Climate Policy and Nonrenewable Resources : The Green Paradox and Beyond [1 ed.] 9780262319836, 9780262027885

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Climate Policy and Nonrenewable Resources

CESifo Seminar Series edited by Hans-Werner Sinn The Evolving Role of China in the Global Economy Yin-Wong Cheung and Jakob de Haan, editors Critical Issues in Taxation and Development Clemens Fuest and George R. Zodrow, editors Central Bank Communication, Decision Making, and Governance Pierre L. Siklos and Jan-Egbert Sturm, editors Lessons from the Economics of Crime Philip J. Cook, Stephen Machin, Olivier Marie, and Giovanni Mastrobuoni, editors Firms in the International Economy Sjoerd Beugelsdijk, Steven Brakman, Hans van Ees, Harry Garretsen, editors Global Interdependence, Decoupling and Recoupling Yin-Wong Cheung and Frank Westermann, editors The Economics of Conflict Karl Wärneryd, editor Climate Policy and Nonrenewable Resources Karen Pittel, Rick van der Ploeg, and Cees Withagen, editors See http://mitpress.mit.edu for a complete list of titles in this series.

Climate Policy and Nonrenewable Resources The Green Paradox and Beyond

edited by Karen Pittel, Rick van der Ploeg, and Cees Withagen

The MIT Press Cambridge, Massachusetts London, England

© 2014 Massachusetts Institute of Technology All rights reserved. No part of this book may be reproduced in any form by any electronic or mechanical means (including photocopying, recording, or information storage and retrieval) without permission in writing from the publisher. MIT Press books may be purchased at special quantity discounts for business or sales promotional use. For information, please email [email protected]. This book was set in Palatino LT Std by Toppan Best-set Premedia Limited, Hong Kong. Printed and bound in the United States of America. Library of Congress Cataloging-in-Publication Data Climate policy and nonrenewable resources: the green paradox and beyond / edited by Karen Pittel, Rick van der Ploeg and Cees Withagen. pages cm. – (CESifo seminar series) Includes bibliographical references and index. ISBN 978-0-262-02788-5 (hardcover: alk. paper) 1. Climatic changes–Government policy. 2. Nonrenewable natural resources. 3. Supply-side economics. I. Pittel, Karen, 1969– II. Ploeg, Rick van der, 1956– III. Withagen, Cees, 1950– QC903.C568 2014 363.738'74561–dc23 2014003839 10

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Contents

Series Foreword

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The Green Paradox: A Mirage? 1 Karen Pittel, Rick van der Ploeg, and Cees Withagen I

Extraction Costs

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Supply-Side Climate Policy and the Green Paradox Michael Hoel

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The Green Paradox as a Supply Phenomenon Julien Daubanes and Pierre Lasserre II

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Technology, Innovation, and Substitutability

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The Green Paradox under Imperfect Substitutability between Clean and Dirty Fuels 59 Ngo Van Long

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Fossil Fuels, Backstop Technologies, and Imperfect Substitution 87 Gerard van der Meijden

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Innovation and the Green Paradox Ralph A. Winter

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Resource Extraction and Backstop Technologies in General Equilibrum 151 Ngo Van Long and Frank Stähler

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Timing, Announcement Effects, and Time Consistency

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Does a Future Rise in Carbon Taxes Harm the Climate? Florian Habermacher and Gebhard Kirchgässner

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The Impacts of Announcing and Delaying Green Policies 211 Darko Jus and Volker Meier

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Going Full Circle: Demand-Side Constraints to the Green Paradox 225 Corrado Di Maria, Ian Lange, and Edwin van der Werf IV

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Empirics and Quantification

Quantifying Intertemporal Emissions Leakage Carolyn Fischer and Stephen Salant Contributors Index 289

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Series Foreword

This book is part of the CESifo Seminar Series. The series aims to cover topical policy issues in economics from a largely European perspective. The books in this series are the products of the papers and intensive debates that took place during the seminars hosted by CESifo, an international research network of renowned economists organized jointly by the Center for Economic Studies at Ludwig-MaximiliansUniversität, Munich, and the Ifo Institute for Economic Research. All publications in this series have been carefully selected and refereed by members of the CESifo research network.

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The Green Paradox: A Mirage? Karen Pittel, Rick van der Ploeg, and Cees Withagen

1.1

Introduction

The interrelation between economics and the environment has posed a number of difficult challenges in the course of history of which climate change is universally considered to be the probably biggest faced so far. The global scale and long-run nature of climate change, the diversity of regional impacts, and the uncertainties involved render policy advice on how to mitigate emissions from fossil fuels efficiently extremely difficult. Due to this complexity a vast literature has developed in the field of climate change not only in natural sciences but also in economics. In the past, economic analysis of climate policy dealt with a variety of aspects and applied numerous methodological approaches. International negotiations and coalition formation were addressed as well as forecasts of costs and benefits of climate policy under different growth, population, and temperature scenarios—to name only a few aspects. The efficiency of policies, however, also depends critically on the reactions of agents to the economic incentives set by climate policies. Until recently most economists focused on the demand side of fossil fuel markets, not only as the starting point for policies but also with respect to the reactions to these politics. Supply-side reactions to climate policies were largely ignored. Interest in the role of energy supply was boosted considerably after the publication of Sinn’s (2008) work on the green paradox, which emphasized the essential role of oil and gas suppliers for the effectiveness of climate policy. 1.2

The Green Paradox and Its Determinants

The concept of the “green paradox” entails that some of the policy instruments that have been favored by economists might be

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counterproductive in the sense that they bring the supply of fossil fuels forward instead of postponing it. Take a specific tax on extracting carbon or on consuming carbon-intensive products. If resource extractors anticipate this tax rate to increase over time at a faster rate than the rate of interest, it will lower the present value of future profits of owners of fossil fuel deposits. The tax will thus give rise to an intertemporal arbitrage effect that shifts extraction to the present and thus speeds up global warming. Oil producers effectively fear the future expropriation of their profits and try to avoid some of it by shifting some oil production from the future to the present. Along the same lines, politicians that rely on subsidizing solar or wind energy rather than imposing a specific tax on fossil fuel use that does not rise faster than the market rate of interest inadvertently create incentives for fossil fuel owners to increase short-term extraction of fossil fuel. Owners of fossil fuel reserves fear that their energy assets will become worthless if solar and wind energy are subsidized and therefore pump up oil and gas more quickly. This speeds up emissions and accelerates global warming. The literature has identified four main channels through which a green paradox can arise: (1) a too rapid tightening of climate policies over time, (2) future reductions in resource demand due to innovations and a fall in prices of green substitutes, (3) lags in the implementation of climate policies, and (4) international carbon leakage due to unilateral climate policies (e.g., van der Werf and Di Maria 2011). In principle, these triggers for paradoxical reactions to climate policies might be avoided by adopting a global cap-and-trade system covering all greenhouse gases from all sectors for all times or a global first-best optimal carbon tax. However, negotiations like those within the United Nations Framework Convention on Climate Change (UNFCCC) clearly indicate that a realistic outcome will not fit these criteria. As has been forcefully argued by Helm (2012), European policy makers congratulate themselves because carbon emissions from production have fallen but fail to mention that carbon emissions from consumption have risen during the last fifteen years. US climate policy might have been more effective. So what has been going on? In Europe there has been a steady process of deindustrialization which has led to the movement of industry to China and elsewhere. The steady decline of manufacturing and growth in services in Europe has led to a reduction in carbon emissions regardless of climate policy. Furthermore manufactured consumption goods

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are imported from low-wage countries such as China which often use more carbon-intensive forms of energy than Europe, such as coal. This explains why carbon emissions from production in Kyoto countries have declined, yet their carbon footprint has not (Aichele and Felbermayr 2012). The latter is unfortunately what matters for global warming. Given this inability to impose a first-best policy, alternative policies have to be evaluated carefully to assess their potential to speed up extraction and to lead to detrimental welfare effects. The literature on the green paradox distinguishes between two concepts, the so-called weak green paradox and the strong green paradox. The former means that resource extraction is reallocated toward the present, the latter arises if climate policy leads to an increase in the net present value of cumulative global warming damages. One factor that is crucial for the net present value of additional emissions is the rate at which future damages are discounted—a discussion that was spawned by the Stern Report (Stern 2008). Yet not only the discount rate but also the degree of intergenerational inequality aversion matters. Furthermore the assumptions made with respect to the shape of damage functions and demographic as well as economic developments affect the estimates of the so-called social costs of carbon (SCC), namely the net present value of a marginal additional unit of emission. Tol (2009) gives an overview of a number of numerical studies that estimate SCCs and finds that uncertainty is very high. While the mean estimate is 105$/tC (with a standard deviation of 243$/tC), the modal estimate amounts to only 13$/tC. Figure 1 in Nordhaus (2011) gives the different damage estimates from Tol (2009) as percentage of output and illustrates the role that the increase in temperature has on these estimates. Social costs of carbon reflect, however, only part of the welfare effects that are induced by additional CO2 emissions. To assess the overall welfare change, all effects—positive as well as negative—would need to be taken into account. If earlier extraction led, for example, to higher investment or faster technological change, the induced boost in output or growth would—at least partially—compensate for higher damages from climate change. One fundamental question that has been discussed extensively concerns the validity of Sinn’s (2008) results in a more general setting. Various papers have extended his relatively simple one-sector, single country, long-run steady-state framework and analyze the scope of the intertemporal arbitrage effect to prevail under different assumptions

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regarding climate policy scenarios as well as under variations of the model setup. Extensions and variations include, for example, the availability of clean or dirty substitutes and the role of stock-dependent extraction costs (e.g., Gerlagh, 2011; van der Ploeg and Withagen 2012a, b) or the international dimension (e.g., Eichner and Pethig 2011).1 This literature has shown that under those more realistic model assumptions the emergence of a strong or even a weak green paradox is by no means certain. For example, a renewables subsidy does not only speed up the process of oil extraction but also induces a quicker transition to the carbon-free era and a bigger stock of carbon to be locked up in the crust of the earth forever. Cumulative carbon emissions are thus reduced and this typically matters much more in the fight against global warming than the temporary acceleration of global warming. As the scope for the green paradox depends crucially on the model assumptions and since no model can ever be more than a simplified, abstract representation of reality, a natural question to be asked is whether empirical evidence for a green paradox exists. Empirical analyses are, however, severely hampered by the relative short history of climate policy. Regarding the oil market, the empirical literature dealing with the Hotelling rule clearly shows the difficulties that empirical analyses encounter on this market. Also climate policies have not necessarily tightened over the last years—as might have been expected. Carbon taxation of oil products in OECD countries, for example, did not increase substantially from 1995 to 2010 (see figure 1.1), in some countries tax rates even decreased. One example relates to the discoveries of Dutch natural gas in the 1960s, since the initial plan was to extract it very fast because it was thought that nuclear energy would become the predominant way of producing electricity. Di Maria, Lange, and van der Werf a (chapter 11 in this volume) attempt to find econometric evidence for green paradox effects. To avoid the problems arising from global markets, they concentrate on the case of coal for which markets are more localized (albeit that scarcity rents will be less substantial than for oil or gas). Specifically, they analyzed a policy whose aim was climate unrelated but affected the profitability of extracting coal nevertheless, the US Clean Air Act of 1995. Scope for a green paradox arose in this case due to the delay between announcement of the policy in 1995 and its implementation in 2000. Whether or not a green paradox arises also depends on how policies affect the fossil energy mix. If, for example, oil were substituted for coal

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700 600 500 400 300 200 100 0 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 Q1 Q3 1995 1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 2010

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Figure 1.1 Excise tax on oil products (based on IEA 2013; OECD 2013; World Bank 2013)

Table 1.1 CO2 emissions from coal and natural gas relative to crude oil Crude oil Coal (anthracite) Natural gas

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due to carbon taxation while the aggregate energy use remained the same, emissions would decrease due to the lower CO2-emission factor of oil (see also table 1.1). Equivalently substitution between different types of coal or different types of oil might matter for emissions and thus for the green paradox. A switch from crude-oil-based liquid fuels to coal-to-liquid would, for example, increase GHG emissions by more than 70 percent (IIASA 2012). Given the limits of econometric analysis and the calls for solid empirical evidence for the green paradox from the political arena, another option might be to employ theoretical models that are calibrated numerically using empirical estimates for elasticities, parameters, and exogenous variables. Examples for such simulation-based analyses of the green paradox are Michielsen (2011) and Bauer et al. (2012). Naturally the results of such analyses depend on the setup of the model as well as the quality of the calibration. This proves to be

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problematic not only with respect to elasticities and parameters that cannot be directly observed but also with respect to supposedly observable data as resource stocks and extraction costs. Take, for example, extraction costs for fossil resources or costs of alternative renewable energies. Extraction costs and their development with increasing cumulated extraction are extremely relevant for the question of whether a green paradox arises or not. Yet even estimates of current extraction costs are difficult to verify as they are usually private information of extracting firms. Future estimates are furthermore subject to technological uncertainty. Especially cost curves for technologies that are in an early stage of large-scale employment—as the case for some renewables—are subject to a high degree of uncertainty. Further examples for data uncertainty are the estimates for reserves and resources of oil, coal, and gas. By giving an overview of different estimates for reserves and resources as well as their development over time, the Global Energy Assessment by IIASA (IIASA 2012) illustrates this uncertainty. Large differences between the estimates result from different reserve and resource delineations and limited access to information from private agents. Also geological data derived from exploration activities are subject to interpretation and judgment (see IIASA 2012). The uncertainty that exists with respect to the aggregate amount of reserves and resources available carries forward to the amounts of fossils that are expected to be economically extractable in the future. While reserves, by definition, can be extracted economically given today’s technologies and prices, matters are more complex with respect to resources that, again by definition, are not economically extractable under current technological and market conditions. So which stocks are counted as reserves and which as resources depend on the level of current market prices? If prices rise, it will be profitable to explore more of the stocks, and consequently reserves increase. Uncertainty about future technology-related costs and prices are further exacerbated by uncertainty about the nature of the resources. With respect to materials such as coal, incentives for further exploration of proven as well as unproven resources are low due to currently high reserve to production ratios. Consequently knowledge about the size of the stocks and the potentially retrievable share is limited. Considering that estimated resources exceed estimated reserves by a substantial amount (see figures 1.2 and 1.3 for oil and hard coal), this

101 - 300

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Conventional and unconventional oil reserves

Figure 1.2 Estimates of conventional and unconventional oil reserves and resources by region (based on IIASA 2012; BGR 2009, 2010)

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Oil reserves and resources

The Green Paradox 7

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Figure 1.3 Estimates of hard coal reserves and resources by region (based on IIASA 2012; BGR 2009, 2010).

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Coal reserves and resources

Coal resources

Coal reserves

8 Karen Pittel, Rick van der Ploeg, and Cees Withagen

The Green Paradox

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high degree of uncertainty makes predictions about future extraction and thus the green paradox even harder. Given the particularly high estimates for coal reserves and resources, the question seems justifiable whether coal should—for all practical purposes—be treated as an exhaustible resource.2 But also the recent discoveries of shale gas in the United States and elsewhere lead some to conclude the end of “peak oil” (e.g., Helm 2012), in which case the green paradox will not be more than a theoretical curiosum. Whether the discovery of new reserves of unconventional energy will be large enough to render the intertemporal inefficiencies highlighted by the literature on the green paradox remains to be seen. It is important to realize that some would argue that the carbon budget is only 300 GtC, which corresponds to the amount of emissions that can occur before the targeted margin of 2 degrees Celsius of global warming is reached. However, proven reserves of listed oil and gas companies are at least two or three times as high as this. This highlights the issue of “stranded assets.” Either stocks of these companies are overvalued so that they will be left with stranded assets when climate policy finally has to be made more aggressive or the carbon budget is made irrelevant due to some technological fix. The key question is thus not how fast fossil fuel is exhausted but also how much fossil fuel is left unexploited in the crust of the earth (e.g., Rezai and van der Ploeg, 2013). With respect to renewables, the data situation is even worse. Taking the theoretical or even technical potentials3 of, for example, wind and sun, both seem to be available in virtually unlimited amounts. Yet to which degree this technological potential translates into “economic” potential is difficult to assess given the still relatively high learning rates with respect to solar and also, to a lesser extent, with respect to wind. Whether or not a green paradox arises due to climate policies depends on all of the issues raised—from the shape and level of the extraction cost curve to the uncertainty about technologies, resource stocks and substitutes. The main aim of this volume is to analyze the conditions under which a green paradox might or might not arise in depth and to draw conclusions from the underlying intertemporal nature of the resource owner ’s decision process. The focus is less on the provision of “ready to use” policy advice but more on the scientific basis for this policy advice.

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1.3

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Contributions to This Volume

The chapters in the volume discuss a number of the above raised issues in detail. First, the role of extraction costs is discussed in the part I and the importance of technologies, innovation, and substitutability for the green paradox is analyzed in part II. Problems regarding timing, announcement effects, and time consistency are addressed in part III, and finally part IV deals with empirics and quantification. In the following we briefly outline the material covered by contributors to this volume and their individual modeling approaches as well as the results obtained. 1.3.1 Extraction Costs The stocks of nonrenewable resources that are available for extraction are not necessarily exogenously given. One reason for this might be that a government might expropriate resources, another reason could be that reserves need to be developed. The first two chapters in this book deal with this phenomenon. One way to interpret the marginal extraction cost function is to look upon it as the marginal cost of extracting from a single well. Alternatively, one could argue that there exists a continuum of wells that can be ordered according to their marginal extraction cost. In the former it is difficult to imagine that part of the stock is expropriated so that only the cheaper part or the more expensive part remains. However, in the latter interpretation this is clearly possible. Michael Hoel, chapter 2 in this book, takes this position. He focuses on the effects of removing resources and shows, among other things, that this always leads to less overall extraction. He also analyzes the consequences of removing high-cost resources compared to low-cost resources if the extraction process itself causes CO2 emissions. Hoel shows that this distinction matters for the occurrence of green paradoxes. Finally he discusses the case of a dirty backstop, which is a particularly grim prospect. In chapter 3 Julien Daubanes and Pierre Lasserre introduce extraction cost together with the exploration cost needed to develop reserves. They use a two-period model to derive supply from a nonrenewable resource that needs to be developed before it can be taken into exploitation. Hence the development decision and the supply strategy both depend on the net present and future market prices as well as on the

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direct extraction cost and the cost of development. They discuss the potential effect of this approach on the occurrence of the green paradox. 1.3.2 Substitutability and Innovation The following set of chapters investigates crucial aspects of renewable and nonrenewable resources, namely substitutability and innovation. These authors show that the innovation and its impact on the degree of substitutability both can lead to counterintuitive results. Most prevailing studies on the green paradox assume that energy from fossil fuels and renewables are perfect substitutes (an exception is Michielsen, 2011). Due intermissions (wind, solar) and other problems (transportation), they are in fact imperfect substitutes. Ngo Van Long investigates the effects of modifying the assumption of perfect substitutability (chapter 4). Long considers the case where there is no decay of CO2 in the atmosphere and where the marginal extraction cost of the fossil fuel are constant and where energy use appears directly in the utility function. The production cost of the renewable is convex. The renewable is subsidized (ad valorem). After defining a measure of substitutability between fossil fuel and renewables in the demand function, the effects of increased substitutability are investigated. He derives necessary and sufficient conditions for increased substitutability to lead to more damage to the environment. Imperfect substitutability and the potential occurrence of a green paradox are also essential in the contribution by Gerard van der Meijden, although the models used differ in several respects. In chapter 5 he studies an endogenous growth model with the following characteristics. A final good is produced using intermediates and energy. Energy is generated by means of oil and renewables, which are assumed to be imperfect substitutes. The production of intermediates requires labor only. Labor is also used for producing renewables, and for the creation of new varieties of the intermediate good. There are spillovers from the knowledge accumulated in the R&D sector to the productivity of oil and renewables. The availability of renewables leads to an increase of initial oil use (weak green paradox). However, also a so-called weak green “orthodox” arises: if an invention increases the degree of substitutability of oil and renewables, initial oil supply is decreased. One of the reasons is that with perfect substitutability there is an initial phase with only oil use, so that with high substitutability supply of energy will predominantly be oil, which is therefore spread more evenly over

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time, than with a low elasticity. Another reason is that a higher elasticity leads to more growth, which increases future demand for energy, and hence also for fossil fuel. Ralph Winter also looks at the development of a clean backstop, and allows for stock-dependent extraction cost of fossil fuel (chapter 6). He shows that prior to a technological breakthrough extraction is higher than in the case without the possibility of innovation. If innovation is possible but does not occur before oil extraction stops, then extraction is always higher, and abandoning oil takes place earlier. If the discovery is made before that moment, extraction is higher as well, but it stops earlier and more oil is left in situ. Winter allows for several feedbacks in the CO2 cycle and finds that it could well be that due to these feedbacks, higher temperatures will come about despite innovation. Hence he makes clear that the effect of innovation is ambiguous and that innovation policy should go hand in hand with appropriate carbon taxation. Ngo Van Long and Frank Stähler, chapter 7, investigate innovation in a backstop. They extend the usual partial equilibrium approach of the green paradox to a general equilibrium one, which allows them to find results that would not appear in a partial equilibrium setting. Long and Stähler construct a two-period model where extraction costs are stock dependent. In every period there is a fixed amount of capital available that can be used to extract oil as well as to produce a composite commodity, for which also oil and a renewable are inputs. The unit cost of producing the renewable is constant. A crucial finding is that the equilibrium interest rate depends on the unit cost of renewables. Different equilibria may arise depending on parameter values. In case of an equilibrium in which both oil and the renewable are used in both periods, a lower unit cost of the renewable will always lead to more extraction initially, whereas it may even lead to more total extraction over the two periods. The latter would not occur if the interest rate were constant. 1.3.3 Timing, Announcement Effects, and Time Consistency Green paradoxes deal with intertemporal effects of policy measures. Hence the timing of such measures matters. It is important to analyze the mere announcement of a future policy and to see whether it is time consistent. The next three chapters deal with such issues. In chapter 8 Florian Habermacher and Gebhard Kirchgässner study the concepts of the weak and the strong green paradox. If due to a

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policy measure the initial supply of fossil fuel increases, we speak of the weak green paradox. The strong green paradox occurs if future discounted damages increase as a result of this measure. Chapter 8 considers these paradoxes in the context of an anticipated introduction of a backstop technology. They vary the expected arrival time of the new technology. It is argued that based on a calibration of extraction cost functions, a strong green paradox will not occur. It takes time to design and implement climate change policy. Moreover it is sometimes deemed right to grant a period of “grace” to consumers and producers (e.g., Smulders et al. 2013). This implies that climate policies are pre-announced, which may have unanticipated effects on the extraction of fossil fuels. Darko Jus and Volker Meier investigate these effects in the context of monopolistic supply of oil (chapter 9). They find that announcing a carbon tax that is implemented only at some future date increases extraction of oil at each instant of time compared to the case of direct implementation of the carbon tax. They also address further delays in the implementation of the policy. 1.3.4 Empirics The literature on the green paradox arose out of concern about the neglect of the supply side of fossil fuels. Hence much of the recent literature focuses on the supply side. Corrado Di Maria, Ian Lange, and Edwin van der Werf, (chapter 10) point at the fact that in turn the demand side runs the danger of being neglected. This is particularly relevant in models with open economies, such as in Eichner and Pethig (2011). More stringent environmental policy in one country leads to fewer emissions in that country, but may lead to more overall emissions only if demand in the other non-abating country increases by more. Hence the size of the green paradox depends on the magnitudes of price elasticities of demand. The authors then continue with the identification of conditions for low responsiveness of demand to price changes. They discuss the fact that in the real world electricity demand by consumers is usually considered inelastic, demand for coal by electricity plants is inelastic as well, and they also mention other reasons for low responsiveness. Most studies on the green paradox are on a high level of aggregation as well as theoretical. In the real world climate change policies are designed by different jurisdictions, mostly in a noncoordinated way. Some jurisdictions don’t have any policy at all. Moreover whether

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green paradoxes arise and whether they are serious or not is oftentimes an empirical question. In the final chapter of this volume, chapter 11, Carolyn Fischer and Stephen Salant set out to fill these important gaps. They consider a world with several fossil fuel suppliers, with different constant marginal extraction costs. Production costs of a backstop are decreasing over time, at a rate that might be affected by policies. World demand for energy is exogenously increasing over time. After calibrating the model, Fischer and Salant investigate the intertemporal carbon leakage and associated green paradox phenomena under several alternative policy measures. They find that in some cases green paradoxes arise but may not be detrimental, whereas strongly negative effects arise for other policies. 1.4

Policy Implications?

The studies collected in this volume illustrate various aspects of the green paradox and implications for climate policy. The aim is not to provide policy makers with a “to do” (or rather “not to do”) list of policy measures but rather to qualify the implications of policy measures in a more detailed (and possibly more realistic) manner than in Sinn’s original publication. Let us first refresh the basic policy implications from Sinn (2008). The following statement summarizes the core of his findings: “If [policy] measures reduce the discounted value of the carbon price in the future more than in the present, the problem of global warming will even be exacerbated because resource owners will have an incentive to anticipate the price cuts by extracting the carbon earlier” (Sinn 2008, p. 388). Traditionally most policy measures that fulfill this criterion aim at reducing the demand for fossil fuels and can be “ranging from taxes on fossil fuel consumption to the development of alternative energy sources” (Sinn 2008, p. 288). Of course, Sinn acknowledges that the demand side exists that could theoretically succeed in reducing the speed of climate change (e.g., decreasing ad valorem taxes, constant unit taxes on carbon consumption, or a global emissions trading system). He, however, emphasizes also that these measures are, more often than not, difficult to implement. It suffices to consider a politician who announces plans to decrease CO2 taxes in the future. This plan seems hardly credibly in a world that increasingly worries about climate change. Beyond demand-oriented policies, Sinn considered

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supply-side policies as potential alternatives for the future (e.g., the subsidization of the stock in situ, safer property rights, or taxing capital assets held by fossil fuel owners). These policies aim directly at flattening resource supply. Within the relatively simple framework that Sinn employs, his policy conclusions are straightforward, but one must examine whether his results hold up under alternative setups. A look in the literature shows quickly that whether and to which extent second-best climate policies will be counterproductive in the sense of accelerating global warming damages depends on numerous factors. If extraction costs are stock dependent, for example, more carbon will be locked in the earth thereby reducing the cumulative amount of carbon emissions. If this cumulative beneficial effect outweighs the adverse short-run green paradox effects, then temperature and green damages will ultimately rise. But also the presence of dirty backstops, the substitutability between the various fossil fuels and renewables, the scarcity of fossil fuels, general equilibrium effects, and strategic considerations can affect the strengths and occurrences of green paradoxes. The contributions in this volume consider a number of these aspects of which many have hitherto been untouched upon in the literature but nevertheless prove highly relevant. Just consider the following examples: Hoel (chapter 2) shows in his contribution that also supply-side policies (very broadly defined as policies that permanently remove some of the carbon resources) might not always be good for the climate. While removing fossil fuel reserves in the absence of emissions from the fuel extraction and in presence of a clean backstop is shown to always contribute to climate protection, this does not necessarily hold anymore if emissions from extracting resources and heterogeneous extraction costs are taken into account. If, for example, low-cost reserves are removed, it cannot be ruled out that early as well as total emissions increase. Or consider the contribution by Long and Stähler (chapter 7) who take a closer look at technological progress that increases the productivity of a backstop technology over time. In Sinn’s paper this is one typical example in which a green paradox arises. Long and Stähler, however, argue that this result depends on the interaction between technological progress and the interest rate. More specifically, they show that a green paradox might not arise if the interest rate is unaffected by technological progress.

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In chapter 4 Long also focuses on the role of technological progress for green paradoxes, but in his analysis technological progress results in changes of the degree of substitutability between fossil and nonfossil fuels. Long finds that if technological progress increases the degree of substitutability over time, this may cause a shift of extraction toward the present and thus faster climate change. Yet he also shows that whether a green paradox arises depends crucially on the degree of substitutability today. While a moderate or high degree of substitutability today makes a green paradox more likely, no green paradox arises if today’s degree of substitutability is close to zero. The examples above only cover a small part of the variety of policy implications derived in this book. The aim of these examples was to provide a first impression on what the reader can expect from the remainder of the volume. The overarching conclusion to be drawn from all the contributions is, however, that climate policy has to be very carefully assessed in order to completely understand its implications— and to avoid all kinds of unintended side effects. Notes Rick van der Ploeg and Cees Withagen gratefully acknowledge financial support from FP7-IDEAS-ERC Grant No. 269788. 1. For an overview of the literature on the green paradox, see van der Werf and Di Maria (2011). 2. In their analysis of the green paradox, van der Ploeg and Withagen (2012b) treat coal accordingly as a backstop whose supply is not limited over time. 3. Following again IIASA (2012), the theoretical potential is defined as the total natural flows of solar, wind, hydro, geothermal energy, and grown biomass. The technical potential “reflects the possible degree of use determined by thermodynamic, geographical, technological, or social limitations without consideration of economic feasibility” (IIASA 2012, p. 123).

References Aichele, R., and G. Felbermayr. 2012. Kyoto and the carbon footprint of nations. Journal of Environmental Economics and Management 63: 336–54. Bauer, N., Hilaire, J., and Bertram, C. 2012. A numerical analysis of the green paradox: The role of fossil fuel supply and demand flexibility. Mimeo. Potsdam Institute for Climate Impact Research. BGR. 2009. Energierohstoffe 2009: Reserven, Ressourcen, Verfügbarkeit. Annual Report. Federal Institute for Geoscience and Natural Resources, Hannover, Germany: BGR.

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BGR. 2010. Reserves, Resources and Availability of Energy Resources. Annual Reports. Federal Institute for Geoscience and Natural Resources, Hannover, Germany: BGR. Eichner, T., and R. Pethig. 2011. Carbon leakage, the green paradox and perfect future markets. International Economic Review 52: 767–805. Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57: 79–102. Helm, D. 2012. The Carbon Crunch: How We’re Getting Climate Change Wrong—and How to Fix It. New Haven: Yale University Press. IEA. 2013. IEA energy prices and taxes statistics. Available at: http://www.oecd-ilibrary .org/energy/data/iea-energy-prices-and-taxes-statistics_eneprice-data-en. IIASA. 2012. Global energy assessment: Toward a sustainable future. Available at: http://www.iiasa.ac.at/ web/home/research/researchPrograms/Energy/Global_Energy _Assessment_FullReport.pdf. Michielsen, T. (2011). Brown backstops versus the green paradox. Discussion paper 2011–110. CentER, Tilburg. Nordhaus, W. 2011. Estimates of the social costs of carbon: Background results from the Rice-2011 model. Working paper 17540. NBER, Cambridge, MA. OECD. OECDiLibrary. Purchasing power parities for GDP and related indicators. Available at: http://www.oecd-ilibrary.org/economics/data/prices/purchasing-power-parities-for -gdp-and-related-indicators_data-00608-en?isPartOf=/content/datacollection/prices -data-en. Rezai, A., and F. van der Ploeg. 2013. Abandoning fossil fuel: How fast and how much? Research paper 123. OxCarre, University of Oxford. Sinn, H.-W. 2008. Public policies against global warming. International Tax and Public Finance 15: 360–94. Smulders, J., Y. Tsur, and A. Zemel. 2013. Announcing climate policy: can a green paradox arise without scarcity? Journal of Environmental Economics and Management 64 (3): 364–76. Stern, N. H. 2008. The Stern Review: The Economics of Climate Change. Oxford: Oxford University Press. Tol, S. J. 2009. The economic effects of climate change. Journal of Economic Perspectives 23: 29–51. van der Ploeg, F., and C. Withagen. 2012a. Is there really a green paradox? Journal of Environmental Economics and Management 64 (3): 342–63. van der Ploeg, F., and C. Withagen. 2012b. Too much coal, too little oil. Journal of Public Economics 96: 62–77. van der Werf, E., and C. Di Maria. 2011. Unintended detrimental effects of environmental policy: The green paradox and beyond. Working paper 3466. CESifo. World Bank. 2013. GDP deflator. Available at: http://data.worldbank.org/indicator/ NY.GDP.DEFL.ZS.

I

Extraction Costs

2

Supply-Side Climate Policy and the Green Paradox Michael Hoel

2.1

Introduction

Unless all countries cooperate on demand-side climate policies such as a carbon tax, it is well-known that the attempts of some countries to reduce carbon emissions through demand-side climate policies, such as a carbon tax or emission quotas, may be undermined by increased emission by other countries, so-called carbon leakage. An early discussion of this was given by Bohm (1993), who also discussed alternative policies that to a less extent were vulnerable to carbon leakage. One of the policies discussed by Bohm was supply-side policies, namely policies aimed at reducing the supply of fossil fuels instead of the use of fossil fuels. This idea was followed up by Hoel (1994), where a cooperating group of countries had to take into account the response of a group of noncooperating countries. It was shown that the optimal policy for the group of cooperating countries was a combination of a tax on their production of fossil fuels and a tax on their use of fossil fuels. The sum of the tax rates was shown to be equal to the Pigovian rate, while the mix between the two tax rates depended on the demand and supply elasticities. More recently Harstad (2012) has shown that a first-best outcome may be achieved if a group of cooperating countries can buy the reserves with the highest costs and keep them permanently out of production. None of the literature above focuses explicitly on the fact that fossil fuels are nonrenewable resources. An early contribution having this focus was given by Sinclair (1992). Sinclair pointed out that “the key decision of those lucky enough to own oil-wells is not so much how much to produce as when to extract it.” Since then, there has been a considerable number of contributions discussing optimal climate policy with explicit attention given to the nonrenewable character of carbon

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Michael Hoel

resources. These contributions either assume a constraint on the amount of carbon in the atmosphere (Chakravorty et al. 2006, 2008, 2012) or explicitly include a climate cost function in the analysis (Ulph and Ulph 1994; Withagen 1994; Tahvonen 1995; Farzin and Tahvonen 1996; Hoel and Kverndokk 1996). One of the insights from the literature is that the principles for setting an optimal carbon tax (or price of carbon quotas) are the same as when the limited availability of carbon resources is ignored: at any time the optimal price of carbon emissions should be equal to the present value of all future climate costs caused by present emissions, often called the social cost of carbon. During the last half decade, there has been a renewed interest in analyzing climate policy with explicit attention given to the nonrenewable character of carbon resources. Much of this later literature discusses the so-called green paradox, a term stemming from Sinn (2008a, b). Sinn argues that some designs of climate policy, intended to mitigate carbon emissions, might actually increase carbon emissions, at least in the short run. Sinn’s point is that if, for example, a carbon tax rises sufficiently rapidly, profit-maximizing resource owners will bring forward the extraction of their resources. Hence, in the absence of carbon capture and storage (CCS), carbon emissions increase.1 A thorough analysis of the effects of taxation on resource extraction was given by Long and Sinn (1985), but without explicitly discussing climate effects. More recently Hoel (2011, 2012) has studied the relationship between carbon taxes and carbon extraction, emphasizing the fact that governments, in practice, cannot commit to future tax rates. A rapidly increasing carbon tax is not the only possible cause of a green paradox. A declining price of a substitute, either because of increasing subsidies or technological improvement, can give the same effect (e.g., see Strand 2007; Gerlagh 2011; Grafton et al. 2012; van der Ploeg and Withagen 2012b). As mentioned above, Sinn used the term “green paradox” to describe a situation where policies intending to mitigate climate change actually increase near-term emissions. Gerlagh (2011) uses the term “weak green paradox” for such a phenomenon, and uses the term “strong green paradox” to describe a situation where policies intending to mitigate climate change increase total climate costs. This distinction is important, since total climate costs depend not only on near-term emissions, but also on all future emissions. One can therefore imagine policies that increase near-term emissions, but that nevertheless reduce future emissions so much that total climate costs decline.

Supply-Side Climate Policy and the Green Paradox

23

While the focus of the green paradox literature described above has been either on demand-side climate policies or on effects of technological changes, the present paper addresses the question of whether there also might be some kind of green paradox related to supply-side policies. To be more precise: Can a permanent removal of some fossil fuel resources increase early emissions, and perhaps even increase total climate costs? This issue is discussed in sections 2.4 and 2.5, after first presenting the model and discussing demand-side policies in the form of a carbon tax in sections 2.2 and 2.3. The short conclusion is that there will no green paradox if supply-side climate policies are aimed at removing high-cost carbon reserves. If instead low-cost reserves are removed, the possibility that both early and total emissions increase cannot be ruled out. Hence “wrong” supply-side climate policies may give a supply-side green paradox. 2.2

Extraction Costs and the Equilibrium Extraction Path

In the simplest Hotelling type models of resource extraction it is assumed that unit extraction costs are constant (often normalized to zero), and that the available amount of the resource is given exogenously. A more interesting and more realistic case is that of the unit cost of extraction, c(A), being increasing in accumulated extraction, where A is accumulated extraction. This is a specification frequently used in the resource literature (e.g., see Heal 1976; Hanson 1980). Notice that the first case mentioned is a special case of this more general description. Constant extraction costs and a fixed amount of resources implies that the cost function c(A) has an inverse-L shape, with the vertical part of the function being at the level A* corresponding to the exogenously fixed amount of the resource. The resource considered may either be interpreted as an aggregate of all fossil fuel resources, or it may be interpreted more narrowly as oil. For most of the present analysis it makes no difference what interpretation is used, but the issue will be discussed further in section 2.6. The resource is henceforth simply called carbon. Extraction of the carbon resource at time t is denoted x(t) and the consumer price is p(t). The demand function is D(p) and is assumed constant over time. It is also assumed that demand is zero if the price is sufficiently high. Formally, there is a choke price p such that D(p) = 0 for p ≥ p , and D(p) > 0 and D′(p) < 0 for p < p .2 There may also be a perfect substitute for the carbon resource that has a unit cost of

24

Michael Hoel

extraction equal to b < p and that is available at a flow rate of at least D(b) and without any limit on cumulative production. Although the existence of such a backstop technology for most of the analysis is of little importance, it will be assumed henceforth, as this makes some of the discussion slightly simpler. Producers are price takers and face an exogenous interest rate r. Producers choose the extraction path x(t) to solve the following optimization problem: ∞

max ∫ e − rt {p (t ) − c ( A (t ))} x (t ) dt

(2.1)

 (t ) = x ( t ) . subject to A

(2.2)

0

Together with the condition that extraction at any time must be equal to demand this gives the following equilibrium conditions (see the appendix for a formal analysis): p (t ) = r [ p (t ) − c ( A (t ))] ,

(2.3)

x(t) = D(p(t)),

(2.4)

p(t*) = b,

(2.5)

c(A(t*)) = b,

(2.6)

where t* is the date at which a switch from carbon extraction to backstop production occurs.3 The equilibrium is illustrated in figure 2.1, where the curve for c(A) and the equilibrium price path p = π(A) for simplicity are drawn linearly, although this will, of course, generally not be the case. (In all figures containing both the cost function c(A) and the price function π(A) the price function is the thinner of the two lines.) The steepness of the equilibrium price path π(A) is given by4

π ′ ( A) = r

p − c ( A) , D ( p)

(2.7)

implying that the path is steeper the larger is p and the lower is A. For any initial price p(0), the development of π(A) follows from (2.2) to (2.4). The equilibrium value of p(0) is determined so that p(t) and c(A(t)) reach b simultaneously. Had we started with a p(0)-value lower than the one drawn in figure 2.1, the curve π(A) would be less steep than the one drawn and would therefore cross the curve for c(A) at a value

Supply-Side Climate Policy and the Green Paradox

25

p

c(A) b

p(0)

0

A*

A

Figure 2.1 Equilibrium price path

of p below b and then start to decline, which violates our equilibrium conditions. Similarly, starting with a p(0)-value above the one drawn in figure 2.1, the curve π(A) would be steeper than the one drawn and would hence reach b at a value of A giving c(A) < b, which also violates our equilibrium conditions. Consider a positive shift in the extraction cost function, namely a change in the cost function from c(A) to c(A) + ε(A), where ε(A) ≥ 0 for all A with a strict inequality holding for some range of A between 0 and A* defined by c(A*) = b. It follows from (2.7) that such a shift in the cost function must shift the whole equilibrium price path π(A) upward. The reason for this is that a hypothetical price path that is anywhere equal to or below the original π(A) would be less steep than the original path for the A-values having ε(A) > 0. This hypothetical price path would hence not reach b, and could therefore not be an equilibrium path. This important result may be formulated as a proposition: Proposition 1 A positive shift of the cost function c(A) for any A < A*, where A* is defined by c(A*) = b, will shift the whole price path p = π(A) upward, and hence delay extraction. Two examples of shifts are given in figures 2.2 and 2.3. In figure 2.2 the cost shift occurs only for low values of A, implying that A* is unaffected

26

Michael Hoel

by the cost shift. The whole price path shifts upward as illustrated in figure 2.2a. The time path for A(t) is illustrated in figure 2.2b. Both before and after the shift the curve for A(t) becomes flatter over time, since Ȧ(t) = D(p(t)) declines as p(t) increases. Since the new price path has a higher value of p for any given value of A after the cost increase than before, the after-shift (dotted) curve in the (t, A) space is flatter (lower Ȧ(t) = D(p(t))) for any given A than the before-shift (fully drawn) curve. At any date prior to the switch from the resource to the backstop cumulative extraction is hence lowered as a consequence of the increase in extraction costs. In figure 2.3 the cost shift occurs only for high values of A, implying that total extraction is reduced to A** as a consequence of the cost shift. As in the previous case, the whole price path shifts upward as illustrated in figure 2.3a. As in the previous case, the curve for A(t) becomes flatter over time; see figure 2.3b. At any date prior to the switch from the resource to the backstop cumulative extraction is hence also in this case lowered as a consequence of the increase in extraction costs. 2.3

Effects of a Carbon Tax

As mentioned at the start of this chapter, the effects of a carbon tax when carbon emissions are explicitly modeled as coming from the use of a scarce carbon resource has been extensively studied in the literature. Some consequences of a carbon tax follow immediately from proposition 1: a constant carbon tax (per unit of carbon emissions) of size θ simply means that the unit cost function will be changed from c(A) to c(A) + θ. From proposition 1 it follows that the whole equilibrium price path will be shifted upward. Moreover total extraction will be reduced from A* ≡ A*(0) to A*(θ), where A*(θ) is defined by c(A*(θ)) + θ = b. According to Allen et al. (2009), the peak temperature increase due to greenhouse gas emissions is approximately independent of the timing of emissions. In the framework of the present model, peak temperature increase thus depends only on A*(θ). However, we would expect this peak temperature increase to occur earlier the more of the emissions occur at an early stage. It also seems reasonable to expect climate costs to be higher the more rapidly the temperature increases, for a given peak temperature increase. Hence it seems reasonable to assume that climate costs are increasing not only in A*(θ) but also in the speed of extraction.

Supply-Side Climate Policy and the Green Paradox

(a)

27

p

c(A) b

p(0)

A*

(b)

A

A

A*

t*

t**

t

Figure 2.2 (a) Shift in the low end of the cost function; (b) change in the time path of A for a shift in the low end of the cost function

28

(a)

Michael Hoel

p

c(A) b

A**

A*

A

(b) A

A*

A**

t**

t*

t

Figure 2.3 (a) Shift in the high end of the cost function; (b) change in the time path of A(t) for a shift in the high end of the cost function

Supply-Side Climate Policy and the Green Paradox

29

For an inverse-L cost function (i.e., c(A) constant for A < A* and becoming vertical at A*), total extraction is independent of a carbon tax (unless it is so high that the resource rent is driven to zero; see Hoel 2012). A main point in the green paradox literature, and emphasized by Sinn (2008a, b), is that if the carbon tax is expected to rise sufficiently rapidly, this may speed up resource extraction, and hence be bad for the climate. Formally (and independent of the cost function) a carbon tax implies that the producers’ optimization problem (2.1) gets an additional term −T, where T is the present value of carbon taxes paid: ∞

T = ∫ e − rtθ (t ) x (t ) dt. 0

If the present value of the tax rate is constant, this can be rewritten as T = θ(0)A*. In the inverse-L case A* is exogenous, implying that producers cannot influence T. A carbon tax rate that is rising at the rate of interest hence has no effect on the extraction path in this case. However, if the present value of the carbon tax rate is rising over time, producers can reduce T by speeding up extraction. In this case the tax is hence detrimental to the climate compared with the case of no tax.5 For the general case of an increasing extraction cost function, a positive carbon tax, whatever way it evolves over time, will reduce total extraction. Consider first a carbon tax for which the present value of the tax rate is constant. For this case producers can only affect T by the choice of how much to totally extract. Let the level of the tax rate be such that the optimal response to the carbon tax is to reduce A from A* to A**. The effect of the carbon tax on the extraction path is therefore identical to imposing an exogenous upper limit A** but having no carbon tax. From proposition 1 it follows that the carbon tax will shift the whole equilibrium price path π(A) upward. Hence a carbon tax rate increasing at the rate of interest will reduce both total extraction and postpone extraction in a similar manner, as shown in figure 2.3b. Such a tax is unambiguously good for the climate; there is no green paradox. Finally, consider a carbon tax that is rising more rapidly than the rate of interest. Also in this case total extraction will decline, say to A**. However, in this case producers have an incentive to speed up extraction compared with a case with no tax and an exogenous upper limit A** on cumulative extraction. If the carbon tax is rising fast enough, this negative effect on p(0) may be stronger than the positive effect on

30

Michael Hoel

p(0) from total extraction being reduced. In the terminology of Gerlagh (2011), there will in this case be a “weak green paradox,” meaning that initial extraction increases as a response to the carbon tax. A “strong green paradox” (Gerlagh 2011), meaning that the total discounted climate costs increase cannot be ruled out but is less likely, the stronger is the effect on total extraction compared with the effect on initial extraction (see Hoel 2012 for a further discussion of these issues). 2.4

Supply-Side Climate Policies

For the reasons given at the start of the chapter, the rest of this chapter studies the effects of supply-side climate policies. By supply-side policies I mean policies that permanently remove some of the carbon resources. To focus on these policies, carbon taxes and other demandside policies are ignored. Removing some of the carbon resources is equivalent to a leftward shift in the cost function c(S). The simplest case is the inverse-L case. Removing some of the resources simply means shifting the vertical part A* leftward. Such a reduction in the available resource supply shifts the whole price path upward. Hence both total and initial resource extraction is reduced, which is unambiguously beneficial to the climate. Also for the more general case removing some of the resources implies a leftward shift in the cost function. This is illustrated in figure 2.4a for the case of a reduction in the resources with the highest costs in the amount A** − A*, and in figure 2.4b for a reduction in the resources with the lowest costs, also here in the amount A** − A*. It follows from proposition 1 that whichever types of resources are removed, the equilibrium price path π(A) must increase. Moreover total extraction must decline. The climate effect of such a supply-side policy is hence unambiguously good. 2.5

Emissions from the Extraction Process

While most of the emissions from fossil fuels come from the end use of the fuels, there are also considerable emissions from the extraction process. Resource extraction requires energy, and a large part of this energy typically comes from the use of fossil fuels. This is the reason why some types of unconventional fuel (e.g., oil sand) often are assumed to be particularly bad for the climate. The emissions from the final oil product, such as gasoline or diesel for transportation, has the same

Supply-Side Climate Policy and the Green Paradox

31

(a) c c(A)

b

A**

A

A*

(b) c

c(A) b

A**

A*

A

Figure 2.4 (a) Removal of high-cost resources; (b) removal of low-cost resources

32

Michael Hoel

amount of carbon emissions as other fuel has. However, the process of extracting and refining oil sands gives much higher carbon emissions than the process of extracting and refining conventional oil. Emissions from the extraction and refining process are modeled as follows. Non-energy extraction costs are given by c ( A ), assumed increasing in A. These costs represent the use of all inputs except the inputs of fossil fuels. In addition to these inputs we need the input of γ (A) units of fossil fuels to extract 1 unit of fossil fuels, and it is assumed that γ ′(A) > 0. In other words, as we move from lower cost to higher cost resources, both non-energy and energy costs are assumed to rise.6 Notice that if there exists a value Ā defined by γ(Ā) = 1 this will be an upper limit to what the endogenously determined total extraction A* can be, since extracting beyond Ā would require more energy than produced. With this modification of the cost assumptions, the producers’ optimization problem is changed to ∞

max ∫ e − rt {p (t )[1 − γ ( A )] − c ( A )} x (t ) dt

(2.8)

0

subject to (2.2). This optimization problem is solved in the appendix. Defining the function c(A) by c ( A) =

c ( A ) , 1 − γ ( A)

we see that the equilibrium conditions (2.3), (2.5), and (2.6) remain valid, while (2.4) is changed to x (t ) =

D ( p (t )) . 1 − γ ( A ( t ))

(2.9)

As before, the equilibrium price will be increasing over time. Regarding emissions, however, it is no longer obvious that they will decline over time. As before, the rising carbon price implies that emissions from final use will be declining over time. However, since γ(A(t)) is increasing over time, this implies that emissions in the extraction process per unit of final use will be increasing. Formally, the ambiguity in the time path of x(t) follows from (2.9), since D(p(t))will be declining while γ(A(t))will be increasing. To focus on the latter aspect, it is assumed in the rest of this and the next section that demand is completely inelastic.7

Supply-Side Climate Policy and the Green Paradox

33

When demand D is constant, it follows from (2.9) that emissions will be increasing until the resource price reaches b at the date t* when the switch from resource extraction to backstop production occurs. This is illustrated by the fully drawn curves in figures 2.5a and 2.5b. To understand the effects of removing some of the resource supply it is useful to consider the two extremes of removing the highest cost resources and removing the lowest cost resources, as illustrated in figures 2.4a and 2.4b, respectively. In both cases total extraction is reduced from A* to A**. When the highest cost resources are removed, the new emission path will be identical to the old one except that emissions will drop to zero at t** instead of t* in figure 2.5a. The total emission reduction is denoted R in this figure, and is unambiguously beneficial for the climate. When the lowest cost resources are removed the cost function will be shifted leftward as illustrated in figure 2.4b. A similar shift will occur for γ(A). This means that the initial emission path will shift upward as illustrated in figure 2.5b until the new (and lower) switch date t**. Emissions will hence increase by an amount I prior to t** in figure 2.5b, but will decline by an amount R after t**. R is larger than I, since total extraction is reduced from A* to A**. But to the extent that early emissions are considered worse for the climate than later emissions, it is not obvious that this type of carbon supply reduction is good from a climate perspective. To summarize, on the one hand, removing the highest cost resources has an unambiguously beneficial effect on the climate. On the other hand, the effect on the climate of removing lower cost resources is ambiguous from a theoretical point of view. 2.6

A Dirty Backstop

So far the backstop has been assumed clean (in the sense that there are no carbon emissions associated with production of the backstop). This is perhaps a natural assumption if the carbon resource is interpreted as all fossil fuels. However, if the carbon resource in the preceding analysis is interpreted more narrowly as oil of various types the assumption of a clean backstop is not obvious. The backstop to oil could, for instance, be biofuel. Although biofuels sometimes are considered “climate neutral,” there is a considerable literature arguing that the production of biofuel will have adverse climate effects.8 Alternatively, one could think of synthetic oil made from coal as a backstop to

34

Michael Hoel

(a) x(t)

R

t**

t*

t

(b) x(t)

I R

t**

t*

t

Figure 2.5 (a) Change in emissions from oil when high-cost resources are removed; (b) change in emissions from oil when low-cost resources are removed

Supply-Side Climate Policy and the Green Paradox

35

oil. In this case the backstop is clearly not clean in the sense described above.9 To see the consequences of the dirty backstop for oil, assume that synthetic oil made from coal is a backstop technology for oil. Assume that coal is available in an unlimited amount at a cost b per unit of carbon, and that each unit of coal extracted requires λb for extraction and transformation to synthetic oil. In this case the cost of synthetic oil per unit end use is b = b (1 − λ b ). The switch from regular oil to synthetic oil from coal will occur when accumulated oil extraction has reached A*, where c(A*) = b, that is, c ( A* ) (1 − λ ( A* )) = b (1 − λ b ) . If, on the one hand, λb > λ(A*), emissions will increase as we switch from regular to synthetic oil. Any removal of the supply of regular oil will in this case advance the date of this transition to increased carbon emissions. Hence in this case any removal of the supply of regular oil will be bad for the climate. If, on the other hand, λb < λ(A*), emissions will decline as we switch from regular to synthetic fuel. If we now remove high-cost reserves, the time point of the switch from regular to synthetic oil will be moved forward from t* to t**, and total emissions will unambiguously decline as illustrated by the area R in figure 2.6a (In this figure E(t) stands for emissions from oil and the dirty backstop). The situation is not so clear if low-cost oil reserves are removed. This case is illustrated in figure 2.6b: As in the case of a clean backstop, we initially get an increase in emissions (I in figure 2.6b) but later a reduction (R in figure 2.6b). With a clean backstop R was higher than I, so total emissions declined. This is, however, not obvious when the backstop is dirty (although it is “less dirty” that the high-cost regular oil). When the backstop is dirty, we may get higher total emissions as a consequence of removing low-cost reserves, and early emissions will certainly increase (at least as long as the price effect on demand is sufficiently low). Removing low-cost oil reserves may therefore be bad for the climate when the backstop is dirty. If the backstop for regular oil is biofuels instead of coal, the analysis is much the same. If the sum of all greenhouse gas emissions from biofuel production are sufficiently large, emissions will increase as we switch from oil to biofuel. It is probably more realistic that emissions from biofuel production are lower than the total emissions from highcost oil reserves. If so, this gives us the cases illustrated by figures 2.6a and 2.6b.

36

Michael Hoel

(a) E(t)

R

t**

(b)

t*

t

E(t)

R I

t** t* t Figure 2.6 (a) Change in emissions from oil and the dirty backstop when high-cost resources are removed; (b) change in emissions from oil and the dirty backstop when low-cost resources are removed

Supply-Side Climate Policy and the Green Paradox

2.7

37

Conclusions

Ignoring emissions from the extraction of fossil fuels and assuming a clean backstop, removing some of the fossil fuel reserves is unambiguously good for the climate. This conclusion is no longer obvious if emissions from extracting the fossil fuels are higher for high-cost reserves than for low-cost reserves. In this case removing low-cost reserves may increase early emissions, although total emissions go down. Removing high-cost reserves is unambiguously good for the climate also in this case, since both total and early emissions decline (including the price effect on early emissions). If the backstop is dirty in the sense that there are greenhouse gas emissions related to the production of the backstop, the climate may be adversely affected even if high-cost reserves of carbon are removed. However, this can only occur if the emissions from the backstop are higher than from the high-cost reserves of carbon resources. If the emissions from the backstop are lower than from the high-cost reserves of carbon resources, total and early emissions go down when high-cost reserves are removed. However, if instead low-cost reserves are removed, the possibility that both early and total emissions increase cannot be ruled out. So the short conclusion is much in line with that of Harstad (2012): if supply-side climate policies are to be used, these policies should be aimed at the high-cost carbon reserves. 2.8

Appendix: Derivation of the Equilibrium

As there are no externalities other than the climate externality, deriving the competitive equilibrium is equivalent to deriving the social optimum. Let the increasing and concave function U(x) be the benefit of using the resource, with U′(x) = p(x) = D−l(x) and U′(0) = b. The equilibrium described by (2.2) through (2.6) in section 2.2 is a special case (with γ = 0) of the equilibrium described in section 2.5. Ignoring the climate externality, the optimization of a social planner for the problem described in section 2.5 is ∞

max ∫ e − rt {U ([1 − γ ( A (t ))] x (t )) − c ( A ) x (t )} dt 0

subject to  (t ) = x (t ) , A (0 ) = 0 , x (t ) ≥ 0 for all t. A

38

Michael Hoel

By defining c ( A ) 1 − γ ( A) and y = [1 − γ(A(t))]x, this problem may be rewritten as c ( A) =

∞

max ∫ e − rt {U ( y (t )) − c ( A (t )) y (t )} dt 0

subject to  (t ) = A

y (t ) , A (0 ) = 0 , y (t ) ≥ 0 for all t 1 − γ ( A ( t ))

(2.10)

The current value Hamiltonian, written so that the shadow price λ is nonnegative, is H ( y , A, λ ) = U ( y ) − c ( A) y − λ

y 1 − γ ( A) ,

and the optimum conditions are (omitting time references where this cannot cause any misunderstanding) ∂H λ (t ) ≤0 = U ′ − c ( A) − ∂y 1 − γ ( A)

with = for y (t ) > 0 ,

∂H λ (t ) = rλ (t ) + , ∂A

(2.11) (2.12)

giving

λ yγ ′ λ = rλ − yc′ − . (1 − γ )2

(2.13)

The tranversality condition for this problem is lim e − rt λ (t ) A (t ) = 0. t→∞

(2.14)

Whenever extraction is positive, it follows from (2.11) and U′ = p that p (t ) = c ( A ) +

λ (t ) . 1 − γ ( A)

Using (2.10) it follows that p (t ) = c ′ ( A )

y λ yγ ′ λ (t ) . + + 1 − γ 1 − γ (1 − γ )3

(2.15)

Supply-Side Climate Policy and the Green Paradox

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Inserting (2.15) and (2.13) into this expression gives us the equilibrium condition for the price path: p (t ) = r [ p (t ) − c ( A (t ))]. From (2.11) it is clear that if extraction stops while λ ≠ 0, the transversality condition will be violated. Similarly, if λ becomes negative, it follows from (2.11) that the transversality condition will be violated. The equilibrium time path of λ(t) therefore must reach zero when extraction stops. In other words, p(t*) = c(A(t*)) at the date t* when extraction stops, and at this date we must have U′(0) = p(t*), meaning p(t*) = b. Notes I am grateful to participants at seminars in Oslo, Venice, and Munich for useful comments. While carrying out this research I have been associated with CREE—Oslo Center for Research on Environmentally friendly Energy. CREE is supported by the Research Council of Norway. 1. Throughout this chapter CCS is ignored; see, for example, Hoel and Jensen (2012) for a discussion of climate policy when there is a possibility of CCS and when the carbon resource scarcity is explicitly taken into consideration. 2. This is a purely technical assumption. If it instead had been assumed that D(p) > 0 for all p but approached zero as p → ∞, it would nevertheless be true that for some high price p (e.g., a million dollars per barrel of oil) demand would be so small that it would be of no practical interest (e.g., 1 barrel of oil per year). 3. If there were no backstop with b < p the conditions p (t ) = p and c ( A ) = p would only be reached asymptotically. 4. This follows from π(A) = p(t(A)), where t(A) is the inverse of A(t). Using (2.2) and (2.3) for A′(t) and p′(t) gives (2.7). 5. Since governments in practice cannot commit to the size of the carbon tax rate for more than a few years, the comparison with “zero tax forever” might not be particularly relevant; see Hoel (2012) for a further discussion. 6. The terms non-energy costs and energy costs should more accurately be called nonfossil fuel costs and fossil fuel costs. If some of the energy is covered by noncarbon energy, this should be included in c ( A ). Notice that it is implicitly assumed that the ratio between the use of carbon energy and other inputs is independent of the price of carbon energy. This assumption of a zero elasticity of substitution is clearly a simplification, but is not important for the results of the analysis. 7. This, of course, means that there must exist a backstop with unit cost b. 8. Adverse climate effects may be caused by fossil fuel use for harvesting, transportation and production, by N2O emissions from fertilizer use and the crop itself (Crutzen et al. 2008), and by direct and indirect land-use changes (Fargione et al. 2008; Searchinger et al. 2008).

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9. See van der Ploeg and Withagen (2012a) for a discussion of optimal carbon taxes for this case.

References Allen, M. R., D. J. Frame, C. Huntingford, C. D. Jones, J. A. Lowe, M. Mein-shausen, and N. Meinshausen. 2009. Warming caused by cumulative carbon emissions towards the trillionth tonne. Nature 458 (7242): 1163–66. Bohm, P. 1993. Incomplete international cooperation to reduce CO2 emissions: Alternative policies. Journal of Environmental Economics and Management 24: 258–71. Chakravorty, U., A. Leach, and M. Moreaux. 2012. Cycles in non-renewable resource prices with pollution and learning-by-doing. Journal of Economic Dynamics and Control 36: 1448–61. Chakravorty, U., B. Magne, and M. Moreaux. 2006. A Hotelling model with a ceiling on the stock of pollution. Journal of Economic Dynamics & Control 30 (12): 2875–2904. Chakravorty, U., M. Moreaux, and M. Tidball. 2008. Ordering the extraction of polluting nonrenewable resources. American Economic Review 98 (3): 1128–44. Crutzen, P., A. Mosier, K. A. Smith, and W. Winiwarter. 2008. N2O release from agrobiofuel production negates global warming reduction by replacing fossil fuels. Atmospheric Chemistry and Physics 8: 389–95. Fargione, J., J. Hill, D. Tilman, S. Polasky, and P. Hawthorne. 2008. Land clearing and the biofuel carbon dept. Science 319: 1235–38. Farzin, Y., and O. Tahvonen. 1996. Global carbon cycle and the optimal path of a carbon tax. Oxford Economic Papers, New Series 48: 515–36. Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57 (1): 79–102. Grafton, R. Q., T. Kompas, and N. V. Long. 2012. Substitution between biofuels and fossil fuels: Is there a green paradox? Journal of Environmental Economics and Management 64: 328–41. Hanson, D. A. 1980. Increasing extraction costs and resource prices: Some further results. Bell Journal of Economics 11 (1): 335–42. Harstad, B. 2012. Buy coal! A case for supply-side environmental policy. Journal of Political Economy 120 (1): 77–115. Heal, G. 1976. The relationship between price and extraction cost for a resource with a backstop technology. Bell Journal of Economics 7 (2): 371–78. Hoel, M. 1994. Efficient climate policy in the presence of free riders. Journal of Environmental Economics and Management 27 (3): 259–274. Hoel, M. 2011. The green paradox and greenhouse gas reducing investments. International Review of Environmental and Resource Economics 5 (4): 353–79. Hoel, M. 2012. Carbon taxes and the green paradox. In R. Hahn, R. and U. Ulph, eds., Climate Change and Common Sense: Essays in Honour of Tom Schelling. New York: Oxford University Press.

