministic trend and the other one where trend is stochastic. The deterministic trend income I t grows with constant rate μ, whereas the relative income it ¼ I t =I t follows the AR1 process: itþ1 ¼ iρt eεtþ1 ,

ð7:38Þ

where ρ 2 (0, 1) and εt + 1 is a normally distributed serially uncorrelated shock with zero mean. In the case of stochastic trend, the relative income follows a geometric random walk: itþ1 ¼ it eμtþ1 μ

ð7:39Þ

with a mean-reverting growth rate: μtþ1  μ ¼ ϕðμt  μÞ þ νtþ1 ,

ð7:40Þ

where ϕ 2 (0, 1) and νt + 1 is a normally distributed serially uncorrelated shock with zero mean.

7.6.4

Price-Stabilizing and -Destabilizing Inventory Behaviour

Dvir and Rogoff simulated the dynamic behaviour of their model in response to income shocks. We will try to interpret their results regarding the effects of random shocks to income level (7.38) and to income growth rate (7.39) on the dynamic

192

7 Commodity Prices and Competitive Storage

behaviour of storage and oil price. Let the supply of oil be inelastic and linked to the trend income, as is assumed in their paper (2009, p. 22): xt ¼ λI t , where λ is a constant. Suppose that the relative income follows the AR1 process (7.38), and let a positive income shock εt > 0 occur in period t. Without storage, supply is equal to consumer demand, xt ¼ Dt, and γthis shock causes an increase of price given by (7.34): pt ¼ ðxt =I t Þγ ¼ λI t =I t . Storage behaviour can mitigate or amplify this effect depending on whether inventories decrease or increase in response to the shock. In the trading regime, the change of storage is defined by equation (7.37). From (7.9), the effective availability is equal to: e at ¼ ðst1 þ xt Þ=I t . Inserting this into (7.37) yields the change of storage in period t:   1 Δst ¼ λI t  I t βE t ptþ1  c γ :

ð7:41Þ

The first term on the right-hand side of this equation is fixed, and the influence of income shock εt > 0 on storage is transmitted through two countervailing effects that appear in the second term. The shock affects storage directly, through an increase of income It, and indirectly, through an increase of the expected future price Etpt + 1. The direct effect causes a reduction of storage, while the indirect effect leads to an increase of storage. The former dominates the latter for the AR1 process (7.38), because the future relative income is expected to be less than the current relative income and any shock εt is expected to dissipate over time. As a result, the positive income shock causes the price upsurge that prompts traders to sell their inventories. The reduction of storage acts as a stabilizing factor in the market. Suppose now that relative income follows the geometric random walk (7.39) and a positive shock to income growth rate νt occurs in period t, so that μt > μ. As in the AR1 case, the shock influences the inventory behaviour through two effects in equation (7.41). Unlike that case, the shock effect on income level is persistent. Therefore, future relative incomes are expected to be higher than the current relative income, and the next-period price is expected to increase relative to the current price. The indirect effect on price expectation in (7.41) can dominate the direct effect on current price through an increase of It, and the income shock can cause an increase of storage (this is the case if βEtpt + 1 is sufficiently close to c, as is seen from (7.41)). The response of traders in such case magnifies the shock effect on demand in anticipation of a higher demand in the future. As a result, the storage behaviour can exert a destabilizing effect on the price dynamic under a stochastic income trend. The model with persistent demand shocks reveals a link between the price volatility amplified by commodity traders and the persistence of price fluctuations. A typical feature of speculative booms is that an increase of volatility leads to an increase of price autocorrelation prompted by actions of speculators. A higher current price leads to higher future prices because it encourages speculative demand driven by expectations of further demand growth. These results of the model are valid only under the assumption of inelastic supply, which is associated with restrictions on the access of market participants to resource

7.6 Stochastic Demand: Persistence of Shocks and Volatility of the Oil Price

193

supplies. Dvir and Rogoff (2009) showed that under a perfectly elastic supply absorbing demand shocks, the magnifying effect of storage behaviour is absent and it is price-stabilizing in both cases of relative income process (7.38), (7.39). This is seen from equation (7.41) with the first term on the right-hand side λI t replaced by xt that denotes a flexible supply absorbing income shocks. Thus, only a combination of regimes of persistent demand shocks and inelastic supply allows storage behaviour to contribute substantially to the oil price volatility.

7.6.5

The Three Epochs of Oil

This inference is relevant to evidence regarding changes in the volatility and persistence of oil price. Figure 7.9 shows the real oil price for the period of available statistical data 1861–2019. A cursory glance reveals stark differences in the oil price behaviour at different time intervals. At least two points of regime change are clearly visible in Fig. 7.9: at the end of the 1870s and at the beginning of the 1970s. Dvir and Rogoff identified three periods of the oil price dynamic marked in this figure with vertical lines: the early high-volatility period 1861–1878, the low-volatility period 1879–1972 and the late high-volatility period since 1973. The oil price series exhibited structural breaks, and the regime switching occurred with 95% confidence in selected turning points. The hypothesis of regime switching in the oil price series accords well with the predictions of the competitive storage model and with historical narratives of the oil market outlined by Dvir and Rogoff. In the first period, 1861–1878, access to oil supply in the United States was restrained by the railroad monopoly seeking rent extraction. Monopolization occurred against the backdrop of industrialization in the United States that experienced growth of industrial output by a factor of three

Fig. 7.9 The real price of oil, 1861–2019. Source: Quandl (2020)

194

7 Commodity Prices and Competitive Storage

between 1860 and 1880. Industrialization encouraged a rapid growth of oil consumption that drove persistent demand-led fluctuations of the oil price. In 1878, the railroad monopoly broke as oil producers launched construction of the bypassing long-distance pipeline Tidewater. In the second period, 1879–1972, the access to oil supplies was unrestricted. The crucial event was the discovery of the giant East Texas Oil Field in 1934 that created an oil glut. To prevent a slump in price, the U.S. government imposed an effective control of the oil supply by setting quotas for producers. The government agencies used regulation of the oil industry to stabilize the market and to increase production during World War II and in a number of post-war crisis events involving the Middle East in the 1940–1960s. This policy resulted in a significant drop of oil price volatility after 1934. As one can see from Fig. 7.9, the oil price volatility was suppressed for four decades between 1934 and 1973. This happened due to the highly flexible supply absorbing external shocks. The third period, 1973–2019, began with the actions of the oil cartel OPEC to restrict the West’s access to oil in the Middle East with its easily extractable reserves. Three years earlier, in 1970, the U.S. oil production reached a peak, and in the subsequent decades, before the boom of shale oil, the U.S. oil producers lacked excess capacities to be able to level out the cartel’s market power. The world economy returned to the regime of restrained access to oil supplies in the period of rapidly increasing world demand for oil brought about by rapid industrialization of the East Asian economies. As in the early epoch of oil, persistent shocks to demand were transmitted by the oil market into highly volatile and persistent fluctuations of the oil price.

7.7

Concluding Remarks

Commodity prices are usually highly volatile, and, in our opinion, there is nothing extraordinary in the high volatility of the crude oil prices observed after the breaking point of 1973. On the contrary, one can view the period of relatively stable oil price, 1934–1973, as an unusual case of tranquil commodity markets. State regulation played a role in stabilizing the oil market, because a competitive supply of resources cannot be perfectly flexible and accommodate all demand shocks. The inelastic supply in periods of “restricted access to supplies” contributed to the high price volatility due to the effects of monopoly power that manifested itself several times in the history of oil and other resource markets. We can summarize the main results presented in this chapter as follows. For the canonical model of competitive storage, the equilibrium price function exists and is unique and is decreasing in the current availability of commodity. This function ensures synthesis of the two regimes of inventory management for a stationary rational expectations equilibrium. Switching between these regimes is determined by the threshold price. Traders choose inventory holding if the current price is below

References

195

the threshold and stock holding covers the carrying costs; if the current price is above the threshold, traders choose the stock-out regime. In trading regime, the intertemporal link in commodity pricing exists, whereas this link is absent under stock-out and the commodity is priced as a perishable good. The dual nature of commodities can also be explained in terms of an embedded timing option. A commodity is priced like an asset if it is stored for future consumption, but it is priced like a consumer good if it is consumed, and the current spot price is the maximum of its current asset and consumption values (Routledge et al. 2000). The process of switching between these two options can be characterized by an average expected time of staying in each regime that, in turn, determines the expected long-term growth rate of the commodity price. The competitive storage model implies that the contribution of speculative activity to the autocorrelation of prices can be significant for weakly autocorrelated supply processes and it is insignificant for strongly autocorrelated processes. However, supply processes are usually not highly autocorrelated and hence a more plausible source of high autocorrelation of commodity prices lies on the demand side. Competitive storage can amplify persistent demand fluctuations caused by shocks of income growth under inflexible supply. Deaton and Laroque (1995, 1996) showed that the competitive storage model with stochastic supply could not explain the high degree of serial correlation observed in the commodity price series. They concluded that the correlations in price series implied by this model are significantly lower than those observed in the actual price series. However, Cafiero et al. (2011) re-estimated their storage model using the same theoretical specification, commodity price dataset and econometric approach. The re-estimation yielded quite different numerical results due to the use of a much finer grid to approximate the equilibrium price function through splines. Application of Deaton and Laroque’s approach, modified only to improve its numerical accuracy, yielded estimates consistent with the observed levels of price variation and autocorrelation for seven of the twelve commodities analysed by Deaton and Laroque (1995, 1996). Furthermore, Cafiero et al. (2011) established that other modifications of their model are able to generate the high levels of serial correlation observed for the prices of major commodities. An obstacle to analysing the competitive storage model is the impossibility of deriving an analytical solution for the equilibrium price function. For this reason, theoretical inferences from this model are based, to a large extent, on numerical simulations. In the next chapter, we will try to address the issue of analytical tractability for a continuous-time modification of the competitive storage model.

References Cafiero C, Bobenreith E, Bobenreith J, Wright B (2011) The empirical relevance of the competitive storage model. J Econ 162:44–45

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Chambers M, Bailey R (1994) A theory of commodity price fluctuations. Univ. Essex, Manuscript, Colchester Chambers M, Bailey R (1996) A theory of commodity price fluctuations. J Polit Econ 104 (5):924–957 Deaton A, Laroque G (1992) On the behaviour of commodity prices. Rev Econ Stud 59(1):1–23 Deaton A, Laroque G (1995) Estimating a non-linear rational expectations commodity price model with unobservable state variables. J Appl Econ 10:S9–S40 Deaton A, Laroque G (1996) Competitive storage and commodity price dynamics. J Polit Econ 104 (5):896–923 Dvir E, Rogoff K (2009) Three epochs of oil. NBER Working Paper No. 14927, p 49 Gustafson R (1958) Carryover levels for grains. A method for determining amounts that are optimal under specified conditions. University of Chicago, Technical bulletin No. 1178, p 92 International Energy Agency (2007) World Energy Outlook 2007, IEA, Paris https://www.iea.org/ reports/world-energy-outlook-2007. Accessed 31 Oct 2020. Quandl (2020) Financial, economic and alternative data. https://www.quandl.com/data/BP/ CRUDE_OIL_PRICES-Crude-Oil-Prices-from-1861. Accessed 30 Oct 2020. Routledge B, Seppi D, Spatt C (2000) Equilibrium forward curves for commodities. J Financ LV (3):1297–1337

Chapter 8

Commodity Trade in Continuous Time, Long-Term Availability and Storage Capacity

Abstract The canonical model of competitive storage is formulated in discrete time, and this brings about a high non-linearity and computational complexity of equilibrium equations. In this chapter, we consider a continuous-time commodity market model utilizing the advantages of stochastic calculus that make this model a more convenient analysis tool. It is shown that the argument of the equilibrium price function is the long-term availability of a commodity instead of the current availability, which does not make sense for continuous time. The equilibrium price function satisfies a second-order differential equation and is given by the saddle path under the standard boundary conditions for switching between the regimes of commodity trade. We refine these conditions for the model extension with an upperboundary constraint on storage capacity and examine the structure of equilibrium price functions in this case.

8.1

Introduction

We offer a modified version of the competitive storage model by introducing continuous time. One of the reasons for this modification is the advantage of stochastic calculus as a convenient instrument of dynamic analysis. In the discretetime models in the previous chapter, highly non-linear functional equations for equilibrium commodity price require substantial involvement of numerical methods. The introduction of continuous time makes possible linearization of these equations that simplifies formal analysis. Within the continuous-time framework, we have to refine the concept of current availability of a commodity, which plays the key role in Chaps. 6 and 7 of this book. In Deaton and Laroque’s model (1992) with serially uncorrelated supply shocks, the current availability is the unique state variable that determines the equilibrium price. In that model, the availability is defined as the sum of storage and commodity supply per period. Although this summing is formally correct, one can question the empirical content of the notion of current availability. To illustrate this matter, let us try to calculate the current availability as the sum of storage and supply per period for a commodity market, for example the U.S. market © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_8

197

198

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

Table 8.1 Availability of crude oil in the U.S. in 2019 for different calendar periods, million barrels

Monthly Annual Daily

Storage 433 433 433

Production 373 4471 12

Import 207 2482 7

Availability 1013 7386 452

Source: U.S. Energy Information Administration (2020a)

for crude oil. We represent supply as the sum of domestic production and import. According to the EIA data, the average monthly U.S. production of crude oil was 373 million barrels in 2019 and the average monthly import was 207 million barrels. Commercial stocks of crude oil amounted to 433 million barrels by the beginning of 2019. Table 8.1 demonstrates these figures. The availability of crude oil per month was equal to the sum of storage and supply, 433 + 373 + 207 ¼ 1013 million barrels. Calculating in the same way the availability per annum yields 433 + 4471 + 2482 ¼ 7386 million barrels, while the estimated average daily availability is only 452 million barrels, as one can see from Table 8.1. Thus, the availability of crude oil notably varies with the length of time period under consideration. This example demonstrates a limitation of the concept of current availability in matching stocks and flows within the discrete-time model framework. A straightforward application of this measure of availability for high-frequency data results in meaningless values, such as the daily availability shown in Table 8.1. Current supply becomes negligible compared to storage for very short calendar periods. Nevertheless, as we will show in this chapter, within the continuous-time framework one can define properly a measure of availability as the state variable for the competitive storage model. This measure defined as the long-term availability proves to be invariant to the length of selected calendar periods. The base model of this chapter is a modified version of the canonical model of competitive storage presented in the previous chapter. We will consider a dynamic stochastic partial equilibrium model with risk-neutral profit-maximizing commodity traders who manage inventories bounded from below at zero. As in the canonical model, the rational expectations equilibrium is characterized by an equilibrium price function mapping the state variable of availability to commodity prices. We will introduce a continuous-time stochastic process for excess supply disturbances and show that the equilibrium price function satisfies a second-order ordinary differential equation implied by the no-arbitrage condition. The saddle path for this equation provides a solution of the model under the standard boundary conditions that specify a regime switching between the regimes of trading and stock-out. As in Deaton and Laroque’s discrete-time model of competitive storage (1992, 1996), the stock-out occurs for high prices exceeding a threshold level. Under high prices, traders reduce their stocks to zero because the expected rate of return on storage is below the interest rate. The saddle-path equilibrium implies the existence and uniqueness of the equilibrium price function, which is analogous to the solution of Deaton and Laroque’s model.

8.1 Introduction

199

Fig. 8.1 Commercial storage of crude oil in the United States and the WTI price. Source: U.S. Energy Information Administration (2020b)

We extend our base model by assuming that storage capacity is bounded from above by an exogenous maximal level. It will be shown that under the storage capacity constraint, the equilibrium price function corresponds to a sub-saddle path of the second-order differential equation. The equilibrium solution changes under this constraint, because we redefine the boundary conditions to allow for regime switching both for high and low prices. The assumption of the upper limit of storage capacity takes into account possible congestion of storage facilities, which can happen in the real world and lead to negative price spikes in the event of oversupply in the market. The situations of abundant oil supply and constrained storage capacity has manifested several times since 2014 due to the boom of shale oil in the United States. Figure 8.1 shows the dynamics of commercial storage of crude oil in the U.S. and the West Texas Intermediate (WTI) price of crude oil in 2010–2020. The most striking case is a recent oil oversupply that occurred in the first half of 2020 as a result of Covid-19 pandemic and the failure of OPEC negotiations on supply curtails in March 2020. The crude oil market proved to be in deep contango, when spot prices were substantially below the prices of futures contracts. The supply glut caused the oil price drop from $60/barrel to less than $20/barrel and threatened to overwhelm storage facilities. Inventories in Cushing, the delivery hub for NYSE crude oil futures, and other storage terminals in the U.S. approached the maximal capacity level. In April 2020, the temporary shortage of available storage capacities even caused the fall of WTI crude oil price below zero for expiring futures contracts. The effects of storage capacity constraint have been examined previously within the theoretical framework of competitive storage. The paper by Atle Oglend and Tore Kleppe (2017) investigates the implications of bounded speculative storage for

200

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

Deaton-Laroque’s model (1992) with independent identically distributed supply shocks. In Oglend-Kleppe’s model, storage is bounded from below at zero and from above by a maximal capacity. These authors showed that a binding capacity mirrors the non-negativity constraint on storage and can lead to negative price spiking. In particular, they considered a non-linear mapping for the equilibrium price functional equation, which extends the main equation in Deaton-Laroque’s model (1992) presented as (7.15) in the previous chapter. They argued that the sequence of price functions obtained by the fixed-point iteration converges to the function that defines the stationary rational expectations equilibrium. However, Guerra et al. (2018) have demonstrated counterexamples showing a divergence of the iteration operator applied by Oglend and Kleppe in their model. Analysis of equilibrium price functions in the model of this chapter does not require consideration of non-linear mappings for determination of the stationary rational equilibrium. Moreover, unlike the Oglend-Kleppe’s (2017) model, we do not bound ourselves by the excess supply process with serially uncorrelated shocks. We assume a positive autocorrelation for the stochastic excess supply, as it is the case in the versions of discrete-time competitive storage model with autocorrelation of supply by Deaton and Laroque (1995, 1996), Chambers and Bailey (1996) discussed in the previous chapter. We will analyse the structure of equilibrium price function under storage capacity constraint for the regimes of stock-out, trading and full capacity. It will be shown that a larger size of storage capacity implies a wider domain of trading under low prices.

8.2

The Model

As we said, the model is a modification of the competitive storage model in the previous chapter. The novel features are continuous time and the altered description of the commodity market. We represent competitive producers and consumers as a homogeneous group of market participants that do not hold storage but generate excess demand covered by competitive homogeneous traders who own storage. The excess demand results from the joint reaction of producers and consumers to price movements and from random disturbances of the market. We consider, first, the commodity market model without traders and then introduce these participants.

8.2.1

Commodity Market

We represent the instantaneous excess demand in a commodity market as the difference between a non-stochastic price-dependent term called the net demand function y( p), where p is the commodity price, and an exogenous random variable x that defines instantaneous excess supply disturbances. Here and henceforth, we do not use a time variable in the model notation if it is not necessary.

8.2 The Model

201

Fig. 8.2 (a) Net demand function. (b) Market-clearing price

The net demand function y( p) is defined for all positive p, twice continuously differentiable, decreasing and convex. This function is positive for low prices and negative for high ones. An example of this function that will be used in what follows is the negative logarithm shown in Fig. 8.2a: y( p) ¼  blnp, where b > 0 is the semi-elasticity of net demand. The excess supply disturbance x follows the mean-reverting process: dx ¼ μxdt þ σdw,

ð8:1Þ

where μ > 0 is the rate of mean reversion, dw is an increment of the standard Wiener process with mean 0 and instantaneous variance 1, and σ is the standard deviation of excess supply fluctuations. The variance of these fluctuations depends linearly on the length of the infinitesimal time interval, E(σdw)2 ¼ σ 2dt. The stochastic equation (8.1) means that in any such interval (t, t + dt), any small change of excess supply dx results from the drift to the long-term zero mean, μxdt, and the impact of serially uncorrelated shocks σdw. In the absence of speculative trade, the price clears the market at any instant, implying that yðpÞ ¼ x:

ð8:2Þ

The market-clearing price satisfying (8.2) is the inverse net demand function of random shocks denoted as P ðxÞ  y1 ðxÞ and shown in Fig. 8.2b. For the logarithmic net demand y( p) ¼  blnp, the inverse net demand is exponential: P ðxÞ ¼ ex=b . The function P ðxÞ is defined for all real numbers. It intersects the vertical axis in Fig. 8.2b at point P ð0Þ ¼ 1, which is the market-clearing price for the long-term mean of excess supply process (8.1).

202

8.2.2

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

Traders and Storage

Suppose that competitive commodity traders are present in the market and consider a continuous-time trader’s problem. A risk-neutral trader maximizes the expected discounted cash flow from trade over an infinite time horizon: Z1 E0

ert pðt Þqðt Þdt,

ð8:3Þ

0

subject to the storage balance equation: dsðt Þ ¼ qðt Þdt,

ð8:4Þ

and the non-negativity constraint: sðt Þ  0,

ð8:5Þ

where r denotes the riskless real interest rate, s storage and q the intensity of trade. Storage is costless and does not deteriorate in time. At any instant, traders observe the current price p and choose the intensity of trade q. They are commodity sellers if q > 0 and buyers if q < 0. For a small time interval (t, t + dt) the volume of trade is qdt and the cash flow is pqdt. According to Eq. (8.4), the change of storage ds equals the volume of trade, qdt. The initial storage is s0  0. At any instant, traders clear the market: q ¼ yðpÞ  x:

ð8:6Þ

Thus, equations of the model with traders include the trader’s problem (8.3)– (8.5), the market-clearing condition (8.6) and Eq. (8.1) specifying the process for excess supply disturbance.

8.3

The Value Function

The value function for the trader’s problem (8.3)–(8.5) is based on Hotelling’s valuation principle that we considered in Chaps. 2 and 4. The asset owned by a trader is storage and the value of this asset at any instant t is the expected cash flow from inventory management:

8.3 The Value Function

203

Z1 V ðt Þ ¼ E t

erτ pðτÞqðτÞdτ,

t

where Et denotes the expectation conditional on information available by time instant t. Hotelling’s valuation principle implies that storage is evaluated through current commodity price: V ðt Þ ¼ pðt Þsðt Þ:

ð8:7Þ

This means that the trader can sell at any instant all the stock he owns at market price. In the trading regime s > 0 and the continuous-time stochastic Bellman equation for the value function V is pqdt þ E t dV ¼ rVdt:

ð8:8Þ

According to this equation, the cash flow obtained in time interval (t, t + dt), plus the expected value gain over this interval, is equal to the riskless return on the stock value. The interest rate defines the opportunity cost of investment in storage. For the value function (8.7), we have it that EtdV ¼ Etpds + Etsdp. At any instant, the change of storage ds is known for the trader, hence Etpds ¼ pds. From the storage balance Eq. (8.4), pds ¼  pqdt. Consequently, the left-hand side of the Bellman equation (8.8) is equal to: pqdt  pqdt + Etsdp ¼ sEtdp. The right-hand side of Eq. (8.8) is rVdt ¼ rpsdt. Dividing both parts of Eq. (8.8) by s > 0, we obtain the dynamic equation for commodity price: E t dp ¼ rpdt:

ð8:9Þ

Thus, in the trading regime the commodity price dynamic is governed by the stochastic Hotelling rule: the expected rate of price growth is equal to the interest rate. In the stock-out regime s ¼ 0, because the expected rate of price growth is below the interest rate: E t dp < rpdt:

ð8:10Þ

The case Etdp > rpdt is ruled out to eliminate arbitrage profits. There is no trade in the stock-out regime, q ¼ 0, and the price fully absorbs any excess supply disturbance x: p ¼ P ðxÞ:

ð8:11Þ

204

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

For the logarithmic net demand function, the price under stock-out is p ¼ P ðxÞ ¼ ex=b .

8.4

Stationary Rational Expectations Equilibrium

For the continuous-time model of storage, define the long-term availability of commodity as the sum of storage and the integral excess supply disturbances expected for the long term: a ¼ s þ x=μ:

ð8:12Þ

The term added to the storage equals the conditional expectation of excess supply integrated over an infinite time horizon. Indeed, for the stochastic process (8.1), the expected lagged excess supply for any time moment t + τ is Etx(t +R τ) ¼ eμτx(t). T Integrating over time interval 0  τ  T, we obtain: E t 0 xðt þ τÞdτ ¼ R T μτ xðt Þ 0 e dτ ¼ xðt Þð1  eμT Þ=μ: For T ¼ 1, this is equal to x(t)/μ. In Deaton and Laroque’s (1992) model, current availability means the amount of commodity available in the current period to consumers and traders. Here the longterm availability is defined as the sum of storage and the expected long-term excess supply of a commodity, which is relevant to intertemporal decisions of traders, who own storage. The long-term availability can be negative if the excess supply disturbance x is negative. The initial long-term availability is a0 ¼ s0 + x0/μ, where x0 is the initial excess supply. Consider a stationary rational expectations equilibrium, for which the stochastic excess supply process (8.1) is stationary. We will show that availability defined as (8.12) is the state variable sufficient to determine the equilibrium price. For the stationary equilibrium, consider the equilibrium price function of availability p(a) defined for all real numbers and twice continuously differentiable. In the trading regime, this function satisfies the no-arbitrage Equation (8.9) rewritten as: EdpðaÞ ¼ rpðaÞdt:

ð8:13Þ

The expectation of price change for the stationary equilibrium is taken with respect to the state variable a that must contain all information relevant to the current price determination. In the stock-out regime, a ¼ x/μ, because s ¼ 0. Let us introduce, for the sake of notational convenience, inverse net demand as the function of availability: PðaÞ  P ðμaÞ: Then condition (8.11) is rewritten for the equilibrium price function as

8.4 Stationary Rational Expectations Equilibrium

205

pðaÞ ¼ PðaÞ:

ð8:14Þ

This condition holds if storage is unprofitable: rP(a)dt > Edp(a). We combine equilibrium conditions (8.13) and (8.14) as the condition of maximum: rpðaÞdt ¼ max ½EdpðaÞ, rPðaÞdt 

ð8:15Þ

and represent conditions of storage balance (8.4) and market clearing (8.6) as: ds ¼ xdt  yðpðaÞÞdt,

ð8:16Þ

implying that for any small time interval the change of storage covers the change of excess supply and net demand. Note that Eq. (8.15) is similar to the condition of stationary rational expectations equilibrium (7.15) in the previous chapter.