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Hoel, M., and S. Jensen. 2012. Cutting costs of catching carbon—Intertemporal effects under imperfect climate policy. Resource and Energy Economics 34 (4): 680–95. Hoel, M., and S. Kverndokk. 1996. Depletion of fossil fuels and the impacts of global warming. Resource and Energy Economics 18 (2): 115–36. Long, N., and H.-W. Sinn. 1985. Surprise price shifts, tax changes and the supply behavior of resource extracting firms. Australian Economic Papers 24 (45): 278–89. Searchinger, T., R. Heimlich, R. Houghton, F. Dong, A. Elobeid, J. Fabiosa, S. Tokgoz, D. Hayes, and T.-H. Yu. 2008. Use of U.S. croplands for biofuels increases greenhouse gases through emissions from land-use change. Science 319: 1238–41. Sinclair, P. 1992. High does nothing and rising and worse: Carbon taxes should be kept declining to cut harmful emissions. Manchester School of Economic and Social Studies 60: 41–52. Sinn, H. 2008a. Das grüne Paradoxon: Plädoyer für eine illusionsfreie Klimapolitik. Berlin: Econ Verlag. Sinn, H. 2008b. Public policies against global warming: A supply side approach. International Tax and Public Finance 15: 360–94. Strand, J. 2007. Technology treaties and fossil fuels extraction. Energy Journal (Cambridge, MA) 28 (4): 129–42. Tahvonen, O. 1995. Dynamics of pollution control when damage is sensitive to the rate of pollution accumulation. Environmental and Resource Economics 5: 9–27. Ulph, A., and D. Ulph. 1994. The optimal time path of a carbon tax. Oxford Economic Papers 46: 857–68. van der Ploeg, F., and C. Withagen. 2012a. Too much coal, too little oil. Journal of Public Economics 96 (1): 62–77. van der Ploeg, F., and C. A. Withagen. 2012b. Is there really a green paradox? Journal of Environmental Economics and Management 64: 342–63. Withagen, C. 1994. Pollution and exhaustibility of fossil fuels. Resource and Energy Economics 16: 235–42.

3

The Green Paradox as a Supply Phenomenon Julien Daubanes and Pierre Lasserre

3.1

Introduction

The so-called green paradox refers to the fact that future, anticipated policies aiming at reducing the demand for an extracted exhaustible resource increase the present or near future equilibrium rate of extraction of that resource. Hans-Werner Sinn (2008) coined the expression, implying “that good intentions do not always breed good deeds” (p. 380). More generally, the credible threat of a “gradual greening of economic policies” (Sinn 2008, p. 360) causes markets to consume stocks of nonrenewable resources more rapidly. This phenomenon has received particular attention in the context of climate change economics and policies. Fossil fuels are responsible for the bulk of greenhouse gas emissions, a pollution that has been labeled “the ultimate commons problem of the twenty-first century” (Stavins 2011). Environmental policies should attempt to slow fossil fuel exploitation. In this context a policy entailing a green paradox would instead accelerate current and near future consumption. Of major practical concern is the possibly undesired effect of a lag in green policies’ implementation1 or of the expectation that green policies will be introduced gradually to be more and more stringent over time. As a policy-induced phenomenon, the green paradox has first been formulated in a market equilibrium setting where resource prices are determined by the combination of supply and demand. The resource literature has studied market equilibria extensively; not surprisingly, the green paradox connects well with well-known results on the incidence of taxes on the extraction path. Since Hotelling’s (1931) seminal contribution on exhaustible resources, such commodities are often thought of as fixed, finite initial

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endowments to be allocated over time. By the Hotelling rule, any profit-maximizing extraction path should equalize the present-value marginal spot profits of extracting at all dates. In this context there exist neutral tax paths that reduce present-value marginal profits in the same proportion at all dates, leaving the intertemporal arbitrage of resource producers unaffected (e.g., Dasgupta et al. 1981).2 Tax paths that are rising faster over time deteriorate the marginal profitability of extracting at distant dates relatively to early dates, thus leading producers to extract less of the stock in the future and to shift production to earlier dates. From Long’s (1975) study, such policies can be interpreted as credible threats of future expropriation. A green paradox arises when a policy is expected to distort relative marginal profitabilities in a way that penalizes future extraction. Sinn argued that such a distortion not only arises from tax policies but may also result from subsidies to substitutes for the resource or from policyinduced improvements in current efficiency.3 In general, a green paradox phenomenon can be described as a substitution between resource units extracted at different dates. In this respect Sinn’s (2008) metaphor of a closed “pneumatic system of various pipes connecting various pistons” (p. 378) is illuminating: “If only one piston is pressed down, the others go up.” In other words, when total cumulative supply is perfectly inelastic, demand-reducing policies only shift quantities away to other dates. As Sinn (2008) suggested, the green paradox arises because of the peculiarities of the supply side in nonrenewable resource markets. More precisely, the pure substitution of resource units supplied at different dates resulting from demand-reducing policies is caused by the perfect inelasticity of the reserves to be exploited, as in Hotelling frameworks. The finiteness of reserves makes nonrenewable resource supply different from the ordinary supply of producible goods. Yet no conventional treatment of nonrenewable resource supply, systematically deriving supplied quantities as functions of exogenous resource prices, has been provided until very recently (Daubanes and Lasserre 2012). Some exceptions are Gray’s (1914) pioneering piece and Burness’s (1976) formal inquiry into the effect on the extraction path of changing the exogenous price, assumed to be constant throughout the extraction period. More recently Sweeney (1993) attempted to reconcile the economics of nonrenewable resources with conventional supply theory,4 by deriving static resource supply as a function of the contemporary producer price, net of the intertemporal rent, that is, as function of the

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marginal extraction cost. In contrast, an abundant literature has investigated the role of the discount rate and the various forms taken by Hotelling’s rule. Moreover the resource literature has often focused on equilibrium quantities, leaving aside the question of characterizing supply independently. Supply theory, like demand theory, is usually considered as a fundamental preliminary to the study of market equilibria. In this chapter we illustrate how a very synthetic nonrenewable resource supply theory allows the study of policy-induced changes like the green paradox as standard resource supply phenomena. We treat the supply of a nonrenewable commodity at various dates as the supply of different goods. We assume the sequence of producer prices to be given and we study the effect of changing any component of that price sequence on resource supply at, or before, that date. When total cumulative extraction is taken as given as in Hotelling models, an exogenous price change occurring at any date modifies the relative marginal profit from extracting at that date relative to other dates and thus entails a pure intertemporal substitution effect, which is reminiscent of the green paradox as initially formulated by Sinn (2008). In the recent literature, authors have departed from the (tight) pneumatic metaphor by arguing that green policies may affect the cumulative quantity of the resource that is to be extracted ultimately. Indeed a change in the resource price path faced by producers may also affect ultimately exploited reserves, both because some existing reserves may become economic or cease to be economic as a result of the price change, and because exploration and discoveries as well as reserve development are affected by the price change. Gerlagh (2011), Hoel (2012), van der Ploeg and Withagen (2012, 2014), and Grafton et al. (2012) use Gordon’s (1967) popular approach where the cost of extracting the resource increases with cumulative extraction. Fischer and Salant (2012) adopt a Herfindahl (1967) type model where several ordered deposits are to be exploited. In all cases some part of the resource may be left unexploited. The total quantity extracted is determined endogenously: marginal extraction costs are constant at each date in the sense that they are independent of current extraction, but depend on past cumulative extraction. Furthermore, a substitute can be produced at some cost. Extraction stops at the date when cumulative extraction is such that the unit cost of extraction overtakes the unit cost of the substitute. Green policies aim at reducing the cost of the substitute, thus reducing the equilibrium total amount of resource extracted

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and increasing the part of the stock that is left unexploited. However, none of these extensions contradicts the green paradox. Leaving some part of the resource unexploited can only be rational if no costs are experienced to develop the resource, or if an unexpected change in the economics of extraction, such as a drop in the price, makes extraction uneconomic. As a matter of fact, reserves to be used for extraction are produced by exploration and development efforts. This is an important aspect of resource supply. We assume that the stock of reserves is produced prior to extraction via exploration and development and is then completely exhausted during the exploitation period, as in Gaudet and Lasserre (1988) and Fischer and Laxminarayan (2005). Exploration and development are sensitive to the rent that accrues to the extractor during the exploitation of the resource and this rent is affected by supply prices. It turns out that the amount of set reserves to be exploited depends on resource prices. This determines what we call the stock effect of a price change, which comes in addition to the intertemporal substitution effect. Using a synthetic two-period model of nonrenewable resource supply, in section 3.2 we compute restricted supply functions, namely supply functions that arise when the stock of reserves is held constant. This characterizes the pure substitution effect of a price change, the mechanism at the heart of the green paradox phenomenon. In section 3.3 we consider reserves to be endogenous, and compute unrestricted supply functions. Besides the substitution effect, the analysis identifies a stock effect of opposite direction. Under standard conditions, the stock effect never more than offsets the substitution effect. In section 3.4 we discuss the peculiarities of resource supply compared with conventional supply and argue that intertemporal policy-induced changes like the green paradox are tantamount to price effects on resource supply. 3.2

A Synthetic Hotelling Resource Supply Model

For many goods the classical theory of supply is static. The nonrenewability of a natural resource makes the intertemporal dimension of extraction decisions crucial. Supplying a resource at several dates is not unlike supplying several goods; the time space in resource supply plays a similar role as the discrete commodity space in classical supply. For our purpose, it is sufficient to assume that there are two periods of

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time, t = 1 and t = 2. Let date 1 represent the present or near future and date 2 represent the distant future. At each date t = 1, 2, assume that a producer of a nonrenewable resource supplies a nonnegative quantity xt of this resource. Following Hotelling, consider that the producer owns a finite initial amount X of exploitable resource, so that x1 + x2 ≤ X.

(3.1)

In this section let us treat these initial reserves as exogenous. The producer faces given prices p1 ≥ 0 and p2 ≥ 0. The cost of extracting a quantity xt at date t = 1, 2 is given by the strictly convex cost function C(xt). Spot exploitation profit at date t = 1, 2 is written ptxt − C(xt). Convex extraction costs imply that marginal spot exploitation profits pt − C′(xt) decrease with contemporaneous extracted quantities xt. The first-order condition characterizing the optimum extraction of a set amount of reserves across several dates is the well-known Hotelling rule: it states that marginal extraction profits should be constant in present value so that each unit of the exploited reserves fetches the same scarcity rent. We will assume that future profits are discounted at the exogenous rate r, and so write Hotelling’s rule as p1 − C ′ ( x1 ) =

p2 − C ′ ( x 2 ) . 1+ r

(3.2)

Further we can assume, as will turn out to be true later on, that the stock of exploitable reserves is always exhausted over the exploitation period. For a given stock of exploitable reserves, the sum of spot supplies x1 and x2 exactly amounts to the total exploitable stock X: the exhaustibility constraint (3.1) holds with equality. Clearly, the Hotelling rule (3.2) and the binding exhaustibility constraint (3.1) jointly determine the producer ’s supplies x1* and x2* , as functions of its price parameters p1 and p2. At this stage these are conventional supply functions. Supplies also depend on the given amount of reserves X. For each t = 1, 2 we can write xt* = x t ( p1 , p2 ; X ) .

(3.3)

How does a price change affect resource supplies in a Hotelling model with fixed reserves? The law of supply is known to always hold in such contexts. Hence supply xt* always increases as a result of a rise in the

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contemporaneous price pt. This positive effect is the own-price effect. As concerns the cross-price effect of, say, p2 on x1* , there is no such general result as the law of supply. In the case of a Hotelling resource with given reserves however, the law of supply determines this crossprice effect. Indeed a rise in p2 causes x2* to increase and thus x1* to decrease by the binding constraint (3.1). Present or near future supply is always negatively affected by the future price. The green paradox was first formulated in a Hotelling model with fixed reserves (Sinn 2008), as in the context of this section. Although the negative cross-price effect shown above only concerns the supply side, it is clearly relevant to the green paradox. The price change, which we have assumed to be exogenous, can just as well be caused by a policy-induced drop in future demand; it causes the reallocation of unchanged exploited reserves from some dates to other dates and should thus be interpreted as the substitution effect mentioned at the start of this chapter: namely that penalizing future extraction results in extraction being shifted away from the future toward the present. 3.3

Endogenizing Reserves

The stock of reserves to be exploited by a mine does not become available without some prior exploration and development efforts. Exploitable reserves are a form of capital, and exploration and development efforts are investment in this capital. It is standard to assume capital to be rigid in the short run and variable in the long run. In this section we integrate into the analysis potential long-term adjustments of the exploitable reserves. Unlike the previous section where it was a parameter, the stock of reserves X now becomes variable. The supply quantities expressed as functions x t characterized in (3.3) of the previous section are clearly not conventional supply functions. Indeed, since they depend on the variable stock of reserves that was assumed to be given, they must be called restricted supply functions (McFadden 1978), a concept meant to represent the short term. Taking into account that optimum exploitable reserves depend on prices, restricted supply functions computed in section 3.2 will become regular, or unrestricted, supply functions, meant to represent the long term. Although exploration and exploitation often take place simultaneously at the aggregate level (e.g., Pindyck 1978; Quyen 1988; see Cairns

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1990 for a comprehensive survey of related contributions); at the microeconomic level of a deposit they occur in a sequence, as in Gaudet and Lasserre (1988) and Fischer and Laxminarayan (2005). Assume that reserves are first developed and are then exploited. This way to model the supply of reserves provides a simple and natural representation that isolates in the problem under study the effect of an anticipated price change on the size of the exploited stock at the firm level. Since the development of reserves is costly, the optimum plans of the producers always bind the exhaustibility constraint. In other words, leaving part of the developed stock ultimately unexploited does not maximize cumulative net discounted revenues. This justifies the assumption made in section 3.2 that the totality of reserves is exhausted when the exploitation of the deposit is over. By the Hotelling rule of equation (3.2), any unit of reserves fetches the same discounted value to the producer regardless of the date it is extracted. This is the valuation by the producer of the marginal unit of reserves. This discounted value is commonly called the unit Hotelling rent, and it can be treated as an implicit price, since it is endogenous to the producer ’s problem unlike price parameters p1 and p2. Prices p1 and p2 positively affect this implicit price. In general, because the law of supply also applies to the case of exploitable reserves, the supply of reserves positively reacts to their implicit price. As a result the optimum amount of reserves X* positively depends on both prices: X* = X* (p1, p2).

(3.4)

In the unrestricted optimum where reserves are variable, supplies x1* and x2* can be obtained by substituting (3.4) into the restricted supply functions (3.3). Let us define xt* = xt* ( p1 , p2 ) ≡ x t ( p1 , p2 ; X * ( p1 , p2 )) .

(3.5)

The law of supply holds whether or not reserves are endogenous. Even when the optimum level X* varies with prices, the own-price effects remain positive: at each date t = 1, 2, the unrestricted supply xt* increases as a result of a rise in the contemporaneous price pt. Unlike the case in section 3.2, this no longer implies negative cross-price effects. Indeed, since x1* = X * − x2* , we now have ∂x1* ( p1 , p2 ) ∂X * ( p1 , p2 ) ∂x2* ( p1 , p2 ) , = − ∂p2 ∂ p2 ∂ p2

(3.6)

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where both terms ∂X * ( p1 , p2 ) ∂p2 and ∂x2* ( p1 , p2 ) ∂p2 on the right-hand side are positive. Thus the effect of the future price p2 on present supply x1* depends on the net effect of the future price on X* and x2* . In section 3.2 the term corresponding to the adjustments of reserves, ∂X * ( p1 , p2 ) , ∂p2 was absent, so we could easily establish the nonambiguously negative cross-price effect ∂x1* ( p1 , p2 ) ∂x2* ( p1 , p2 ) =− < 0. ∂p2 ∂p2 In contrast, the adjustment of reserves, ∂X * ( p1 , p2 ) , ∂p2 causes a stock effect of opposite direction. The total cross-price effect is thus ambiguous. Yet, when the returns to scale to exploration and development activities are decreasing, and extraction costs are strictly convex as assumed above, it is always the case that the future price p2 negatively affects present supply x1* (see Daubanes and Lasserre 2012). In the long run the returns to scale to the production of new reserves do diminish. Of course, new reserves may be found, but their current and future development is subject to severe limitations: the finiteness of exploration prospects is a fixed factor in the production process. Reserves are not like conventional goods: they cannot be produced under constant returns to scale. Because of the scarcity of exploration prospects, marginal exploration and development costs must be rising. In a long-term perspective, indefinitely producible conventional goods can be produced under constant returns to scale. Were that the case for

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nonrenewable resources, stocks would perfectly adjust to meet spot supplies and no cross-price effect would materialize. Moreover, in contrast to the long-run production of conventional goods, decreasing returns to scale in the production of new reserves imply that the supply of reserves is not infinitely elastic. Other things equal, the deeper or the more remote a deposit, the higher the cost of development. Since that marginal reserve unit must be financed by the unit present-value rent that it will earn during the exploitation period, the optimum level of reserves and their marginal value must go hand in hand. Explicitly taking into account decreasing returns to scale in the development of scarce resources, Daubanes and Lasserre (2012) showed that the pure substitution effect ∂x2* ( p1 , p2 ) ∂p2 not only counteracts but always dominates the stock effect ∂X * ( p1 , p2 ) . ∂p2 In resource supply, cross-price effects are generally negative.5 The own-price effect now consists of two partial effects working in the same direction: by Le Châtelier principle, the positive effect highlighted in section 3.2 is completed by the positive stock effect of developing greater reserves because long-term supply reactions are of a greater magnitude than short-term reactions. 3.4

Discussion

Resource supply differs from regular supply. Exhaustible resource supply presents an analogy with classical demand theory: resource producers allocate a stock of resource to different dates in a way that is comparable to the way consumers allocate their income to different expenditures on different goods. The time space in resource supply plays a similar role as the good space in demand, while the stock constraint in resource supply is not unlike the budget constraint in demand theory; both constraints depend on prices. The price-effect decomposition obtained by our resource supply analysis of the substitution effect and a stock compensation effect may be viewed as the counterpart of

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Slutsky decomposition in resource supply. Yet such analogy is not an isomorphism. For example, the fundamental law of supply, which states that the supply of a good increases if its price rises,6 has no demand equivalent: for a Giffen good, unlike an ordinary good, the demand increases if its price rises. There is no such paradox in resource supply. The supply of a conventional good or service is independent from the price of another good or service as long as their production costs are independent. By the Hotelling rule, the supply of a nonrenewable resource at one date is affected by its price at another date even when the cost of extraction at one date is independent from the cost of extraction at all other dates. This is because the so-called augmented marginal extraction cost includes a resource scarcity rent, the opportunity cost of extracting the scarce resource, that connects extraction costs at all dates to each other: a change in the price of the resource at any date affects the rent at all dates, which in turn affects the supply at all dates. Questions about intertemporal substitution and compensation effects do not arise in standard supply theory. In the case of nonrenewable resources, substantial research efforts have been devoted to analyze these issues; currently, for example, much research activity revolves around the green paradox, both at the policy and the theoretical levels. However, while it is customary to disentangle supply from demand in conventional partial equilibrium analyses, this is usually not done in the case of nonrenewable resources because the standard theory of supply has not been systematically extended to nonrenewable resources. A systematic identification and examination of resource supply issues is likely to facilitate and clarify both theoretical and policy analysis. Policy-induced changes are more complex than the analysis of supply for one main reason: the policy usually affects prices indirectly through the demand for the resource. For example the green paradox is often described as the effect on current or near future resource supply of policies, reducing resource demand over some future period via various forms of assistance to alternative energy sources. Say the prices in the analysis above are endogenously determined on competitive markets by the intersection of resource supply with standard demand schemes. Then reductions in future demand induced by the policy negatively will affect the value of the resource at all dates, that is, not only the future price but also the present price. Yet future demand reductions will more negatively affect the future price of the resource than the present price. It can be shown that supply reacts to these

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policy-induced price changes in the same way as it reacts to a exogenous price change solely occurring in the future (Daubanes and Lasserre, 2012). Intertemporal policy-induced changes like the green paradox are tantamount to price effects in resource supply. Notes We thank participants at the CESifo Venice Summer Institute workshop on “The Theory and Empirics of the Green Paradox.” Particular thanks go to Michael Hoel, Ian Lange, Ngo Van Long, Karen Pittel, Rick van der Ploeg, and Cees Withagen. Financial support from the Social Science and Humanities Research Council of Canada, the Fonds Québécois de recherche pour les sciences et la culture, the CIREQ, and the CESifo is gratefully acknowledged. 1. See Smulders et al. (2010). 2. A constant tax rate applied to cash flows is neutral because it amounts to a conventional profit tax on the total-discounted-profit objective. Absent the extraction cost, a constant-present-value levy on the resource is formally identical to a constant cashflow tax. Even with nonzero extraction costs, there exists a continuum of neutral tax paths (Dasgupta et al. 1981). All those extraction taxes are neutral for the same reason that they affect the equilibrium price in such a way that the equilibrium Hotelling rule remains satisfied without any further readjustments of quantities; in particular, they leave the producers’ profit-maximizing extraction unchanged. Tax trajectories that are rising more rapidly than those neutral ones cause a more rapid extraction (see also the comprehensive analysis of Gaudet and Lasserre 1990). 3. See Daubanes et al. (2013) for a recent discussion. 4. Sweeney (1993) noticed that for depletable resources, “the quantity supplied at any time must be a function of prices at that time and prices and costs at all future times. Thus static supply functions, so typical in most economic analysis, are inconsistent with optimal extraction of depletable resources” (p. 780). 5. Following Gerlagh (2011), van der Ploeg and Withagen (2012, 2014) not only consider effects of environmental policies on resource extraction quantities but also cumulative environmental damages. They point out that the environment may not then be worsened, even in regard to the green paradox. 6. “The law of supply holds for any price change. Because, in contrast with demand theory, there is no budget constraint, there is no compensation requirement of any sort...” (Mas-Colell et al. 1995, p. 138).

References Burness, H. S. 1976. On the taxation of nonreplenishable natural resources. Journal of Environmental Economics and Management 3: 289–311. Cairns, R. D. 1990. The economics of exploration for non-renewable resources. Journal of Economic Surveys 4: 361–95. Dasgupta, P. S., G. M. Heal, and J. E. Stiglitz. 1981. The taxation of exhaustible resources. Working paper 436. NBER.

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Daubanes, J., A. Grimaud, and L. Rougé. 2013. Green paradox and directed technical change: The effect of subsidies to clean R&D. Working paper 4334. CESifo. Daubanes, J., and P. Lasserre. 2012. Non-renewable resource supply: Substitution effect, compensation effect, and all that. Working paper 2012s-28. CIRANO. Fischer, C., and R. Laxminarayan. 2005. Sequential development and exploitation of an exhaustible resource: Do monopoly rights promote conservation? Journal of Environmental Economics and Management 49: 500–15. Fischer, C., and S. W. Salant. 2012. Alternative climate policies and intertemporal emissions leakage: Quantifying the green paradox. Discussion paper 12-16. RFF. Gaudet, G., and P. Lasserre. 1988. On comparing monopoly and competition in exhaustible resource exploitation. Journal of Environmental Economics and Management 15: 412–18. Gaudet, G., and P. Lasserre. 1990. Dynamiques comparées des effets de la taxation minière. L’Actualite Economique 66: 467–97. Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57: 79–102. Gordon, R. L. 1967. A reinterpretation of the pure theory of exhaustion. Journal of Political Economy 75: 274–86. Grafton, R. Q., T. Kompas, and N. V. Long. 2012. Substitution between biofuels and fossil fuels: Is there a green paradox? Journal of Environmental Economics and Management 64: 328–41. Gray, L. C. 1914. Rent under the assumption of exhaustibility. Quarterly Journal of Economics 28: 466–89. Herfindahl, O. C. 1967. Depletion and economic theory. In M. Gaffney, ed., Extractive Resources and Taxation. Madison: University of Wisconsin Press, 63–90. Hoel, M. 2012. Carbon taxes and the green paradox. In R. W. Hahn and A. Ulph, eds., Climate Change and Common Sense: Essays in Honour of Tom Schelling. New York: Oxford University Press, 203–24. Hotelling, H. 1931. The economics of exhaustible resources. Journal of Political Economy 39: 137–75. Long, N. V. 1975. Resource extraction under the uncertainty about possible nationalization. Journal of Economic Theory 10: 42–53. Mas-Colell, A., M. D. Whinston, and J. R. Green. 1995. Microeconomic Theory. New York: Oxford University Press. McFadden, D. L. 1978. Duality of production, cost, and profit functions. In M. A. Fuss and DL. McFadden, eds., Production Economics: A Dual Approach to Theory and Applications. Vol. I: The Theory of Production. Amsterdam: North-Holland, 2–109. Pindyck, R. S. 1978. The optimal exploration and production of nonrenewable resources. Journal of Political Economy 86: 841–61. van der Ploeg, F., and C. Withagen. 2012. Is there really a green paradox? Journal of Environmental Economics and Management 64: 342–63.

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van der Ploeg, F., and C. Withagen. 2014. Growth, renewables and the optimal carbon tax. International Economic Review 55: 283–311. Quyen, N. V. 1988. The optimal depletion and exploration of a nonrenewable resource. Econometrica 56: 1467–71. Sinn, H.-W. 2008. Public policies against global warming: A supply side approach. International Tax and Public Finance 15: 360–94. Smulders, S., Y. Tsur, and A. Zemel. 2010. Uncertain climate policy and the green paradox. Discussion paper 93129. Department of Agricultural Economics and Management, Hebrew University of Jerusalem. Stavins, R. N. 2011. The problem of the commons: Still unsettled after 100 years. American Economic Review 101: 81–108. Sweeney, J. L. 1993. Economic theory of depletable resources: An introduction. In A. V. Kneese and J. L. Sweeney, eds., Handbook of Natural Resource and Energy Economics, vol. 3. Amsterdam: Elsevier, 759–854.

II

Technology, Innovation, and Substitutability

4

The Green Paradox under Imperfect Substitutability between Clean and Dirty Fuels Ngo Van Long

4.1

Introduction

The burning of fossil fuels generates emissions that harm the environment not only in the present but also in the future, since emissions add to a pollution stock that decays only very slowly. As has been pointed out by many authors, the first best policy measure is to impose at each point of time a tax on emissions that equals the capitalized value of the stream of marginal damages of emissions (e.g., see Hoel 2011a; van der Ploeg and Withagen 2012). When the first best measure cannot be implemented because of political constraints, there are a variety of policy measures that at first sight might seem to approximate the first best measure. However, in some cases a careful analysis would reveal that some policies that seemingly would do the job will actually turn out to have an adverse effect on the environment, contrary to the good intention of the policy makers. This outcome is known as the green paradox (Sinn 2008a, b, 2012). The possibility of a green paradox outcome has been shown to exist under a wide variety of circumstances. Sinn (2008a, b) pointed out that the announcement of a steeply rising path of carbon tax can induce owners of oil and coal reserves to extract their resources more quickly, resulting in a worse climate outcome in the short and the medium term.1 Hoel (2008) demonstrated that technical progress in the backstop technology that produces at constant cost a clean energy which is a perfect substitute for fossil fuels would have a similar effect on the supply behavior of resource-extracting firms.2 Grafton et al. (2012) showed that an increase in a time-independent ad valorem rate of subsidy on biofuels can result in a green paradox outcome if the supply curve for biofuels is sufficiently concave. Van der Ploeg and Withagen (2012) showed that whether a green paradox arises may depend on

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whether extraction costs increase sharply as the size of the remaining stock diminishes. Long and Stähler (2012) demonstrated that, if both fossil fuels and non-fossil fuels are being used concurrently, a fall in production cost of non-fossil fuels may generate income effects leading to a fall in the interest rate, which in turn induces greater current extraction rate of fossil fuels, and possibly greater cumulative extraction.3 All the models above assume that if several types of fuels are available, they are perfect substitutes. While perfect substitution is often a useful assumption to simplify the analysis, one must admit that at the present level of technology, biofuels cannot entirely replace petroleum in a number of uses, such as in aviation. Efforts are being made to improve the substitutability of biofuels for petroleum. This chapter therefore relaxes the assumption of perfect substitutability.4 This allows us to ask a number of interesting questions. Does a technological change that makes biofuels a closer substitute to petroleum benefit or harm the environment in the near term? If it harms the environment, we call this a technology-induced green paradox (as distinct from tax/ subsidy induced green paradox).5 A related question is whether imperfect substitutability increase or reduce the likelihood of a green paradox outcome induced by raising the subsidy rate on biofuels or by increasing the base-year rate of an ad valorem tax on fossil fuels. When the prices for fossil fuels and non-fossil fuels are not identical because of imperfect substitutability, the analysis can become tedious, because now there are several prices to consider and these prices will be changing over time. To facilitate the analysis, this chapter defines the concept of a “reduced-form demand function” for fossil fuels. This function incorporates the parameters of the demand and supply functions of the clean energy. Using this reduced-form demand function, we are able to analyze the possibility of a green paradox outcome caused by a technological change by identifying its direct effect (usually “pro-green”), and its indirect effect (usually “anti-green”). The direct effect is defined as the change in the quantity demanded keeping the price of fossil fuels constant (while allowing the price of non-fossil fuels to change to clear that market). The indirect effect arises from intertemporal optimization behavior of owners of fossil fuel stocks. It works through the change in the equilibrium price path of fossil fuels. We assume that the non-fossil energy is produced under increasing marginal cost. This assumption reflects the reality that renewable substitutes such as biofuels are produced using different grades of land (Chakravorty et al. 2011). In fact, by moving up the supply curve of

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land, the greater use of biofuels may increase the unit cost of production of the renewable substitutes. By and large, the existing analyses of biofuel subsidies use a static framework. In the static context, many authors have identified mechanisms for increased carbon emissions that could result from biofuel subsidies. For example, a common argument is that the production process of biofuels is not environmentally friendly because it involves the use of inputs with high carbon contents. The present study, by showing how the equilibrium price path of fossil fuels responds to increased substitutability, complements the existing literature with a dynamic mechanism that arises even if a technical change in favor of substitutability occurs only once. The main contribution is a delimitation of cases when a green paradox outcome occurs and when it does not.6 It will be shown that for the case of a system of linear demand functions, an increase in substitutability can result in a green paradox outcome if the existing degree of substitutability is already high. However, in starting from a very low degree of substitutability, a small increase in substitutability cannot generate a green paradox outcome. I also examine the case where the fossil fuel producers form a cartel and act as a Stackelberg leader while biofuel producers are followers. It is found that for certain range of parameter values, there is a green paradox outcome induced by an increase in substitutability. 4.2

A Brief Review of the Related Literature

There is a large literature that connects the dynamic analysis of nonrenewable fossil resources with climate change damages. Sinclair (1992, 1994) pointed out that climate change policies must aim at delaying the extraction of oil, and argued that the carbon tax, expressed in ad valorem terms, must decline over time to encourage owners of fossil fuels to defer extraction. This result, however, depends in part on the assumptions that (1) damages appear multiplicatively in the production function, (2) the pollution stock does not decay, and (3) capital and oil are substitutable inputs in a Cobb–Douglas production function.7 Recently Groth and Schou (2007) confirm Sinclair ’s declining tax result using a similar model, allowing endogenous growth. Ulph and Ulph (1994), specifying damages as an additive term in the social welfare function, and assuming exponential decay of pollution, find that the time profile of the optimal per unit carbon tax has

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an inverted-U shape. Hoel and Kverndokk (1996) specify a model of economic exhaustion that includes rising extraction costs. They show that carbon tax peaks before the peak in atmospheric carbon. Consistent with Ulph and Ulph (1994), they find that in the optimal long run the carbon tax approaches zero. They also consider the case where there is a backstop technology that produces a substitute at constant cost. Farzin and Tahvonen (1996), assuming the decay rate of pollution to be nonlinear in the stock, show that the optimal carbon tax can take a variety of shapes. Similarly Tahvonen (1997) obtains eleven different tax regimes, depending on initial sizes of the stock of CO2 concentration and the stock of fossil fuels. Contrary to the models above that focused on the optimal carbon tax, the key point of the green paradox literature is that the optimal carbon tax cannot be implemented given the political economy that exists in most countries. In particular, this literature argues that climate change policies that superficially may seem to be second-best measures could cause environmental damages, if the response of owners of fossil fuel stocks is to hasten their extraction (e.g., to avoid high future carbon taxes). Models that depict this adverse response to anticipation of taxes or substitute production include Sinn (2008a, b), Hoel (2008), Di Maria et al. (2008), Gerlagh and Liski (2008), and Eichner and Pethig (2010).8 According to Strand (2007), a technological international agreement that makes carbon redundant in the future may increase current emissions. Hoel (2008, 2011b) assumes that carbon resources remain cheaper than the substitute and analyses the situation where different countries have climate policies of different ambition levels. He shows that in the absence of an efficient global climate agreement, climate costs may increase as a consequence of improved technology of substitute production. Gerlagh (2011), in studying the effect of an improvement in the backstop technology, makes a distinction between a weak green paradox and a strong green paradox. The former is said to arise when current emissions increase as a result of an improvement in the backstop technology. The latter arises when the net present value of the stream of all future damages increases. He shows that both the weak and the strong green paradox arise in the benchmark model with constant extraction cost and unlimited supply of the backstop energy. Assuming linear demand, he finds that increasing extraction costs counteract the strong green paradox, while a rising marginal cost of the substitute may reduce the likelihood of both the weak and the strong

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green paradox. Van der Ploeg and Withagen (2012) address the case of stock-dependent marginal extraction costs while retaining the assumption that the substitute is available in unlimited supply at a constant marginal cost. They show that if first-best policies are not feasible, a green paradox occurs if the cost of backstop decreases, provided that the backstop remains expensive such that the nonrenewable resource stock is eventually exhausted. By contrast, if the backstop becomes so cheap that physical exhaustion will not take place, then there is no green paradox outcome. Grafton et al. (2012) emphasize the facts that biofuels are already available, but the expansion of biofuel output is possible only with increasing costs. They assume that biofuels and fossil fuels are perfect substitutes. Using a framework where both types of fuels are simultaneously consumed in the first phase, they find conditions under which a green paradox outcome will not occur, as well as conditions under which it will occur. Their main focus was on the effect of a biofuel subsidy on the date of exhaustion of the fossil fuel resources. A striking result was that in the case of a linear demand for energy, together with (1) a zero extraction cost for fossil fuels and (2) an upward-sloping linear marginal cost of biofuels, a biofuel subsidy will have no effect on the amount of fossil fuels extracted at each point of time, even though it will reduce fuel prices. The increase in quantity of fuel demanded is exactly matched by an increased output of biofuels, leaving the extraction rate unchanged. Thus the time at which the stock of fossil fuels is exhausted is unchanged. Their result stands in sharp contrast to the inevitable green paradox in the model of Hoel (2008), which assumes that the supply curve of the renewable resource is horizontal. When the assumption of zero extraction cost is replaced by the assumption of a positive constant marginal extraction cost, a biofuel subsidy will result in a longer time over which the fossil fuel stock is exhausted. In this second case there is no green paradox outcome: the subsidy works as intended; it delays the exhaustion time. Finally Grafton et al. consider the case where the marginal cost of biofuel production is strictly increasing and strictly concave. In this situation along the equilibrium price path, as the price of energy rises gradually, the rate of the supply increase (per dollar increase in energy price) is greater when the price is higher. Consequently fossil fuel firms, in anticipation of the greater expansion of the substitute in the later stage, respond by increasing their extraction at an earlier date. In this case a green paradox outcome is obtained.

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4.3 A Model of Imperfect Substitutability between Fossil Fuels and Renewable Energy We consider an economy with three goods: fossil fuels, denoted by x, renewable energy, denoted by y, and a numéraire good, denoted by z. Assume that y is a perfectly clean source of energy. In contrast, the consumption of fossil fuels generates emissions which contribute to a stock of pollution. For simplicity, we assume that the demands for these goods come from the utility maximization of a representative consumer. Goods x and y are imperfect substitutes and thus command different prices. The stock of pollution at time t is denoted by St. The rate of change in St is assumed to be equal to xt. This simplifying assumption may be justified on the ground that the rate of natural decay of GHG pollution is very slow. Then t

St = S0 + ∫ xτ dτ . 0

Fossil fuels are extracted from a resource stock Rt: R t= –xt,

R(0) = R0, Rt ≥ 0.

(4.1)

Then cumulative emissions from time zero to time t is t

∫ xτ dτ = R(0) − R

t

0

and the stock of pollution is linearly related to the stock of exhaustible resources, St = S0 + R(0) – Rt. Note that Rt will be falling over time, causing St to rise over time, until the resource stock is exhausted. We assume that the representative consumer has a separable net utility function U(xt, yt, zt) – C(St),

(4.2)

where C(St) represents the damages caused by pollution. Because of the pollution externalities embodied in the net utility function (4.2), it is widely thought that policies that encourage replacing fossil fuels by a clean energy should be promoted. One often hears arguments in favor

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of the subsidization of public and private R&D activities that would increase the degree of substitutability of clean energy for fossil energy. In this chapter we do not model R&D activities. We wish instead to find out whether an exogenous technical progress that increases the substitutability is good or bad for the environment, given that first best policies are not available. 4.3.1 Assumptions on Demand For simplicity, we abstract from the income effect, and assume that U is quasi-linear: U(xt, yt, zt) = u(xt, yt) + zt. This assumption implies that any income change will impact only the demand for the numéraire good. As usual, it is assumed that income is sufficiently large such that the consumption of the numéraire good is strictly positive. The function u(xt, yt) is strictly concave. We assume that the marginal utility of any good is finite even when its consumption is zero. Let P1t and P2t denote the consumer prices of good xt and yt respectively. Consumer ’s maximization then leads to the following first order conditions that characterize an interior maximum: u1 ( xtd , ytd ) = P1t , u2 ( xtd , ytd ) = P2t . where u1 and u2 stand for the partial derivatives of u with respect to x and y, respectively. The superscript d in xtd and ytd indicate that these are quantities demanded. From the FOCs, we obtain the demand functions xtd = X d (P1t , P2t , μ ),

(4.3)

ytd = Y d (P1t , P2t , μ ),

(4.4)

where we have used μ as a parameter representing the degree of substitutability. We assume that each of these demand functions is decreasing in its own price, and increasing in the price of the other good (its imperfect substitute). Furthermore we assume that an increase in substitutability will reduce the demand for a good if its price is higher than the price of the substitute: ∂xtd < 0 if P1t > P2t , ∂μ

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∂ytd < 0 if P2t > P1t . ∂μ These assumptions are satisfied for the standard linear quadratic preferences. 4.3.2 Assumptions on the Supply of Renewable Energy Let yts denote the quantity of renewable energy supplied at time t. We assume that marginal cost is positive and increasing in output level, yts. The producers of renewables are price takers. They produce at the level that equates their marginal cost to the producer price P2ft: MC( yts ) = P2ft . The superscript f indicates that this is the price the firms receive per unit sold. The difference between the producer price and the consumer price, P2ft − P2t , is the subsidy on renewables. Let us assume that there is a constant ad valorem subsidy rate, denoted by s ≥ 0, so that P2ft = (1 + s) P2t ≡ hP2t . We call h the “subsidy factor.” Note that h ≥ 1. We can then derive the supply function for renewables: yts = G ( hP2t ) , where G in the inverse of the marginal cost function MC. Clearly, G′ > 0 because we assumed an upward sloping MC function. 4.3.3 Supply of Fossil Fuels and the Choke Price We assume that there are a large number of identical resource-extracting firms, each owning a deposit of fossil fuels. The deposits are homogeneous and are of identical size. The aggregate fossil resource stock at time zero (the present time) is denoted by R0. Resource-extracting firms operate under perfect competition. They perfectly forecast the price path of the fossil fuel. Assume that the marginal cost of extraction is constant, c ≥ 0. Define the net price of the fossil fuel as P1t – c. As long as the net price rises at the rate of interest r, individual fossil fuel producers are willing to supply any amount. In equilibrium, however, the industry’s supply of the fossil fuel at any time t must equal the demand for it. There will be a time T when the price of fossil fuel is so high that the demand for fossil fuel becomes zero, and from that time onward

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only renewable energy is consumed, in quantity y and at the price P2. Such a high price for fossil fuel is called the “choke price” for fossil fuel, and we denote it by P1. The price P1 is implicitly defined by the following three equations, which determined (P1 , P2 , y ): u1 (0, y ) = P1 , u2 (0, y ) = P2 , y = G ( hP2 ) . Let T be the time at which the aggregate fossil resource stock is exhausted. Then P1T = P1. From the Hotelling rule, the present value of the net price is the same at all points of time in the interval [0, T]:

(P1t − c ) e − rt = (P1 − c ) e − rT

for all t ≤ T .

This means that if we know T, we can calculate P1t as follows: P1t = c + ( P1 − c ) e − r(T −t) ≡ φ ( P1 , T , t ) . Then the time of exhaustion T must satisfy the following equation, which requires that total consumption of fossil fuel from time 0 to time T must be equal to the total resource stock: T

∫ X (P d

1t

, P2t , μ ) dt = R0 ,

0

where P1t is given by the function φ ( P1 , T , t ) specified above. But what about P2t? Since the market must clear, the demand for the renewable energy at any time t must equal its supply. Thus Y d (φ ( P1 , T , t ) , P2t , μ ) = G ( hP2t ) . This equation shows that P2t can be expressed as a function of φ ( P1 , T , t ), which is the equilibrium P1t along the Hotelling path. This observation leads us to a simple reformulation, using the concept of the “reducedform demand function for fossil fuel.” This will be made clear in the next section. 4.3.4 The Reduced-Form Demand Function for Fossil Fuels We assume that the market for renewables clears at each point of time: the quantity demanded equals the quantity supplied. Then Yd(P1t, P2t, μ) = G(hP2t).

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This relationship allows us to solve for the equilibrium consumer price function for renewable energy, which we denote by π2(.), P2t = π2(P1t, μ, h).

(4.5)

In other words, given the subsidy factor h and given the fossil fuel price P1t, we can deduce the price P2t that would clear the market for renewables. Clearly, ∂π 2 > 0 and ∂P1t

∂π 2 < 0, ∂h

where the first inequality reflects the fact that the two goods are substitutes rather than complements. The second inequality simply means that an increase in the subsidy factor for renewables will reduce the equilibrium consumer price for renewables, at any given price of fossil fuels. Substituting (4.5) into equation (4.3), we obtain the demand function for fossil fuels, given that the market for renewables clears: xtd = X d ( P1t , π 2 ( P1t , μ , h ) , μ ) ≡ Dr ( P1t , μ , h ) .

(4.6)

We call Dr(P1t, h, μ) the reduced-form demand for good x. Notice that the function Dr does not contain the price P2t as an argument. This does not mean that the demand for fossil fuels is independent of the price of non-fossil fuels. Rather, the market-clearing P2t which is conditional on P1t, has been used. Clearly Dr is decreasing in P1t: ∂D r ∂X d ∂X d ∂π 2 = + < 0. ∂P1t ∂P1t ∂P2t ∂P1t We assume that Dr is decreasing in μ, at least for those values of P1t high enough so that P1t > P2t = π2(P1t, μ, h). In particular, assume that at any P1 near the choke price P1, a small increase in substitutability will reduce the demand for fossil fuels: ∂Dr ( P1 , μ , h ) < 0. ∂μ

(4.7)

This assumption can be verified in the linear quadratic utility function case.9 From our earlier definition of the choke price for fossil fuels, P1, it holds that

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0 = Dr ( P1 , h , μ ) . From the properties of the function Dr, we deduce that an increase in μ will reduce the choke price: dP1 [∂Dr (P1 , μ , h)] ∂μ < 0. =− ∂Dr ∂P1 dμ 4.3.5 The Two Phases We consider an equilibrium path consisting of two phases: Phase 1: Both fossil fuels and renewable energy are simultaneously supplied. Phase 2: Only renewable energy is supplied, because the fossil fuel stock has been exhausted. Let T be the time at which phase 1 ends and phase 2 begins. At time T the fossil fuel stock is just exhausted. We must determine T endogeneously. At time T we have x(T) = 0. Thus T must satisfy the equation T

∫ D (P r

1t

, μ , h ) dt − R0 = 0 ,

0

where P1t = φ ( P1 , T , t ) . We wish to show that there is a range of value for μ such that, at any given μ in this range, a small increase in it will lead to an earlier exhaustion date. The following fact is obvious: Fact 1 If there exists a real interval (μ*, μ**) such that the exhaustion time T is decreasing in μ then a small increase in μ in this interval will cause the pollution stock to be higher for all t < T. 4.4 Effect of an Increase in Substitutability on the Exhaustion Time What are the forces that determine the net effect of an increase in the substitutability parameter μ on the time of exhaustion T? We have assumed (and this can be verified for the linear quadratic case) that an increase in μ will lower the fossil fuel choke price P1. At the same time, the inequality (4.7) indicates that an increase in μ rotates the (reduced-form) demand curve for fossil fuel in the

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counterclockwise direction. Does an increase in μ generate, average, greater demand for fossil fuels over the time interval To answer this question, it is useful to decompose the effect of increase in μ on fossil fuel consumption into a direct effect and indirect effect.

on T? an an

Direct Effect The counterclockwise rotation of the (reduced-form) demand curve for fossil fuels implies that at any given price P1t sufficiently high, the quantity demanded is smaller than before. This direct effect is captured by the term Dμ which is generally negative, at least for high P1t. The direct effect is generally “pro-green”: an increase in substitutability reduces demand for fossil fuels, at any given sufficiently high price P1t. Indirect Effect Since the increase in μ (substitutability) lowers the fossil fuel choke price P1, it follows that, holding T constant, the price P1t must fall for all t < T. A fall in P1t increases the quantity demanded. The indirect effect, captured by the term ∂Dr ∂P1t , ∂P1t ∂μ is positive, imeaning it is “anti-green.” The total effect on fossil fuels consumption at any time t is the sum of the direct effect and the indirect effect at that time. In general, if the function Dr(ϕ, μ, h) is not restricted beyond the usual assumption that the demand curve is downward sloping, the total effect can be positive at some points of time and negative at some other points of time. Therefore, to find the effect of an increase in μ on the exhaustion time T, one has to compute the cumulative total effect, over the interval [0, T]. If this cumulative total effect is positive, it means that the exhaustion time is brought closer to the present, which is a green paradox outcome. A more formal analysis follows. To find the net effect of a change in μ on the exhaustion time T, we define the function T

Ω (T , μ , h ) = ∫ D ( P1t , μ , h ) dt − R0 = 0 , 0

where P1t = φ ( P1 ( μ ) , T , t ) .

(4.8)

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Then T⎛ ∂P ⎞ − ∫ ⎜ DPr1 1t + Dμr ⎟ dt 0 ⎝ ⎠ Ωμ dT ∂μ =− = . T P ∂ dμ ΩT D r P , μ + ⎛ D r ( 1 ) ∫0 ⎝ P1 ∂T1t ⎞⎠ dt

(4.9)

The denominator is positive. Thus we can state the following proposition. Proposition 1 (Necessary and sufficient condition for a green paradox) A small increase in substitutability will bring the resource exhaustion date closer to the present if and only if the cumulative sum of the indirect effect (anti-green) outweighs the cumulative sum of the direct effect (generally pro-green): T

⎛

∫ ⎜⎝ D 0

r P1

∂P1t ⎞ + Dμr ⎟ dt > 0. ⎠ ∂μ

(4.10)

Without a more explicit specification of the reduced-form demand function Dr, we cannot determine whether a green paradox outcome will arise from an increase in substitutability. 4.4.1 Parameterizing Substitutability: The Linear Quadratic Case For illustrative purposes, consider the following linear-quadratic formulation. Assume that b b U ( x , y , z ) = θ x − x 2 + θ y − y 2 − μ xy + z , 2 2 where μ ≥ 0, θ > 0 and b > 0. To ensure that U is concave, we assume that b > μ. In the limiting case where μ = b, the two types of fuels are perfect substitutes. The utility function above implies, on the one hand, that for given (x, y), if the goods become closer substitute (μ increases) then the utility decreases.10 On the other hand, for environmental reasons, an increase in substitutability gives the economy the potential to increase welfare by a well-designed change in the composition of demand to reduce environmental damages. At each point of time t, the consumer faces the budget constraint P1txt + P2tyt + zt = Mt,

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where Mt is the total expenditure allocated to period t. Assume that Mt is sufficiently great, so that zt is always positive. Then the consumer ’s FOCs with respect to xt and yt are θ – bxt – μyt – P1t ≤ 0, xt ≥ 0, xt[θ – bxt – μyt – P1t] = 0,

(4.11)

θ – byt – μxt – P2t ≤ 0, yt ≥ 0, yt[θ – byt – μxt – P2t] = 0.

(4.12)

It is straightforward to derive the demand functions ⎧ (θ − P1t ) b − (θ − P2t ) μ ⎫ xtd = max ⎨0 , ⎬, b2 − μ 2 ⎩ ⎭

(4.13)

⎧ (θ − P2t ) b − (θ − P1t ) μ ⎫ ytd = max ⎨0 , ⎬. b2 − μ 2 ⎩ ⎭

(4.14)

We will focus on the phase where both types of fuels are demanded in positive amounts. Note that if P1t ≥ P2t, an increase in substitutability will reduce the demand for fossil fuels:11 2 ∂xtd − (b − μ ) (θ − P2t ) − 2μb ( P1t − P2t ) = . ∂μ ( b 2 − μ 2 )2

Renewable energy is produced by perfectly competitive firms, under increasing marginal cost. Its supply is denoted by ys. Assume that the marginal cost of producing ys is A + Bys, where 0 ≤ A < θ and B > 0. Then the supply of renewable energy satisfies the condition that marginal revenue, hP2t, is equated to marginal production cost, A + Bys, hP2t = A + Byts , which gives yts =

hP2t − A >0 B

(4.15)

if hP2t ≥ A.

Recall our assumption that the market for renewables clears at each point of time. Equating ytd with yts, we obtain the equilibrium P2t for a given P1t

(θ − P2t ) b − (θ − P1t ) μ b2 − μ 2

=

hP2t − A . B

It is convenient to define

λ=

A < 1. θ

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Then P2t =

(b − μ )[Bθ + (b + μ ) λθ ] bB + (b − μ ) h 2

2

+

Bμ P1t ≡ π 2 ( P1t , μ , h ) , bB + (b 2 − μ 2 ) h

(4.16)

where π2 is increasing in P1t, decreasing in h, and decreasing in μ: ∂π 2 Bθ h (b − μ )2 + 2μ AbB2 < 0. =− ∂μ [bB + (b2 − μ 2 ) h]2

(4.17)

Thus, for a given P1t, an increase in substitutability reduces the market clearing price P2t. Remark Along the Hotelling path, P1t will be rising, and so will P2t, as equation (4.16) indicates. ∂P2t Bμ . = ∂P1t bB + (b 2 − μ 2 ) h

(4.18)

In particular, this response is greater, the higher is degree of substitutability. We record this result as fact 2: Fact 2 The increase in the price of renewables in response to the increase in the price of the exhaustible resource along the Hotelling path is itself an increasing function of the substitutability parameter μ. Notice that given market clearance, the gap between the renewable energy price and the fossil fuel price can be expressed as follows: P2t − P1t =

B (b − μ ) (1 + (b + μ ) λ )θ ⎡ h (b − μ )2 + (b − μ ) B ⎤ −⎢ ⎥ P1t . 2 2 bB + (b 2 − μ 2 ) h ⎣ bB + (b − μ ) h ⎦

(4.19)

Thus the price gap P2t – P1t decreases as P1t increases. Substituting (4.16) into (4.13), we obtain the “reduced-form demand function” for fossil fuels: xtd = W − VP1t ≡ Dr ( P1t , μ , h ) ,

(4.20)

where V ( h, μ ) ≡

bh + B

(b 2 − μ 2 ) h + bB

W ( h, μ ) ≡

> 0,

(b − μ ) hθ + μλθ + Bθ

(b 2 − μ 2 ) h + bB

> 0.

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Let us find the choke price for fossil fuels, given that non-fossil fuels are available. Setting Dr(P1t, μ, h) = 0, we obtain the “choke price” for fossil fuels: P1 =

W (b − μ ) hθ + μλθ + Bθ μ (h − λ ) ⎤ = . = θ ⎡⎢1 − V bh + B bh + B ⎥⎦ ⎣

Note that P1 is decreasing in μ and increasing in B, where 1/B measures the steepness of the marginal cost curve of renewable fuels. Equation (4.20) can be used to draw the reduced-form demand curve for fossil fuels in a Marshallian-type diagram where P1t is measured along the vertical axis, and xt on the horizontal axis. Here W is the intercept on the horizontal axis, while vertical intercept, W/V, is the “choke price” for fossil fuels. The slope of this demand curve is 1/V. Since h ≥ 1 > λ, an increase in substitutability will lower the choke price P1, as expected. When P1t equals the choke price P1, then x1d = 0, and the equilibrium price of non-fossil fuels then attains its steady-state price P2 as defined below:12 P2 =

Bθ + bλθ . bh + B

How does an increase in substitutability affect the reduced-form demand function for fossil fuels? Since Vμ =

2μ h (bh + B)

[ h (b 2 − μ 2 ) + bB]

2

≥ 0,

it follows that an increase in substitutability makes the residual demand curve flatter. We now show that there exists a unique threshold m, where 0 < m < b, such that an increase in μ will move the quantity intercept of the reduced-form demand curve to the left if μ ∈ (0, m) and to the right if μ ∈ ( m, b ). From the definition of W(h, μ), we obtain Wμ =

− ( hθ − λθ )[ h (b 2 − μ 2 ) + bB] + 2μ h [(b − μ ) hθ + μλθ + Bθ ]

[ h (b2 − μ 2 ) + bB]2

The sign of the numerator is the sign of the expression g(μ) = –h(h – λ)μ2 + 2h(bh + B)μ – b(bh + B)(h – λ).

.

(4.21)

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75

Since h ≥ 1 > λ, the function g(μ) is quadratic and concave in μ. It is negative at μ = 0 and positive at μ = b. Consequently there exists a unique value m in (0, b) such that for all μ ∈ (0, m), a marginal increase in μ will shift the quantity intercept to the left, and for all μ ∈ ( m, b ), a marginal increase in μ will shift the quantity intercept to the right. The threshold value m is given by ⎡

m=

(bh + B) − (bh + B) ⎢bh + B − ⎣ h−λ

b ( h − λ )2 ⎤ ⎥ h ⎦

.

(4.22)

It follows from equation (4.21) that keeping P1t constant, a marginal increase in μ will unambiguously reduce the quantity of fossil fuels demanded (a pro-green effect) if μ is in (0, m). In contrast, if μ is in (m, b ), then Wμ > 0 and a marginal increase in μ will increase the quantity of fossil fuels demanded if P1t is low, and reduce it if P1t is high. Therefore, for any given μ ∈ ( m, b ), there exists a corresponding positive “pivot price” P * * such that for at any given price P1t below this pivot price, a marginal increase in μ will increase the demand for fossil fuels.13 For μ ∈ ( m, b ), the pivot price is P** ≡

Wμ ( μ ) g ( μ )θ = < P1 . Vμ ( μ ) 2μ h (bh + B)

(4.23)

Thus, at any P1t in the interval (P * *, P1), an increase in μ will reduce the demand for fossil fuels. 4.5 Sufficient Conditions for a Green Paradox Outcome Caused by an Increase in Substitutability Let us investigate the possibility of a green paradox outcome when substitutability increases, under the assumption of linear demand arising from a linear quadratic utility function. There are two ways to proceed with the analyisis. The first method makes use of proposition 1 (in section 4.4) and consists of determining whether the necessary and sufficient condition (4.10) is satisfied. The second method consists of evaluating the market clearing equation (4.8) directly, which allows us to solve for T as a function of μ. The first approach is useful because it is applicable also to the case of nonlinear reduced-form demand functions.

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4.5.1 Approach 1: Evaluating the Direct Effect and the Indirect Effect Let us first evaluate the direct effect at any given point of time. For the case of linear reduced-form demand, the direct effect is Dμr = Wμ − Vμ P1t

= Wμ − Vμ {c + [ P1 ( μ ) − c ] e − r(T −t) } .

This term is negative if μ < m. If μ > m, this term is positive (pro-green) for low values of P1t and negative (anti-green) for high values of P1t. On the other hand, the indirect is unambiguously anti-green: DPr1

∂P1t ( h − λ )θ = e − r (T − t ) 2 > 0. ∂μ (b − μ 2 ) h + bB

Combining the direct effect and the indirect effect, and defining γ = c/θ, we obtain the following proposition: Proposition 2 Under the linear quadratic formulation, the necessary and sufficient condition for a green paradox outcome (a marginal increase in substitutability hurts the environment) is that μ is sufficiently high such that f(μ) = –h(h – λ)μ2 + 2h(bh + B)(1 – γ)μ – b(bh + B)(h – λ) > 0. Remark The function f(μ) is quadratic and concave in μ. It is negative at μ = 0 and positive at μ = b provided that γ is sufficiently small such that B(h + λ) + 2bhλ > 2h(bh + B)γ.

(4.24)

Therefore, if this condition holds, there exists a unique value μ* in (0, b) such that for all μ ∈ (0, μ*), a marginal increase in μ will reduce the exhaustion time, bringing climate change damages closer to the present. Note that 0 < m < μ *. In fact14 ⎡

μ *=

(bh + B) (1 − γ ) − (bh + B) ⎢(bh + B) (1 − γ )2 − ⎣ h−λ

b ( h − λ )2 ⎤ ⎥ h ⎦

.

Numerical Examples Setting h = 1, b = 1, λ = 0.5, γ = 0.1, B = 0.1, and r = 0.05. Then μ *  0.30. Thus a green paradox outcome will occur when there is a marginal

The Green Paradox under Imperfect Substitutability

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increase in substitutability, where μ ∈ (0.30, 1). For example, if initially μ = 0.50, and R0/θ = 417.88 then competitive firms will take 500 years to exhaust the stock. Let there be a technological progress such that the substitutability increases by 4 percent, meaning the new μ is 0.52. We find that the time of exhaustion falls by 1.2 percent, meaning T is now 494 years (making exhaustion occur 6 years earlier). Let us consider a smaller γ. Say γ = 0.02. Then μ *  0.27. Thus a smaller extraction cost facilitates a green paradox outcome What about the subsidy factor h? Keeping all other parameters as specified in the base line scenario, but let h = 1.5. Then μ *  0.43, meaning a green paradox outcome is less likely. Consider now a higher marginal cost intercept of renewables, A. An increase in A is equivalent to an increase in λ. Let the new λ be λ = 0.6. Then μ *  0.23, meaning a green paradox outcome occurs for a wider range of μ. An increase in B (the steepness of the supply curve) also makes a green paradox outcome is more likely. If B = 5, then μ *  0.28. 4.5.2 Approach 2: Direct Computation of the Exhaustion Time Using the linear reduced-form demand function Dr, we can compute the exhaustion time directly from the equation T

∫ (W − VP

1t

) dt = R0 ,

0

where W P1t = c + ( P1 − c ) e − r(T −t) = c + ⎛ − c⎞ e − r(T −t) . ⎝V ⎠ Then the condition that total demand for fossil fuels over the time interval [0, T] must equal the total supply reduces to a simple equation: T

∫ (W − VP

1t

⎛ ⎝

) dt = (W − cV ) ⎜ T −

0

1 − e − rT ⎞ ⎟ = R0 . r ⎠

(4.25)

This equation allows us to compute T, once W, V, r, and R0 are specified. Since W and V are dependent on μ, we can determine how T responds to an increase in μ. Define G(μ) = W(μ) – cV(μ),

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Ngo Van Long

F (T ) = T −

1 − e − rT . r

Note that G(μ) > 0 because we have assumed that c is lower than the choke price. Then the exhaustion time T can be obtained from G(μ)F(T) = R0. Proposition 3 The effect of a marginal increase in substitutability on the exhaustion time is dT − F (T ) G′ ( μ ) = , dμ G ( μ ) F ′ (T ) and the necessary and sufficient condition for a marginal increase in substitutability to reduce exhaustion time is G′(μ) > 0, namely Wμ – cVμ > 0. Proof

(4.26)

Use the facts that

F′(T) = 1 – e–rT > 0 and G(μ)F′(T)dT + F(T)G′(μ)dμ = dR0. Remark The condition (4.26) is identical to f(μ) > 0. Since Vμ > 0, a necessary condition for a green paradox outcome is Wμ > 0. Condition Wμ – cVμ > 0 is equivalent to Corollary If condition (4.26) holds, the stock of pollution at each point of time is increasing in the subsitutability parameter μ for μ ∈ (μ*, b). Proof

The equilibrium time path of price is

P1t = c + ( P ( μ ) − c ) e − r(T (μ )−t) . Since xt(μ) = W – VP1t(μ) and t

St = S0 + ∫ xτ dτ , 0

it follows that dSt ( μ ) dx = ∫ τ dτ , dμ dμ 0 t

The Green Paradox under Imperfect Substitutability

79

where dP1t ( μ ) dP1t ( μ ) dxτ = DPr = −V dμ dμ dμ dP ⎡ ⎛ dT ⎞ ⎤ = −V ⎢ e − r(T (μ )−t) + ( P ( μ ) − c ) ⎜ − r ⎟ ⎥ < 0. ⎝ dμ ⎠ ⎦ dμ ⎣ Therefore the stock St increases with μ if μ ∈ (μ*, b). 4.6

Monopoly

What happens if the fossil firm is a monopolist? The monopolist chooses the price path of price P1t to maximize its stream of discounted profits, knowing the reduced-form function xt = Dr(P1t). This formulation implies that the monopolist is a leader in the market for fuels. He knows that the price of non-fossil fuels will adjust to his announced price of fossil fuel so that the market clears. Formally, let Tm denote the monopolist’s exhaustion time. The monopolist seeks the terminal time Tm and the price path P1t defined over [0, Tm] to maximize

∫

Tm

0

(P1t − c ) Dr (P1t ) e − rt dt

subject to ˙ t = –Dr(P1t) R and RTm ≥ 0. The following results are obtained: The monopolist’s planned exhaustion time Tm is determined by 1 1 − e − rTm ⎞ ⎛ (W − Vc ) ⎜ Tm − ⎟⎠ = R0 . ⎝ r 2

(4.27)

Thus from equations (4.25) and (4.27) we can see that 1. the threshold level of substitutability beyond which a green paradox outcome occurs is the same under monopoly as under perfect competition and

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2. the relationship between the monopolist’s exhaustion time Tm and the competitive exhaustion time Tc satisfies the following equation: Tm − [(1 − e − rTm ) r ] = 2. T − [(1 − e − rT ) r ] We can state the following proposition: Proposition 4 The threshold level of substitutability beyond which a green paradox outcome occurs is the same under monopoly as under perfect competition. The monopolist takes a longer time to exhaust the stock R0, and the response of Tm to an increase in substitutability when μ ∈ (μ*, b) is of the same sign as the response of Tc to an increase in substitutability. In absolute value, a given increase in μ decreases the monopolist’s exhaustion time by more than under perfect competition. A Numerical Example We set h = 1, b = 1, λ = 0.5, γ = 0.1, B = 0.1, and r = 0.05. Then a green paradox outcome will occur if μ ∈ (μ*, 1), where μ *  0.30. Let R0/θ = 417.88. We have found that that if μ = 0.50, then under perfect competition, competitive firms will take 500 years to exhaust the stock. Under the same parameter values, the monopolist will exhaust the stock in 980 years. Suppose there is a technological progress such that substitutability increases by 4percent, meaning the new μ is 0.52. Under perfect competition, the time of exhaustion falls by 1.2 percent, meaning Tc is now 494 years (making exhaustion occur 6 years earlier). Under monopoly, the time of exhaustion falls by 1.22 percent (Tm falls from 980 to 968, making exhaustion occur 12 years earlier). 4.7

Concluding Remarks

This chapter explores the possibility of a green paradox associated with an increase in the extent to which non-fossil fuels can be substituted for fossil fuels. I have shown that a technological change that increases marginally the degree of substitutability may cause fossil fuels producers to anticipate lower demand in the future, and to react by increasing current extraction, leading to higher near-term emissions and accelerating climate change damages. Such a green paradox outcome is more likely to occur if the existing degree of substitutability is moderate or high. In fact, if the current degree of substitutability is near zero, then

The Green Paradox under Imperfect Substitutability

81

there will be no green paradox outcome associated with a marginal increase in substitutability. Appendix: Proof of Proposition 4 Let ψt denote the co-state variable. The Hamiltonian is H = (P1t – c)Dr(P1t) – ψtDr(P1t). The necessary conditions are

(P1t − c − ψ t ) DPr + Dr = 0 and the co-state variable must rise at the rate equal to the interest rate. Simple manipulation yields (Pt – c – ψ0ert)(–V) + (W – VPt) = 0, W + V(c + ψ0ert) = 2VPt. Then Pt =

W 1 + (c + ψ 0 e rt ) . 2V 2

When P = P = W V , we have W + V(c + ψ0erTm) = 2W. So c + ψ 0 e rTm =

W , V

W ψ 0 = ⎛ − c⎞ e − rTm , ⎝V ⎠ Pt =

W 1 W 1 ⎡ ⎛ W ⎞ − rTm rt ⎤ + (c + ψ 0 e rt ) = + c+ − c e e ⎥. ⎠ 2V 2 2V 2 ⎢⎣ ⎝ V ⎦

Then exhaustion implies that Tm

Tm

∫ (W − VP ) dt = ∫ t

0

0

{

W−

}

W V ⎡ ⎛ W ⎞ − rTm rt ⎤ − c+ − c e e ⎥ dt = R0 . ⎠ 2 2 ⎢⎣ ⎝ V ⎦

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Thus 1 1 − e − rTm ⎞ ⎛ (W − Vc ) ⎜ Tm − ⎟⎠ = R0. ⎝ r 2 Notes For helpful discussions on the green paradox, I am indebted to Gérard Gaudet, Reyer Gerlagh, Michael Hoel, Chuck Mason, Karen Pittel, Rick van der Ploeg, Rüdiger Pethig, Hans-Werner Sinn, Sjak Smulders, Frank Stähler, and Cees Withagen. I am grateful to Ifo Institute for providing a stimulating research environment. 1. For some early analyses of responses of intertemporal extraction plan to anticipations of taxation or expropriation, see Long (1975) and Long and Sinn (1985). For an overview of recent contributions to the green paradox literature, see van der Werf and Di Maria (2011). 2. See also Strand (2007). Though early models of substitute production of Heal (1976) and Hoel (1978,1983) did not deal with CO2 emissions, they contained all the ingredients from which one can deduce a green paradox result. Welsch and Stähler (1990) provided an early treatment of the dynamic supply response of the green paradox type. 3. There is a large literature on technological changes in the context of exhaustible resources. See, for example, Pittel and Bretschger (2011), and Acemoglu et al. (2012). However, Long and Stähler (2012) were the first to focus on the endogenously generated interest rate effect. 4. Our model is consistent with empirical facts concerning biofuel production. The main producing countries for transport biofuels are the United States, Brazil, and the European Union. Brazil and the United States produced 55 and 35 percent, respectively, of the world’s ethanol in 2009 while the European Union produced 60 percent of the total biodiesel output. There are wide-ranging policies that encourage the substitution away from fossil fuels, especially for motor vehicles. Government mandates for blending biofuels into vehicle fuels have been introduced in no less than 17 countries, and in many states and provinces within these countries. Typical mandates require blending 10 to 15 percent ethanol with gasoline or blending 2 to 5 percent biodiesel with diesel fuel. Recent targets have encouraged greater adoption rate of biofuels in various countries. 5. Subsidies and tax exemptions have complemented quantity-based policies such as setting targets and blending quotas. In the United States, the total value of biofuels supports in 2008 was estimated to be between $9.2 and 11.07 billion. These include consumption mandates, tax credits, import barriers, investment subsidies and general support to the sector such as public research investment. See Koplow (2007, pp. 29–31). 6. Fischer and Salant (2012) examined the green paradox in the presence of a subsidy for renewable resources. However, unlike our model, they assumed perfect substitutablity and constant unit cost of renewable in any given period. They allow unit cost to fall over time through knowledge accumulation. 7. Heal (1985) and Sinn (2008a, b) also model damages from GHGs emissions as a negative externality in production. Most studies, however, specify damages as an additive term in the social welfare function.

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8. As pointed out in Hoel (2008), prior to 2008, “there is little work making the link between climate policies and exhaustible resources when policies are non-optimal or international agreements are incomplete.” He mentioned a few exceptions: Bohm (1993), Hoel (1994) in a static framework. 9. It can also be shown that Dr in increasing in h. This implies that a subsidy for renewables will, at constant P1t, reduce the demand for fossil fuels. 10. One could add a scale parameter that depends on μ, so that when μ increases, the scale parameter also change in such a way that, keeping current consumption constant, utility rises. I refrained from doing this in order to keep the analysis simple. 11. Note, however, that if P1t is much lower than P2t, then an increase in μ will increase xt. 12. Interestingly P2 is independent of μ. 13. It can also be shown that for any given P1t ≥ 0, an increase in the subsidy rate for renewables will result in a lower demand for fossil fuels. It follows that an increase in the subsidy rate for renewables will reduce the demand for fossil fuels at any given price, and make the slope of the demand curve flatter. 14. The condition that f(b) > 0 is sufficient for the root μ* to be real.

References Acemoglu, D., P. Aghion, L. Bursztyn, and D. Hermous. 2012. The environment and directed technical change. American Economic Review 102 (1): 131–66. Bohm, P. 1993. Incomplete international cooperation to reduce CO2 emissions: Alternative policies. Journal of Environmental Economics and Management 24: 258–71. Chakravorty, U., M.-H. Hubert, M. Moreaux, and L. Nostbakken. 2011. Will biofuel mandates raise food prices? Working paper 2011–1. University of Alberta. Di Maria, C., S. Smulders, and E. van der Werf. 2008. Absolute abundance and relative scarcity: Announced policy, resource extraction, and carbon emissions. Working paper 92.2008. FEEM. Eichner, T., and R. Pethig. 2010. Carbon leakage, the green paradox and perfect future markets. Working paper 2546. CESifo. Farzin, Y. H., and O. Tahvonen. 1996. Global carbon cycle and the optimal time path of a carbon tax. Oxford Economic Papers 48 (4): 515–36. Fischer, C., and S. Salant. 2012. Alternative climate policies and intertemporal emissions leakages. Discussion paper DP 12–16. RFF. Gerlagh, Reyer. 2011. Too much oil. CESifo Economic Studies 57 (10): 79–102. Gerlagh, Reyer, and M. Liski. 2008. Strategic oil dependence. Working paper 72.2008. FEEM. Grafton, Q., T. Kompas, and N. V. Long. 2012. Substitution between biofuels and fossil fuels: Is there a green paradox? Journal of Environmental Economics and Management 64 (3): 328–41.

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Groth, Christian, and Poul Schou. 2007. Growth and non-renewable resources: The different roles of capital and resource taxes. Journal of Environmental Economics and Management 53 (1): 80–98. Heal, G. M. 1976. The relationship between price and extraction cost for a resource with a backstop technology. Bell Journal of Economics 7 (2): 371–78. Heal, G. M. 1985. Interaction between economy and climate: A framework for policy design under uncertainty. In V. Smith and A. White, eds., Advances in Applied Microeconomics. Greenwich, CT: JAI Press, 151–68. Hoel, M. 1978. Resource extraction, substitute production, and monopoly. Journal of Economic Theory 19: 28–77. Hoel, M. 1983. Monopoly resource extractions under the presence of predetermined substitute production. Journal of Economic Theory 30: 201–12. Hoel, M. 1994. Efficient climate policy in the presence of free riders. Journal of Environmental Economics and Management 27 (3): 259–74. Hoel, M. 2008. Bush meets Hotelling: Effects of improved renewable energy technology on greenhouse gas emissions. Working paper 2492. CESifo. Hoel, M. 2011a. The green paradox and greenhouse gas reducing investment. International Review of Environmental and Resource Economics 5: 353–79. Hoel, M. 2011b. The supply side of CO2 with country heterogeneity. Scandinavian Journal of Economics 113 (4): 846–65. Hoel, M., and S. Kverndokk. 1996. Depletion of fossil fuels and the impacts of global warming. Resource and Energy Economics 18 (2): 115–36. Koplow, D. 2007. Biofuel—At what cost? Government support for ethanol and biodiesel in the United States: 2007 update. Prepared for the Global Subsidies Initiative (GSI) of the International Institute for Sustainable Development, Geneva, Switzerland. Available at: http://www.globalsubsidies.org/files/assets/Brochure_-_US_Update .pdf. Long, N. V. 1975. Resource extraction under the uncertainty about possible nationalization. Journal of Economic Theory 10 (1): 42–53. Long, N. V., and H.-W. Sinn. 1985. Surprise price shift, tax changes and the supply behaviour of resource extracting firms. Australian Economic Papers 24 (45): 278–89. Long, N. V., and F. Stähler. 2012. Resource extraction and backstop technologies in general equilibrium. Paper presented at the CESifo Venice Summer Institute “The Theory and Empirics of the Green Paradox.” Pittel, Karen, and Lucas Bretschger. 2011. The implications of heterogeneuos resource intensities on technical change and growth. Canadian Journal of Economics. [Revue Canadienne d’Economique] 43 (4): 1173–97. van der Ploeg, F., and C. Withagen. 2012. Is there really a green paradox? Journal of Environmental Economics and Management 64 (3): 342–63. Sinclair, P. J. N. 1992. High does nothing and rising is worse: Carbon taxes should keep falling to cut harmful emissions. Manchester School 60: 41–52.

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Sinclair, P. J. N. 1994. On the optimal trend of fossil fuel taxation. Oxford Economic Papers 46: 869–77. Sinn, H.-W. 2008a. Public policies against global warming: A supply-side approach. International Tax and Public Finance 15 (4): 360–94. Sinn, H.-W. 2008b. Das grüne Paradoxon: Plädoyer für eine illusionsfreie Klimatpolitik. Berlin: Econ Verlag. Sinn, H.-W. 2012. The Green Paradox. Cambridge: MIT Press. Strand, J. 2007. Technology treaties and fossil fuels extraction. Energy Journal (Cambridge, MA) 28: 129–42. Tahvonen, O. 1997. Fossil fuels, stock externalities, and backstop technology. Canadian Journal of Economics. Revue Canadienne d’Economique 30 (4): 855–74. Ulph, A., and D. Ulph. 1994. The optimal time path of a carbon tax. Oxford Economic Papers 46: 857–68. van der Werf, E., and C. Di Maria. 2011. Understanding detrimental effects of environmental policy: The green paradox and beyond. Working paper 3466. CESifo. Welsch, H., and F. Stähler. 1990. On externalities related to the use of exhaustible resources. Journal of Economics 51: 177–95.

5

Fossil Fuels, Backstop Technologies, and Imperfect Substitution Gerard van der Meijden

5.1

Introduction

Technical progress and backstop technologies are now generally considered to be the solution to the sustainability problem raised by the Club of Rome in their alarming report about the limits to growth on our finite planet (Meadows et al. 1972): the economic consequences of the finite availability of natural resources can be mitigated by increasing the productivity of these resources or by finding substitutes that can replace them. However, although this putative panacea releases the economy from the physical scarcity problem, it may also have adverse effects on sustainability by affecting environmental quality. In particular, the literature on the “green paradox” has shown that the introduction of a backstop technology might lead to faster depletion of natural resources, like fossil fuels, and therefore to an increase in current environmental pollution (see Sinn 2008, 2012). For reasons of simplicity these studies assume that the backstop technology is capable of producing a perfect substitute for fossil fuels. The current chapter contributes to the literature by generalizing the analysis to the more realistic case in which the backstop technology delivers a good, but imperfect substitute for nonrenewable natural resources. In so doing, I was able to scrutinize the role of the degree of substitutability on the consequences that the introduction of a backstop technology has for production growth and resource extraction. While this chapter does not incorporate global warming on environmental damages directly, it does lay a foundation for a better understanding of the dynamic pattern of fossil fuel extraction in economies that are undergoing a transition toward the use of alternative energy sources. This foundation is essential because the dynamic pattern of fossil fuel extraction coincides with the pattern of carbon emissions.

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There are currently no technologies available that will be able to provide a perfect substitute for fossil fuels on an economywide level. New electricity production techniques like nuclear fission, nuclear fusion, solar power, hydro power, and wind power all suffer from a relatively low energy concentration: the storage of the generated electricity uses much more space than fossil fuels would to carry the same amount of energy, which makes them less suitable for the transport sector (Sinn 2008, 2012, p. 177). Wind and solar power have the additional problem of being less reliable than fossil fuels because of their intermittent energy supply. At the moment, biofuels are the closest substitute for fossil fuels. Biofuels, however, cannot replace fossil fuels entirely. In aviation, for instance, biofuels have to be blended with conventional petroleum because otherwise they break down and leave deposits under the high temperatures of aircraft fuel systems (Hileman et al. 2009, p. 65). Moreover the energy supply capacity of biofuels is limited, and the production costs are convex in the level of energy generated (Sinn 2008, 2012, p. 177). To put it into perspective, satisfying the current global energy demand from the transport sector alone purely with biofuels would already require the total agricultural area available on earth (see International Energy Agency 2006, p. 289). The transition from extraction of fossil fuels to alternative energy technologies is inevitable given the finiteness of reserves. This transition has started already, but only to a small degree—not enough to provide us with answers to fundamental economic questions: Will the transition to clean energy be abrupt as predicted by existing models? To what extent will the time path of innovation be affected? How important is the degree of substitutability between the backstop technology and the nonrenewable resource? This chapter addresses these questions in the simplest possible model. By taking the imperfect substitutability between new technologies and fossil fuels into account, I was able to analyze the consequences of this feature for the energy transition. My main findings are as follows. If the elasticity of substitution is large enough, the future introduction of the backstop technology will take place abruptly, and thus the outcomes of the model are in line with the results obtained in models with perfect substitution. If substitution possibilities are more limited, however, any transition from fossil fuels to the backstop technology is gradual. The lower the elasticity of substitution between fossil fuels and the backstop technology, the more prolonged will be the period during which a nonnegligible amount of both energy sources is used simultaneously. In line with the literature

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on the green paradox, the availability of a backstop technology leads to more aggressive extraction of the resource in the short run. Using the terminology of Gerlagh (2011), my model thus gives rise to a “weak green paradox.”1However, I also find a “weak green orthodox”: an invention that increases the substitutability between the backstop technology and the nonrenewable resource leads to a short-run decrease in resource extraction. Furthermore I find that the time profile of innovation is nonmonotonic if the elasticity of substitution between the resource and the backstop technology is large enough: innovation first decreases slightly over time, it increases during the early part of the energy transition and then declines to a lower long-run level as the energy transition is completed. Finally, I find that the long-run outcomes of the model are not affected by the substitution possibilities in the energy sector as long as the elasticity of substitution exceeds unity.2 My analysis builds on the so-called Dasgupta–Heal–Solow–Stiglitz (DHSS) model, which integrates nonrenewable resources into the neoclassical growth framework. It includes contributions of Dasgupta and Heal (1974), Solow (1974a, b), and Stiglitz (1974a, b).3 In the analysis of Dasgupta and Heal (1974) and in the related work of Heal (1976), Hoel (1978), and Dasgupta and Stiglitz (1981), the available backstop technology is assumed to provide a perfect substitute for the resource. As a result energy generation will initially rely completely on the resource. Over time the relative price of the resource compared to the backstop technology increases and the backstop is adapted once prices are equalized. More recent contributions also assume perfect substitutability between the resource and the backstop technology (e.g., Tsur and Zemel 2003, 2005; Valente 2011; Van der Ploeg and Withagen 2013). By assuming that the backstop technology is characterized by increasing instead of constant marginal production costs, Hung and Quyen (1993), Tahvonen and Salo (2001), and Chakravorty, Leach, and Moreaux (2012) obtain simultaneous use of the resource and the substitute in their theoretical models. However, these analyses still rely on perfect substitution between the resource and the backstop technology. There are two notable exceptions to the ubiquitous perfect substitution assumption in the literature on the transition from resources to backstop technologies.4 The first is Michielsen (2011) who studies climate policy in a framework of imperfect substitution between a nonrenewable resource and two backstop technologies: a clean and a dirty one. His focus, however, is on the effects of climate policy consisting of taxes on fossil fuels and cost reductions of the clean backstop.

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Moreover the analysis takes place in a partial equilibrium setting, leaving no room for output growth and changes in the energy demand function over time. The other study that takes imperfect substitution into account is Long (2014). He shows that if the existing degree of substitutability between the resource and the backstop technology is moderate or high, a technological change that further increases the degree of substitutability may cause fossil fuel producers to anticipate lower demand in the future, which encourages them to increase extraction immediately. Accordingly, Long (2012) predicts a weak green paradox. The difference from the weak green orthodox that I obtain occurs because Long (2012) uses a partial equilibrium analysis and imposes linear demand functions for the resource and the backstop technology. In this chapter, I develop the simplest possible general equilibrium model that incorporates poor substitution between energy and humanmade factors of production on the one hand, and imperfect substitution between nonrenewable resources and the backstop technologies on the other.5 A dynamic general equilibrium setting is required to account for the linkages between investment in innovation, expenditure on a backstop energy technology, extraction of fossil fuels, and consumption over time. Moreover, by allowing for investment in R&D, I extend the analysis beyond the simple “cake-eating problem” therewith introducing a possibility of obtaining long-run growth in output. In contrast to the DHSS model, I choose for investment in knowledge instead of in physical capital to orient my analysis toward the long run, when technical change rather than capital accumulation is the determinant of output growth. Hence the essential trade-off between current and future consumption that is at the heart of modern growth theory is captured in our model by the allocation of a primary input, namely labor, over production of consumption goods and investment in innovation and by the trade-off between using more fossil fuel today versus leaving more resources underground to extract in the future. For simplicity, the final good and factor markets are characterized by perfect competition. However, the market for intermediate goods is assumed to be monopolistically competitive, because the nonrivalry of knowledge would lead to zero R&D under perfect competition (Romer 1990). The structure of the model is as follows. Final output is produced with intermediate goods and energy, according to a constant elasticity of substitution (CES) specification. The production of intermediate goods requires labor. Energy is derived from a nonrenewable natural resource that can be extracted at zero costs and from a costly backstop

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technology that uses labor. The elasticity of substitution between the resource and the backstop technology is assumed to be larger than unity. In line with the empirical evidence, the elasticity of substitution between energy and intermediate goods is assumed to be smaller than unity (Koetse, de Groot, and Florax 2008). Technological progress in the model is driven by labor allocated to R&D directed at the invention of new intermediate goods. I assume that there are knowledge spillovers from the stock of invented intermediate goods to the resource sector and the backstop sector. I solve analytically for the steady state of the model and I develop a graphical apparatus to study its transitional dynamics. Finally I calibrate and simulate the model to study the behavior of the economy for different degrees of substitutability between the resource and the backstop technology. Throughout the chapter, my focus will be on the decentralized market equilibrium. Although I do not explicitly include pollution from the combustion of fossil fuels, the market equilibrium does not coincide with the social optimum. The reasons are (1) the monopolistic competition in the intermediate goods sector, leading to lower than optimal production of existing varieties, and (2) the intertemporal knowledge spillover, implying a suboptimally low level of investment in the invention of new varieties. The remainder of this chapter is structured as follows. Section 5.2 describes the model. Section 5.3 discusses the solution procedure. Section 5.4 characterizes the transitional dynamics and describes the calibration of the model. Section 5.5 discusses the main results, and section 5.6 concludes. 5.2

The Model

This section describes the model in detail. It first discusses the production and energy generation sectors. Subsequently the process of knowledge generation through research and development will be specified and market equilibrium conditions will be presented. Finally, the behavior of households in the model will be described. Figure 5.1 gives a schematic representation of the goods, factor, and knowledge flows in the model. 5.2.1 Final Good Sector Final output Y is produced with energy E and an intermediate input M, according to the following constant elasticity of substitution (CES) specification:

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Labor

Public knowledge

Nonrenewable resource

L

N

R

LR

Innovation G

LH

Backstop technology

Energy

H

E Final output Y

LK

Differentiated intermediate goods M

Figure 5.1 Schematic representation of goods, factor, and knowledge flows

Y = [θ E(σ −1) σ + (1 − θ ) M (σ −1) σ ]

σ ( σ − 1)

,

(5.1)

where 0 < σ < 1 denotes the elasticity of substitution between intermediate inputs and energy, and 0 < θ < 1 regulates the relative productivity of the two inputs. To simplify the analysis, let us abstract from the accumulation of physical capital. The intermediate input M is a CES aggregate of different varieties of machines k: M=

(∫

N

0

k βj dj

)

1β

,

(5.2)

where N > 0 is the mass of intermediate input producers. The elasticity of substitution between the different varieties is equal to 1/(1 − β). Each firm in the intermediate input sector produces one specific machine variety. Machine producers are assumed to be identical, so that in equilibrium the producer of each machine variety j will choose the same amount of output, so that (5.2) reduces to

Fossil Fuels, Backstop Technologies, Imperfect Substitution

M = NϕK,

93

(5.3)

where K ≡ Nk represents the total input of intermediates and ϕ ≡ (1 − β)/β is a measure of the gains from specialization: while keeping aggregate intermediate goods K constant, the intermediate input M rises with the number of varieties N because of increased specialization possibilities in the use of intermediate goods (e.g., Ethier 1982; Romer 1987, 1990). Final goods producers maximize profits in a perfectly competitive market. They take their output price pY, the prices of intermediate goods pK j , and the energy price pE as given. Relative demand for intermediate goods and energy is therefore given by σ

σ

K ⎛ pE ⎞ ⎛ 1 − θ ⎞ = N −φ (1−σ ) . E ⎜⎝ pK ⎟⎠ ⎜⎝ θ ⎟⎠

(5.4)

5.2.2 Energy Generation Sector The generation of energy E uses the nonrenewable resource R and the backstop technology H, according to the following production function: γ ( γ − 1) γ + (1 − ω )( AR R)(γ −1) γ ⎤⎦ ⎪⎧ ⎡⎣ω ( AH H ) E=⎨ ω 1−ω ⎪⎩( AH H ) ( AR R)

( γ − 1)

if γ ≠ 1, if γ = 1,

(5.5)

where 0 < γ 0 and AR > 0 are productivity indexes. Let us focus on the case where the resource and the backstop technology are gross substitutes, which corresponds with γ > 1. This condition implies that the resource and the backstop can be substituted relatively easily for each other so that the expenditure share of the resource shrinks if the relative supply of the resource R/H goes down. Moreover in this case the resource is not necessary for production.6 The market for energy is characterized by perfect competition. Therefore suppliers of energy take the prices of the nonrenewable resource and the substitute as given. Relative demand from the energy sector for both inputs is given by γ

R ⎛ 1 − ω ⎞ γ ⎛ pH ⎞ ⎛ AR ⎞ = ⎜ ⎟ H ⎝ ω ⎠ ⎜⎝ pR ⎟⎠ ⎝ AH ⎠

γ −1

,

(5.6)

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where pH and pR denote the prices of the backstop technology and the nonrenewable resource, respectively. 5.2.3 Intermediate Goods Sector Each firm in the intermediate goods sector produces a unique machine variety. Before a firm is allowed to produce a certain machine, it first has to buy a patent on the market. The production function for intermediate goods is given by k j = lK j ⇒ K = LK ,

(5.7)

where lK j denotes labor demand by firm j, which is equal for each firm so that lK j = lK, and LK ≡ NlK is aggregate labor demand by the intermediate goods sector. The different machine varieties are imperfect substitutes for each other. As a result, the intermediate goods market is characterized by monopolistic competition. Each producer maximizes profits and faces a demand elasticity of 1/(1 − β) so that all firms charge the same price of a markup 1/β times marginal cost, which equals the wage rate w: pK =

w . β

(5.8)

Because of this markup, firms in the intermediate goods sector make profits, which are used to cover the costs of obtaining a patent. Combining (5.7) and (5.8), profits for each firm are given by

π = pK k − wk = φ

wK . N

(5.9)

In equilibrium the price of a patent will be equal to the present discounted value of the profits generated by the corresponding machine variety. 5.2.4 Backstop Technology Sector Firms in the backstop technology sector use labor to produce the substitute for the nonrenewable resource, according to the simple production function H = ηLH,

(5.10)

where H and LH denote aggregate output of and labor demand from the backstop technology sector. The market for the substitute is

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perfectly competitive. Therefore the price of one unit of the substitute equals its marginal cost: pH =

w . η

(5.11)

5.2.5 Research and Development By undertaking research and development (R&D), intermediate firms invent new intermediate good varieties. Following Romer (1990), I assume that the stock of public knowledge evolves in accordance with the number of invented intermediate goods. New varieties are created according to 1 N = LR N , a

(5.12)

where LR denotes labor allocated to research and a is a productivity parameter. The right-hand side of (5.12) features the stock of public knowledge, to capture the “standing on shoulders effect”: researchers are more productive if the available stock of public knowledge is larger (see Romer, 1990). We can define the innovation rate as g≡

N . N

(5.13)

Free entry of firms in the research sector implies that whenever the cost of inventing a new variety, wa/N, is lower than the market price of a patent, pN, entry of firms in the research sector will take place until the difference is eliminated. As a result free entry gives rise to the following condition: aw ≤ pN N

with equality (inequality) if g > 0 (g = 0).

(5.14)

As argued before, the market price of a patent will be equal to the present discounted value of the profits generated by the corresponding machine variety: z

pN (t ) = ∫ π ( z ) e ∫t ∞

t

r ( s)ds

dz ,

(5.15)

where r denotes the nominal interest rate. Differentiating (5.15) with respect to time, we find the following Hamilton–Jacobi–Bellman equation:

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i

π + pN = rpN ,

(5.16)

which can be interpreted as a no-arbitrage condition that requires investors to earn the market interest rate on their investment in patents. Combining (5.8), (5.9), (5.13), (5.14), and (5.16) obtains an expression for the return to innovation: r =φ

K ˆ if g > 0 , −g+w a

(5.17)

where the hat denotes a growth rate. The return to innovation depends positively on K because of a market size effect and negatively (positively) on g (ŵ) because fast innovation (high wage growth) implies a rapidly decreasing (increasing) patent price. The parameters a and β both have a negative effect on the return to innovation because they are related negatively to the productivity of researchers and the markup on the price of intermediate goods, respectively. The increase of the number of varieties enhances the aggregate productivity of intermediate goods, as shown by (5.3). I assume that this process of knowledge accumulation also generates spillovers to the backstop sector and the resource sector, by using the following specifications for the productivity indexes of the energy inputs: AR = NϕR,

(5.18a)

AH = NϕH.

(5.18b)

Below I will show that the economy will asymptotically converge to a regime in which energy generation relies exclusively on the backstop technology. Given that we are not interested in this regime per se, but merely in the transition from the nonrenewable resource to the backstop technology, we can simplify the analysis by imposing ϕH = ϕ, so that the final regime will be a steady state in which the innovation rate and the income shares are constant. Furthermore, to be on the conservative side, we will assume only moderate knowledge spillovers to the resource sector, by imposing ϕR < ϕ. Assumption 1 summarizes the discussion about knowledge spillovers. Assumption 1 Knowledge accumulation generates spillovers to the resource and backstop sector. Spillovers to the backstop sector are strong: ϕH = ϕ, and spillovers to the natural resource sector are weak: ϕR < ϕ.

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The results of the model depend on this assumption. Nevertheless, the assumption is in accordance with the analysis in Van der Meijden and Smulders (2012a), who use a model with directed technical change to show that the economy converges to a regime in which ϕR = 0. 5.2.6 Factor Markets Equilibrium on the labor market requires that aggregate labor demand from the intermediate goods sector, the backstop technology sector, and the research sector equals the fixed labor supply of L: LK + LH + LR = K +

H + ag = L. η

(5.19)

Resource extraction depletes the resource stock S according to S (t ) = − R (t ) , S (0 ) = S0 , R (t ) ≥ 0 , S (t ) ≥ 0 ,

(5.20)

which implies that total extraction cannot exceed the initial resource stock. 5.2.7 Households The representative household dynasty lives forever, derives utility from consumption of the final good according to a logarithmic specification, and inelastically supplies L units of labor at each time. It owns the resource stock with value pRS and it is the owner of all equity in intermediate goods firms with value pNN. The household maximizes lifetime utility7 ∞

U (t ) = ∫ ln Y ( z ) e − ρ(z −t) dz

(5.21)

t

subject to the flow budget constraint8 V = r (V − pRS) + p RS + wL − pY Y

(5.22)

and a transversality condition lim λ ( z )V ( z ) e − ρz = 0 , z→∞

(5.23)

where ρ denotes the pure rate of time preference, V ≡ pNN + pRS total wealth, and λ the shadow price of wealth. The optimization problem of the representative household gives rise to the familiar rules Yˆ = r − pˆ Y − ρ ,

(5.24)

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pˆ R = r.

(5.25)

The first one, (5.24), is the Ramsey rule, which requires the growth rate of consumption to equal the difference between the real interest rate and the pure rate of time preference. Equation (5.25) is the Hotelling rule, which ensures that owners of the resource stock are indifferent between selling an additional unit of the resource to earn interest at rate r and conserving it to earn a capital gain at rate pˆ R. Although there is no physical capital in the model, the real interest rate is still determined on the market for savings (and investment). The supply of savings is governed by the Ramsey rule (5.24), which can be interpreted as the rate of interest that consumers require for a given rate of consumption growth. The demand for savings wLR = wag results from (5.17) that links the equilibrium innovation rate to the rate of interest. 5.3

Solving the Model

This section describes the procedure that we perform to solve the model. We will first condense the model to a four-dimensional dynamic system. Subsequently we determine its steady state. In the next section we will derive isoclines for the three state variables and perform a numerical analysis to determine the saddle path along which the economy converges to its steady state. 5.3.1 Deriving the Dynamic System The model described in section 5.2 constitutes a dynamic system with two predetermined (state) variables: N and S. Therefore the analysis and the visualization of the dynamics of the dynamic system are complex. However, we are able to condense the model to a four-dimensional block-recursive system of differential equations in the energy income share θ, the backstop expenditure share ω, the innovation rate g, and the reserve-to-extraction rate y ≡ S/R. Beyond simplifying the mathematical analysis, this re-expression of the model also helps clarify the economics behind our results. The variables of the dynamic system have a clear interpretation as they are indicators of energy scarcity, fossil fuel addiction, technical progress, and physical resource scarcity. The income and expenditure shares are defined as follows:

θ≡

pE E pK K ⇒ 1−θ = , pY Y pY Y

(5.26a)

Fossil Fuels, Backstop Technologies, Imperfect Substitution

ω≡

pH H pR R ⇒ 1−ω = . pE E pE E

99

(5.26b)

The system is block recursive in the sense that the system of θ, ω, and g can be solved independently from y. All growth rates in the model can be expressed in terms of θ, ω, and g. Subsequently the differential equation for y can be used to solve for the initial reserve-to-extraction rate, which pins down the initial levels of all variables in the model. In this section we analyze the dynamic (θ, ω, g)-subsystem described in proposition 1, and we relegate the solution to the differential equation for y to appendix (A.6). Proposition 1 The dynamics of the model are described by the following three-dimensional system of first-order nonlinear autonomous differential equations in the variables θ(t), ω(t), and g(t):

θ (t ) = θ (t )[1 − θ (t )][1 − ω (t )](1 − σ )[ r (t ) − wˆ (t ) + ν g (t )] ,

(5.27)

ω (t ) = ω (t )[1 − ω (t )](γ − 1)[ r (t ) − wˆ (t ) + ν g (t )] ,

(5.28)

L ˆ (t ) ) ] , g (t ) = ⎛ − g (t )⎞ [ ρ − Γ 1 (t ) g (t ) − Γ 2 (t ) ( r (t ) − w ⎝a ⎠

(5.29)

with Γ1 ≡ νθ(1 − ω)[(1 − σ)(1 − θ)(ω − β)λ−1 − ω(1 − γ)], Γ2 ≡ θ(1 − θ)(1 − ω)(1 − σ)(ω − β)λ−1 + [1 − ω(1 − ω)θ(1 − γ)], λ ≡ β(1 − θ) + ωθ > 0, ν ≡ ϕ − ϕR > 0, where, at an interior solution, the term r(t) − ŵ(t) is a function of θ(t), ω(t), and g(t): ˆ (t ) = r (t ) − w Proof

L 1 ⎡ (1 − β ) (1 − θ (t )) − {[1 − θ (t )] + ω (t )θ (t )} g (t )⎤⎥ . ⎢ a λ (t ) ⎣ ⎦

See appendix A.3. 䊏

5.3.2 Steady State A steady state of the model is defined as a combination of θ, ω, and g such that θ = ω = g = 0. The only attainable internal steady state of the

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system in proposition 1 that satisfies transversality condition (5.23) is given by9 L g * = (1 − β ) (1 − θ * ) ⎛ + ρ⎞ − ρ , ⎝a ⎠

(5.31a)

ω* = 1.

(5.31b)

Hence, in line with intuition, the steady-state innovation rate depends positively on the maximum attainable innovation rate L/a and the price markup in the intermediate goods sector 1/β, and negatively on the rate of time preference ρ. The steady state value of the innovation rate also depends on the income share of energy in the steady state, θ*. We have to determine this income share separately. In the steady state with ω = 1, the resource will not be used anymore so that R = 0, and the energy generation function (5.5) boils down to E = ωγ

( γ − 1)

AH H .

(5.32)

Substituting (5.32) into (5.4) and using (5.8), (5.11), (5.18), and (5.26) gives the steady-state income share of energy: ⎡ ⎛ 1−θ ⎞σ ⎛ γ θ * = ⎢1 + ⎜ ⎟ ⎜ω ⎣ ⎝ θ ⎠ ⎝

( 1−γ )

β⎞ η ⎟⎠

σ −1 −1

⎤ ⎥ . ⎦

(5.33)

The steady-state energy income share thus depends negatively on the backstop technology’s productivity parameter ω , and positively on β/η, which is the price of a unit of energy generated with the backstop technology relative to the price of intermediate goods. 5.4

Transitional Dynamics

In order to visualize the transition of the economy to the steady state, in this section we derive isoclines for each of the variables θ, ω, and g, along which these variables are constant over time. Subsequently we calibrate the model, develop a graphical apparatus to visualize the transitional dynamics of the model, and numerically solve for the saddle path along which the economy converges to the steady state by using the relaxation algorithm put forward by Trimborn, Koch, and Steger (2008). 5.4.1 Isoclines Imposing θ = 0 in (5.27) and ω = 0 in (5.28), we find that the isoclines for θ and ω coincide:

Fossil Fuels, Backstop Technologies, Imperfect Substitution

g θ =0 = g ω =0

L β (1 − β ) (1 − θ ) a = 2 . β (1 + φR ) (1 − θ ) − [1 − β ( 2 + φR )]θω

101

(5.34)

Appendix A.5 discusses the properties of these isoclines. The innovation isocline is obtained by imposing g = 0 in (5.29), yielding

g g =0

L (1 − β ) (1 − θ ) Λ1 − λ 2 ρ a = , [(1 − θ ) + ωθ ] Λ1 + Λ 2

(5.35)

where we have defined Λ1 ≡ θ(1 − θ)(1 − σ)(ω − β)(1 − ω) + λ[1 − (1 − γ)θ(1 − ω)ω], Λ2 ≡ [1 − β(1 + ϕR)]θ(1 − ω)λ{(1 − θ)(1 − σ)(β − ω) + (1 − γ)ωλ}β−1. Because (5.35) is a complex function of θ and ω, it is cumbersome to construct a phase portrait of the dynamic system for the general case. Therefore we first calibrate the model and then develop a graphical apparatus to visualize the transitional dynamics for the calibrated model. 5.4.2 Calibration Empirical evidence suggests that the elasticity of substitution between energy and human-made factors of production is less than unity. Koetse, de Groot, and Florax (2008) conduct a meta-analysis and find a point estimate for the cross-price elasticity between capital and energy in Europe of 0.338 in the short run and 0.475 in the long run. We take the average of these values to obtain σ = 0.4. According to the estimation results of Roeger (1995), the markup of prices over marginal cost in the manufacturing sector of the US economy over the period 1953 to 1984 varied from 1.15 to 3.14. We impose β = 0.8, which is at the top of the range implied by the estimates for the markup.10 We set the production function parameters θ , ω , and the rate of pure time preference ρ to 0.1, 0.9, and 0.01, respectively. By imposing ϕR = 0.05, we ensure that knowledge spillovers to the resource extraction sector are small. Labor supply L and the initial knowledge stock N(0) are normalized to 1 and 0.1, respectively. In our benchmark calibration, we assume that the nonrenewable resource and the backstop technology are good substitutes, by imposing γ = 50. We will also study scenarios with lower elasticities of substitution.

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The initial resource stock is chosen to get an initial share of resource expenditures in GDP θ(0) of 8.8 percent, to match the average US energy expenditure share in GDP over the period 1970 to 2009 (US Energy Information Administration 2011).11 We use the research productivity parameter a to obtain an initial consumption growth rate Ĉ(0) of 1.7 percent, which is equal to the average yearly growth rate of GDP per capita in the United States over the period 1970 to 2010 (Conference Board 2011). By setting the backstop production parameter η equal to 3.35, the implied initial ratio between the per unit of energy price of the backstop technology and the resource amounts to 3.12 According to US Energy Information Administration (2012), the reserve-to-production ratios for oil, natural gas, and coal in 2008 were 44, 58, and 127, respectively. Our implied initial reserve to extraction rate of 50 lies within this range.13 5.4.3 Graphical Apparatus The dynamic system described in section 5.3.1 is three dimensional. Drawing and analyzing a three-dimensional phase portrait, however, is complex. To simplify the analysis, we therefore show the dynamics of the model in two-dimensional (θ, g)-space while fixing ω at three different values, namely at its minimum (ω = 0), medium (ω = 12 ), and maximum (ω = 1) value. Because the locations of the isoclines (5.34) and (5.35) depend on ω, we obtain figures with three different isoclines for the energy income share θ and three different isoclines for the innovation rate g. Figure 5.2 shows the income share and innovation isoclines in the (θ, g) plane. The dashed, dotted, and solid lines correspond to ω = 0, ω = 0.5, and ω = 1, respectively. Panel a depicts the income share isoclines and panel b shows the innovation isoclines for the calibrated model. The horizontal arrows in panel a correspond to the direction of the income share development over time and the vertical arrows in panel b indicate the direction of change of the innovation rate. If the backstop technology is not used at all, namely when ω = 0, the isoclines in figure 5.2, which are depicted by the dashed lines, coincide with the ones that were obtained in the resource regime in Van der Meijden and Smulders (2012b), where the resource and the backstop technology are perfect substitutes. At the other extreme, if only the backstop technology is used for energy generation, the isoclines are depicted by the solid lines in the figure and the innovation isoline coincides with the one that is obtained in the backstop regime of Van der Meijden and Smulders

Fossil Fuels, Backstop Technologies, Imperfect Substitution

(a)

103

g

0.10

θ = 0 |ω = 0 θ = 0 |ω = 0.5

0.08

0.06

θ = 0 |ω = 1 0.04

0.02

0.0

(b)

0.0

0.2

0.4

0.6

0.8

1.0

θ

g

0.10

0.08

g = 0 |ω = 0.5

0.06

g = 0 |ω = 0 0.04

g = 0 |ω = 1 0.02

0.0

0.0

0.2

0.4

0.6

0.8

1.0

θ

Figure 5.2 Income share and innovation rate isoclines: (a) Energy income share; (b) innovation rate. The dashed lines correspond to the isoclines for ω = 0, the dotted lines to the isoclines for ω = 0.5, and the solid lines to the isoclines with ω = 1. The horizontal arrows indicate the direction of the income share dynamics for each isoline; the vertical arrows indicate the direction of the innovation rate dynamics for each isocline.

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(2012b). The income share isocline in panel a is linear at these two extremes (flat at ω = 0 and downward sloping at ω = 1) but concave and downward sloping for the intermediate case with ω = 0.5. The innovation isocline in panel b is linear and downward sloping at the backstop extreme with ω = 1. It is concave and downward sloping for the other cases. Its intersection with the vertical line θ = 1 is decreasing in the value of ω, but the vertical intercept is independent of ω. Figure 5.3 shows both isoclines (again for three different values of ω) and the saddle path along which the economy converges from point A to the steady state, which is indicated with point B.14 The saddle path is located below all depicted income share isoclines, implying that the income share of energy is increasing over time. From (5A.13) and (5A.15) it is clear that sgn ω = sgn θ , so that the backstop expenditure share is increasing over time during the transition to the steady state. g

0.10

θ = 0 |ω = 0 0.08

θ = 0 |ω = 0.5

g = 0 |ω = 0.5

0.06

θ = 0 |ω = 1 g = 0 |ω = 0 0.04

g = 0 |ω = 1 0.02

0.0

0.0

0.2

0.4

0.6

0.8

1.0

Figure 5.3 Transitional dynamics. The gray and black lines are the isoclines for the income share and the innovation rate, respectively. The dashed lines correspond to the isoclines for ω = 0, the dotted lines to the isoclines for ω = 0.5, and the solid lines to the isoclines with ω = 1. The horizontal arrows indicate the direction of the income share dynamics for each income share isoline that is depicted. The vertical arrows indicate the direction of the innovation rate dynamics for each innovation rate isocline that is depicted. The fat dots represent the saddle path leading to the steady state at point B.

θ

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The first part of the saddle path is located below the relevant innovation isocline, so that the innovation rate is initially decreasing over time. However, after the saddle path has crossed the innovation isocline, the innovation rate starts to increase over time. During the convergence to the steady state, the backstop expenditure share increases, so that the innovation isocline initially becomes more concave (see the difference between the black dashed and the black dotted lines). Hence, during the phase of increasing extraction, both the actual innovation rate that moves along the saddle path and the value of the innovation rate attached by the innovation isocline to the current actual income share of energy move upward. As soon as the isocline passes the point on the saddle path at which the economy is located, the innovation rate starts to decline again and converges to point B at the innovation isocline corresponding to ω = 1. Therefore, whereas during the transition, when 0 < ω < 1, both the resource and the backstop technology are used, over time ω increases and in the long run the economy will converge to a regime with ω = 1 in which only the backstop technology will be used for energy generation. We will interpret the dynamic behavior of the energy income share, the backstop technology expenditure share, and the innovation rate in the next section. 5.5

Results

This section discusses our simulation results. We determine the time paths of the innovation rate, resource extraction, energy generation with the backstop technology, and the growth rate of consumption numerically. Our focus will be on the effect of the ease with which the backstop technology is able to replace the nonrenewable resource. Therefore we simulate the model for various levels of the elasticity of substitution between the two energy sources. Moreover we compare our results to those of Van der Meijden and Smulders (2012b), who analyze the extreme cases of perfect and no substitution between the resource and the backstop technology. Figure 5.4 shows the time profiles of g, R, H, and Ĉ for different scenarios: perfect, good, intermediate, moderate, and no substitution possibilities. The solid black lines represent the outcomes of the first scenario, where the resource and the backstop technologies are good substitutes (γ = 50). The dotted line represents the intermediate substitution scenario (γ = 10), and the dashed black line gives the time path for the moderate substitution scenario (γ = 5). The gray lines depict the

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(a)

(b)

Figure 5.4 Transitional dynamics with varying substitution possibilities. The five different lines represent cases with a different substitution elasticity between the backstop technology and the nonrenewable resource. The solid black line represents the scenario with γ = 50, the dotted black line represents the scenario with γ = 10, the dashed black line represents the scenario with γ = 5, the solid gray line represents the scenario with γ → ∞, and the dotted gray line represents the scenario with γ = 0. The underlying parameter values are a = 2.5, β = 0.8, ϕR = 0.05, η = 0.215, ρ = 0.01, σ = 0.4, θ = 0.9, ω = 0.9, and L = 1. The backstop productivity parameter η equals 3.35 in scenario 1. In the other scenarios, η is adjusted to obtain the same θ* as in scenario 1. The initial knowledge stock N0 equals 0.1. The initial resource stock S0 equals 275 to obtain θ0 = 0.912 in scenario 1.

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(c)

(d)

Figure 5.4 Continued

107

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extreme cases: the solid gray lines correspond to the scenario in which the backstop and the resource are perfect substitutes and the dotted gray lines represent the time paths in economies without a backstop technology.15 Panel a of figure 5.4 gives the time profiles of the innovation rate.16 In the good substitutes scenario (see the solid black line), though less pronounced, we obtain the same nonmonotonicity as in the model with perfect substitution. The reason is that the same mechanism is at work. During the run-up to the backstop technology in the perfect substitutes model, the innovation rate increases to prevent a downward jump in consumption at the moment that the economy switches completely to the backstop technology: the innovation rate necessarily jumps down to free labor as soon as energy generation with the backstop technology jumps up. In our imperfect substitutes model the change to the backstop technology occurs less abruptly, but households still want to smooth consumption over time. As a result, by increasing savings (and thus innovation) before generating energy with the backstop technology will absorb a substantial part of the economy’s productive capacity, households effectively transfer part of their resource wealth to the backstop era during which energy generation comes at cost of production. For lower levels of the elasticity of substitution, the transition to the backstop technology occurs more gradually. As a result the nonmonotonicity in the time profile of the innovation rate becomes weaker in the intermediate scenario and even disappears completely in the scenario with only moderate substitution possibilities (see the dotted and dashed black lines). Intuitively, if the generation of energy with the backstop is not going to increase from almost zero to its positive longrun value very quickly, there is no need to smooth consumption over time by increasing investment considerably to compensate for the fall in output just after the quick introduction of the backstop technology. It is clear from panel a that the good substitutes scenario is close to the perfect substitutes case. The intermediate and moderate substitutes scenarios differ considerably from the perfect substitutes case during the transition to the backstop technology. In the short and long run, however, all scenarios with an elasticity of substitution exceeding unity lead to identical innovation rates, in line with the observation that the nonmonotonicity in the time profile is a transitional result of consumption smoothing between the resource and the backstop era. The model without a backstop technology (see the dotted gray line) generates a

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monotonically declining innovation rate that starts below the innovation rates in the scenarios with a backstop technology. Panel b shows a similar record for the growth rate of consumption. Consumption growth declines substantially just before the introduction of the backstop technology in the perfect and good substitution scenarios. The reason is the increase in innovation during the run-up to the backstop technology, which comes at cost of consumption possibilities. The nonmonotonicity in the time profile becomes weaker for smaller values of the elasticity of substitution between the backstop technology and the resource. In the long run, consumption growth is the same in all scenarios in which there exists a backstop technology. The time profiles of energy generation with the backstop technology in Panel c confirm that the switch from the resource to the backstop technology occurs more gradually if substitution possibilities between the two energy sources are limited. In the good and perfect substitutes scenario, there is a clear distinction between a resource era where the backstop technology is not used (or only to a very small extent) and a backstop era where the resource is effectively not used anymore. This clear distinction disappears when the elasticity of substitution between the backstop technology and the resource becomes smaller, as this results in a prolonged period of simultaneous use of both energy sources. Panel d shows that the time profile of resource extraction may be increasing temporarily. This is a consequence of the imperfect substitutability between energy and intermediate goods. To see this, we decompose the income share of the resource into the income share of energy and the fossil fuel share in total energy expenditures: pR R ω = (1 − ω )θ ⇒ Rˆ = − ωˆ + θˆ − ρ , 1−ω pY Y where the second equality uses the rules of Ramsey and Hotelling (5.24) and (5.25). Hence, if the energy income share θ increases fast enough, and the backstop expenditure share ω increases not too fast, resource extraction grows over time. Panel d furthermore reveals that in line with the literature on the green paradox, the availability of a backstop technology leads to frontloading of resource extraction. At the same time, however, we also find a “weak green orthodox”: an invention that makes the backstop technology a closer substitute to the nonrenewable resource leads to an

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immediate decrease in resource extraction. The reason is that an increase in the elasticity of substitution between the backstop technology and the resource lowers the initial use of the backstop technology and postpones the moment at which the usage of the backstop technology grows beyond a negligible amount.17 As a result there will be a longer period during which energy generation relies heavily on the resource, so initial extraction must go down. In the long run the outcomes are the same in the four scenarios with a backstop technology available: the economy will asymptotically approach a regime in which only the backstop technology will be used for energy generation. To obtain this long-run neutrality of the elasticity of substitution between the resource and the backstop technology, we endogenously vary the backstop productivity parameter η so that the long-run energy income share is the same in each scenario; see (5.33).18 Summarizing the results and linking them to the different features of the model, we first find a nonmonotonicity of innovation and consumption growth for high values of the elasticity of substitution, due to consumption smoothing of the households. Second, the model generates simultaneous use of the resource and the backstop technology because those two inputs are imperfect substitutes. Third, the time path of resource extraction can be temporarily upward sloping because energy and intermediate goods are gross complements. Fourth, the introduction of the backstop technology leads to a weak green paradox because of the Hotelling rent on the nonrenewable resource. Finally an increase in the substitutability between energy inputs gives rise to a weak green orthodox because of the gross complementarity of intermediate goods and energy, the effect on the initial importance of the backstop technology and therefore on the subsequent increase in its use, and the effect of changes in future energy demand due to output growth. 5.6

Conclusion

This chapter has investigated the effect of different degrees of substitutability on the transition from a nonrenewable resource to a backstop technology. For this purpose I have constructed a general equilibrium endogenous growth model in which growth is driven by R&D and energy can be generated by a nonrenewable resource and a backstop technology. The resource and the backstop technology are good, but imperfect substitutes for each other. My analysis takes into account that

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energy generation with the backstop technology is costly and that energy and human-made factors are poor substitutes. The steady-state equilibrium is determined analytically. I calibrate and simulate the model to visualize its transitional dynamics. Contrary to the literature that simplifies the analysis by assuming perfect substitution, I do not find different regimes of energy generation. The economy gradually shifts from mainly using the nonrenewable resource to mainly using the backstop technology for energy generation. In other words, I find a prolonged period of simultaneous use of both energy sources. The lower the elasticity of substitution between the resource and the backstop technology, the longer is the period during which a nonnegligible amount of both energy sources is used simultaneously. If the elasticity of substitution between the inputs in energy generation is large enough, my results come close to those obtained in models with perfect substitution that are otherwise similar. In particular, I find a strong increase in investment during the transition to the backstop technology, similar to what I reported in Van der Meijden and Smulders (2012b). This result disappears if substitution possibilities are more modest. The availability of a backstop technology results in front-loading of resource extraction, in line with the literature on the green paradox. At the same time, however, I find a weak green orthodox: an invention that increases the substitutability between the backstop technology and the nonrenewable resource leads to a short-run decrease in resource extraction. Although the transition to the backstop technology thus crucially depends on the ease with which the resource can be replaced by the backstop technology, the long-run outcomes of the model are not affected by the substitution possibilities in the energy sector as long as the elasticity of substitution exceeds unity. Because of its effect on resource extraction, the value of the elasticity of substitution in the energy sector is relevant for the strength of the weak green paradox. However, to address the role of imperfect substitution for the strong green paradox, the model needs to be extended with stock-dependent extraction costs and feedback effects of pollution from resource combustion on either production or utility so that the discounted value of environmental damages can be calculated. Another direction for future research would be to endogenize the direction of technical change, in order to investigate the interaction between the backstop technology and the direction and pace of technological progress.

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Appendix This appendix contains the derivations of the mathematical results in the main text of the chapter. A.1 Energy Price Index The first-order conditions for the allocation of expenditure between the resource R and the backstop H in the energy sector are given by 1 ( γ − 1) ∂E = ⎡ω ( AH H )(γ −1) γ + (1 − ω )( AR R)(γ −1) γ ⎤⎦ ω ( AH H )−1 γ AH = pH , ∂R ⎣

(5A.1) 1 ( γ − 1) ∂E (1 − ω )( AR R)−1 γ AR = pR . = ⎡⎣ω ( AH H )(γ −1) γ + (1 − ω )( AR R)(γ −1) γ ⎤⎦ ∂R

(A.2) Combining (5A.1), (5A.2), the first row of (5.5), and pEE = pHH + pRR obtains an expression for the energy price index: 1 ( 1−γ )

1−γ 1−γ ⎧ ⎡ pH ⎤ ⎡ pR ⎤ ⎫ + − pE = [ω (1 − ω )]−1 ⎨ω ⎢(1 − ω ) ( 1 ω ) ω ⎢⎣ AR ⎥⎦ ⎬ AH ⎥⎦ ⎩ ⎣ ⎭

.

(5A.3)

Converting (5A.3) into growth rates, we get ⎛ pH ⎞ pˆ E = ω γ ⎜ ⎝ AH pE ⎟⎠

−γ

(

ˆ H + (1 − ω )γ ⎛ pR ⎞ pˆ H − A ⎜⎝ A p ⎟⎠ R E

)

1−γ

(1 − ω ) ( pˆ R − Aˆ R ) .

(5A.4)

By using (5.6) into the first row of (5.5), we can rewrite (5A.4) to obtain (5A.12). A.2 Flow Budget Constraint In this section we derive the flow budget constraint of the households (5.22). Total wealth is equal to V = pNN + pRS, so that the change in wealth is given by V = p N N + pN N + p RS + pRS = p N N + pN N + p RS − pR R,

(5A.5)

where the second equality uses (5.20). Nominal GDP can be written as pY Y = pK K + pR R + pH H = π N + wLK + pR R + pH H = rpN N − p N N + wLK + pR R + pH H ,

(5A.6)

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where the second and third equality use (5.9) and (5.16), respectively. Using (5A.6) to substitute for pRR in (A.5), we obtain V = pN N + p RS − pY Y + rpN N + wLK + pH H = rpN N + p RS + wL − pY Y , (5A.7) where we have used (5.10), (5.1), (5.14), and (5.19) for the second equality. Using the definition of wealth again, we get (5.22). A.3 Dynamic System In this section we will derive the differential equations for θ, ω, and g. Because each of these will feature the rate of return to innovation, we start by rewriting (5.17) in terms of these three variables. Using the definitions in (5.26), we can express the output of the substitute as follows H=

ωθ pK ωθ η = , 1 − θ pH 1 − θ β

(5A.8)

where the second equality uses the pricing equations (5.8) and (5.11). Substitution of (5A.8) into the labor market equilibrium (5.19) yields K=

(1 − θ ) β (L − ag ) . ωθ + (1 − θ ) β

(5A.9)

We use this expression to substitute for K in (5.17), so that the return to innovation becomes ˆ ≥ r−w

1 ⎡(1 − β ) (1 − θ ) L − (1 − θ ) + ωθ g ⎤ , [ ] ⎥ a ωθ + (1 − θ ) β ⎢⎣ ⎦

(5A.10)

with equality if g > 0. When we substitute definition (5.26) into (5.4), the relative factor income share and its growth rate can be written as σ

⎛ pE ⎞ θ ⎛ θ ⎞ N φ (1−σ ) ⎜ ⎟ =⎜ ⎟ ⎝ ⎠ ⎝ pK ⎠ 1−θ 1−θ

1−σ

⇒ θˆ = (1 − θ ) (1 − σ )[ pˆ E − ( pˆ K − φ g )] , (5A.11)

Consequently the income share of energy increases if the effective relative price of energy and intermediates increases (the relative productivity change is captured by the term ϕg) because we have assumed that σ < 1, implying that energy an intermediates are bad substitutes. In appendix A.1 we show that energy price changes according to

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pˆ E = ω ( pˆ H − φ H g ) + (1 − ω ) ( pˆ R − φR g ) .

(5A.12)

Substitution of (5A.12) into the final result of (5A.11), we obtain differential equation (5.27) in the main text, which governs the evolution of the income share of energy:

θ = θ (1 − θ ) (1 − σ ) (1 − ω )[ pˆ R − φR g − ( pˆ K − φ g )] ˆ + ν g], = θ (1 − θ ) (1 − σ ) (1 − ω )[ r − w

(5A.13)

where ν measures the bias in technical change and we have used the pricing equations (5.8), (5.11), and the Hotelling rule (5.25). This expression reveals that the income share of energy goes up if the effective relative price of the resource and intermediates increases. By using (5.18) and definition (5.26) in the relative factor demand function (5.6), we get the following expression for the expenditure share in the energy sector:

ω ω ⎞ γ ν (γ −1) ⎛ pR ⎞ N =⎛ ⎜⎝ p ⎟⎠ 1−ω ⎝ 1−ω⎠ H

γ −1

.