8.4.1

The Equilibrium Price Function

In the trading regime, the state variable a is driven by the stochastic Ito process: da ¼ yðpðaÞÞdt þ ðσ=μÞdw

ð8:17Þ

that results from combining the excess supply process (8.1) with storage balance under market clearing (8.16): da ¼ ds + dx/μ ¼ xdt  y( p(a))dt + (μxdt + σdw)/ μ ¼  y( p(a))dt + (σ/μ)dw. The process for long-term availability (8.17) has the non-linear drift rate y( p(a)) and the instantaneous standard deviation σ/μ. The standard deviation of the stochastic term in Eq. (8.17) is the ratio of σ to the rate of mean reversion μ. This is the standard deviation of the stochastic process for x adjusted for autocorrelation. The lower the coefficient μ, the higher the autocorrelation of x, and the higher is σ/μ, the standard deviation for a. If μ is small, the excess supply disturbances are persistent and the instantaneous variance of long-term availability is large, as well as the variance of long-term excess supply (for the mean-reverting process (8.1) the long-term variance is lim Var t xðt þ τÞ ¼ σ 2 =2μ). τ!1

In the stock-out regime, a ¼ x/μ and da ¼ dx/μ. From (8.1), the differential for the state variable is: da ¼ xdt þ ðσ=μÞdw,

ð8:18Þ

which is identical to Eq. (8.17), because x  yðP ðxÞÞ ¼ yðPðaÞÞ ¼ yðpðaÞÞ. It is important that the excess supply term x vanishes from the right-hand side of Eq. (8.17) and does not influence the differential da directly. This is the consequence of our choice of availability measure as a ¼ s + x/μ. The stochastic process for

206

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

availability (8.17) or (8.18) incorporates conditions of equilibrium: storage balance (8.4) and market clearing condition (8.6) for instantaneous excess supply x and net demand y( p). The long-term availability a is indeed the state variable containing all information sufficient for the formation of price expectations and equilibrium price determination according to the intertemporal equilibrium condition (8.15). Applying Ito’s Lemma for the stochastic process for availability (8.17), one can express the expected price differential as the first-order series expansion involving the second-order derivative of the equilibrium price function: 1 EdpðaÞ ¼ p0 ðaÞyðpðaÞÞdt þ ðσ=μÞ2 p00 ðaÞdt: 2

ð8:19Þ

The expected price change results from the effect of net demand and the effect of the long-term availability variance. Substituting the no-arbitrage condition of trading regime (8.13) for the expectation term in Eq. (8.19), and dividing both sides by dt, yields the second-order non-linear differential equation for p(a): 1 ðσ=μÞ2 p00 ðaÞ  p0 ðaÞyðpðaÞÞ ¼ rpðaÞ: 2

ð8:20Þ

The equilibrium price function p(a) is the solution of this equation subject to properly defined boundary conditions. The non-linear term in the left-hand side of Eq. (8.20) captures the feedback effect of net demand on the price change, which is taken into account by commodity traders. Note that the similar feedback effect in the discrete-time competitive storage model in the previous chapter is a cause for a high non-linearity of the key functional equation of that model (7.15). Thus, at any instant, the equilibrium price p(a) in the continuous-time model is determined by the state variable a that evolves according to stochastic differential Eqs. (8.17) or (8.18). The change of equilibrium storage is found at any instant as ds ¼ [x  y( p(a))]dt, according to Eq. (8.16). Consequently, the information about instantaneous excess supply x is relevant to the change of storage, but it is redundant per se for determination of the equilibrium price function p(a). We will consider, following Deaton and Laroque (1992), a “synthetic” equilibrium price function p(a), which is decreasing and consisting of two pieces pasted together at the switching point a. This function is shown in Fig. 8.3. The threshold price p ¼ p(a) separates the zone of stock-out that takes place for a < a such that p (a) ¼ P(a) > p from the zone of trading that takes place, including the switching point, for a  a such that p(a) ¼ p(a) and P(a)  p(a)  p, where p(a) is a solution of Eq. (8.20). The dashed curve in the figure shows the off-equilibrium continuation of the inverse net demand function P(a) in the zone of trading. Note that the threshold availability shown in Fig. 8.3 is negative, a < 0. A negative threshold availability means that switching to trading occurs under the negative excess supply term x ¼ μa < 0. In this case, the trading zone is wide in the

8.4 Stationary Rational Expectations Equilibrium

207

Fig. 8.3 The equilibrium price function

sense that the unconditional stationary probability that traders hold inventories at any instant is high: Pr(x  x) > 1/2.

8.4.2

Free-Boundary Conditions

Figure 8.3 is similar to Fig. 7.2 in the previous chapter that illustrated regime switching in the discrete-time model of commodity price. The stationary stochastic equilibrium in continuous time has the properties of real option. In particular, the threshold availability a and the threshold price p ¼ p(a) should meet the freeboundary conditions of value matching and smooth pasting. These conditions establish a connection between the unknown equilibrium price function p(a) and the known inverse net demand function P(a). The former is found as a solution of the second-order differential equations (8.20) satisfying the two free-boundary conditions. The value-matching condition requires the equilibrium price function to be continuous at the switching point a: pða Þ ¼ Pða Þ:

ð8:21Þ

At this point, the trader should be indifferent between trading and stock-out. From the maximum condition (8.15), Edp(a) ¼ rP(a) ¼ rp(a) implying (8.21). The smooth-pasting condition means that the equilibrium price function is differentiable at the switching point:

208

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

p0 ða Þ ¼ P0 ða Þ:

ð8:22Þ

Both pieces of this function should meet tangentially at a, as shown in Fig. 8.3. In Appendix A.1 we present a formal explanation for the smooth-pasting condition (8.22) based on the exposition by Avinash Dixit and Robert Pindyck (1994). We show that if p0(a) 6¼ P0(a), the no-arbitrage condition (8.13) is violated for a small zone around the switching point. Thus, unlike the discrete-time storage model in the previous chapter, the equilibrium price function in the continuous-time model has no kink at the switching point. The boundary conditions (8.21), (8.22) establish a real-option structure of equilibrium that greatly simplifies our analysis. We do not need to care about the maximum condition (8.15) at any instant, because the probability of regime switching over an infinitesimal interval of time is zero at any point a beyond an infinitesimal interval around a. For the continuous-time stochastic process for availability (8.17) or (8.18), the existing regime of inventory management is maintained over a sufficiently small period of time.

8.5

The Saddle-Path Solution

Consider the second-order differential equation for the equilibrium price function (8.20). One can represent it as the two-dimensional system of first-order differential equations for the equilibrium price function and its derivative: p0 ð aÞ ¼ z ð aÞ

ð8:23Þ

1 ðσ=μÞ2 z0 ðaÞ ¼ zðaÞyðpðaÞÞ þ rpðaÞ: 2

ð8:24Þ

Equation (8.23) presents variable z(a) as the derivative of equilibrium price with respect to the state variable, and Eq. (8.24) is identical to Eq. (8.20). A pair of price function p(a) and its derivative z(a) are the phase variables satisfying the system (8.23), (8.24). We analyse this system in Appendix A.2 for the logarithmic net demand y ( p) ¼  blnp. Figure 8.4 depicts its phase plane. Curved arrows show paths of price p(a) and price derivative z(a) under increasing argument a. We search for an equilibrium price function p(a) and focus attention on the fourth quadrant of the phase plane, where the price is positive and the price derivative is negative, since the equilibrium price function we are looking for should be decreasing. As one can see in this figure, there are three types of trajectories in the phase plane of the system (8.23), (8.24). The first type, drawn with the dotted curved arrow B, is characterized by a non-monotonic price dynamic. The price is decreasing for z < 0 and, after intersecting the horizontal axis, z ¼ 0, is increasing for z > 0. The price growth under increasing availability is caused by a speculative bubble generated by

8.5 The Saddle-Path Solution

209

Fig. 8.4 The phase plane and the saddle path

perpetual storage accumulation, which is driven by expectations of further price growth. Thus, each B-type path in Fig. 8.4 represents a bubble solution. The second type of trajectory in Fig. 8.4 drawn with the dotted curved arrow C demonstrates paths with a decreasing price and non-monotonic price derivative. Curve Z in the figure is the locus of the zero second-order derivative, z0(a) ¼ 0, which is given by the function z ¼  rp/y( p). For the logarithmic net demand y( p) ¼  blnp, this function, specified as Z( p) ¼ rp/bln( p), is decreasing and strictly concave. As any type- C path crosses curve Z, the price derivative z starts to decrease, and the price falls thereafter at an accelerating pace. The third type of trajectory in Fig. 8.4 is represented by the saddle path S, which is drawn with the bold curved arrow. The saddle path separates the trajectories of types B and C and converges to the origin O as a tends toward infinity. The saddle path is the unique path converging to the origin. Line P in Fig. 8.4 depicts the locus of inverse net demand in the phase plane. Each point of this locus is a combination of P(a) and P0(a) for any a. For logarithmic net demand, locus P is indeed given by the line: z ¼ pμ=b,

ð8:25Þ

because PðaÞ  P ðμaÞ ¼ eaμ=b, P0(a) ¼  P(a)μ/b. Note that the saddle path S and the path of type C drawn in Fig. 8.4 intersect the line of inverse net demand.

210

8.5.1

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

Existence and Uniqueness of Equilibrium

The equilibrium price function is given by a path on the phase plane in Fig. 8.4 satisfying the maximum condition (8.15) and the free-boundary conditions (8.21), (8.22). Paths of types B and C do not satisfy equilibrium conditions. We rule out type-B paths that are bubble solutions: traders over-accumulate storage along this path and, as a result, the price function pB(a) corresponding to a type-B path begins to grow with a, as one can see from Fig. 8.4. We also rule out type-C paths, because for any price function pC(a) corresponding to this type we have: p(a) < P(a) for some a > a, and condition (8.15) is violated. Traders sell out their stocks that are reduced to zero, because the price function pC(a) intersects line P, as one can see in Fig. 8.4, and drops to zero. The only trajectory on the phase plane that satisfies the conditions of equilibrium is the saddle path S. The point of its intersection with the locus of inverse net demand function P in Fig. 8.4 is the switching point Π ¼ ( p, z), where p ¼ p(a), z ¼ p0(a). The value-matching and smooth-pasting conditions (8.21), (8.22) are met at point Π. It is shown in Appendix A.2 that the slope of the saddle path of the system of differential equations (8.23), (8.24) is zero in the origin and tends toward 1 for price tending toward infinity. Therefore, the saddle path S in the phase plane satisfies Inada conditions and intersects the line of inverse net demand P at some point Π, as is shown in Fig. 8.4. This property implies the existence and uniqueness of the synthetic equilibrium price function defined by the saddle path S and satisfying the free-boundary conditions (8.21), (8.22). The synthetic equilibrium path is depicted in Fig. 8.4 as the bold curve consisting of two pieces: 1) the piece of inverse net demand line P corresponding to the stockout regime such that P(a) > p for all a < a; and 2) the piece of saddle path S between the threshold point Π and the origin O corresponding to the trading regime (including the switching point) such that P(a)  p for all a  a. The dashed piece of line P is the off-equilibrium continuation of inverse net demand in the zone of trading. The dashed-and-dotted curve in Fig. 8.4 is the continuation of saddle path in the zone of stock-out. Let pS(a) denote the price function corresponding to the saddle path. Then the synthetic equilibrium price function is defined as ( p ð aÞ ¼ S

pS ðaÞ, a  a PðaÞ, a < a

Figure 8.5 demonstrates the equilibrium price function pS(a) and non-equilibrium price functions pB(a) and pC(a) drawn with dotted curves and corresponding respectively to the diverging paths of types B and C in Fig. 8.4. The saddle-path price function pS(a) satisfies the equilibrium condition pS(a)  P(a) for all a  a, and converges to zero for a tending toward infinity. The bubble price function pB(a) is

8.6 The Storage Capacity Constraint

211

Fig. 8.5 The price functions of different types

non-monotonic: it is decreasing for lower a and increasing for higher a. The price function pC(a) falls below P(a) at point ah exceeding a and drops to zero. The dashed curve in Fig. 8.5 shows the continuation of the inverse net demand function P(a) in the zone of trading, a  a. The dashed-and-dotted curve is the off-equilibrium continuation of the saddle-price function pS(a) in the zone of stockout, a < a. For this curve, the no-arbitrage condition (8.20) is fulfilled, but storage is negative and “accumulated” under decreasing long-term availability (as indicated by a more rapid growth of price pS(a) relative to P(a) with a decrease of a for a < a). As a result, we have demonstrated that the synthetic equilibrium price function pS(a) generated by the saddle path S is unique. This is because only this path does not represent a price bubble and matches the equilibrium condition that p(a)  P(a) for any a  a. Other solutions of the system of differential equations (8.23), (8.24) have been ruled out: B-type paths are bubble solutions, while C-type paths are “sell-off” solutions that do not satisfy this equilibrium condition.

8.6

The Storage Capacity Constraint

The structure of solutions for the equilibrium price function will change, if we introduce a capacity constraint on storage. We have assumed thus far that storage is unlimited in size and trade occurs for all a  a and for arbitrary low prices p (a)  p(a). Let us consider a modified trader’s model (8.3)–(8.5) with the two-sided constraint on storage size instead of non-negativity constraint (8.5):

212

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

0  s  s,

ð8:26Þ

where s denotes the storage capacity. If the capacity constraint s  s is binding, speculative trade does not occur. In the full-capacity regime, the long-term availability is equal to the sum: a ¼ s þ x=μ. This regime mirrors the stock-out regime in the sense that the equilibrium price function is defined by the inverse net demand function as pðaÞ ¼ PðaÞ  P ðμða  sÞÞ: In the full-capacity regime, the price is below a threshold level, under which speculative commodity trade is unprofitable. Under low prices below this threshold level, traders benefit from holding storage s ¼ s without sales because the expected growth rate of price exceeds the interest rate: EdPðaÞ > rPðaÞdt, as will be shown below. Therefore, the equilibrium price function under capacity constraint p(a) includes the upper and lower thresholds of price. The upper price threshold is the point of switching from trading to stock-out, while the lower price threshold is the point of switching to the full capacity regime. Consider the two-sided free-boundary conditions on the equilibrium price function p(a) with lower and upper boundaries for the trading zone: al and ah, respectively, such that  1 < al < ah  1 (al ¼ xl/μ, ah ¼ s þ xh =μ, where xl and xh are the lower and upper boundaries in terms of excess supply). As above, the freeboundary conditions on p(a) are value-matching:         p al ¼ P al , p a h ¼ P a h ,

ð8:27Þ

p0 ðal Þ ¼ P0 ðal Þ, p0 ðah Þ ¼ P0 ðah Þ:

ð8:28Þ

and smooth-pasting:

The stock-out occurs under high prices that are above the upper boundary for price p(al). The full capacity takes place under low prices that are below the lower boundary p(ah). In the stock-out regime: p(a) ¼ P(a) for the zone of availability a < al, in the trading regime: PðaÞ  pðaÞ  PðaÞ for al  a  ah, and in the fullcapacity regime: pðaÞ ¼ PðaÞ for a > ah. For the full-capacity regime, the locus of inverse net demand in the phase plane is given by the same line P, which is drawn in Fig. 8.4. Indeed, each point of this line for any a > ah is a combination of PðaÞ and P0 ðaÞ, satisfying Eq. (8.25): z ¼ pμ=b, because for the exponential inverse net demand PðaÞ ¼ eμðasÞ=b , P0 ðaÞ ¼ PðaÞμ=b.

8.6 The Storage Capacity Constraint

213

Fig. 8.6 The phase plane for the equilibrium with capacity constraint

The two-sided boundary conditions (8.27), (8.28) define the structure of equilibrium price function in the phase plane demonstrated in Fig. 8.6. The saddle path S is drawn in this figure with the thin curved arrow that intersects line P, the locus of the inverse net demand function, at the threshold point Π ¼ ( p, z), as in Fig. 8.4. However, the saddle path cannot be an equilibrium solution for the model with limited storage capacity, because it does not include a full-capacity zone for low prices. The solution is defined by a “sub-saddle” path of type C that was ruled out above for the model without capacity constraint. This path in Fig. 8.6 intersects line P twice: at points Π2 ¼ ( p2, z2) and Π1 ¼ ( p1, z1) such that p > p2 > p1 and z < z2 < z1. Conditions of value-matching (8.27) and smooth-pasting (8.28) are satisfied at these points for the price function pC(a) corresponding to the C-type path. The synthetic equilibrium path of this type is drawn in Fig. 8.6 as the bold kinked curve C consisting of three pieces: (a) line P for the stock-out zone p > p2, where p2 ¼ pC(al); (b) the piece of path C for the trading zone p1  p  p2, where p1 ¼ pC(ah); and (c) line P for the full-capacity zone p < p1. The trading zone is given by the piece of path C between the points of switching Π1 and Π2. The zones of stock-out and full capacity are given by the solid pieces of line P outside the dashed interval (Π1, Π2). The dashed-and-dotted curved arrow in Fig. 8.6 is the off-equilibrium continuation of the C-type trajectory for pC(a) < p1. Thus, under boundary conditions (8.27), (8.28), the “sub-saddle” C-type path defines the trading zone for availability [al, ah] so that the stock-out takes place for prices higher than p2 ¼ pC(al) and the full capacity occurs for prices lower than p1 ¼ pC(ah). The synthetic equilibrium price function is given by

214

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

Fig. 8.7 Equilibrium price function under storage capacity constraint

pC ð aÞ ¼

8 > > < > > :

PðaÞ, a < al pC ðaÞ, al  a  ah PðaÞ, a > ah

This function is drawn in Fig. 8.7 as the bold solid curve. The lower boundary of the trading zone al is the point of tangency of the C-type price function pC(a) with the inverse net demand function under stock-out P(a). Similarly, the higher boundary ah is the point of tangency of pC(a) with the inverse net demand function under full capacity PðaÞ. The dashed curves in Fig. 8.7 show the continuations of the inverse net demand functions P(a) and PðaÞ in the trading zone al  a  ah. The dashed-anddotted curve is the continuation of pC(a) in the full-capacity zone a > ah. As one can see in Fig. 8.6, there is a continuum of C-type price paths crossing the line of inverse demand P at two points similar to Π1 and Π2. It is possible to select among these paths a unique one, corresponding to the storage capacity s and characterized by the interval between the points of intersection (the dashed interval on line P in Fig. 8.6). This interval defines the trading zone [al, ah] in Fig. 8.7, which, in turn, determines the unique capacity-constrained equilibrium price function pC(a) for the storage capacity s. The larger this capacity, the wider is the trading zone [al, ah], because the inverse demand function under full capacity PðaÞ in Fig. 8.7 shifts to the right with an increase of s. At the limit, as s tends to infinity, the interval [al, ah] widens and tends to the trading zone for the saddle-path solution [a, 1), whereas the capacityconstrained equilibrium price function pC(a) converges to the saddle-path function pS(a), which is drawn in Fig. 8.7 for the case of unlimited storage capacity. The lower boundary al cannot be below a, the switching point for the saddle-path

8.6 The Storage Capacity Constraint

215

solution, because otherwise the price function tangent to P(a) at point al would be a bubble price function pB(a) shown in Fig. 8.5.

8.6.1

Storage Capacity and Trading Zones

Any synthetic equilibrium price function pS(a) or pC(a) generates a stochastic process for commodity price. With Ito’s Lemma, the price differential for the stochastic process for availability (8.17) is the sum of expected drift and stochastic term: dp(a) ¼ Edp(a) + (σ/μ)p0(a)dw. In the trading regime, the expected price growth is Edp(a) ¼ rp(a)dt and the price satisfies the stochastic differential equation: dpðaÞ ¼ rpðaÞdt þ ðσ=μÞp0 ðaÞdw:

ð8:29Þ

This is a stochastic version of Hotelling’s rule: the rate of price growth is equal to the interest rate plus the random disturbance. The variance of disturbance term is high if μ is low, implying that the effect of traders’ activity on price volatility is strong under a high persistence of excess supply shocks. In the stock-out regime, the price differential for the logarithmic net demand is: 1 1 dpðaÞ ¼ p0 ðaÞda þ ðσ=μÞ2 p00 ðaÞdt ¼ ðμ=bÞpðaÞda þ ðσ=bÞ2 pðaÞdt, 2 2 because a ¼ x/μ and p0(a) ¼  p(a)μ/b, p00(a) ¼ p(a)(μ/b)2 for p(a) ¼ P(a) ¼ eaμ/b. Combining this with dx ¼  μxdt + σdw and rearranging terms gives the stochastic equation for price: h i 1 dpðaÞ ¼ aμ2 =b þ ðσ=bÞ2 pðaÞdt  ðσ=bÞpðaÞdw: 2

ð8:30Þ

The similar equation is fulfilled for the full-capacity regime, because for pðaÞ ¼ PðaÞ ¼ eμðasÞ=b we have it that da ¼ dðs þ x=μÞ ¼ dx=μ , p0(a) ¼  p(a)μ/ b, p00(a) ¼ p(a)(μ/b)2. In the absence of h speculative trading, i the expected growth rate of price for (8.30) 2 2 1 is EdpðaÞ=pðaÞ ¼ aμ =b þ 2 ðσ=bÞ dt. Hence, the condition of no-arbitrage, Edp (a)/p(a) ¼ rdt, is fulfilled only for one point e a such that e aμ2 =b þ 12 ðσ=bÞ2 ¼ r: e a¼

2b2 r  σ 2 : 2bμ2

ð8:31Þ

We show in Appendix A.3 that for s > 0 this point is located between the upper and lower boundaries of trading zone:

216

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

Fig. 8.8 The trading zones and mean-reverting dynamics of price

al < e a < ah :

ð8:32Þ

Under zero storage capacity, s ¼ 0, trading is absent and the expected growth rate of price is below the interest rate, Edp(a) < rp(a)dt, for a < e a, and above this rate, Edp (a) > rp(a)dt, for a > e a. This is the consequence of the mean-reverting property of the supply process (8.1). The trading zones for prices satisfying the no-arbitrage condition Edp(a) ¼ rp(a)dt are shown for s  0 in Fig. 8.8. This zone shrinks to the point pC ðe aÞ for s ¼ 0 and widens with the size of storage capacity s > 0. As we have seen, at the limit, as s tends to infinity, pC(ah) goes to zero, pC(al) converges to pS(a), and the trading zone for prices [pC(ah), pC(al)] converges to the interval [0, pS(a)] defined by the one-sided boundary condition for unlimited storage capacity: a  a. Arrows in Fig. 8.8 illustrate the mean-reverting dynamics of prices in the absence of trading: Edp(a) > rp(a)dt for high a and low prices and Edp(a) < rp(a)dt for low a and high prices. As a result, the commodity price exhibits a mean-reverting behaviour, if the variable of stochastic disturbance x falls into the tails of its distribution. If downward or upward movements of this variable push the market into the stock-out or fullcapacity regimes, the price tends to return to the trading zone, where it evolves according to the stochastic Hotelling rule given by Eq. (8.29).

8.7

Concluding Remarks

Discrete-time models of competitive storage considered in Chap. 7 are suitable for the analysis of commodity prices influenced by calendar cycles. The relevant case is the annual dynamic of agricultural commodity prices, for which unexpected supply shocks are naturally linked to realizations of stochastic harvests. In contrast, a continuous-time model of competitive storage examined in this chapter is applicable for the analysis of the impact of storage behaviour on commodity prices in the very short term. For example, fluctuations of oil prices reflect information about current changes in commercial oil storage. Daily data series are available for the WTI crude

8.7 Concluding Remarks

217

price and weekly data series—for commercial storage of crude oil in the United States. Two points are essential in the analysis of the continuous-time model of competitive storage: first, the choice of state variable that determines the equilibrium price at any instant; second, the specification of boundary conditions for the trading regime that define the structure of equilibrium solutions. For the base model with storage bounded from below by zero, we have demonstrated the existence and uniqueness of the equilibrium price function satisfying the one-sided boundary conditions of regime switching. This function corresponds to the saddle path of the second-order differential equation implied by the no-arbitrage condition. In connection with the first point, one should emphasize the role of market anticipations in the definition of long-term availability as the sum of storage and the expected excess supply over an infinite time horizon. Brian Wright (2001, p. 832) noted that a common feature of all storage activity is that stocks are constrained to be non-negative, hence it is impossible at the margin to borrow from the future if current stocks are zero. We have shown that even though traders cannot borrow stocks from the future, they manage their inventories by taking into account the excess supply disturbances expected in the future. This expectation determines the state variable of long-term availability of commodity and is absorbed, through actions of traders, into the current commodity price. The effect of traders’ expectations on price volatility is strong under a high persistence of excess supply fluctuations. We provided the graphical illustration for the existence and uniqueness of equilibrium price function in Fig. 8.4 for the case of exponential function of inverse net demand. The locus of this function in the phase plane of price and price derivative is the line originating from zero and intersecting the saddle path in one point. This fact establishes the saddle-path solution for the model with one-sided constraint on storage. One can extend this result to other cases of net demand including, for example, the linear function and the power functions with intercept: y( p) ¼ A  Bpb with A, B, b > 0 or A, B, b < 0. In general, the dependence between the price derivative P0(a) and the price P(a) should be monotonic and not very steep to ensure that the curve of inverse net demand in the phase plane intersects the saddle path in one point implying the existence and uniqueness of the equilibrium price function. Introduction of the capacity constraint into the competitive storage model implies that the solution for this function is defined by the two-sided boundary conditions of regime switching. We demonstrated the existence of C-type equilibrium price functions for the model with storage capacity constraint, in addition to the saddlepath solution for the model with unlimited storage capacity. Each such function can be selected in accordance with the size of storage capacity. The two-sided boundary condition on availability defines an interval of trading, which widens with the capacity size. Stock-out is temporarily preferable for traders, when prices are high and commodity is too expensive for profitable storage. Similarly, the full-capacity regime is preferable under a supply glut, when the commodity price is low and expected to grow at a rate faster than the rate of interest.

218

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

Appendices A.1 Smooth-Pasting Condition (8.22) Our elucidation of this condition replicates the one given by Dixit and Pindyck (1994, pp. 130–132). Consider a small zone around the point of pasting of the functions P(a) and p(a), a ¼ x/μ satisfying the value-matching condition (8.21), P (a) ¼ p(a). Suppose that the smooth-pasting condition (8.22) does not hold, P0(a) 6¼ p0(a). Then a is the kink point shown in Fig. 8.9a, b. The differential for availability at threshold point a is given by Eqs. (8.17) and (8.18): da ¼ yðpða ÞÞdt þ ðσ=μÞdw

ð8:33Þ

da ¼ x dt þ ðσ=μÞdw

ð8:34Þ

for da  0 and

for da < 0. The drift rates coincide at point a, because y( p(a)) ¼ x. Consider the discrete-time random walk approximation of processes (8.33), (8.34) near point a with discrete-time increments Δt and upward and downward pffiffiffiffiffi steps of size Δa ¼ ðσ=μÞ Δt. The upward and downward binomial probabilities for this approximation are given by:     1 a μ pffiffiffiffiffi 1 a μ pffiffiffiffiffi 1þ Δt , 1  π ¼ 1 Δt : π¼ 2 σ 2 σ The expected small change of the equilibrium price for a close to a is

Fig. 8.9 (a) P0(a) > p0(a). (b) P0(a) < p0(a)

ð8:35Þ

Appendices

219

EΔpða Þ ¼ πpða þ ΔaÞ þ ð1  π ÞPða  ΔaÞ  pða Þ πp0 ða ÞΔa  ð1  π ÞP0 ða ÞΔa, since P(a) ¼ p(a). From Eq. (8.35):     pffiffiffiffiffi 1 a μ pffiffiffiffiffi 0  1 a μ pffiffiffiffiffi 0  1þ Δt p ða ÞΔa ¼ 1þ Δt p ða Þðσ=μÞ Δt 2 σ 2 σ   pffiffiffiffiffi 1 0  ¼ p ða Þ ðσ=μÞ Δt þ a Δt : 2

πp0 ða ÞΔa ¼

Similarly,   pffiffiffiffiffi 1 ð1  π ÞP0 ða ÞΔa ¼  P0 ða Þ ðσ=μÞ Δt  a Δt : 2 Consequently, pffiffiffiffiffi 1 1 EΔpða Þ ¼ ðp0 ða Þ  P0 ða ÞÞðσ=μÞ Δt þ ðp0 ða Þ þ P0 ða ÞÞa Δt: 2 2

ð8:36Þ

If p0(a) 6¼ P0(a), the first term on the right-hand side of Eq. (8.36) dominates the pffiffiffiffiffi second one for Δt close to zero, because Δt goes to zero faster than Δt . For the same reason, the expected price change EΔp(a) dominates in absolute value the opportunity cost of storage: |EΔp(a)| > rp(a)Δt. Storage is unprofitable for a small interval of time in the case p0(a) < P0(a) shown in Fig. 8.9a or it allows for arbitrage opportunities in the case p0(a) > P0(a) shown in Fig. 8.9b. As a result, the smoothpasting condition (8.22) must hold at the threshold point a, because otherwise the no-arbitrage condition (8.13) is violated, Edp(a) 6¼ rp(a)dt.