(5A.14)

Converting (5A.14) into growth rates and multiplying the result by ω, we obtain differential equation (5.28) in the main text, which describes the evolution of the expenditure share of the backstop technology in the energy sector:

ω = ω (1 − ω ) (γ − 1)[ pˆ R − φR g − ( pˆ H − φ g )] ˆ + ν g], = ω (1 − ω ) (γ − 1)[ r − w

(5A.15)

where the second equality uses the pricing equation (5.11) and the Hotelling rule (5.25) again. Equation (5A.15) shows that the expenditure share on the backstop technology goes up if the relative price of the backstop technology decreases because γ > 1, implying that the resource and the backstop technology are good substitutes. In order to derive the differential equation for the innovation rate g, we first convert the labor market equilibrium condition (5A.9) into growth rates: Kˆ = −

g 1 ⎡ ωθ θˆ + ωθωˆ ⎤ − . ⎥ ⎢ β (1 − θ ) + θω ⎣ 1 − θ ⎦ ( L a) − g

(5A.16)

The income share definition (5.26) implies that 1−θ 1−θ ˆ ˆ )⎤⎦ , ⎡ pˆ K + Kˆ − pˆ Y + Yˆ ⎤⎦ = − ⎡ K + ρ − (r − w θˆ = − θ ⎣ θ ⎣

(

)

(5A.17)

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where the second equality uses the pricing equation (5.8) and the Ramsey rule (5.24). Substituting of (5A.10) and (5A.16) into (5A.17), we find the differential equation (5.29) in the main text. A.4

Steady States

Proposition 2 The only attainable internal steady state that satisfies the transversality conditions is given by (5.31). The other four steady states of the model satisfy g* =

L > 0, ω * = 1 > 0, a

(5A.18a)

g* =

L > 0, θ * = 0, ω * = 0, a

(5A.18b)

g* =

L (1 − β ) − βρ , θ * = 0, ω * = 0, a

(5A.18c)

βρ < 0, 1 − β (1 + φR ) L (1 − β )[1 − β (1 + φR )] + β 2 (1 + φR ) ρ a * θ = . L (1 − β )[1 − β (1 + φR )] + ρ [β 2 (1 + φR ) + ω [1 − β (2 + φR )]] a g* = −

(5A.18d)

Proof The derivation of (5.31) is provided in the text. The first two steady states (5A.18a) and (5A.18b) do not satisfy the transversality condition, since substitution of K* = L − ag* = 0 into (17) implies that (r − ŵ)* = −g* < 0 and the transversality condition (5.23) in growth rates requires that ˆ (t ) − r (t ) ≤ 0 ⇒ lim r (t ) − w ˆ (t ) ≥ 0 , lim = pˆ N (t ) + N t→∞

t→∞

(5A.19)

where the second equality uses (5.14) and (5.25). Hence the two steady states with (r − ŵ)* = −g* < 0 do not satisfy the transversality condition. Steady state (5A.18c) is located at the intersection of the innovation locus with the θ = 0 line, and below the income share locus in (θ, g) space. It is immediately clear from the dynamics in figure 5.3 (θ > 0 ) that this steady state cannot be attained. The economy can only be situated here if there is an infinite amount of oil available from the beginning (so that θ* = 0), which is impossible. Steady state (5A.18d) cannot be an internal equilibrium, because this requires g > 0. 䊏

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A.5 Properties of Differential Equations and Isoclines The first-order derivative of the income and expenditure share isoclines with respect to θ is given by ∂ ( g θ =0 ) ∂θ

L (1 − β ) βω [1 − β (2 + φR )] ∂ ( g ω =0 ) . = =− 2 a ∂θ [β (1 + φR ) (1 − θ ) − [1 − β (2 + φR )]θω ]2

(5A.20)

Hence we have β ( 2 + φR ) > 1 ⇒ ∂ ( g θ =0 ) ∂θ < 0 , and the isocline has a vertical asymptote at θ = 1 − β2(1 + ϕR) − ω[1 − (2 + ϕR)β] ∈ (0, 1)

if β(2 + ϕR) < 1. (5A.21)

The first-order derivatives of the income and expenditure share differential equations are given by

θ (1 − θ ) (1 − σ ) (1 − ω )[ β 2 (1 + φR ) (1 − θ ) − [1 − β ( 2 + φR )]θω ] ∂θ , (5A.22) =− ∂g βλ

(γ − 1) (1 − ω )ω [β 2 (1 + φR ) (1 − θ ) − [1 − β (2 + φR )]θω ] ∂ω . =− ∂g βλ

(5A.23)

Hence sgn ∂θ ∂g = sgn ∂ω ∂g and β ( 2 + φR ) > 1 ⇒ ∂θ ∂g < 0 , ∂ω ∂g < 0. A.6 Initial Condition To determine the initial point [θ(0), ω(0), g(0)], we exploit the fact that total resource extraction over time should be equal to the initial resource stock. From the definition (5.26) we have E = pRR/[pE (1 − ω)]. If we additionally define y ≡ S/R as the reserve-to-extraction rate, and use both definitions in (5.4), we obtain

θ θ ⎛ pE (1 − ω ) yK ⎞ = ⎟⎠ S 1 − θ 1 − θ ⎜⎝ pR

( 1−σ ) σ

.

(5A.24)

Combining (5A.11), (5A.14), (5.8), and (5.11) obtains pE η ⎛ θ ⎞ 1 (1−σ ) ⎛ θ ⎞ = ⎜⎝ 1 − θ ⎟⎠ pR β ⎝ 1 − θ ⎠

σ ( σ − 1)

1 ( 1−γ ) γ ⎛ ω ⎞ ⎛ ω ⎞ ⎝ 1−ω⎠ ⎝ 1−ω ⎠

( 1−γ )

Nν .

(5A.25)

Substitution of (5A.25) into (5A.24), we obtain an expression for y in terms of the system variables θ, ω, g, and the state variables N and S:

ω ⎞ y=⎛ ⎝ 1−ω⎠

γ ( γ − 1)

γ ⎛ ω ⎞ ⎝ 1−ω⎠

( 1−γ )

[θω + (1 − θ ) β ]SN −ν θωη ( L − ag )

.

(5A.26)

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For given values of N(0) and S(0), (5A.26) gives a relationship between y and the initial values of θ, ω, and g. Converting (5A.24) into growth rates, and using (5.8), (5.11), (5.25), (5.26), (5A.12), (5A.13), (5A.15), and Ŝ = −y−1, we obtain a differential equation for y: ˆ + ν g ) + ρ y − 1. y = − [(1 − ω ) (1 − σ ) (1 − θ ) + ω (1 − γ )]( r − w

(5A.27)

Substitution of (5A.10) into (5A.27) gives y = − [(1 − ω ) (1 − σ ) (1 − θ ) + ω (1 − γ )]

(1 − θ ) + ωθ ⎞ ⎫ ⎧ (1 − β ) (1 − θ ) L ⎛ g ⎬ y + ρ y − 1. + ⎜ν − ⎨ ⎝ a θω 1 θ β θω + ( − ) + (1 − θ ) β ⎟⎠ ⎭ ⎩

(5A.28)

Imposing y = 0 in (5A.28) gives the reserve-to-extraction rate isocline. The equality between total extraction and the initial resource stock, which necessarily follows from (5.20) and the positive marginal product of energy, requires y to converge to this isocline. Because the time paths of θ, ω, and g are already determined, the y-isocline pins down the steady state value of y. The differential equation (5A.28) can subsequently be used to construct the saddle path of y through (θ, ω, g, y) space. Because at t = 0, y must be located at this saddle path and y must also satisfy (5A.26), the intersection point of (5A.26) and the saddle path of y determines the initial point [θ(0), ω(0), g(0), y(0)]. Notes The author gratefully acknowledges financial support from FP7-IDEAS-ERC Grant No. 269788. Moreover the author would like to thank Rick van der Ploeg, Sjak Smulders, Ralph Winter, Cees Withagen, and participants of the CESifo Workshop on the Theory and Empirics of the Green Paradox (Venice, July 2012) for helpful comments. 1. In the terminology of Gerlagh (2011), a weak green paradox arises if “[the anticipation of] a cheaper clean energy technology increases current emissions.” 2. Long-run outcomes are dramatically affected if the elasticity of substitution between the resource and the backstop drops below unity, as the model would then converge to a different steady state. 3. Recently Benchekroun and Withagen (2011) have developed a technique to calculate the closed-form solution to the DHSS model. 4. Smulders and van der Werf (2008) also allow for imperfect substitution in a model with resource extraction, but in their analysis both resources are nonrenewable. 5. For the perfect substitutes case, see Van der Meijden and Smulders (2012b). 6. Following Dasgupta and Heal (1979), a factor of production is called “necessary” if output would be zero without this factor, that is, if Y = F(R, X) and F(0, X) = 0, where Y is output, F(·) the production function, R the necessary factor, and the vector X represents

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all other factors of production. A necessary exhaustible resource needs to be distinguished from an “essential” exhaustible resource: an exhaustible resource is essential if, due to its necessity, feasible consumption must necessarily decline to zero in the long run (Dasgupta and Heal 1979, pp. 197–98). 7. Note that final output cannot be stored and there is no physical capital accumulation in the model, so that consumption equals output, namely C = Y. Households are still able to intertemporally allocate consumption possibilities by choosing resource extraction and investment in R&D firms. 8. Appendix A.2 derives the flow budget constraint of the households. 9. Appendix A.4 discusses the other four steady states of the model and shows why they cannot be equilibria. 10. The results for a value of β at the bottom of the implied range are available from the author on request. 11. We attribute energy expenditure entirely to resource expenditure, although part of the energy expenditure in the data consists of factor costs. Taking this distinction into account would imply a smaller initial resource expenditure share, without affecting the dynamics of the model. 12. From (5A.14) this ratio is given by [ω (0 ) (1 − ω (0 ))]

1 ( 1−γ )

.

13. The initial expenditure on quality improvement as a share of GDP is equal to 15 percent. Given that expenditure on innovation is the only investment possibility in the model, this number should be interpreted as the aggregate investment share in the economy. 14. The start point of the saddle path, point A in figure 5.3, is determined by imposing that total resource extraction equals the initial resource stock, as shown in appendix A.6. 15. The current model nests the model of Van der Meijden and Smulders (2012b): by taking the limits γ → ∞ and ω → 0, we obtain the specification with perfect and no substitutes, respectively. 16. Note that this is not innovation in the backstop technology, but in intermediate goods. A reasoning that the pattern in panel a of figure 5.4 can be explained by the fact that investment in clean energy should be concentrated around the time of implementation of the backstop technology therefore does not apply. 17. Recall the extreme situation of no backstop use until the regime shift in the case of perfect substitutes that Van der Meijden and Smulders (2012b) describe. 18. When keeping η constant, the long-run effects will differ only marginally between the scenarios. The qualitative results of the model remain unchanged.

References Benchekroun, H., and C. Withagen. 2011. The optimal depletion of exhaustible resources: A complete characterization. Resource and Energy Economics 33 (3): 612–36. Chakravorty, U., A. Leach, and M. Moreaux. 2012. Cycles in nonrenewable resource prices with pollution and learning-by-doing. Journal of Economic Dynamics and Control 36 (10): 1448–61).

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Conference Board. 2011. Total Economy Database. New York: Conference Board. Dasgupta, P., and G. Heal. 1974. The optimal depletion of exhaustible resources. Review of Economic Studies 41: 3–28. Dasgupta, P., and G. Heal. 1979. Economic Theory and Exhaustible Resources. Cambridge, UK: Cambridge University Press. Dasgupta, P., and J. Stiglitz. 1981. Resource depletion under technological uncertainty. Econometrica 49 (1): 85–104. Ethier, W. J. 1982. National and international returns to scale in the modern theory of international trade. American Economic Review 72 (3): 389–405. Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57 (1): 79–102. Heal, G. 1976. The relationship between price and extraction cost for a resource with a backstop technology. Bell Journal of Economics 7 (2): 371–78. Hileman, J. I., D. S. Ortiz, J. T. Bartis, H. M. Wong, P. E. Donohoo, M. A. Weiss, and I. A. Waitz. 2009. Near-Term Feasibility of Alternative Jet Fuels. Santa Monica: RAND Corporation. Hoel, M. 1978. Resource extraction, substitute production, and monopoly. Journal of Economic Theory 19 (1): 28–37. Hung, N. M., and N. V. Quyen. 1993. On R&D timing under uncertainty: The case of exhaustible resource substitution. Journal of Economic Dynamics and Control 17 (5–6): 971–91. International Energy Agency. 2006. Energy Technology Perspectives. Paris: IEA Publications. Koetse, M. J., H. L. de Groot, and R. J. Florax. 2008. Capital-energy substitution and shifts in factor demand: A meta-analysis. Energy Economics 30 (5): 2236–51. Long, N. V. 2014. The green paradox under imperfect substitutability between clean and dirty fuels. Mimeo. CESifo Venice Summer Institute. Meadows, D. H., D. L. Meadows, J. Randers, and W. W. Behrens, III. 1972. The Limits to Growth. New York: Universe Books. Michielsen, T. 2011. Brown backstops versus the green paradox. Discussion paper 2011110. CentER, Tilburg University. Roeger, W. 1995. Can imperfect competition explain the difference between primal and dual productivity measures? Estimates for U.S. manufacturing. Journal of Political Economy 103 (2): 316–30. Romer, P. M. 1987. Growth based on increasing returns due to specialization. American Economic Review 77 (2): 56–62. Romer, P. M. 1990. Endogenous technological change. Journal of Political Economy 98 (5): S71–S102. Sinn, H.-W. 2008. Public policies against global warming: A supply side approach. International Tax and Public Finance 15 (4): 360–94. Sinn, H.-W. 2012. The Green Paradox: A Supply-Side Approach to Global Warming. Cambridge: MIT Press.

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Smulders, S., and E. van der Werf. 2008. Climate policy and the optimal extraction of high- and low-carbon fossil fuels. Canadian Journal of Economics. Revue Canadienne d’Economique 41 (4): 1421–44. Solow, R. M. 1974a. The economics of resources or the resources of economics. American Economic Review 64 (2): 1–14. Solow, R. M. 1974b. Intergenerational equity and exhaustible resources. Review of Economic Studies 41: 29–45. Stiglitz, J. 1974a. Growth with exhaustible natural resources: Efficient and optimal growth paths. Review of Economic Studies 41: 123–37. Stiglitz, J. E. 1974b. Growth with exhaustible natural Rresources: The competitive economy. Review of Economic Studies 41: 139–52. Tahvonen, O., and S. Salo. 2001. Economic growth and transitions between renewable and nonrenewable energy resources. European Economic Review 45 (8): 1379–98. Trimborn, T., K.-J. Koch, and T. M. Steger. 2008. Multidimensional transitional dynamics: A simple numerical procedure. Macroeconomic Dynamics 12 (03): 301–19. Tsur, Y., and A. Zemel. 2003. Optimal transition to backstop substitutes for nonrenewable resources. Journal of Economic Dynamics and Control 27 (4): 551–72. Tsur, Y., and A. Zemel. 2005. Scarcity, growth and R&D. Journal of Environmental Economics and Management 49 (3): 484–99. US Energy Information Administration. 2011. Annual Energy Review. Washington, DC: GPO. US Energy Information Administration. 2012. International Energy Statistics. Washington, DC: GPO. Valente, S. 2011. Endogenous growth, backstop technology adoption, and optimal jumps. Macroeconomic Dynamics 15 (3): 293–325. van der Meijden, G., and J. Smulders. 2012a. Backstop technologies and directed technical change. Mimeo. VU University Amsterdam and Tilburg University. van der Meijden, G., and J. Smulders. 2012b. Resource extraction, backstop technologies, and growth. Mimeo. VU University Amsterdam and Tilburg University. van der Ploeg, F., and C. Withagen. 2013. Growth, renewables and the optimal carbon tax. International Economic Review 55 (1): 283–311.

6

Innovation and the Green Paradox Ralph A. Winter

6.1

Introduction

A recent economics literature has joined the theory of exhaustible resources with the dynamics of global warming.1 This theoretical literature shows that government policies directed at climate change can have perverse consequences. The anticipation of a future policy promoting clean energy can increase current emissions of carbon. The announcement of a future tax on carbon, for example, lowers the anticipated value of retaining fossil fuel reserves for sale in the future. This drop in the opportunity cost of selling reserves today reduces current fuel prices, which raises the consumption of fossil fuels and current carbon emissions. Thus the anticipation of future carbon taxes raises current carbon emissions. We see perverse short-run reactions to the announcement of future government policy throughout the economy. If a tax on home transactions is announced for next year, the number of transactions this year will rise. When the government announces plans to subsidize particular types of investment as part of a stimulus package, investment will drop until the subsidy package is ready. The announcement of an increase in the US capital gains tax rate to be implemented in 1987 had the effect of increasing the realization of capital gains in 1986 (Auerbach and Hines 1987). President Obama’s attempts to regulate assault weapons were, paradoxically, a tremendous boost for the gun industry. Other tax and regulatory effects are similar. The green paradox—the perverse current impact of anticipated future clean energy policies—is an example of what we see in any market with intertemporal substitution. The green paradox logic applies to a second class of perverse consequences, however, that is specific to the context of carbon emissions.

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This is the impact on global warming of innovation in clean energy. One would like to think that innovation in clean energy helps the battle against global warming. But consider the discovery of a low-cost clean energy substitute. Suppose that the current price of oil is 100 dollars per barrel and that an innovation in clean energy suddenly yields a perfect substitute for oil at a cost equivalent to, say, 60 dollars per barrel. This innovation will lead immediately drop in price of oil below 60 dollars. This drop in price will raise the demand for fossil fuels and therefore increase carbon emissions. The stock of all oil reserves with extraction costs below 60 dollars will be sold more quickly, and the carbon emissions from this stock released into the atmosphere earlier, than without the innovation. The impact is an increase in current emissions and possibly an increase in the present value of damages from global warming. The impact may even include a permanent increase in the path of global temperatures, as we shall see. This chapter offers a high-level, relatively nontechnical overview of the paradoxical impact of clean energy innovation. We can distinguish among at least seven versions of the innovation green paradox: 1. The impact of clean energy innovation in increasing current carbon emissions. This could be called the “weak green paradox” of clean energy innovation, following Gerlag (2010). 2. The impact of innovation in increasing the present value of damages from global warming—defined by Gerlag as a decrease in “green welfare.” This type of effect is referred to in the literature as a “strong green paradox.” 3. An even stronger paradox with an impact of innovation decreasing not just green welfare but total welfare. If the increase in present value of damages more than offsets the gains from savings in energy costs, then total welfare falls. It is this strongest paradox that is relevant for policy in the sense of determining whether energy innovation—in a second-best world, without carbon pricing—is even beneficial. 4. An ex ante green paradox that results from the possibility of innovation, as opposed to the realization of innovation. The mere threat of innovation leads to more rapid extraction of fossil fuels and therefore accelerated carbon emissions and higher global warming in the near term. 5. A long-term green paradox. In the models usually adopted in the literature, clean energy innovation must help in the long run as innovation reduces total emissions. Even the effect of accelerating emission helps in the long run in these models, when the stock of

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atmospheric carbon is assumed to decline exponentially. When we allow for positive feedback effects in climate dynamics, however, the acceleration of carbon emissions can push us over a tipping point. Innovation can lead to permanently higher temperature. 6. The impact of clean energy innovation in a first-best world in which carbon taxes enable a social planner to implement the optimal allocation of resources. Such innovation must improve overall welfare, since extraction costs and global warming costs are lowered for each consumption plan by the social planner. Yet even in this first-best world, clean energy innovation can make global warming worse. Paradoxically, even the social planner may respond to the sudden availability of a clean energy substitute by allowing the global warming problem to worsen. 7. The impact of innovation on optimal carbon taxes in a first-best world. One might think that clean energy innovation reduces the need for high carbon taxes. With a drop in the cost of the clean energy backstop, however, we will show that the implementation of the firstbest may require an increase in the optimal carbon tax. Innovation in clean energy, that is, may raise the marginal social cost of emissions even at the first-best optimum. Under which sets of circumstances do each of these innovation green paradoxes arise? I address this general question in a series of models joining (1) the Hotelling world of an exhaustible fossil fuel with (2) simple climate dynamics and (3) innovation in clean energy. I start in the next section by reviewing the central points in van der Ploeg and Withagen (2012), which is a careful analysis of the innovation green paradox in a model of certainty. In section 6.3 I apply the Dasgupta and Stiglitz (1981) model of uncertain innovation in a backstop technology to the innovation green paradox. I explore in section 6.4 various extensions : the presence of exogenous sources of carbon emissions beyond fossil fuel; innovation not just in a clean energy backstop but also in carbon abatement; innovation in efficiency in the use of oil rather than in substitution for oil (leading to the integration of the green paradox on the supply side of the energy market with the Jevons paradox on the demand side of energy markets); and an extension to incorporate feedback effects. Winter (2014) develops the model of uncertain innovation and feedback effects, demonstrating that clean energy innovation can increase not just current emissions but the entire path of temperature into the future.

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Discussing such a wide range of models and questions in a short chapter necessarily involves a large dose of conjectures and open questions. My aim in this chapter is to outline a set of questions that should be explored in more depth. In the conclusion I step back from the models to consider the implications for policy. The implication of the paradoxically negative impact of clean energy innovation is not that innovation should be discouraged. With carbon pricing available as an instrument, the welfare impact innovation is always positive. The policy message is that carbon pricing and innovation are complementary instruments. Carbon pricing is essential in tackling global warming even when clean energy innovation is likely to be successful. Indeed, carbon pricing is more important in an environment with innovation than without. 6.2

Van der Ploeg and Withagen (2012)

Our starting point is the important contribution of van der Ploeg and Withagen. The van der Ploeg–Withagen paper asks a wide range of questions in a wide range of models. We focus here on the central question, captured in the title of the paper, “Is there really a green paradox?” Van der Ploeg and Withagen adopt a continuous time model with the following assumptions. A constant demand for energy is derived from an instantaneous utility function, U, which is quasi-linear, increasing and concave. Energy can be provided by fossil fuel, extracted at a rate q, or by a clean backstop at a rate x and available a constant price per unit, b. The marginal extraction cost of the fossil fuel is an increasing, smooth function c(S) where S is the cumulative amount of the fossil fuel extracted.2 (The function c(S) 6reflects heterogeneity in the extraction cost of fossil fuel and the well-known principle that lower cost fuel is extracted first.) The flow of environmental damages is a function of the stock of emissions to date, D(E(t)). In other words, the model does not allow for decay of the stock of atmospheric carbon. The discount rate is given by a constant interest rate, r. 6.2.1 Green Paradoxes at the First-Best Optimum Van der Ploeg and Withagen first consider the first-best optimum. How does the social planner react to a one-time reduction in b, the cost of the clean energy backstop? Under the assumptions stated, the social planner chooses fossil fuel consumption q(t) and clean energy consumption x(t) to maximize

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∞

125

∫ e [U (q (t ) + x (t )) − c (S (t )) q (t ) − bx (t ) − D (E (t ))] dt. − rt

(6.1)

0

The maximization of (6.1) is subject to S (t ) = q (t ) and E˙(t) = q(t), and the resource constraint that S (t ) ≤ S , the exogenous amount of fossil fuel. The authors show that under the assumptions described, the social planner uses the fossil fuel only until a date T at which point the planner terminates extraction of fossl fuels and switches entirely to the clean backstop technology. Define “green welfare,” Λ, as the negative of the present value of global warming damages. Given a termination date T, Λ can be expressed as ∞

T

0

0

Λ = − ∫ e − rt D (E (t )) dt = − ∫ e − rt D (E (t )) dt −

e − rt D (E (t )) . ρ

If c (S ) > b, the termination date T satisfies c (S (T )) +

D ′ ( E (T )) = b. r

(6.2)

Equation (6.2) states that at T the marginal social cost of using fossil fuel, which is the extraction cost plus the present value of damages from an additional unit of emissions, must equal the marginal cost, b, of using the backstop technology.3 Consider the impact of an exogenous reduction in b, say from a value b0 to b1.4 The social planner may well respond to the cheaper source of perfectly clean energy with a plan that increases the damages from global warming. In this sense even the first-best optimum carries a paradoxical effect. Within the van der Ploeg–Withagen model, a sufficient condition for green welfare to drop with cheaper clean energy at the first-best optimum is c (S ) + D′ (S ) r < b (van der Ploeg and Withagen 2012, prop. 4). This is the condition that the extraction cost and the marginal environmental damages both be small enough that full extraction of the stock of fossil fuels be optimal even at the lower cost b1. (Call this the “full extraction condition.”) The response of the social planner to a drop in the cost of clean energy is to accelerate the extraction of the full stock of available fossil fuels, making the global warming problem worse. Consider the social planner implementing the first-best optimum with an time-varying carbon tax. Under the full extraction condition,

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one can show that the optimal response of the social planner to a reduction in b, is to raise the carbon price. One might expect that with cheaper clean energy, a high carbon tax would be unnecessary. Paradoxically, the optimal carbon tax increases with innovation in clean energy. The first-best response to innovation thus involves two paradoxes. Damages from global warming may increase. And the optimal carbon tax may increase. A sufficient condition for these effects is the full extraction condition. The economic intuition underlying these paradoxical effects is simple. It can be illustrated with the canonical static model of the socially optimal level of an activity with externalities. Let the level of the activity be a, with benefit b(a), and with private cost c(a) and external cost e(a). Assume that b(a) is increasing and concave and that c(a) and e(a) are increasing and convex. The social planner implements the first-best optimum, a*, defined as the solution in a to b(a) − c′(a) − e′(a) = 0.

(6.3)

The optimum a* can be implemented via a tax on the activity equal to t = e′(a*). Suppose that there is a shift downward in the marginal private cost curve, c′(a). The social planner, implementing the optimum (6.3), responds to the drop in marginal cost by increasing a, since at the original level of the activity the marginal social benefit now exceeds the marginal social cost. But the increase in a involves an increase in both the total externality, e(a*), and (from the convexity of e) an increase in the marginal externality, e′(a*). The pollution gets worse and the optimal tax rises. In the dynamic model of fossil fuel exhaustion, the private opportunity cost of extracting early falls with innovation—this cost including the now lower Hotelling rent. The planner responds by accelerating the extraction of fossil fuel, which makes the global warming problem worse. The social planner tolerates an increase in global warming damages in responding to the drop in the opportunity cost of fuel by raising the extraction rate. The optimal carbon tax must rise because the marginal external cost of emissions also rises with earlier emissions. 6.2.2 The Green Paradox without a Carbon Tax In the first-best world considered above, innovation in clean energy may lead to increased global warming, but the environment may be so

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tightly controlled that global warming is hardly a major problem. The more important question is whether in reality—which does not include significant carbon taxes—innovation in clean energy helps or hurts the battle against global warming. We start by posing this question in the van der Ploeg–Withagen framework of a one-time change in the backstop cost at time 0. This framework includes the standard assumption that carbon emissions are a complete externality in the sense that they ignored by market participants. The assumption that global warming is a complete externality allows us to define a competitive equilibrium independent of global warming dynamics. A competitive equilibrium is a price path p(t) and a date of extraction for each incremental unit of fossil fuel (of a particular cost) such that (1) the date is optimal for the owner of the unit of the resource and (2) the market clears at each date t in the sense that the total units extracted at t equals the demand for energy at p(t). Because of the complete externality property of emissions (and the separability of energy demand from global warming, in this partial equilibrium model) we can use the fundamental theorem of welfare economics to characterize the competitive equilibrium path as the solution to the following maximization problem: ∞

∫ e [U (q (t ) + x (t )) − c (S (t )) q (t ) − bx (t )] dt. − rt

(6.4)

0

The competitive market path of extraction maximizes (6.4) subject to S (t ) = q (t ) and Ė(t) = q(t), and the resources constraint that S (t ) ≤ S , the exogenous amount of fossil fuel. This is identical to (6.1) except that the environmental damage term is omitted. The competitive equilibrium price path, for the case c(0) < b, is characterized by a period of extraction of fossil fuel followed by the use of only the backstop.5 The competitive price path for energy over the extraction phase is (from the first-order conditions in the maximization of (6.4)) characterized by d [ p (t ) − c (S (t ))] = r [ p (t ) − c (S (t ))] − c′ (S (t )) dt

(6.5)

with the boundary condition c(S(T)) = b. It is possible that a drop in the cost of the backstop technology from a value b0 to b1 < b0 will reduce the present value of environmental ∞ damages, ∫ 0 e − rt D (E (t )) dt. If, at the extreme, b1 < c(0), then fossil fuels

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are eliminated and damages must fall. But if b1 > c(0), the drop in b involves two effects. All fossil fuel with extraction cost in (b1, b0) remains in the ground instead of being extracted. This is a positive effect on the environment. But the fossil fuel that is extracted is extracted more rapidly: the entire price path governed by (6.5) drops as the boundary condition changes from c(S(T)) = b0 to c(S(T)) = b1. This is a negative environmental effect because carbon enters the atmosphere earlier, adding to damages from an earlier date. If the finite stock S of fossil fuel all involves extraction costs less than b1 then the entire stock will be extracted with or without the innovation. The first effect disappears and we are left with a negative impact of clean energy innovation on the environment: the innovation green paradox. In the case where c (S ) > b1, it is still possible that the negative effect of innovation on global warming dominates. If so, we are left with the paradoxical effect that innovation in clean energy reduces green welfare. Innovation must help in the long run when c (S ) < b1, however, because under this condition the cumulative emissions must eventually fall by the amount c−1(b0) − c−1(b1). The green paradox cannot hold in the long run with the simple climate dynamics of this model. Eventually the environment must benefit from clean energy innovation. 6.3

Uncertain Innovation in Clean Energy Technology

The van der Ploeg–Withagen model considers the impact of once-andfor-all change in the cost of the backstop technology at date 0. I explore in this section an alternative issue: the impact of uncertain innovation. An uncertain innovation if and when it occurs drops the backstop energy cost from b0 to b1. Recognizing that innovation is neither immediate nor certain is realistic. Even more important is the extension that this assumption allows tothe consideration of green paradox effects from the ex ante impact of the mere threat of innovation. The benchmark model retains the basic structure of the model discussed in the previous section. A stationary demand for energy is derived from a utility u, which is increasing and concave. Energy can be provided by fossil fuels, for which the extraction cost c(S) is an increasing function of the cumulative extraction, S, or by a clean technology at cost per unit b0. Fossil fuels emit carbon at a rate of one unit of carbon for each unit of energy used. The stock of carbon, E, in the atmosphere leads to a flow of damages, D(E), and atmospheric carbon decays at a rate a. The welfare from an energy consumption path qt and path of atmospheric greenhouse gas concentrations, Et, is

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∞

e − rt [U ( q (t ) + x (t )) − c (S (t )) q (t ) − b0 x (t ) − D (E (t ))] dt , where q and x are the flows of fossil fuel and clean energy production. The dynamics of atmospheric carbon are described by Ė = q − λE. Regarding the equilibrium, the damages from global warming are a pure externality, ignored by participants in the competitive market for energy. We add to this model the assumption of an exogenous probability, ρdt, of discovery of a new, more efficient clean backstop in any small interval of time, dt. The unit cost of energy under the new technology is b1 < b0. One the new clean energy technology is discovered, no further innovation takes place. A positive amount of fossil fuel reserves have extraction costs between b1 and b0, so additional fossil fuels will be displaced under the new technology if it is discovered before all fossil fuels with extraction costs less than b0 are extracted. The economic assumptions, drawn from Winter (2014), extend the Dasgupta and Stiglitz (1981) model to the case of positive and heterogeneous fossil fuel extraction costs. (The economics outlined below are developed more fully in Winter 2014.) The assumptions outlined support a Markov model with three state variables: S, E, and an indicator variable ϕ(t) = 0 or 1 depending on whether innovation has occurred before t. At any time these three state variables summarize all that is relevant from the history of the market and climate for predicting the future. (The initial values of the state variables, S0 and E0, are exogenous; and ϕ(0) = 0.) The key to the tractability of this model is a separability: the economic equilibrium at any time t in terms of prices and extraction rates depends only on S(t), but not on E(t). The economic equilibrium is easily summarized here. Figure 6.1 presents the equilibrium price paths under three scenarios. The path labeled “innovation impossible” in the figure, with ρ = 0, satisfies (deleting arguments of functions)

∫

0

p = r [ p − c (S)] ,

(6.6)

d ( p − c (S)) dt d (c′ (S)) dt c ′ (S ) ⋅ q ( p ) =r− =r− . p − c (S ) p − c (S ) p − c (S )

(6.7)

The second of these equations (from which the first follows) states that the Hotelling rent, which is price minus extraction cost, increases at the rate of interest minus a term reflecting the rate of change of extraction cost. The boundary conditions in the innovation-impossible case are

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Innovation unsuccessful

Price b0

b1

Innovation sucessful

τ

Innovation impossible

Td(τ)

T0

t T

Figure 6.1 Fossil fuel price paths for three cases: innovation impossible, innovation unsuccessful, and innovation successful at date τ

p(Tm) = c(S(Tm)) = b0 for some date Tm. At date Tm, production of energy switches entirely from fossil fuels to the backstop technology. Emissions stop at Tm. If innovation is sufficiently early, the price path after innovation is realized at date τ is determined by the identical differential equation, but the boundary condition changes to p(Tτ) = c(S(Tτ)) = b1, where Tτ denotes the termination date of extraction of fossil fuels after discovery at date τ. Let ptd (t )be the price path, for t ≥ τ, after discovery at date τ. The price path when innovation is possible (ρ > 0) but has not yet occurred satisfies d ( p − c (S)) dt c ′ (S ) ⋅ q ( p ) ⎛ p − ptd (t ) ⎞ =r− + ρ⎜ , ⎝ p − c (S) ⎟⎠ p − c (S ) p − c (S )

(6.8)

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where pt (t ), by definition, is the level to which price falls instantaneously with innovation at date t. To understand this equation, note that when innovation is possible but has not yet occurred, there is a reduction in the opportunity cost of holding onto the oil for extraction and sale in the future: the risk of a capital loss with innovation. The final term in (6.8) represents this reduction in opportunity cost. The boundary condition for the innovation-possible-but-not-discovered price path is p(Tn) = c(S(Tn)) = b0 for some date Tn. The competitive equilibrium in this model thus yields three types of price paths, depending on the date of innovation and the probability of innovation, ρ. We have defined three terminal dates for extraction, Tm, Tn, and Tτ, depending on whether innovation is impossible; innovation is possible but occurs after Tn; or innovation occurs at date τ < Tn.  Next define t as the date at which the extraction cost under the preinnovation scenario exactly matches b1, the cost of clean energy under the (undiscovered) new technology. That is, t is defined by c (S (t )) = b1 along the pre-innovation path. If innovation occurs at a date τ ∈[ 0, t ), the price follows the pre-innovation path until the date of innovation, then drops discontinuously (as depicted in figure 6.1). The price path then continues according to the differential equation (6.6) until it reaches the new clean energy cost, b1. If innovation occurs in the region [t , Tn ], then extraction of fossil fuels terminates immediately upon innovation. Innovation is unambiguously helpful. If innovation occurs in the region [Tn, ∞), then extraction of fossil fuels has already terminated, and innovation simply involves a switch to the lower cost clean fuel. In this case the realized innovation has no impact on the path of greenhouse gases (although the threat of innovation has had an impact before that date). When innovation is impossible, welfare, Wm, can be expressed as the present value of surplus from energy, minus the present value of damages: d

Tm

Wm =

∫ [ u ( q ( t )) − q (t ) ⋅ c ( s ( t ) ) ] e 0

− rt

dt +

e − rTm [u (q (b0 )) − b0 q (b0 )] r

∞

− ∫ D (E (t )) e − rt dt , 0

where q(pm(t)) is summarized as q(t), and the dynamics of pm, E and S are governed by Ė = q − aE, and s = q.

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The realized welfare in the event that the new benchmark technology is discovered at date τ is analogous; the dynamics of p in the preinnovation period is described by (6.8) and in the post-innovation period by either (6.6) or p = b1 depending on whether τ is early or late. If we denote this realized welfare by W(τ), the overall expected welfare at the outset is W = EW(τ), where the expectation is with respect to the exponential distribution of the date of discovery. That ∞ is, W ≡ ∫ 0 W (τ ) ρe − ρτ dτ . 6.3.1 Green Paradoxes in This Model The green paradox questions in this model involve comparisons of global warming damages under the three scenarios depicted in figure 6.1: innovation impossible; innovation possible but not discovered until after Tn, when extraction has terminated; and innovation discovered early. The weak green paradox refers to an increase in current emissions with innovation, which follows from the discontinuous drop in price with the realized innovation. The strong green paradox refers to a possible drop in the expected present value of damages with possible innovation compared to innovation impossible. (Here expectations taken with respect to the single random variable, the date of discovery.) The weak and strong green paradoxes are ex post in that they refer to the effect of the realization of innovation. We begin, however, with discussion of an ex ante green paradox. 6.3.2 The Ex ante Strong Green Paradox As discussed, we can disentangle two dimensions of the strong green paradox. The first is the ex ante strong green paradox. If we move from the innovation-impossible scenario with the innovation-possible-butnot-realized scenario, is there an increase or a decrease in the present value of global warming damages? The impact of this shift, namely the impact of the threat of innovation is to move the entire distribution of emissions, c−1(b0), closer in time—and to increase the average intensity of emissions over the extraction phase. It is possible that moving emissions closer to the present can reduce the present value of global damages. This is yet another surprising result in global warming dynamics. Consider the movement of a single unit of emissions from the future to the present, and ask if this can reduce the present value of damages. Suppose that (1) the stock of atmospheric carbon is rising over time as emissions accumulate, until reaching a peak at or before the end of the extraction period (the stock

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must then decline after the extraction period under our assumptions because the stock decays at a constant rate), and (2) that the damage function is extremely convex. Under these two suppositions, the main impact of a unit of emissions, whenever it occurs, is its contribution to the stock of atmospheric carbon near the peak of the stock. A unit of carbon emitted near the peak will have a bigger impact than the same unit emitted much earlier, because of the rate of decay in the unit once it is added to the stock. Thus moving a unit of emissions from the future (near the peak of carbon concentrations) to the present will, under these conditions, reduce the present value of damages. This example, however, relies on a significant rate of decay of atmospheric carbon. It is the decay that makes early emissions at the margin less costly than later emissions in the example. With a sufficiently low rate of decay, earlier emissions are more costly and therefore the lower price under the innovation we get an ex ante strong green paradox: merely the threat of increased innovation is enough to exacerbate the global warming problem because it leads to a faster rate of the accumulation of atmospheric carbon. 6.3.3 The Ex post Strong Green Paradox Figure 6.1 reveals as well the trade-off involved in determining whether realized innovation helps or hurts total welfare. One effect is the increase in emissions immediately instead of in the future. This increases the present value of damages if the rate of decay, λ, is sufficiently small. The second effect is the reduction in total emissions (which, for an early innovation, equals c−1(b0) − c−1(b1)). This second effect is beneficial, but if the amount of fuels remaining to be exhausted is high, the beneficial effect is not felt until well into the future. It is possible, if innovation occurs at a sufficiently early date, that total welfare falls with innovation. The immediate positive impact on emissions may overwhelm the longer run welfare benefits. If innovation occurs sufficiently late, even over the period in which fossil fuel is being extracted, there can be no strong green paradox. 6.4 Extensions: The Impact of Additional Factors on the Likelihood of a Strong Green Paradox 6.4.1 Other Sources of Emissions The strong green paradox arises in the benchmark model of uncertain innovation because emissions are brought forward in time, which may

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increase the present value of the resulting damages, and because emissions increase in intensity as the period over which emissions occur is shortened. But the benchmark model, like other models in the literature, yields an awkward prediction: with a > 0 in the long run, atmospheric gases disappear. Atmospheric greenhouse gases and damages are zero in the limit, because emissions drop to zero after the extraction of fossil fuels end yet are reabsorbed at a constant rate, a. The most prominent contributions to the theory of clean energy innovation and global warming get around this awkward implication in different ways. Van der Ploeg–Withagen assume that a equals zero, so that greenhouse gases and the flow of damages remain constant after the date at which the use of fossil fuels end. And the van der Ploeg– Withagen model incorporates the optimistic assumption, common to a number of models, that the stock of atmospheric carbon stops growing after the exhaustion of fossil fuels. Hoel and Kvemdokk (1996), in an alternative approach, assume that once all fossil fuels are extracted, atmospheric carbon and damage levels will converge back to the postindustrial level. Most theoretical models of exhaustible resources and global warming assume that the market being modeled is the sole source of atmospheric carbon. An alternative is to recognize that there are sources of carbon emissions (some anthropogenic, others natural) apart from the exhaustible fossil fuel market being modeled. Emissions from the burning of coal must be included in the list of other emissions, if we are applying the fossil fuel model to the extraction of oil and gas. Coal reserves are enough to 900 years of consumption at current rates and therefore Hotelling rents are currently negligible for coal; the green paradox can apply to oil and gas but cannot apply to coal, since the Hotelling rents are too small for the argument to be relevant. In a competitive market for coal the market price is close to extraction cost, which does not fall when clean energy is discovered. Oil and gas account for more current emissions that coal: oil, gas, and coal recently accounted for 36 percent, 20 percent, and 35 percent of CO2 from fossil fuel consumption.6 But these figures will change, in the absence of regulation or carbon pricing, as oil and gas are depleted relative to coal. CO2 emissions from fuel account for about 75 percent of the emissions of anthropogenic greenhouse gases, in CO2-equivalent units, other sources being agriculture (methane and nitrous oxide), forest-clearing, and so on. This means that oil and gas (fuel markets with significant Hotelling rents and therefore subject to the green

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paradox) account for just over 40 percent of anthropogenic greenhouse emissions, and a much smaller proportion of total carbon emissions, including natural emissions such as carbon released from decaying plant material and methane from oceans. The proportion of GHG emissions accounted for by oil and gas will decline as the reserves diminish in the future. In short, there is a solid empirical basis for incorporating emissions from other sources into models of the green paradox. Given these other sources of emissions, we consider for simplicity the case where the innovation, b0 − b1, is relatively small. We can then take (with some degree of approximation) the time series of the marginal social cost of emissions as given, independent of whether and when the innovation is realized.7 Recall that innovation in the benchmark model has the potentially costly impact of shifting carbon emissions closer in time, and the benefits of reducing total emissions (and reducing the cost of energy). The environmental cost and benefit of innovation, in this extended model, can be measured by the change in emissions over time multiplied by the marginal social cost of emissions, in present value terms. To the extent that the marginal social cost of emissions is rising, the net effect on welfare of bringing emissions forward in time will be reduced by consideration of other sources of GHG emissions. Suppose that the marginal social cost of emissions is c initially and rises at a rate k over the entire extraction period. The present value of environmental damages arising from the possibility of innovation in clean energy, is given by the difference between the present value of damages under the innovation-possible regime, Dn, and the present value of damages under the innovation-impossible regime, Dm. Under the assumption of an exogenous and rising marginal social cost of emissions, this difference is Tm

Dn − Dm = ∫ ce kt [ q ( pn (t )) − q ( pm (t ))] e − rt dt 0

Tm

=

∫ c [q ( p

n

(t )) − q ( pm (t ))] e (k − r )t dt.

(6.9)

0

Note that the flow of damages from the market is a function of the flow of emissions (and therefore of fossil fuel consumption), not of the stock of atmospheric gases as in the benchmark model. The change in our assumptions on the nature of the externalities from extraction of fossil fuels has no impact on the economic equilibrium, since all carbon emissions are ignored by the market. This impact depends on the relative

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size of k and r, the effect of the growth rate k is equivalent to discounting the flow of damages by r − k instead of r. If k < r, the impact of the threat of innovation is to increase the present value of damages—the ex ante strong green paradox. This is because innovation shifts forward in time the cumulative emissions. An ex post strong green paradox, an increase in the present value of damages as a result of a discovery of clean energy technology, is made less likely by the introduction of other sources of greenhouse gases and a rising marginal social cost of carbon. The gain from innovation in terms of the environment is the reduction in future emissions. This gain is higher, relative to the cost of greater near-term emissions, once we incorporate the fact that marginal social costs of carbon will be higher in the future. Estimates of the marginal social cost of emissions over time in an unregulated market (our context) are unavailable. This rate of increase, however, should be much larger than the rate of increase of the marginal social cost of carbon if the optimal carbon tax were to be imposed, because the stock of atmospheric greenhouse gases will grow at a higher rate and the social cost of these gases is convex in the stock. Estimates of the rate of increase of the marginal social cost of carbon (i.e., the optimal tax rate) under taxation vary widely, but the estimates almost all involve rapidly increasing taxes. Nordhaus, for example, estimates an optimal tax rate that increases at a rate between 2 and 3 percent per year (Nordhaus 2008, p. 16). If the growth rate of the marginal social cost of carbon in a world without taxes is significantly higher than this, then the difference between the real interest rate (generally taken to be 4 or 4.5 percent) and the growth rate is very small. The growth rate largely offsets discounting. The cost side of innovation, bringing emissions forward in time, is small, and the benefits of reduced total emissions via substitution to clean energy in the future is large. Nordhaus’s model does not attach value to increased risk of loss of life or reduced life span in the future (population is exogenous in the DICE model). Taking into account the very high damages that must be attached to risks to future lives means that the estimate rate of increase in the marginal social cost of carbon is even higher than in Nordhaus’s model. And the consideration of the potential cost to earlier generations of the loss of species in the future increases any reasonable estimate of the growth rate of the marginal social cost of carbon even further. With these considerations there is little basis for assuming that

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the growth rate, k, in the marginal social cost of carbon is less than the real interest rate, r. When k ≥ r, a green paradox is impossible. In summary, the recognition of other sources of emissions—if this is the only extension adopted to the benchmark model of uncertain clean energy technology—makes a green paradox less likely. 6.4.2 Innovation in Carbon Abatement Innovation in clean energy substitutes for fossil fuels, such as wind and solar power, is one of a portfolio of innovation types designed to reduce carbon emissions over time. Innovation in the abatement of carbon emissions from the burning of fossil fuels is another. The most prominent abatement technology is carbon sequestration, or carbon capture and storage (CCS), captures carbon dioxide from exhaust of fossil fuel burning. Carbon dioxide would be stored in liquid form in underground caverns, including those left from extraction of fossil fuels. The views on the prospects of CCS vary widely. Those supporting fossil fuel industries express optimism, while others are more pessimistic.8 I do not commit to a particular view on CCS, I simply take an arbitrary rate of technical progress as given in analyzing the impact on the likelihood of a green paradox.9 The question posed here is about an interaction effect: How does the presence of steady innovation in abatement of carbon emissions affect the impact of a sudden innovation in clean energy, and the likelihood of a green paradox? To capture innovation in carbon abatement, suppose that the extraction of fossil fuel, q(t), at date t results in emissions e−αtq(t).Here α represents a constant improvement in the percentage of carbon (1 − e−αt) that can be captured and stored rather than released. Adopting the assumptions of the benchmark model, the present values of damages under the innovation-impossible scenario and under the innovation-possible scenario are then ∞

DmA = ∫ D (Em (t )) e − rt dt , 0

∞

DnA = ∫ D (En (t )) e − rt dt , 0

where the atmospheric carbon, E(t), in each model, is determined as in the base model by its initial value, E(0) and Ė(t) = 1 − e−αtq(t) − aE(t). An expression for the present value of damages under the realization of an innovation in clean energy at date τ is similar.

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The presence of the steady innovation in carbon abatement has no impact at all on the economic equilibrium, including the extraction path, of the benchmark model. Emissions are a pure externality, not taken into account by any economic actor. The effect of a positive rate of improvement in abatement technology is to make a strong green paradox (ex ante or ex post) more likely. This is because the benefits of innovation, the reduction of future emissions, are reduced by the improvement in abatement technology, relative to the costs of increased near-term emissions. The point is simple, but an understanding of the power of the green paradox outcome for clean energy innovation requires incorporating other dimensions of clean innovation. 6.4.3 Innovation in the Efficiency of Fossil Fuel Energy Use A more prominent and more promising dimension of innovation in the energy field than carbon abatement is the improvement in efficiency of machines that use fossil fuels. Fuel efficiency in cars has improved dramatically over time, and in the US regulation has passed that requires US automobile producers to attain an average of 54.5 miles per gallon (4.3 liters per 100 km) for their fleets of 2025. Electricity generation has also increased in efficiency as older plants have been replaced by newer ones, but at a lower rate.10 In this subsection I speculate on paradoxical effects related to the interaction among three forms of innovation: 1. clean energy innovation; 2. innovation in the efficiency of energy production from fossil fuel; and 3. innovation in the efficiency of the production of services, such as automobile transportation, from energy. The first of these has been analyzed to this point in the chapter. The second and third are innovations at the two stages of the fossil fuel– energy-service supply chain. Global Warming Impact of Increased Combustion Efficiency in Producing Energy Consider first the impact of an innovation of the second type, an increase in the combustion efficiency of oil to produce energy. One might think that increasing combustion efficiency reduces the need for oil and therefore reduces global warming. In fact the issue is more complicated.

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In considering this increase in efficiency, we hold constant both the demand curve for energy and the cost of a clean energy alternative, in dollars per Watt-hour (or joule). Suppose that the combustion efficiency in producing energy from oil doubled. Oil contains 1,700 Wh of energy per barrel. If the efficiency in capturing this energy doubles from (say) 30 percent to 60 percent, then the amount of useful energy obtained from a gallon of oil has, obviously, doubled. If it was initially worthwhile to extract all oil with extraction cost up to 150 dollars per barrel before turning to the clean energy alternative, after the increase in efficiency it will be worthwhile to extract all oil up to a cost of 300 dollars per barrel. An increase in combustion efficiency unambiguously increases the size of the oil reserves that will eventually be extracted, as well as the total carbon that will eventually be emitted. The extraction of a greater amount of fossil fuels means an eventual increase in cumulative emissions. If a is 0 or close to 0, then the inevitable impact of the increase in combustion efficiency is a higher accumulation of atmospheric carbon in the long run. If it is the long run that warrants substantial focus, then the net impact of efficiency in the production of energy from oil may well be to reduce welfare. Offsetting this negative environmental impact of combustion efficiency is a possible decrease in the quantity of oil demanded, and therefore a spreading-out of emissions over time. But whether this beneficial effect emerges depends upon the elasticity of demand. Global Warming Impact of Increased Efficiency in the Use of Energy Turning to innovation in the second stage of the fuel–energy-service supply chain, we ask what is the impact of a steady increase in energy efficiency in machines on the impact of clean energy innovation. (Assume that the same increase in efficiency applies whether the source of the energy is oil or a clean substitute.) The answer depends on the elasticity of demand for energy. Jevons’s paradox states that an improvement in energy efficiency of machines may result in an increase in the demand for energy (Jevons 1865).11 Taking the example of automobiles, suppose that demand for automobile miles driven depends on the cost per mile and that (for simplicity) the only cost is fuel. Then the demand for transportation as a function of cost per mile is d(p/e), where e is the miles per gallon and p is the price per gallon of fuel. The demand for energy, q(p), is the ratio of miles driven to miles per gallon, or q(p) = d(p/e)/e. The impact of a 1 percent increase in efficiency on demand

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Figure 6.2 Jevons’s paradox: increasing energy efficiency can increase the demand for energy

for energy is −d ln[q(p)]/d ln p − 1.12 In other words, if the demand for energy is elastic (equivalently, the demand for miles driven is elastic), then an increase in efficiency has a positive effect on the demand for energy. If demand is inelastic, the effect of increased energy efficiency is to reduce energy demand. Thus an increase in efficiency causes the demand for energy to rotate about the point on the demand curve where the elasticity is equal to 1, as in figure 6.2. The region of elastic demand is the region where Jevons’s paradox holds: an increase in energy efficiency has the effect of increasing demand for energy. Available evidence is that the elasticity of demand for energy is less than 1.13 This means that a steady increase in efficiency in the use of energy leads to a decrease over time in demand for energy. Let us assume that demand satisfies limp→0 qt (p) = ∞ at any t, so that the energy demand curve has no choke point. Consider the impact of a drop in demand over time on equilibrium in our benchmark model. In order to isolate the impact of increased energy efficiency, consider the

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benchmark model in the case m where innovation in clean energy is impossible. The differential equation in pricing (6.5) is unchanged by the introduction of falling demand over time; this equation still follows from the first-order condition, on the supply side of the market, for optimal date of extraction. However, the rate of change of S at any value of p and S is reduced, since S˙ = qt(p(t)), which is smaller when qt is declining over time. We compare the equilibrium under scenario m with a scenario—denoted by e—in which demand drops over time due to increasing energy efficiency. It is helpful to consider the two equilibria in the state space (p, S) (figure 6.3). Figure 6.3 depicts the line p = b0 as well as the marginal extraction cost curve c(S). Under both the m and the e scenarios, the terminal point is the intersection of the p = b0 line and p

c(s)

b0 Pm Pd˄τ Innovation impossible

Pn

b1

Paths after innovation realized

Pdτ

Innovation possible but unrealized

s sτ

s˄τ

Figure 6.3 Economic state space: early innovation at τ or late innovation at tˆ

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c(S). But the p(S) path is steeper under e than under m. To see this, let qt(p) be the demand curve; qt(p) is stationary under m and initially identical but falling under e. Under either path we have dp dp dt r [ p − c (S)] = = . dS dS dt qt ( p )

(6.10)

Since qt(p) is lower under e for all t > 0, the slope dp/dS is greater under e. Given the identical boundary condition, p(S0) = c(S0) = b0 in both scenarios, this means that p(S0) = p0, the price at t = 0, is lower under e.14 Suppose that demand were elastic. Then the fact that p0 is lower under e would mean that taking the current demand curve as given, the prospect of increasing energy efficiency always leads to a weak green paradox. The prospect of increased energy efficiency lowers the initial price, thereby increasing demand and emissions. The increased early emissions from a regulatory mandate of increased efficiency are offset by benefits of reduced rate of future emissions: demand is higher in the future under scenario e, so that the effect of increased energy efficiency is not only to reduce future emissions but to spread out the future emissions over time. I do not explore the trade-off formally here, but the clear consequence is a greater benefit than simply a reduction in future benefits. Mandated regulations of increased energy efficiency over time are likely to produce greater benefits than costs, as compared to other forms of innovation. Increased energy efficiency has no impact, however, on the long-run cumulative carbon emissions in the benchmark model, since all oil with an extraction price of b0 or lower is extracted under the m or e scenarios. To summarize the implications for long-run carbon emissions of the various types of innovations considered in this chapter, clean energy innovation necessarily reduces cumulative emissions in the long run, since it results in reserves with extraction costs in (b1, b0) being left in the ground. Increased efficiency in the production of energy from of oil results in increased cumulative emissions, since with this innovation oil of higher extraction cost becomes economic. And increase efficiency in energy use has no impact on long-run cumulative carbon emissions. When a is close to zero and either discount rates are low or the marginal social cost of emissions is rising quickly over time, it is the long-run cumulative emissions that dominate in determining whether or not there is a strong green paradox.

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Global Warming Impact of Clean Energy Innovation, Taking as Given a Pattern of Increasing Energy Efficiency The final question of this section is about an interaction effect. What does increased energy efficiency (as an assumed background condition) imply for the possibility of a strong green paradox outcome from innovation in clean energy? Clean energy innovation, recall, involves an increase in short-run emissions and an even greater decrease in long-run emissions. A condition of increasing efficiency over time, and therefore decreasing demand over time (taking the realistic case of inelastic demand for energy), means a shift of the reserves of fossil fuels to the present from the future in which the release of carbon is spread out over time. The implications for a strong green paradox from increased energy efficiency depend on this trade-off. This trade-off clearly depends on key parameters such as the rate of reabsorbtion a and the discount rate. The key point is that the green paradox effect is a transfer of the stock of fossil fuels from the future, where the rate of extraction is low, to the present, where the rate of extraction is relatively high. The cost of this transfer, increased current emissions, is high because emissions are already adding to atmospheric carbon at a high rate and the stock of this carbon is subject to a convex damage function. Since low demand in the future leads to a low rate of carbon emissions and a a > 0 allows the carbon to be reabsorbed before adding substantially to the stock of green house gases, the benefit of the green paradox transfer we would expect to be small. While the analytics remain to be developed, the likelihood is high for a strong green paradox from a regulatory mandate of increased energy efficiency. The suggestion is that a strong green paradox is more likely with increased energy efficiency: the future externalities involved in leaving oil in the ground at any date, for extraction later, are low because the future now contains greater fuel efficiency, a lower demand, and emissions that are more spread out over time. With carbon settling back to the oceans from the atmosphere, a lower rate of emissions in the future maintains temperature at a lower level, and given the convexity of the damage function and the delay in emissions the present value of damages is reduced. A lower present value of the external cost from leaving oil in the ground, means that the impact of innovation in bringing extraction forward in time is more likely to be, on net, costly. The future benefits of lower emissions with innovation are reduced since the emissions are spread out even without the innovation.