A.2 System (8.23), (8.24) For y( p) ¼  blnp, the stationary state of this system is the origin, because it is the intersection of the two curves in the phase plane in Fig. 8.4, corresponding to zero derivatives: p0(a) ¼ 0 and z0(a) ¼ 0. The first curve is z ¼ 0 and the second one is given by the function Z( p) ¼ rp/blnp, which is drawn with curve Z in this figure. The characteristic equation of linearized system (8.23), (8.24) is: λ gzy0 ðpÞ þ gr

¼ λ2  gyðpÞλ  gðzy0 ðpÞ þ r Þ ¼ gyðpÞ  λ 1

λ2 þ gðblnpÞλ  gðr  bz=pÞ ¼ 0,

ð8:37Þ

220

8 Commodity Trade in Continuous Time, Long-Term Availability and Storage. . .

where λ is the characteristic root, g ¼ 2(μ/σ)2. Near the origin, this equation has two real roots of a different sign, since z/p  0, hence the stationary state is the saddle point. Consider the function Z( p) ¼ rp/bln( p). It has derivative Z 0 ð pÞ ¼

r ðlnp  1Þ , bðlnpÞ2

which tends to zero in the origin. One can see from Fig. 8.4 that the slope of the saddle path near the origin is above the slope of Z( p) and below zero. Consequently, the slope of the saddle path and the negative root of Eq. (8.37) tend to zero as a! + 1. Consider the saddle path for a!  1. Rewrite Eq. (8.24) as z 0 ð aÞ pð aÞ ¼ gblnpðaÞ þ gr 0 : p0 ð aÞ p ð aÞ The right-hand side of this equation tends toward 1 for p!1, because lim lnp ¼ þ1 and p(a)/p0(a) < 0. Consequently, the slope of the saddle path

p!1

tends toward infinity for a!  1.

A.3 Inequalities (8.32) Since p(a) > P(a) for any a 2 (al, ah), the price differential is higher for p(a) at point al: dp(al) > dP(al). Due to the smooth-pasting condition (8.28), this implies the inequality for second-order derivatives (for a second-order approximation):     p00 al > P00 al :

ð8:38Þ

The value-matching condition (8.27) is equivalent to    y p al ¼ μal ,

ð8:39Þ

         because y p al ¼ y P al ¼ y P μal ¼ μal . For the logarithmic net       l l demand function, we have: P al ¼ P μal ¼ ea μ=b , P0 al ¼ ðμ=bÞea μ=b and   l P00 al ¼ ðμ=bÞ2 ea μ=b . From the value-matching and smooth-pasting conditions (8.27), (8.28), we find that     l l p al ¼ ea μ=b , p0 al ¼ ðμ=bÞea μ=b : Inserting these conditions and (8.39) into differential equations (8.20) implies

References

221

  l l 1 ðσ=μÞ2 p00 al þ ðμ=bÞea μ=b μal ¼ rea μ=b , 2 which is rearranged as ðσ=μÞ2 b 00  l  p a þ μ2 al ¼ br 2eal μ=b and yields     μ2 al ¼ br  σ 2 p00 al =2bP00 al ,

ð8:40Þ

  l because P00 al ¼ ðμ=bÞ2 ea μ=b . Inequality (8.38) implies that σ 2p00(al)/ 2bP00(al) > σ 2/2b, hence Eqs. (8.31) and (8.40) imply: e a¼

2b2 r  σ 2 > al : 2bμ2

One can show similarly that e a < ah .

References Chambers M, Bailey R (1996) A theory of commodity price fluctuations. J Polit Econ 104 (5):924–957 Deaton A, Laroque G (1992) On the behaviour of commodity prices. Rev Econ Stud 59(1):1–23 Deaton A, Laroque G (1995) Estimating a non-linear rational expectations commodity price model with unobservable state variables. J Appl Econ 10:S9–S40 Deaton A, Laroque G (1996) Competitive storage and commodity price dynamics. J Polit Econ 104 (5):896–923 Dixit A, Pindyck R (1994) Investment under uncertainty. Princeton University Press, Princeton, NJ, p 468 Guerra E, Bobenreith E, Bobenreith J (2018) Comments on: “On the behavior of commodity prices when speculative storage is bounded” Oglend A, Kleppe T (2017) On the behavior of commodity prices when speculative storage is bounded. J Econ Dyn Control 75:52–69 U.S. Energy Information Administration (2020a) U.S. crude oil supply and disposition. https:// www.eia.gov/dnav/pet/PET_SUM_CRDSND_K_M.htm. Accessed 7 July 2020 U.S. Energy Information Administration (2020b) Petroleum & other liquids data: prices, crude reserves and production. https://www.eia.gov/petroleum/data.php. Accessed 7 July 2020 Wright B (2001) Storage and price stabilization. In: Handbook of agricultural economics, vol 1. Elsevier, New York, pp 817–861

Part III

Resource Cartel

Chapter 9

Cartel Behaviour in an Exhaustible Resource Industry

Abstract Under low-elasticity demand, a resource monopoly is supposed to be constrained either by the presence of competing participants in the market or by the existence of a substitute for the natural resource. In the presence of a competitive fringe, the cartel’s activity in an exhaustible resource industry violates the Herfindahl principle presented in Chap. 2 that an advantageous oil region depletes its resource stock before a disadvantageous region begins extraction. In the cartel-fringe models of resource industry considered in this chapter, the time sequencing of production dramatically changes and permits the case where the low-cost cartel and the highcost fringe produce simultaneously. In the presence of a backstop technology, a perfect substitute provides a ceiling on the exhaustible resource price. In the model with such technology, the fringe fully exhausts its resource before the cartel becomes a monopoly that sells the resource at the backstop price. The cartel initially makes a strategic choice of resource allocation over time between the transition phase and the backstop phase.

9.1

Introduction

The previous chapters of this book are based on the premise of perfect competition between resource market participants. This assumption neglects the essential features of international resource markets that resulted from the existence of cartels in these markets. The most influential international resource cartel is OPEC, the Organization of the Petroleum Exporting Countries. It was established in 1960 and has changed the nature of the world oil market. In Chap. 7, we referred to empirical evidence indicating that the activity of this cartel contributed to the increase of the oil price volatility. In this chapter, we will focus on some theoretical issues concerning resource cartel behaviour. The oil cartel demonstrated its market power for the first time in 1973–1974, when the price of imported oil nearly quadrupled over a quarter because of the oil export embargo imposed by OPEC. At various times after the first oil price shock caused by the embargo, OPEC has displayed apparent collusive behaviour through publicly concluded agreements between its member states about oil production and © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_9

225

226

9 Cartel Behaviour in an Exhaustible Resource Industry

price levels. The oil cartel has affected the world oil price by setting production quotas for individual cartel members to restrict the oil supply by the cartel and to keep the oil price above a competitive level. The market power of the oil cartel OPEC is ensured by its significant advantage in terms of the quantity and accessibility of oil reserves. The cartel holds the vast majority of world oil reserves that are easily extracted from onshore deposits. Due to this natural advantage, the cartel members incur substantially lower extraction costs than oil-producing regions outside the cartel. This advantage is especially relevant to the Middle East as the core region of OPEC and its most resource-rich countries such as Saudi Arabia, Kuwait, Iraq and the United Arab Emirates. The oil-rich OPEC members have obtained systematically high profits in excess of the scarcity rents, which could not be so high under a competitive market, in view of the plentiful oil resources of these countries. Significant and persistent excess profits have been demonstrated, for example, by the accumulation of substantial wealth by these countries. The coexistence of self-imposed constraints on production with longterm excessive benefits indicates OPEC’s anti-competitive behaviour in the world oil market. In this chapter, we analyse the consequences of a resource market cartelization from the perspective of exhaustible resource economics. We discuss the question of how the cartel behaviour influences the price dynamics and the behaviour of non-cartel producers. This analysis is based on the representation of the resource cartel as a cohesive group of producers operating in the market as a single entity. This assumption is questionable because of the incentives of OPEC members to exceed production above the quotas unilaterally. However, the assumption of unanimity of cartel members in their collective choice is necessary for understanding the role of the resource cartel as a dominant player in the market. In such a role, the cartel can influence the market price and take account of this influence in making its production decisions. We will assume in this chapter that the cartel owns a limited resource stock that must be fully exhausted over a finite or infinite time horizon. We thus return to Hotelling’s model framework presented in Chap. 2 of this book and consider a cartel exercising its monopoly power by exploiting a finite resource stock. The cartel does not reduce supply, like a static monopoly, once and forever to raise the resource price above a competitive level, but instead it reallocates resource extraction over time. Therefore, a temporary reduction of extraction intensity must be offset by the equivalent increase of this intensity in the future. The cartel extracts the same total amount of resource as competitive suppliers in case they have controlled the same resource stock, but it uses a different time profile of extraction. This outcome takes place for the extreme case of a pure monopoly examined by Hotelling in his paper (1931). He derived a rule for the marginal revenue of the monopoly, similar to his rule for competitive price: the marginal revenue net of the extraction costs must grow at a rate equal to the rate of real interest. Hotelling mentioned that “monopolistic exploitation of an exhaustible asset is likely to be protracted immensely longer than competition would bring about or a maximizing of social welfare would require. This is simply part of the general tendency of

9.1 Introduction

227

production to be retarded under monopoly” (1931, p. 152). Hotelling’s model of resource monopoly was revised in the 1970s by Joseph Stiglitz (1976), Michael Hoel (1978) and other authors. The essential assumption underlying the models of pure resource monopoly is that the price elasticity of demand is above one in equilibrium. A high-elasticity demand reduces more rapidly in percentage terms than the price rises, hence a profitmaximizing monopolist cannot set the price arbitrarily high. However, the realworld price elasticity of demand is normally below unity for natural resources. Under low-elasticity demand, a monopoly (in a partial equilibrium framework) gains from raising the price indefinitely and equilibrium fails to exist. In this case, it is reasonable to take into account that monopolistic activity in resource markets is constrained either by the existence of some substitute for the natural resource or by the presence of competing participants in the market. In the energy sector, a renewable energy can serve as a substitute for fossil fuels produced from non-renewable natural resources. The energy price dynamic depends, to a large extent, on the degree of substitution between conventional and alternative resources. In Chap. 2, we considered a backstop technology as a perfect substitute for an exhaustible resource. In Chap. 4, the backstop technology provided an imperfect substitute for a conventional resource, and the resource price was determined in the process of gradual transition to alternative energy. In this chapter, the price of a perfect substitute is supposed to be the ceiling on the cartel price that bounds its monopoly power. This power can also be bounded by competitive suppliers acting in the industry outside the cartel. This case of industry structure is relevant to the world oil production: one can regard OPEC as a dominant oil company and non-OPEC producers as a “competitive fringe” of oil companies. The market share of each independent oil company outside the cartel is relatively small. One can see this from the diagram in Fig. 9.1 showing the market shares of OPEC and of the ten biggest oil companies in 2007. OPEC’s share in the world oil production was 42.2 per cent, while the individual share of the biggest non-OPEC oil companies varied between 2 and 3 per cent. James Smith characterized these “super-majors” as “small fry” relative to the size of the world oil market (Smith 2009, p. 160). A suitable formal description of such an industry structure is given by the models including a cartel seeking to make monopoly profits and a competitive fringe, a set of small identical firms acting competitively. Stephen Salant (1976) offered a cartelfringe model of the world oil industry, where the cartel and the fringe have the same extraction costs and differ only in the size of their reserves. The cartel in that model behaves as a dominant supplier that can influence the resource price, while competitive firms are price takers. In a Cournot-Nash equilibrium, each firm from the competitive fringe takes as given the time path of price set by the cartel and chooses the extraction path to maximize the sum of discounted rents. The cartel takes the extraction path of the fringe as given and chooses the price path supported by its sales to maximize discounted profits. The sides know perfectly well each other’s resource stocks and technology parameters. The selection of extraction paths

228

9 Cartel Behaviour in an Exhaustible Resource Industry

Fig. 9.1 Cartel and fringe: the shares of OPEC and the ten biggest non-OPEC oil companies in the world oil production in 2007, per cent. Source: Smith (2009, p. 148); BP Statistical Review of World Energy – 2020

includes the choice of time horizons of extraction, which is absent in the non-resource oligopoly models of industrial organization. Distinguishing between a cartel acting as a cohesive and dominant producer and the rest acting as a competitive fringe is an attractive approach that provides simple solutions for industry equilibrium. David Newberry (1981) noted that the advantage of this approach is that it avoids the “implausible extremes” of pure monopoly and perfect competition without running into formal difficulties specific for dynamic oligopolistic models. In the case of resource-extracting industry, such models, as, for example, the one suggested by Tracy Lewis and Richard Schmalensee (1980), entail analytical difficulties that are eliminated within the cartel-fringe analytical framework. We will outline in this chapter the cartel-fringe models for the cases of linear or zero extraction costs that allow for simple analytical solutions. These models can represent a world industry for an exhaustible resource (e.g. crude oil) consisting of two regions of resource extraction. A cartel is supposed to be formed in the region with advantage in the costs of extraction. Under a competitive market, the Herfindahl principle holds: the regions extract resource sequentially and the advantageous region depletes its resource stock first. In the cartelized market, the time sequencing of production dramatically changes. The extraction cost differential across the regions affects the time patterns of extraction, including cases when the cartel or the fringe produces alone or both produce simultaneously. The most interesting (and relevant to the world oil market) is the case of simultaneous production by the cartel and the fringe. It requires compatibility of two of Hotelling’s price rules: for the net price growth and for the net marginal revenue growth. The rule for the net price must hold if the fringe produces, while the

9.2 The Base Competitive Model

229

rule for the net marginal revenue must hold if the cartel produces. Under these price rules, the cartel and the fringe are indifferent to intertemporal allocation of production and select extraction paths to ensure compatibility of these rules. In the presence of a backstop technology, a perfect substitute provides a ceiling on the exhaustible resource price. We have seen in Chap. 2 that under a competitive market the resource is fully exhausted by the time the price reaches the backstop level. In the cartel-fringe model, only the fringe fully exhausts its resource by this time, while the cartel becomes a monopoly that sells the resource at the backstop price. The resource extraction path hence consists of two periods: the transition phase when the cartel is constrained by the presence of the fringe and the backstop phase when the cartel is constrained only by the substitute. The cartel initially makes a strategic choice of resource allocation over time between these two periods. This choice determines the initial resource price and the cartel market share during the transition phase.

9.2

The Base Competitive Model

Let us return to the base model of a competitive industry depleting an exhaustible resource that we considered in Chap. 2. Homogeneous producers use a linear technology of resource extraction with the time-constant marginal cost of production c. A firm takes the price path p(t) as given (time is continuous) and chooses the time path of extraction q(t) to maximize the present value of future rents: ZT V¼

ert ½pðt Þ  cqðt Þdt,

ð9:1Þ

0

subject to the intertemporal resource constraint ZT qðt Þdt  S,

ð9:2Þ

0

where r is the discount rate, T is the time of extraction termination and S ¼ S(0) is the initial resource stock. The demand for resource is y( p(t)) and the market clears at any instant, q(t) ¼ y( p(t)). The present-value Lagrangian for the competitive firm problem (9.1)–(9.2) can be represented as:

230

9 Cartel Behaviour in an Exhaustible Resource Industry

ZT L¼

ert ½pðt Þ  c  ert v0 qðt Þdt þ v0 S,

0

where v0 is the dual variable related to the resource constraint (9.2). The marginal value of the resource at time t is v(t) ¼ v0ert. As was shown in Chap. 2, this marginal value equals the resource rent, v(t) ¼ p(t)  c, which grows exponentially with rate r. The competitive price satisfies the Hotelling rule: p_ ðt Þ ¼ r ðpðt Þ  cÞ:

ð9:3Þ

The necessary first-order condition for a finite termination date results from differentiating the Lagrangian L with respect to T: ½pðT Þ  c  vðT ÞqðT Þ ¼ 0: Extraction at the termination date is zero if demand is zero at this date, q(T ) ¼ y( p (T )) ¼ 0.

9.3

The Pure Monopoly Model

Suppose that all suppliers in the industry have formed a monopoly controlling the initial resource S. At any instant t, the monopoly supplies x(t) units of resource and sets a market-clearing price P(t) taking into account the reaction of competitive consumers. The monopoly faces a price schedule P ðxðt ÞÞ given by the inverse demand function P ðyðt ÞÞ under the condition of a cleared market, x(t) ¼ y(t). The monopoly chooses the termination date TM and the extraction path x(t) for t  TM to maximize the present value of future profits: ZT M W¼

ert ½P ðxðt ÞÞ  cxðt Þdt,

ð9:4Þ

0

subject to the resource constraint: ZT M xðt Þdt  S: 0

At any instant, the market-clearing monopoly price is Pðt Þ ¼ P ðxðt ÞÞ.

ð9:5Þ

9.3 The Pure Monopoly Model

9.3.1

231

Dynamic of Marginal Revenue

The present-value Lagrangian for the monopoly problem (9.4)–(9.5) is represented as: ZT M LM ¼

ert ½P ðxðt ÞÞ  c  ert w0 xðt Þdt þ w0 S,

ð9:6Þ

0

where w0 is the dual variable related to the resource constraint (9.5), which is binding under the linear extraction costs. The marginal revenue associated with selling x(t) units is the price less the marginal loss due to the reduction in price. Let ρ ¼ ρ(x) denote the marginal revenue: ρðxÞ ¼

∂ðP ðxÞxÞ ¼ P ðxÞ þ P 0 ðxÞx ∂x

and w(t) ¼ ertw0 denote the current marginal value of resource for the monopoly at time t. Then the first-order condition for positive x(t) is ∂L/∂x ¼ 0: wðt Þ ¼ ρðxðt ÞÞ  c:

ð9:7Þ

It implies the Hotelling rule for the marginal monopoly rent: ρ_ ¼ r ðρ  cÞ,

ð9:8Þ

because w_ ðt Þ ¼ rw (t). Thus the marginal resource value w(t) equals the marginal monopoly profit ρ(x(t))  c, which grows at a rate equal to the interest rate. If the choke price is finite, P ð0Þ < 1, the terminal date condition results from maximizing the Lagrangian (9.6) with respect to TM: ½P ðxðT M ÞÞ  c  wðT M ÞxðT M Þ ¼ 0: The expression in brackets is positive for P 0 ðxÞ < 0 due to the condition (9.7), hence x(TM) ¼ 0. The termination time TM is infinity only if lim P ðxÞ ¼ 1. x!0

9.3.2

Solution for Linear Demand

Consider the case of linear price schedule faced by the monopolist:

232

9 Cartel Behaviour in an Exhaustible Resource Industry

P ðxÞ ¼ α  βx

ð9:9Þ

with choke price P ð0Þ ¼ α > c and slope coefficient β. The marginal revenue is ρðxÞ ¼ P ðxÞ  βx ¼ 2P ðxÞ  α , and the growth of marginal revenue is ρ_ ðxÞ ¼ 2P_ ðxÞ. Hence, the dynamic equation for marginal revenue (9.8) is represented as the dynamic equation for monopoly price P(t):   αþc P_ ðt Þ ¼ r Pðt Þ  : 2

ð9:10Þ

We show in Appendix A.1 that the equilibrium solution of this equation is Pðt Þ ¼ c þ

αc þ w0 ert , 2

ð9:11Þ

where w0 ¼ 12 ðα  cÞerT M is the initial marginal rent. The monopoly rent per unit, P (t)  c, is thus the sum of excess monopoly profit, (α  c)/2, which would apply in statistical analysis of monopoly and the marginal resource rent, w0ert. Inserting P(t) into Eq. (9.9) yields the solution for monopoly extraction: xð t Þ ¼

  αc 1  erðtT M Þ : 2β

ð9:12Þ

Inserting x(t) into the binding resource constraint (9.5) and integrating implies the equation for termination time TM:  α  c rT M þ erT M  1 ¼ S: 2βr

ð9:13Þ

The time of termination is increasing with the initial resource stock, as well as the monopoly extraction x(t) for any t  TM. Consider now the solution for the competitive problem (9.1)–(9.2) under linear demand. The time path of competitive price satisfying the Hotelling rule (9.3) is: pðt Þ ¼ c þ v0 ert ,

ð9:110 Þ

where v0 ¼ (α  c)erT is the initial marginal rent. As was shown in Chap. 2, the competitive extraction and the equation for termination time are:   αc 1  erðtT Þ β   αc rT þ erT  1 ¼ S: βr qðt Þ ¼

ð9:120 Þ ð9:130 Þ

Inverting equations (9.13) and (9.130 ) yields the time of termination as the increasing function of initial resource stock for competitive market and monopoly:

9.3 The Pure Monopoly Model

233

Fig. 9.2 Monopolistic and competitive price paths

Fig. 9.3 Monopolistic and competitive extraction paths

T ¼ T ðSÞ, T M ¼ T ð2SÞ:

ð9:14Þ

Figures 9.2 and 9.3 portray the price and extraction paths for the monopoly and the competitive market, corresponding to Eqs. (9.11), (9.110 ) and (9.12), (9.120 ), respectively. Solutions (9.14) imply that the time horizon of extraction is longer under monopoly. Competitive and monopoly paths intersect each other at some time moment T0. In the early phase, before T0, the extraction is lower and the price is higher for the monopolized industry. In the later phase, after T0, the extraction is lower and the price is higher for the competitive industry. This configuration of solutions results from the identity of initial resource stocks S in both cases. Note that although the competitive price is above the monopoly price after time moment T0, the

234

9 Cartel Behaviour in an Exhaustible Resource Industry

monopoly cannot switch to the competitive price path, because its resource stock remaining at this moment is larger than it would be for the competitive solution.

9.3.3

Conservationism of Resource Monopoly

The monopolistic and competitive paths coincide under a constant demand elasticity, σ(x) ¼ σ, and zero extraction costs, c ¼ 0, as was shown by Joseph Stiglitz (1976). In this case the monopoly price schedule is P ðxÞ ¼ x1=σ and the marginal revenue is proportional to price, ρðxÞ ¼ P ðxÞð1  1=σ Þ: The price growth rates are, therefore, _ _ ¼ P=P identical for competitive market and monopoly, p=p ¼ r. (Here and in what follows, the time argument is omitted if possible.) The time horizon of extraction in both cases is infinite, because lim x1=σ ¼ 1 . The competitive market and the x!0

monopoly are described by the same equations, and the terminal conditions are the same. One can see from Eq. (2.16) in Chap. 2 that in this case the initial marginal resource values are also the same. Hence, the competitive and monopolistic prices are identical at any instant and extraction paths coincide. As Stiglitz pointed out in his paper (1976, p. 655), “in some other cases there is some tendency for a monopolist to be more ‘conservation minded’ than a competitive market would be”. The time paths of monopoly price and extraction portrayed in Figs. 9.2 and 9.3 are more conservationist in the sense that they are flatter than the competitive time paths. The monopoly price in Fig. 9.2 is increasing slower than the competitive price, while the monopoly extraction in Fig. 9.3 is decreasing slower. We show in Appendix A.2 that for c ¼ 0 the rate of monopoly price growth is below r at any time t  TM if the price elasticity of demand is strictly decreasing: σ 0 ðxÞ < 0:

ð9:15Þ

This condition is fulfilled for the linear inverse demand (9.9), in which case σ ðxÞ ¼ 1 þ

α : βx

Condition (9.15) is sufficient for the marginal revenue to be declining in output, ρ0(x) < 0, as is shown in Appendix A.2. A possible justification for the assumption that the demand elasticity is decreasing with consumption is that under higher prices consumers have stronger incentives to use alternative resources. For instance, oil and coal dominated as energy sources in the epoch of cheap energy. The growth of energy prices raised the demand for natural gas and non-conventional sources such as biofuels, solar and wind energy. As a result, the demand for conventional energy has become more elastic at higher prices due to the appearance of new substitutes for this energy.

9.4 Perfect Competition of Two Regions

9.3.4

235

The Case of Low-Elasticity Demand

The essential assumption of the pure monopoly model (9.4)–(9.5) is that the price elasticity of demand is above unity in equilibrium, because the marginal revenue ρðxÞ ¼ P ðxÞ þ P 0 ðxÞx is positive if, and only if, σ ð xÞ ¼ 

P ð xÞ >1 xP 0 ðxÞ

(for the domain of x such that P 0 ðxÞ < 0). Otherwise, the monopoly would have gained from raising the price indefinitely and the problem (9.4)–(9.5) would have had no solution. However, the assumption of high-elasticity demand is at odds with empirical evidence on resource markets, where the price elasticity of demand is normally below unity. For the oil market, the estimates of short-term price elasticity vary in the interval between 0.03 and 0.3 (Smith 2009, p. 150). Therefore, the pure monopoly model should be modified to incorporate the case of low-elasticity demand, σ  1. First, the pure monopoly model can be extended to include two groups of producers. The first group forms a cartel, while the second group, a competitive fringe, consists of competitive suppliers outside the cartel. Suppose that the cartel produces xa, the competitive fringe produces xb and the resource market clears, xa + xb ¼ y. Let the cartel take the fringe’s output as given. Then the marginal revenue of the cartel, ρðxa , xb Þ ¼ P ðxa þ xb Þ þ P 0 ðxa þ xb Þxa , is positive if P =P 0 > xa or: σ ð yÞ >

xa : xa þ xb

Therefore, the price elasticity of demand σ( y) can be below one in the presence of a competitive fringe that constrains the market power of the resource cartel. Second, this elasticity can be below one in the presence of a backstop technology. If the demand for resource displays low elasticity, but there exists a perfect substitute for this resource, the monopoly cannot raise the price indefinitely and should set it infinitesimally below the backstop price. If, in addition, the monopoly is constrained by a competitive fringe, the solution includes a phase of gradual transition to the backstop price, as will be shown below.

9.4

Perfect Competition of Two Regions

Now we extend the base competitive model of Sect. 9.2 to the case of a competitive resource industry consisting of two regions that have the same discount rate r, but differ in the marginal extraction costs ca and cb and in the initial resource stocks Sa

236

9 Cartel Behaviour in an Exhaustible Resource Industry

Fig. 9.4 Competitive extraction paths in the two-region world

and Sb. Region a is assumed to be advantageous in the sense that its marginal extraction cost is lower than that of region b, ca  cb. Suppose that demand is linear. The market-clearing price in the two-region model is p ¼ α  βðqa þ qb Þ, where qa and qb denote outputs by region a and b, respectively. As we have shown in Chap. 2 in our discussion on the Herfindahl principle, under constant marginal costs the regions cannot produce simultaneously. The advantageous region a depletes its resources first, and the disadvantageous region b extracts its resource thereafter. The competitive outcome is characterized by the two periods of production such that only region a produces in the first period, qa(t) > 0, qb(t) ¼ 0, and only region b in the second period, qa(t) ¼ 0, qb(t) > 0. These periods are designated as: 0  t  Ta and Ta  t  Tb, where Ta and Tb are the times of termination for region a and b, respectively. The extraction paths qa(t) and qb(t) are shown in Fig. 9.4, which replicates Fig. 2.8 from Chap. 2. For region b, the termination price is pb(Tb) ¼ α and hence the extraction path is given by formula (9.120 ):   qb ðt Þ ¼ ðα  cb Þ 1  erðtT b Þ =β: Region b produces as the single resource supplier in the industry, and the presence of region a in the earlier phase does not matter for its production plan. The extraction path for sum of the term q0a ðt Þ ¼   region a can be 00represented as rthe rðtT a Þ ðtT a Þ =β and the term qa ðt Þ ¼ qb ðT a Þe ð α  ca Þ 1  e :

9.5 The Cartel-Fringe Model

237

qa ðt Þ ¼ q0a ðt Þ þ q00a ðt Þ: The term q0a ðt Þ does not depend explicitly on region b0s activity in the later phase.

9.5

The Cartel-Fringe Model

The time sequence of extraction in the resource industry changes substantially under imperfect competition on the part of producers. Let the market fundamentals be the same as in the previous section. Suppose that producers from region a have formed a cartel that can influence the price level, whereas region b represents a fringe of competitive producers. In this section, we will use the cartel-fringe model with an exhaustible resource examined by Ulph and Folie (1980). The cartel behaviour in this model is assumed to satisfy the conditions of the intertemporal Cournot-Nash equilibrium with simultaneous movements of the cartel and the fringe. At any instant, the cartel takes the output plans of the fringe as given and sets prices (hence makes output decisions) to maximize the net present value of profits subject to the condition of market clearing. Competitive producers take the prices set by the cartel as given, and select outputs to maximize their profits. At each moment in time, the cartel behaves as a pure monopolist with respect to the residual demand defined as consumer demand net of the fringe’s supply. The inverse demand is P ðxa þ xb Þ ¼ α  βðxa þ xb Þ and the cartel’s marginal revenue is ρa ¼ P  βxa ¼ 2P  α þ βxb ,

ð9:16Þ

where xa and xb denote outputs by the cartel and the fringe, respectively, and P is the cartel price. On the one hand, from Eq. (9.8), the marginal revenue grows as: ρ_ a ¼ r ðρa  ca Þ:

ð9:17Þ

Equation (9.16) implies that ρ_ a ¼ 2P_ þ βx_ a . Combining this with (9.17) yields the equation for cartel price growth:   α þ ca βxb βx_ P_ ¼ r P  þ  b: 2 2 2

ð9:18Þ

On the other hand, the price growth equation for region b is given by the Hotelling rule: P_ ¼ r ðP  cb Þ:

ð9:19Þ

238

9 Cartel Behaviour in an Exhaustible Resource Industry

If both the cartel and the fringe are present in the market, the price growth Eqs. (9.18) and (9.19) must coincide, implying that xb satisfies the following dynamic equation:   x_ b ¼ r xb  β1 ðα  cb  ΔcÞ ,

ð9:20Þ

where Δc ¼ cb  ca  0 is the marginal cost differential. Thus, if both regions produce simultaneously, the price grows according to the Hotelling rule (9.19) for high-cost region b. The cartel and the fringe produce to meet the market-clearing condition: P ¼ α  β ð xa þ xb Þ

ð9:21Þ

and the condition of price growth compatibility (9.20).

9.5.1

Dynamics of Outputs

The time sequence of production under a cartelized market is more complex than under a competitive market. Ulph and Folie (1980, p. 651) specified five possible patterns of time sequencing of production for the cartel-fringe model with an arbitrary unit-cost differential Δc: I. II. III. IV. V.

Cartel alone; simultaneous; fringe alone Simultaneous; fringe alone Simultaneous Simultaneous; cartel alone Fringe alone; simultaneous; cartel alone.

These cases are listed in decreasing order of cost advantage to the cartel measured by the cost differential Δc. Figure 9.5 illustrates the order of these production patterns with regard to the size of Δc. We focus here on cases II and III shown in Fig. 9.5, where the cartel has a significant cost advantage to the fringe. In these cases, the cartel and the fringe produce simultaneously until the cartel exhausts its reserves (in case II), and the fringe remains alone in the market. In case III, the cartel and the fringe produce simultaneously and their reserves are exhausted on the same date.

Fig. 9.5 Patterns of production in the cartel-fringe model

9.5 The Cartel-Fringe Model

239

The cost differential for these two cases is constrained by the upper and lower bounds Δ1c and Δ2c that are shown in Fig. 9.5 and will be determined below. The termination dates for these cases are Ta and Tb such that Ta  Tb and both regions produce simultaneously in period 0  t  Ta, whereas region b produces alone in period Ta  t  Tb in case II. The solution for price P(t) satisfying the Hotelling rule (9.19) and the marketclearing condition (9.21) for 0  t  Tb is: Pðt Þ ¼ cb þ vb ð0Þert

ð9:22Þ

with vb ð0Þ ¼ ðα  cb ÞrT b , as follows from (9.110 ). To find the solution for extraction paths, we make the following guess. Suppose that during the period of simultaneous production, 0  t  Ta, the cartel acts as if it was a price taker and had the same marginal extraction cost as the fringe, cb > ca. We can, therefore, regard both regions as a population of price-taking producers with the marginal extraction cost cb and the initial resource stock Sa + Sb. Denote the industry supply as q(t) ¼ xa(t) + xb(t). For the price path (9.22), the industry supply meets the demand at any instant 0  t  Tb if   qðt Þ ¼ ðα  cb Þ 1  erðtT b Þ =β,

ð9:23Þ

as follows from Eqs. (9.21), (9.22) and (9.120 ). Then the cartel’s output is given by   xa ðt Þ ¼ Δc 1  erðtT a Þ =β

ð9:24Þ

for 0  t  Ta, as we show in Appendix A.3. Hence the fringe’s output is ( xb ð t Þ ¼

qðt Þ  xa ðt Þ, 0  t  T a qðt Þ, T a  t  T b

:

ð9:25Þ

For the period of simultaneous production, 0  t  Ta, the solution (9.25) is given by:     βxb ðt Þ ¼ ðα  cb Þ 1  erðtT b Þ  Δc 1  erðtT a Þ :

ð9:26Þ

It satisfies differential Eq. (9.20) ensuring the compatibility of price dynamics because, from Eq. (9.26), βx_ b ðt Þ ¼ r ðα  cb ÞerðtT b Þ þ rΔcerðtT a Þ ¼ r ðβxb ðt Þ  ðα  cb  ΔcÞÞ: ð9:27Þ The cartel’s output (9.24) is proportional to the extraction cost differential. The fringe operates as a residual supplier (9.25) to meet the consumer demand that

240

9 Cartel Behaviour in an Exhaustible Resource Industry

Fig. 9.6 Extraction paths for the cartel-fringe model, Δ1c  Δc  (α  cb)erΔT

Fig. 9.7 Extraction paths for the cartel-fringe model, (α  cb)erΔT < Δc  Δ2c

governs the industry output path (9.23). Both price growth equations for the cartel (9.18) and for the fringe (9.19) are fulfilled along this path, because the Hoteling-rule solution (9.22) holds for the fringe and the condition of price compatibility (9.20) is fulfilled. Thus, the solution (9.23)–(9.25) satisfies equilibrium conditions, and we made the right guess that the low-cost cartel is mimicking the high-cost competitive fringe. Figures 9.6 and 9.7 show the time profiles of outputs for both regions given by the solution (9.23)–(9.25). The solid curve depicts the industry supply q(t). The dashed curve shows the fringe’s supply xb(t). It is non-increasing in time, x_ b ðt Þ  0 , if

9.5 The Cartel-Fringe Model

241

Δc  (α  cb)erΔT with ΔT ¼ Tb  Ta, as follows from Eq. (9.27). Figure 9.6 portrays the case Δ1c  Δc  (α  cb)erΔT such that xb(t) is non-increasing for the whole period of resource exploitation, 0  t  Tb. Figure 9.7 illustrates the case (α  cb)erΔT < Δc  Δ2c with xb(t) increasing for the period of simultaneous production, 0  t  Ta, and decreasing for Ta  t  Tb. As a result, the cartel’s behaviour dramatically changes the time sequence of extraction. One can compare Figs. 9.6 and 9.7 with Fig. 9.4 showing the time profiles of supply for the competitive market. The higher-cost resource supplier in the cartelfringe model is involved in production from the beginning of resource exploitation. The fringe extracts the resource more intensively in the early phase, as shown in Fig. 9.6, even though it has a cost disadvantage. The Herfindahl principle is, therefore, violated due to the cartel’s distorting impact on the intertemporal and cross-regional allocation of production. The cost differential Δc > 0 determines, apart from the allocation of production between the regions, the cartel’s excess profit at any time 0  t  Ta:   ðPðt Þ  ca  va ð0Þert Þxa ðt Þ ¼ Δc½Δc þ ðvb ð0Þ  va ð0ÞÞert  1  erðtT a Þ =β, as results from Eqs. (9.22) and (9.24). Consequently, the cartel’s excess profit is similar in its nature to the differential rent obtained from superior land pieces in the Ricardian model of land rent that was presented in the introduction to this book.

9.5.2

The Bounds of the Cost Differential

The resource constraint for the cartel follows from the output time profile (9.24):  Δc  rT a þ erT a  1 ¼ Sa βr with the solution given by Ta ¼ T



 α  cb Sa , Δc

ð9:28Þ

where T(.) is the solution of Eq. (9.130 ). The time of termination for region b is found from the resource constraint for the whole industry:  α  cb  rT b þ erT b  1 ¼ Sa þ Sb , βr which follows from the industry supply (9.23) and implies that

242

9 Cartel Behaviour in an Exhaustible Resource Industry

T b ¼ T ðSa þ Sb Þ:

ð9:29Þ

Now we can determine the bounds of the cost differential for the pattern of production II, Δ1c < Δc < Δ2c, shown in Fig. 9.5. The lower bound Δ1c is found from solutions (9.28), (9.29) and the condition that the termination dates for the cartel and the fringe coincide, Ta ¼ Tb: Δ 1 c ¼ ð α  cb Þ

Sa < α  cb : Sa þ Sb

The coincidence of termination dates corresponds to case III of simultaneous production for the whole period of industry resource depletion, Δc ¼ Δ1c, shown in Fig. 9.5. For Δc < Δ1c, we have cases IV and V such that Ta > Tb and the fringe exhausts resources before the cartel. The upper bound Δ2c is the maximal cost differential such that the fringe produces initially together with the cartel. This bound is found from Eq. (9.26) and the condition that xb(0) ¼ 0: Δ2 c ¼ ðα  cb Þ

1  erT b > α  cb : 1  erT a

Inequality on the right-hand side holds since Tb > Ta. We thus have shown that Δ1c < Δ2c. For Δc > Δ2c, we have case I with the cartel producing alone in the early phase of resource extraction because of its high cost advantage.

9.6

Cartel and Fringe in the Presence of a Backstop Technology

Suppose that a backstop technology can provide a perfect substitute for an exhaustible natural resource. Let the price of the substitute be F ¼ const and consumer demand for the resource be given by the isoelastic function y(P) ¼ Pσ with price elasticity less than one in magnitude, 0 < σ < 1. Under low-elasticity demand, the pure monopoly would set the price initially at (infinitesimally below) the backstop level F and deplete its resource stock afterwards at this constant price. The outcome differs if the resource cartel is constrained by a competitive fringe. In this section, we use the cartel-fringe model with a backstop technology suggested by Richard Gilbert (1978). Suppose that the cartel is formed by region’s producers and the competitive fringe is represented by region b. We assume, following Gilbert (1978, p. 390), that the extraction costs of both regions are zero. The regions differ only in the initial resource stocks denoted as Sa and Sb. In the presence of a competitive fringe, the resource price path consists of the two periods shown in Fig. 9.8. In the first period, 0  t  Tb, which we call the transition phase, both regions produce simultaneously. The price is set initially below the

9.6 Cartel and Fringe in the Presence of a Backstop Technology

243

Fig. 9.8 The cartel price and the time sequencing of extraction

backstop level F and increases until time Tb, when it reaches this level. At this time, the competitive fringe’s resource runs out and the cartel becomes a pure monopoly selling the resource at the backstop price. Otherwise, if P( Tb) < F, the resource price will jump instantaneously to F and create arbitrage opportunities. In the second period, Tb  t  Ta, called the backstop phase, the cartel sells the resource at price F until termination date Ta. On this date, the cartel’s resource is exhausted and consumers switch to the substitute good. The time path of the cartel price P(t) is drawn with the solid kinked curve in Fig. 9.8. In this model, the cartel chooses at initial time t ¼ 0 the extraction path xa(t) and the termination date Ta to maximize the present value of rents: ZT b Wa ¼

e

rt

ZT a Pðt Þxa ðt Þdt þ

ert Fxa ðt Þdt,

ð9:30Þ

Tb

0

subject to the resource constraint:

RT a

xa ðt Þdt ¼ Sa. The fringe chooses the extraction

0

path xb(t) and the termination date Tb to maximize ZT b Wb ¼ 0

subject to

RT b 0

xb ðt Þdt ¼ Sb .

ert Pðt Þxb ðt Þdt,

ð9:31Þ

244

9 Cartel Behaviour in an Exhaustible Resource Industry

The resource price in the transition phase is given by the inverse demand function, P ¼ ðxa þ xb Þ1=σ . The sides move simultaneously at any instant, and the cartel’s marginal revenue is ρa ¼ P þ P 0 x a ¼ P  P

  xa ψ , ¼P 1 σ σ ð xa þ xb Þ

ð9:32Þ

where ψ ¼ ψ(t) denotes the cartel’s market share during the transition phase: ψ ðt Þ ¼

xa ð t Þ : xa ð t Þ þ xb ð t Þ

In equilibrium, the cartel’s market share must be below demand elasticity, ψ(t) < σ, for the marginal revenue (9.32) to be positive. During the transition phase, the Hotelling rule must hold for price and marginal revenue: P_ ¼ rP

ð9:33Þ

ρ_ a ¼ rρa :

ð9:34Þ

These two equations are compatible if the cartel market share is time-constant: ψ ¼ const:

ð9:35Þ

Otherwise ρa would not grow at rate r, as one can see from (9.32). From Eq. (9.33), the price path is Pðt Þ ¼ P0 ert :

ð9:36Þ

and the outputs of regions for 0  t  Tb are given by rσt rσt xa ðt Þ ¼ ψPσ , xb ðt Þ ¼ ð1  ψ ÞPσ : 0 e 0 e

ð9:37Þ

Consequently, the extraction paths are determined by the time-constant cartel’s market share ψ and the initial resource price P0.

9.6.1

The Cartel’s Strategic Choice

At time t ¼ 0 the cartel chooses the initial price P0 and the market share for transition phase ψ. The choice of the cartel’s market share results from its strategic decision about intertemporal resource allocation between transition and backstop phases. Let

9.6 Cartel and Fringe in the Presence of a Backstop Technology

245

Q denote the cumulative industry resource to be extracted during the transition phase: ZT b Q¼

ðxa ðt Þ þ xb ðt ÞÞdt: 0

Due to Eqs. (9.37), this is equal to

Q¼

Pσ 0

ZT b

erσt dt ¼ Pσ 0

1  erσT b  QðP0 , T b Þ: rσ

ð9:38Þ

0

During the backstop phase, Tb  t  Ta, the cartel’s output is time-constant and xa(t)  Fσ . The amount of resource assigned for this period is B ¼ Fσ ΔT, where ΔT ¼ Ta  Tb. The amount of industry resource remaining after the transition phase is Sa + Sb  Q, hence the industry resource balance implies that B ¼ Sa + Sb  Q. We show in Appendix A.4 that the cartel’s problem (9.30) is reduced to the strategic choice at the initial time of the initial price P0 and the backstop resource B maximizing the present value of rents:  rBF σ rT b 1  e F W a ¼ max P0 ðSa  BÞ þ e , rF σ P0 , B

ð9:39Þ

subject to the resource balance: B ¼ Sa þ Sb  QðP0 , T b Þ:

ð9:40Þ

The objective function (9.39) is the discounted sum of the rent obtained from selling resource Sa  B during the transition phase and the rent from selling resource B during the backstop phase.

9.6.2

Cournot-Nash Equilibrium

The cartel’s strategic problem (9.39)–(9.40) includes the variable of termination time Tb selected initially by the fringe. The Hotelling rule (9.35) implies the link between initial and backstop prices: P0 ¼ erT b F that gives a simple decision rule for termination time:

ð9:41Þ

246

9 Cartel Behaviour in an Exhaustible Resource Industry

Tb ¼ r

1



 F ln : P0

ð9:42Þ

Under the Cournot-Nash equilibrium for strategic choice, the cartel takes Tb as given. Inserting (9.40) into (9.39) and differentiating with respect to P0 yields the first-order condition for the cartel’s problem that accounts for the link (9.41): Sa  B þ P0

∂Q ∂Q rBF σ ¼ P0 e : ∂P0 ∂P0

ð9:43Þ

The left-hand side of this equation is the marginal gain of the cartel from a price increase over the transition phase, while the right-hand side is the marginal gain over the backstop phase. Note that in the original paper by Gilbert (1978, p. 391), the term on the right-hand side of the equation similar to Eq. (9.43) is absent due to the assumption adopted in that paper that the cartel’s initial resource stock Sa is arbitrarily large, implying that the backstop phase ΔT ¼ BFσ is arbitrarily long. The resource selected by the cartel for the transition phase is equal to Q ¼ Pσ 0

ð1  ðF=P0 Þσ Þ Pσ  F σ 1  erσT b Pσ ¼ 0 , ¼ 0 rσ rσ rσ

ð9:44Þ

as follows from Eqs. (9.38) and (9.41). The marginal effect of the initial price on this resource is Pσ  F σ ∂Q 1  erσT b Q ¼ σPσ1 ¼ σ , ¼ 0 0 P0 rσ rP0 ∂P0

ð9:45Þ

provided that the cartel takes Tb as given. Using Eqs. (9.44) and (9.45), the first-order σ condition (9.43) can be written as Sa  B  σQ ¼ σQerBF or, taking into account the resource balance (9.40): 

σ 1  σ þ σerBF Q ¼ Sb :

ð9:46Þ σ

Thus, the share of fringe in the resource for transition is Sb =Q ¼ 1  σ þ σerBF . Note that we could have assumed that the cartel behaves in making a strategic choice at time t ¼ 0 as a Stackelberg leader that takes account of the fringe’s reaction to this choice. In this case, the cartel is supposed to play a more active role as the dominant firm. It internalizes the effect of the initial price on the termination time selected by the fringe, Tb ¼ r1 ln (F/P0). Then this effect is accounted for in differentiating the objective function (9.39) and in calculating the derivative ∂Q/ ∂P0. However, one can check that the outcome for the Stackelberg equilibrium is the same as for the Cournot-Nash equilibrium, because the direct effect of the initial price P0 on the termination time Tb and the indirect effect of P0 on the industry resource for transition Q exactly offset each other.

9.6 Cartel and Fringe in the Presence of a Backstop Technology

9.6.3

247

The Equilibrium Resource Allocation

The equilibrium resource allocation is found from Eqs. (9.40) and (9.46), which we rewrite for the sake of convenience as B ¼ Sa þ Sb  Q Q¼

Sb σ : 1  σ þ σerBF

ð9:47Þ ð9:48Þ

These equations determine the intertemporal allocation of industry resource Sa + Sb between transition and backstop phases, Q and B. The first equation is the resource balance, while the second one determines the proportion of allocation as shown in Fig. 9.9. Line B(Q) corresponds to Eq. (9.47) and curve Q(B) to Eq. (9.48). The Cournot-Nash solution (QN, BN) exists and is unique because B(Q) is decreasing and Q(B) is increasing. The initial equilibrium price is found from condition (9.44):  1=σ PN0 ¼ F σ þ rσQN :

ð9:49Þ

The time of transition Tb is found from Eq. (9.42), and the cartel’s market share is obtained from the first-order condition (9.46):

Fig. 9.9 The equilibrium resource allocation

248

9 Cartel Behaviour in an Exhaustible Resource Industry

Fig. 9.10 Extraction paths in the cartel-fringe model with backstop price

ψN ¼

  Q N  Sb rBN F σ ¼ σ 1  e : QN

ð9:50Þ

The equilibrium cartel’s market share is below demand elasticity, hence the marginal revenue ρa is positive during the transition phase, as follows from Eq. (9.32). Figure 9.10 portrays the time paths of regions’ production. They decrease during the transition phase due to Eq. (9.37). A jump of the cartel’s output xa(t) occurs at time Tb, when the fringe’s resource is completely exhausted and the cartel captures the whole market.

9.6.4

The Cartel’s Distorting Effects

In the absence of a cartel, under competitive allocation of production, the backstop phase does not exist and ΔT ¼ B ¼ 0. The initial resource stock of the industry Sa + Sb must be fully exhausted by the time of switching to the backstop technology T, as is shown by the dashed curves p(t) and q(t) in Figs. 9.8 and 9.10, respectively (the terminal condition is q(T ) ¼ Fσ ). The competitive resource allocation implies that Q ¼ Sa + Sb. Substituting this for QN in the initial equilibrium price (9.49) gives the socially efficient price for an economy endowed with the initial resource stock Sa + S b: P0 ¼ ðF σ þ rσ ðSa þ Sb ÞÞ1=σ :

9.7 Concluding Remarks

249

This formula, derived by Dasgupta and Stiglitz (1975), was presented in Chap. 2 of this book. Remember that the efficient resource price P0 results from the maximization of the discounted sum of producer rent and consumer surplus. The cartel’s activity brings about welfare losses by consumers. Moreover, the higher the cartel’s initial resource share is, the more substantial is the distorting effect of the cartel on resource allocation. Let the total industry resource be normalized to one, Sa + Sb ¼ 1. Then a higher cartel’s resource share means a lower Sb, the nominator of Eq. (9.48) characterizing resource allocation in the Cournot-Nash equilibrium. For a lower Sb, this allocation moves away from the optimal point Q ¼ Sa + Sb, as is illustrated by the dashed curve in Fig. 9.9. The equilibrium allocation shifts to the new point (QN0, BN0) such that QN0 < QN and BN0 > BN. Thus, the cartel with a higher resource share allocates less resource to the transition phase and more to the backstop phase. One can see from Eqs. (9.49), (9.50) that for the new allocation (QN0, BN0) the cartel’s market share and the initial resource price are higher than for (QN, BN), indicating an increase of the cartel’s market power due to its larger initial resource share. As a result, the higher the cartel’s resource share, the more significant is the deviation of the equilibrium resource allocation from the optimum. In particular, if the cartel’s resource share is high, then the amount of backstop resource is large and the duration of the backstop phase is long. In the case of the energy sector, this means that the market power of a resource-rich cartel can lead to a significant delay of the time of switching to renewable energy as a substitute for fossil fuels.

9.7

Concluding Remarks

We have considered the extensions of Hotelling’s model to imperfect competition. In these extensions, a cartel is supposed to control a stock of exhaustible resource and act as the dominant supplier in the presence of a fringe of competitive producers. The cartel-fringe models can be viewed as an intermediate case between the extremes of pure competition and pure monopoly. The merits of these models are analytical tractability and the relevance to the world oil market, including the oil cartel OPEC and a fringe of competitive suppliers from non-OPEC countries. The resource cartel in the cartel-fringe models behaves as an imperfect monopoly weighing its full unit cost, which is the sum of extraction and opportunity costs, against marginal revenue. A cartel, such as OPEC, is able to exercise market power because it has a significant cost advantage and abundant resources that guarantee a sufficient market share to influence the reaction of consumers. The latter are supposed to be myopic in the sense that they do not make intertemporal decisions regarding resource consumption. Therefore, consumer demand is a function of current price and does not depend on price expectations. Unlike consumers, competitive producers outside the cartel control resource stocks and plan outputs for the whole time horizon of extraction. Their choice of extraction paths depends on the

250

9 Cartel Behaviour in an Exhaustible Resource Industry

whole price path selected by the cartel. Hence, the behaviour of a competitive fringe cannot be expressed by a reaction function similar to the static demand function of consumers. Nevertheless, the cartel influences the time profile of extraction selected by the fringe, and this influence is analysed within the cartel-fringe model framework. It was shown in this chapter that under the cartelized market, there exists a phase of simultaneous production by the high-cost and low-cost regions, contrary to the Herfindahl principle that a lower-cost region should extract a resource before a higher-cost one begins extraction. The low-cost cartel exercises its monopoly power by mimicking the high-cost competitive fringe: it acts as if it was a price taker and had the same marginal extraction cost as the fringe. Such behaviour ensures simultaneous production by both sides, which requires the compatibility of Hotelling rules for the cartel’s marginal revenue and for the resource price. In the presence of a backstop technology, the cartel allocates its resource between the transition phase, when it competes with the fringe, and the backstop phase, when the cartel is constrained only by the presence of a substitute for a conventional resource. The cartel’s market share in the transition phase and the initial resource price are determined as the result of this intertemporal resource allocation. The higher the cartel’s initial resource share, the more significant is the deviation of equilibrium resource allocation from the optimum. In particular, the activity of the oil cartel with a higher resource share results in a more substantial delay of the time of switching to renewables as a substitute for fossil fuels. The resource monopoly can be regarded as a conservationist because it depletes its resource stock more slowly than the competitive market. The resource cartel sets the price above the full unit cost of production and selects a more conservationist extraction path than the competitive market does. The cartel thereby acts as if it has imposed a tax on consumers to discourage current resource consumption and to retard resource extraction. However, the revenue from this “tax” only benefits the cartel, unlike the carbon taxes on fossil fuel consumption imposed by governments involved in climate policy implementation to reduce the emission of greenhouse gases into the atmosphere. Furthermore, in addition to the retarding effect on the depletion of fossil resources, the cartel’s market power may result in a retarding effect on the transition to non-fossil energy. We have not analysed the issue of gains and losses by producers and consumers from the resource cartel formation. Salant (1976) showed that the competitive fringe’s profits can rise proportionally more, relative to a competitive market, than those of the cartel. This finding raises the issue of cartel stability, because the cartel members have incentives to leave it. By contrast, Ulph and Folie (1980) demonstrated that the formation of a cartel reduces the profits of the competitive fringe if the latter has a significant cost disadvantage. A possible implication of this result is that OPEC gains at the expense of both consumers and high-cost competitors. In the next chapter, we will study in detail the issue of gains and losses from resource industry cartelization for consumers and producers for a cartel-fringe model with reserves development.