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6.4.4 Positive Feedback Effects Winter (2012) joins the economic assumptions of the benchmark model with climate change assumptions that reflect feedback effects within the carbon cycle. The benchmark model analyzed in section 6.3 contains the following elements: a constant demand for energy, a stock of fossil fuels of varying extraction costs that can supply this energy, an existing clean energy technology that can produce energy with zero carbon emissions at a cost b0, and a constant probability ρ of discovery over time of a new clean energy technology with cost b1 < b0. A state variable A, the total carbon in the atmosphere decays at a rate a as in the benchmark model. But A increases not just with emissions e(t) but also because of feedback effects drawing carbon from the earth’s surface. The higher the atmospheric carbon, the higher the temperature and the greater the rate of release, F(A), of carbon from the earth’s surface. The dynamics of atmospheric carbon are given by  = e + F ( A ) − aA. A

(6.11)

The shape of F is logistic, reflecting evidence that feedbacks are small at low temperatures and the proportion of GHG’s in the atmosphere must be bounded below 1. The economic equilibrium is completely unchanged from our benchmark model: since greenhouse gases are a pure externality. Again, the emissions are equal to the fossil fuel extracted at any date, in the equilibrium of the economic model. The resulting model is Markov with three state variables: ϕ(t) = 0 or 1 depending on whether innovation has occurred before t (ϕ(t) is exogenous), S(t) is again the cumulative stock of fossil fuel extracted by date t, and A(t). The positive feedback effects represent those discussed by climate scientists. There are at least seven: the release of methane from the permafrost in the tundra as the higher temperature melts the tundra, the release of methane from methyl hydrate crystals in the ocean, the greater average concentration in the atmosphere of water vapor (an important greenhouse gas) as temperature rises, the albedo effect in the loss of reflective ice fields as temperature rises, the release of CO2 with increased respiration from bacteria in the soil, the death of vegetation in regions such as the Amazonian rain forests, and the reduced ability of oceans to absorb carbon dioxide as the ocean becomes more acidic. The effect of the positive feedback assumptions is that innovation, by raising the demand for fossil fuels through the weak green paradox

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effect, increases the intensity of emissions and the atmospheric gases, gtA. Through the feedback effect this elicits the release of greenhouse gases from the earth’s surface and causes a further temperature increase. If the feedback effects are strong, then innovation can lead to a permanently higher temperature path. The focus is on the long-run steady-state level of A and hence the steady state of temperature. Under the dynamics given by (6.11), once extraction has terminated a steady-state value of A is (from equation 6.11) a solution to F(A) − aA = 0. Given the logistic shape of F, there are three steady states for the most interesting parameters. Two of these are stable: a low temperature/ carbon steady state, AL* , which is interpreted as a long run in which global warming is under control, and a high-temperature/carbon state, AH* , representing runaway global warming. Depending on the economic and innovation dynamics generating the time path of emissions, the steady state will be AL* or AH* . The middle, unstable steady state, ATP, is a tipping point: once A passes ATP, F(A) − aA remains positive and runaway global warming is inevitable. The key insights of this model are captured in the mapping from the date τ of innovation to the long-run steady-state temperature. Let Φ be the set of dates τ that map into runaway global warming, which is the steady state AH* . Runaway global warming may be impossible, with Φ = ϕ. Or runaway global warming may be inevitable. For the most interesting sets of parameters, however, Φ is nonempty and consists of two disjoint subsets of [0, ∞). One subset of Φ is [t2, ∞), for some t2. If clean energy innovation has not taken place before t2, then it is too late to be of much help in helping global warming by inducing the economy to leave fossil fuels (with extraction costs greater than b1) in the ground. We get runaway global warming results. The second subset of Φ is [0, t1], for some t1 < t2. If innovation is too early, then the green paradox kicks in. The extraction cost c(S) is small because S is small, with the result that the drop in fossil fuel price, p, with innovation is very sharp. Extraction and carbon emissions rise substantially, pushing atmospheric carbon over the tipping point. Again, we get runaway global warming. The first result of this model is thus a green paradox from realized innovation that is too early. This one could call the long-run ex post innovation green paradox. The impact of innovation in this model is not

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just on current emissions, as in previous models. The feedback dynamics lead to the long-run effect of runaway global warming as a consequence of innovation. Global warming is a long-run problem, and it is the long-run consequences of clean energy innovation that are most important. The second main result of the model is the following. Denote the two subsets of Φ by Φ1 = [0, t1] and Φ2 = [t2, ∞). Now let the probability of innovation, ρ, approach zero. The early subset, Φ1, representing the disastrous consequences of early innovation, changes slightly but remains. But for some parameters the higher subset, Φ2, disappears as ρ → 0. The probability of not discovering the new innovation until very late increases, of course, but the consequences of late discovery evaporate. This result captures the ex ante long-run innovation green paradox: when the probability of innovation ρ is high, the rate of extraction is high because price is low—as reflected in equation (6.8) of our benchmark model. The low price reflects the low opportunity cost of extraction due to the threat of innovation, not the realization of innovation. As ρ → 0, this effect disappears, the rate of emissions drops, and runaway global warming is avoided. 6.5 Conclusion: The Innovation Green Paradox and Implications for Climate Change Policy This chapter re-examines the conditions giving rise to the innovation green paradox. It starts with the van der Ploeg–Withagen model of the impact of known innovation and then moves to a benchmark model of uncertain innovation. These models illustrate the possible negative impact of realized innovation. The discovery of a new, perfectly clean energy technology may increase the damages from global warming. Even the threat of such innovation may increase damages. We extend the benchmark to explore the innovation green paradox in a variety of circumstances. A green paradox effect of innovation in clean energy is less likely when there are other sources of emissions resulting in an exogenous rate of increase in the marginal social cost of carbon. The assumption of an exogenous rate of innovation in carbon abatement, on the other hand, makes the green paradox effect more likely. We consider innovation not only in clean energy but in efficiency in the use of energy, including innovations in the two stages of the fuel—energy—service supply chain. Innovations in the

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efficiency of production of energy, in and of themselves, lead to greater cumulative emissions in the long run. Reserves of high-cost oil that would be noneconomical to extract before such innovations are extracted once efficiency innovations have increased the value of these reserves. Increases in efficiency in the production of energy from fuel or in the production of services (e.g., transportation) from energy can lead to a decrease or an increase in energy demand and emissions. Our discussion on this point brings together the green paradox on the supply side of energy markets with Jevons’s paradox on the demand side. Evidence suggests that demand for energy is inelastic, meaning that innovations in energy efficiency reduce demand, spreading out emissions over a longer time. This is a benefit to innovation in energy efficiency, offsetting the cost of greater total emissions. We also discussed the interaction of two types of innovation: the prospect of a green paradox from clean energy innovation given a background of exogenous efficiency innovation. Finally, I introduced positive climate feedback effects in the carbon cycle into the theory, in outlining a model discussed elsewhere (Winter 2012). With feedback effects, early innovation in clean energy substitutes can lead to runaway long-run global warming. And yet a failure to innovate early enough may mean that low-cost clean energy is not available to replace the extraction of much high-cost oil. More emissions result and the failure to innovate leads to runaway global warming. Innovation realized in an intermediate time frame may offer the Goldilocks solution of a low-temperature steady state—neither too early to set off the green paradox effects nor too late to be of little use in displacing oil with high extraction cost. My conclusion is that there is analytical support for the strong, innovation green paradox: the possibility of net negative welfare consequences of innovation in low-cost clean energy substitutes. This is an important conclusion because of the policy thrust in some jurisdictions away from carbon pricing toward subsidies of clean energy innovation. Such innovation might not even be of positive value. The ultimate policy conclusion, however, is not that clean energy innovation should be abandoned. Innovation is always beneficial when a carbon tax is available as a policy instrument. It is the absence of carbon taxes that renders innovation of possible negative value. The policy implication of the innovative green paradox theoryis to underscore the complementarity of carbon pricing and clean energy innovation. These instruments are not substitutes. Carbon pricing is even

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more important when innovation in clean energy sources is active, to ensure the optimal gains from such innovation, and even to ensure a positive value of innovation. Notes I am grateful to Werner Antweiler, Gerard van der Meijden and Rick van der Ploeg for very helpful comments. 1. The prominent papers in this literature include van der Ploeg and Withagen (2012), Sinclair (1994), Ulph and Ulph (1994), Hoel and Kverndokk (1996), Sinn (2008), Strand (2007), Gerlagh (2010), and Farzin and Tahvonen (1996). 2. I have adopted as a state variable the cumulative amount of fossil fuel extracted, to be consistent with the most common formulation. In van der Ploeg and Withagen’s formulation, the state variable S is the amount of fossil fuel remaining in situ. 3. Note that the van der Ploeg–Withagen assumption that all carbon emissions remain in the atmosphere rather than decay (settle to the earth’s surface) is necessary for the result that the social planner switches from using only fossil fuels to using only the backstop. To see this, assume that the rate of decay, λ, is positive, namely that Ė(t) = q(t) − λE(t). Suppose that there were some date T that involved a complete and permanent switch from fossil fuels to clean energy. At this date, the marginal social cost of using fossil fuels (including the marginal impact on the present value of future damages) must equal b. But after T, the decay in E would lead to a reduction in the marginal environmental cost of fossil fuel extraction, implying that the marginal social cost of using fossil fuels would fall, leaving this marginal cost below b. A switch back to fossil fuels would improve welfare, contradicting the supposition. 4. I consider a discrete drop in b to be consistent with models discussed in the next section of this paper. Van der Ploeg and Withagen consider a marginal drop in b. 5. Unlike the analogous proposition for the social optimum, this does not require a zero rate of decay in atmospheric carbon. 6. Raupach et al. (2007). 7. This brings the approach close to that of Gerlagh (2010), who considers an exogenous social cost of carbon. 8. Liquid CO2 is difficult to store. Liquid CO2 takes up much more volume than the carbon (fossil fuel) burned, and in addition Ehli-Economides and Economides (2010) conclude that only 1 percent of available pore space can be used. The ratio of CO2-storage space needed to space created by extraction of coal (for example) is greater than 50, according to Ehlig-Economides and Economides. 9. CCS involving capture of carbon from the atmosphere is also being researched, but this technology is somewhat more speculative at this point. 10. In Europe the average energy efficiency of conventional thermal electricity production in the EU-25 improved over the period 1990 to 2004 by 3.2 percentage points to 38.2 percent. Including useful heat, as well as electricity in calculating efficiency, this results in an increase of 6.4 percentage points to 47.8 percent over the same time period (European Environmental Association 2007).

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11. Jevons wrote The Coal Question in response to calls for increased efficiency of machines, so that coal, which fueled the Industrial Revolution in Britain, would last longer. The belief at the time was that British coal reserves were small and represented a potential constraint on the continued strength of the British Empire. Increased efficiency in the use of coal, Jevons pointed out, would not necessarily extend the life of the coal reserves. 12. To derive this expression, note that d ln[d(p/e)/e]/d ln e = d ln[d(p/e)]/d ln e − 1 = −d ln[d(p/e)]/d ln p − 1 = −d ln[d(p/e)/e]/d ln p − 1 = −d ln[q(p)]/d ln p − 1. 13. For example, Epsey (1996) in a meta study found an average estimate of long-run demand for gasoline of 0.6. 14. We adopt the assumption that demand remains inelastic at the lower initial price under e. We also assume that b0 is below the price at which elasticity equals 1 in demand at any date, adopting the usual assumption that elasticity of demand is increasing in price along a demand curve.

References Auerbach, Alan J., and James R. Hines Jr. 1987. Anticipated tax changes and the timing of investment. In Taxes and Capital Formation. Cambridge, MA: NBER, 85–92. Dasgupta, P., and J. Stiglitz. 1981. Resource depletion under technological uncertainty. Econometrica 49 (1): 85–104. Ehlig-Economides, Christine, and Michael J. Economides. 2010. Sequestering carbon dioxide in a closed underground volume. Journal of Petroleum Science Engineering 70: 123–30. European Environment Agency. 2007. Efficiency of conventional thermal electricity production. Report. European Environment Agency, Copenhagen, April 1. Farzin, Y. H., and Tahvonen. 1996. Global carbon cycle and the optimal time path of a carbon tax. Oxford Economic Papers 48 I(4): 515. Gerlagh, R. 2010. Too much oil. CESifo Economic Studies 57 (1): 79–102. Hoel, Michael, and Snorre Kvemdokk. 1996. Depletion of fossil fuels and the impacts of global warming. Resource and Energy Economics 18: 115–36. Jevons, W. Stanley. 1865. The Coal Question: An Enquiry Concerning the Progress of the Nation, and the Probable Exhaustion of Our Coal Mines. London: Macmillan. Nordhaus, William. 2008. A Question of Balance. New Haven: Yale University Press. Raupach, M. R., et al. 2007. Global and regional drivers of accelerating CO2 emissions. Proceedings of the National Academy of Sciences USA 104 (24): 10288–293. Sinclair, P. J. N. 1994. On the optimum trend of fossil fuel taxation. Oxford Economic Papers 46: 869–77. Sinn, H. W. 2008. Public policies against global warming: A supply side approach. International Tax and Public Finance 15: 360–94.

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Strand, J. 2007. Technology treaties and fossil fuels extraction. Energy Journal (Cambridge, MA) 28 (4): 129–42. Ulph, A., and D. Ulph. 1994. The optimal time path of a carbon tax. Oxford Economic Papers 46: 857–68. van der Ploeg, Frederick, and Cees Withagen. 2012. Is there really a green paradox? Journal of Environmental Economics and Management 64 (3): 342–63. Winter, Ralph A. 2012. Innovation and the dynamics of global warming. Journal of Environmental Economics and Management, forthcoming.

7

Resource Extraction and Backstop Technologies in General Equilibrium Ngo Van Long and Frank Stähler

7.1

Introduction

In this chapter we study the impact of technological progress of a backstop technology in a general equilibrium model where owners of finite resource stocks adjust their extraction paths to maximize their wealth, and where the rate of interest is endogenously determined. Using this framework, we investigate the possibility of a green paradox outcome. The term “green paradox” was coined by Sinn (2008), and it refers to the possibility that due to an intertemporal supply response, any effect reducing resource uses in a static model might have the opposite effect in the dynamic model. This phenomenon of adverse intertemporal adjustment has been further explored in a series of papers, including Eichner and Pethig (2011), Gerlagh (2011), Grafton, Kompas and Long (2012), Hoel (2008, 2011), van der Ploeg and Withagen (2012), among others, but in fact the concept was introduced in an early paper by Welsch and Stähler (1990). These papers investigated different channels through which green paradox outcome might arise. While these papers have either used a partial equilibrium framework in which the interest rate is fixed or a general equilibrium framework in which the interest rate does not change over time, we endogenize the determination of the interest rate in our model. In doing so, we are able to identify a new channel: technological progress, as this can indirectly induce a change in the equilibrium interest rate that in turn induces a change in the time path of extraction in the opposite direction. In particular, technological progress of a backstop technology can result in more global resource extractions, or it may bring resource extractions closer to the present. In our model, resources, either provided by an exhaustible stock or by the technology, are used in production together with a composite factor. We also consider the

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role of the composite factor and demonstrate that factor augmentation plays a much more favorable role than green technological progress in our setup. Accordingly, the remainder of the paper is organized as follows: section 7.2 introduces the two period general equilibrium model, section 7.3 deals with the case of no exhaustion, section 7.4 deals with the case of exhaustion, and section 7.5 concludes the chapter. 7.2

The Model

We employ a two-period model of resource extraction. In order to focus on the role of the backstop technology, we assume that all markets are perfectly competitive. In any period t, t ∈ {1, 2}, a composite good Yt is produced using energy, Et, and a composite factor (capital), denoted by Kyt, where the subscript denotes the use of the factor. Energy can be obtained from an exhaustible resource Rt (extracted from a resource stock St) and from renewable resources, such as wind, solar, or biofuels, denoted by xt, and both are perfect substitutes such that Et = Rt + xt. The backstop technology is able to convert the composite good into renewable resources, and thus we have to distinguish between gross and net output. The gross output of the composite good is Yt = F (Et, Kyt), and we assume that F (Et, Kyt) is a neoclassical production function and homogeneous of degree 1. Defining the energy intensity et ≡ Et/Kyt, we can rewrite production and marginal productivities as follows: Yt = K yt f ( et ) , FEt = f ′ ( et ) , FK yt = f ( et ) − et f ′ ( et ) . The composite good is the numéraire in our model. The backstop technology allows one to convert the composite output into (renewable) energy, denoted by x. To produce one unit of x in period t, one must use λ units of the composite good, where λ is the technological parameter. Its inverse, 1/λ is a measure of the efficiency of the backstop technology.1 If both x and the exhaustible resource are used by firms, λ is also the price of the exhaustible resource, in terms of the composite good. Consequently the net output of the composite good, intended for consumption and denoted by Q is equal to Qt = Yt − λxt = F(Rt + xt, Kyt) − λxt. We assume that the production function is of Cobb–Douglas type such that f ( et ) = etβ with 0 < β < 1. We are interested in the general equilibrium effects of the backstop technology on resource extraction, and we

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also want to take into account that resource extraction is subject to rising extraction costs. In our general equilibrium context, to extract the quantity Rt, the representative extracting firm must use the composite factor, and we denote by K Rt the use of this factor in resource extraction. In order to model potential rising extraction costs, we suppose that to extract any amount R2 > 0, the amount of the composite factor required is dependent on how much was extracted in period 1. This assumption is meant to capture the salient features of fossil fuels: deposits are not homogeneous and lower cost layers are extracted first. The input requirements for the two periods are given by K R1 = K R2

1 2 R1 , 2

1 = θ R1R2 + R22 , 2

(7.1)

where θ ∈ [0, 1] measures the strength of the increase in capital input requirements due to extractions in the first period. We close the supply side of model by the factor market-clearing condition for the composite factor and the stock-flow condition that the sum of extractions over the two periods must not exceed the available stock S: K yt + K Rt = K , R1 + R2 ≤ S, where K is a constant in both periods. The demand side of the model is given by the behavior of a representative consumer who maximizes her intertemporal utility U (c1 ) +

1 1 U (c2 ) subject to c1 + c2 = W0 , 1+ ρ 1+ i

where W0 denotes the initial wealth (the capitalized value of her income stream) and ρ > 0 denotes the rate of time preference. U is the per period utility for which we assume b U (ci ) = aci − ci2 2 such that U′ > 0 in the relevant range and U″ = −b < 0.2 In our model the interest rate i is determined endogenously. To keep the model simple and keep other effects on the interest rate out of our model, we do not allow capital formation (K is fixed) or storing the consumption good which implies that ct = Qt. Utility maximization leads to

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U ′ (Q1 ) a − bQ1 1 + i = . = U ′ (Q2 ) a − bQ2 1 + ρ

(7.2)

The model can now be solved by scrutinizing the optimal behavior of resource owners and consumers over time. Clearly, dependent on parameter values, it is possible that in equilibrium resources are never extracted or the backstop technology is never used. In order to deal with realistic cases, we confine the analysis to the case in which both the resource and the backstop technology are used in both periods. Note carefully that this does not mean that the resource will be completely exhausted or not, so we have to distinguish these two cases. In both cases the final good producers use xt as an input in both periods t = 1, 2, and consequently it must be the case that the marginal product of xt is equated to its price, λ: FEt ≡ f ′ ( et ) = β e

β −1 t

=λ

⎛ λ⎞ for t = 1, 2 ⇒ et = e = ⎜ ⎟ ⎝ β⎠

1 ( β − 1)

.

Note that the existence of a backstop technology fixes the resource price and the energy intensity. Furthermore a decrease in λ is equivalent to an increase in ē. The constant energy intensity also implies that the factor price of the composite factor, denoted by w, is identical in both periods: wt = f ( et ) − et f ′ ( et ) = (1 − β ) e β = w. Since the resource is used simultaneously with the backstop technology, it must be the case that the final good producers are paying the same price for both energy inputs, that is, Pt = FEt ≡ f ′ ( e ) = λ . Given the Cobb–Douglas technology, the unit cost function for the composite good is C (λ , w ) = Aλ β w1−β , where 1 A≡ β . β (1 − β )1−β Since the backstop technology will be used in both periods, it works like a limit price for resource owners.3 In general, each resource owner takes the price path (P1, P2) = (λ, λ) and the path of the factor prices of the composite factor (w1, w2) = (w, w) as given, and chooses R1 and R2 to maximize the discounted sum of profit

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1 (λ R2 − wK R2 ) 1+ i subject to (7.1) , R1 + R2 ≤ S and Rt ≥ 0.

λ R1 − wK R1 +

Let η ≥ 0 be the Lagrange multiplier associated with the constraint S − R1 − R2 ≥ 0 and assume R1 > 0 and R2 > 0. The first-order conditions imply the Hotelling rule

λ − wR1 −

θ R2 1 w= [λ − w (θ R1 + R2 )] = η. 1+ i (1 + i )

(7.3)

This condition states that the present value of marginal profit from extraction in period 2 must be equal to the marginal profit from extraction in period 1 net of its impact on period 2 extraction cost. Before we discuss the two different cases, complete exhaustion of the resource or no exhaustion, let us consider the determination of the interest rate in more detail. Let us define the input price ratio

δ≡

λ λ = , w (1 − β ) ⎡(λ β )1 ( β −1) ⎤ β ⎣ ⎦

dδ 1 = > 0. dλ (1 − β ) w

(7.4)

Thus an increase in the efficiency of the backstop technology (a fall in λ) can also be measured equivalently by a decrease in δ. We can also rewrite w to see the income effect technological progress has on the factor price of the composite factor, w (δ ) = (1 − β )(1−β ) β β δ − β ,

dw w = −β , dδ δ

(7.5)

which clearly demonstrates that an increase in the backstop technology’s productivity has a positive income effect the size of which depends on the importance of the composite factor as measured by β. Using ē = (xt + Rt)/Kyt, the net output in each period, we can express Qt ≡ Yt − λxt as Qt = [ K − K Rt ] f ( e ) − λ [ e ( K − K Rt ) − Rt ] = [ K − K Rt ][ f ( e ) − e f ′ ( e )] + f ′ ( e ) Rt

(7.6)

= ( K − K Rt ) w + λ Rt = wK + (λ Rt − wK Rt ) , This equation says that net output is equal to the payment to the composite factor plus the pure profit of the resource-extracting sector. We can now rewrite consumer behavior (7.2) as Ω ≡ (1 + i )U ′ [ w ( K + δ R2 − K R2 )] − (1 + ρ )U ′ [ w ( K + δ R1 − K R1 )] = 0.

(7.7)

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Equation (7.7) determines the interest rate, and we are now interested how a change in the efficiency affects the interest rate. Suppose that resource extractions stay constant. Differentiation of (7.7) yields di dδ

dR1 = dR2 = 0

= U ′′

(1 + ρ ) (wδ R1 − βQ1 ) − (1 + i ) (wδ R2 − βQ2 ) . δ U ′ (Q2 )

(7.8)

This change gives the direct effect on i of a change in the efficiency of the backstop technology. The sign of the denominator is unambiguously positive, and the sign of the numerator depends on the relative resource extraction and the income effects across the two periods. If resource extractions do not change, an increase in the efficiency of the backstop technology (i.e., a fall in δ) will increase the interest rate if and only if ⎛ 1+ i ⎞ . wδ R1 − βQ1 > ( wδ R2 − βQ2 ) ⎜ ⎝ 1 + ρ ⎟⎠ Suppose that wδR2 − βQ2 > 0. Then the condition above can be stated as wδ R1 − βQ1 ⎛ 1 + i ⎞ . > wδ R2 − βQ2 ⎜⎝ 1 + ρ ⎟⎠

(7.9)

(If wδR2 − βQ2 < 0, then the inequality above must be reversed.) This condition can also be expressed as (λ R1 Q1 ) − β ⎛ 1 + i ⎞ Q2 > , (λ R2 Q2 ) − β ⎜⎝ 1 + ρ ⎟⎠ Q1 where λ Rt Qt is the value share of exhaustible resource input in national income, and β is the value share of energy in gross output. This is, of course, only the direct effect, which has not been taken into account, that resource extractions will respond to the change in δ and the change in the interest rate. We now scrutinize the optimal resource extraction plans and their dependence on the efficiency of the backstop technology and the interest rate. We will consider two cases of simultaneous use of exhaustible and renewable energy in both periods. In the first case, the stock of resource is relatively large, so exhaustion does not occur. The marginal resource rent is zero in that case. In the second case, the sum of extractions in the two periods is equal the initial resource stock S.

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In what follows, we are going to prove the following: Proposition 1 In any equilibrium with simultaneous use of the backstop technology and extraction from the exhaustible resource, the effect of technological progress of the backstop technology on the extraction path is ambiguous. In the case of exhaustion, technological progress may lead to a larger resource extraction in the first period. In the case of nonexhaustion, technological progress may also lead to a larger resource extraction in the first period and even to larger aggregate extraction. Thus we show that a development which will clearly have a resource saving effect in a static environment may lead to opposite results in a dynamic context. 7.3

Nonexhaustion of the Resource

If R1 > 0 and R2 > 0 and the resource is not exhausted, the marginal resource rent must be zero. This implies that in the second period λ − w(θR1 + R2) = 0 ⇒ θR1 + R2 = δ. Note that the second-period extraction depends on the first-period extraction because a larger extraction in the first period increases the input requirement for extracting the resource in the second period. For the first period the assumption that R1 + R2 < S implies that

λ − wR1 −

θ R2 θ w = 0 ⇒ δ − R1 − [δ − θ R1 ] = 0, 1+ i 1+ i

where we have used the second period optimal extraction. Solving for extractions, we obtain 1+ i −θ ⎞ R1 = δ ⎛ , ⎝ 1+ i −θ2 ⎠

(7.10)

1 + i − θ − iθ ⎞ R2 = δ ⎛ . ⎝ 1+ i −θ2 ⎠

(7.11)

Notice that R1 = R2 = δ if θ = 0, implying that Q1 = Q2 and i = ρ, so that consumption levels in the two periods are equal. If θ = 1, then R2 = 0. In what follows we focus on the interesting case where 0 < θ < 1, and we find: Lemma 1 Assume 0 < θ < 1. A nonexhaustion equilibrium, with simultaneous use of the backstop technology and resources in both periods, exists if the following conditions are met:

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2 (1 + i − θ ) − iθ ⎤ S > δ ⎡⎢ , ⎦⎥ ⎣ 1+ i −θ2 K > max

{

(7.12)

}

1 2 1 2 R1 , R2 + θ R1R2 , 2 2

R1 ≤e K − K R1

and

R2 ≤ e, K − K R2

(7.13) (7.14)

where R1 and R2 satisfy (7.10) and (7.11). Proof

The proof follows immediately from R1 + R2 < S. 䊏

Note that the conditions above are satisfied if K and S are sufficiently large. Let us assume for a start that the interest rate stays constant. It can then easily be seen from equations (7.10) and (7.11) that technological progress of the backstop technology, that is, a decrease in δ, will lead to a decrease in both R1 and R2. So the good news is that, at constant interest rate, per period and aggregate resource use will decline with an increase in efficiency of the backstop technology. However, we already know from equation (7.8) that the interest rate will change with a change in the efficiency, and this effect will have repercussions on resource extractions. Differentiation of (7.10) and (7.11) yields ∂R1 δθ (1 − θ ) > 0, = ∂i 1 ( + i − θ 2 )2

δθ 2 (1 − θ ) ∂R2 < 0, =− ∂i (1 + i − θ 2 )2 ∂R1 ∂R2 δθ (1 − θ )2 + > 0, = ∂i ∂i (1 + i − θ 2 )2 which shows that an increase in the interest rate (1) will increase period 1 extraction, (2) decrease period 2 extraction, and (3) increase aggregate extraction. Let us now summarize the effects: for a given resource extraction plan, an increase in the efficiency of the backstop technology must imply an increase in the interest rate if equation (7.9) is fulfilled. At the same time both periods’ extractions decline (by the same proportion) due to the better performance of the backstop technology, potentially moderating the effect on the interest rate given constant resource extractions. Furthermore an increase in the interest rate increases resource

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extraction in period 1 and decreases resource extraction in period 2, and increases aggregate extraction. We will now demonstrate that the effects of technological progress are ambiguous even in our simple model. First, we consider how output levels compare in the two periods which has a clear implication for the interest rate. Lemma 2 Proof

In case of nonexhaustion, Q1 > Q2 and i < ρ.

From (7.6), we have output levels given by

1 Q1 = w ⎛ K + δ R1 − R12 ⎞ , ⎝ 2 ⎠ 1 Q2 = w ⎛ K + δ R2 − θ R1R2 − R22 ⎞ . ⎝ 2 ⎠ Let Π1 and Π2 denote the profit of the exhaustible resource sector: ∏t = λ Rt − wK Rt . Then Q1 > Q2 iff Π1 > Π2. Compare Q1 and Q2. Q1 divided by w and net of K is equal to Π1/w, which is 1 + i − θ ⎞ 1 ⎛ 1 + i − θ ⎞ 2 δ 2 (1 + i − θ ) (1 + i + θ − 2θ 2 ) δ ⎛δ . − δ = 2 ⎝ 1+ i −θ2 ⎠ 2 ⎝ 1+ i −θ2 ⎠ 2 (1 + i − θ 2 ) Similarly Q2 divided by w and net of K is equal to Π2/w, which is 1 + i − θ − iθ ⎞ 1 + i − θ ⎞ ⎛ 1 + i − θ − iθ ⎞ − θ ⎛δ δ ⎛δ δ ⎝ 1+ i −θ2 ⎠ ⎝ 1+ i −θ2 ⎠ ⎝ 1+ i −θ2 ⎠ 2 2 1 1 + i − θ − iθ ⎞ 2 (1 + i ) δ 2 (1 − θ ) − ⎛δ = . 2 2 ⎝ 1+ i −θ2 ⎠ 2 (1 + i − θ 2 ) Hence Q1 > Q2 if and only if (1 + i − θ) (1 + i + θ − 2θ2) − (1 + i)2(1 − θ)2 = θ (2(1 + i)2 − (2 + i)2θ + 2θ2) > 0. Consider the function Ψ(θ) = (2(1 + i)2 − (2 + i)2θ + 2θ2) with Ψ′(θ) = −(2+i)2 +4θ < 0 so that Ψ has a minimum at θ = 1 in the relevant range. Since Ψ(θ = 1) = i2 > 0, it follows that Ψ(θ) > 0 for all θ ∈ (0, 1). Thus Q1 > Q2. Finally Q1 > Q2 implies i < ρ because of U'' = −b < 0. 䊏 We can now scrutinize the potential ambiguity of technological progress on the resource extraction path. We rewrite the equilibrium conditions as

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f (⋅) = (1 + i − θ 2 ) R1 − δ (1 + i − θ ) = 0 ,

g (⋅) = (1 + i − θ 2 ) R2 − δ (1 + i ) (1 − θ ) = 0 , 1 h (⋅) = (1 + i )U ′ ⎡⎢ w ⎛ K + δ R2 − θ R1R2 − R22 ⎞ ⎤⎥ ⎝ 2 ⎠⎦ ⎣ 1 − (1 + ρ )U ′ ⎡⎢ w ⎛ K + δ R1 − R12 ⎞ ⎤⎥ = 0. 2 ⎠⎦ ⎣ ⎝ Total differentiation yields ⎡ f R1 ⎢ 0 ⎢ ⎢⎣ hR1

0 f R1 0

f i ⎤ ⎡ dR1 ⎤ ⎡ − fδ ⎤ gi ⎥ ⎢ dR2 ⎥ = ⎢ − gδ ⎥ dδ , ⎥ ⎥⎢ ⎥ ⎢ hi ⎦⎥ ⎣⎢ di ⎥⎦ ⎢⎣ − hδ ⎥⎦

where the elements of matrix A on the left-hand side can be computed as f R1 = g R2 = 1 + i − θ 2 > 0 ,

f i = R1 − δ < 0 ,

gi = R2 − δ (1 − θ ) > 0

because R1 < δ and

(1 + i ) (1 − θ )

> δ (1 − θ ) ⇔ 1 + i > 1 + i − θ 2 , 1+ i −θ2 fδ = − (1 + i − θ ) < 0 , gδ = − (1 + i ) (1 − θ ) < 0 ,

R2 = δ

hR1 = − wbθ R2 (1 + i ) − wb (δ − R1 ) (1 + ρ ) > 0 , because U'' = −b < 0, hR2 = wb (δ − θ R1 − R2 ) (1 + i ) = 0 , because θR1 + R2 = δ, hi = U ′ (Q2 ) = a − bQ2 > 0 , βQ2 ⎞ βQ (1 + i ) − ⎛ wR1 − 1 ⎞ (1 + ρ )⎤⎥ . hδ = −b ⎡⎢⎛ wR2 − ⎝ ⎝ ⎠ δ δ ⎠ ⎦ ⎣ Note that hδ is ambiguous in sign. This is exactly the direct effect of a technology change on the interest rate (see equation 7.8): di dδ

=− dR1 = dR2 = 0

hδ . hi

What is the intuition for this ambiguity? When δ declines, both Q1 and Q2 change in complex ways even for given resource extractions. First, for given resource extractions, the partial effect is that both decline via

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the effect that resource extractions have become relative less productive. Second, the factor price of the composite factor increases, and the strength of this income effect depends on the importance of this factor for production as measured by β. The sign of wRt −

βQt λ Rt − βQt = δ δ

is ambiguous for both periods and may even differ across both periods. From Cobb–Douglas, we know that the share of all energy input from gross production stays constant, that is, λEt = βYt, but Et > Rt and Yt > Qt due to the backstop technology. In any case, this effect is positive (negative) if the income effect is weak (strong).4 Let us now turn to the equilibrium and its properties in more detail. Expanding the matrix A along the first row yields the determinant det ( A ) = f R1 ( f R1 hi − f i hR1 ) > 0 , which proves that the equilibrium is unique. Define ⎡ − fδ A1 = ⎢ − gδ ⎢ ⎢⎣ − hδ

0 f R1 0

fi ⎤ ⎡ f R1 ⎥ gi , A2 = ⎢ 0 ⎥ ⎢ ⎢⎣ hR1 hi ⎥⎦

− fδ − gδ − hδ

fi ⎤ ⎡ f R1 ⎥ gi , A3 = ⎢ 0 ⎥ ⎢ ⎢⎣ hR1 hi ⎥⎦

0 f R1 0

− fδ ⎤ − gδ ⎥ ⎥ − hδ ⎥⎦

so that the changes with δ are given by dR1 det ( A1 ) = , dδ det ( A )

dR2 det ( A2 ) = , dδ det ( A )

di det ( A3 ) = dδ det ( A )

according to Cramer ’s rule. Since det(A) > 0, the signs of the changes are given by sign (dR1/dδ) = sign (det(A1)), sign (dR2/dδ) = sign (det(A2)), sign (di/dδ) = sign (det(A3)), where the determinants det ( A1 ) = f R1 ( − fδ hi + f i hδ ) ,

det ( A2 ) = f R1 ( − gδ hi + gi hδ ) + fδ ( − gi hR1 ) + f i gδ hR1 ,

det ( A3 ) = f R1 ( − hδ f R1 + hR1 fδ ) ,

are all ambiguous in sign. In particular, if the direct effect on the interest rate for given resource extractions, measured by hδ, and the responsiveness of first period resource extractions to the interest rate, measured by fi, are sufficiently strong, first-period extraction will go up with technological progress, that is, dR1/dδ < 0. In this case the interest rate effect is so strong that it overcompensates the second countervailing

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effect which is due to the increased productivity of the backstop technology. If hδ > 0, however, we also observe that dR2/dδ > 0 because the interest rate has the opposite effect on the response of second-period extractions to the interest rate (gi < 0). Thus, while first-period extraction may increase with technological progress, second-period extraction will decrease with technological progress if hδ > 0. Since we consider the case of nonexhaustion, we may now also explore whether aggregate resource extraction (which is less than S) may increase with technological progress. Could it be the case that the aggregate R1 + R2 declines with δ? Consider det(A1) + det(A2), which has the same sign as d(R1 + R2)/dδ and is ambiguous because det ( A1 ) + det ( A2 ) = f R1 hδ ( f i + gi ) + f R1 ( − fδ ) hi

+ f R1 ( − gδ ) hi + fδ ( − g i hR1 ) + f i gδ hR1 ,

where the first term on the right-hand side is negative if hδ > 0 and the sum of the remaining four terms is positive: f i + gi = R1 + R2 − δ − δ (1 − θ ) = −

δ (1 − θ )2 θ < 0. 1+ i −θ2

(7.15)

Thus we find that even aggregate resource extraction may increase with technological progress if hδ is sufficiently large. Given our specification, we can now present the results of a simulation. Table 7.1 demonstrates a case in which technological progress unambiguously increases the interest rate due to the implied income effect. This income effect can be observed by the increase in w. In this simulation the interest rate effect is sufficiently strong as to increase first period extractions in the range of high δ’s. For sufficiently low δ’s a further technological progress decreases first period extractions as the interest rate effects is no longer strong enough. Thus the simulation clearly demonstrates the possible nonmonotonicity of resource extractions with technological progress. As is shown before, resource extractions in the second period decline when they increase in the first period, but in this simulation the interest rate effect is not sufficiently strong to make aggregate extractions increase with technological progress. We can also scrutinize the effect of an increase in the composite factor on the resource extraction path. Total differentiation of the equilibrium conditions yields ⎡ dR1 ⎤ ⎡ 0 ⎤ A × ⎢ dR2 ⎥ = ⎢ 0 ⎥ dK , ⎢ ⎥ ⎢ ⎥ ⎢⎣ di ⎥⎦ ⎢⎣ − hK ⎥⎦

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Table 7.1 Simulation of technological progress δ

R1

R2

R1 + R2

Q1

Q2

I

w

K R1

K R2

2.10 2.08 2.06 2.04 2.02 2.00 1.98 1.96 1.94 1.92 1.90

1.6206 1.62919 1.63622 1.6416 1.64531 1.64733 1.64772 1.64655 1.64391 1.6399 1.63462

1.2897 1.2654 1.24189 1.2192 1.19735 1.17633 1.15614 1.13672 1.11804 1.10005 1.08269

2.9103 2.8946 2.87811, 2.8608 2.84265 2.82367 2.80386 2.78328 2.76196 2.73995 2.71731

4.75607 4.74148 4.72618 4.71024 4.69371 4.67668 4.65924 4.64147 4.62344 4.60522 4.58689

3.91191 3.89481 3.87877 3.86377 3.8498 3.83682 3.82478 3.81363 3.80331 3.79375 3.78488

0.345113 0.40348 0.465252 0.530132 0.597764 0.66776 0.739723 0.813264 0.888018 0.963653 1.03988

0.670805 0.671447 0.672096 0.672752 0.673416 0.674086 0.674764 0.675449 0.676142 0.676843 0.677552

1.31317 1.32713 1.33861 1.34743 1.35351 1.35685 1.3575 1.35557 1.35122 1.34463 1.33599

1.87671 1.83142 1.78715 1.74394 1.70182 1.66079 1.6208 1.58191 1.54399 1.5070 1.471

Note: Simulation uses β = 0.1, ρ = 5, K = 5, θ = 0.5, a = 10, and b = 2.

where the derivatives on the LHS are as before and hK = wU''(i − ρ) > 0, since i < ρ, U'' = −b < 0.5 Define ⎡ 0 C1 = ⎢ 0 ⎢ ⎢⎣ − hK

0 f R1 0

fi ⎤ ⎡ f R1 ⎥ g i , C2 = ⎢ 0 ⎥ ⎢ ⎢⎣ hR1 hi ⎦⎥

0 0 − hK

fi ⎤ ⎡ f R1 ⎥ g i , C3 = ⎢ 0 ⎥ ⎢ ⎢⎣ hR1 hi ⎦⎥

0 f R1 0

0 ⎤ 0 ⎥ ⎥ − hK ⎥⎦

so that the changes with K are given by dR1/dK = det(C1)/det(A), dR2/dK = det(C2)/det(A), di/dK = det(C3)/det(A). We find that det (C1 ) = f R1 f i hK < 0 , det (C2 ) = f R1 gi hK > 0 , and det (C3 ) = − f R21 hK < 0 are all unambiguous in sign. The interest rate decreases with an increase in K, and consequently resource extraction will decrease in the first period and increase in the second period. The intuition is straightforward: if K increases, both Q1 and Q2 increase for given R1 and R2. Since U'' is constant, the interest rate must decrease as to guarantee that h(·) = 0 holds. The difference to technological progress is that a change in endowment has the same marginal effect on Q1 and Q2. Thus factor augmentation has much more favorable effects on resource extractions over time and cannot lead to an increase in

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first-period extractions. Furthermore we find that the change in aggregate extraction, the sign of which is given by det (C1 ) = f R1 hK ( fi + gi ) < 0 , is clearly negative because fi + gi is negative (see equation 7.15). Thus an increase in the composite factor endowment decreases aggregate resource extraction unambiguously. 7.4

Exhaustion of the Resource

Let us turn to the case of exhaustion, with positive extraction in both periods. In this case aggregate extraction is fixed, but per period extraction will change with technological progress. According to equation (7.3), the optimal resource extraction plan is given by R1 =

S (1 − θ ) + iδ S (1 + i − θ ) − iδ , R2 = S − R1 = . 2 (1 − θ ) + i 2 (1 − θ ) + i

(7.16)

This happens if S is not too large (so that exhaustion will happen) and not too small (so that period 2 extraction is positive). Lemma 3 An equilibrium with simultaneous use of the backstop technology and resources in both periods and exhaustion exists if iδ 2 (1 + i − θ ) − iθ ⎤ . < S ≤ δ ⎡⎢ 1+ i −θ ⎦⎥ ⎣ 1+ i −θ2

(7.17)

Proof For R2 to be positive, we must have S(1 + i − θ) > iδ. The other constraint simply follows from lemma 1. 䊏 Note that if iθ ≥ 0 and θ > 0, then condition (7.17) implies that S < 2δ. In the case of exhaustion we have R1 > R2 provided that i > 0: R1 − R2 =

i ( 2δ − S) . 2 (1 − θ ) + i

Again, we can show that consumption in period 1 is higher than consumption in period 2 and the equilibrium rate of interest is smaller than the rate of time preference: Lemma 4

In case of exhaustion, Q1 > Q2 and i < ρ.

Proof Due to (7.6), output levels are given by lemma 2. Compare Q1 and Q2. Q1 divided by w and net of K is equal to

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165

δ 2 (1 + i − θ ) (1 + i + θ − 2θ 2 ) 2 (1 + i − θ 2 )

2

Similarly Q2 divided by w and net of K is equal to

[(1 + i) (S − 3δ ) + (3 + 2i)δθ − Sθ 2 ][(1 + i) (−S + δ ) − δθ + Sθ 2 ] . 2 2 (1 + i − θ 2 ) Hence Q1 > Q2 if χ(S) ≡ 2Sδ(−2(1+i)+(2+i)θ)(1+i−θ2)+S2(1+i−θ2)2 +2δ2(2(1+i)2−(1+i)(3+i)θ + θ3 > 0. We find that χ'(S) = 2 (1 + i − θ2)2 > 0, and thus χ has a minimum at 2 (1 + i − θ ) − iθ ⎤ S = δ ⎡⎢ , ⎦⎥ ⎣ 1+ i −θ2 which is the upper bound of the resource stock for the existence of an exhaustion equilibrium (see lemma 3). Furthermore

χ ⎡⎢S = δ ⎛ ⎝ ⎣

2 (1 + i − θ ) − iθ ⎞ ⎤ = δ 2θ ⎡⎣ 2 (1 + i )2 − ( 2 + i )2 θ + 2θ 2 ⎤⎦ > 0, 1 + i − θ 2 ⎠ ⎥⎦

which completes the proof. 䊏 For the case of exhaustion, we have to consider only the first period; opposite changes hold for the second period. Making use of (7.16) and S < 2δ, we find that i ∂R1 = > 0, ∂δ 2 (1 − θ ) + i

∂R1 (1 − θ ) ( 2δ − S) > 0, = ∂i [2 (1 − θ ) + i ]2

(7.18)

so these derivatives have the same signs as in the nonexhaustion case. First, we observe that, at constant interest rate, an increase in the efficiency of the backstop technology, that is, a decline in δ, will lead to less extraction in period 1 (and an increase in the extraction in the second period). This result differs from the literature on a backstop technology where the backstop is not used in the current period (e.g., see Hoel 2008; Gerlagh 2011). Hoel (2008) shows that lowering the cost of the (future) backstop will lead to more current extraction. However, equation (7.18) gives only partial effects. To find the full effect of a

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decline in δ on R1, we must take into account the adjustment of the equilibrium interest rate. For this purpose we rewrite the equilibrium conditions as F (⋅) = ( 2 (1 − θ ) + i ) R1 − iδ − (1 − θ ) S = 0 , G (⋅) = R1 + R2 − S = 0 , 1 H (⋅) = (1 + i )U ′ ⎡⎢ w ⎛ K + δ R2 − θ R1R2 − R22 ⎞ ⎤⎥ ⎝ 2 ⎠⎦ ⎣ 1 − (1 + ρ )U ′ ⎡⎢ w ⎛ K + δ R1 − R12 ⎞ ⎤⎥ = 0. ⎝ 2 ⎠⎦ ⎣ Total differentiation yields ⎡ FR1 ⎢ 1 ⎢ ⎢⎣ H R1

0 1 H R2

Fi ⎤ ⎡ dR1 ⎤ ⎡ − Fδ ⎤ 0 ⎥ ⎢ dR2 ⎥ = ⎢ 0 ⎥ dδ , ⎥⎢ ⎥ ⎢ ⎥ H i ⎥⎦ ⎢⎣ di ⎥⎦ ⎢⎣ − Hδ ⎥⎦

where the elements on the left-hand side of the equation above can be computed as FR1 = 2 (1 − θ ) + i > 0 , Fi = R1 − δ < 0 , Fδ = −i because R1 < δ, H R1 = − wθ R2 (1 + i )U ′′ − w (δ − R1 ) (1 + ρ )U ′′ > 0 because U'' = −b < 0, H R2 = w (δ − θ S) (1 + i )U ′′ < 0 because U'' = −b < 0, and due to the existence condition for exhaustion, 2 (1 + i − θ ) − iθ δ < δ ⇔ 2θ − θ 2 < 1 ⇔ (1 − θ )2 > 0 , 1+ i −θ2 H i = U ′ (Q2 ) > 0 , βQ2 ⎞ βQ Hδ = ⎛ wR2 − (1 + i )U ′′ − ⎛ wR1 − 1 ⎞ (1 + ρ )U ′′. ⎝ ⎝ δ ⎠ δ ⎠

θS = θ

Note that Hδ is ambiguous in sign also in the case of exhaustion. Expanding along the first row yields the determinant det (B) = FR1 H i + Fi ( H R2 − H R1 ) > 0 , which proves that the equilibrium is unique. Since R2 = R1 − S, we may confine the comparative statics results to the effects of a change in δ on R1 and i. Define

Resource Extraction and Backstop Technologies

⎡ − Fδ B1 = ⎢ 0 ⎢ ⎢⎣ − Hδ

0 1 H R2

Fi ⎤ ⎡ FR1 0 ⎥ , B3 = ⎢ 1 ⎥ ⎢ ⎢⎣ H R1 H i ⎥⎦

0 1 H R2

167

− Fδ ⎤ 0 ⎥ ⎥ − Hδ ⎥⎦

so that the signs of the changes are given by sign (dR1/dδ) = sign (det(B1)), sign (di/dδ) = sign (det(B3)) due to det(B) > 0. We find that det(B1) = −Fδ Hi + FiHδ and det (B3 ) = FR1 ( − Hδ ) − Fδ ( H R2 − H R1 ) are both ambiguous in sign, and thus we have the similar effects as for the case of no exhaustion. The interest effect depends on the size of the income effect.6 If the interest rate increases with technological progress and this effect is sufficiently strong, then first-period extraction will increase and second-period extraction will decrease. We can also scrutinize the effect of an increase in the composite factor on the resource extraction path. Total differentiation of the equilibrium conditions yields ⎡ 0 D1 = ⎢ 0 ⎢ ⎢⎣ − H K

0 1 H R2

Fi ⎤ ⎡ FR1 0 ⎥ , D3 = ⎢ 1 ⎥ ⎢ ⎢⎣ H R1 H i ⎥⎦

0 1 H R2

0 ⎤ 0 ⎥ ⎥ − H K ⎥⎦

so that the changes with K are given by dR1/dK = det(D1)/det(B), di/dK = det(D3)/det(B) according to Cramer ’s rule. We find that det(D1) = fiHK < 0 and

(7.19)

det (D3 ) = − f R21 H K < 0

(7.20)

are both unambiguous in sign, since the interest rate declines with an increase in K, and thus we find also for the case of exhaustion that factor augmentation leads to a more favorable and unambiguous outcome as first period extractions decline. Since the resource is exhausted, an increase in K will shift resource uses from the first to the second period. 7.5

Concluding Remarks

This chapter has presented a simple dynamic general equilibrium model of resource extractions when a backstop technology is available. We investigated the possibility of a green paradox arising from the technological progress of a backstop technology. We identified a new channel, by which technological progress influences the extraction path

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indirectly with a change in the equilibrium interest rate. We scrutinized the possibility of a green paradox arising in a simple two-period general equilibrium model with endogenous interest rate determination. We obtained an interesting result: technological progress of the backstop technology would not result in a green paradox outcome if the interest rate does not change, but it may lead to a green paradox outcome if the interest rate increases sufficiently. Technological progress leads to an increase in factor prices, and if the interest rate stays constant, resource extraction in the first period will decline and increase in the second period in the case of exhaustion. This would tend to decrease period 1’s net output, and increase period 2’s net output. To induce consumers to increase the ratio of second-period consumption to firstperiod consumption, the interest rate must rise. If this increase in interest rate is large enough, the indirect effect will outweigh the direct effect, resulting in a green paradox. We demonstrated also that the strength of these effects can be so strong that these results also hold for both the case of exhaustion and the case of nonexhaustion. Furthermore it is possible that aggregate extraction increases with technological progress in the case of nonexhaustion. Our model has been kept simple in order for us to be able to focus on the general equilibrium repercussions of technological progress. We also showed that our effects depend on the importance of resource inputs for production. If the value of resource input as a share of the output value is relatively small, the green paradox will not materialize. However, by definition, this would also mean at the same time that the problem under consideration is of minor importance. Thus we conclude that the effects described in this chapter are of potential relevance when the problem at hand is of practical importance. We could expect similar results if we do not consider technological progress but the use of (static) policy instruments. Furthermore we kept the determination of the interest rate simple; there is no investment in our model, and the endowment with the composite factor is kept constant over the two periods. Investment would allow consumers to smooth the consumption path, so the income effect could be mitigated. At the same time, however, it will increase the marginal productivity of resource inputs. As may be obvious, introducing investment will not make the analysis less complex, since the change of interest rate and its implications are already ambiguous without investment. We leave the analysis of whether investment may add to or mitigate a potential green paradox to future research.

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Notes We wish to thank Christian Beermann and the participants of the conference for useful comments on an earlier version of this chapter. The usual disclaimer applies. 1. Note that we do not assume any technological progress from one period to the other, which alone could give rise to a green paradox. Any timing effect is not necessary for our results. 2. In Long and Stähler (2014) we demonstrate that our results also hold for a general utility function with the usual properties. 3. Hence it would not matter whether the resource industry is perfectly competitive or monopolistic, as long as the capital input requirements for resource extractions are the same for a monopolistic and a competitive industry. 4. Additionally the marginal utilities change across both periods. In a model in which utility were not linear-quadratic, the strength of this effect in each periods would depend on how the second derivative of the utility function changes as consumption changes, that is, on the third derivative of the utility function. This effect is not present in our model. 5. This effect would be ambiguous if we were not using a linear-quadratic specification. In general, the effect depends on the third derivative of the utility function as well. 6. In the case of a more general utility function U, it would also depend on the third derivative of the utility function, which measures the relative change of marginal utilities.

References Eichner, T., and R. Pethig. 2011. Carbon leakage, the green paradox, and perfect future markets. International Economic Review 52: 767–805. Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57: 79–102. Grafton, Q., T. Kompas, and N. V. Long. 2012. Substitution between biofuels and fossil fuels: Is there a green paradox? Journal of Environmental Economics and Management 64 (3): 328–41. Hoel, M. 2008. Bush meets Hotelling: Effects of improved renewable energy technology on greenhouse gas emissions. Working paper 2492. CESifo. Hoel, M. 2011. The green paradox and greenhouse gas reducing investment. International Review of Environmental and Resource Economics 5: 353–79. Long, N. V. and F. Stähler. 2014. General equilibrium effects of green technological progress. Working paper 2014s-04. CIRANO, Montreal. Sinn, H.-W. 2008. Public policies against global warming: A supply-side approach. International Tax and Public Finance 15: 360–94. van der Ploeg, F., and C. Withagen. 2012. Is there really a green paradox? Journal of Environmental Economics and Management 64 (3): 342–63. Welsch, H., and F. Stähler. 1990. On externalities related to the use of exhaustible resources. Journal of Economics 51: 177–95.

Announcement Effects, and Time III Timing, Consistency

8

Does a Future Rise in Carbon Taxes Harm the Climate? Florian Habermacher and Gebhard Kirchgässner

8.1

Introduction

In his seminal contribution Pearce (1991) discussed the advantages of a carbon tax as an efficient policy instrument to reduce carbon dioxide emissions. He considered only the demand side, implicitly assuming a fixed, exogenous energy supply. Today a large fraction of climate economics research still exhibits the same limitation, reducing the supply side of the energy market to a static process. However, at least since the contribution of Sinn (2008), there is a growing awareness that supply-side effects can be crucial in assessing carbon emission reduction strategies. According to the claim that Sinn entitled “green paradox,” a realistic carbon tax, which for political reasons deviates from the optimal tax1 and is introduced at a low initial level but rapidly increasing over time, might be counterproductive for the climate; rather than delaying or reducing the exploitation of limited resources, the tax could accelerate their combustion. Sinn does not claim that the optimal carbon tax would rise at a rate higher than the interest rate. Instead, the relevance of the rapidly rising tax is explained in terms of the politically feasible. Past experiences have confirmed that realistically, governments are not willing or not able to impose carbon taxes with a high initial level. In addition it is not necessarily implausible that the tax may rise rather rapidly over time, as governments seek ways to increase their revenues and as popular resistance against the tax may fall once it is introduced and its general principle becomes accepted. We therefore here start from the assumption that the tax would indeed rise at the postulated high rates, even if, as van der Ploeg (2013) shows, the optimal tax would rise less rapidly. That is, we clearly consider a second-best world. In this case the green paradox occurs if, in the early periods, owners of fuel

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anticipate the tax to be higher in the future, causing them to sell more of their fuels today rather than on the highly taxed future markets. While controversial, Sinn’s analysis has impressively demonstrated the importance of supply-side effects for the assessment of greenhouse gas policies. A growing body of literature addresses the mentioned counterproductive effects of climate policies. Gerlagh (2011) examines the impact of supplier anticipation on the climate benefits of cheaper future backstop technologies. A similar approach is taken by van der Ploeg and Withagen (2012), who also show that a specific tax that is not rapidly increasing could be beneficial for the climate, but do not discuss effects of other, suboptimal taxes. Polborn (2011) concludes that intensifying research on carbon capture and storage has the advantage of reversing the negative anticipation effects that research on backstop technologies would have in terms of near-term carbon emissions. We use the notions of the weak and the strong green paradox for emission damages, going back to Gerlagh (2011). In addition we propose to distinguish a very strong version of the paradox: Definitions A weak green paradox arises if the carbon tax increases initial damages. A strong green paradox arises if the carbon tax increases the net present value of all future damages. A very strong green paradox arises if the carbon tax (weakly) increases the damages at all future times. We assume damages to strictly increase in cumulative emissions. The weak version of the paradox is thus equivalent to increased initial emissions, and the very strong version to (weakly) increased cumulative emissions at each period. Sinn (2008) and subsequent contributors often considered the case where a fixed amount of resources would be extracted at low, potentially constant marginal costs. This simplified setting supported the very strong green paradox: the tax may (weakly) increase cumulative emissions at each period. Such reserve homogeneity is, however, in stark contrast with reality. The fuel reserve structure is better described as a continuum of resources that may be extracted at increasing, up to prohibitively high, costs. For example, the ultimately exploited fraction of the hydrocarbons stored in a specific oil field depends positively on the oil price (e.g., IEA 2008), with the fraction of the hydrocarbons remaining underground never reaching zero. Extraction cost

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175

projections such as Rogner (1997) and IEA reports (e.g., IEA 2008) predict convex marginal long-run extraction costs as a function of the amount of fuels extracted. Hoel (2010) shows how, when increasing extraction costs are considered, any tax can reduce total extractions (and thus emissions); that is, there is no very strong green paradox. Because a tax thus reduces expected long-term emissions but may increase early emissions, it becomes an empirical question which effect—early emission increases or long-run cumulative emission reductions—quantitatively dominates the net present-value (NPV) change of the future damages. So far the existing, mostly theoretical contributions leave this question open (see table 8.1). The aim of the present paper is to shed light on this question. We model the effect of a—potentially rapidly increasing—carbon tax on medium- and long-term climate damages. We use two distinct but related analyses that suggest that the relevant strong green paradox is unlikely. The first considers that, independent of the decision on the current tax, other climate-relevant developments may take place in future. The analyses by Sinn and subsequent contributors assumed a world in which the debated policy would be the only potential relevant climate measure and that this would hold today and forever. Abstaining from a carbon tax today, however, does not imply that neither a carbon tax nor any alternative climate relevant development such as backstop technologies, global fuel demand cartels à la Kyoto, or carbon capture and storage systems will materialize in the future. Rather, absent substantial measures today, the unlimited growth of the climate threat may increase the urgency of future measures, requiring even more stringent future measures than if the carbon tax were introduced today. Hoel (2010) considers this issue. He notes that purposely avoiding the introduction of a current tax influences only the probability of having a tax in the medium- or long-term future, rather than strictly preventing any potential future tax.2 In a stylized two-period model with an endogenous carbon tax in the second period he finds that the impossibility of long-term commitments that typifies current politics increases the (environmental) desirability of introducing a carbon tax today. Sinn (2008) himself also pointed out a number of potential climate change remedies that humankind should attempt to apply in future, some of which could be used to reduce also cumulative longterm extractions or emissions. He clearly believes that certain measures may be possible in future, as we writes:

Hoel 2010 Haberma-cher and Kirch-gässner (here)

BAU BAU or future switch to alternative measure Future optimal tax

unlikely

Yes

Yes

Yes

No

No

unlikely

Unlikely

?

Yes

No

No

Strong

no

No

No

Yes

No

No

Very strong

Note: A partial overview including analyses for delayed taxes and renewable energy subsidy schemes is in van der Werf and Di Maria (2012).

Two periods, constant intra-period tax

Hoel 2010

Sinn 2008

Fixed reservoir, constant (or low) extraction cost Finite choke price, increasing extraction costs Finite choke price, increasing extraction costs, calibration to IEA projections Increasing extraction costs

BAU

High initial level Low initial level

Rapidly in-creasing tax

van der Ploeg 2013; van der Ploeg and Withagen 2012 Hoel 2010

BAU or future backstop

Optimal tax (rises more slowly than interest rate)

Example

Weak

Baseline

Tax path

Resources

Form of tax/paradox

Results

Analyzed framework

Table 8.1 Green paradox for currently Introduced carbon tax

176 Florian Habermacher and Gebhard Kirchgässner

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177

[T]he good thing about the Kyoto Protocol is that it did show that world-wide cooperative agreements are possible. Integrating the three big countries mentioned [India, China, and the United States] and Australia, which recently announced that it wants to sign, would mean that another 45% of carbon consumption, in total three quarters of world consumption, would be captured. This share in itself would be substantial, and there could be hope that the remaining quarter could also be disciplined by political means. If the world acts quickly, before the resource owners have time to react, it might be possible to establish a world-wide trading system without loopholes. (Sinn 2008, p. 386)

He has, however, not considered how, if implemented in future, such measures may alter the conclusion about the effect of current taxes or measures. We assess the impact of current carbon taxes given that even if a tax is currently avoided, other climate measures, such as backstop technologies, global fuel demand cartels à la Kyoto, carbon capture and storage systems, and alternative carbon taxes, may be introduced at some point in the future; that is, the relevant baseline scenario is not a perpetual business as usual (BAU). To keep the model tractable, we assume that these future measures will be introduced independently of the current tax. We explain that considering the endogeneity of such measures could strengthen our results. In the case of a future regime change, such as the emergence of a backstop technology, any presently implemented positive tax path that bridges the time until the future measure becomes effective will reduce cumulative emissions not only in the long term but already in the medium term, suggesting that the strong version of the green paradox may not hold. According to recent estimates, the warming effect of emissions in the current century will remain almost unchanged over at least the next 1,000 years (Solomon et al. 2009). This suggests that in terms of mediumterm emissions, the cumulative emissions are of primary importance and that the exact path of the emissions across the decades is only of limited additional relevance. Thus, by reducing cumulative mediumterm emissions, the tax is very likely to have favorable effects on the climate. If the point of the backstop introduction is stochastic rather than fixed and known, even the weak version of the green paradox does not necessarily hold; taxes that increase at rates higher than the real interest rate not only can reduce cumulative emissions for some future period but reduce current and near-term emissions as well.

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The second, numerical analysis accounts for the worldwide fuel demand structure and for estimated extraction cost curves to quantify the likely effect of different carbon taxes on the emission path and damages. The medium- and long-term emission reductions induced by theoretically even “green paradox facilitating” tax paths appear to outweigh the potential early emission increases even for very high emission discount rates; the strong green paradox does not materialize. This is robust to a variety of model parametrizations, and even if we abstract from the future measures we assumed in the first part. While we implicitly assume linear damages in the main analysis, we extend the analysis to a damage function that is convex in cumulative emissions, and show that this further reduces the likelihood of the strong green paradox. In the following, section 8.2 presents the analytical exercise concerning the future measures, section 8.3 the numerical simulations that quantify the green paradox, and section 8.4 concludes. 8.2

Future Alternative Developments

This section examines the scope for a strong green paradox to occur if the baseline scenario, within which the introduction of a carbon tax is considered, is not a business-as-usual (BAU) scenario but instead is a baseline scenario (BS) already containing future climate relevant developments that may take place even if the tax is currently abstained from. 8.2.1 Model Considering the different categories of fossil fuels as one resource, we assume a world in which consumers’ instantaneous demand rate rt, which equals the extraction rate, is a continuous, strictly decreasing, and potentially time-varying function of the resource price, pt. Thus we have the demand curve, rt (pt), as well as its inverse, pt (rt), as two strictly decreasing functions, rt′(⋅) < 0, pt′ (⋅) < 0, where the strict inequalities may only cease to apply if the values of r or p reach their respective upper or lower boundaries, should these exist. Instantaneous extraction rates integrate to cumulative extractions, At, t which are normalized to zero at the starting time, A0 ≡ 0, At = ∫ 0 rsds. Extraction costs, c, are assumed to be strictly increasing in the cumulative extractions: c′(A) > 0. This relationship implies that the most easily extractable resources are extracted first—a standard assumption that

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has been shown to be a necessary condition for the optimality of an extraction path (Herfindahl 1967). The resource owners maximize the present value of expected total net revenues, applying a positive discount rate ρ. Given a specific carbon tax path τt, the revenue flow for a specific seller i at time t is rt,i ∙ (pt − ct − τt), where rt,i is seller i’s extraction rate, and the suppliers’ maximization problem can thus be written as t

U i = max ∫ e− ρt rt ,i ⋅ ( pt ( rt ) − c ( At ) − τ t ) dt rt ,i

t

 t = rt and A0 = 0 , subject to A

(8.1)

t

At = ∫ rsds and rt = ∑ rt ,i . 0

i

In the competitive (“comp”) case, suppliers’ individual rates are so small that each considers the market price as independent of his own supply, while the monopolistic (“mono”) supplier will take the effect of his extraction rate on prices into account because the total rate equals his own supply rate, rt ≡ rt,i. Defining Pt as the considered rate of change of the gross sales revenues, rt ∙ pt (rt) in equation (8.1), we thus have Pt ,mono ( rt ) = pt ( rt ) + rt pt′ ( rt ) and Pt,comp (rt) = pt (rt). This yields the following current-value Hamiltonian and its first-order conditions: H = rt ⋅ ( pt ( rt ) − c ( At ) − τ t ) − λt rt , ∂H = 0 : Pt ( rt ) = c ( At ) + τ t + λt , ∂rt ∂H λ t = ρλt + : λ t = λt ρ − ct , ∂At

(8.2) (8.3) (8.4)

where we defined ct ≡ c(At) and used the fact that rt =

∂At ∂t

to develop ct ≡

∂c ( At ) ∂c ( At ) ∂At ∂c ( At ) = = rt , ∂t ∂t ∂At ∂At

and where λt is the shadow value at time t for a marginal unit of resource stock after the cumulative extraction of At previous units. This multiplier λt is a nonnegative value; with a larger resource stock, the

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producer ’s future extraction costs will be reduced and the future achievable profit will therefore be (weakly) higher. The forward-looking explicit solution for the multiplier in equation (8.4) becomes, for any t < t < t , t

λt = eρ(t − t )λ t + ∫ eρ(t − s)csds.

(8.5)

t

The primary assumptions on which we will base our analysis of the supply behavior implicitly defined by the maximization problem are as follows: • Property 1: p(0) > c(0). In the absence of a tax, there will be a strictly positive extraction rate, at least at the start. • Property 2: p(0) < ∞. The choke-price is finite. This assumption is intuitive, notably as surrogates such as renewable wood or plant oils lend themselves as natural substitutes. • Property 3: c(A) < p(0) ⇒ 0 < c′(A) < ∞. As long as some resources are profitably extractable, the rate of increase of the extraction costs is strictly positive and finite. • Property 4: limr→∞p(r) = 0. When the supply rate tends to infinity, the demand price becomes zero. • Property 5. Single crossing in the first-order conditions for the monopolistic supplier: the marginal revenue of a monopolist’s resource sales at a specific period is decreasing in the current rate of extraction, or ∂ [ p (r ) + p′ (r ) r ] 0 but λT,BS = 0 for the baseline case where all fuel sales are prevented after T), and have lower opportunity costs for selling fuels during a specific pre-T period. That is, fuels will be sold for lower prices. As the market prices during the time approaching T were already lower than the demand choke price (otherwise, no fuel would have been sold then), and the shortening of the sales horizon

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further lowers market prices, we have fuels sold at prices strictly below the choke price during the periods immediately preceding the final period T. Therewith fuel owners do not extract fuels that have extraction costs of the level of the choke price, as else the extraction costs would exceed the gross sales revenues. Note that while lemma 2 is provided for the case where the time-T measure leads to λT = 0, it is intuitive, and known from lemma 1, that, if the fuels keep a certain value after period T, λT > 0, the amount of fuel left underground at time T must be at least as large as for λT = 0. Cap-and-Trade Scheme For an illustration of a post-T scheme that does not prevent all fuel sales, we consider a worldwide cap-and-trade scheme à la Kyoto from time T, imposing a fixed upper limit on the fuel consumption rate, r . We abstract from extraction costs and assume that a fixed stock S is extracted under perfect competition. We further consider an isoelastic demand, rt = d ⋅ ptε , ε < 0, and the demand price to be limited by the price of a perfect backstop available for a constant price pb. For a given stock ST of resources remaining at T, the cap reduces the value of the marginal resource for post-T sales, λT, as can be seen as follows: optimization by the fuel owners implies that present-value profits are constant over time, so that λt = λ0eρt, independently of the capping scheme. Because extraction costs are zero, the producer price equals the resource rent, pt = λt. In the absence of a cap, this results in the consumption path rt = d ∙ (λ0eρt)ε. With the cap, this remains the same, except that for periods where the binding cap drives a wedge, πt > 0, between the seller and the consumer price, we have, for the same λ0, a reduced consumption, rtcap = d ⋅ (λt + π t )ε < d ⋅ (λt )ε . As, however, the same post-T stock, ST, must overall be consumed during the post-T phase, the consumption must increase in some of the other periods where the cap is not binding, that is, there exist some t for which rtcap > rt. As the consumption in periods without binding cap is directly determined by λ (and decreasing in it), this is only possible if the resource value λ is lowered (for AT taken as a given). This tendency of the cap to lower the value of resources for a given pre-T extraction AT, as well as the tendency of lower final resource rents, λT, to increase pre-T sales, imply that the cap leads to an increase in the consumption prior to the introduction of the cap. This increase can only be mitigated if additional policies, such as an early emission tax, make the extraction less lucrative in the pre-T period.

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Numerical Example A numerical example illustrates this effect. We consider a normalized initial stock, S = 1, a unitary elastic demand, r = p−1, a discount rate ρ = 5 percent, and a backstop price pb = 20. The capping scheme is assumed to start in period T = 3 and to limit consumption to r = 0.06. Time t , when consumption stops, is given by λ t = pb , implying that t =

log ( pb λ0 ) . ρ

The equilibrium resource rent (and thus also price) path for which the demand just exhausts the stock in the undistorted market (no cap) t equilibrium is given by ∫ 0 r (λt ) dt = S. Solved for λ0, this becomes λ0 = (1/pb + ρS)−1 = 10, and we have λt = 10e0.05t which implies a stopping time t = 20 log 2 = 13.86. Until T, cumulative extraction is T AT = ∫ 0 λt−1dt = 0.28. Now we consider the introduction of the cap. At the time the cap just stops to be binding, t*, the seller price must yield rt* = r , implying that λt* = r −1. Therewith, exhaustion from undistorted post-t* t consumption, ∫ t* r (λt ) dt = S*, with S* the stock remaining at t*, solves to S* = ( r − pb−1 ) ρ = 0.2. The corresponding cumulative extraction thus amounts to A* = 0.8. The binding cap from T through t* implies that t * = T + ( At* − AT ) r . Constant NPV profit thus yields * − ρ ⎡T + A* − AT ) r ⎤⎦ λ0 = λt* e − ρt = r −1e ⎣ ( . Therewith cumulative pre-T consumpT tion, AT = ∫ 0 r (λt ) dt , becomes AT =

ρ * r Tρ ( A − AT ) e − 1] e r [ ρ .

Numerically this solves to AT = 0.30. The implied NPV resource rent is λ0 = 9.42. The cap stops to be binding at t* = 11.4, and extraction stops at t =

log ( pb λ0 ) = 15.05. ρ

We thus see that, as explained above, the future cap lowers the resource rent and induces the fuel owners to sell more fuel up to T, compared to the case where the cap will not be introduced. Figure 8.1 illustrates the two situations. 8.2.3 Introducing a Tax before the Backstop Here we consider the case for a present tax when the baseline contains a backstop technology introduced in the medium-term future at time

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Extraction

Resource rent

0.12

22 No cap Cap r (pb )

20

0.1

18 0.08

0.06

λ

Rate

16

14 0.04 12

0.02

0 0

10

T

5

10 t* Time

15

8 0

T

5

10 t* Time

Figure 8.1 Illustration post-T cap and trade

T. The Hamiltonian formulation with the corresponding first-order conditions for the dynamic problem is given in equations (8.2) through (8.4) in section 8.2.1. We assume the backstop, when emerging at time T, such as due to a technological breakthrough, to be cheaper than the resource (extraction) at that time. In this case, the resource owners cannot profitably extract fuel anymore after T. Accordingly, the multiplier, indicating the economic value of an additional reserve, becomes zero, λT = 0 (e.g., see also Dasgupta and Heal 1974 who have studied this case of the emergence of a backstop for the resource). Next, section 8.2.4 examines the case where, instead of this cheap backstop, a different technological or political change occurs in future. As we will explain, in this case the fuel owner ’s profit maximization may imply that λT remains positive and can vary with the amount of fuel remaining underground at T.

15

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As the tax generally reduces the possible net revenues from resource sales, it seems intuitive that positive tax rates will lead to reduced cumulative extractions, given that the fuel owners freely choose the amount of fuel to be sold, and how much to be left underground in the pre-T period.5 This is emphasized in proposition 1 and shown analytically in Habermacher and Kirchgässner (2014, annex E). Proposition 1 If at a specific time T > 0 a breakthrough, such as a cheap enough backstop, emerges and prevents all profitable post-T fuel sales, profit maximization by the fuel owners implies that the marginal resource rent at T becomes zero, λT = 0, and that any scheme of positive carbon taxes up to time T, with strictly positive rates as time approaches T, leads to a reduction of cumulative emissions up to time T. This extends to the hypothetical case where a breakthrough changes the post-T situation in a way such that the fuel owner ’s maximization leads to a marginal post-T resource rent that is positive but independent of cumulative extractions up to T, λT(AT) = λT = constant. If a regime change, such as the introduction of a backstop technology, is anticipated, then a carbon tax yields a decrease of total medium-term consumption up to time T, largely independently of the form of the tax path or of the demand and production cost structure. According to our arguments above, reducing cumulative medium-term emissions is of primary importance compared to the exact path of the emissions, as long as relatively limited time spans are considered. Thus, under the assumption of a future backstop in the baseline scenario, quite any path of nonnegative tax rates seems beneficial for the climate, at least in the case where the backstop is cheaper than the extraction of fuels from time T on. The case where the future measure allows for some fuel use after T is addressed in the next section. The hypothetical situation considered in the second part of proposition 1, of a post-T situation that leads to a marginal resource rent at T that does not depend on the amount of fuel in situ, is, while theoretically imaginable, surely of a more theoretical nature; we include it here because it will be useful for an extension to alternative future measures in the next section. Importantly, this extension itself will not depend on how realistic the here considered hypothetical situation is in the first place. Further, the first part of proposition 1, on the effect of the tax in presence of a future backstop, is valid also independently of the other, hypothetical situation of the second part. The numerical examples in section 8.3 confirm that future backstop technologies decrease the likelihood of a green paradox to occur.

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8.2.4 Alternative Future Schemes Proposition 1 has established that, when the resource stock loses its value at a given time T—such as because an emerged cheap backstop prevents future fuel sales—a positive tax before this period T reduces total pre-T fuel sales. In this case the prevention of future sales implies that the marginal value of unexploited reserves at time T, λT, becomes zero, independently of the amount of fuel that remains underground, that is, independently of AT: λT corresponds to the value of the resource to the fuel suppliers, and resources that cannot economically be sold have zero value. For the following analysis, it is important to bear in mind that proposition 1 further applies to the—admittedly somewhat more abstract—case where instead of being zero independently of AT, the value of the marginal reserve available for post-T sales would, while still independent of AT, be positive rather than zero. In reality this value of the marginal remaining resource unit for postT sales, λT, is, if above zero at all, unlikely to be independent of the size of the stock of remaining resources. That is, absent a (perfect) backstop, we generally expect λT to vary with the cumulative extractions at time T, rather than being constant. This can, for example, be expected if the post-T scheme is a demand cartel or an extremely high tax that still allows for a certain amount of extractions: If the post-T regime does not prohibit all lucrative sales of the resource, fuel owners will derive profits from sales, which, for the marginal resource, correspond to λT and can positively or negatively depend on the amount of resources left underground. Satiation tends to decrease the marginal profit derived from additional resources, but the lower extraction costs for added remaining resources tends to increase them. Thus, without making further assumptions about the exact nature of the post-T resource market framework or about extraction costs or the demand function, it cannot be known a priori whether λT (AT) is upward or downward sloping. Using lemma 1, we show that one can rule out one possible case for the relationship between λT and AT in the region of the optimally chosen amount of cumulative extractions, AT* . We then discuss why the derived restriction on the relationship between λT and AT signifies that proposition 1 extends to cases with a flexible implied final multiplier λT (AT). First, a few additional words on the relationships between the marginal value and the amount of cumulative exploitations at the time of the introduction of the new regime, λT and AT, seem useful. We have introduced the function λT (AT) as the value of a marginal additional

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unexploited unit of resource at time T that is available for the post-T period. This value is defined as the additional (expected) profit the resource owner can make in the post-T future if he has a marginally increased stock of remaining exploitable resources at time T, and it thus depends on how exactly the post-T fuel market framework looks like. Conversely, the function AT (λT) designates the cumulative amount of pre-T sales the resource owner chooses when the value of a marginal unit left underground at T is λT. The function therefore corresponds to the amount of pre-T sales for which the sale of an additional marginal unit in the pre-T period would yield exactly λT additional corresponding units of pre-T profits (ignoring the influence on the post-T situation). To maximize his overall profits, the resource owner will choose an amount AT* of pre-T sales for which the marginal additional pre-T profit for another sold marginal unit in the pre-T period just equates the marginal forgone profit from post-T sales due to the increase of the pre-T exploitations. In other words, if AT* denotes the chosen (optimal) amount of pre-T sales, the following condition is satisfied !

λT ( AT* ) = ATinv ( AT* ) , where ATinv (⋅) is the inverse function of AT (λT). Let λTpre ( AT ) ≡ ATinv ( AT ), whose simple interpretation is the marginal pre-T profit from additional pre-T sales given AT units sold until T. For clarity, let λTpost ( AT ) ≡ λT ( AT ). Recall from lemma 1 that AT (λT) is decreasing in λT. For the optimal amount of cumulative exploitations AT* , the condition ∂λTpre ( AT* ) ∂λTpost ( AT* ) ≤ ∂AT ∂AT

(8.7)

must hold, as otherwise it would be lucrative for the resource owner to increase AT* : the change in overall discounted profits, Π = Πpre + Πpost, can be approximated as Π ( AT* + ε ) − Π ( AT* ) = Π pre( AT* + ε ) + Π post ( AT* + ε ) − Π ( AT* ) ≈ ελTpre( AT* ) + ≈

pre post ε 2 ∂λT ( AT* ) ε 2 ∂λT ( AT* ) − ελTpost ( AT* ) − 2 ∂AT 2 ∂AT

pre post ε 2 ⎡ ∂λT ( AT* ) ∂λT ( AT* ) ⎤ − ⎢ ⎥ ∂AT 2 ⎣ ∂AT ⎦

(8.8)

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for small deviations from AT* . Clearly, if equation (8.7) does not hold, equation (8.8) would imply profits that increase for any small value of ε; that is, AT* would not be a profit-maximizing choice. This result is illustrated graphically in figure 8.2, where the pluses indicate regions in which it would be optimal for the resource owner to increase pre-T sales, and minuses indicate where it would be optimal for him to decrease sales. Second, recall from proposition 1 that the tax unambiguously reduces pre-T sales for any given fixed λT. As the function AT (λT) remains the same here as when λTpost ( AT ) was constant, we thus know that in a diagram with AT on the horizontal axis, AT,tax must lie strictly to the left of AT,no in all relevant ranges, as is shown in Fig. 8.3. In the case where λTpost′ ( AT* ) > 0, it is implied that the tax reduces the optimal amount of pre-T sales AT* . This prediction is illustrated in figure 8.4. As λTpost′ ( AT* ) > 0, λTpre′ ( AT ) < 0 and AT,tax (λT) < AT,no (λT), we have * AT ,tax < AT* ,no.

λT

λT(AT)

AT(λT)

Local profit Minimum

AT Local profit Maximum

No optimum

Local profit Minimum No optimum

Potential optimum Figure 8.2 Possible equilibrium situations with flexible λT (AT)

AT

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By a similar argument and using equation (8.7), it becomes clear that even if λTpost′ ( AT ) < 0, AT* ,tax < AT* ,no holds. This situation is depicted in figure 8.5. Therefore the proposition from the previous section extends to the case of a flexible final multiplier, λT* = λT ( AT* ) , which we summarize in proposition 2. Proposition 2 If at a specific future time T an alternative climate measure, such as a Kyoto-like demand capping or a mandatory carbon capture and storage scheme, is introduced, implying that the marginal value of resources for post-T sales is a continuous differentiable function of the cumulative extractions up to T, λT = λT (AT), any scheme of positive CO2 taxes up to time T leads to a reduction in cumulative emissions up to time T. For an independent political or technological development replacing a potential initial tax at an exogenously given time T, the debated pre-T tax thus yields a decrease of total medium-term consumption up to time T, generally independent of the extent to which fuel owners will be able to make use of their resources left underground after time T. While we leave it open which—if any—exact future measure will lead to a regime switch in the future—after all, many different technical or political developments are theoretically possible, and, as history

λT

AT(λT), Tax

AT(λT), No tax

λconst

1

λbackstop = 0

AT(λbackstop), AT(λbackstop), Tax No tax Figure 8.3 Tax reduces pre-T emissions AT for constant λT

At

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λT

AT(λT), Tax

λT(AT)

AT(λT), No tax

λ*T, No tax λ*T, Tax

A*T, Tax

At

A*T, No tax

Figure 8.4 Tax reduces pre-T emissions AT for flexible λT (AT) when λT′ ( AT ) > 0

λT

AT(λT), Tax

AT(λT), No tax

λ*T, Tax λ*T, No tax λT(AT)

A*T, Tax

A*T, No tax

Figure 8.5 Tax reduces pre-T emissions AT for flexible λT (AT) when λT′ ( AT ) < 0

At

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teaches us, even not yet thought-about developments may become relevant rather abruptly—a rather generally applicable reasoning explains why a fuel saving up to time T implies a sustained strict reduction of cumulative emissions for a substantial time beyond T, potentially perpetually. First, as the net demand for fossil fuels is finite even without the tax or any additional carbon-limiting measure, it will require a nonmarginal period of time until the amount of emissions saved up to time T could be offset by increases in the post-T era. Second, for some stylized scenarios of different conceivable alternative measures, cumulative emissions may (1) fully converge to the BAU emissions only after the time when all fuels would have been used in the BAU scenario without pre-T tax, or they may (2) not converge at all after T, or they may (3) converge only partially overall. Cases 2 and 3 imply eternal overall emission savings. Case 1 is the natural outcome in a framework with a global cap-and-trade scheme after time T with exogenous and constant allowances, or if we had a backstop from time T on, supplied infinitely elastically at a fixed price b below the choke price of a demand which we assume to be constant over time. In this case, it is easy to see that it would take a strictly positive time Δ after T until cumulative emissions in the case with the tax correspond to the emissions until T in the no-tax baseline scenario. As this would naturally mean that the emissions path after time T + Δ in the pre-T-tax case corresponds to the emissions path after time T with a constant shift of Δ periods, it is clear that cumulative emissions would converge only once all fuel extraction has stopped in the tax scenario (which would in this case be Δ periods later than when extraction would have stopped in the no-tax baseline). Given that marginal damages are lower for lower atmospheric carbon stocks and that the social value of extraction is higher if extraction costs are lower, one may expect the emission allowances in a global cap-and-trade scheme to be larger in the case with the pre-T tax. If the emission allowances are adapted dynamically according to the social value of emissions as a function of cumulative emissions and extractions, it is still the case that, in the simplest world, it would require a nonmarginal time Δ after T until cumulative emissions in the case with the tax reach the level of emissions up to T in the no-tax baseline, and that afterward the emission (and allowance) paths would be the same, shifted by Δ in time, still leading to strictly lower cumulative emissions for each period of time from T up to beyond the baseline duration of extractions. In a more dynamic world with technological progress that could reduce the social value of emissions (for

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a specific amount of cumulative emissions and extractions) over time, one could even expect the tax-induced delay of fuel consumption from the pre-T phase to result in total emission reductions for all times after T (case 3). Finally, if the future measure is a globally enforced carbon capture and storage mechanism with allowances that decrease over time, post-T emissions would not have to be larger in the case with the pre-T tax than in the baseline case, implying that no convergence of emissions after T takes place (case 2). Figure 8.6 gives a numerical example illustrating proposition 2 for the case when a cap-and-trade scheme with a constant global cap replaces the tax after 50 years.6 The right plot shows how the tax increases cumulative emission for more than 100 years in the BAU scenario, while the same reduces cumulative emissions already after Emission path

Tax-induced emission change, cumulative

30

0.4 No cap, no tax No cap, tax Cap, no tax Cap, tax

0.2 Normalized cumulative, tax-induced emission change

Emission rate (GtCO2/yr)

25

No cap Cap

20

15

10

5

0

−0.2

−0.4

−0.6

−0.8

0 0

T

200

−1 0

Year

Figure 8.6 Tax in BAU versus tax with future cap-and-trade scheme

T

200 Year

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20 years when the baseline contains already a cap-and-trade scheme starting after 50 years. The plot also illustrates how the emission reductions up to T are carried onward into the longer term future. 8.2.5 Extensions This section analyzes the stochastic emergence of a regime switch and discusses the case when an alternative technology is introduced at an endogenous time. Habermacher and Kirchgässner (2014) also show that the main results extend to limited regional taxes. Stochastic Introduction of the Future Scheme It cannot be predicted with certainty which climate change mitigation policy may prevent the release of the remaining carbon stored in fossil fuels into the atmosphere. It would be even more unrealistic to pretend to know when exactly such a breakthrough will occur. Additionally the change may come gradually, over several years, rather than at one specific point in time, and there may be substantial uncertainty about the time of the future regime change. Finally, it could also be the case that there will be no real regime change at all. To account for these uncertainties, a stochastic model has to be considered, complicating the analysis considerably. An analytical investigation of the stochastic case may be possible to a certain extent, especially with a backstop, at the emergence of which the resources left underground at time T will lose all their value. In such a case the stochastic end time can readily be accounted for by augmenting the discounting rate ρ by an appropriate term ψt and otherwise using the deterministic model, as has been shown by Dasgupta and Heal (1974).7 For simplicity, we here consider the case where the probability of the emergence of a backstop, conditional on no prior occurrence (further called periodic probability), is constant. The additional discounting factor, ψ, which equals this periodic probability, inherits this constancy, namely ψt = ψ. This result implies that the analytical structure of the model does not differ from the deterministic case at all. Note that the underlying (unconditional) probability density for the introduction of the backstop at date t is then f(t) = ψe−ψt. The additional discount factor from the possible emergence of the backstop alters the conclusion about the taxes’ impact on the emissions. While in the case where no backstop was considered, the green paradox would hold to a certain extent, implying that a tax rising more rapidly

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than with the real interest rate could lead to larger current emissions, this finding is not valid anymore in the case of the possible backstop: in this case taxes that exponentially rise at any rate lower than ρ + ψ necessarily imply reductions of current emissions and lower cumulative emissions at any future time period. We emphasize this claim with proposition 3, and the analytical proof is given in Habermacher and Kirchgässner (2014, annex G). Proposition 3 Any positive tax exhibiting a rate of increase, θ, that figures between 0 and the sum of the real interest rate, ρ, and the periodic probability of the emergence of a backstop technology, ψ, leads to a reduction in the expectancy of the cumulative emissions and, notably never yields increased potential cumulative emissions. The assumption that the backstop is a perfect substitute and, after the breakthrough, cheaper than the fuel, is relaxed in van der Ploeg (2013). Focusing on R&D subsidies rather than on a tax, they use a value function approach and show that the subsidy can increase initial emissions but reduce expected long-run emissions. Section 8.3 also contains numerical examples that confirm that a stochastically emerging backstop technology reduces the probability of a green paradox to materialize. Endogenous Future Regime Change For alternatives such as advancements in backstop or carbon capture and storage technologies, for which the time of the (potentially gradual) introduction will depend on the consumer price of the fuels, the tax path will have a direct influence on the time of the regime switch (which, for the case of a gradual introduction, may be considered as the time at which the new technology makes the standard combustion completely or almost completely redundant). We thus have T as a function of the tax path τt. We omit the proof, but it seems clear that in this case, the results above would even be strengthened: in the late periods before T (where λ approaches 0 even in the case without the tax), the positive tax increases the consumer price of the conventional fuel combustion, meaning the alternative technology becomes economic—and thus replaces the conventional fuel combustion—earlier, that is, the tax leads to an earlier Ttax < TBAU, leaving less time to sell fuels overall, thus reducing cumulative emissions already before the time of the introduction of the alternative technology in the BAU scenario TBAU. In this sense, the endogeneity of the time when the future development

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becomes effective makes the strong green paradox even less likely to occur than in our simplified analysis above. 8.2.6 Interpretation This section shows that if the baseline scenario contains future, alternative climate-relevant measures, which cannot only be cheap enough backstop technologies but also, for example, efficient global cap-andtrade systems, that replace the tax in the medium-term future, the current tax reduces not only long-run but already cumulative mediumterm emissions. Reducing the time during which a tax may lead to increased emissions, these future developments reduce the potential relevance of anticipation effects of a green paradox type with respect to future tax rate rises; increases of the NPV of all discounted future emissions become less likely. At the current rate of consumption, the well-assessed, worldwide oil reserves last for another 46 years. Given past growth rates of the worldwide fuel consumption, it is plausible that, absent any relevant political or technological developments, the large majority of the oil resources, which exceed the reserves by a factor of around three, would be burned well before the end of the twenty-first century. Gas and coal reservesto-production ratios exceed those for oil, but growth rates of their consumption have even exceeded those for oil in the last decades.8 It is therefore foreseeable that in a BAU future, a large fraction of the overall extractable hydrocarbon reserves will be transformed into atmospheric carbon dioxide before the end of the century, with potentially devastating warming effect of several degrees. If there exists some hope for the climate to be saved from this scenario, this hope must be based on stringent climate-protecting measures becoming effective well within the current century. No ready-made solution is at hand right now and some pessimism may be justified given the small fruits efforts of the last decades have produced. However, the hope to find some solutions exists—else the money and political and personal efforts spent all around the world to find solutions to the climate problem would hardly be acceptable. As the quote in the introduction shows, this valid hope is even explicitly acknowledged by Sinn in his seminal contribution on the green paradox. At least when a measure such as a future global Kyoto is introduced, the resource owners will not sell a strictly fixed quantity of fuels in the time prior to the global Kyoto; instead—and this is what our analysis emphasizes—any path of positive taxes prior to that medium-term

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measure will strictly reduce the amount of fuels sold in the medium term. For a stylized, analytically tractable illustration, we here assume the medium-term change to occur with certainty. We thus adopt a position fully opposite of the framework within which the green paradox has originally been brought forward and where the probability of future alternative developments has implicitly been assumed to be zero. Reality lies between these two extremes and our results can be considered as illustrating the way the green paradox results change if the positive probability of future measures is accounted for. 8.3

Numerical Analysis of the Green Paradox

This part uses a small, dynamic numeric and calibrated fuel market model to assess the scope for the strong green paradox to occur, given the worldwide fuel demand as well as long-run fuel extraction cost curves. 8.3.1 Model A dynamic exhaustible fuel market model, closely corresponding to the model developed in Habermacher (2011) to assess temporal fuel leakage, is employed. For simplicity, the here used model is restricted to a single resource, oil. This restriction can be considered as conservative in terms of refuting the green paradox, since the relevant resource scarcity rents seem to play a smaller role in coal owners’ extraction decisions than in oil owners’ decisions.9 The salient features of the model are as follows: Resource owners ration their fuel sales over time such as to maximize expected present discounted net fuel sales revenues. They sell their fuels on a market with a demand that is downward sloping in the fuel consumption price. The carbon tax corresponds to a wedge between the sales and the consumer price of the fuel. Fuel sales revenues are considered net of marginal extraction costs which are smoothly increasing in the cumulative amount of extractions. For simplicity, fuel owners are assumed to operate competitively (see the alternative setups in section 8.3.5 for results from a variant of the model with a monopolistic fuel owner). The model allows for a backstop whose price per unit of energy may decrease exponentially over time until a lower price bound is reached, from which on the price remains fixed. The supply of the backstop is

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assumed infinitely elastic at its price.10 The tax is assumed to grow exponentially over time before it may reach a (high) fixed upper tax limit. While the general model setup corresponds to that described in detail in Habermacher (2011), the fuel demand is here modeled as an isoelastic curve, calibrated to the current worldwide consumption and market price. This demand is assumed to grow over time. 8.3.2 Main Setup The main setup uses the following parameters, which in general are chosen in a conservative manner in terms of refuting the green paradox: Initial tax level τini = 6 $/tCO2.11 Annual rate of increase of the tax level gτ = 6 percent.12 Upper tax limit τup = 800 $/tCO2.13 Backstop: initial price pb,0 = 400$/bbl-eq.; rate of backstop price decrease kb = 0.5 percent; lower bound on backstop price (once reached, the price does not decrease further) pb,1ow = 200$/bbl-eq.14 Discount rate of oil owners ρo = 3 percent.15 Demand elasticity ε = −0.85.16 Demand growth rate gd = 1 percent.17 The main setup assumes perfect competition among fuel owners. The oil extraction cost curve is a third order polynomial fitted to the IEA (2005) oil supply cost curve, shown in figure 8.7. The fitted curve compares favorably also to alternative extraction cost projections, by IEA (2008) and Rogner (1997). 8.3.3 Note on Interpretation of Results We calculate the net present value (NPV) of emission changes induced by the climate tax as ΔENPV ≡ Σ t (Et ,tax − Et ,no tax ) e − ρet , where Et stands for the cumulative emissions up to year t. That is, we implicitly assume damages that are linear in the cumulative emissions (section 8.3.6 considers convex damages). We calculate the NPV of the emission changes for a range of different emission discount rates ρe. As the initially low and rapidly rising tax will tend to increase emissions in early periods,

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Cumulative extraction (Gbbl)

7,000 6,000 5,000 4,000

Rogner 1997 IEA 2008 IEA 2005 Fitted curve A(c)

3,000 2,000 1,000 0

0

50

100

150

200

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Marginal cost ($[2012]/bbl) Figure 8.7 Fitted and original oil extraction cost curves. The curves are inflation adjusted to $2012.

for large enough emission discount rates ρe this NPV will generally be a positive value, that is, for large ρe, overall climate damage is considered to increase due to the tax. The tax will, however, reduce the absolute amount of cumulative long-run emissions. The NPV of the emission change will thus be negative for low enough discount rates. This implies a positive climate effect of the tax for low enough emission discount rates and a negative climate effect for large discount rates. The crucial point is, whether the discount rate for which the climate effect turns from positive to negative is large enough for being deemed unrealistically high. We define ρe* as the threshold emission discount rate required for the initial emission increases to just offset the future emission savings. That is, higher (lower) discount rates ρe > ρe* ( ρe < ρe* ) imply that the tax introduction does, overall, exacerbate (alleviate) the climate problem in terms of how we evaluate it. An annual discount rate is an abstract measure. For a more intuitive understanding of the threshold discount rates, we report an implied “half-value time” of emissions, t1*/2 ≡ t1/2 ( ρe* ) : it expresses by how many years into the future emissions would have to be postponed for them to be valued only as much as half of their amount in terms of emissions occurring today according to the discount rate. That is,

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{

}

t1*/2 ≡ t|(1 − ρe* ) = 0.5 . t

8.3.4 Result Main Setup Figure 8.8 illustrates the impact of the carbon tax on the emission path (middle plots) in the standard scenario (left), containing a backstop whose price (dashed-dotted line in bottom plot) decreases over time, and for the case without backstop (right). The two upper plots show how the NPV of emissions is affected as a function of the applied emission discount rate ρe. The bottom plots show the evolution of the gross fuel price (squares), extraction cost (crosses), and the resource shadow value (circles), for the case with the tax (solid lines) and in the baseline without tax (dashed). The NPV emission change graphs in the top plots show that the impact of the tax is positive for low discount rates and worsens when higher discount rates are applied, corresponding to the fact that initial emissions grow but cumulative long-run emissions decrease. The discount rate for which the net effect of the tax on discounted emissions is just zero, ρe*, is 3.7 percent in both cases, corresponding to half-value times t1*/2 of only 18 to 19 years. Therewith, a rather unusually strong taste for emission delays (a very rapid discounting of the future) is required for the tax to be considered as counterproductive in terms of climate protection. The difference between the main setup and the case without backstop is straightforward to understand: with a backstop, extraction stops as soon as the extraction costs (in $/CO2-eq.) plus the carbon tax become as high as the backstop cost. The marginal value of resources converges to zero up to that time, while in the case without backstop it reaches zero only toward the end of the simulation period, independently of the tax. Thus only in the presence of the backstop does the tax shorten the fuel-sales horizon, in the numerical example from 132 down to 75 years, implying a dramatic reduction of cumulative emissions (left middle plot). In the absence of a backstop, and assuming an isoleastic fuel demand, the fuel demand increase is dampened by increasing extraction costs but consumption does not stop even with a tax. The tax does, however, reduce cumulative emissions for all times in the long run, even though, similarly to the case with backstop, the rapid initial rate of increase of the tax implies an increase in the emissions in the first few years or decades (these initial increases are so low that one can barely see them in the graphs for either of the two variants;

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compare the middle plots). In both baseline scenarios, the introduction of the tax lowers the resource shadow value λ at all times. 8.3.5 Alternative Setups This section examines the robustness of the main result with respect to departures from a number of key assumptions in the model. Table 8.2 lists the model changes and their impact on the threshold half-value time and threshold discount factor. Figure 8.9 plots the impact on the NPV of the emissions for a range of emission discount rates, for all scenarios. Most variants are self-explained by their names in table 8.2; except for the changes indicated by the variant names, the model (parametrization) corresponds to the main setup from the previous section, “no backstop” in variants 1 and 2 stands for the absence of the nonperfect18 backstop included in the main version of the model. “Cheap backstop” in variants 11 and 12 signifies the (possible) emergence of an additional, perfect backstop that is either assumed to emerge deterministically at a specific point in time (variant 11) or stochastically with a specific annual probability of introduction (variant 12). The reason for looking at variant 2 in addition to variant 1 is that in the absence of a (general, nonperfect) backstop, the upper tax limit is no longer irrelevant to the model results: if there exists no backstop at all, the demand for oil may remain nonzero even for taxes above the tax limit of the main version of the model. Modeling the behavior of a monopolistic fuel supplier in the presence of an endogenously emerging (nonperfect) backstop would have required larger adaptations in the model used here. Therefore the monopolistic case (variant 13), is only modeled in the absence of a backstop. It is also modeled with a somewhat amplified elasticity of demand ε, of −1.25, as for any ε ≥ −1, the monopolist as modeled here would have supplied only a marginal quantity of oil at an arbitrarily high price, which can be attributed to the simplification in the model of the absence of a fringe supply which in reality would lead a quasimonopolist to supply fuel at more than a marginal rate. The results (table 8.2) are interesting in several ways. First, the only two variations which increase (to a notable extent) the chances of a green paradox to occur correspond to scenarios which are probably less realistic than the main scenario: real interest rates observed over the past decades, as well as uncertainty about property rights in many oil rich regions make oil owner discount rates of only 2 percent or less

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Table 8.2 Threshold half-value time and discount rate for emissions in model variants

1 2 3 4 5 6 7 8 9 10 11 12 13

Variant

t1*/2 ( yrs )

ρe*

Main setup No backstop No upper tax limit, No backstop Higher tax (τini = 12 $/tCO2) Lower tax (τini = 3 $/tCO2) More rapidly increasing tax (gτ = 9%) Less rapidly increasing tax (gτ = 4%) Patient oil owners (ρo = 2%) Impatient oil owners (ρo = 6%) Less elastic demand (ε = −0.5) More elastic demand (ε = −1.2) Cheap backstop after 50 years Stochastic cheap backstop (2%) Monopolistic, no backstop

19 18 20 5%

Note: Dark gray {rows} are changes that substantially increase the likelihood of a green paradox to materialize; light gray {rows} are changes that substantially reduce the likelihood of a green paradox to materialize.

annually (variant 7) unlikely. Similarly, despite the substantial burdens regarding the introduction of a climate tax, an initial level of only 3 $/ tCO2 or less (variant 4) seems very low. Note that even in these two variants the discounting required to neutralize (revert) the beneficial effect of the tax would be very strong, with half-value times equal to (lower than) 22 and 29 years. Second, not only less rapidly (variant 6) but even more rapidly (variant 5) increasing taxes are less likely to lead to a green paradox than the tax raising at 6 percent! While it is not astonishing that the less rapidly increasing tax does not lead to a green paradox,19 the fact that the more rapidly increasing tax with gτ = 9 percent is less likely to lead to a green paradox than the tax in the main scenario, may initially surprise. This finding is, however, easily understood by recognizing that a tax increasing rapidly enough becomes soon prohibitive for fuel sales independently of possible anticipation effects, therefore decreasing present discounted emissions even for large initial fuel sales.20 Third, if a monopolist controls the fuel supply (variant 13), the likelihood of a green paradox to materialize seems strongly reduced as well. A possible explanation for this result is that the monopolist takes into account that increasing extractions in early periods reduces the price

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of the (large) amount he sells during these periods, making him more reluctant to increase initial extraction rates than perfect competitors who, as price takers, disregard this direct price effect. Finally, as suggested in the analytical analysis in section 8.2, the presence of independent alternative developments that (may) take place in the future also strongly increases the desirability of the current carbon tax (variants 11 and 12). Note that the results confirm that the nonperfect backstop, as well as the upper bound of the tax of 800 $/ tCO2, considered in the main model setup, seem to barely influence the model results (variants 1 and 2). In all variants, the implied threshold emission half-value times are substantially below 40 years. If one agrees that within a potential emission scenario it is probably undesirable to prevent some current emissions to just emit twice that amount before 40 years have passed, the

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carbon taxes would be desirable for the climate—there exists in this sense no (strong) green paradox for those taxes. 8.3.6 Convex Damages Climate damages are generally considered to rise more than proportionally with cumulative emissions. Taking this convexity into account, emission changes are weighted relatively more if they occur when a lot of previous emissions have occurred. Early emission increases are thus weighted relatively less and emission reductions later on relatively more. This increases the threshold discount rate up to which the strong green paradox does not occur. To illustrate this, we compute, for the main model setup, the NPV damages with and without the rapidly increasing tax used above, under the assumption of purely quadratic damages. Given that already around half a trillion tons of carbon (TtC) have been emitted today (Allen et al. 2009), the marginal damages D′(E) after the cumulative emission E from today on are thus proportional to D′(E) ∝ (0.5 TtC + E)2.21 With this damage form, the threshold discount rate increases to ρe* = 3.9% (t1*/2 = 17 years ), from the value of 3.7 percent (19 years) for the case of linear damage. 8.3.7 Interpretation Given the amount of fuel available and the level of worldwide demand, a (rapidly) rising worldwide carbon tax is likely to reduce the mediumterm and longer term fossil fuel consumption so drastically that corresponding future emission reductions strongly outweigh potential near-term fuel consumption increases. This is only reversed if future emissions are discounted at an unrealistically rapid pace (half-value times of 20 years or less). That is, while it is possible that the tax leads to short-run emission increases, the relevant, strong green paradox seems unlikely. 8.4

Conclusions

This chapter provides two distinct analyses, one in a purely analytical and one in a numerical framework, that qualify the green paradox; they show that the claim that carbon taxes with rapidly increasing rates would exacerbate the climate problem rather than alleviate it cannot be sustained as generally as has been suggested. The analytical part details two primary limitations with regard to the claims proposed with the green paradox, based on the fact that the

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perpetual business as usual is the wrong baseline scenario against which the tax must be compared: even if we were to abstain from introducing a carbon tax today, other future climate-related developments may influence the resource market in the future and thus the carbon emission path. Such possible developments encompass not only technological innovations driven by increasing fossil fuel extraction costs, but also political movements driven by ever-increasing emissions and climate damages. Potential measures include backstop technologies, demand cartels, carbon capture and storage systems or prohibitively high future carbon taxes. Given such possible future measures, a currently introduced carbon tax may be more favorable for our climate than has been predicted by the green paradox: First, if some of the previously mentioned future climate regime switches were to materialize at a specific, foreseeable time in the medium term, then the cumulative emissions may become more relevant than the detailed evolution of the emission path, and any path of positive taxes can be expected to reduce these cumulative emissions up to the time of the regime switch. Second, if a future regime switch (e.g., the introduction of a backstop technology) is stochastic, our model suggests that even the weak version of the green paradox does not hold: even current emissions can be reduced by carbon taxes whose levels increase more rapidly than at the real interest rate. The analytical analysis in section 8.2 has its shortcomings. It assumes the future regime switch to be exogenous to the decision on the tax, while in reality, many possible future developments will depend on gross and net prices of the fuels. Also, while the analysis provides some reason why the longer run emission savings may dominate potential initial emission increases, it does not provide a detailed NPV comparison of emission paths. In the calibrated numerical model in section 8.3 we partially address these issues. The simulations reveal that a carbon tax seems, for a large variety of parameter specifications, very likely to reduce the present value of emissions, that is, no strong green paradox occurs. Notes 1. If other distortions are controlled for separately, the optimal climate tax does obviously not worsen the climate outcome overall, as else it could not be optimal in comparison to the no-tax baseline. Van der Ploeg (2013) shows that the optimal tax does not rise at a level rapidly enough for the counterproductive green paradox effects to occur.

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2. Van der Ploeg and Withagen (2011) discuss the effect of dirty and clean backstops on optimal carbon taxation; however, they focus on backstops that are already available today and, more importantly, consider only the choice of optimized or prohibitive tax paths, leaving aside the arbitrarily increasing taxes of the green paradox. 3. This assumption seems largely unproblematic; an extended note on it is provided in Habermacher and Kirchgässner (2014, annex A). 4. See section 8.2.5 for a discussion of the assumption that the switching time is exogenous, and why its relaxation would generally strengthen our results. In the numerical analysis in section 8.3, the backstop introduction time is endogenous. 5. Recall from lemma 2 that a strictly positive amount of the theoretically exploitable fuels is left underground until time T. 6. The example is based on an adapted version of the model used in section 8.3, for a constant oil demand and a tax starting at 1 $/tCO2 and rising at an annual rate of 6 percent, and a cap of 15 Gbbl of oil. 7. See also Strand (2007), where the stochastic introduction of an alternative technology is shown to augment the overall discount rate considered by the resource owners. 8. For details about past, current and projected fuels consumption see the World Energy Outlook 2010 (IEA 2010, p. 84, fig. 2.4). The outlook reports reserve-to-production ratios for oil, gas and coal of 46, 58, and 150 years. 9. For example, see van der Ploeg and Withagen (2011) and Burniaux and OliveiraMartins (2012) for works stressing the relative abundance of coal in comparison to oil. Habermacher (2011) has found much smaller relative scarcity rents for coal than for oil and, correspondingly, negligible medium-term exhaustibility effects for coal compared to those for oil. This suggests that green paradox anticipation effects play a more important role for the oil market than for the coal market. The simulations in the extensions of section 3.6 confirm this intuition. 10. That is, backstop supply is zero for a fuel market price lower than the exogenous backstop price, and for a price equal to the backstop price any demand not matched by fuel supply will be covered by backstop supply. Technically we thus have b ≥ 0 ⊥ pb ≥ p, where ⊥ stands for complementary slackness between the two inequalities, b is backstop supply, pb is the exogenous backstop price, and p the market energy price. 11. This is chosen as a conservative value: even in the current circumstances with important political resistance against stringent climate measures, once a genuine carbon tax would be agreed upon, it would likely start at a level larger than 6 $/tCO2. In this case fuel owners would have even less time to sell their resources before the tax reaches high levels, making the relevant green paradox even less likely. 12. We consider a rate of increase of 6 percent not only a rather realistic value but notably one, one might choose if she were to try to “elicit” a materialization of the paradox: a value much lower than 6 percent seems less likely to lead to the paradoxical counterproductive results because it could be only marginally above the fuel owners’ discount rate. A value much higher than 6 percent would likely cause the tax level to rise so rapidly that the fuel owners would hardly have time to sell their fuel before the tax reaches prohibitive levels, making a relevant, strong green paradox unlikely as well. 13. This upper limit is imposed for illustrative purposes. It is irrelevant for the results; with a CO2 intensity of oil at 0.43 tCO2/bbl, even given its initial price of 400 $/bbl, the

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backstop (see below) would replace oil already for taxes lower than this upper tax limit, given the nonnegligible extraction costs. 14. The lower price bound avoids incredibly low backstop prices in the longer run. It corresponds to the view that technical progress is likely to reduce the costs of known alternative forms of energy to some extent in future, but that the costs will not eventually approach zero. 15. Given historic real interest rates of 5 percent and higher (e.g., see Nordhaus 2008 for an overview and discussion), plus conceivable substantial time-discounting due to political stability and property right issues in many oil-rich regions (e.g., Sinn 2008), this value is a conservative (low) estimate of the earning impatience the major oil owners may exhibit. As the green paradox can only occur if the tax rate of increase exceeds the oil owner discount rate by a large enough margin, higher discount rates would tend to reduce the likelihood of a green paradox. 16. This value follows Golombek et al. (1995) who use elasticities of −0.9 for the OECD and −0.75 for the rest of the world. 17. This is a compromise between current low growth rates in the OECD and the higher growth rates in the emerging world which are thought to be converging to lower growth rates as their economies mature. 18. By nonperfect, we mean that while it is also clean and available at any desired quantity, its fixed price is not necessarily low; see the model description in section 8.3.2. 19. It is clear (and indirectly proved in Habermacher and Kirchgässner 2014, annex G) that in the basic framework in which the green paradox has been brought forward, a constant or a very slowly increasing tax will reduce cumulative emissions even at every point in time, and it is has been pointed out by Sinn (2008) that rapidly increasing taxes can lead to a green paradox. 20. One may think of this as follows: consider a nearly infinitely high rate of increase of the tax level. Then the tax level alone will prohibit almost any fuel sales from the second year after the initial imposition of the tax. Since, in the first year, the fuel owners would try to sell maximally the amount of fuels that would reduce the demand price to the initial extraction costs, such a rapidly increasing tax would leave almost all fuels in the ground forever, despite the anticipation effects. 21. Roughly half of the emitted carbon is rapidly absorbed and the other half stays in the atmosphere for hundreds of years. As this applies equally to the 0.5 TtC of historic emissions as to future emissions E, the proportionality is not affected by this factor of one-half.

References Allen, M. R., D. J. Frame, C. Huntingford, C. D. Jones, J. A. Lowe, M. Meinshausen, and N. Meinshausen. 2009. Warming caused by cumulative carbon emissions towards the trillionth tonne. Nature 458 (7242): 1163–66. Burniaux, J. M., and J. Oliveira-Martins. 2012. Carbon leakages: A general equilibrium view. Economic Theory 49: 473–95. Dasgupta, P., and G. Heal. 1974. The optimal depletion of exhaustible resources. Review of Economic Studies 41: 3–28.

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Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57: 79–102. Golombek, R., C. Hagem, and M. Hoel. 1995. Efficient incomplete international climate agreements. Resource and Energy Economics 17 (1): 25–46. Habermacher, F. 2011. Optimal fuel-specific carbon pricing and time dimension of leakage. Economics working paper 1144. HSG St. Gallen. Habermacher, F., and G. Kirchgässner. 2014. Climate effects of carbon taxes, taking into account possible other future climate measures. Economics working paper 2011–10. HSG St. Gallen. Herfindahl, O. C. 1967. Depletion and economic theory. In M. H. Gaffney, ed., Extractive Resources and Taxation. Madison: University of Wisconsin Press, 63–69. Hoel, M. 2010. Is there a green paradox? Working paper 3168. CESifo, Munich. IEA. (International Energy Agency). 2005. Resources to Reserves: Oil and Gas Technologies for the Energy Markets of the Future. Paris: OECD/IEA. IEA. (International Energy Agency). 2008. World Energy Outlook 2008. Paris: OECD/IEA. IEA. (International Energy Agency). 2010. World Energy Outlook 2010. Paris: OECD/IEA. Nordhaus, W. 2008. A Question of Balance: Weighing the Options on Global Warming Policies. New Haven: Yale University Press. Pearce, D. 1991. The role of carbon taxes in adjusting to global warming. Economic Journal 101: 938–48. Polborn, S. 2011. The green paradox and increasing world energy demand. Mimeo. University of Aarhus. Rogner, H.-H. 1997. An assessment of world hydrocarbon resources. Annual Review of Energy and the Environment 22: 217–62. Sinn, H.-W. 2008. Public policies against global warming: A supply side approach. International Tax and Public Finance 15: 360–94. Solomon, S., G. K. Plattner, R. Knutti, and P. Friedlingstein. 2009. Irreversible climate change due to carbon dioxide emissions. Proceedings of the National Academy of Sciences USA 106: 1704–1709. Strand, J. 2007. Technology Treaties and Fossil-Fuels Extraction. Energy Journal (Cambridge, MA) 28: 129–41. van der Ploeg, R., and C. Withagen. 2011. Optimal carbon tax with a dirty backstop: Oil, coal, or renewables? Working paper 3334. CESifo, Munich. van der Ploeg, R., and C. Withagen. 2012. Is there really a green paradox? Journal of Environmental Economics and Management 64 (3): 342–63. van der Ploeg, R. 2013. Cumulative carbon emissions and the green paradox. Annual Review of Resource Economics 5: 281–300. van der Werf, E., and C. Di Maria. 2012. Imperfect environmental policy and polluting emissions: The green paradox and beyond. International Review of Environmental and Resource Economics 6 (2): 153–94.

9

The Impacts of Announcing and Delaying Green Policies Darko Jus and Volker Meier

9.1

Introduction

The green paradox can be interpreted as a story about the announcement effects of “green” policies, and in particular, taxes on the extraction or the use of fossil resources. In a static setting, introducing a specific consumption tax or employing some similar measure always reduces equilibrium output. This conventional wisdom turns out to be misleading if applied to a dynamic framework. When it comes to extracting a nonrenewable resource, timing issues are of crucial importance. The distinctive element of a problem involving a nonrenewable resource is that the action at one point in time affects the set of possible actions in the future. More specifically, once a unit of the resource is extracted, it is not available for extraction in the future anymore. And announcing higher taxes somewhere in the future makes it comparatively more attractive to extract and sell output today. Consequently announcing higher taxes on fossil fuel consumption may induce more extraction and consumption today, resulting in higher cumulative extraction until any given point in time. As extraction and consumption of fossil resources is strongly positively correlated with carbon dioxide emissions, and since the greenhouse effect can safely be understood as an increasing function of cumulative emissions, announcing environmental taxes may even accelerate the process of global warming. The scientific debate about the relevance of the green paradox has centered around identifying conditions under which green policies are indeed detrimental for the climate. The two main research questions are whether climate policy can (1) reduce “final” cumulative emissions by inducing resource owners to keep part of the resource in ground forever and (2) change the timing of resource extraction by providing incentives for slowing down the relevant path.

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As regards the final cumulative emissions, the benchmark consideration is whether following a path realizing Cournot quantities at each point in time would lead to a complete exhaustion of the initial stock of the resource. Such an extraction path would mirror static optimization where instantaneous marginal profit would always be equal to zero. By contrast, dynamic optimization generally yields positive marginal profit levels as slowing down extraction is associated with a higher present value of profits. It is necessary and sufficient for a full exhaustion of the stock that following a Cournot path would also not leave any stock in the ground. Final cumulative extraction under a Cournot path may fall short of the initial stock if either tax rates become prohibitive before the stock is exhausted, or if a so-called backstop technology would make any further extraction unprofitable. For the application of oil, both scenarios do not seem plausible. It is hardly conceivable that governments are willing to set prohibitive taxes, or that a new technology turns out to be superior to oil for any profitable use. Should a superior technology arise, it is much more likely that the worldwide transition for replacing fossil fuels takes decades, proceeding much slower in developing countries, and that oil remains superior in several applications in the chemical industry. Therefore our main focus lies on timing issues, assuming that the Cournot strategy exhausts the initial stock of the resource in full. The discussion has shown that the time path of extraction crucially depends on the growth rate of a specific tax. In general, resource extraction will speed up when the unit tax increases at a rate exceeding the interest rate (or the relevant discount rate of the resource owner), and slow down if the specific tax grows at a rate that falls short of the interest rate, stays constant or declines over time (Dasgupta and Heal 1979, ch. 12; Sinn 2008; Edenhofer and Kalkuhl 2011). Following these insights, we investigate the consequences of announcing a unit tax to start at some point in future that grows at the interest rate. Such a policy is shown to increase cumulative extraction until any given point in time compared to a baseline zero tax policy. We proceed by considering an unexpected delay of the introduction of the tax at the date at which it was originally announced to start. While the message of the green paradox literature usually states that lowering taxes implies less cumulative emissions, this is not true for postponing the introduction of the tax. Re-optimizing resource owners facing a period with unexpectedly low taxes will speed up extraction during the period with the laxer policy and correspondingly reduce their

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extraction later on. As a consequence cumulative extraction is higher at any given point in time. Moreover cumulative extraction under the announcement-cum-delay policy will also be higher than if the ultimately realized policy is already announced at the outset. This is true because creating misleading expectations of an intermediate period with higher environmental taxes speeds up extraction in early periods. When the delay is announced, the remaining policy path would be identical to the one under a less ambitious policy. However, the stock of the resource is already lower at that moment. The remainder of our contribution is organized as follows. After discussing the relevance of climate policy implementation delays in section 9.2 and reviewing the related literature in section 9.3, we introduce our modeling approach in section 9.4. The main results are presented and interpreted in section 9.5. Section 9.6 discusses possible generalizations and policy implications of our analysis. 9.2

Climate Policy Delays

Announcements of climate policy actions and subsequent delays can frequently be observed on different policy layers. The most prominent example is the process toward a successor of the Kyoto Protocol. The Kyoto Protocol of 1997 with its provisions for lower emissions during the “first commitment period” in the years 2008 to 2012 has created the expectation of high green taxes during this period and afterward. The naming “first commitment period” suggests that already in 1997 the idea of having further commitment periods with additional emission reductions existed. This is also documented in the Kyoto Protocol itself, which mentions “commitments for subsequent periods” several times. It also defines when the negotiations of the successor were supposed to begin: “The Conference of the Parties … shall initiate the consideration of such commitments at least seven years before the end of the first commitment period” (United Nations 1998, art. 3). This announcement has been made repeatedly, for example, also at the Conference of the Parties in Bonn in 1999, where the Executive Secretary spoke about “the continuation of the Protocol into the second and future commitment periods without a break” (United Nations 1999, art. IX). Negotiations toward a successor protocol have taken place in several rounds, starting with the 2007 United Nations Climate Change Conference (UNCCC) in Bali. At the 2011 UNCCC in Durban it became clear

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that no legally binding successor of the Kyoto Protocol will be agreed on soon. Instead, an Ad hoc Working Group was assigned to develop a proposal for a new protocol until the 2015 UNCCC, which is then supposed to be implemented from 2020. Thus, after the first commitment period, which ends in 2012, there will be a period of at least eight years without an internationally binding contract. At the same time it transpires that many countries are about to fail the original Kyoto targets. Taken these observations together, the suspicion is raised that negotiations will culminate into a new agreement, specifying the old goals for a new commitment period. This likely projection can be interpreted as delaying the originally announced measures to some later date. Moreover the histories of several national climate action plans in the United States, Australia, South Korea, and Japan exhibit similar structural features. After some initial announcement, the expectation of an early implementation of a tax and other measures has been created. And again, due to delays in the process of legislation, effective policy action has been postponed to some later dates (see Jus and Meier 2012). 9.3

Related Literature

Our contribution is closely related to the literature on the green paradox. The latter states that carbon dioxide emissions may increase as a reaction to green policies, if those become sufficiently stricter over time (see Sinn 2008, 2012). The argument is based on the theory of exhaustible resources as introduced by Hotelling (1931) and further developed by Dasgupta and Heal (1979), among others. In a very basic setup with only one exhaustible resource and no backstop technology, a specific unit tax that grows at the interest rate is neutral for the extraction path. A tax that rises more quickly leads to the green paradox as it makes early extraction relatively more profitable (Dasgupta and Heal 1979, ch. 12; Sinn 2008; Edenhofer and Kalkuhl 2011). Considering the existence of a backstop technology in such a model and studying the effect of a reduction of its costs in the future would also provoke the green paradox provided that eventually the entire carbon stock will be extracted (Hoel 2008; Gerlagh 2011). This becomes even more likely if the backstop technology constitutes an only imperfect substitute for the use of fossil resources (Hoel 2008). These results, obtained for zero or constant extraction costs, do not necessarily fully

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carry over to the situation in which extraction costs rise as the stock of the resource declines. In such a situation it depends on whether the entire stock will be consumed. It may occur that a green paradox still arises in the sense that more is being extracted earlier in time while at the same time total extraction declines, that is, more of the resource remains nonextracted forever (Hoel 2008; Gerlagh 2011; van der Ploeg and Withagen 2012). In contrast to these results that largely confirm the existence of the green paradox, a tax that is sufficiently high always leads to an immediate reduction of carbon emissions, independent of the expectations about future carbon taxes. This tax must be larger than the initial resource rent, thereby making the extraction of the initial stock unprofitable as such (Hoel 2012). In this case, however, the maximization problem of the resource owner mirrors Cournot behavior, and it is questionable whether such a tax would be politically feasible. Furthermore alternative policy instruments can yield more differentiated outcomes. Di Maria et al. (2012) analyze the announcement of a carbon cap that will be binding for some period in the future. While such a policy increases extraction before the implementation of the cap and thus has a green paradox component, it also increases extraction when the cap no longer binds, indicating a reduction of cumulative extraction and emissions in the long run. In addition Di Maria et al. (2012) point to a different channel generating a green paradox outcome. If the existing fossil resources differ in their carbon content, an announcement of green policies leads to an increased early use of the resource with high carbon content in order to preserve the low-carbon resource that becomes more valuable after the implementation. Moreover announcing a carbon tax may lead to more fossil fuel consumption until the date of implementation due to dynamic optimization on the demand side. For example, households may fear a drop in consumption at the implementation date and therefore accumulate capital stock more quickly in order to smooth their consumption path. A higher capital stock is, however, associated with more fossil fuel consumption already at the earlier stage (Smulders et al. 2012). The same argument holds when households anticipate the introduction of a carbon tax at some uncertain date in the future. In this case a discontinuous jump in the consumption path may occur whenever the expectations of households are updated. Thus a credible announcement will generally have some impact on the course of emissions even when it already has been anticipated with some probability (Smulders et al.