Appendices

251

The cartel’s influence on the time profile of extraction by the competitive fringe depends on the extraction cost differential, as we have seen for Ulph and Folie’s model (1980). In that model, the fringe extracts a resource more intensively in the early phase only if its cost disadvantage is not high or it has a cost advantage relative to the cartel. We will show in Chap. 11, for a model with an economically recoverable resource stock, that the cartel can encourage an early and rapid resource exhaustion by the competitive fringe with a high cost disadvantage.

Appendices A.1 Formula (9.11) The solution for differential equation (9.10) is Pðt Þ ¼

αþc αc þ w0 ert ¼ c þ þ w0 ert : 2 2

ð9:51Þ

The terminal condition x(TM) ¼ 0 implies that P(TM) ¼ α, hence, from Eq. (9.51), α¼

αþc þ w0 erT M 2

and w0 ¼ 12 ðα  cÞerT M .

A.2 Condition (9.15) The demand elasticity is σ ðxÞ ¼ P ðxÞ=xP 0 ðxÞ for the domain of x such that P 0 ðxÞ < 0. The derivative of demand elasticity is σ 0 ð xÞ ¼ 

xP 0 ðxÞ2  P ðxÞðP 0 ðxÞ þ xP 00 ðxÞÞ : ðxP 0 ðxÞÞ2

It is negative if, and only if, xP 0 ðxÞ xP 00 ðxÞ , 0, P 0 ð xÞ

ð9:53Þ

because xP 0 ðxÞ=P ðxÞ < 1 for ρ(x) > 0. Consider the equation for the marginal revenue growth (9.8) with c ¼ 0. On the one hand, ρ_ ðxÞ ¼ P_ ðxÞ þ P 0 ðxÞ_x þ xP 00 ðxÞ_x . On the other hand, x_ ¼ P_ ðxÞ=P 0 ðxÞ . Hence, ρ_ ðxÞ ¼ P_ ðxÞð2 þ xP 00 ðxÞ=P 0 ðxÞÞ and Eq. (9.8) is equivalent to P ðxÞ þ xP 0 ðxÞ P_ ðxÞ ¼ r : 2 þ xP 00 ðxÞ=P 0 ðxÞ Condition P_ ðxÞ < rP ðxÞ holds for x > 0 if, and only if, P ðxÞ þ xP 0 ðxÞ < P ðxÞð2 þ xP 00 ðxÞ=P 0 ðxÞÞ, which is equivalent to (9.52), the necessary and sufficient condition for σ 0(x) < 0. Hence, condition σ 0(x) < 0 is necessary and sufficient for P_ < rP . Condition (9.53) is necessary and sufficient for ρ0 ðxÞ ¼ 2P 0 ðxÞ þ xP 00 ðxÞ < 0, hence σ 0(x) < 0 is sufficient for ρ0(x) < 0.

A.3 Equation (9.24) The general solution of differential equation (9.20) is xb ðt Þ ¼ D  kert , where D ¼ β1(α  cb  Δc) and k is an unknown constant. From (9.23) we have:   βxa ðt Þ ¼ βðqðt Þ  xb ðt ÞÞ ¼ ðα  cb Þ 1  erðtT b Þ  βD þ βkert ¼ βkert  ðα  cb ÞerðtT b Þ þ Δc: The termination condition for the cartel is xa(Ta) ¼ 0, implying that βkerT a ¼ ðα  cb ÞerðT a T b Þ  Δc or βkert ¼ ðα  cb ÞerðtT b Þ  ΔcerðtT a Þ : Consequently, βxa ðtÞ ¼ Δc  ΔcerðtT a Þ ¼ Δcð1  erðtT a Þ Þ.

References

253

A.4 The Objective Function (9.39) During the transition phase, t  Tb, the cartel produces xa ¼ y(P)  xb ¼ Pσ  xb. The initial present value of the cartel’s rent during this phase is ZT b

ert Pðt Þxa ðt Þdt ¼ P0

0

ZT b

ðPðt Þσ  xb ðt ÞÞdt ¼ P0 ðQ  Sb Þ ¼ P0 ðSa  BÞ

0

due to (9.36) and the resource balance B ¼ Sa + Sb  Q. During the backstop phase, Tb  t  Ta, the cartel’s output is xa(t)  Fσ , and the initial present value for this phase is ZT a e

rt

ZT a Fxa ðt Þdt ¼ F

Tb

F σ ert dt ¼ erT b F

σ

1  erΔT 1  erBF ¼ erT b F , σ rF rF σ

Tb

since ΔT ¼ BFσ . Thus, the cartel’s objective function (9.30) transforms to  σ 1  erBF W a ¼ max P0 ðSa  BÞ þ erT b F : rF σ P0 , B

References BP Statistical Review of World Energy – 2020 edition. https://nangs.org/analytics/bp-statisticalreview-of-world-energy. Accessed 18 Nov 2020 Dasgupta P Stiglitz J (1975) Uncertainty and the rate of extraction under alternative arrangements. Institute of Mathematical Studies in the Soial Sciences (IMSSS), Techical report no. 179, Sep 1975 Gilbert R (1978) Dominant firm pricing policy in a market for an exhaustible resource. Bell J Econ 9:385–395 Hoel M (1978) Resource extraction, substitute production, and monopoly. J Econ Theory 19:28–37 Hotelling H (1931) The economics of exhaustible resources. J Polit Econ 39:137–175 Lewis T, Schmalensee R (1980) On oligopolistic markets for non-renewable natural resources. Q J Econ 95(3):475–491 Newberry D (1981) Oil prices, cartels, and the problem of dynamic inconsistency. Econ J 91:617–646 Salant S (1976) Exhaustible resources and industrial structure: a NashCournot approach to the world oil market. J Polit Econ 84(5):1079–1094 Smith J (2009) World oil: market or mayhem. J Econ Perspect 23:145–164 Stiglitz J (1976) Monopoly and the rate of extraction of exhaustible resources. Am Econ Rev 66 (4):655–661 Ulph AM, Folie GM (1980) Exhaustible resources and cartels: an intertemporal Cournot-Nash model. Can J Econ 13:645–658

Chapter 10

The Oil Cartel and Misallocation of Production

Abstract The oil cartel’s activity causes distortions of the oil industry structure resulting from the spatial misallocation of production. In the long term, production shifting from the low-cost cartel to the high-cost competitive fringe brings about a misallocation of global oil resources. In this chapter we consider a two-region model of resource extraction with investment in reserves development, in which the global allocation of oil resources is determined endogenously. We show that if the cartel’s advantage in production costs over the competitive fringe is considerable, the distorting effects of the cartel are substantial, and the efficiency losses on the industry level may be significant. The cartel captures a part of consumer benefits at the cost of significant losses from resource misallocation and deadweight losses. These losses, however, can be reduced due to technological changes improving the competitiveness of the fringe.

10.1

Introduction

In the previous chapter, we considered the effects of a resource cartel’s activity on the intertemporal allocation of an exhaustible oil resource. Here we focus on the structural effects of this activity under a premise of heterogeneity of oil producers located in different regions of the world. Extraction costs vary between cheap onshore oil resources concentrated in the Middle East, the core region of OPEC, and expensive offshore fields in the Northern Sea or tar sands of Alberta in Canada or shale formations of North Dakota in the United States. It is shown in this chapter that the cartel affects the allocation of production between the low-cost and high-cost oil regions, exerting a distorting influence on the industry structure. We will consider the effects of misallocation of production caused by the cartel’s market power. The short-term effects result from the production shifting by the low-cost cartel to the high-cost competitive fringe. In the long term, these effects manifest themselves through misallocation of global oil resources caused by underinvestment in the development of new producing reserves by the cartel and This chapter is based on the unfinished paper by Barry Ickes and Georgy Trofimov. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_10

255

256

10

The Oil Cartel and Misallocation of Production

Fig. 10.1 OPEC and non-OPEC crude oil production, 1970–2019, million barrels per day. Source: BP Statistical Review of World Energy, 2020

overinvestment outside the cartel. These processes imply that the effective resource share of high-cost producers (measured by their share of producing reserves) increases, while this share of low-cost producers decreases. Empirical evidence on the world oil industry supports this view on the structural effects of monopoly power. Since OPEC began to exercise its market power in the 1970s, production has shifted from low-cost to high-cost suppliers. Figure 10.1 shows, for example, that OPEC’s crude oil production did not return to its early 1970s level until the late 1990s, while non-OPEC production more than doubled. Production and investment were thus directed toward high-cost regions, including European countries, while OPEC fell behind these regions. This shift in supply implied the development of high-cost oil resources and a loss in the economic efficiency of production in the long term. James Smith (2009) outlined OPEC’s tendency to underinvest in the development of producing reserves. By 2008, OPEC had accounted for only 10 per cent of the petroleum industry’s upstream capital investment, although it produced nearly half of the global output. The crude oil production capacity of OPEC was virtually unchanged between 1973 and 2008, while non-OPEC producers, working mostly in more expensive petroleum areas, expanded their capacity by 69 per cent during the same period. In 2008, OPEC initiated new projects that amounted to $40 billion per year, whereas the five largest international oil companies spent about $75 billion during 2007 on developing a new production capacity. In 2007, the super-majors reinvested 25 per cent of their gross production revenues to expand capacity, whereas OPEC members invested only about 6 per cent of their net export revenues in such projects (Smith 2009, pp. 152–153).

10.1

Introduction

257

In this chapter we consider the issue of misallocation of production for a model of the world oil industry with opportunities for resource development. The model is built on the premise that a cartel, exercising a monopoly power, is not the only supplier in the industry. There is also a competitive fringe of price-taking firms that “fill the niche” if the cartel reduces its output. In the cartel-fringe models of the previous chapter, the cartel behaves as a dominant supplier that can influence the resource price. Those models extended Hotelling’s model (1931) of exhaustible resource extraction with no account of investment in resource extension. Here we utilize a version of Robert Pindyck’s model (1978) of resource extraction and development of oil reserves that was considered in Chap. 3. We present first a base model of the competitive world oil industry with two oil-producing regions that differ in terms of productivity of extraction and investment. For the sake of simplicity, it is assumed that there is no effect of resource depletion, and in this case a competitive equilibrium is easily calculated as a stationary path. The competitive allocation of production is counterfactual in the sense that oil is produced only in the advantageous region with low production costs. However, the competitive equilibrium serves as a benchmark to demonstrate the cartel’s effects on misallocation of production and the distribution of benefits from oil extraction. Moreover, one can suggest that an outcome with all oil production concentrated in the lowest-cost regions (e.g. in the Middle East) would be optimal for an ideal case of a perfectly integrated world economy, in which all possible barriers to trade, geopolitical constraints and national energy security considerations were absent. The base model of a competitive oil industry is then modified to introduce a cartel—a sole producer in the low-cost oil region. The cartel exercises a monopoly power by reducing supply and shifting production to the high-cost oil region. On the one hand, such a structural shift in the industry enables the cartel to set a price margin above its production costs in the presence of a competitive fringe. On the other hand, the cartel sets the price at the level of marginal production cost of the high-cost region, hence the cartel price margin is equal to the production cost differential. We demonstrate that the cartel pricing causes a reallocation of resources between the regions. The effective resource shares of the cartel and the fringe are determined in equilibrium by the production cost differential. Production shifting and the reallocation of resources are more profitable for the cartel if it has a higher advantage in production costs over competitors. If this advantage is considerable, the distorting effects of the cartel are substantial and the efficiency losses for the industry may be significant. As a result of resource reallocation, the cartel captures some of the consumer benefits they enjoy under a competitive market at the cost of losses incurred by the industry from misallocation of production and deadweight losses. These distorting effects can be mitigated due to a technological change materialized in a reduction of the production cost differential. Such a change brings about a new allocation of resources that weakens the cartel’s power to set the price margin. It is important to note that the increase of output by high-cost competitive producers caused by the technological change can improve efficiency, because it prompts a reduction of the cartel’s value gain and of deadweight losses by the industry.

258

10

The Oil Cartel and Misallocation of Production

Though the model in this chapter is dynamic, the equilibrium paths prove to be stationary due to the assumption that the depletion effect is absent. We thus neglect the important feature of resource supply and obtain an equilibrium with a timeconstant oil price. On the one hand, the possibility of such an outcome should not be excluded, as one can see from Fig. 7.9 in Chap. 7 showing the history of the real oil price. It was relatively stable within a range of $10 to $20 per barrel in 2018 U.S. dollars during the half century between 1923 and 1973, due to flexible supply and abundant oil resources. On the other hand, by assuming the absence of resource depletion, we can greatly simplify the calculation of equilibria that are stationary and defined by static equations. Stationary equilibria are especially suitable for simple welfare comparisons of long-term gains and losses from the industry cartelization. The model, therefore, can be regarded as a convenient tool for demonstrating the allocative inefficiencies caused by the resource cartel’s activity.

10.2

A Model of the Competitive Global Oil Industry

Consider a model of the global oil industry consisting of two oil-producing regions denoted as a and b. Each region is populated with homogeneous competitive firms that extract oil and can invest to develop producing reserves of oil. Time is discrete in this model. We can write the one-period net profit of a firm in region j ¼ a, b in period t as
 π jt ¼ pt xjt  c j xjt , sjt  ijt , where pt is the world oil price, xjt is extraction, cj(xjt, sjt) is expenditure on intermediate inputs in the production, sjt is the beginning-of-period reserve stock of producing reserves exploited by a firm and ijt is investment in resource development via drilling new wells. As in the model of Chap. 3, the extraction cost function cj(xjt, sjt) is increasing in extraction and decreasing in reserve stock, hence a higher oil stock reduces in-period extraction costs. We specify the extraction cost function as quadratic in xjt and linearly homogenous in both arguments:
 c j xjt , sjt ¼ x2jt =2δ j sjt , where δj is a parameter of productivity of oil extraction in region j. Each firm in region j selects a sequence of extraction xjt and investment in reserve addition ijt to maximize the net present value of profits, which is given by:

10.3

The Competitive Oil Price and the Allocation of Production

Vj ¼

1 X

βt π jt ,

259

ð10:1Þ

t¼0

subject to the reserve stock equation: sjtþ1 ¼ sjt  xjt þ ijt =z j ,

ð10:2Þ

where β is the discount factor and zj is the time-constant marginal cost of reserve addition. We assume that the model parameters ensure that in any period the feasibility constraint xjt  sjt holds and is non-binding. The initial resource stock in region j ¼ a, b is equal to zero, sj0 ¼ 0. A firm invests in reserve creation in time period 0, but begins extraction in period 1, so xj0 ¼ 0 and the stock equation (10.2) for t ¼ 0 is s

j1

¼ i j0 =z j :

The parameter of extraction productivity δj and the marginal cost of reserve addition zj are region-specific. This will capture the fact that it is easier to extract oil and to augment reserves in some regions than in others. In particular, δa > δb and za < zb imply that region a is more advantageous in terms of the extraction from exploited fields and the development of new fields. The number of firms in each region is unity, and the market-clearing condition in period t is xat þ xbt ¼ D  μpt ,

ð10:3Þ

where D  μpt is the linear world demand for oil, D is the intercept and μ is the slope of the demand function.

10.3

The Competitive Oil Price and the Allocation of Production

We now consider a solution for the equilibrium oil price in each period and the allocation of production across regions. The Lagrangian for the firm’s problem (10.1)–(10.2) is: Lj ¼

1 h X
i βt pt xjt  x2jt =2δ j sjt  it þ βvjtþ1 sjt  xjt þ ijt =z j  sjtþ1 : t¼0

ð10:4Þ

260

10

The Oil Cartel and Misallocation of Production

The dual variable vjt indicates the marginal value of resource in region j ¼ a, b in period t. The marginal extraction cost in region j is ∂cj/∂xjt ¼ xjt/δjsjt  ξjt. This is the ratio of extraction to the productivity-weighted oil stock of the firm. The internal first-order condition for extraction is ∂Lj/∂xjt ¼ 0 or, from Eq. (10.4), pt ¼ ξjt þ βvjtþ1 :

ð10:5Þ

A price-taking firm is indifferent between extracting a barrel of oil and receiving an operating profit per barrel, pt  ξjt, on the one hand, and holding it in the ground and being rewarded by the opportunity cost of extraction per barrel, βvjt + 1, on the other. The oil price (10.5) is the sum of the marginal extraction cost and the opportunity cost of extraction. If, for example, the opportunity cost is high, the firm shrinks current extraction to reduce the marginal extraction cost. The first-order condition for oil stock is ∂Lj/∂sjt ¼ 0, implying the dynamic equation for the marginal value of oil in the ground: vjt ¼

x2jt 2δ j s2jt

þ βvjtþ1 :

ð10:6Þ

This is the sum of the marginal effect on the extraction cost of an oil stock increase, ∂c j =∂sjt ¼ x2jt =2δ j s2jt , and the discounted next-period marginal value. The marginal value vjt thus differs across regions, because the marginal cost of extraction is region-specific, as the quality of the deposits, characterized by the parameters of extraction productivity δj and the stock levels sjt, differ. Finally, the first-order condition for positive investment in period t  0 in region j is ∂Lj/∂ijt ¼ 0 or, from Eq. (10.4): βvjtþ1 ¼ z j :

ð10:7Þ

The opportunity cost of extraction is equal to the marginal cost of reserve addition. If (10.7) holds in period t, then, from Eq. (10.5), the extraction is:
 xjt ¼ δ j sjt pt  z j :

ð10:8Þ

The extraction is equal to the productivity-weighted oil stock δjsjt multiplied by the firm’s income per barrel, measured as pt  zj, the oil price less the marginal cost of reserve addition. Let Et ¼ δasat + δbsbt + μ denote the slope of the excess supply function, Et ¼ ∂(xat + xbt  (D  μpt))/∂pt, from Eq. (10.8). This is the sum of the productivity-weighted world stock of oil, δasat + δbsbt, and demand slope μ. Combining the supply functions (10.8) with the market-clearing condition (10.3) yields the equilibrium price:

10.3

The Competitive Oil Price and the Allocation of Production

pt ¼ σ at za þ σ bt zb þ σ dt α:

261

ð10:9Þ

where α ¼ D/μ is the choke price, σ jt ¼ δjsjt/Et for j ¼ a, b, σ dt ¼ μ/Et. The equilibrium price (10.9) is the weighted sum of the regions’ marginal costs of reserve addition zj and the choke price α. The weights σ jt in the equilibrium price formula (10.9) indicate the effective resource shares of the regions that characterize the contributions of their opportunity costs (which are equal to zj in equilibrium) to the oil price. The weight σ dt is the demand share characterizing a contribution of the choke price.

10.3.1 The Stationary Competitive Equilibrium An equilibrium path with positive investment in any period is time-stationary. To show it, consider the dynamic equation for the marginal value of oil in the ground (10.6). From the first-order condition for investment (10.7), this value is constant, vjt  zj/β for j ¼ a, b, and the marginal value Eq. (10.6) is represented as rz j ¼

x2j 2δ j s2j

:

ð10:10Þ

where r ¼ β1  1 is the discount rate. Here and in what follows we omit the time subscript in the notation of stationary equilibria. Equation (10.10) implies that the marginal extraction cost is constant and equal to
1=2 ξ j ¼ 2rz j =δ j :

ð10:11Þ

for j ¼ a, b. The first-order condition (10.5) implies that the equilibrium price is also constant: p ¼ ξ j þ z j:

ð10:12Þ

The marginal costs of extraction and reserve addition are higher for the high-cost region, ξb + zb > ξa + za, because zb > za and δb < δa. Hence, condition (10.12) cannot hold for both regions, and in any time period production in one of the regions must be equal to zero (we assume that resources in both regions are economic and the choke price is above the full marginal costs: α > ξj + zj for j ¼ a, b). In the efficient stationary equilibrium, production takes place only in the low-cost oil region a, whereas the reserve stock and extraction in the high-cost region b are zero. From the market-clearing condition (10.3), the output of region a is constant under a constant price, xat  xa, and, from Eq. (10.10), the reserve stock sa is constant.

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10

The Oil Cartel and Misallocation of Production

Proposition 10.1 In the stationary competitive equilibrium with xb ¼ σ b ¼ 0, the oil price is p ¼ ξa þ za :

ð10:13Þ

The region a’s effective resource share is αp α  za

ð10:14Þ

ξa : α  za

ð10:15Þ

σa ¼ and the demand share is σd ¼

Equations (10.14) and (10.15) are checked by inserting σ a and σ d into the marketclearing price Eq. (10.9) for σ b ¼ 0. Equation (10.13) states that the competitive price p is equal to the full marginal cost of production in the low-cost region. The equilibrium resource share σ a ensures the compatibility of price given by Eq. (10.13) with the market-clearing price (10.9).

10.3.2 Competitive Production We can find from Proposition 10.1 the level of production for the effective resource share σ a. For sb ¼ 0, we have: σ a ¼ δasa/(δasa + μ), and the productivity-weighted oil stock in the low-cost region is δ a sa ¼

μσ a αp ¼μ ξa 1  σa

due to Eqs. (10.13), (10.14). Then the output (10.8) is equal to xa ¼ δa sa ξa ¼ μðα  pÞ,

ð10:16Þ

because ξa ¼ p  za. The equilibrium extraction is thus proportional to the marginal consumer surplus, α  p.

10.3.3 The Firm’s Value and the Social Benefit of Extraction In any period t  1, the reserve addition is balanced with extraction implying that

10.3

The Competitive Oil Price and the Allocation of Production

i a ¼ z a xa

263

ð10:17Þ

and due to Eq. (10.13) the net profit π a ¼ pxa  ca(xa, sa)  ia is equal to: π a ¼ ðξa þ za Þxa  x2a =2δa sa  za xa ¼ ξa xa  x2a =2δa sa ¼ x2a =2δa sa :

ð10:18Þ

In period t ¼ 0, the firm invests i0a ¼ zasa to create the reserve stock sa. The firm’s net present value at the initial time is Va ¼

1 X

βt π a  i0a ¼ r 1 π a  za sa :

ð10:19Þ

t¼1

We show in Appendix A.1 that the initial net present value is zero: V a ¼ 0,

ð10:20Þ

because the present value of future rents r1π a matches the initial investment in reserve creation (this was shown in Chap. 3). The integral social benefit is the sum of the present value of consumer surplus Q and the net present value of profits at date 0: W ¼ Q þ V a, where Q¼

1 X

βt ðuðxa Þ  pxa Þ ¼ ðuðxa Þ  pxa Þ=r

t¼1

and uðxa Þ ¼ αxa  x2a =2μ is the linear-quadratic utility per period. Since Va ¼ 0, the social benefit equals the present value of consumer surplus:
 W ¼ Q ¼ αxa  x2a =2μ  pxa =r: Due to Eqs. (10.13), (10.16), the social benefit is proportional to the squared output: W ¼ ðα  xa =2μ  ξa  za Þxa =r ¼ x2a =2μr:

ð10:21Þ

Figure 10.2 illustrates the distribution of social benefit from resource development and extraction. The horizontal axis shows extraction by region a and the vertical axis shows the price and the unit production costs. The line connecting the choke price α with the intercept of demand D is the inverse demand line. Line a is the locus of marginal costs or inverse supply: za + x/δasa. The intersection of inverse

264

10

The Oil Cartel and Misallocation of Production

Fig. 10.2 Distribution of social benefits under competitive equilibrium

demand and supply lines gives the equilibrium point E with coordinates xa and p ¼ za + ξa. The extraction cost ca ðxa , sa Þ ¼ x2a =2δa sa is represented by triangle za EM , the reserve addition investment ia ¼ zaxa by rectangle OzaMxa and the net profit π a ¼ x2a =2δa sa by triangle za pE. Consumer surplus per period Qr ¼ x2a =2μ is given by triangle pαE. Since Va ¼ 0, the net present value of profits covers the initial investment in reserves creation. Therefore, the area of net profit triangle za pE in Fig. 10.2 equals the per-period return on initial investment in reserve creation, π a ¼ rzasa, hence consumer surplus shown by the triangle pαE coincides with social benefit W. As a result, consumers get all benefits from oil production in competitive equilibrium. The net present value of profits is zero—first, because of the absence of depletion in resource development, and second, because the extraction cost function cj(xjt, sjt) is linearly homogeneous in its arguments. It is quadratic in extraction, implying that under a fixed oil stock, the returns on intermediate inputs are diminishing and rents are positive in any period. However, in the initial period, the oil stock is not fixed and the present value of profits is equal to the initial investment because of the linear homogeneity of extraction technology. Third, consumers get all benefits from resource development and extraction due to the allocative efficiency of the competitive market. The outcome is different under the presence of a cartel in the industry.

10.4

10.4

Equilibrium in the Global Oil Industry with a Cartel

265

Equilibrium in the Global Oil Industry with a Cartel

We modify the basic model and assume that the cartel has been organized in the advantageous region, while firms in the disadvantageous one are competitive. The cartel affects the world oil price by taking into account the reaction of consumers and price-taking producers. Thus, the cartel behaves as the Stackelberg leader with respect to consumers and the competitive fringe. The notation of regions’ indices is changed for variables of the model with a cartel by using the uppercase letters: J ¼ A, B instead of j ¼ a, b. The notation is not changed for the region-specific technology parameters δa, δb, za, zb, which are the same in both models. Firms in both regions solve the problem (10.1)–(10.2) under the cleared oil market (10.3). The cartel established in region A exercises the monopoly power by reducing its output until the point of equalization of the marginal revenue and the marginal cost of production. The Lagrangian for firms in both regions is the same as (10.4). The first-order condition for extraction by the cartel, ∂LA/∂xAt ¼ 0, implies. Pt þ xAt  ∂Pt =∂xAt ¼ ξAt þ βvAtþ1 ,

ð10:22Þ

where Pt denotes the oil price under a cartel, and ξAt ¼ xAt/δasAt is the marginal extraction cost of the cartel. The first-order condition for price-taking firms in region B is the same as (10.5): Pt ¼ ξBt þ βvBtþ1 ,

ð10:23Þ

where ξBt ¼ xBt/δbsBt is the marginal extraction cost in region B. The first-order condition for positive investment is the same as (10.7): βvJtþ1 ¼ z j :

ð10:24Þ

Under this condition, the supply by region B is given by Eq. (10.8), which we rewrite as xBt ¼ δb sBt ðPt  zb Þ: Then the market-clearing condition (10.3) implies: Pt ¼ ½D  ðxAt þ xBt Þ=μ ¼ ½D  ðxAt þ δb sBt ðPt  zb ÞÞ=μ: This yields the price schedule faced by the cartel

ð10:25Þ

266

10

Pt ¼

The Oil Cartel and Misallocation of Production

D  xAt þ δb sBt zb : μ þ δb sBt

ð10:26Þ

We can use the fact that ∂Pt/∂xAt ¼  1/(μ + δbsBt) to derive (in Appendix A.2) the first-order condition for extraction by the cartel: Pt ¼

ξAt þ za , σ Bt þ σ Dt

ð10:27Þ

where σ Jt ¼ δ j sJt =ECt are the effective resource shares for J ¼ A, B and σ Dt ¼ μ=ECt is the demand share, and ECt ¼ δa sAt þ δb sBt þ μ is the slope of excess supply under the cartel. Equation (10.27) differs from the first-order condition for the competitive market (10.13), because the cartel sets a price that indicates a mark-up 1/(σ Bt + σ Dt) over the marginal extraction cost in region A. We show in Appendix A.2 that the marketclearing cartel price is Pt ¼ θAt za þ θBt zb þ θDt α,

ð10:28Þ

where θAt ¼

σ At σ Bt σ Dt ,θ ¼ ,θ ¼ 1 þ σ At Bt 1  σ 2At Dt 1  σ 2At

are modified resource and demand shares. The equilibrium cartel price (10.28) is the weighted sum of the opportunity costs and the choke price. The modified weights θAt, θBt and θDt are tilting the cartel price away from the opportunity cost of the cartel, which equals za, towards the opportunity cost of the competing region, which equals zb, and the choke price α, as compared to the competitive price (10.9). It is important to note that the model assumption that the cartel behaves as the Stackelberg leader with respect both to consumers and to the competitive fringe is justified, because the reaction functions of these two groups are similar. These reaction functions are given by consumer demand and the fringe’s supply. Consumer demand is the decreasing linear function of the current price, while the fringe’s supply is the increasing linear function of this price (10.25). It does not depend explicitly on the next-period marginal resource value vJt + 1 due to the first-order condition for investment (10.24). Therefore, the cartel faces the price schedule given by Eq. (10.26) and internalizes the marginal effect of its supply on current price.