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2010). Unlike our supply-side analysis, this demand-side channel discussed by Smulders et al. does not require exhaustibility of the natural resource. Thus incorporating adaptive reactions of firms and consumers on the demand side would strengthen the tendency of higher cumulative extraction due to announcements and delays of green tax policies. The survey article by van der Werf and Di Maria (2012) discusses further arguments in favor and against the emergence of the green paradox scenarios. Finally, one could argue that overly hasty emission reduction policies increase the compliance costs by being more distortive than if more time was available for the adjustment. This would indeed be a reason for announcing climate policy as early as possible. For the case of Canada, Kennedy (2002) finds that the costs of compliance could be up to 17 percent higher with early action compared to the derived optimum. 9.4

The Model

Restricting our attention to a particularly simple and transparent framework, we analyze the behavior of a resource-extracting monopolist aiming to maximize the present value of his profits. The strategy of the monopolist consists of choosing a time path q(t) for extracting a nonrenewable resource, where t denotes time, which is considered as a continuous variable. The stock of the resource at date t is denoted by x(t) with an initial value of x0 > 0. To keep matters simple, the instantaneous profit function without taxation is time-invariant and denoted by π0(q). The profit function is assumed to be well-behaved, that is, strictly concave, such that marginal profit decreases in the extraction rate q. Moreover, to ensure that there is always an incentive to shift some extraction to the future, the marginal profit increases above all finite bounds when extraction goes to zero. In the appendix we present an example that induces a profit function with the desired properties. At date t = T1, a unit tax path θ(t; T1) is introduced, being meant to reduce extraction and thus also the carbon dioxide emissions. It should be noted that the analysis is similar if the tax is replaced by a cost to satisfy a new environmental standard or a price of a permit to extract or sell the resource. Since the tax shifts the marginal cost curve upward, marginal profit changes to π1′(q, t; T1) = π0′(q) − θ(t; T1). If we impose a structure in which the time path of the unit tax itself does not give any incentives for decelerating or accelerating extraction, the tax grows at

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the interest rate r > 0, which is assumed to be time invariant: θ (t ; T1 ) = θ 0 e r (t −T1 ) . In order to ensure that only the time path of extraction matters, whereas ultimately the full stock of the resource will be exhausted, we assume that pursuing the strategy to always extract the static Cournot quantity will exhaust the initial stock after some finite period. Thus the whole stock will be depleted at some instant if the resource owner behaves like a static monopolist who always sets output to a level at which instantaneous marginal profit is zero. For simplicity, repercussions of lower profits and redistributed tax proceeds on demand are ignored. In the basic model the policy switch is announced at date t = 0. The law of motion governing the evolution of the stock of the resource is given by x = − q. The monopolist is assumed to maximize the present value of his profits given the announced evolution of the tax path. Hence the dynamic optimization problem reads as follows: ∞ T ⎪⎧ 1 ⎪⎫ max ⎨ ∫π 0 ( q (t )) e − rt dt + ∫π 1 (q (t ))e − rt dt ⎬ q( t ) T1 ⎩⎪ 0 ⎭⎪

with

π 1 ( q (t )) = π 0 ( q (t )) − θ 0 e r (t −T1 ) q(t) subject to the equation of motion x = − q and the initial condition on the stock of the resource x (0 ) = x0 > 0. The dynamic optimization model is solved by standard methods of optimal control theory, where the technical details are described in Jus and Meier (2012). As our assumptions rule out the possibility that the monopolist pursues the path to always extract the static Cournot quantity, marginal profit will be positive at each point in time. Moreover the marginal profit path is continuous and grows at the interest rate r before the announced tax is introduced and afterward. This arbitrage condition makes the resource owner indifferent between earning the interest income by extracting the marginal unit immediately and waiting for the price increase of the resource in ground. The same reasoning holds at the introduction point of the tax: the marginal profit grows continuously at rate r, since otherwise the present value of profits can be increased by shifting extraction across time. As the marginal profit is assumed to fall in the output level, extraction decreases over time during the zero tax period. When the announced tax is actually introduced at T1, a discontinuous fall in extraction occurs. This happens

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because the marginal profit curve shifts downward at this instant while the marginal profit on the optimal path still increases continuously. When the tax is steadily increasing afterward, extraction declines over time because two forces work in the same direction. First, due to the concavity of the profit function at any given tax level, a steadily increasing marginal profit requires a continuously decreasing extraction level. Second, rising taxes shift the marginal profit curve down, such that even keeping marginal profit constant would be associated with declining extraction levels. At the same time, as there is always an incentive to leave some stock in ground, cumulative extraction converges to the initial stock only as time goes to infinity. The optimal extraction path is depicted by the dashed line in figure 9.1. 9.5

The Consequences of Delaying Green Policies

First of all, we can show that cumulative extraction at any given point t

in time t > 0, defined as Q (t ) = ∫ q ( s) ds, will exceed the levels of the 0

zero-tax baseline scenario if the regime switch is announced at t = 0 for date T1 and implemented accordingly. While the previously discussed announcement effect works, there are no further surprising tax changes at this stage of our analysis. Compared to the zero-tax baseline case, q

T1 Figure 9.1 Extraction paths with and without tax

t

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the initial stock is devalued owing to the tax that is implemented from the date T1 onward. At unchanged behavior, marginal profits in that later period will be lower. Consequently an adjusted optimal extraction path is adopted. This new path still displays the property of marginal profits rising at the interest rate. However, owing to the downward shifts of the marginal profit curve from T1 onward, the new marginal profit level at any given point in time will be lower than before. Since the instantaneous profit curve at zero tax is concave, this obviously implies higher extraction levels at any given point in time up to date T1. This is illustrated in figure 9.1 by the solid line, which is the zero-tax extraction path, and the dashed line, which corresponds to the case of an announcement of a tax from T1 onward. It follows that the stock of the resource in the ground at date T1 is smaller in the situation when a tax has been announced. For the remaining time after T1, on average the instantaneous extraction with the tax must fall short of the levels in the zero-tax baseline case. It can be shown that this property also holds at any given point in time. As ultimately the remaining stock converges to zero, this in turn implies that cumulative extraction until any given point in time is higher in the scenario with the tax. This confirms the message of the green paradox: announcing green policies accelerates extraction. Thus we can state this as a proposition: Proposition 1 Announcing at t = 0 a green policy with a unit tax θ(t; T1) to start at T1 > 0 increases cumulative extraction Q(t) at any given point in time t∈(0, ∞). We proceed by once again considering the tax policy announced for T1, where now its implementation is delayed until T1 once the originally scheduled switching date T1 is reached. Hence the postponement occurs surprisingly. Accordingly the optimal extraction path until the surprising delay is unaffected, which is also true for the remaining stock of the resource at T1. The term “delay” here means that the originally announced tax rates are applied from T2 onward, whereas they are set to zero between T1 and T2. Denoting the solution to the original optimal extraction path problem by q*(t), the new optimization problem after the announcement of the delay at date T1 is given as follows: ∞ ⎧⎪T2 ⎫⎪ max ⎨ ∫ π 0 ( q (t )) e − rt dt + ∫ π 1 (q (t ))e − rt dt ⎬ q( t ) ⎪⎩ T1 ⎪⎭ T2

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with π 1 ( q (t )) and the equation of motion defined as above and the new T1 initial condition x (T1 ) = x0 − ∫ q* (t ) dt > 0. 0 The new tax path with the delay is more favorable for the monopolist than the initially announced path, thus increasing the value of the stock. For the period until T2, the marginal profit curve lies at the zerotax level, which is higher than initially expected. Consequently, on the adjusted optimal extraction path, all resulting marginal profit levels after T1 will be higher than before. It turns out that a policy of delaying the implementation date of a previously announced tax path additionally increases cumulative extraction until any given point in time after announcing the delay. Our insight is summarized in the following proposition: Proposition 2 Announcing at t = 0 the implementation of the green policy at T1, but delaying it until T2 > T1 at date T1, further increases cumulative extraction Q(t) at any given point in time t> T1. The intuition behind this result is as follows: When the delay is announced, the monopolist reoptimizes the remaining extraction path. The value of any unit of the stock increases as within the period between T1 and T2 the instantaneous marginal profit curves shift upward. At the same time the optimal path needs to display the property that the marginal profit increases at the interest rate. Due to the higher value of the stock at T1, the marginal profit levels on the optimal path will be higher everywhere. This in turn implies, as depicted by the dotted curve in figure 9.2, that all instantaneous extraction levels will be lower during the period in which the tax policy is as originally announced, namely after T2. Hence the remaining stock of the resource is smaller with the announcement-cum-delay policy at T2. Since the new tax path makes it more attractive to extract during the new zero-tax phase between T1 and T2, both instantaneous—as illustrated for the dotted curve in figure 9.2—and cumulative extraction will be higher in that intermediate interval. The optimal instantaneous extraction path in the announcement-cum-delay scenario jumps twice: first upward at T1 because the surprise delay induces a shift of the extraction forward, then downward at T2 when the tax is finally implemented. Proposition 3 states that the announcement-cum-delay policy will also lead to higher cumulative emissions until any given point in time than if the government pursues the less ambitious policy with announcing the later implementation date at the outset.

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q

T1

T2

t

Figure 9.2 Impacts of delaying the green policy on the extraction path

Proposition 3 Announcing at t = 0 the implementation of the green policy at T1, but delaying it until T2 > T1 at date T1 increases cumulative extraction until any given point in time t∈(0,∞) compared to announcing the same policy at t = 0 to start at T2 and implementing it accordingly. When comparing the two alternative policy announcements at date t = 0, the announcement-cum-delay tax path (wrongly) appears less favorable since a tax is announced in the period between T1 and T2. Consequently the value of a marginal unit of the stock appears to be lower at the outset. This is translated into lower marginal profit levels and higher instantaneous extraction on the optimal path during the zero-tax phase up to date T1. At date T1 the true remaining tax path is revealed, which turns out to be identical under both policies. The only difference lies in the fact that the stock of the resource at T1 is smaller when the tax policy has been announced as stricter at the outset. Since regardless of the tax level all instantaneous profit curves are concave, a lower stock implies a higher value of marginal profit at each instant from T1 on. This in turn means a lower instantaneous extraction at any given date. Putting these pieces together shows that cumulative extraction will be smaller at each point in time if the tax path that is eventually realized is already announced at the outset.

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Conclusions

Our analysis deals with the effects of announcements of green policies that are delayed later on. We assume that realistic policy measures can only influence the timing of extraction, while being unable to achieve that part of the fossil resource remains in ground forever. The literature then argues that decelerating the speed of extraction can be induced by a carbon tax that becomes laxer over time, at least in real terms, that is, with declining discounted marginal tax levels. Given the general need to announce tax policy paths well in advance, any path with higher tax rates in the future will imply more extraction until the announced measures are actually implemented. In terms of climate protection, this can be perceived as a cost that cannot be avoided. However, we emphasize that it is more costly to first engage in an ambitious announcement in terms of a strict policy path, knowing that this will be readjusted and relaxed because being too demanding for many participants in the greenhouse gas reduction game. Such a strategy, which describes the Kyoto process well, is likely to increase global emissions twice: first in the early years of the laxer policy, then in the newly arising transition period after the delay in which higher profits can be achieved by shifting extraction forward once more. The mechanisms driving our results are robust to a number of modifications bringing the model closer to realistic scenarios. In particular, our simplifying view of extraction cost levels being independent of the remaining stock may be replaced by extraction costs which increase in cumulative extraction. Such a modification is certainly plausible as extraction tends to start where extraction cost levels are lowest, moving on to more and more difficult and costly environments. In such a case marginal profit no longer grows at the interest rate along the optimal extraction path because the deterioration of future cost structures by current extraction has to be taken into account. However, as unit taxes growing at the interest rate remain neutral with respect to the optimal extraction path in general, using a stock-dependent extraction cost function does not change the analysis substantially. If economic exhaustion is reached in finite time due to a perfect backstop technology the results can look quite different. We can distinguish two cases. First, assuming constant extraction costs and an exogenous endpoint of extraction, the optimal path typically coincides with the Cournot path. In that event, there is no announcement effect of a tax. It simply reduces the Cournot extraction rates, thus reducing ‘final’

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cumulative emissions. And delaying the introduction of the tax again raises emissions during the unexpected zero tax phase. Second, allowing for an endogenous endpoint of extraction by combining the availability of an initially more expensive backstop with extraction costs rising in cumulative extraction will generally lead to more cumulative extraction before the announced tax is implemented, but less final cumulative emissions. The delay of the tax again increases cumulative extraction at any point in time after announcing it. Finally, the results presumably carry over to most alternative market structures of the resource extraction industry. While alternative frameworks with imperfect competition make the analysis more complicated due to various sources of strategic interaction, the main logic of our benchmark framework survives. Optimal extraction paths for individual firms still need to exhibit the structure of increasing marginal profits over time, largely irrespective of the impact of that firm on current and future market prices. Hence, although the presented framework may be regarded as simplistic, its policy implications are quite robust. Appendix Profit Function Examples of the profit function used in the analysis can be derived as follows: Consider a linear cost function C ( q) = cq, with c > 0, and an inverse demand function p(q). The instantaneous profit function is π ( q) = p ( q) q − cq. First and second derivatives are then given by π ′ ( q) = p ′ ( q) q + p ( q) − c and π ′′ ( q) = p ′′ ( q) q + 2 p ′(q). The profit function is strictly concave if p ′ ( q) q / p ′(q) < −2 holds everywhere. Using a constant elasticity of demand specification p ( q) = Bq−(1/ε ) with B > 0 yields π ′ ( q) = (1 − 1 / ε )Bq−(1/ε ) ) − c and π ′′ ( q) = [(1 − ε ) / ε 2 ] Bq−(1+1/ε ) . The profit function then exhibits the desired properties for any ε > 1. References Dasgupta, P. S., and G. M. Heal. 1979. Economic Theory and Exhaustible Resources. Cambridge, UK: Cambridge University Press. Di Maria, C., S. Smulders, and E. van der Werf. 2012. Absolute abundance and relative scarcity: Environmental policy with implementation lags. Ecological Economics 74: 104–19.

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Edenhofer, O., and M. Kalkuhl. 2011. When do increasing carbon taxes accelerate global warming? A note on the green paradox. Energy Policy 39: 2208–12. Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57: 79–102. Hoel, M. 2008. Bush meets Hotelling: Effects of improved renewable energy technology on greenhouse gas emissions. Working paper 2492. CESifo, Munich. Hoel, M. 2012. Carbon taxes and the green paradox. In R. W. Hahn and U. Ulph, eds., Climate Change and Common Sense: Essays in Honour of Tom Schelling. Oxford: Oxford University Press, 203–24. Hotelling, H. 1931. The economics of exhaustible resources. Journal of Political Economy 39: 137–75. Jus, D., and V. Meier. 2012. Announcing is bad, delaying is worse: Another pitfall in well-intended climate policy. Working paper 3844. CESifo, Munich. Kennedy, P. 2002. Optimal early action on greenhouse gas emissions. Canadian Journal of Economics. Revue Canadienne d’Economique 35: 16–35. Sinn, H.-W. 2008. Public policies against global warming: A supply side approach. International Tax and Public Finance 15: 360–94. Sinn, H.-W. 2012. The Green Paradox. Cambridge: MIT Press. Smulders, S., Y. Tsur, and A. Zemel. 2010. Uncertain climate policy and the green paradox. Discussion paper 1.10. Hebrew University of Jerusalem. Smulders, S., Y. Tsur, and A. Zemel. 2012. Announcing climate policy: Can a green Pparadox arise without scarcity? Journal of Environmental Economics and Management 64: 364–76. United Nations. 1998. Kyoto Protocol to the United Nations Framework Convention on Climate Change. Paris: OECD. United Nations. 1999. Report of the Conference of the Parties on Its Fifth Session. Paris: OECD. van der Ploeg, F., and C. Withagen. 2012. Is there really a green paradox? Journal of Environmental Economics and Management 64: 342–63. van der Werf, E., and C. di Maria. 2012. Imperfect environmental policy and polluting emissions: The green paradox and beyond. International Review of Environmental and Resource Economics 6: 153–94.

10

Going Full Circle: Demand-Side Constraints to the Green Paradox Corrado Di Maria, Ian Lange, and Edwin van der Werf

10.1

Introduction

Hans-Werner Sinn coined the term “green paradox” to indicate the possibility that climate policies, such as carbon taxation and subsidies to renewable sources of energy, might induce resource owners to increase fossil fuel supplies in the short run, and hence increase current greenhouse gas emissions and climate damages (Sinn 2008). Sinn’s argument is that current emission reduction policies share a common focus on the demand side, while ignoring behavioral responses on the part of fossil fuels suppliers. This demand-side focus, which— according to Sinn—is common among policy makers, may prove counterproductive if policies are not at the same time designed to provide correct incentives to resource owners to reduce current supply. Sinn (2008, 2012) goes on to suggest alternative supply side remedies to the challenge of climate change that would not suffer from this criticism. Sinn’s work has spawned a large theoretical literature, which discusses numerous mechanisms that might lead to a green paradox. Overall, these contributions present stylized theoretical analyses rather than attempts to assess the policy relevance of the effects suggested by Sinn (2008, 2012) and others. Very recently, however, Di Maria et al. (2013) presented the first study that empirically assesses the emergence of a green paradox. Their analysis focuses on the passing into law of the Clean Air Act Amendments of 1990. The empirical results of Di Maria et al. (2013) provide mixed evidence as to the empirical relevance of the green paradox hypothesis. More precisely, their study suggests that, while green-paradox-like effects can be observed in time series of coal prices in the United States, the data provide little evidence that this change in price has led to an increase in the actual amount of coal used. The authors suggest several reasons why this might be the case,

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focusing on factors that might limit the ability of resource users to benefit from the price drop. In a way, then, the debate seems to have now come full circle, pointing to the necessity of taking into account demand factors when assessing the risk of a green paradox. In this chapter we first present a brief discussion of the literature on the green paradox, and point out that this literature has largely ignored the demand side of the resource market. We next argue that the (potential) magnitude of a green paradox will depend on the characteristics of the demand side and use a simple model to show that the size of the green paradox indeed depends on the flexibility of demand. According to the theoretical literature, the magnitude of the green paradox should be determined using data on scarcity rents of exhaustible resources before and after a policy change. Section 10.3 discusses several reasons why it is hard to follow such an empirical strategy, and then describes the recent findings of Di Maria et al. (2013) on the potential green paradox effects following the announcement of the cap on sulfur dioxide emissions in the United States in the 1990s. We discuss how the demand side of the resource market may be constrained in its response to price changes in section 10.4. In section 10.5 we discuss what these restrictions may imply for the four imperfect climate policy designs that have been studied in the literature on the green paradox. We conclude in section 10.6. 10.2 Theories of the Green Paradox and the Demand for Resources In his seminal paper Sinn (2008) observed that current climate policy approaches focus on reducing the demand for fossil fuels, while ignoring the supply side of the fossil fuel market. However, fossil fuels are derived from nonrenewable resources and polluting emissions emerge as the result of both the production and the consumption of such fuels. Moreover both the overall level and the timing of greenhouse gas emissions matter in determining the extent of climatic change and the costs it imposes on the aggregate economy. Hence what matters when judging the effectiveness of climate policy is the overall time path of extraction of fossil energy sources. If environmental policy reduces future demand for nonrenewable resources but does not reduce cumulative supply, the only result will be a lower price for fuels in the short run. Consequently short-term emissions will increase, while the cumulative level will remain unchanged, all else equal. Climate

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change mitigation policies should to the contrary aim at reducing the discounted value of profits for resource owners in the short term, thereby inducing them to postpone extraction (Hoel and Kverndokk 1996). Sinn (2008), however, argues that currently implemented demand-side policies would induce an increase, rather than a decrease, in current extraction and may thereby increase current emissions and speed up global warming, a result he termed a green policy paradox or green paradox for short. 10.2.1 Is There Really, Really a Green Paradox? Sinn’s work prompted several authors to study the effects of climate policy on the supply of nonrenewable resources using both theoretical and numerical models of resource extraction. An early article on the topic is Gerlagh (2011), which introduces the notions of a weak and a strong green paradox. Gerlagh defines a weak green paradox as an increase in current emissions in response to climate policy, while a strong green paradox occurs when the net present value of cumulative damages from global warming increases. Studying the effects of cheaper low-carbon energy sources, he finds that a green paradox may or may not occur depending on the exact model under scrutiny. For the simplest model, in which marginal extraction costs for the resource are constant over time and independent of the resource stock, he finds that a drop in the price of a perfectly substitutable clean technology (a socalled backstop) will induce both a weak and a strong green paradox. However, he is quick to point out that this result is far from general. When marginal extraction costs are linear in cumulative supply—and assuming a linear demand—a weak green paradox still occurs but a strong paradox can no longer be found, as the reduction in cumulative supply that follows from the increased competitiveness of the backstop offsets the increase in climate damages arising from the short-term increase in emissions that is common in green paradox models. Finally, Gerlagh (2011) also studies the case where the two energy sources are imperfect substitutes, and marginal extraction costs are constant. In that case a cheaper substitute to fossil energy eats away resource rents and lengthens the period of resource use, without increasing current resource supply, such that also the weak paradox disappears. Similar results have been found by van der Ploeg and Withagen (2012) and Grafton et al. (2012). Support for alternative energy sources is not the only type of policy that affects the behavior of fossil fuel suppliers. Van der Werf and Di

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Maria (2012) identify three more classes of policy measures that have been shown in the literature to have the potential to generate a green paradox. Since it is the profile of discounted instantaneous profits for resource owners that determines their extraction path, a carbon tax that increases over time—like the one briefly implemented in Australia (Australian Government 2014) —may not be sufficient to postpone emissions, a point already made clearly by Sinclair (1992), for example. Indeed a carbon tax that grows at a rate higher than the discount rate will, in a world with constant marginal extraction costs and in the absence of a substitute fuel, induce an increase in current extraction. Hoel (2011, 2012a) shows, however, that including increasing extraction costs to this simplistic model makes for a larger range of carbon tax growth rates that have the ability to postpone emissions. Additionally he shows that in the presence of endogenous investment in alternative energy sources, a green paradox may not occur at all provided that extraction costs do not rise too fast in cumulative extraction. Another type of policy design that might lead to the emergence of a green paradox is the unilateral implementation of demand curbing measures. Indeed carbon tax policies, and direct measures supporting renewable energy—such as subsidies, feed-in tariffs, or renewable energy mandates—are not implemented on a global scale but rather adopted unilaterally by small groups of countries. While such policies have been widely studied using models without nonrenewable resource (see the discussion in section 5 of van der Werf and Di Maria 2012), it was only with the emergence of the interest in the green paradox that scholars started to study this type of policy using models with intertemporal resource scarcity. Using a two-period, three-country model, Eichner and Pethig (2011) show that the tightening of a unilateral cap in the first period causes a green paradox (i.e., in their context, an increase in global emissions in the first period) if the intertemporal elasticity of substitution is sufficiently low, or the demand elasticity for the resource is sufficiently high, or both. The intuition behind this result is straightforward. The tightening of the cap reduces the resource price in both periods as the scarcity rent drops but increases the consumer price of the resource in the country with the unilateral cap. An increase in first-period global emissions occurs only if the non-abating region absorbs more of the resource than the drop in consumption in the abating region, which will only happen if the abating region does not shift too much of its consumption to the (unconstrained) second period

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and the non-abating region is sufficiently willing to respond to the lower resource price. Finally, environmental policies in general, and especially climate change mitigation policies, are usually announced some years before they are actually implemented. Indeed it took more than ten years before first commitment period of the Kyoto Protocol—which was signed in 1997 and came into power in 2005—started in 2008. The US Acid Rain Program was signed into law in 1990, while the first trading phase only started in 1995, and the full implementation of the program happened in 2000. Di Maria et al. (2012) first use a simple model with a single resource and constant extraction costs to show that announcing a policy that is implemented with some delay induces an increase in extraction in the interim period, namely the period between announcement and enforcement of the policy, hence giving rise to a green paradox. Furthermore they show that this result may be amplified in the case of multiple resources that differ in their pollution intensity, as besides the increase in resource use, there may also be a tendency to substitute toward dirtier resources ahead of implementation (see Di Maria et al. 2012 for the precise conditions under which this effect emerges). Jus and Meier (2012) show that further postponing the implementation of the policy, once the announced implementation date is reached, induces a green paradox as well. Likewise Eichner and Pethig (2011) use their three-country, two-period model to also study the announcement of a tighter unilateral cap. In their model a green paradox does not occur if the intertemporal elasticity of substitution is sufficiently low or the resource demand elasticity is sufficiently high, or both. In all these cases in fact resource use in the first period does not respond sufficiently to the lower resource price to lead to a paradox. This brief overview of the green paradox literature indicates that the green paradox emerges rather starkly from traditional models of optimal physical exhaustion of resources, while this type of outcome becomes less likely in models where stock dependent extraction costs à la Heal (1976), in the presence of a backstop, lead to economic rather than physical exhaustion. Quite naturally, in this type of model, increased extraction early on in the planning period comes at the cost of a more rapid switch to the alternative source of energy, and hence proves less attractive. As a general point, one might consider the class of models with economic rather than physical exhaustion as more realistic, and conclude that a green paradox becomes less likely with the increasing realism of the analysis. The possibility of a (weak) green

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paradox as a result of imperfect environmental policies, however, cannot be ruled out on these grounds. The second takeaway point from this short review is that none of the authors mentioned above discuss the potential magnitude of green paradox effects. Indeed the literature has so far only focused on the use of analytical models (both theoretical and numerical) and lacks empirical evidence. The final point is that apart from recognizing the important role of demand elasticities, the literature virtually ignores the demand side of the resource market. For example, the literature discussed above does not make a distinction between different types of fossil fuels beyond their marginal extraction costs. Yet each type of fossil fuel has specific uses: most coal is used to generate electricity, whereas oil is mostly used for transportation (IEA 2012). This affects the path of extraction both with and without climate policy (Chakravorty and Krulce 1994; Chakravorty et al. 1997). As a consequence each type of fuel is subject to different policies. For example, the demand for oil is affected by mandates requiring the blending of biofuels into gasoline, while the relative demand of coal and gas for electricity generation hinges upon the level of carbon prices (either taxes or permit prices) in regions implementing such policies, and the overall demand for electricity generated from fossil fuels depends on subsidies to renewable energy sources. In the remainder of this section we argue that the size of the increase in initial extraction and emissions depends strongly on the demand for resources. 10.2.2 The Demand for Resources and the Size of the Green Paradox As correctly pointed out by Sinn (2008), currently implemented climate change mitigation policies focus on the reduction of the demand for fossil fuels. In line with current thinking in environmental economics, most such policies are based on market instruments, be they tradable emissions permits (e.g., EU Emissions Trading Scheme and Regional Greenhouse Gas Initiative in the Northeastern United States), carbon taxes (as implemented by Sweden, Australia, etc.), or market-based measures supporting alternative energy sources (feed-in tariffs, subsidies to R&D for clean energy technologies, etc.). Since these policies cover neither all emitting sectors of the economy nor all points in time, there is ample scope for the demand side to respond to changes in fossil fuel prices beyond what is envisioned by policy makers. The logical next question, then, is how does the demand for fossil fuels respond to

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changes in fuel prices, across sectors of the (global) economy and over time, following the announcement or implementation of emission reduction policies? As noted above, in the theoretical literature, with few exceptions, the demand side does not matter for the materialization of a green paradox. In closed-economy models, whether a paradox emerges is determined by the exact design of the policy such as the growth rate of the carbon tax or the characteristics of the resource sector, especially the marginal extraction costs of the resource. Resource demand is usually modeled through a concave utility function, or directly using a (possibly linear) demand function. These functions then incorporate both the willingness of the demand sector to respond to lower prices at a given point in time as well as the willingness to shift consumption over time. As long as the demand elasticity along either dimension is positive, resource use will increase in response to a lower user price. In open-economy models, in contrast, a positive demand elasticity is not sufficient for a weak green paradox. Eichner and Pethig (2011) split the demand for resources into two parts. First, there is a sector that produces the final good using the resource via a strictly concave production technology. Next, consumers in both the abating and the non-abating country have to decide how much of the final good to consume in each period. In this way the intertemporal elasticity of substitution is separated from the elasticity of demand for the resource. In a multi-country model with abating and non-abating regions, then, it is not sufficient to have a positive demand elasticity in order to get a green paradox in response to a tightening of the first-period cap on the part of the abating region. The reduction in resource demand in the first period in the constrained region could be shifted to either the second period, or to the non-abating region. For a green paradox to materialize (i.e., for global extraction and emissions to increase in the first period), the increased first-period resource use in the abating region must more than offset the demand reduction on the other region, thus leading to an emissions leakage rate in excess of 100 percent. Not only should the demand elasticity in this country be sufficiently high, but also the intertemporal elasticity of substitution should be sufficiently low, or too much extraction would be shifted to the second period for a green paradox to materialize. This discussion suggests that the characteristics of the demand side of the resource market may affect the magnitude of the green paradox. Given that an imperfectly designed policy may induce an increase in

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initial extraction and emissions, it is both the willingness to absorb additional resources at each point in time (i.e., the demand elasticity) and the willingness to shift resource consumption to earlier points in time (i.e., the intertemporal elasticity of substitution) that determine by how much current resource use increases in response to the demand reduction policy. Numerical results in Grafton et al. (2012) hint in this direction as well. They study the effects of subsidizing alternative energy on the extraction path of a nonrenewable resource. They report that the lower the value of the demand elasticity is, the smaller the change in the exhaustion date of the resource stock, and the smaller the increase in initial extraction. We now provide a simple proof of the importance of the demand side of the resource market for the size of the green paradox, using a simplified version of the model of Di Maria et al. (2012). Competitive resource owners sell their finite resource stock S0 to a consumption sector. The representative consumer has a constant relative risk aversion utility function, U (t ) =

R (t )1−η − 1 , 1−η

(10.1)

so that 1/η measures both the inter-temporal elasticity of substitution and the intratemporal price elasticity of the demand for resources. The dynamics of the resource stock S are given by S (t ) = − R (t ) , R (t ) ≥ 0 , S (0 ) = S0 .

(10.2)

In the absence of any restriction on emissions, the economy is a simple cake-eating economy and resource extraction declines over time at rate ρ/η, where ρ is the utility discount rate (see the appendix). We illustrate this path of extraction with the dashed line in figure 10.1, with initial level of extraction R (0 ) . We assume that one unit of resource consumption, R, is associated with one unit of polluting emissions; figure 10.1 also presents the time path of emissions. The green paradox features of this model emerge when we assume that at time t = 0 it is announced that starting at t = T, a cap (R) will be imposed on emissions, that will be enforced forever. Formally, we have R (t ) ≤ R

∀t ≥ T > 0.

(10.3)

Even a cursory look at the constrained extraction path, drawn as a solid line in figure 10.1, confirms that a green paradox arises, since

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R(t) R(0) ~ R(0)

R

T

TX TH

t

Figure 10.1 Extraction paths for the laissez faire economy (dashed line) and for the economy with an announced emissions constraint (solid line).

extraction is higher at the beginning of the planning horizon in the constrained case, compared with the unconstrained case. It is instructive to understand why this green paradox emerges by building toward it in steps. First, just like in the unconstrained cake-eating problem, a simple variational argument informs us that it is optimal to follow an extraction path along which the present value of marginal utility is constant as long as this is feasible. This implies that extraction will decrease at rate ρ/η before the cap is binding—namely for all t ∈ [0, T) —and after the constraint ceases to bind—namely for every t ∈ [TH, +∞), where TH is defined as the (endogenous) instant when the cap on emissions ceases to be binding. Second, during the constrained phase overall extraction must be less than what would emerge in the unconstrained case, or the cap would not be binding. Third, unless the resource price outside of the constrained phase is lower than in the cake-eating problem, some of the stock of resource could not be extracted. Hence the optimal extraction path that emerge is one like the

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solid line in figure 10.1: initial extraction is higher following the policy announcement, relative to an economy that never faces a constraint. Hence, the announcement of future emission reductions, in an attempt to slow down global warming, increases initial emissions: a clear example of a green paradox (see Di Maria et al. 2012, prop. 1).1 We show in the appendix to this chapter that the initial level of extraction along the constrained path (R(0)) must solve S0 −

η ⎛η ⎞ η R (0 ) (1 − e−( ρ η )T ) − ⎜ (ln R (0 ) − ln R) R − TR⎟ − R = 0. ⎝ρ ⎠ ρ ρ

(10.4)

Using this expression, we can easily calculate the effect of a change in the elasticity of demand for the resource 1/η on the initial resource extraction R(0), and hence on the size of the green paradox. Thus we have Proposition 1 Suppose that an economy is described by equations (10.1) and (10.2) and a cap on emissions is announced at t = 0 to become effective at t = T > 0, as in equation (10.3). Then the lower the elasticity of resource demand 1/η, the smaller is the upward jump in initial extraction due to announcement and the smaller the size of the green paradox. Proof

See the appendix. 䊏

This much simplified model allows us to transparently show that the increase in extraction (and hence emissions) that follows the policy announcement is positively correlated to the price elasticity of the demand for resources. One implication of this is that one would expect that empirically the emergence of a green paradox would be dampened by any factor that would reduce the price elasticity of demand. With this in mind, we now turn to the empirics of the green paradox and the role of the demand side. We first discuss why it might be hard to carry out empirical analyses of the green paradox, and then use the results from the only empirical study we are aware of, to argue that indeed constraints imposed by the demand side of the resource market are crucial to the emergence of a green paradox. 10.3 Empirical Evidence of a Green Paradox: Announcement of the US Acid Rain Program Empirically testing whether the green paradox occurs in practice is difficult for many reasons. First, to directly test the green paradox one

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would ideally obtain mine- or well-level data on scarcity rents before and after the implementation of a policy (or after its announcement in case of delayed policy implementation). Since scarcity rents are unobservable in practice, however, and given that most extraction firms are vertically integrated, one would need to be able to separate the costs of extraction, milling, refining, and other processing that is done before the resource is sold (Krautkraemer 1998; Slade and Thille 2009). Because the costs of those processes are usually proprietary information, it is extremely hard to back out those rents from data on reserve sales or resource prices, which usually include such costs (Krautkraemer 1998; Slade and Thille 2009). As always, when the theory is not directly testable, an alternative, albeit indirect way to subject the green paradox to empirical test, is to investigate whether the implications of the theory can be shown to match observable data. In the case of the green paradox, one would need to verify that a decrease in the price of the resource actually materialized after the policy introduction, and that the use of the resource has increased. Moreover provided there is a differential in the emissions intensity of the resource (e.g., coal vs. natural gas, or different types of coal with different pollution intensities) one could verify whether a switch to the dirtier or lower quality resource has indeed taken place. A further issue that complicates matters for petroleum, the main resource discussed in the green paradox literature, is that it is traded as a commodity in what is practically a global market. Any policy instituted by one country or small group of countries is unlikely to be big enough to alter scarcity rents sufficiently to be observed in the data. Next, establishing causality, that is, ensuring that any change in behavior found after the policy can be attributed to the green paradox, requires identifying a subset of countries (or firms) that may plausibly be construed as representing the “control” group in a semi-experimental design. An ideal control group would be countries or firms that are unaffected by the policy, but that are otherwise similar to the “treated” group. Since, as mentioned, the oil market is a global one, it is unlikely that a control group exists that is unaffected by the policy.2 For the purposes of testing a green paradox, it is much more likely that a suitable control group be found for policies affecting coal or natural gas prices, given that most observers consider their markets to be regional. Another issue in testing the implications of the green paradox (i.e., increased resource use and a switch to the subset of dirtier resources)

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is whether the resource is consumed by only one sector of the economy. In the case of natural gas—which is broadly used both in power generation and in the residential heating sector—an analysis of the residential heating sector ’s natural gas consumption after the policy is in effect would have to control for changes in the electricity sector as well, for example. A final problem that affects the possibility to test the green paradox, regardless of the resource in question, is that testing requires firm- or household-level micro data before and after the policy shock. In many countries, such micro data do not exist, are of insufficient quality, or are difficult for researchers to access. The United States has a rich set of micro data for electric power plants over an extended period of time. Both the United States and most EU countries have large micro datasets on households, which include questions on energy use. Given the long list of caveats listed above, the dearth of empirical analyses on the green paradox is hardly surprising. Indeed the only empirical test of the green paradox that the authors are aware of is the one in Di Maria et al. (2013). This paper uses the announcement of the Acid Rain Program in the United States, which restricted the emissions of sulfur dioxide (SO2) for the last remaining set of unregulated coalfired power plants in the United States, to test whether there was an increase in the use of coal and an increase in the use of higher sulfur coal, following the announcement.3 They use plant-level panel data for the price, quantity and quality of coal used by electric utilities in the 1980s and 1990s. Another advantage of this particular policy case study is that the US coal market is and was largely self-contained as only around 2 percent of coal transacted was imported or exported. Moreover power plants are by far the largest demander: during the period under scrutiny, they acquired more than 90 percent of the coal supplied on the US coal market. As a consequence a policy shock affecting a large part of the US coal-fired power plants would also affect the prices on the coal market and potentially affect the extraction path for this resource. As the data do also provide details on the quality of coal, Di Maria et al. (2013) are also able to study potential shifts among coal of different SO2 content. By 1990 power plants that were built during the late 1970s and the 1980s were subject to existing environmental regulation, whereas older plants (known as phase I plants) were not. The Acid Rain Program was signed into law in 1990 and imposed a cap on the SO2 emissions of phase I plants from January 1995 onward, while adding (nearly) all

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other power plants from 2000 onward. Hence the signing into law acted as a signal to coal mine owners that it would become harder to sell their product in the future, especially for high-sulfur mines. Still phase I power plants were unregulated in the interim of the regulation of their SO2 emissions, so the policy falls into the class of policy designs investigated analytically in Di Maria et al. (2012): an announced cap on polluting emissions with a time lag between the date of announcement and the date of implementation. Di Maria et al. (2013) study the behavior of phase I plants in the period 1986 to 1994 and use previously regulated plants as a control group. The latter group was unable to respond to any coal price changes that might have occurred as a result of the announcement of the SO2 cap (due to the type of environmental regulation applied to them), whereas the former group was in a position to benefit from such price changes in the period between the announcement and the implementation of the cap. The authors first use a hedonic price model for the price of coal delivered to power plants to test whether coal prices indeed dropped after the announcement. Next they utilize a difference-indifference approach to test how phase I plants altered their coal consumption relative to non–phase I plants, after it became clear that a future cap would be implemented. Furthermore they use a triple difference-in-difference analysis to determine whether market structure or concurrent regulation altered the way unregulated firms responded to the policy announcement. Di Maria et al. (2013) find that coal prices were overall lower after the announcement than before, and especially for coal with higher sulfur content: not only did coal mine owners in general face a deterioration of their future market, this was even more so for mines with high sulfur content. This finding affirms the “price part” of the green paradox theories. The question then is whether the lower prices triggered an increase in coal consumption, and in particular, consumption of dirtier coal, in the interim period. The results reveal that the announcement of the policy had no statistical impact on coal consumption of phase I plants as a group. However, using their triple difference-in-difference analysis, the authors are able to identify one group that increased its heat input: phase I plants that obtained a large share of their input on the spot market. During the 1980s and 1990s a large share of coal purchased was obtained through long-term contracts while only 10 to 20 percent was purchased on the spot market (Kozhevnikova and Lange 2009). Still the percentage of coal purchased on the spot markets varied

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considerably between utilities, and power plants that were not largely “stuck” in long-term contracts with previously agreed prices were able to exploit lower prices on the spot market and so increased their coal consumption. Finally Di Maria et al. (2013) study how the sulfur content of coal acquired by power plants changed after the announcement of the cap on sulfur dioxide emissions. Contrary to what the green paradox literature predicts, the authors find that the sulfur content of coal decreased. This finding is in line with what has been found before in the literature on the US coal market: in anticipation of the implementation of the Acid Rain Program, utilities reduced the sulfur content of their coal in order to comply with the upcoming regulation (Ellerman and Montero 1998). The authors confirm this finding in their triple difference-indifference analysis, where they find that plants in states that required pre-approval of compliance utilities’ plans reduced the sulfur intensity of their coal. 10.4

Demand-Side Responses to Changes in Supply

The empirical analysis of Di Maria et al. (2013) shows little evidence of a green paradox in response to the announcement of a cap on sulfur dioxide emissions in the United States, despite the drop in coal prices. The additional results in that paper hint at potential reasons why the demand side of the resource market may not be very responsive to lower resource prices. In what follows, we build on these empirical findings and discuss potential constraints to the ability of firms to take advantage of a drop in the price of fossil fuels. The first reason why demand responses may be muted is to be found in the nature of the goods and services that fossil fuels are instrumental in providing. The demand for electricity and heating services—both mostly produced by burning coal or natural gas—and for gasoline—which is obtained from the distillation of petroleum—is commonly considered to be very inelastic, especially in the short to medium run. For electricity, recent studies, such as Lijesen (2007) and Alberini et al. (2011), estimate the short-run demand elasticity to be significantly smaller than one. As a consequence a large fall in the price of electricity is needed in order for a large increase in coal demand to occur. Similar reasoning applies to the use of natural gas for residential heating (Alberini et al., 2011). The demand for gasoline for transport can increase either through an increase in miles driven per car, or through an increase in the number

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of cars. Both types of demands are quite irresponsive to changes in the price of gasoline, however (Brons et al. 2008). Furthermore Dargay and Gately (2010) show that the demand for crude oil products (gasoline vs. residual fuel oil) is becoming less price-sensitive over time. Considering directly the behavior of electricity generators, the responsiveness of coal use (and to a lesser extent of natural gas) to price changes may also be restricted by the structure of the electricity sector. Since there exist a large number of different technologies and fuels that compete on the electricity market to supply electricity, and since their marginal operating costs vary across fuel types, the electricity supply curve is generally well described as a step function.4 The lowest step of the supply curve represents the marginal cost of renewables like wind, solar, and hydro that have very low marginal costs. One step up in the supply function is usually nuclear as it has a very low cost of generation but is extremely complicated to stop and start up. Next often comes coal, but can be natural gas if the price is low enough. Natural gas is generally easier to ramp up and down, and thus coal tends to be lower on the step than natural gas in most countries. Given this setup, a decrease in the price of coal may lead it to out-compete natural gas, but it is very unlikely that the price would fall low enough for coal to compete with nuclear. Hence coal demand by power stations tends to be quite inelastic. Another point to consider is that as discussed, for example, by Bushnell and Wolfram (2005), (coal-fired) power plants enjoy significant returns to scale in production, and achieve their highest fuel efficiency when they are running close to full capacity. As a result most coal plants run above 80 percent of capacity, as can be inferred by the solid line in figure 10.2. This implies that capacity is likely to be a significant constraining factor to an increase in demand, at least as refers to base-load power plants and in the short run. Capacity constraints also seem to impose similar restrictions to gasoline refining. Indeed data from the US Energy Information Administration show that average refining capacity utilization between 1985 and 2011 was at 89 percent. As a result short-term increases in gasoline supply may be limited by the availability of refining capacity (see figure 10.3). Although capacity may be a restriction on a firm’s ability to respond to lower fuel prices, firms may expand their capacity through investment in new capital stock. Still these investments are very costly and take time to materialize, and hence a firm’s cost–benefit analysis will take into account whether there is ample time to get a sufficiently high

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Figure 10.2 Average (dashed) and median (solid) percentage of the yearly operating capacity of US coal-fired power plants utilized, 1986 to 2005. Source: Authors’ calculations using EIA Form 767.

Figure 10.3 Percentage of the US refining capacity utilized, 1985 to 2011. Source: EIA Monthly, Petroleum Marketing Monthly.

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rate of return on the investment. Given the time it takes to build new capacity, this requires that the period during which the firm can benefit from lower fuel prices is sufficiently long. In case of an announced policy this implies that the implementation lag should be sufficiently long. Indeed for phase I plants it was most likely not profitable to invest in new capacity, given that the period between announcement and implementation of the Acid Rain Program only covered some four years. Yet, even when there is sufficient time to make a profitable investment in new capacity, firms may be restricted in their investment behavior by regulatory agencies and local governments. Before a firm can expand capacity on its premises or start building a new plant on a newly obtained site, it needs to obtain several licenses. In many countries local governments (e.g., city councils) have to provide planning permission. Even though new plants may generate new employment, “not in my backyard” attitudes toward emissions to air and water may induce local authorities not to give planning consent (e.g., see Levinson 1999). In addition new plants often need to obtain licenses from environmental regulatory agencies. For example, the burning of coal produces emissions of particulates, nitrogen oxides, sulfur dioxide, and mercury, among other pollutants. These pollutants may be highly regulated, either directly or indirectly through (local) ambient air quality regulations, and firms will have to show convincingly that their new capacity will be in line with existing (local) environmental regulations. Preexisting environmental regulations will also limit plants’ ability to increase activity levels and thus utilize existing capacity, so as to not provoke additional monitoring by environmental regulators. For example, coal-fired power plants and oil refineries have to be careful not to violate preexisting national- or state-level regulations that restrict their ability to increase the total amount of fuel used or its pollution content or both. Indeed regulated firms tend to be risk averse with respect to compliance. In the Acid Rain Program, many plants operated with a “self-sufficient” strategy, ensuring that they did not need to purchase permits in the market (Rose 1997). Indeed Borenstein et al. (2012) show that regulated natural gas firms are unlikely to make profitable trades when there is concern that regulators may ask questions about the prudence of the decisions. Similar restrictions hold for the use of petroleum by the transport sector. Many US states and countries have instituted regulations on

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automobiles to limit the health damages from air pollution. The principally regulated emissions from the transportation sector are volatile organic compounds, nitrogen oxides and particulate matter. The United States has National Ambient Air Quality standards for pollutants related to transportation emissions, a host of other on road-specific regulations, as well as state-level regulations (e.g., California’s Clean Cars law). To control particulate matter emissions, over 150 cities in 9 European countries have implemented Low Emission Zones (LEZ) (Wolff and Perry 2010). Any automobile entering a LEZ must be classified as a low particulate matter emitter, and these cars are also generally more fuel efficient. In addition car producers are restricted in the characteristics of the cars they produce, be it through fleet-level policies (Corporate Average Fuel Economy CAFE in the United States) or at the vehicle level (Euro 5 vehicle emission standards in the European Union). One final restriction that might limit firms’ ability to increase fuel consumption in response to lower fuel prices, and thereby generate a green paradox, stems from the contractual arrangements prevailing in the fuel markets. In the United States and European Union, coal-fired power plants tend to procure their coal on the basis of long-term forward contracts for two main reasons. The first one being that quantity certainty is important when the plant operates most of the time (base-load plants). There is a large literature concerning the complexity of coal contracts and the risk sharing involved, such as Joskow (1985, 1990). The second reason is that coal is of quite heterogeneous quality, and there are costs for a plant if they burn fuel with different characteristics than the boiler was designed for (Crio and Condren 1984). These costs could be reduced efficiency of the generation process, or more maintenance of the boiler. As a result long-term contracts are signed to ensure a supply of the optimal type of coal. Deliveries of natural gas are often agreed upon in long-term contracts as well, as suppliers want to make sure that their investment in the pipelines required for deliveries is profitable and hence prefer long-term contracts with their customers (Hirschhausen and Neumann 2008). As mentioned in the previous section, Di Maria et al. (2013) find that the subset of plants for which a green paradox seemed to have materialized in response to the announcement of the cap on sulfur dioxide emissions under the Acid Rain Program consists of utilities with relatively flexible coal provision arrangements, as indicated by their propensity to purchase coal on the spot market.

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While the green paradox literature has largely ignored the demand side of the fossil fuel market, the discussion in this section clearly shows how that side of the market may be constrained in its ability to respond to lower fuel prices through increased fuel demand, and hence higher emissions. Among other things, the inelastic nature of the demand for fossil fuel based energy, preexisting regulation and the characteristics of the fuel market may all strongly mitigate the potential magnitude of a green paradox. 10.5

Demand-Side Restrictions and Imperfect Climate Policy

How might the demand side restrictions identified above affect the potential magnitude of a green paradox for each of the four imperfect policy designs identified in section 10.2? To answer this question it is important to establish the nature and size of the market that gets affected by the policy. An important part of greenhouse gas emission reduction policies is aimed at reducing emissions arising from the conversion of fossil fuels into electricity and heat by the power sector. The lessons we derived above from the empirical study of Di Maria et al. (2013) are directly applicable to these type of policies, and they provide important guidance on the likely implication of demand constraints for the emergence of a green paradox. From the discussion in section 10.2, we know that announcements of a future restriction on emissions, support for alternative energy sources, as well as tax paths that rise “too fast” all may induce responses as identified in the green paradox literature. Yet, as discussed in the preceding section, many power plants would not be in a position to fully exploit the likely fall in scarcity rents that characterizes the green paradox. In countries with preexisting (environmental) regulation, with low elasticity of demand, and with structural (in terms of both the available capacity, and the competitive and regulatory framework) constraints in the electricity sector, one would expect that the utilities’ behavioral responses would be rather muted. Rigidity in supply chains, such as the widespread use of long-term contracts for the delivery of fossil fuels, would play a role as well, although the impact of this type of rigidities seems to be diminishing. Indeed, not only did the spot market for coal deepen over time, especially in the United States, but also natural gas markets show a clear tendency to globalize as more countries develop reserves of shale gas, and the

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increasing diffusion of LNG infrastructure makes it easier to ship natural gas across continents. The single most important type of policy aimed at emission reduction from transport is regulation of the design of new automobiles. Although this type of policy has not been studied explicitly in the green paradox literature, it could induce responses that have been identified in that literature. Automobile regulation is usually announced in advance (CAFE regulation in the United States for model years 2017 to 2025 and Euro 6 standards in the European Union), often gets tightened over time, and may include regulation on blending of biofuels or a minimum share of low emission or electric vehicles. If the scale of such regulation is sufficiently large, it may induce resource owners to shift extraction forward in time, resulting in a lower equilibrium oil price. Higher demand for petroleum in response to lower oil prices should come from refineries and the previous section has identified several reasons why this sector is, directly or indirectly, restricted in its response to a lower resource price. Directly, the rate of capacity utilization is usually high, petroleum companies in industrialized countries seem to have underinvested in refinery capacity, and it takes considerable time to build a new refinery. Indirect restrictions come from low demand elasticities for gasoline as well as preexisting regulation. It is important to note that the policies discussed in this section are all unilateral in nature and mostly implemented by industrialized countries. The unilateral nature of the policies may induce other countries to increase their emissions in response, a mechanism called carbon leakage (e.g., see Eichner and Pethig 2011; Hoel 2012b; van der Werf and Di Maria 2012), through two channels. First, lower international resource prices, resulting from unilateral demand-reducing policies, may induce countries to increase their resource demand and thereby their emissions. This is called the “energy market channel of carbon leakage.” Whether global resource prices are affected by unilateral policies depends on the size of the policy and the nature of the resource market. Naturally, if the resource affected by the policy is not traded on a global market, unilateral policy is unlikely to affect resource prices in other countries, and even if the resource is traded internationally in large volumes, both the policy and the country or group of countries affected by the policy should be sufficiently large in order to have a noticeable effect on global resource prices. The second channel of international carbon leakage is the so-called terms-of-trade channel. When the prices of energy-intensive goods produced in countries with climate

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polices increase—either directly due to a domestic price on greenhouse gas emissions or indirectly through obligatory use of expensive renewable energy technologies—production of such goods may shift toward other countries. To the extent that these countries are industrialized countries, firms in these countries may be restricted in their ability to increase emissions due to reasons explained above. As we will argue below, firms in developing countries may have more scope to increase their emissions. Besides incentives through the energy market channel and terms-oftrade channel of carbon leakage, firms in developing countries may face incentives to increase their emissions in response to domestic policies. Moreover, as policy makers in developing countries start to introduce demand-reducing policies for fossil fuels, they too may induce a green paradox insofar these policies suffer from the imperfections discussed in section 10.2. To what extent firms in developing countries are restricted in their ability to increase emissions is largely an open question. In terms of lower fossil fuel prices, demand elasticities for electricity and gasoline are likely to be low in both industrialized and developing countries, although the exact size of these elasticities for developing countries is still quite unknown. A greater potential for an increase in emissions from developing countries may therefore come from increased production for the export market. Restrictions on production through preexisting regulation, especially environmental regulation, are likely to play a much smaller role in developing countries than in industrialized countries. Capacity expansions are easier to implement when policy makers value (development through) job creation over environmental amenities. Lower relative prices for energy-intensive goods from developing countries (through the terms-of-trade channel of carbon leakage or through domestically induced green paradoxes in these countries) may induce demanders of such goods to shift to goods produced in developing countries. Depending on the country under scrutiny, it may not be preexisting environmental regulation that restricts such expansion of production but (lack of) openness to trade and security of energy or electricity supply. It should be noted, however, that such a shift to production from developing countries may come along with a transfer of clean technologies, which in turn mitigates the potential size of a green paradox (Di Maria and van der Werf 2008). Similarly increased demand for transport in developing countries due to lower fuel prices may be partially offset by increased use of cleaner cars. As the

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automobile market in developing countries is dominated by Western (including Japanese) manufacturers, and since economies of scale play an important role in this market, it is unlikely that these manufacturers will develop dirtier but cheaper models for sale in emerging economies. 10.6

Discussion

In this chapter we have argued that the literature on the green paradox has largely ignored the demand side of the resource market, and that this side of the market may mitigate the size of an emissions increase in response to imperfect climate policies. We have supported these claims using the recent empirical findings of Di Maria et al. (2013) on the response of US coal-fired power plants to the drop in coal prices after announcement of the cap on sulfur dioxide emissions in 1990. Furthermore we have argued that similar restrictions exist for the response of the demand for petroleum after a drop in the price of oil. Last we have identified emerging economies as potentially the biggest threat to the ineffectiveness of greenhouse gas emission reducing policies. What lessons can we draw for policy makers? The first and obvious advice would be to repair current emission reduction policies and thereby reduce the potential for a green paradox. However, vested interests and lobbying make it hard for policy makers to simply set an emissions price that reflects marginal damages (not to mention the large uncertainties surrounding the social price of carbon; e.g., see Tol 2009). We further learned that a large part of the response to demandreducing policies could occur in countries that did not implement those policies, and this shows the importance of including as many countries as possible in an international agreement to reduce greenhouse gas emissions. Unfortunately, recent international conferences on the topic do not give much reason for optimism. Imposing border carbon tariffs on imports from other countries may be effective in reducing potential carbon leakages but may be hard to implement (Böhringer et al. 2012). The case study of Di Maria et al. (2013) suggests few potential policies to prevent a green paradox from domestic emission sources. US power plants located in states with strict preexisting policies, especially policies that required pre-approval of compliance plans of the announced policy, were less able to exploit lower coal prices. If a domestic demand-reducing policy specifies milestones or asks the

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regulated firms to submit compliance plans in advance of the policy’s implementation, a green paradox seems less likely to occur. Other restrictions on firms in the case study seem to have stemmed from factors beyond a policy maker ’s control (at least on competitive markets). Clearly, more research on potential demand-side responses to emission reduction policies as well as on potential responses by policy makers is necessary. Appendix: A Simple Analytical Model From equations (10.1) to (10.3) we can formulate the following Lagrangian: L (⋅) = U ( R (t )) − λ (t ) R (t ) + I (t ) τ (t ) ( Z − R (t )) ,

(10.5)

where I(t) = 0 ∀ t < T and I(t) = 1 ∀ t ≥ T. First-order conditions are (R(t))−η = λ(t) + I(t)τ(t),

(10.6)

λ (t ) = ρλ (t ) .

(10.7)

The complementary slackness condition for the emission constraint is

τ (t ) ≥ 0 , R − R (t ) ≥ 0 , τ ( t ) ( R − R (t ) ) = 0 ,

∀t ≥ T ;

(10.8)

the transversality condition for the resource stock reads lim λ (t ) S (t ) e− ρt = 0. t→∞

(10.9)

As long as climate policy is not binding, τ (t) = 0. For this case, taking the time derivative of (10.6), combining it with (10.7), dividing by (10.6), and rearranging terms gives

ρ Rˆ (t ) = − . η

(10.10)

Integrating this expression, we obtain that the level of extraction along the optimal path is R (t ) = R (0 ) e − ρ η. Using this expression in the integral of (10.2), and taking into account that the resource stock needs to be completely exhausted over time—as implied by (10.7) and (10.9)— gives instantaneous extraction and emissions when the economy is never constrained (i.e., when T → ∞) as

ρ ρ R (t ) = S (t ) = S0 e−( ρ η )t . η η

(10.11)

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Let us now turn to the case of the announced binding cap, and let us divide the planning horizon in three phases, the interim phase (0, T), the constrained phase [T, TH), and the unconstrained phase [TH, ∞). This last phase starts at the instant when the emission constraint ceases to be binding, TH. Hence we can write T

TH

∞

0

T

TH

S0 = ∫ R (t ) dt +

∫ R (t ) dt + ∫ R (t ) dt.

(10.12)

At time TH, extraction is just equal to the cap: R (TH ) = R. Since (10.10) must hold for any t outside the interval [T, TH), we find that R (0 ) = Re( ρ η )TH .

(10.13)

This can be solved to find TH as a function of initial extraction: TH =

η (ln R (0) − ln R). ρ

(10.14)

Furthermore we can use (10.10) to solve the first integral in (10.12) in terms of initial extraction. From t = TH on, the economy is unconstrained, hence at this instant R (TH ) = R. Using (10.10) and the fact that the stock gets exhausted in infinite time, we find S (TH ) = (η ρ ) R, which we can substitute for the last integral. Noting that R (t ) = R ∀t ∈[T , TH ] for the second part of (10.12), we can rewrite it as S0 =

η η η R (0 ) (1 − e−( ρ η )T ) + (ln R (0 ) − ln R) R − TR + R. ρ ρ ρ

(10.15)

After rearranging terms, we have with this equation an expression that solves for the initial level of extraction, R0. A simple application of the implicit function theorem gives

ρ (T ( R (0 ) e−( ρ η )T − R) − ρS0 ) dR (0 ) =− > 0, d (1 η ) 1 − e − ( ρ η ) T + ( R R ( 0 ))

(10.16)

which proves the proposition in the main text. Notes 1. A more detailed discussion of this simple model is presented in the appendix to this chapter. The detailed proof can be found in Di Maria et al. (2012). 2. Furthermore oil is much more linked to macroeconomic outcomes than coal or natural gas, as emphasized by the wealth of literature on oil and the macroeconomy (see Hamilton 2008 for a recent summary) compared to that on the dearth of coal—or gas—and

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the macroeconomy. Because of this, there are many more general equilibrium effects that would be difficult to disentangle in an analysis of the green paradox that looked at oil consumption reducing policies. 3. Notice that although this policy is not aimed at mitigating climate change, it definitely has the potential to trigger a green paradox, since it affects the extraction path of a nonrenewable resource. Moreover, in terms of both policy design and compliance options, it is a close fit to current climate policy, implying that the results of Di Maria et al. (2013) are likely to carry over to climate policy. 4. For an illuminating example of an actual electricity supply curve, see Mansur (2007, p. 668), who uses data from the PJM interconnector.

References Alberini, A., W. Gans, and D. Velez-Lopez. 2011. Residential consumption of gas and electricity in the U.S.: The role of prices and income. Energy Economics 33: 870–81. Australian Government. 2012. An overview of the Clean Energy Legislative Package. Available at: www.comlaw.gov.au/Series/C2011A00131. Accessed February 14, 2014. Böhringer, C., B. Bye, T. Fæhn, and K. E. Rosendahl. 2012. Alternative designs for tariffs on embodied carbon: a global cost-effectiveness analysis. Energy Economics 34: S143–53. Borenstein, S., M. Busse, and R. Kellogg. 2012. Career concerns, inaction, and market inefficiency: Evidence from utility regulation. Journal of Industrial Economics 60 (2): 220–48. Brons, M., P. Nijkamp, E. Pels, and P. Rietveld. 2008. A meta-analysis of the price elasticity of gasoline demand. A SUR approach. Energy Economics 30 (5): 2105–22. Bushnell, J., and C. D. Wolfram. 2005. Ownership change, incentives and plant efficiency: The divestiture of U.S. electric generation plants. Working paper 140. Center for the Study of Energy Markets, University of California at Berkeley. Chakravorty, U., and D. L. Krulce. 1994. Heterogenous demand and order of resource extraction. Econometrica 62 (6): 1445–52. Chakravorty, U., J. Roumasset, and K. Tse. 1997. Endogenous substitution among energy resources and global warming. Journal of Political Economy 105 (6): 1201–34. Crio, M., and A. Condren. 1984. Which coal at which cost? Public Utilites Fortnightly 113 (16): 32–36. Dargay, J. M., and D. Gately. 2010. World oil demands shift toward faster growing and less price-responsive products and regions. Energy Policy 38: 6261–77. Di Maria, C., and E. van der Werf. 2008. Carbon leakage revisited: Unilateral climate policy with directed technical change. Environmental and Resource Economics 39: 55–74. Di Maria, C., I. Lange, and E. van der Werf. 2013. Should we be worried about the green paradox? Announcement effects of the acid rain program. European Economic Review. forthcoming. http://dx.doi.org/10.1016/j.euroecorev.2013.03.010 Di Maria, C., S. Smulders, and E. van der Werf. 2012. Absolute abundance and relative scarcity: environmental policy with implementation lags. Ecological Economics 74: 104–19.

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Eichner, T., and R. Pethig. 2011. Carbon leakage, the green paradox and perfect future markets. International Economic Review 52 (3): 767–805. Ellerman, A. D., and J.-P. Montero. 1998. The declining trend in sulfur dioxide emissions: Implications for allowance prices. Journal of Environmental Economics and Management 36 (1): 26–45. Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57 (1): 79–102. Grafton, R. Q., T. Kompas, and N. V. Long. 2012. Substitution between biofuels and fossil fuels: is there a green paradox? Journal of Environmental Economics and Management 64: 328–41. Hamilton, J. 2008. Oil and the macroeconomy. In S. Durlauf and L. Blume, eds., New Palgrave Dictionary of Economics, 2nd ed. London: Palgrave Mcmillan, 201–28. Heal, G. 1976. The relationship between price and extraction cost for a resource with a backstop technology. Bell Journal of Economics 7 (2): 371–78. Hirschhausen, C., and A. Neumann. 2008. Long-term contracts and asset specificity revisited: An empirical analysis of producerimporter relations in the natural gas industry. Review of Industrial Organization 32 (2): 131–43. Hoel, M. 2011. The green paradox and greenhouse gas reducing investments. International Review of Environmental and Resource Economics 5 (4): 353–79. Hoel, M. 2012a. Carbon taxes and the green paradox. In . R. W. Hahn and A. Ulph, eds., Climate change and common sense: essays in honor of Tom Schelling. New York: Oxford University Press, 203–24. Hoel, M. 2012b. The supply side of CO2 with country heterogeneity. Scandinavian Journal of Economics 113 (4): 846–65. Hoel, M., and S. Kverndokk. 1996. Depletion of fossil fuels and the impacts of global warming. Resource and Energy Economics 18: 115–36. IEA. 2012. Key World Energy Statistics, 2012 Edition. Paris: International Energy Agency. Joskow, P. 1985. Vertical integration and long-term contracts: The case of coal-burning electric generating plants. Journal of Law Economics and Organization 1 (1): 33–80. Joskow, P. 1990. The performance of long-term contracts: Further evidence from coal markets. Rand Journal of Economics 21 (2): 251–74. Jus, D., and V. Meier. 2012. Announcing is bad, delaying is worse: Another pitfall in well-intended climate policy. Working paper 3844. CESifo, Munich. Kozhevnikova, M., and I. Lange. 2009. Determinants of contract duration: Further evidence from coal-fired power plants. Review of Industrial Organization 34 (3): 217–29. Krautkraemer, J. A. 1998. Nonrenewable resource scarcity. Journal of Economic Literature 36 (4): 2065–2107. Levinson, A. 1999. NIMBY taxes matter: the case of state hazardous waste disposal taxes. Journal of Public Economics 74 (1): 31–51. Lijesen, M. G. 2007. The real-time price elasticity of electricity. Energy Economics 29 (2): 249–58.

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Mansur, E. T. 2007. Do oligopolists pollute less? Evidence from a restructured electricity market. Journal of Industrial Economics 55 (4): 661–89. Rose, K. 1997. Implementing an emissions trading program in an economically regulated industry: Lessons from the SO2 trading program. In R. F. Kosobud and J. Zimmerman, eds., Market Based Approaches to Environmental Policy: Regulatory Innovations at the Fore. New York: Van Nostrand Reinhold. Sinclair, P. 1992. High does nothing and rising is worse: carbon taxes should keep declining to cut harmful emissions. Manchester School 60 (1): 41–52. Sinn, H.-W. 2008. Public policies against global warming. International Tax and Public Finance 15 (4): 360–94. Sinn, H.-W. 2012. The Green Paradox: A Supply-Side Approach to Global Warming. Cambridge: MIT Press. Slade, M. E., and H. Thille. 2009. Whither Hotelling: Tests of the theory of exhaustible resources. Annual Review of Resource Economics 1 (1): 239–59. Tol, R. S. J. 2009. The economic effects of climate change. Journal of Economic Perspectives 23 (2): 29–51. van der Ploeg, F., and C. Withagen. 2012. Is there really a green paradox? Journal of Environmental Economics and Management 64: 342–63. van der Werf, E., and C. Di Maria. 2012. Imperfect environmental policy and polluting emissions: The green paradox and beyond. International Review of Environmental and Resource Economics 6 (2): 153–94. Wolff, H., and L. Perry. 2010. Trends in clean air legislation in Europe: Particulate matter and low emission zones. Review of Environmental Economics and Policy 4 (2): 293–308.