10.4

Equilibrium in the Global Oil Industry with a Cartel

267

10.4.1 Resource Reallocation As under a competitive market, the equilibrium path under a cartel is stationary because of the first-order condition for investment (10.24), implying that the opportunity cost is constant and equal to zj. The dynamic equation for vJt is the same as (10.6) and is represented for both regions similarly to (10.10): rz j ¼

x2J : 2δ j s2J

This condition gives the stationary marginal extraction cost, which is the same as in the competitive market case (10.11):
1=2 ξJ ¼ ξ j ¼ 2rz j =δ j :

ð10:29Þ

From Eq. (10.23) we obtain the cartel price as the sum of marginal extraction and reserve addition costs in region B: P ¼ ξ b þ zb :

ð10:30Þ

Thus, we have derived three equations for the cartel price: Eqs. (10.27), (10.28) and (10.30). The first two equations are rewritten for the stationary equilibrium as P¼

ξa þ za : 1  σA

P ¼ θA za þ θB zb þ θD α:

ð10:31Þ ð10:32Þ

Equation (10.30) represents the cartel price as the marginal production cost in the high-cost region. Equation (10.31) relates to setting the cartel price above the production cost in the low-cost region, and Eq. (10.32) is the condition of market clearing. These three equations must be fulfilled in equilibrium and their compatibility is ensured by the effective resource shares σ A, σ B. Unlike the competitive equilibrium, in which the resource share is positive only for the low-cost oil production, in equilibrium with the cartel both resource shares σ A and σ B can be positive. Proposition 10.2 In the stationary equilibrium, the cartel’s effective resource share is σA ¼

ΔP , ξa þ ΔP

ð10:33Þ

where ΔP ¼ P  p is the cartel price margin. The region B’s effective resource share is

268

10

The Oil Cartel and Misallocation of Production

σ B ¼ ð1  σ A Þϕ

ð10:34Þ

σ D ¼ ð1  σ A Þð1  ϕÞ,

ð10:35Þ

and the demand share is

where
 2 PP ϕ¼ α  zb: and P¼

αþp : 2

Proof in Appendix A.3 Proposition 10.2 determines the equilibrium resource and demand shares as functions of technology and demand parameters. The cartel price margin ΔP is defined as the difference between the cartel price P ¼ ξb + zb and the competitive price p ¼ ξa + za. Hence, the cartel price margin is equal to the production cost differential: ΔP ¼ Δz þ Δξ,

ð10:36Þ

where Δz ¼ zb  za and Δξ ¼ ξb  ξa. Proposition 10.2 implies that the cartel pricing causes a reallocation of resources between the regions. The larger the production cost’s differential ΔP, the higher is the cartel’s resource share (10.33). According to (10.34), the region B’s effective resource share is positive if, and only if, ϕ > 0 or P < P:

ð10:37Þ

The cartel price should be below the upper bound P , which is the arithmetic average of the choke price and the competitive price. In other words, the unit production cost of the competitive fringe ξb + zb ¼ P should not be too high for the fringe to be able to compete with the cartel. Condition (10.37) is equivalent to: ΔP < α  P, which means that the cartel price margin ΔP is less than the marginal consumer surplus, α  P, as shown graphically in Fig. 10.3. The demand share in Eq. (10.35) is positive because
 α  zb  2 P  P P þ ΔP  zb 1ϕ¼ ¼ α  zb α  zb and the nominator is equal to: P + ΔP  zb ¼ ξb + ΔP > 0.

10.4

Equilibrium in the Global Oil Industry with a Cartel

269

Fig. 10.3 Illustration of condition that σ B > 0

10.4.2 The Pure Monopoly Case The condition of region B’s resource share positivity (10.37) is not fulfilled if the marginal production cost of this region is above the upper bound of cartel prices: ξb þ zb  P ¼

α þ ξ a þ za 2

In this case σ B ¼ 0, σ D ¼ 1  σ A and the equilibrium resource share σ A ensures the compatibility of two cartel price Eqs. (10.31), (10.32) rewritten as: ξa þ za 1  σA

ð10:310 Þ

P ¼ θA za þ θD α,

ð10:320 Þ

P¼

where θA ¼

σA σD 1  σA 1 ,θ ¼ ¼ ¼ 1 þ σ A D 1  σ 2A 1  σ 2A 1 þ σ A

The third cartel price Eq. (10.30) does not hold. We show in Appendix A.4 that the equilibrium resource share of the cartel is σA ¼

αp α  p þ 2ξa

The cartel price is found by inserting this σ A into Eq. (31.10 ): P¼

α  p þ 2ξa αp αþp þ ξa þ za ¼ ¼P þ za ¼ 2 2 2

As a result, if the unit production cost of the competitive fringe is above the upper bound P, the cartel sets the monopoly price at this bound. The same price can be obtained for a pure monopoly model with only one oil-producing region and without consideration of interregional resource allocation.

270

10

The Oil Cartel and Misallocation of Production

10.4.3 A Numerical Example Consider a numerical example illustrating the stationary equilibria of the model and the effect of resource reallocation. Let the choke price be: α ¼ $250/barrel, the reserve addition costs: za ¼ $10/barrel, zb ¼ $25/barrel and the productivities of extraction: δa ¼ 0.004, δb ¼ 0.001. Suppose that the discount rate is r ¼ 0.05. The marginal extraction costs are calculated using formula (10.29): ξa ¼ (2 ∙ 0.05 ∙ 10/0.004)1/2 ¼ $15.8/barrel, ξb ¼ (2 ∙ 0.05 ∙ 25/0.001)1/2 ¼ $50.0/ barrel. The competitive equilibrium oil price is p ¼ za þ ξa ¼ 10 þ 15:8 ¼ $25:8=barrel: From Proposition 10.1, region a’s effective resource share under a competitive market is σa ¼

α  p 250  25:8 ¼ 0:93 ¼ α  za 250  10

and the demand share is σ d ¼ 1  σ a ¼ 0.07. The cartel price is P ¼ zb þ ξb ¼ 25 þ 50 ¼ $75=barrel and the cartel price margin is ΔP ¼ P  p ¼ 75  25:8 ¼ $49:2=barrel: The condition of region B’s positive resource share (10.37) is fulfilled, since P ¼ $75/barrel < P ¼ ð250 þ 25:8Þ=2 ¼ $137:9=barrel. From Proposition 10.2, the cartel’s resource share is σA ¼

ΔP 49:2 ¼ 0:76, ¼ ξa þ ΔP 15:8 þ 49:2

the competitive fringe’s resource share is σ B ¼ ð1  σ A Þϕ ¼ ð1  σ A Þ

2ðP  PÞ 2ð137:9  75Þ ¼ 0:13 ¼ 0:24  α  zb 250  25

and the demand share is σ D ¼ 1  σ A  σ B ¼ 0:11:

10.5

Production Shifting and Welfare Losses

Table 10.1 The oil price and the regions’ effective resource shares

Region A(a) Region B(b)

271 zj 10 25

ξj 15.8 50.0

p 25.8 ___

P ___ 75.0

σj 0.93 0

σJ 0.76 0.13

Thus, as a result of industry cartelization, the low-cost region’s effective resource share decreases from 0.93 to 0.76, while the high-cost region’s share increases from 0 to 0.13. The demand share increases from 0.07 to 0.11. Table 10.1 summarizes the numerical example and shows equilibrium outcomes under a competitive market and a cartel. The last two columns demonstrate the effective resource shares. Note that in this example, the cartel’s resource share is quite high relative to region’s resource share because both these shares relate to productivity-weighted reserve stocks. In our example, the productivity of extraction δa is four times higher than δb. Furthermore, the cartel’s resource share given by Eq. (10.33) is high, because the production cost differential is high: ΔP ¼ $49.2/barrel.

10.5

Production Shifting and Welfare Losses

Consider how the cartel exercises its power by shifting part of its production to the competing region. The cartel sets the price margin ΔP over the competitive price by decreasing its supply by ΔxA ¼ xA  xa and by taking into account that the oil demand is reduced by ΔD ¼ D(P)  D( p) ¼  μΔP. Production shifts to region B, which increases its supply by ΔxB ¼ xB  xb ¼ xB, and xB ¼ ΔD  ΔxA

ð10:38Þ

due to the market-clearing condition (10.3). Figure 10.4a demonstrates production shifting under cartel pricing. The horizontal axis shows overlapping outputs of both regions and the vertical one—the oil price and unit production costs. The cartel sets the price P by decreasing the supply from xa to xA sо that region B’s supply increases from xb ¼ 0 to xB and the market clears: xB  ΔD ¼  ΔxA (the bold interval on the horizontal axis depicts ΔD). Line a shows the marginal costs function for region a: za + x/δasa. Lines A and B show the inverse supply functions for regions A and B, respectively: za + x/(1  σ A)δasA and zb + x/δbsB. These functions result from the cartel price Eqs. (10.30) and (10.31). The intersection of line a with competitive price p gives competitive supply xa. The intersections of lines A and B with cartel price P gives supplies under a cartel, xA and xB, such that xA < xB, as is shown in Fig. 10.4a. Figure 10.4b portrays the case with xA > xB . The slopes of lines A and B in Fig. 10.4a, b are determined in equilibrium to ensure that the length of interval (0, xB  ΔD) is equal to the length of interval (xA, xa), to meet the condition of production shifting (10.38). The slopes of lines A and

272

10

The Oil Cartel and Misallocation of Production

Fig. 10.4 (a) Production shifting under cartel pricing, xA < xB. (b) Production shifting under cartel pricing, xA > xB

B in these figures depend on the productivity-weighted stocks δasA and δbsB that determine equilibrium outputs under a cartel, xA ¼ δasAξa and xB ¼ δbsBξb. The outputs can be expressed as the functions of resource and demand shares, as the following proposition states. Proposition 10.3 The equilibrium outputs are equal to: xA ¼

σA σ μξ , x ¼ B μξ : σD a B σD b

ð10:39Þ

Proof in Appendix A.5 The volume of the cartel’s production increases with its resource share σ A. The higher this share, the lower the volume of production shifting that is required to set the cartel price. From Proposition 10.2, the cartel resource share increases with the production cost differential ΔP. Thus, the more advantageous region A in terms of production costs, the more significant is the cartel’s power to set the price by shifting production to the disadvantageous region B.

10.5.1 Cartel’s Value Gain Consider gains and losses of producers and consumers caused by production shifting and cartel pricing. In the stationary equilibrium, the reserve stocks in both regions are constant for t  1, and reserve addition matches extraction implying that z j xJ ¼ iJ for J ¼ A, B. The net profit of a competitive firm in region B for t  1 is:

ð10:40Þ

10.5

Production Shifting and Welfare Losses

273

π B ¼ PxB  cb ðxB , sB Þ  iB ¼ ðξb þ zb ÞxB  x2B =2δb sB  zb xB ¼ ξb xB =2, since P ¼ ξb + zb and due to Eq. (10.40). In period t ¼ 0, the firm invests i0B ¼ zbsB to create the oil stock sB. The initial net present value of this firm is VB ¼ r1π B  zbsB, and, as we have shown, it equals zero: V B ¼ 0: The cartel’s net profit in period t  1 is π A ¼ PxA  ca(xA, sA)  iA, and the initial net present value of profits is VA ¼ r1π A  i0A ¼ r1π A  zasA. We show in Appendix A.6 that the cartel’s net profit is πA ¼

1 þ σA ðξ x =2Þ ð1  σ A Þ a A

ð10:41Þ

and the cartel’s initial value is V A ¼ xA ΔP=r:

ð10:42Þ

The cartel’s net profit (10.41) equals the competitive net profit ξaxA/2 multiplied by the rate of excess profit, (1 + σ A)/(1  σ A), which is the increasing convex function of the cartel’s resource share. The cartel’s value gain is equal to its value (10.42), because its competitive value is zero: ΔV A ¼ V A  V a ¼ V A : The cartel can transfer this gain only from consumers, who are the sole beneficiaries of oil production under a competitive market.

10.5.2 Welfare Losses The present value of consumer surplus under the cartel is QC ¼ μðα  PÞ2 =2r:

ð10:43Þ

The social benefit is W C ¼ QC þ V A þ V B ¼ QC þ V A and the decrease of this benefit is ΔW ¼ ΔQ + ΔVA, where ΔQ ¼ QC  Q and ΔVA ¼ xAΔP/r, as follows from Eq. (10.42). We show in Appendix A.7 that

274

10

The Oil Cartel and Misallocation of Production

ΔW ¼ xB ΔP=r  μðΔPÞ2 =2r:

ð10:44Þ

The welfare loss, ΔW, is the sum of the loss from misallocation of production: ΔW 1 ¼ xB ΔP=r

ð10:45Þ

ΔW 2 ¼ μðΔPÞ2 =2r:

ð10:46Þ

and the deadweight loss:

As a result, production shifting and shifting of benefits from consumers to the cartel leads to the welfare losses rising non-linearly with the cartel price margin.

10.5.3 The Graphical Illustration Figure 10.5 illustrates the cartel market equilibrium and the distribution of benefits and welfare losses. The horizontal axis shows sequentially outputs of regions A and B. The vertical axis shows the cartel price P, the competitive price p and the unit production costs. As in Fig. 10.2 for competitive equilibrium, the inverse demand line connects the choke price α and the intercept of demand D. Line A is the inverse

Fig. 10.5 Cartel market equilibrium and the distribution of gains and losses

10.5

Production Shifting and Welfare Losses

275

supply for region A, za + x/(1  σ A)δasA. Line B is the inverse supply for region B defined for x  xA: zb + (x  xA)/δbsB. The inverse supply lines are implied by the cartel price Eqs. (10.30), (10.31). The intersection of inverse supply line A with the horizontal line of cartel price P gives the cartel’s supply, xA. The intersection of the inverse supply line B with the horizontal line P is the equilibrium point E C , which gives supply by region B, xB, and total supply, xA + xB. The point of competitive equilibrium is E with coordinates xa and p. Line a in Fig. 10.5 is the marginal costs line for region A, za + x/δasA. It intersects the horizontal line of competitive price p at point G corresponding to the cartel’s supply xA so that xA/δasA ¼ ξa, as follows from condition (10.29).The extraction cost of region A is represented by triangle zaGK, the reserve addition cost by the rectangle denoted IA and the net competitive profit by triangle zapG. The extraction cost of region B is given by triangle LE C M, the reserve addition cost by rectangle IB and the net competitive profit by triangle LFE C . The present values of net competitive profits of both regions cover the initial investments in resource creation (this is shown for the cartel in the derivation of Eq. (10.41) in Appendix A.6). The consumer surplus under the cartel in Fig. 10.5 is given by triangle PαE C , which is a part of the competitive consumer surplus shown by triangle pαE. The loss of consumer surplus is portrayed by the trapezoid pPE C E . It includes the cartel’s value gain, ΔVA ¼ xAΔP/r, shown by the dashed rectangle pPFG, the welfare loss from misallocation of production ΔW1 ¼ xBΔP/r shown by the shaded rectangle GFE C N and the deadweight loss ΔW2 ¼ μ(ΔP)2/2r shown by the shaded triangle NE C E.

10.5.4 Continuation of the Numerical Example To demonstrate the shifts in benefit distribution for our numerical example, we have to specify the slope of demand function μ. It is possible to normalize this parameter to one if we consider only relative gains and losses. Such a normalization means a specification of units of measurement for equilibrium outputs (10.39), for which μ is the scale coefficient. Under the competitive market xb ¼ 0 and, from Eq. (10.16), xa ¼ μðα  za  ξa Þ ¼ 250  10  15:8 ¼ 224:2 for μ ¼ 1. Proposition 10.3 implies that equilibrium outputs under a cartel are xA ¼

σA 0:76  15:8 ¼ 109:2 μξ ¼ σ D a 0:11

276

10

Table 10.2 Cartel’s value gain and welfare losses

The Oil Cartel and Misallocation of Production

VA/W, % 21.4

xB ¼

ΔW1/W, % 11.6

ΔW2/W, % 4.8

σB 0:13  50 ¼ 59:1: μξ ¼ σ D b 0:11

The cartel’s value is found from Eq. (10.42): V A ¼ xA ΔP=r ¼ 109:2  49:2=0:05 ¼ 107, 453: since ΔP ¼ 49.2 for our example. This value indicates benefit shifting from consumers to the cartel. In relative terms, this is 21.4 per cent of the optimal social benefit W coinciding with consumer surplus: W ¼ x2a =2μr ¼ ð250  25:8Þ2 =ð2  0:05Þ ¼ 502, 656: From Eq. (10.44), the welfare loss is equal to: ΔW ¼ ðxB ΔP þ ðμ=2ÞðΔPÞ2 Þ=r ¼ ð59:1  49:2 þ 0:5  49:22 Þ=0:05 ¼ 82, 360: In relative terms, the welfare loss amounts to 16.4% of the optimal social benefit W. The loss from misallocation of production is ΔW1 ¼  xBΔP/r ¼  59.1 ∙ 49.2/ 0.05 ¼  58,154 or 11.6% of social benefit. The deadweight loss is ΔW2 ¼  μ(ΔP)2/2r ¼  0.5 ∙ 49.22/0.05 ¼  24,206 or 4.8% of social benefit. Table 10.2 shows the cartel’s gain and the welfare losses for our example. The total loss of consumer surplus is the sum of the cartel’s value gain and welfare losses. One can see from Table 10.2 that the total loss in percentage terms is: 21.4 + 11.6 + 4.8 ¼ 37.8 per cent of the optimal social benefit. Such a high share of losses is caused by the significant disadvantage of the high-cost competitive fringe. A reduction of this disadvantage can weaken the cartel’s market power and partially offset the losses incurred by consumers.

10.6

The Effects of Technological Change

As has been shown, under a competitive market extraction takes place only in the low-cost region. An opportunity of high-cost oil production is granted by the cartel, which creates a market niche for the high-cost competitive fringe. Nevertheless, this opportunity is constrained by the cartel’s market power, which is determined by the cartel’s cost advantage. The higher this advantage, the more pronounced are the

10.6

The Effects of Technological Change

277

distorting effects of production shifting and the narrower is the niche for the highcost oil producers. This niche widens due to a technological change materialized in a reduction of production costs in the high-cost region. As a result of this change, the effective resource share of the cartel decreases, while this share of competitive producers increases. Such a shift of resource shares weakens the cartel’s power to set the price margin and mitigates its distorting influence on the allocation of resources and the distribution of benefits from oil production. To demonstrate this effect we specify the technical change occurring in region B as an increase of the parameter of extraction productivity δb. Such an increase causes a decrease of the marginal extraction cost ξb(δb) ¼ (2rzb/δb)1/2, as follows from Eq. (10.29). The marginal extraction cost influences the model variables through
the cartel price given by Eq. (10.30) as P(δb) ¼ ξb(δb) + zb and defined for  interval p, P . The upper bound of the cartel price P ¼ ðα þ pÞ=2 determines the lower bound of productivity δ from condition: P ¼ ξb ðδb Þ þ zb as δb ¼ b
2 2rzb = P  zb . A competitive fringe presents in the market if δb  δb . Consider the comparative static dependences of the cartel’s value gain and welfare losses on the extraction productivity δb. We derive in Appendix A.8 the following value functions of δb measured in relative terms as shares of optimal benefit W ¼ μ(α  p)2/2r: V A ðδ b Þ ðα  zb ÞðPðδb Þ  pÞ2 ¼ W ðPðδb Þ  ez  0:5ξa Þðα  pÞ2
 2ξb ðδb Þ P  Pðδb Þ ðPðδb Þ  pÞ ΔW 1 ðδb Þ ¼ W ðPðδb Þ  ez  0:5ξa Þðα  pÞ2 ΔW 2 ðδb Þ ðPðδb Þ  pÞ2 ¼ , W ð α  pÞ 2

ð10:47Þ ð10:48Þ ð10:49Þ

where ez ¼ ðza þ zb Þ=2 denotes the average marginal cost of reserves addition. Figure 10.6 demonstrates the welfare loss functions (10.48), (10.49) for δb 2 ðδb , δa Þ. The model parameters used for calculating these graphs are the same as above: za ¼ 10, δa ¼ 0.004, zb ¼ 25, α ¼ 250, r ¼ 0.05. The upper bound of the cartel price is P ¼ $137:9 /barrel, as we have shown. The lower bound of productivity 2 is δ ¼ 2rzb =ðP  zb Þ ¼ 2  0:05  25=ð137:9  25Þ2 ¼ 0:000196. In Fig. 10.6, the b

function of welfare loss from misallocation of production, ΔW1(δb)/W, has the maximum point, and the deadweight loss function, ΔW2(δb)/W, is decreasing. The loss from misallocation of production ΔW1(δb)/W ¼ xBΔP/rW is increasing for low δb, because production by region B is increasing with δb. This effect is partially offset by a decline of the cartel price Pb(δb) with δb. As one can see from Fig. 10.6, for δb above the maximum point, the loss from misallocation of production is decreasing quite slowly. This loss is persistent under technology improvements,

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Fig. 10.6 The relative welfare losses ΔW1(δb)/W and ΔW2(δb)/W

Fig. 10.7 The value gain VA(δb)/W, the welfare loss ΔW(δb)/W and the loss from misallocation of production ΔW1(δb)/W

because xB is increasing with δb, while ΔP is decreasing. In contrast, the deadweight loss ΔW2(δb)/W decreases quite rapidly with the increase of productivity δb. Figure 10.7 shows the cartel’s value gain VA(δb), the total welfare loss ΔW (δb) ¼  (ΔW1(δb) + ΔW2(δb)) and the loss from misallocation of production. The

10.6

The Effects of Technological Change

279

value gain and the total welfare losses are decreasing in δb. In the figure, these functions converge to each other with growth of the productivity parameter. It is worth noting that for the lower bound of productivity parameter δb ¼ 0:000196 shown in Fig. 10.7, the value gain equals V A ðδb Þ=W ¼ 0:5 and the welfare loss equals ΔW ðδb Þ=W ¼ 0:25. At this point, we have the pure monopoly case: the competitive fringe does not produce, xB ¼ 0, and the loss from misallocation of production is zero, ΔW 1 ðδb Þ ¼ 0. The welfare loss coincides with the deadweight loss, which is equal to 25 per cent of the optimal social benefit because, as we have shown, P ¼ P for the pure monopoly case, hence ΔW 2 ðδ Þ ¼ μðΔPÞ2 =2r ¼ b

2

μðP  pÞ =2r ¼ μðα  pÞ2 =8r ¼ 0:25  W . Similarly, the consumer surplus is 2 QC ðδ Þ ¼ μðα  PÞ =2r ¼ μðα  pÞ2 =8r ¼ 0:25  W , hence the total loss of social b

benefit at the lower bound of productivity parameter δb is 75 per cent. To demonstrate more concretely the effects of technological change on the distribution of benefits suppose that the base parameter of extraction productivity is δb ¼ 0.001, as in the above example, and it increases to 0.002 as a result of this change. Then the new marginal extraction cost of region B is calculated for δb ¼ 0.002 from Eq. (10.29): ξb ¼ ð2rzb =δb Þ1=2 ¼ ð2  0:05  25=0:002Þ1=2 ¼ $35:4=barrel: The cartel price decreases from $75 /barrel to P ¼ zb þ ξb ¼ 25 þ 35:4 ¼ $60:4=barrel and the cartel price margin also decreases: ΔP ¼ P  p ¼ 60:4  25:8 ¼ $34:6=barrel: From Proposition 10.2, the cartel’s effective resource share is σA ¼

ΔP 34:6 ¼ 0:687, ¼ ξa þ ΔP 15:8 þ 34:6

the fringe’s effective resource share is σ B ¼ ð1  σ A Þϕ ¼ ð1  σ A Þ

2ðP  PÞ 2ð137:9  60:4Þ ¼ 0:277 ¼ 0:313  α  zb 200  25

and the demand share is σ D ¼ 1  σ A  σ B ¼ 0:036:

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Table 10.3 The effects of technological change

Base technology New technology

ξb $50.0

P $75.0

σA 0.76

σB 0.13

VA/W, % 21.4

ΔW1/W, % 11.6

ΔW2/W, % 4.8

ΔW/W, % 16.4

$35.4

$60.4

0.69

0.28

15.3

10.8

2.4

13.2

The value gains and losses under technological change are found from formulae (10.47)–(10.49): ΔVA(0.002)/W ¼ 0.153, ΔW1(0.002)/W ¼ 0.108, ΔW2(0.002)/ W ¼ 0.024. Table 10.3 summarizes the numerical results for the base technology with δb ¼ 0.001 and for the new technology with δb ¼ 0.002. This table demonstrates the decrease of the cartel’s effective resource share from 76 to 69 per cent and the increase of the fringe’s share from 13 to 28 per cent. A reduction of the cartel’s resource share leads to a decrease of the cartel’s value gain and welfare losses. The cartel’s value gain drops from 21.4 per cent of the optimal social benefit to 15.3 per cent, while the deadweight loss falls from 4.8 per cent to 2.4 per cent. The total consumer loss from the cartel activity decreases in relative terms from 37.8 per cent to 15.3 + 13.2 ¼ 28.5 per cent of the optimal social benefit. A noteworthy feature of these effects is the insignificant change of the loss from misallocation of production, from 11.6 to only 10.8 per cent. The persistence of this loss results from the structural effect of production shifting, which is getting more pronounced under technological improvement. Although an extension of output by the high-cost competitive fringe implies persistent misallocation of production, it proves to be welfare-improving, because it brings about a reduction of the cartel price.

10.7

Concluding Remarks

The main inference made from the models with an exhaustible resource considered in Chap. 9 is that the oil cartel OPEC tends to be more conservative than the competitive market. Robert Solow noted in his seminal article on resource economics (Solow, 1974, p. 8) that “if a conservationist is someone who would like to see resources conserved beyond the pace that competition would adopt, then the monopolist is the conservationist’s friend”. Higher prices set by the resource monopolist lead to lower current consumption implying stretching out of finite resource extraction in the future. However, a resource cartel can exert a distorting influence on investments in resource development bringing about long-term welfare losses.

10.7

Concluding Remarks

281

In the cartel-fringe model in this chapter, the main factor of the cartel’s distorting impact on the industry is its advantage over competitors in terms of production costs. If this advantage is significant, the cartel’s market power is strong due to its high effective resource share. Strong market power means in our model that the cartel can set a high price margin by transferring an insignificant amount of supply to the competitive fringe. In this case, the cartel behaves more like a pure monopolist capturing a substantial value gain at the cost of high deadweight losses incurred by the industry. Under a lower cartel’s cost advantage, the distorting impact occurs merely through a reallocation of resources that brings about welfare losses from misallocation of production in addition to deadweight losses. In a recent article, John Asker et al. (2017) confirmed the importance of market power in the misallocation of world oil production. Their empirical study is based on rich microdata on the production costs and reserves of 11,455 oil fields covering the global market for crude oil from 1970 to 2014. The authors construct a counterfactual scenario, in which all fields are exploited by competitive price-taking producers with no market power. A comparison of actual microdata with the benchmark competitive allocation provides evidence of substantive production inefficiency due to the market power exercised by the oil cartel. Asker et al. (2017, p. 31) point out that in the analysis of welfare benefits and losses it is important to focus on the rectangle indicating “the welfare loss due to misallocation of production, given the quantity produced”. This is in contrast to the traditional focus on the triangle indicating “the welfare loss due to the quantity reduction implied by market power”. We have illustrated in Fig. 10.5 the relationship between the rectangle of misallocation losses and the triangle of deadweight losses for our model. We have shown that the cartel captures a value gain from consumer surplus at the cost of welfare losses that can be significant. However, the deadweight losses are markedly decreasing under a technological shift favourable for the competitive fringe, because it diminishes the cartel’s cost advantage and effective resource share. Though the losses from misallocation of production are persistent, such a technological shift improves the efficiency of equilibrium outcome, because production shifting becomes a less costly burden for consumers. This is indicated by the drop in the cartel price margin and by the reduction of total welfare losses. Besides diminishing the cartel’s market power and mitigating its distorting effects on the industry, a technical change in favour of a competitive fringe can substantially reduce the cartel value. The decline of the expected gains from cartelization weakens the economic incentives of oil producers in the low-cost region to form a cartel. We assumed in the model in this chapter no resource depletion and therefore could focus on stationary equilibria. This approach makes possible a simple analysis of comparative static effects of the cartel on the reallocation of production and welfare losses. We disregarded the issue of the cartel’s influence on the timing of extraction, which is important for a depletable resource. This issue requires consideration of transition equilibrium paths, which will be examined for a model of economically recoverable resource in the next chapter.

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Appendices A.1 Condition (10.20) Investment in creation of the initial reserve stock is ia ¼ za sa ¼ za xa =δa ξa ¼ za xa ð2δa rza Þ1=2 ¼ xa ð2rza =δa Þ1=2 =2r ¼ xa ξa =2r, since ξa ¼ xa/δasa and due to (10.11). From Eq. (10.18), the firm’s net present value (10.19) is V a ¼ r 1 π a  ia ¼ x2a =2rδa sa  xa ξa =2r ¼ xa ðξa  ξa Þ=2r ¼ 0:

A.2 Equations (10.27), (10.28) From Eqs. (10.22), (10.24), the first-order condition for the cartel is: Pt þ xAt

∂Pt x  At  za ¼ 0: ∂xAt δa sAt

We use the fact that ∂Pt/∂xAt ¼  1/(μ + δbsBt) and rearrange the left-hand side: Pt þ xAt

∂Pt x xAt x  At  za ¼ Pt   At  za δ s μ þ δ s δ ∂xAt a At b Bt a sAt ðμ þ δa sAt þ δb sBt ÞxAt xAt ¼ Pt   za ¼ Pt   za ðμ þ δb sBt Þδa sAt ðσ Bt þ σ Dt Þδa sAt ξAt ¼ Pt   za : σ Bt þ σ Dt

This yields (10.27). Consider Eq. (10.28). From Eqs. (10.8), (10.27), the supply functions are: xAt ¼ δa sAt ð1  σ At ÞðPt  za Þ, xBt ¼ δb sBt ðPt  zb Þ: Inserting these functions into the market-clearing condition (10.3) implies the equation for Pt: σ At E Ct ð1  σ At ÞðPt  za Þ þ σ Bt E Ct ðPt  zb Þ ¼ μðα  Pt Þ, because δ j sJt ¼ σ Jt ECt for J ¼ A, B, D ¼ αμ. This equation can be rewritten as

Appendices

283

σ At ð1  σ At ÞðPt  za Þ þ σ Bt ðPt  zb Þ ¼ σ Dt ðα  Pt Þ: and the market-clearing cartel price is equal to Pt ¼

σ At ð1  σ At Þza þ σ Bt zb þ σ Dt α σ At ð1  σ At Þza þ σ Bt zb þ σ Dt α ¼ σ At ð1  σ At Þ þ σ Bt þ σ Dt 1  σ 2At ¼ θAt za þ θBt zb þ θDt α:

A.3 Proposition 10.2 The cartel price Eqs. (10.30) and (10.31) are compatible if σ A satisfies: ξb þ zb ¼

ξa þ za 1  σA

or 1  σ A ¼ ξa/(ξb + zb  za), implying (10.33): σA ¼ 1 

ξa ΔP ¼ , ξb þ zb  za ξa þ ΔP

since ΔP ¼ Δξ + Δz, where Δξ ¼ ξb  ξa, Δz ¼ zb  za. Equation (10.31) is compatible with (10.32) represented as P¼

σ A ð1  σ A Þza þ σ B zb þ σ D α , 1  σ 2A

if the following equation is fulfilled: σ ð1  σ A Þza þ σ B zb þ σ D α ξa þ za ¼ A : 1  σA 1  σ 2A Rearrange both sides:
 ð1 þ σ A Þξa þ 1  σ 2A za ¼ σ A ð1  σ A Þza þ σ B zb þ σ D α ð1 þ σ A Þξa þ ð1  σ A Þza ¼ σ B zb þ σ D α or, since σ D ¼ 1  σ A  σ B,

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ð1 þ σ A Þξa  ð1  σ A Þðα  za Þ ¼ σ B ðzb  αÞ: From Eq. (10.31), ξa ¼ (1  σ A)(P  za), hence σ B ¼ ð1  σ A Þ

α  za  ð1 þ σ A ÞðP  za Þ : α  zb

We have: ð1 þ σ A ÞðP  za Þ ¼

ξa þ 2ΔP ðP  za Þ ¼ ξa þ 2ΔP ¼ za  p þ 2P ξa þ ΔP

due to (10.33) and since p ¼ ξa + za. Consequently σ B ¼ ð1  σ A Þ


 2 PP α þ p  2P ¼ ð1  σ A Þ : α  zb α  zb

A.4 The Pure Monopoly Case Equating the right-hand sides of Eqs. (10.310 ) and (10.320 ) yields the following equation on σ A: ξa σA α þ za ¼ z þ : 1  σA 1 þ σA a 1 þ σA Rearrange the terms: ξa α  za ¼ 1  σA 1 þ σA ð1 þ σ A Þξa ¼ ð1  σ A Þðα  za Þ and obtain: σA ¼

α  za  ξa αp ¼ : α  za þ ξa α  p þ 2ξa

Appendices

285

A.5 Proposition 10.3 We have it from condition (10.29) that ξJ ¼ xJ/δjsJ ¼ ξj implying that xJ ¼ δjsJξj for J ¼ A, B and j ¼ a, b. This yields xJ ¼ (σ J/σ D)μξj because σ J ¼ δjsJ/EC and σ D ¼ μ/ EC.

A.6 Equations (10.41), (10.42) From Eqs. (10.31), (10.40): 

 x2 x2 ξa ξ x þ z a xA  A  z a xA ¼ a A  A 1  σA 2δa sA 1  σ A 2δa sA   ξ x ξ x 1 1 1 þ σ A ξ a xA ¼ a A  a A¼  ξ x ¼  : 1  σA 2 a A 1  σA 2 1  σA 2

π A ¼ PxA  ca ðxA , sA Þ  iA ¼

The cartel value in period 0 is VA ¼ r1π A  zasA. Since xA ¼ δasAξa and from Eq. (10.29) we have: za sA ¼ za

xA za x A ¼ : δa ξa ð2δa rza Þ1=2

ð10:50Þ

Hence VA ¼ r

1

! 1 þ σ A ξa za r   x =r 1  σ A 2 ð2δa rza Þ1=2 A !   1=2 1 þ σ A ξa ð2rza =δa Þ 1 þ σ A ξa ξa x =r xA =r ¼     ¼ 1  σA 2 2 1  σA 2 2 A   2σ ξ σ ξ 1 þ σA  1 xA ξa =2r ¼ A a xA =2r ¼ A a xA =r ¼ xA ΔP=r, ¼ 1  σA 1  σA 1  σA

1 þ σ A ξ a xA z a xA  ¼  1  σA 2 ð2δa rza Þ1=2

because, from Eq. (10.33), ΔP ¼

σ A ξa : 1  σA

Equations (10.50), (10.29) imply that i0A ¼ za sA ¼ za xA ð2δa rza Þ1=2 ¼ xA ð2rza =δa Þ1=2 ð2r Þ1 ¼ r 1 xA ξa =2

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The Oil Cartel and Misallocation of Production

¼ r 1 xA ðp  za Þ=2: This means that the initial investment by the cartel equals the present value of the cartel’s net competitive profit shown by triangle zapG in Fig. 10.5.

A.7 Equation (10.44) The loss of consumer surplus due to cartel pricing is   ΔQ ¼ QC  Q ¼ μ ðα  PÞ2  ðα  pÞ2 =2r ¼ μðp  PÞððα  PÞ þ ðα  pÞÞ=2r ¼ ΔPðDðPÞ þ DðpÞÞ=2r: ð10:51Þ From Eqs. (10.42), (10.51) and (10.3): ΔW C ¼ ΔQ þ ΔV A ¼ ðDðPÞ þ DðpÞÞΔP=2r þ xA ΔP=r ¼ ðDðPÞ þ DðpÞ 2xA ÞΔP=2r ¼ ðxB þ xa  xA ÞΔP=2r ¼ ðxB  ΔxA ÞΔP=2r: From Eq. (10.38), ΔxA ¼ xB  ΔD and, consequently, ΔWC ¼  (2xB  ΔD)ΔP/2r ¼  (xB + (μ/2)ΔP)ΔP/r ¼  (xBΔP + (μ/2) (ΔP)2)/r, since ΔD ¼ μΔP.

A.8 Equations (10.47)–(10.49) From Proposition 10.2, the demand share is σ D ¼ ð1  σ A Þð1  ϕÞ ¼ ð1  σ A Þ

ξ þ ΔP P þ ΔP  zb ¼ ð1  σ A Þ b : α  zb α  zb

Equations (10.42), (10.39), (10.33), (10.52) imply that ΔV A ¼ xA ΔP=r ¼ μ ¼μ

σ A ðα  zb Þ σA ξ ΔP ¼ μ ξ ΔP rσ D a r ð1  σ A Þðξb þ ΔPÞ a

ΔPðα  zb Þ μðα  zb ÞðΔPÞ2 ξa ΔP ¼ r ð2ξb þ zb  ξa  za Þ rξa ðξb þ ΔPÞ

ð10:52Þ

References

287

¼

μðα  zb ÞðΔPÞ2 μðα  zb ÞðΔPÞ2 ¼ : 2r ðξb þ zb  0:5ξa  0:5ðzb þ za ÞÞ 2r ðP  ez  0:5ξa Þ

From Eq. (10.21), W ¼ μ(α  p)2/2r, hence ðα  zb ÞðP  pÞ2 ΔV A : ¼ W ðP  ez  0:5ξa Þðα  pÞ2 From Eqs. (10.45), (10.34), (10.39), (10.52) we have it that

 2 PP μ P  P ξb ΔP σB ΔW 1 ¼ xB ΔP=r ¼ μ ξ ΔP ¼ μ ξ ΔP ¼ : rσ D b r ðξb þ ΔPÞ b r ðP  ez  0:5ξa Þ Consequently
 2ξb P  P ðP  pÞ ΔW 1 : ¼ W ðP  ez  0:5ξa Þðα  pÞ2 From Eqs. (10.46), (10.21), ΔW2 ¼  μ(ΔP)2/2r and ðP  pÞ2 ΔW 2 : ¼ W ð α  pÞ 2

References Asker J, Collard-Wexler A, Loecker JD (2017) Market power, production (mis)allocation and OPEC. NBER Working Paper No. 23801, National Bureau of Economic Research, INC, September 2017 BP Statistical Review of World Energy – 2020 edition. https://nangs.org/analytics/bp-statisticalreview-of-world-energy. Accessed 25 Nov 2020 Hotelling H (1931) The economics of exhaustible resources. J Polit Econ 39:137–175 Pindyck R (1978) The optimal exploration and production of non-renewable resources. J Polit Econ 86:841–861 Smith J (2009) World oil: Market or mayhem. J Econ Perspect 23:145–164 Solow R (1974) The economics of resources or the resources of economics. Am Econ Rev 64 (2):1–14

Chapter 11

Anti-Conservationist Effects of the Conservationist Oil Cartel

Abstract A resource monopoly behaves as a “conservationist” in models of exhaustible resources. It extracts less today and keeps more resource under the ground than the competitive market does. In this chapter, we consider a model of economically recoverable resources and show that, in the presence of a high-cost competitive fringe, the cartel’s “conservationism” turns into a too early and too intensive resource extraction by the fringe. The “anti-conservationism” of the equilibrium extraction path selected by the fringe proves to be a robust property of the model. As a result, the cartel acts as if it was a strategic player planning its longterm moves to accelerate the depletion of competitors’ resources and to aggravate their competitive disadvantages.

11.1

Introduction

In the previous chapter we focused on the oil cartel’s effects on the allocation of resources and production across oil-producing regions. It was shown that the cartel’s activity encourages overinvestment in resource development in the high-cost region that results in an abundance of expensive resources. As we have shown, this outcome can bring about significant losses for consumers in the long term. In this chapter we examine the effects of the cartel on time profiles of extraction paths. The main inference of Hotelling’s model (1931) extensions to imperfect competition reviewed in Chap. 9 is that the resource cartel selects a conservationist extraction path with a slower rate of resource depletion than the competitive market does. Here we question this inference and consider a cartel-fringe model with an economically recoverable resource. As will be shown, the extraction path is conservationist for the cartel, but anti-conservationist for high-cost non-cartelized producers that are forced by the cartel’s behaviour to extract resources more rapidly than under a competitive market. The difference in extraction rates across cartelized and non-cartelized oil-producing regions is supported by empirical evidence. Figure 11.1 shows the extraction rates measured as the ratio of annual production to proven reserves for OPEC and non-OPEC countries. This ratio for non-OPEC oil producers was four © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 A. Vavilov, G. Trofimov, Natural Resource Pricing and Rents, Contributions to Economics, https://doi.org/10.1007/978-3-030-76753-2_11

289

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Anti-Conservationist Effects of the Conservationist Oil Cartel

Fig. 11.1 OPEC and non-OPEC extraction rates. Source: BP Statistical Review of World Energy (2020)

times higher than for OPEC in 2019. However, as one can see from Fig. 11.1, the extraction rate tends to decrease over time for non-OPEC producers, whereas this rate has increased by two thirds, since the end of the 1980s, for Middle-Eastern OPEC members. One can suggest that the extraction rates should be higher in the non-OPEC countries, because local oil suppliers are competitive price takers producing at near full capacity. Moreover, the more intensive extraction by many non-OPEC oil companies is based on the usage of advanced technologies of resource finding and extraction, including the development of non-conventional oil resources. However, technical innovations mitigate natural disadvantages of the high-cost non-OPEC producers, but do not eliminate the issue of resource depletion. The diagram in Fig. 11.2 shows the ranking of extraction costs for large onshore fields, deep-water and shallow-water fields, oil sands and tight oil. It is important that the majority of low-cost conventional resources are located in large onshore fields, particularly those in the Middle East that are under the cartel’s control. Figure 11.3 shows the dynamics of OPEC’s and Middle-Eastern OPEC’s market shares. Both fell dramatically in the early 1980s, but have grown since. In particular, the MiddleEastern OPEC’s market share has increased from 17.2% in 1985 to 30.5% in 2019. One of the reasons for these structural shifts is the resource depletion in high-cost regions. The majority of non-OPEC producers face increasing extraction costs, particularly in the development and exploitation of expensive conventional resources. As a result, the low-cost producers have gained in terms of their competitive advantages and got additional opportunities to increase their market share at the expense of high-cost producers. A possible reason for this outcome is that the highcost non-OPEC producers had developed too early their oil resources, which have, to

11.1

Introduction

291

Fig. 11.2 Costs of supply and resources of oil. Source: BP Energy Outlook (2017, p. 52)

Fig. 11.3 OPEC’s and Middle-Eastern OPEC’s market shares. Source: BP Statistical Review of World Energy (2020)

a large degree, already been depleted. The oil cartel, meanwhile, has maintained cheap resources due to its conservationist strategy of the four preceding decades. Benchekroun et al. (2017) emphasized the importance of the extraction sequence effect for the world oil industry. These authors showed, for a cartel-fringe model with

292

11

Anti-Conservationist Effects of the Conservationist Oil Cartel

an exhaustible resource, linear extraction costs and a backstop technology, that imperfect competition can violate the Herfindahl principle (Herfindahl 1967). Remember that this is the principle of extracting cheap resources before the more expensive ones. As Benchekroun et al. have demonstrated, the competitive fringe exhausts its reserve stock of expensive oil earlier than the cartel exhausts its reserves of cheap oil, if the latter are sufficiently large. We call the effect of cartel pricing on early overproduction by high-cost competitive producers production shifting over time. In this chapter, we will analyse this effect by using the model of economically recoverable resource presented in Chap. 3. The resource stock in that model is not given initially, but is determined endogenously from conditions of long-term equilibrium. The ultimate recoverable resource is defined as a resource that can be recovered at prices that consumers can afford to pay in the long term. As will be shown below, the cartel activity transforms the time profiles of extraction rates, which are constant under a competitive market. The extraction rate of the competitive fringe proves to be decreasing in time under a cartelized market, since this rate is high initially. In contrast, the cartel’s extraction rate is increasing over time, since it is low initially. The cartel behaves like a conservationist resource monopoly in Hotelling’s model, whereas competitive producers are forced to be anti-conservationists due to the cartel’s behaviour. The anti-conservationist extraction path accelerates resource depletion and the growth of extraction costs of high-cost competitive producers. At the initial time, the cartel lets competitors increase their market share, which is lost thereafter to a larger degree than it was gained initially. The cartel acts as if it was a strategic player planning its long-term moves to deplete the resources of competitors and to aggravate their competitive disadvantage. This “strategy” of strengthening the cartel’s monopoly power in the long term is just a consequence of the short-term cartel pricing. In the next section we present the base model of a competitive market with an endogenous resource stock. In the two subsequent sections, this model is modified to introduce a cartel and is extended to the model with two oil-producing competitive regions. In Sect. 11.5 we consider the main two-region model with a resource cartel and present the results.

11.2

The Base Model of a Competitive Market

The base model of Farzin (1992) with an ultimate recoverable resource was presented in Chap. 3. Competitive producers extract oil from a resource stock in the ground, which is unlimited but depletable. Resource depletion causes an increase of the extraction cost with cumulative extraction denoted as Q(t). The unit cost is supposed to be the linear function of cumulative extraction:

11.2

The Base Model of a Competitive Market

293

cðQðtÞÞ ¼ c0 þ c1 QðtÞ, where c0  0 is the initial unit cost, c0 ¼ c(Q(0)), and c1 > 0 is the slope of the unit cost function. The initial cumulative extraction is Q(0) ¼ 0. Time is continuous, and the time argument t is dropped when possible. Producing firms select the extraction path to maximize the present value of future rents over an infinite extraction horizon: Z1

ert ½pðt Þ  cðQðt ÞÞxðt Þdt,

ð11:1Þ

0

subject to the equation for cumulative extraction: Q_ ðt Þ ¼ qðt Þ,

ð11:2Þ

where p is the resource price, r is the discount rate and q is the instantaneous intensity of extraction. Each firm takes into account the cumulative effect of extraction on the unit production cost c(Q). The market-clearing condition is fulfilled at any instant: q ¼ y,

ð11:3Þ

where y is instantaneous demand. The inverse demand function is linear: p ¼ α  βy

ð11:4Þ

with α denoting the choke price and β the slope of inverse demand. The choke price is assumed to be above the initial unit cost, α > c0. We have shown in Chap. 3 that the equilibrium extraction path is represented by state variables p and Q satisfying the dynamic equations for price and cumulative extraction: p_ ¼ r ðp  c0  c1 Q Þ

ð11:5Þ

Q_ ¼ ðα  pÞ=β:

ð11:6Þ

Figure 11.4 demonstrates the phase plane of this system with the saddle-path solution for a competitive market drawn with the solid arrow. The saddle path converges to the steady state given by: p ¼ α

294

11

Anti-Conservationist Effects of the Conservationist Oil Cartel

Fig. 11.4 The saddle-path solutions in the one-region model

Q ¼

α  c0 : c1

The steady-state price p equals the choke price, while the economically recoverable resource Q is proportional to the difference between the choke price and the initial unit cost. We have also shown in Chap. 3 (Sect. 3.2.3) that the saddle-path solution of (11.5), (11.6) is pðt Þ ¼ α  βθQ eθt   Qðt Þ ¼ Q 1  eθt ,

ð11:7Þ ð11:8Þ

where θ¼

  ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ 4rc1 =β  r 2

ð11:9Þ

is the equilibrium extraction rate. The saddle-path solution (11.7), (11.8) converges to the steady state p, Q and satisfies the initial condition Q(0) ¼ 0. The initial competitive price p0 ¼ p(0) is equal to 

p0 ¼ α  βδQ ¼



 βθ βθ 1 α þ c0 : c1 c1

ð11:10Þ

11.3

The Pure Monopoly Model

295

The initial price is the linear combination of choke price α and initial unit cost c0 with weight βθ/c1 < 1, as was shown in Chap. 3 for Eq. (3.19).

11.3

The Pure Monopoly Model

Suppose that all producers are organized into a cartel that fully monopolizes the market and accounts for the effects of supply changes on the market-clearing price schedule, P ðxÞ ¼ α  βx (as in the previous two chapters, we use a different notation for output and price under a competitive market and a cartel). The cartel chooses the extraction intensity path x(t) and sets at any instant t the monopoly price Pðt Þ ¼ P ðxðt ÞÞ to maximize the present value of future rents over an infinite extraction horizon: Z1

ert ½P ðxðt ÞÞ  cðX ðt ÞÞxðt Þdt,

ð11:11Þ

0

subject to the equation for cumulative extraction X(t): X_ ðt Þ ¼ xðt Þ:

ð11:12Þ

As above, X(0) ¼ 0. For the linear inverse demand, the marginal revenue of the cartel is ρðxÞ ¼ P ðxÞ þ P 0 ðxÞx ¼ α  2βx:

ð11:13Þ

The present-value Hamiltonian for the pure monopoly is H ¼ ert ½ðP ðxÞ  cðXÞÞx  wx, where -w is the costate variable related to Eq. (11.12). Maximizing with respect to x and X gives the first-order conditions: ρðxÞ  cðXÞ ¼ w w_ ¼ rw þ

∂H ¼ rw  c0 ðXÞx: ∂X

ð11:14Þ ð11:15Þ

Inserting (11.14) into (11.15) implies ρ_ ¼ r ðρðxÞ  cðX ÞÞ,

ð11:16Þ

296

11

Anti-Conservationist Effects of the Conservationist Oil Cartel

since w_ ¼ ρ_  c0 ðXÞx. Equation (11.14) means that the marginal resource value of the cartel, w, equals the marginal revenue less the marginal cost. For linear demand, _ For linear marginal cost, c the marginal revenue is: ρ ¼ 2P  α, hence ρ_ ¼ 2P. (X) ¼ c0 + c1X, we have the dynamic system for price and cumulative extraction, as implied by Eqs. (11.16), (11.12): r P_ ¼ ð2P  α  c0  c1 X Þ 2 X_ ¼ ðα  PÞ=β:

ð11:17Þ ð11:18Þ

The steady state of this system is the same as for the competitive market: P ¼ α α  c0 X ¼ : c1 Thus, the cartel does not influence the steady state, but affects the transition path to this state. The saddle-path solution of system (11.17), (11.18) is drawn in Fig. 11.4 with the dashed arrow. We show in Appendix A.1 that this solution is Pðt Þ ¼ a  βδX  eδt   X ðt Þ ¼ X  1  eδt ,

ð11:19Þ ð11:20Þ

where δ¼

  ffi 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ 2rc1 =β  r 2

ð11:21Þ

is the equilibrium extraction rate under a cartel. It is below this rate under a competitive market, θ. At any instant, the intensity of extraction is the constant fraction of the remaining recoverable resource stock: x(t) ¼ δ(X  X(t)). From Eq. (11.19), we obtain the initial cartel price P0 ¼ P(0):   βδ βδ P0 ¼ α  βδX ¼ 1  α þ c0 : c1 c1 

ð11:22Þ

The weight of the initial unit cost βδ/c1 is below this weight βθ/c1 for the initial competitive price (11.10), since δ < θ. Therefore, at the initial time the cartel price P0 is above the competitive price p0, as shown in Fig. 11.4.

11.3

The Pure Monopoly Model

297

11.3.1 A Numerical Example Consider a numerical example that illustrates the model. Let the interest rate be r ¼ 0.05, the choke price α ¼ $250 per barrel and the slope of inverse demand β ¼ 10. Suppose that parameters of the unit cost function are: c0 ¼ $5 per barrel, c1 ¼ 0.1. As in the linear case of the model in Chap. 3, we can introduce arbitrarily (for a numerical example) a scale parameter N for a unit of quantity measurement so that the inverse demand is P ðyÞ ¼ N ðα  βyÞ, and let N ¼ 106. Then the economically recoverable resource is Q ¼ X ¼ (α  c0)N/c1 ¼ 245 ∙ 106/0.1 ¼ 2,450 ∙ 106 barrels. The competitive extraction rate is calculated from Eq. (11.9): θ ¼ 0.0085, and the initial competitive price of a barrel of oil is found from Eq. (11.10): p0 ¼ α  βθQ ¼ 250  10 ∙ 0.0085 ∙ 2,450 ¼ $40.7. The monopoly extraction rate is found from Eq. (11.21): δ ¼ 0.0046. The initial monopoly price of a barrel, from Eq. (11.22), is P0 ¼ α  βδX ¼ 250  10 ∙ 0.0046 ∙ 2,450 ¼ $137.8, and the initial monopoly surplus per barrel is P0  p0 ¼ 137.8  40.7 ¼ $97.1.

11.3.2 Comparison of Transition Paths Figures 11.5 and 11.6 show the dynamics of extraction and price for the competitive market and the monopoly cartel for the numerical example. The solid curves depict extraction and price paths for the monopolized market and the dashed curves for the competitive market. Both paths flatten under the cartel. The cartel’s extraction is

Fig. 11.5 Extraction paths under competitive market and monopoly

298

11

Anti-Conservationist Effects of the Conservationist Oil Cartel

Fig. 11.6 Resource price under competitive market and monopoly

lower than the competitive one at the early stage and higher at the later stage, while the cartel price is higher at the early stage and lower at the later stage.

11.4

The Two-Region Model with Competitive Market

Suppose that there are two oil-producing regions labelled a and b that differ in extraction technologies. The unit extraction cost is the linear function of cumulative extraction, cj(Qj) ¼ c0j + c1jQj, j ¼ a, b. Firms in each region are competitive and maximize the present value of rents: Z1

 

ert pðt Þ  c j Q j ðt Þ q j ðt Þdt,

0

subject to the equation for cumulative extraction: Q_ j ðt Þ ¼ q j ðt Þ: The initial cumulative extraction is Qj(0) ¼ 0, j ¼ a, b. The market-clearing price is given by the inverse demand function:

11.4

The Two-Region Model with Competitive Market

299

p ¼ α  βðqa þ qb Þ: Region a has an advantage in terms of production costs: the initial unit cost and the slope of this cost are lower or equal for this region, c0a  c0b, c1a  c1b.

11.4.1 The Marginal Costs Equalization We can proceed over the same steps as in Sect. 11.2 to obtain the equation for oil price growth for j ¼ a, b:   p_ ¼ r p  c0j  c1j Q j :

ð11:23Þ

Both regions produce at any instant only if the unit costs are equalized: c0a þ c1a Qa ¼ c0b þ cba Qb :

ð11:24Þ

Taking the time derivatives in Eq. (11.24) implies a proportional link between the intensities of extraction: qb ¼

c1a q: c1b a

The market-clearing price is p ¼ α  β ð qa þ qb Þ ¼ α  β

c1a þ c1b qa , c1b

hence qa ¼ φðα  pÞ=β,

ð11:25Þ

where φ¼ is the market share of region a.

qa c1b ¼ qa þ qb c1a þ c1b

ð11:26Þ

300

11

Anti-Conservationist Effects of the Conservationist Oil Cartel

11.4.2 The Equilibrium Extraction Path The equilibrium extraction path satisfies Eqs. (11.23) and (11.25), which can be represented as the dynamic system for price and cumulative extraction of region a: p_ ¼ r ðp  c0a  c1a Qa Þ

ð11:27Þ

Q_ a ¼ φðα  pÞ=β:

ð11:28Þ

The steady-state solution is: p ¼ α α  c0a Qa ¼ c1a: and for region b: Qb ¼

α  c0b c1b

due to the equalization of marginal costs (11.24). Let the initial unit costs be the same, c0a ¼ c0b. We show in Appendix A.2 that the solution for system (11.27), (11.28) is   pðt Þ ¼ α  βθ Qa þ Qb eθt   Q j ðt Þ ¼ Qj 1  eθt ,

ð11:29Þ ð11:30Þ

where j ¼ a, b and θ¼

  1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi r 2 þ 4rc1a φ=β  r 2 2

ð11:31Þ

is the equilibrium extraction rate, the same for both regions:   q j ðt Þ ¼ θQj eθt ¼ θ Qj  Q j ðt Þ

ð11:32Þ

for j ¼ a, b. As above, the intensity of equilibrium extraction is the constant fraction of the remaining resource stock. Consider our numerical example with c0a ¼ c0b ¼ $5 per barrel, c1a ¼ 0.1 and c1b ¼ 0.3. The market share of region a is φ ¼ 0.3/(0.1 + 0.3) ¼ 3/4, from Eq. (11.26), and the extraction rate is θ ¼ 0.0066, from Eq. (11.31). With the scale parameter N ¼ 106, the steady-state cumulative extraction is Qa ¼ ðα  c0a ÞN=c1a ¼ ð250  5Þ  106 =0:1 ¼ 2, 450  106 barrels and Qb ¼ ðα  c0b ÞN=c1b ¼ ð250  5Þ 

11.5

The Two-Region Model with a Cartel

301

Fig. 11.7 Extraction paths for the two-region competitive model

106 =0:3 ¼ 817  106 barrels. Figure 11.7 shows the competitive extraction paths qa(t) and qb(t) given by Eq. (11.32) for this example.

11.5

The Two-Region Model with a Cartel

Now suppose that producers in region a have established a cartel that influences the resource price, whereas producers in the competitive fringe, region b, are price takers. Producers in the cartelized market behave as Cournot players: the cartel takes the fringe’s output as given, while a firm in region b takes the price as given. The cartel sets the price Pðt Þ ¼ P ðxa ðt Þ þ xb ðt ÞÞ and takes into account the effects of its supply changes on the price schedule, P ðxa þ xb Þ ¼ α  βðxa þ xb Þ:

ð11:33Þ

The cartel’s marginal revenue is ρ ¼ P  βxa. We can fulfil the same steps as in Sects. 11.3 and 11.4 to obtain price growth equations similar to Eqs. (11.17) and (11.23): r P_ ¼ ðP  βxa  c0a  c1a X a Þ 2 P_ ¼ r ðP  c0b  c1b X b Þ:

ð11:34Þ ð11:35Þ

It is shown in Appendix A.3 that the price growth Eqs. (11.34), (11.35) are compatible if the following condition is fulfilled:

302

11

Anti-Conservationist Effects of the Conservationist Oil Cartel

  βxb ¼ c1a X a þ X a  2c1b X b  2Δc0 ,

ð11:36Þ

where Δc0 ¼ c0b  c0a denotes the differential of initial unit costs. This condition determines the output of region b as the function of cumulative extractions Xa and Xb.

11.5.1 The Dynamic System The state variables of the model are P, Xa and Xb. We combine Eqs. (11.33), (11.35) and (11.36) to obtain the system of linear differential equations for the state variables: P_ ¼ r ðP  c0b  c1b X b Þ   βX_ a ¼ α  P  c1a X a þ X a þ 2c1b X b þ 2Δc0   βX_ b ¼ c1a X a þ X a  2c1b X b  2Δc0 :

ð11:37Þ ð11:38Þ ð11:39Þ

The steady state is P ¼ α, X a ¼

α  c0a  α  c0b , Xb ¼ : c1a c1b

ð11:40Þ

We show in Appendix A.4 that the characteristic equation for system (11.37)– (11.39) is βλ3 þ ðrβ  ðc1a þ 2c1b ÞÞλ2 þ r ðc1a þ 2c1b Þλ þ rc1a c1b =β ¼ 0:

ð11:41Þ

This cubic equation has two real-valued negative roots, λ1 and λ2, if the intercept is sufficiently small. Indeed, if the intercept were equal to zero, Eq. (11.41) would have three real-valued roots: λ2 < 0, λ1 ¼ 0 and λ3 > 0. A small shift of the cubic polynomial upwards would imply that the middle zero root moves to the left and becomes negative, λ1 < 0. Figure 11.8 illustrates the solution of characteristic Eq. (11.41) with two negative roots. Consider again our numerical example with r ¼ 0.05, α ¼ 250, β ¼ 10, c1a ¼ 0.1 and c1b ¼ 0.3. The negative roots of Eq. (11.41) are: λ1 ¼ 0:0042, λ2 ¼ 0:0681: Note that for this example (and other plausible examples) the intercept in Eq. (11.41) is sufficiently small compared to other coefficients of this equation and, consequently, two negative real-valued roots exist (solutions with complexvalued roots are ruled out because cumulative extraction cannot be oscillating). As

11.5

The Two-Region Model with a Cartel

303

Fig. 11.8 Two negative characteristic roots

we will see in what follows, the existence of two negative characteristic roots is the key property of the cartel-fringe model of this chapter. This property predetermines the qualitative differences in time profiles of extraction rates between the regions.

11.5.2 The Solution The saddle-path solution of the dynamic system (11.37)–(11.39) converges to the steady state. This path must satisfy the initial conditions, Xj(0) ¼ 0, and the terminal conditions, lim X j ðt Þ ¼ X j , for j ¼ a, b. These conditions are fulfilled for the t!1

cumulative extraction paths:   X a ðt Þ ¼ X a 1  σ a eθ1 t  ð1  σ a Þeθ2 t   X b ðt Þ ¼ X b 1  σ b eθ1 t  ð1  σ b Þeθ2 t ,

ð11:42aÞ ð11:42bÞ

implying that the intensity of extraction is the linear combination of negative exponents: x j ðtÞ ¼ X_ j ðtÞ ¼ X j ðσj θ1 eθ1 t þ ð1  σ j Þθ2 eθ2 t Þ, j ¼ a, b,

ð11:43Þ

where θ1 ¼  λ1, θ2 ¼  λ2 and σ a, σ b are the weights of exponential terms corresponding to the root λ1 with minimal absolute value. Inserting xa(t), xb(t) into Eq. (11.33) yields the equilibrium cartel price:   Pðt Þ ¼ α  β X a þ X b ½ψσ a þ ð1  ψ Þσ b θ1 eθ1 t þ½ψ ð1  σ a Þ þ ð1  ψ Þð1  σ b Þθ2 eθ2 t  

¼ α  β X a þ X b ξθ1 eθ1 t þ ð1  ξÞθ2 eθ2 t , where

ð11:44Þ

304

11

Anti-Conservationist Effects of the Conservationist Oil Cartel

ψ¼

X a

X a þ X b

is the cartel’s initial resource share and ξ ¼ ψσ a þ ð1  ψ Þσ b : The weights σ a and σ b are found from matching the solution (11.43), (11.44) with differential Eqs. (11.37) and (11.39) at the initial time. We show in Appendix A.5 that σa ¼ ψ

1

θ22 þ rθ2  ωa    ð1  ψ Þσ b 2 θ2 þ rθ2  θ21 þ rθ1

!

θ 2  ωb θ2  θ1   with ωa ¼ rc1b ð1  ψ Þ=β, ωb ¼ c1a X a  2Δc0 =βX b . σb ¼

ð11:45Þ ð11:46Þ

11.5.3 Time Profiles of Extraction Rates and the Cartel’s Market Share Let us return to the numerical example. Suppose that the initial unit costs are c0a ¼ $5 per barrel, c0b ¼ $50 per barrel and c1a ¼ 0.1, c1b ¼ 0.3. The differential of initial unit costs is Δc0 ¼ $45 per barrel. The initial stocks of recoverable resources are X a ¼ ð250  5Þ  106 =0:1 ¼ 2, 450  106 barrels and X b ¼ 6 6 ð250  50Þ  10 =0:3 ¼ 667  10 barrels. The cartel’s initial resource share is ψ¼

2, 450 ¼ 0:786: 2, 450 þ 667

The weights σ a and σ b are found from Eqs. (11.45), (11.46): σ a ¼ 1:066, σ b ¼ 0:702: for θ1 ¼ 0.0042, θ2 ¼ 0.0681, obtained above. Unlike the cases considered in the previous sections, the extraction rates are timedependent. The time profiles of these rates are derived from Eqs. (11.42a, b), (11.43) for j ¼ a, b:

11.5

The Two-Region Model with a Cartel

305

Fig. 11.9 Time profiles of the regions’ extraction rates Θa(t) and Θb(t)

  σ j θ1 eθ1 t þ 1  σ j θ2 eθ2 t x j ðt Þ   Θ j ðt Þ ¼  ¼ X j  X j ðt Þ σ j eθ1 t þ 1  σ j eθ2 t   σ j θ1 þ 1  σ j θ2 eðθ2 θ1 Þt   ¼ : σ j þ 1  σ j eðθ2 θ1 Þt One can take the derivative dΘj(t)/dt to check that the extraction rate Θa(t) is increasing for our numerical example because σ a > 1, while Θb(t) is decreasing because σ b < 1. Both paths of Θa(t) and Θb(t) converge to the characteristic root θ1. The time profiles of extraction rates for the numerical example are shown in Fig. 11.9. The increasing solid curve is Θa(t) and the decreasing dashed curve is Θb(t). The dashed-and-dotted horizontal line shows the characteristic root θ1 ¼ 0.0042, to which both curves converge. The dotted horizontal line shows the time-constant extraction rate θ ¼ 0.0066, which was obtained for the two-region competitive model. Figure 11.9 shows that for the early phase of resource extraction, the extraction rate of the high-cost region Θb(t) is substantially higher than that of the low-cost region Θa(t) and the competitive extraction rate θ. Figure 11.10 demonstrates the time profile of the cartel market share: Φðt Þ ¼

xa ð t Þ : xa ð t Þ þ xb ð t Þ

This share is drawn with a solid curve in the figure. The time-constant competitive market share of region a, φ ¼ c1b/(c1a + c1b), is drawn with a dashed horizontal line.

306

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Anti-Conservationist Effects of the Conservationist Oil Cartel

Fig. 11.10 The cartel’s market share

The cartel’s market share increases in time and converges to the upper limit calculated from (11.43) and depicted with a dotted horizontal line in Fig. 11.10: lim ΦðtÞ ¼

t!1

ψσ a 0:786  1:066 ¼ 0:848: ¼ ψσ a þ ð1  ψÞσ b 0:786  1:066 þ 0:214  0:702

As one can see in this figure, in the early phase of resource extraction the cartel’s market share is considerably below the competitive market share of region a in the numerical example of the previous section, φ ¼ 0.3/(0.1 + 0.3) ¼ 0.75.

11.5.4 Production Shifting Over Time In Fig. 11.11 we show the extraction paths of the regions under a cartelized market xa(t) and xb(t) given by Eq. (11.43) and calculated for our numerical example. One can compare these paths with extraction paths under the competitive market qa(t) and qb(t) shown in Fig. 11.7. The production of the low-cost region a shifts over time to the later stage, whereas the production of the high-cost region b shifts to the early stage. The time profile of the cartel’s extraction in Fig. 11.11 has a peak because the weight σ a is above unity and xa(t) is non-monotonic. Note that in the numerical example in this section, the high-cost competitive region has disadvantages both in the slope of the unit cost function and in the initial unit cost. We assumed that this initial unit cost is quite high, c0b ¼ $50 per barrel, to avoid analysis of corner solutions. If c0b were below this level, the cartel production would be zero at the early stage of resource extraction. In such a case, we would have

11.6

Concluding Remarks

307

Fig. 11.11 Extraction paths under cartelized market

to consider the time sequencing of production, as in Ulph and Folie’s model (1980) presented in Chap. 9. In one of the cases of that model, the low-cost fringe produces alone at the early stage and then both the fringe and the high-cost cartel produce simultaneously. In the model in this chapter, it is possible that the high-cost fringe produces alone at the early stage, but we do not consider such cases. As a result, the conservationist behaviour of the cartel accelerates resource depletion in the high-cost region, which causes an increase of extraction costs in this region and a more rapid subsequent fall of production than under the competitive market. The cartel in the model can only affect the current price, but it acts as a strategic player aiming to strengthen its monopoly position in the future by exhausting the resource base of the competitive fringe.

11.6

Concluding Remarks

We have shown that cartel pricing can result in production shifting over time that encourages early depletion of a high-cost resource. This effect is caused by the choice of the conservationist extraction path by the low-cost cartel and of the anti-conservationist path by the high-cost competitive fringe. This inference rests on the simple model of an economically recoverable resource. We assumed, for the sake of simplicity, that the unit extraction costs are linear functions of cumulative extraction. This assumption allows for an explicit solution of equilibrium dynamic systems. The model extensions to non-linear extraction cost functions could capture more realistically the features of resource depletion, but would require numerical

308

11

Anti-Conservationist Effects of the Conservationist Oil Cartel

integration of differential equations, which means a loss in analytical tractability of the model. The effects of the cartel on the extraction rates of oil-producing regions are robust to variations of the model parameters. For the cartel-fringe model in Sect. 11.5, the characteristic cubic equation for an equilibrium dynamic system (11.41) retains two negative characteristic roots under variations of the model parameters. As we have shown, the existence of two negative roots implies qualitative differences in the dynamics of extraction by the regions under competitive and cartelized markets. We have demonstrated that under a cartelized market, the initial extraction rate can be significantly higher for the high-cost competitive fringe than for the low-cost cartel. The time path of extraction rate decreases for the fringe and increases for the cartel, whereas the cartel’s market share increases over time. These results conform to the real-world tendencies of the global oil market presented in the introduction: the extraction rate is higher for non-OPEC producers than for OPEC, decreasing for non-OPEC countries and increasing for Middle Eastern OPEC members. Moreover, the market share of OPEC tends to increase, while the extraction costs of non-OPEC producers grow more rapidly than these costs for the oil cartel. We have not analysed in this chapter the welfare implications of the cartel’s activity. The main point could be a misallocation effect and the deadweight losses that result from the front-loading over time of high-cost oil production demonstrated in Fig. 11.11. It is also possible to examine the effects of the cartel on welfare losses from the negative environmental impacts that can be included in the model. We could assume, following the paper by Benchekroun et al. (2017) mentioned in the introduction and other papers, that “dirty” (e.g. unconventional) oil is produced in the high-cost region and “clean” oil in the low-cost region. There may be two effects of the cartel on the environment. A conservation effect takes place due to the cartel’s conservationist pattern of extraction shown in Fig. 11.11 and implying that using clean oil shifts to the future. A sequence effect is caused in our model by the anticonservationist pattern of extraction by the high-cost region implying a more intensive early impact on the environment. The contribution of the sequence effect to welfare losses may be significant, as Benchekroun et al. (2017) have demonstrated in their study.

Appendices A.1 Solution (11.19), (11.20) The characteristic equation for system (11.17), (11.18) is

Appendices

309

rλ

1=β

rc1 =2

¼ λ2  rλ  rc1 =2β ¼ 0 λ

ð11:47Þ

and the negative root is  pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ ¼ r  r 2 þ 2rc1 =β =2 ¼ δ: Let us check that time paths (11.19), (11.20) satisfy differential Eqs. (11.17), (11.18). On the one hand, inserting (11.19), (11.20) into (11.17) implies    P_ ¼ 0:5r ð2P  α  c0  c1 X Þ ¼ 0:5r 2α  2βδX  eδt  α  c0  c1 X  1  eδt   ¼ 0:5r α  c0  c1 X   2βδX  eδt þ c1 X  eδt ¼ 0:5r ðc1  2βδÞX  eδt : On the other hand, differentiating (11.19) yields P_ ¼ βδ2 X  eδt : These two equations are identical if βδ2 ¼ 0:5r ðc1  2βδÞ: This is the case for δ ¼  λ satisfying the characteristic Eq. (11.47).

A.2 Solution (11.29), (11.30) The characteristic equation for system (11.27), (11.28) is

rλ

φ=β

rc1a

¼ λ2  rλ  rφc1a =β ¼ 0: λ

ð11:48Þ

 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi The negative root is λ ¼ r  r 2 þ 4rφc1a =β =2 ¼ θ . Solution (11.29), (11.30) satisfies (11.28):   Q_ a ¼ θQa eθt ¼ θφ Qa þ Qb eθt ¼ φðα  pÞ=β,   since c0a ¼ c0b and Qa ¼ φ Qa þ Qb due to condition (11.26). Insert Eqs. (11.29), (11.30) into (11.27):

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    p_ ¼ r ðp  c0a  c1a Qa Þ ¼ r α  βθ Qa þ Qb eθt  c0a  c1a Qa ¼           r α  c0a  c1a Qa  βθ Qa þ Qb eθt ¼ r c1a Qa  Qa  βθ Qa þ Qb eθt ¼     r c1a Qa eθt  βθ Qa þ Qb eθt ¼ r ðc1a  ðβθ=φÞÞQa eθt :     The time derivative of Eq. (11.29) is: p_ ¼ βθ2 Qa þ Qb eθt ¼ βθ2 =φ Qa eθt . This coincides with (11.27) for λ ¼  θ satisfying (11.48).

A.3 Equation (11.36) Equalization of time derivatives of price for Eqs. (11.34), (11.35) implies: r ðP  βxa  c0a  c1a X a Þ ¼ rðP  c0b  c1b X bÞ: 2 Rearrange the terms in this equation and take into account (11.33): P þ βxa þ c0a þ c1a X a ¼ 2c0b þ 2c1b X b α  βðxa þ xb Þ þ βxa þ c0a þ c1a X a ¼ 2c0b þ c1b X b α  c0a  βxb þ c1a X a ¼ 2c0b  2c0a þ c1b X b   βxb ¼ c1a X a þ X a  2Δc0  c1b X b , because α  c0a ¼ c1a X a from Eq. (11.40).

A.4 Characteristic Equation (11.41) The characteristic equation for system (11.37)–(11.39) is

Appendices

311





c1a 2c1b

  λ 2c1b

1 rc1a c1b

β β

þ 

β

¼ ðr  λÞ

β β

c 2c 1a 1b

  λ

2c1b

β β   λ β   c þ 2c1b rc c ¼ ðr  λÞ λ2 þ 1a λ þ 1a2 1b ¼ β β   c þ 2c c þ 2c1b rc c 1b λ2 þ r 1a λ þ 1a2 1b ¼ 0, λ3 þ r  1a β β β

r  λ 0

c

1  1a  λ

 β

β

c1a

0

β

rc1b

which implies: βλ3 þ ðrβ  ðc1a þ 2c1b ÞÞλ2 þ r ðc1a þ 2c1b Þλ þ rc1a c1b =β ¼ 0:

A.5 Formulae (11.45), (11.46) (a) Consider Eq. (11.39) for t ¼ 0: βX_ b ð0Þ ¼ c1a X a  2Δc0 since Xa(0) ¼ Xb(0) ¼ 0. From (11.43): X_ b ð0Þ ¼ X b ðσ b θ1 þ ð1  σ b Þθ2 Þ hence σ b θ1 þ ð1  σ b Þθ2 ¼

c1a X a  2Δc0 βX b

and σb ¼ where

ωb  θ 2 θ 2  ωb ¼ θ1  θ2 θ2  θ1

ð11:49Þ

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Anti-Conservationist Effects of the Conservationist Oil Cartel

ωb ¼

c1a X a  2Δc0 βX b

which yields (11.46). (b) Consider Eq. (11.37) for t ¼ 0: P_ ð0Þ ¼ r ðPð0Þ  c0b Þ

ð11:50Þ

From Eq. (11.44), the time derivative of the cartel price is:  

P_ ðt Þ ¼ β X a þ X b ξθ21 eθ1 t þ ð1  ξÞθ22 eθ2 t , implying that  

P_ ð0Þ ¼ β X a þ X b ξθ21 þ ð1  ξÞθ22 From Eq. (11.44), we also have it that   Pð0Þ ¼ α  β X a þ X b ½ξθ1 þ ð1  ξÞθ2  Combining the last two equations with (11.50) implies:   

  β X a þ X b ξθ21 þ ð1  ξÞθ22 ¼ r α  c0b  β X a þ X b ½ξθ1 þ ð1  ξÞθ2  Rearrange the terms: r ðα  c Þ ξθ21 þ ð1  ξÞθ22 þ r ðξθ1 þ ð1  ξÞθ2 Þ ¼   0b  β Xa þ Xb 

 rc1b X  rc ð1  ψ Þ θ21 þ rθ1  θ22  rθ2 ξ þ θ22 þ rθ2 ¼   b   ¼ 1b , β β Xa þ Xb

since α  c0b ¼ c1b X b from (11.40). We have:   ωa  θ22 þ rθ2   ξ¼ 2 θ1 þ rθ1  θ22 þ rθ2 where ωa ¼ rc1b(1  ψ)/β, implying that

References

313

σa ¼ ψ

1

θ22 þ rθ2  ωa    ð1  ψ Þσ b 2 θ2 þ rθ2  θ21 þ rθ1

!

since ξ ¼ ψσ a + (1  ψ)σ b. (c) Let us check that Eq. (11.38) is fulfilled as identity for t ¼ 0: βX_ a ð0Þ ¼ α  Pð0Þ  c1a X a þ 2Δc0

ð11:51Þ

From Eqs. (11.43), (11.44), we have it that βX_ a ð0Þ ¼ βX a ðσ a θ1 þ ð1  σ a Þθ2 Þ,   Pð0Þ ¼ α  β X a þ X b ½ξθ1 þ ð1  ξÞθ2  Insert these two equations into (11.51):   βX a ðσ a θ1 þ ð1  σ a Þθ2 Þ ¼ β X a þ X b ½ξθ1 þ ð1  ξÞθ2   c1a X a þ 2Δc0   Divide both sides by β X a þ X b and rearrange: c1a X a  2Δc0   ¼ ψ ðσ a θ1 þ ð1  σ a Þθ2 Þ β X a þ X b c1a X   2Δc0  ¼ ψ ðσ a θ1 þ ð1  σ a Þθ2 Þ þð1  ψ Þðσ b θ1 þ ð1  σ b Þθ2 Þ   a β X a þ X b

ψ ðσ a θ1 þ ð1  σ a Þθ2 Þ ¼ ξθ1 þ ð1  ξÞθ2 

because, due to (11.49): ð1  ψ Þðσ b θ1 þ ð1  σ b Þθ2 Þ ¼ ð1  ψ Þ

c1a X a  2Δc0 c1a X a  2Δc0  ¼   βX b β X a þ X b

Consequently, (11.51) holds as identity.

References Benchekroun H, Meijden G, Withagen C (2017) OPEC, shale oil, and global warming: on the importance of the order of extraction. Tinbergen Institute Discussion Paper – 104/VII BP Energy Outlook, 2017 edition. bp.com/energyoutlook # BPstats https://www.bp.com/content/ dam/bp/business-sites/en/global/corporate/pdfs/energy-economics/energy-outlook/bp-energyoutlook-2017.pdf. Accessed 30 Nov 2020 BP Statistical Review of World Energy, 2020 edition. https://nangs.org/analytics/bp-statisticalreview-of-world-energy. Accessed 30 Nov2020

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Farzin A (1992) The time path of scarcity rent in the theory of exhaustible resources. Econ J 102 (413):813–830 Herfindahl O (1967) Depletion and economic theory. In: Gaffney M (ed) Extractive resources and taxation. Madison University of Wisconsin Press, Madison, pp 63–90 Hotelling H (1931) The economics of exhaustible resources. J Polit Econ 39:137–175 Ulph A, Folie G (1980) Exhaustible resources and cartels: an intertemporal NashCournot model. Can J Econ 13:645–658