IV Empirics and Quantification

11

Quantifying Intertemporal Emissions Leakage Carolyn Fischer and Stephen Salant

11.1

Introduction

Reducing emissions of the greenhouse gases (GHGs) that contribute to global climate change is the greatest collective action problem of our time. According to the Intergovernmental Panel on Climate Change (IPCC), avoiding the largest risks of climate change will require that global emissions of carbon dioxide (CO2) stop rising within the next two decades (IPCC 2007). At the same time, however, the current United Nations Framework Convention on Climate Change, under the principle of “common but differentiated responsibilities,” requires no mandatory action on the part of developing countries, including major emerging economies that are large emitters. Furthermore, in the absence of a binding successor to the Kyoto Protocol, not even developed countries are committed to emissions targets, although the Copenhagen Accord does call on countries to make individual pledges of action. In this context of largely uncoordinated activities, several countries are taking significant steps to reduce their own GHG emissions. However, their unilateral actions may be partly or even completely undermined by the actions of others. There are two channels through which carbon might “leak”: spatial and intertemporal. With spatial leakage, one government’s efforts to raise the cost of fossil fuel use may cause economic activity to shift to countries with fewer regulations and lower costs. Furthermore the reduction in demand for fossil fuels in one regime causes global energy prices to fall, expanding fuel consumption in nonregulating countries. A variety of studies using static computable general equilibrium (CGE) models have analyzed spatial leakage. Most of these studies find leakage rates in the range of 10 to 30 percent (e.g., see Babiker and Rutherford 2005; Boehringer, Balistreri, and Rutherford 2012), representing both changes

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in fossil fuel markets and the shifting of other economic activities. These spatial leakage estimates have been shown to be highly sensitive to fossil fuel supply elasticities (e.g., Burniaux and Martins 2000; Mattoo et al. 2009; Boehringer, Balistreri and Rutherford 2012). Here we focus on intertemporal leakage as oil suppliers respond to a government’s attempts to reduce fossil fuel emissions. Since fossil fuels are in finite supply, current extraction decisions depend not only on current prices but also on future prices. If climate policies will make it less attractive to sell fossil fuels tomorrow, suppliers may decide to sell more today. Indeed current price–cost margins for some large fossil fuel reserves are ample enough to permit significant price reductions if clean substitutes eventually become cheaper and begin luring consumers away. Early studies of this phenomenon, rooted in the “green paradox” literature, assumed that intertemporal leakage would reach 100 percent, and the only option available to policy makers is thus to influence the time path of emissions and therefore the present value of damages. However, according to the IPCC Fourth Assessment Report, reaching a stabilization target of 450 parts per million (ppm) would require cumulative emissions over the twenty-first century to be in the range of 1,370 to 2,200 GtCO2 (or 375 to 600 GtC; IPCC 2007, p. 67).1 In comparison, Kharecha and Hansen (2008) estimated that there remain 70 to 140 GtC of natural gas, 120 to 250 GtC of conventional oil, 500 to 1,000 GtC of coal, and 150 to 1,000 GtC of unconventional oil from sources like tar sands and shale. Especially if the upper range of reserve estimates holds, complete exhaustion of all proven resource pools, regardless of the time scale, would constitute a flagrant disregard for the GHG concentration targets. We thus shift focus from the time path of emissions, the emphasis of the prior literature on the green paradox, to the effectiveness in generating cumulative reductions. The reasons are threefold. First, it is not clear that the social cost of carbon rises more slowly than the discount rate, which is necessary to prefer delaying emissions.2 Second, given the longevity of carbon effects, the stabilization targets that are the focus of international negotiations are tantamount to a cumulative carbon budget over the fossil fuel era.3 Third, most climate policy models do not consider fossil fuel suppliers dynamic responses to policy changes,4 yet it is important to know the amount of intertemporal leakage.

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To gauge the potential magnitude of intertemporal leakage and emissions acceleration, we construct a calibrated model of global oil markets.5 The model is based on the classic Hotelling framework (as is common in the green paradox literature) but incorporates increasing cumulative extraction costs by reflecting the different costs of accessing oil from five major categories of sources, both conventional and unconventional. Costs, reserves, and demand parameters are all drawn from official sources and the broader literature. Furthermore we distinguish among the emissions intensities of different sources. We then model the effects of five climate policies: 1. Lowering the cost of a carbon-free backstop technology 2. Taxing emissions 3. Improving energy efficiency 4. Mandating a blend or portfolio ratio with the green backstop technology 5. Mandating a rate of carbon capture and sequestration We require each policy to meet a cumulative emissions target and compare the effects of the different policies on two summary measures: (1) the time interval before green technology replaces fossil fuels and (2) the degree of intertemporal leakage.6 The first metric relates to the green paradox and the time profile of emissions; other things equal, policy makers may prefer longer time intervals to adjust to a given level of cumulative emissions. We show that regardless of the number of pools assumed and their sizes and costs, the first four policies can be ranked unambiguously in this dimension: for any given level of cumulative emissions, the green backstop policy results in the least time to adapt, followed by the emissions tax, while the energy efficiency and blend mandate policies actually have identical effects and give society the longest time to adapt.7 On the one hand, the calibrated simulations reveal that the difference in timing between the tax and backstop policies is minor and unlikely to influence the present value of damages associated with a given cumulative emissions target. On the other hand, conservation policies might postpone emissions over a long enough horizon to have an effect. For our second metric, we define the intertemporal leakage rate in a similar manner to the conventional spatial leakage rate: what is the change in emissions resulting from the rent adjustment, as a share of the reductions that would occur in the absence of rent adjustment? This

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metric allows us to correct standard climate policy forecasts by taking into account the supply responses that the static models neglect. We conclude that intertemporal leakage, though potentially substantial, can be considerably less than 100 percent, particularly as the stringency of the policies increases. Furthermore the alternative energy policy is no more subject to intertemporal leakage than the emissions tax. The chapter proceeds as follows: In section 11.2, we review the related literature on the green paradox. In section 11.3, we describe the model and the climate policies under consideration. In section 11.4, we explain the calibration of the model. In section 11.5, we compare the consequences of these policies with respect to the time to transition to the backstop technology. In section 11.6, we quantify intertemporal leakage rates associated with each policy. Section 11.7 discusses some limitations to conducting welfare analysis, and section 11.8 concludes the chapter. 11.2

Literature Review

Asked by Foreign Policy, “How can we stop climate change?” Bjorn Lomborg (the “Skeptical Environmentalist”) replied, “By being smart and investing in research to make green energy cheap instead of trying to make oil unaffordable” (Dickinson 2010). Although such a policy might address some spatial leakage concerns, the prescription has been criticized in studies of the “green paradox.” Notably Sinn (2008), who coined the term, argues that alternative energy strategies are particularly likely to accelerate rather than slow emissions over time. This acceleration could not only obviate any emissions reductions in the long run but also increase the present discounted value of damages (the “strong” version of the green paradox). In contrast, extraction taxes can at least be designed to slow fossil fuel consumption. But Sinn argues more generally that policies to promote energy efficiency or to expand the use of clean substitutes are destined to speed global warming, whereas carbon sequestration is one of the few useful options for slowing it. Other authors are more or less pessimistic about the prospects for clean energy policies. Strand (2007) makes a similar point about the indirect effects of reducing the cost of substitute technologies. Winter (forthcoming) notes that with positive feedback effects between atmospheric carbon and the release of terrestrial carbon, innovation in clean energy technology can lead to a permanently higher temperature path.

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Grafton, Kompas, and Long (2012) find that subsidies to biofuels that are ongoing substitutes for fossil fuels may accelerate or delay extraction, depending on the relative cost parameters. Chakravorty et al. (2011) show that greater potential for learning-by-doing in the substitute technology results in lower equilibrium energy prices, which deter innovation, leading to increased resource extraction and greenhouse gas emissions. Other studies have combined the analysis of intertemporal and spatial emissions leakage. Hoel (2011) extends this analysis by assuming that countries differ in their taxation of fossil fuel use. Fischer and Salant (2014) assume that an exogenous fraction of world demand is unregulated. Eichner and Pethig (2011) use a two-period model with separate abating, non-abating, and fossil fuel—supplying countries to explore the conditions under which tightening the emissions cap in the abating country accelerates global emissions. In nearly all of these models the nonrenewable resource is ultimately exhausted, albeit at different rates; the cumulative carbon emissions are thus constant and the intertemporal leakage rate is 100 percent. Cumulative extraction is invariant in these models because of a combination of assumptions made about the extraction technology and green substitute, and because the policies considered fall within a range that would not choke off fossil fuel demand. In reality, extraction costs will rise as fossil fuels become increasingly scarce, so reasonable climate policies can cause low-value resources to be left in the ground. In principle, this may be modeled either by positing a functional form for extraction costs that includes cumulative extraction as one argument or by assuming that different pools of oil have different per unit costs. Gerlagh (2011) and van der Ploeg and Withagen (2012) adopt the first approach. Gerlagh assumes that extraction costs are linear in cumulative extraction and finds that lowering the cost of a green substitute decreases cumulative emissions and can decrease the present value of damages, but it still increases initial emissions—an effect he terms a “weak green paradox.” Van der Ploeg and Withagen also assume that extraction costs increase with depletion and posit a range of costs for clean backstop technologies; they find that the green paradox still holds with cost reductions in expensive backstop technologies, but it need not arise with cost reductions in relatively cheap backstops. In this chapter we adopt the second approach. We assume that oil pools differ in their per unit cost of extraction, and hence extraction costs will rise over time as higher cost pools are accessed. This

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framework retains intuitive characteristics of the early studies and also allows us more flexibility when calibrating our model than if we had assumed a particular form of the cost function. In addition it allows us to take account of the different emissions factors associated with the different types of fossil fuels, previously ignored in the literature. 11.3

Climate Policies in a Model with Multiple Pools

We develop and later parameterize a Hotelling-style model reflecting the five major types of oil, which we list in order of increasing costs: low-cost Middle East and North African (MENA) conventional oil (denoted with subscript “L”); medium-cost conventional oil from other sources (subscript “M”); enhanced oil recovery (EOR) and deepwater drilling (subscript “D”); heavy oil bitumen, including oil sands (subscript “H”); and oil shales (subscript “S”). We find that this level of disaggregation will be sufficient to capture the main effects of cost and reserve heterogeneity. We describe the basics of the market equilibrium below and note that the formal set of equations defining all of the endogenous variables for a general multi-pool model is described in the appendix of Fischer and Salant (2012). Each pool of oil (indexed by i) has a fixed stock of reserves (Si) with a constant per unit extraction cost (ci) and an emissions intensity ( μi , in tons of CO2 per unit extracted and sold). Thus the flow of emissions from a given pool equals the quantity of oil produced (qi(t)), multiplied by the emissions factor ( Mi (t) = μi qi (t)). Resource owners are assumed to be price takers operating in a competitive market. Since oil can be stored in the ground, they can choose to extract now, from which the proceeds can earn the interest rate (r), or hold the oil and sell it later. Therefore, as long as the owner of a given type of oil is extracting, the present discounted value of profit per unit must be constant. (Technically, if the price of oil at time t is p(t), it must be that p(t) = ci + λi e rt for qi (t) > 0 , where λi represents the value of a barrel of oil in the ground.) In addition to the fossil-fuel sources, a carbon-free backstop technology is available in unlimited capacity at a constant marginal cost (B) that evolves over time as a function of the rate of technological change.8 Initially too expensive to warrant consideration by consumers, the marginal cost of this backstop declines exogenously over time toward a long-run cost (specifically, B(t ; z) = BLR + (B0 − BLR ) e − zt ). We assume that this longrun cost is ultimately lower than the lowest extraction cost among the resource pools (BLR < cL ). Furthermore the convergence rate (z) can be increased by government policy. In the baseline scenario ( z = z0 > 0), we

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assume that this per unit cost declines slowly enough that the pool of oil is completely exhausted before the backstop is utilized. In this competitive market, where the per unit costs of extraction differ across types of oil, the pools will be extracted in order of their extraction costs (Herfindahl 1967). In the market equilibrium a pool with a lower extraction cost will have higher discounted profits per unit (i.e., a higher scarcity rent). The intuition is that with perfect competition, if two fossil energy sources are producing simultaneously, it must be that the costs (including the scarcity values) are the same. Otherwise, the pool with lower costs is more competitive and the higher cost pool will wait to be accessed until the lower cost source is fully exhausted. That transition from the lowest cost source currently available to the next lowest cost source occurs at the point in time when the costs are identical; thus the pool with the lower extraction cost must have a higher rent and thus an opportunity cost that rises faster over time than that of a pool with higher extraction costs. (Technically, let xk denote the date of transition from pool k – 1 to the pool k; at this point in time p( xk ) = ck −1 + λ k −1e rxk = ck + λ k e rxk ). Total extraction ends when the backstop becomes the least-cost alternative. The transition to the backstop occurs when the cost of the last pool that is accessed equals the cost of the backstop. (Formally, let xB denote the date when the backstop replaces oil; if m is the last pool, this occurs when p( xB ) = cm + λ m e rxB = B( xB ; z)). Thereafter the market price must equal the backstop cost (formally, if qB (t) > 0, then p(t) = B(t , z)). In sum, the overall price path will be the minimum of the costs of all the options over time (i.e., p(t) = min i∈{ L , M ,D , H ,S} [ci + λi e rt , B(t ; z)]). Consumer demand (D) is a function of the price of oil and time, reflecting changes in energy service technologies and demographics. (Specifically, we assume a linear demand function such that D( p(t), t) = y0 e gt − mp(t), where y0 is the baseline intercept or choke price, g is the growth rate in demand, and m is the slope.) In a market equilibrium, supply must equal demand at all points in time (so qi (t) = D( p(t), t) for xi ≤ t ≤ xi+1, for i = {L, M , D, H , S, B}). Furthermore we have the stock constraints that cumulative extraction cannot exceed the xi +1 reserves of a given pool ∫ qi (t)dt ≤ Si .

(

xi

)

We then have two potential regimes for each pool, regarding when it is the last pool to be utilized. In regime a, the pool of oil is fully exhausted (all of the oil eventually ends up above ground). In this case, the per unit value of oil in the ground is strictly positive for the marginal pool (λ m > 0), as well as for all those exhausted before it, and cumulative demand must equal the resource stock.

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In regime b, the pool of oil is incompletely extracted (some oil remains below ground). In this case the shadow value of oil for the marginal pool must be zero (λ m = 0), and the share extracted, θ , is determined by cumulative demand during production of that pool up to the switchover point

(∫

xB

xm

)

D( p(t), t) = θ Sm . The transition to the

backstop occurs when the backstop cost falls to the marginal extraction cost of the last pool (B( xB ; z) = cm ). Suppose that we start with an equilibrium of complete exhaustion of all pools in the absence of any policy intervention. As we gradually tighten any one of the climate policies we will consider, the equilibrium will fall successively into each of 10 qualitative regimes RSa , RSb , ..., RLa , RLb . In any regime a, every pool that is utilized will ultimately be exhausted and the associated scarcity rents will each be strictly positive. Strengthening the policy in a regime a causes the rents to decline until the lowest rent reaches zero. Further strengthening of the policy moves the equilibrium into a regime b. In such regimes the last pool utilized has a zero scarcity rent and will (except for the boundary case) be only partially exhausted. Further strengthening of the policy within regime b crowds out extraction of the marginal pool; scarcity rents for the inframarginal pools can be further eroded by policies that influence demand. When the resource pool with the zero rent ceases to be utilized at all (θ m = 0), the equilibrium falls into the next a regime, and so on. Next we describe each of the policy options we consider. To simplify, we assume that each policy is exogenously imposed and fully anticipated. Thus we ignore questions of induced technological change and focus on the direct effects of each policy. These effects can be quite different in the b regimes than the a regimes that are the standard focus of the green paradox literature. 11.3.1 Accelerating Backstop Cost Reductions Our first policy accelerates cost reductions in green backstop technologies. (B(0 ; z) is constant regardless of z, but ∂B(t ; z)/ ∂z < 0 for t > 0.) An increase in the innovation rate causes the backstop marginal cost to decline faster. If the price path did not change, the transition to the clean technology would occur sooner. But then the oil would not be exhausted, a disequilibrium. For equilibrium to be restored, the scarcity rent must decline, and as a result, the entire price path falls. A more rigorous policy thus lowers the rents on the resource and results in an

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earlier transition ( xB falls). In any regime a, the green paradox arises: faster reductions in the unit cost of the green technology do not reduce cumulative emissions (100 percent leakage), but the policy shortens the time until fossil fuels are exhausted and therefore raises the annual rate of emissions during the remainder of the fossil fuel era. If the innovation rate becomes large enough that the rents of the marginal pool are driven to zero, we enter regime b. Oil from the last pool is sold at its marginal cost of extraction until the backstop enters. Faster innovation does not alter the price but does hasten the transition to the green technology (smaller xB). It therefore increases the stock of reserves that are left in the ground rather than transformed into greenhouse gases. Note that in regime b, increasing z does not alter the rate of extraction, so the green paradox disappears. 11.3.2 Emissions Tax An emissions tax levies a cost τ (t) per unit of emissions at time t. We assume that extractors pay the tax; however, since the policy is assumed to be global, the incidence would be the same as if buyers pay the tax. Let p(t) denote the price that consumers pay. After paying the tax, extractors retain p(t) − τ (t)μi . Thus, for the present discounted value of oil profits to remain constant while pool i is producing, we have p(t) = c + τμi + λi e rt. The transition to the next pool, and ultimately the backstop, occurs when the costs (inclusive of both scarcity values and emissions payments, as well as extraction costs) are equal. The prior literature has given much attention to the influence of the time path of emissions taxes on the green paradox in regime a (e.g., Sinn 2008; Hoel 2012). We note here that an emissions tax that rises at the rate of interest causes a fully offsetting shift in oil rents, resulting in no change in the transition time to the backstop technology. An emissions tax that rises at less than the rate of interest tilts the path of consumer prices for oil, resulting in higher prices early on and lower prices later, delaying the transition to the backstop.9 If the tax path is large enough to drive the scarcity rent of the marginal pool to zero, regime b has been reached. The price consumers pay while the marginal pool is being exploited remains at cm + τ (t)μ m until backstop becomes cheaper and they switch. The higher the tax, the higher is the price that consumers pay for oil, and the sooner they switch to the clean backstop, leaving more of the last pool in the ground. Thus in regime b the effect of the emissions tax is more similar to that of the alternative energy policy.

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11.3.3 Improvements in Energy Efficiency An alternative to promoting green substitutes or taxing emissions is to reduce the demand for oil by increasing the efficiency with which it is utilized. Examples include retrofit programs for buildings, energy efficiency standards for appliances, and fuel economy standards for motor vehicles. We consider the effects of a mandate that energy efficiency be improved by a certain percentage ( f (t)), which can be either an immediate and permanent improvement or a growing share over time.10 Consumers demand energy services and purchase oil to provide those services. An improvement in efficiency has two countervailing effects: (1) it reduces the number of barrels required to obtain any level of energy services; but (2) it increases the level of energy services demanded by lowering their effective price—what has been termed the rebound effect. (Formally, the effect of the mandate is to shift demand for oil such that D( p(t), t) = (1 − f (t)) ( y0 e gt − m(1 − f (t))p(t)).) Whether an improvement in efficiency decreases the demand for oil depends on whether the first effect dominates the second effect. The empirical literature looking at transportation fuel demand suggests that the rebound effect is likely to be smaller than 10 percent.11 Therefore improved efficiency causes the demand for oil to shift inward at any price. This policy does not alter the producer profit conditions but rather the equilibrium conditions that cumulative demand equals cumulative extraction from each pool. The transition time to the backstop is governed by when the backstop cost falls to the marginal value (the marginal cost of extraction plus any rent) of the last pool to be utilized. In regime a, better energy efficiency decreases demand for oil and lowers the price path uniformly. The intuition is that if the price path did not fall, the cumulative demand up until the switchover point would be less than the stock. Thus, to continue to exhaust the last resource pool, scarcity rents fall and delay the transition to the backstop. Emissions are postponed, but exhaustion still occurs. If the efficiency improvements are sufficiently large, we reach regime b, where the scarcity rent is driven to zero. Here further strengthening of the energy efficiency mandate has effects distinct from those of the previous policies. Since the transition to the backstop occurs when the backstop price falls to the marginal cost of extraction, the transition time is unaffected by additional efficiency improvements. This result lies in contrast to the tax and backstop policies, which accelerated the transition times in region b. In the case of the backstop, since demand is unaffected in region b, it is this accelerated transition that reduces cumulative

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consumption. Energy efficiency improvements nevertheless result in less cumulative usage of the oil stock by reducing the rate of utilization of fuel without altering the date when it is replaced. 11.3.4 Blend Mandate A blend mandate is similar to a renewable fuel standard or renewable portfolio standard: it would require a certain minimum percentage (β (t)) of a green substitute for oil in every unit of fuel. The policy combines some of the effects of the emissions tax—paid in the form of a cost premium for the mandated share of energy from the backstop source—and some of the effects of the energy efficiency policy, since fossil fuels are being displaced in a given level of energy services with the backstop. The blend mandate affects the producer profit conditions in the following way. An extractor must blend one barrel of fossil fuel with β /(1 − β ) barrels of the backstop, and then sell the resulting 1 /(1 − β ) barrels of the blended product at price pt per barrel of blend to obtain pt /(1 − βt ) per unit extracted. So, while the extractor is operating, the price must itself equal a weighted average of the two energy source costs (i.e., p(t) = (1 − β (t))(c + λ e rt ) + β (t)B(t ; z0 )). Since the backstop costs more, the mandate functions like a tax; since the backstop cost falls over time, the implicit tax also falls unless the mandate becomes sufficiently more stringent over time. As with the emissions tax policy, we consider a policy path that ensures that the implicit tax does not rise faster than the rate of interest. However, the blend mandate does not change the transition conditions like the emissions tax. For the last pool utilized, the transition to the backstop still occurs when the marginal cost of that source (including any rents) equals the backstop cost, because that is the same point at which the cost of the blend equals the cost of the backstop. Similarly, for the transition from one pool to the next, the cost of the blend with one source equals the cost of the blend with the next source when the costs of the two fossil fuel sources are the same. This point is important, since the transition conditions are then identical to those of the energy efficiency mandate. The mandate also affects the equilibrium demand conditions, much like the energy efficiency mandate. At any given price only a fraction of the demand for barrels of blend is fulfilled by the fossil energy source, so qt = (1 − βt )D( pt , t); the other βt D( pt , t) units are provided by the backstop component of the blend. Thus larger blend requirements decrease demand for oil by displacing it.

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In regime a, larger blend requirements also raise the initial price, in the same way that an emissions tax does (when it does not rise faster than the interest rate). Also the price path cannot lie uniformly above the no-policy path, else cumulative extraction would be less than the stock. Hence the new price path must be tilted to cross the previous (less stringent) policy path. Consequently the switch to the backstop must occur later. As with the energy efficiency policy, emissions are postponed, but exhaustion still occurs. Once the blend mandate become stringent enough, we reach regime b, where the scarcity rent of the marginal pool is just driven to zero. Here the transition to the backstop occurs when the backstop marginal cost equals the marginal extraction cost of the last pool utilized, exactly as in the case of the energy efficiency policy. In regime b, as with the efficiency policy, further strengthening of the blend mandate reduces cumulative emissions without affecting the transition time. 11.3.5 Carbon Capture and Sequestration Carbon capture and sequestration (CCS) has been considered one of the few viable options for addressing the green paradox, since it reduces emissions even while oil continues to be extracted and consumed (Sinn 2008). We model CCS policy as a mandate that a share ( ρ ) of emissions from fuels be captured and stored. The actual emissions from fuel combustion need not be captured and sequestered, a particularly unrealistic idea for transportation fuels; rather, we assume the mandate merely requires an equivalent amount of emissions to be sequestered. Compliance could be achieved either directly (e.g., by capturing emissions from oil sands upgrading) or indirectly by purchasing offsets or CCS credits (e.g., for the capture of emissions from coal-fired or gasfired electricity generation or even aforestation credits). Let us assume that CCS costs κ per unit, so per unit fuel costs are then c + (κρ)μ + λ e rt. The mandate thus has the exact same effect on the extraction path, price path, and transition times as a carbon tax of level τ = κρ . The two policies would not generate the same cumulative emissions, however. The CCS policy would indeed generate 1 − ρ times the cumulative emissions. 11.4

Parameterizing the Model

Estimates of oil reserves and costs vary widely. The Energy Information Administration (EIA) currently estimates global proven reserves to be

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about 1,200 billion barrels (including conventional and some unconventional, like Canadian oil sands). Kharecha and Hansen (2008) report reserves estimates in GtC, which if converted to billion barrels of oil equivalent (BBOE) range from 1,000 to 2,100 BBOE of conventional oil and 1,300 to 8,500 BBOE of unconventional oil. Aguilera et al. (2009) include projections of future reserve growth, leading to estimates of conventional oil reserves of 6,000 to 7,000 billion barrels available at prices as low as $5 a barrel, heavy oil reserves of 4,000 billion barrels at $15 per BOE, oil sands reserves of 5,000 billion barrels at $25 per BOE, and up to 14,000 billion barrels of oil shale that could be tapped at $35 per BOE. For our purposes, we draw rough estimates from International Energy Agency (IEA) (2010), which gives a range of production costs and available reserves by oil type. Our specific reserves and cost assumptions are given in table 11.1. CO2 emissions (right column) are based on the US Environmental Protection Agency’s conversion rate—that a barrel of oil contributes 0.43 tons12 of CO2—and adjusted for different unconventional sources’ larger emissions factors relative to conventional oil.13 We assume that the backstop fuels are nonemitting.14 Although the cost estimates are based on biofuels, future backstops could include hydrogen or clean electricity for plug-in vehicles, among other possibilities15 The assumed initial backstop marginal cost for biofuels is in line with the IEA estimates: sugarcane ethanol and other conventional biofuels are currently cheaper, but cellulosic ethanol and biodiesel— second-generation fuels that have greater potential for the larger scale supplies needed to function as backstop technologies—have higher costs.16 Here we assume that backstop costs start at $100 and will ultimately asymptote to $10 (i.e., be lower than conventional oil in the far Table 11.1 Reserves and cost assumptions

Oil reserve source MENA conventional Other conventional EOR, deep water Heavy oil, oil sands Oil shale Biofuels, backstop technology

Available reserves (BBOE)

Production cost

Relative emissions factor

CO2 (Gt)

900 940 740 1,780 880 Unlimited

$17 $25 $50 $60 $70 $100

1 1 1.105 1.27 2 0

387 404 352 972 757 0

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future), following a modest no-intervention cost reduction rate of 0.25 percent per year of the excess over the long-run cost (z = 0.0025). We assume a real interest rate of r = 2 percent. Combined, these cost assumptions ensure that without policy interventions, all oil resources will be fully exhausted by the end of the century. CCS cost estimates vary widely, according to the source of the carbon stream being captured (coal-fired power plants being cheaper than industrial sources), the transportation costs, and the sink being used (geological sequestration being cheaper than ocean sequestration or mineral carbonization), as well as monitoring and verification costs (IPCC 2010). CCS from oil sands upgrading is likely to be on the costlier end; furthermore it is limited to the energy used for upgrading, so a mandate of any larger magnitude would require purchasing sequestration credits from other sources. For our purposes we assume a constant and fixed cost of $100 per ton sequestered, which falls within the admittedly large range of estimates. For the demand side of the simulation model, we parameterize our linear demand function of D( p(t), t) = y0 e gt − mp(t). According to EIA, global annual oil consumption has been roughly 86 million barrels per day in recent years, or an annual consumption of 31.4 billion barrels.17 Meanwhile EIA’s International Energy Outlook 2010 projects an increase in global demand (primarily from developing countries) of 49 percent from 2007 to 2035, or about 1.45 percent per year. Thus g = 0.0145. We assume an effective elasticity of –0.25. This value roughly corresponds to the median estimate of a global oil demand elasticity from Kilian and Murphy (2013). On the one hand, earlier estimates of the price elasticity of demand for gasoline (primarily in the United States) find short-term demand elasticities of about –0.25 and long-run elasticities of about –0.6 (Espey 1996; Goodwin et al. 2004). On the other hand, Cooper (2003) and Dargay and Gately (2010) find much lower price elasticities of demand (–0.15 and smaller) when considering a broader array of countries, particularly non-OECD countries, and more recent time periods. However, Kilian and Murphy (2013) warn that most studies of such elasticities using dynamic models have been econometrically flawed by not accounting for price endogeneity. We position the initial demand curve such that (1) it passes through the quantity 31.4 BBOE, (2) it has a point elasticity at that quantity of –0.25, and (3) the initial price on the equilibrium price path in the base (no policy) case induces a quantity demanded of 31.4 BBOE. This solution implies a slope m = 0.1914 and an initial choke price y0/m = 205.

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$BOE 80

60

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Other conventional

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Figure 11.1 No policy price path with five pools

The simple Hotelling model predicts that the competitive market price starts at $41 per barrel. It does not explain the simultaneous exploitation of high-cost resources alongside low-cost ones or current $75 per barrel prices.18 However, our modest additions make the model more realistic while still allowing for green paradoxes as conventionally modeled. Figure 11.1 displays the no-policy price path indicated by the fivepool model. We see that differentiating among more pools leads to a smoother price path. Demand growth outpaces price growth, so corresponding consumption rises smoothly over time and fossil fuels are exhausted after 83 years. 11.5

Comparing Trade-offs in Emissions and Backstop Transition

The time required to transition to the backstop while achieving a given cumulative emissions target is one indicator of the speed of emissions along the equilibrium path. Within any regime a, it is straightforward

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to show that the backstop policy accelerates the transition, the emissions tax policy may slow it somewhat, and the two conservation policies (the energy efficiency and blend mandates) necessarily delay the transition. Within any regime b, we have seen that both the backstop and emissions tax policies accelerate the transition but the conservation policies have no effect on the transition time. Fischer and Salant (2012) show that a clear ranking of these four policies can be derived analytically for a given level of cumulative extraction: the timing of the transition to the backstop fuel is soonest for the backstop policy, then the tax policy, and last for the conservation policies, which have identical effects on that timing. The question is how significant these differences might be in practice. Figure 11.2 plots the relationship between cumulative emissions and the length of time to switch to the backstop for the five-pool model. The emissions tax is time varying, rising at the interest rate.19 The transition trade-off curves for the conservation policies are independent of the policy growth path. (Recall that in regime b the transition time is driven solely by the baseline backstop cost function and the marginal extraction costs for each pool.) Notice that the difference between the backstop and emissions tax policies is rather small. Indeed average emissions are so similar that an appreciable difference in the present value of damages under these two policies appears highly unlikely. On the one hand, the difference in transition times between these policies and the CCS policy is small for modest emissions reductions, growing to a decade or more for larger reduction targets. On the other hand, the energy efficiency and blend mandate policies greatly delay the transition to the backstop, on the order of centuries. For more stringent targets, the extraction horizon is extended well beyond the current “fossil era,” and the associated change in emissions patterns over time could significantly influence the present value of damages from emissions. The fundamental problem of the green paradox is the acceleration of consumption that arises from falling scarcity rents. The transition trade-off curves compare average annual emissions during the period of exploitation, a measure of the speed of emissions along the path. Although this measure takes into account the effects of scarcity rent equilibration, it also includes policy-induced changes in the emissions path that would occur in the absence of rent adjustments. Indeed, since all policies reduce scarcity rents, all policies also suffer from a weak version of the green paradox, as initial emissions are higher than they

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3,000

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would be if rents did not adjust. The difference is, for all but the green backstop policy, the direct effects of the climate policies on consumption outweigh the effects of the rent adjustment. 11.6

Comparing Intertemporal Leakage Rates

Next we isolate the effects of intertemporal leakage—that is, the change in emissions that can be attributed to market adjustments in scarcity rents. Climate policy models often omit suppliers’ responses to anticipated changes in future fossil fuel demand; intertemporal leakage rates indicate whether such models overestimate the policy effects and underestimate the costs of reaching emissions targets. To measure each policy’s susceptibility to intertemporal leakage, we first calculate the emissions reductions that would occur in the absence of rent adjustment (i.e., holding the shadow values of the pools fixed). Leakage is defined as the extra emissions that occur after rents adjust compared with their level in the absence of a rent adjustment. The leakage rate under a given policy is the leakage as a percentage of the emissions reduction that the policy would induce in the absence of rent adjustment.20 (This definition is intended to mimic the common definition of spatial emissions leakage.) Thus, if a model that takes no account of the change in scarcity rents predicts that policy i will cut emissions by 20 tons and the simulated leakage rate for that policy is 60 percent, then after scarcity rents equilibrate, we predict the policy will cut emissions by only 8 tons (40 percent of 20). Figure 11.3 illustrates the extent of intertemporal leakage for the case of the (optimal) emissions tax in the five-pool model. The solid line shows the predicted cumulative emissions reductions induced by the policy after rents re-equilibrate. The dashed line indicates the consequences of the policy if rents remain fixed at the no-policy level (no intertemporal leakage). Leakage is the vertical difference between the lines. The leakage rate divides that amount by the height of the dashed line. Leakage increases in the a regimes and decreases in the b regimes. 11.6.1 Leakage with Cumulative Emissions Targets From these simulations we calculate the average intertemporal leakage rates associated with a given level of cumulative emissions. We do this for a range of time-varying and stationary policies. Although policy paths have little influence on the backstop transition trade-offs, they can affect the leakage trade-offs, particularly for the conservation

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Reductions (% of baseline emissions)

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Leakage 40

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0 0

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Initial emissions tax ($ per ton CO2) With rent adjustment

No rent adjustment

Figure 11.3 Emissions reductions as function of policy stringency with and without rent adjustment

policies. Figures 11.4 and 11.5 depict these leakage trade-off curves. The horizontal segments in the diagram arise because an interval of stringencies inducing the same cumulative emissions will induce different leakage rates. Policies to the left have less leakage, on average. Initially, leakage is 100 percent for all policies, but the rate declines as cumulative emissions fall. Figure 11.4 compares the backstop policy with an emissions tax that rises at the rate of interest and an emissions tax that rises more slowly, at the rate of demand growth. The emissions tax policies have less leakage than the backstop policy initially, in part because of their ability to differentiate among higher emissions intensity pools. However, for more dramatic reductions, the backstop has lower leakage rates than the tax policies. Meanwhile the slower tax path that is associated with somewhat more delay in the backstop transition has a consistently (though not greatly) higher leakage rate than the emissions tax rising at the rate of interest. Particularly after the extraction of the highest cost, highest emitting pool is eliminated, the leakage rate differences among all three fall within 5 percentage points of each other. Figure 11.5 compares the leakage trade-off curves for two variants of each conservation policy, with the backstop policy as a reference.

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Cumulative emissions (% of baseline)

100

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40

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0 0%

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Leakage (% of reductions with no rent adjustment) Figure 11.4 Leakage trade-off curves for backstop and tax policies

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With “EE fixed” and “blend fixed,” we simulate the policies as described in the one-pool model: the mandates require an immediate and permanent improvement in energy efficiency (or similarly, a blend ratio). These policies have nearly identical effects and are associated with consistently higher leakage rates than all other policies. With “EE growing” and “blend growing,” we assume that the mandates require an annual rate of improvement in efficiency or the backstop blend.21 We find that delay in raising the stringency of the conservation policies causes less intertemporal leakage. Although their leakage rates are still higher at more modest targets, they outperform the backstop and emissions tax at more ambitious reduction targets. Although CCS does induce more emissions reductions than the equivalent carbon tax, its susceptibility to intertemporal leakage is not that different in magnitude. It performed better in the a regimes, however, particularly between oil shale and oil sands. 11.6.2 Leakage with Present-Value Emissions Targets Ideally a policy would minimize the present value of emissions damages, net of the costs. Our focus on cumulative emissions targets

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is reasonable in this context if marginal damages are rising at the discount rate, or more generally if one is concerned about potential threshold effects and long lags in climatic response that require setting a cumulative emissions budget for the time period. Of course, perceptions vary as to the validity of these assumptions. The evolution of the social cost of carbon over time, and therefore the benefits or costs of delay, is uncertain. The more thorough of recent attempts to quantify it find the social cost of carbon to be essentially proportional to gross domestic product (Golosov et al. 2014). Although real GDP growth may typically be lower than real interest rates, it can be higher than the social discount rates used in climate valuation. Nordhaus (2007), using a subjective discount rate of 1.5 percent, calculates a social cost of carbon that rises at 5.5 percent per year (i.e., faster than the discount rate, meaning delay would be costly). Anthoff et al. (2011) find that the social cost of carbon increases by 1.3 to 3.9 percent per year, with a central estimate of 2.2 percent (i.e., similar to the discount rate). Most, if not all, the green paradox literature assumes that the social cost of carbon rises more slowly than the discount rate, implying that accelerating emissions can increase the present value of damages. Hoel (2012) gives assumptions that lead to the social cost of carbon being basically constant over time. In this spirit we also consider the same policy effects when the target is expressed in discounted cumulative emissions.22 Comparing present-value emissions against cumulative emissions, we find, as shown in figure 11.7, that the CO2 tax and green backstop policy have nearly identical trade-offs, but the conservation policies have much lower present-value emissions. In particular, the conservation policies that mandate immediate and permanent improvements have the lowest present-value emissions for a given cumulative emissions target. Meanwhile the CCS mandate performs nearly as well as these conservation policies by this metric. As for leakage rates, the qualitative results are similar, as shown in figure 11.8. The relative backstop and emissions tax leakage rates change little, but the conservation policies perform better. (To avoid clutter, we leave out the blend mandates, which perform similarly to the energy efficiency mandates.) In particular, the fixed mandates have the lowest present-value leakage rates when the policies are relatively modest and crowd out only oil shale. Leakage rates from CCS policies lie in between.

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11.6.3 Sensitivity Some additional parameters in the numerical simulations merit sensitivity analysis. First, we lower the elasticity of demand for fuel from –0.25 to –0.1, closer to short-run elasticity estimates. This slightly delays the initial switchover times and compresses the difference between the backstop and tax policies in the transition trade-off curves. However, the leakage rate trade-off curves are indistinguishable from the higher elasticity baseline. Second, we vary our assumptions about resource scarcity, as we noted a wide variation in estimates of oil reserves. If all reserves prove to be 50 percent larger than in the baseline, initial prices would be lower, transition horizons longer, and leakage rates smaller for modest targets, although the qualitative results are unchanged. 11.7

Limitations for Welfare Analysis

Without knowing the costs of hastening the competitiveness of alternative fuels and permanently improving energy efficiency, we cannot conduct meaningful welfare or cost-effectiveness analyses. However, a rough estimate of the policies needed to meet mitigation goals suggests the magnitudes of the potential costs. Table 11.2 compares the levels of policy stringency required to achieve given levels of extraction in the simulation model, depending on whether one accounts for intertemporal leakage. For example, when rents adjust, avoiding emissions from oil sand and shale reserve extraction requires (1) a $17/ton CO2 tax (within the range of the European Union’s Emission Trading Scheme allowance prices over the past year), (2) an increase in the rate of cost reductions in cellulosic biofuels by 1 percent per year, (3) a 2.9 percent annual improvement in energy efficiency, or (3) a 2.7 percent annual reduction in the share of fossil sources in the fuel blend. Ignoring leakage, one would expect these stringency levels to be one-third to one-half lower. Even with reliable cost estimates, however, a proper welfare analysis would require more certainty about the shape of the damage function, among other things. 11.8

Conclusion

Because GHGs decay quite slowly, stabilizing their atmostpheric concentrations requires something akin to a limit on cumulative emissions

Source: Fischer and Salant (2012).

No policy No shale rents Shale unexploited No oil sands rents Oil sands Unexploited No deepwater rents Deepwater Unexploited No conventional Rents Conventional Unexploited No MENA rents MENA unexploited

If no leakage $0 $0 $1 $1 $9 $9 $20 $20 $57 $57 $137

With leakage $0 $3 $4 $10 $17 $27 $36 $63 $94 $105 $193

With leakage 0.25% 0.50% 0.55% 0.81% 1.12% 1.57% 2.00% 4.66% 8.28% 12.10% Infinite

If no leakage 0.25% 0.25% 0.31% 0.31% 0.64% 0.64% 1.00% 1.00% 2.85% 2.85% Infinite

CO2 tax (rising at discount rate)

Backstop (annual improvement rate)

Table 11.2 Levels of policy stringency required

With leakage 0.0% 1.6% 1.9% 2.1% 2.9% 2.9% 3.5% 3.5% 5.7% 5.7% Infinite

If no leakage 0.0% 0.0% 0.7% 0.7% 2.1% 2.1% 3.0% 3.0% 5.5% 5.5% Infinite

Energy efficiency (annual improvement rate) With leakage 0.0% 1.5% 1.8% 2.1% 2.7% 2.7% 3.3% 3.3% 5.0% 5.0% Infinite

If no leakage 0.0% 0.0% 0.6% 0.6% 1.8% 1.8% 2.6% 2.6% 4.7% 4.7% Infinite

Blend mandate (annual reduction in fossil share)

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over the next century. One “green paradox” is that efforts to reduce GHG emissions may be undone in part or in whole by emissions leakage, not only across countries but also over time, given that the major sources of GHGs are exhaustible resources. Not only may emissions leak over time, but some efforts to spur a transition to clean energy may accelerate emissions in such a way that the present value of the damages of climate change may actually increase. Our study reinforces earlier findings that, unlike demand-reducing strategies, accelerating cost reductions in a clean backstop technology tends also to accelerate extraction of nonrenewable resources. However, in a reasonably parameterized model of oil markets, the differences in outcomes between a green technology policy and an emissions tax seem unimportant: given a cumulative emissions target, the two policies exhibit very similar backstop transition times and present value of emissions. By contrast, energy efficiency improvements and clean energy blend mandates do not accelerate adoption of the backstop technology and can significantly delay both the transition and emissions. In our simulations the delay that occurs as a result of rent adjustments is of a magnitude that the present value of emissions is likely to be affected if the social cost of carbon rises slowly. Another concern is the degree of intertemporal leakage—that is, the magnitude of the additional emissions reductions under each policy that would arise if rents did not re-equilibrate but instead remained fixed at their no-policy levels. This alternative perspective has practical value, since one can adjust the forecasts of the predicted reduction in cumulative emissions made by models that do not account for adjustments in scarcity rents. In the simulations, on the one hand, we see that all policies suffer from intertemporal leakage, and the rankings can be quite different from those of the transition trade-offs. For example, in comparing the intertemporal leakage rates for cumulative emissions, we find the backstop policy can actually outperform other policy alternatives, including an emissions tax. On the other hand, conservation policies can suffer from considerable leakage rates, particularly if they mandate stringent action early on. Although all policies considered can reduce cumulative emissions under the right circumstances, the carbon capture and storage mandate is the only policy that does it in all circumstances—yet it, too, entails intertemporal leakage. For all the policies, leakage rates are highest when policies are weak, but as reduction targets become more ambitious, leakage rates tend to fall.

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We made some important simplifications, noted above, in our analysis. Even with the given policies, additional assumptions would be needed to address the relative cost effectiveness of meeting a given cumulative emissions target. We have not explicitly represented the costs of energy efficiency improvements, backstop technology policy, or carbon capture and storage. For example, although an energy efficiency policy may look attractive because it delays emissions, ultimately it will also depend on the costs of achieving large-scale efficiency improvements; indeed, in our simple simulation model, rapid reductions in energy demand are required to achieve anything more than modest improvements. An emissions tax would be efficient in the absence of market failures, but a comparison of its costs and benefits with those of the other policies requires taking those market failures and barriers into account (e.g., see Fischer and Newell 2008). Nor in our parsing of the policy effects did we allow for emissions prices or energy price changes to induce investments in backstop or energy efficiency improvements or in carbon capture. In reality, climate policy will be a portfolio of options and responses to this long-term problem. The research on intertemporal leakage indicates that this portfolio may well need to be ambitious to reach emissions goals, but the efforts are not likely to be undone to the extent indicated by earlier studies. Notes This chapter is a nontechnical summary based on Fischer and Salant (2012), “Alternative Climate Policies and Intertemporal Emissions Leakage,” currently under submission. Fischer would like to acknowledge support from the Mistra Foundation ENTWINED program and the Norwegian Research Council Clean Energy for the Future Program (RENERGI). Fischer and Salant are grateful to the University of California Center for Energy and Environmental Economics (UCE3). Additional thanks to CESifo and workshop participants. 1. Stabilization of atmospheric concentrations would precede climate stabilization. According to IPCC (2007, p. 67), “For most stabilisation levels global average temperature is approaching the equilibrium level over a few centuries.” 2. See discussion in section 11.6.2. 3. In a recent review of studies of the atmospheric lifetime of CO2, Archer et al. (2009) find a “strong consensus” across models of global carbon cycling that “the climate perturbations from fossil fuel–CO2 release extend hundreds of thousands of years into the future.” They further cite evidence that “the radiative impact of a kilogram of CO2 is nearly independent of whether that kilogram is released early or late in the fossil fuel era.” 4. For example, the above-mentioned CGE models focus attention on spatial leakage but ignore intertemporal leakage. Most integrated assessment models, including DICE, have

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nonscarce carbon fuels; exceptions are RICE-99 (Nordhaus and Boyer 2000), which looks only at carbon tax paths, and MERGE (Manne et al. 1995), which fixes production as a share of reserves and constrains the rate of new discoveries. 5. We focus on emissions from the extraction and use of oil because this fuel arguably has the greatest potential for the rent adjustments that lead to intertemporal leakage. GHG emissions from coal are potentially much larger, but the resource is also considered much less (or non-) scarce. 6. Since the cost of implementing some of these (e.g., the first and third policies) is unknown, as is the damage resulting from a given path of cumulative emissions, assessing the welfare consequences of the different policies is impossible. 7. Fischer and Salant (2012) prove that these rankings hold analytically. Furthermore the same rankings persist even if we assume that each conservation policy changes over time or that the emissions tax rises over time at the rate of interest. 8. This assumption abstracts from several issues with transport fuel substitutes. Biofuels, hydrogen, and electrification are not necessarily carbon free; furthermore marginal costs may increase with production levels at a large scale because of, for example, land pressure. 9. The logic is as follows: if the initial price on the new path were unchanged or smaller, the scarcity rent would have to be strictly smaller, but then the remainder of the path would lie strictly below the old path and cumulative demand would exceed the unchanged stock. The initial price therefore must be higher on the new path. To induce the same cumulative demand, the new path must cross the old one from above and the backstop will enter at a later date (larger xB). 10. Fischer and Salant (2012) define ϕ t = 1 /(1 − ft ) as energy services per barrel of oil equivalent. 11. Kilian and Murphy (2013), Espey (1996), Goodwin et al. (2004), Hughes et al. (2008), and Small and van Dender (2007). 12. http://www.epa.gov/greenpower/pubs/calcmeth.htm. 13. See table 3-2 of the California technical analysis of the low-carbon fuel standard, http://www.energy.ca.gov/low_carbon_fuel_standard/UC_LCFS_study_Part_1 -FINAL.pdf. 14. We acknowledge that the actual emissions factors for biofuels, particularly those associated with land-use changes, are controversial. 15. Of course, synthetic fuels derived from coal or natural gas could also be substitutes, but we assume fossil-based backstops are precluded. 16. In 2007, the US Department of Agriculture estimated cellulosic ethanol production costs at $2.65 per gallon, compared with $1.65 for corn-based ethanol. 17. See http://www.eia.gov/cfapps/ipdbproject/iedindex3.cfm?tid=5&pid=54&aid=2. 18. Gaudet et al. (2001) show how to generalize the Hotelling model to the case where the location of demanders (as well as reserve deposits) is exogenously distributed. In such a model, resources pools are sometimes accessed simultaneously by spatially distributed users even though the pools differ in extraction costs. Despite its greater realism, we declined to use this spatial Hotelling model in our preliminary investigation, since

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the nonspatial model has been used by all the other contributors to the green paradox literature. 19. We also simulated a tax that rises at the rate of demand growth, but the difference was insignificant. For a given cumulative emissions, the slower growth tax path delays the switchover by less than 2 percent. We did not consider a fixed tax, since with the parameterized demand growth, the tax levels required to meet reduction targets would choke off demand in the early years. 20. Technically, Li = (Ei − EiNL )/(ENP − EiNL )), where EiNL are cumulative emissions under policy i without rent adjustment, Ei are emissions with rent adjustment, and ENP are emissions in the absence of any climate policy. 21. Specifically, βt = 1 − e − bt and ft = 1 − e − ht. 22. We define present-value emissions as the discounted flow of emissions, that is, xB

PVE = ∫ M(t)e − rt dt. The present-value leakage rate is then defined analogously to the 0

cumulative emissions intertemporal leakage rate: PVLi = (PVEi − PVEiNL )/(PVENP − PVEiNL ).

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Eichner, T., and R. Pethig. 2011. Carbon leakage, the green paradox and perfect future markets. International Economic Review 52 (3): 767–805. Espey, M. 1996. Explaining the variation in elasticity estimates of gasoline demand in the United States: A meta-analysis. Energy Journal 17 (3): 49–60. Fischer, C., and R. G. Newell. 2008. Environmental and technology policies for climate mitigation. Journal of Environmental Economics and Management 55 (2): 142–62. Fischer, C., and S. Salant. 2012. Alternative climate policies and intertemporal emissions leakage. Discussion paper 12–16. Resources for the Future, Washington, DC. Fischer, C., and S. Salant. 2014. Limits on limiting greenhouse gases: Intertemporal leakage, spatial leakage, and negative leakage. Discussion paper 14-09. Resources for the Future, Washington, DC. Gaudet, G., M. Moreaux, and S. Salant. 2001. Intertemporal depletion of resource sites by spatially distributed users. American Economic Review 91 (4): 1149–59. Gerlagh, R. 2011. Too much oil. CESifo Economic Studies 57 (1): 79–102. Golosov, M., J. Hassler, P. Krusell, and A. Tsyvinski. 2014. Optimal taxes on fossil fuel in general equilibrium. Econometrica 82 (1): 41–88. Goodwin, P., J. Dargay, and M. Hanly. 2004. Elasticities of road traffic and fuel consumption with respect to price and income: A review. Transport Reviews 24 (3): 275–92. Grafton, R. Q., T. Kompas, and N. V. Long. 2012. Substitution between biofuels and fossil fuels: Is there a green paradox? Journal of Environmental Economics and Management 64 (3): 328–41. Herfindahl, O. C. 1967. Depletion and economic theory. In M. Gaffiney, ed., Extractive Resources and Taxation. Madison: University of Wisconsin Press, 68–90. Hoel, M. 2011. The supply side of CO2 with country heterogeneity. Scandinavian Journal of Economics 113 (4): 846–65. Hoel, M. 2012. Carbon taxes and the green paradox. In R. Hahn and A. Ulph, eds., Climate change and common sense: Essays in honor of Tom Schelling. New York: Oxford University Press. Hughes, J. E., C. R. Knittel, and D. Sperling. 2008. Evidence of a shift in the short-run price elasticity of gasoline demand. Energy Journal 29 (1): 93–114. Intergovernmental Panel on Climate Change (IPCC). 2007. Climate change 2007: Synthesis report. In R. K. Pachauri and A. Reisinger, eds., Contribution of Working Groups I, II and III to the Fourth Assessment Report of the Intergovernmental Panel on Climate Change. Core Writing Team. Geneva: IPCC. Available at: https://www.ipcc.ch/pdf/assessment-report/ ar4/syr/ar4_syr.pdf. Intergovernmental Panel on Climate Change (IPCC). 2010. IPCC special report: Carbon dioxide capture and storage (technical summary). Available at: http://www.ipcc.ch/ pdf/special-reports/srccs/srccs_technicalsummary.pdf. International Energy Agency (IEA). 2010. Resources to reserves. Paris: IEA. Kharecha, P. A., and J. E. Hansen. 2008. Implications of ‘‘peak oil’’ for atmospheric CO2 and climate. Global Biogeochemical Cycles 22: 3012.

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Kilian, L., and D. P. Murphy. 2013. The role of inventories and speculative trading in the global market for crude oil. Journal of Applied Econometrics. DOI: 10.1002/jae.2322 (Article published online April 10, 2013). Manne, A., R. Mendelsohn, and R. Richels. 1995. MERGE: A model for evaluating regional and global effects of GHG reduction policies. Energy Policy 23 (1): 17–34. Mattoo, A., A. Subramanian, D. van der Mensbrugghe, and J. He. 2009. Reconciling climate change and trade policy. Policy research working paper 5123. World Bank Development Research Group Trade and Integration Team, Washington, DC. Nordhaus, W. D. 2007. To tax or not to tax: The case for a carbon tax. Review of Environmental Economics and Policy 1 (1): 26–44. Nordhaus, W. D., and J. Boyer. 2000. Warming the World: Economic Models of Global Warming. Cambridge: MIT Press. Sinn, H.-W. 2008. Public policies against global warming: A supply side approach. International Tax and Public Finance 15: 360–94. Small, K., and K. van Dender. 2007. Fuel efficiency and motor vehicle travel: The declining rebound effect. Energy Journal (Cambridge, MA) 28: 25–51. Strand, J. 2007. Technology treaties and fossil fuels extraction. Energy Journal (Cambridge, MA) 28: 129–42. van der Ploeg, F., and C. Withagen. 2012. Journal of Environmental Economics and Management 64 (3): 342–63 (November special issue: 2010 Monte Verita Conference on Sustainable Resource Use and Economic Dynamics (SURED)). Available at: http://www.eaere .org/section-events/monte-verit%C3%A0-conference-sustainable-resource-use-and -economic-dynamics-sured-2014. Winter, R. A. Forthcoming. Innovation and the dynamics of global warming.

Contributors

Julien Daubanes Center of Economic Research at ETH Zurich Corrado Di Maria University of East Anglia Carolyn Fischer Resources for the Future, Gothenburg University, and CESifo Research Network Florian Habermacher University of St. Gallen and CESifo Research Network Michael Hoel University of Oslo Darko Jus University of Munich Gebhard Kirchgassner Institute for Advanced Study, Berlin, University of St. Gallen: SIAW, Leopoldina, CESifo and CREMA Ian Lange Colorado School of Mines Pierre Lasserre Université du Québec à Montréal, CIRANO, and CIREQ

288

Volker Meier University of Munich Karen Pittel University of Munich and Ifo Institute Stephen Salant University of Michigan and Resources for the Future Frank Stähler University of Tübingen and CESifo Research Network Gerard van der Meijden VU University Amsterdam Rick van der Ploeg Oxford University Edwin van der Werf Wageningen University Ngo Van Long CESifo and Department of Economics, McGill University Ralph A. Winter Sauder School of Business, UBC Cees Withagen VU University and Tilburg University

Contributors

Index

Abatement, carbon, 137–38 Aggregate resource extraction, 162–64 Alberini, A., 238 Allen, M. R., 26 Anthoff, D., 276 Augmented marginal extraction cost, 52 Aviation and biofuels, 88 Backstop technology. See also General equilibrium model of resource extraction baseline scenario and, 181–82 climate policy and, 214–16 comparing trade-offs in emissions and transition to, 269–72 cost drop, 127–28, 262–63 Cournot path and, 212 cumulative emissions targets and, 272–75 dirty, 33–35, 36, 207n2 end of total extraction and cost of, 261 energy price index and, 112 future regime change in baseline scenario and, 180–84 innovation isoclines and, 100–105 introducing tax before, 184–86 partial equilibrium analysis, 90 sector, DHSS model, 94–95 stochastic introduction of future scheme and, 194–95 supply and price, 207n10 sustainability and, 87–89 technological improvements, 62, 207n4 Baseline scenario (BS) alternative future schemes, 187–94 endogenous future regime change and, 195–96 future regime change in, 180–84

introducing tax before backstop in, 184–86 model, 178–80 stochastic introduction of future scheme and, 194–95 Bauer, N., 5 Bertram, C., 5 Binding exhaustibility constraint, 47 Biofuels in aviation, 88 as closer substitute for petroleum, 60 dirty backstop and, 33, 35, 36 increased marginal cost of producing, 60–61 model of imperfect substitutability between fossil fuels and, 64–69 production, 82n4 subsidies, 63, 82n5–6, 259 tax exemptions for, 82n5 Blend mandate, 265–66 Bohm, P., 21 Borenstein, S., 241 Burness, H. S., 44 Bushnell, J., 239 Business-as-usual (BAU), 178, 206 Busse, M., 241 “Cake-eating problem,” 90 Calibration, DHSS model, 101–102 Cap-and-trade scheme, 183–84 Capital gains taxes, 121 Carbon abatement, 137–38 Carbon capture and storage (CCS), 22, 39n1, 137–38, 268, 276 backstop transition and, 270 sequestration and, 266

290

Carbon emissions abatement innovation, 137–38 atmospheric lifetime, 281n3 clean energy innovation with increasing energy efficiency impact on, 143 comparing trade-offs in backstop transition and, 269–72 consequences of delayed climate policy, 218–21 convex damages, 205 dirty backstop and, 33–35, 36 dynamic exhaustible fuel market model and, 197–205 from extraction process, 30–33 green paradox and, 3–4 increased energy efficiency impact on, 139–42 intertemporal leakage, 256–58, 272–78 long-term warming effect of, 177 marginal social cost of, 135–36, 148n3 rising consumer and falling industrial, 2–3 sources of, 133–37, 148n8 targets, cumulative, 272–75 targets, present-value, 275–77, 278 timing, 26 Carbon leakage, 21 Carbon taxes, 4–5, 22 alternative future schemes, 187–94 baseline scenario, 178–80 blend mandate and, 265–66 cap-and-trade scheme, 183–84 development of green paradox and, 174–75, 208n19 dynamic exhaustible fuel market model and, 197–205 effects of, 26–30, 207n2 endogenous future regime change and, 195–96 exemptions for biofuels, 82n5 exponential decay of pollution and, 61–62 faster extraction and low, 212–13 future change in baseline scenario and, 180–84 gradually increasing, 228 green paradox without, 126–28 impact of clean energy innovation on, 123 introduced before backstop, 184–86 multiple pools and, 263 optimal, 123, 125–26, 173

Index

rapidly increasing taxes and, 208n19 stochastic introduction of future scheme, 194–95 two period model, 175–76 zero tax forever, 39n5 Chakravorty, U., 259 Choke price for fossil fuels, 66–67, 74, 140–41 Clean Air Act Amendments of 1990, 225 Clean backstop technology, 207n2 Clean energy innovation, 122–24, 280 impact on global warming, given pattern of increasing energy efficiency, 143 prospects for, 258–60 uncertain, 128–33 Climate change agricultural processes and, 39n8 convex damages, 205 economic analysis of, 1 ex ante green paradox and, 122, 132–33 ex post strong green paradox and, 133 first-best optimum and, 124–26 greenhouse gases (GHGs) targets and, 255–56 green paradox of, 1–9 green paradox without carbon taxes, 126–28 impact of clean energy innovation with pattern of increasing energy efficiency on, 143 impact of increased combustion efficiency in producing energy, 138–39 impact of increased efficiency in use of energy, 139–42 innovation green paradox implications for policy on, 122–24 positive feedback effects and, 144–46 Closer substitution, 60 Club of Rome, 87 Coal, 6–9, 148n8, 149n11, 248–49n2 abundance of, 207n9 climate policy announcements and, 237–38 current emissions levels, 134–35 dirty backstop and, 33–35 price changes and use of, 239 Coal Question, The, 149n11 Cobb–Douglas production function, 61, 161

Index

Combustion efficiency, 138–39 Computable general equilibrium (CGE) models, 255, 281–82n4 Constant elasticity of substitution (CES), 90–92 Consumption growth, 207n8 backstop technology and, 109 exhaustion of resources and, 196 Convex damages, 205 Cooper, J. C. B., 268 Cournot path, 212, 222–23 Cramer ’s rule, 161 Cumulative emissions targets, 272–75, 283n19 Damages, convex, 205 Dargay, J. M., 239, 268 Dasgupta, P., 89, 123, 129, 194, 214 Dasgupta–Heal–Solow–Stiglitz (DHSS) model, 89, 90–91 backstop technology sector, 94–95 calibration, 101–102 dynamic system, 98–99, 113–15 energy generation sector, 93–94, 109 factor markets, 97 final good sector, 91–93 graphical apparatus, 102–105 households, 97–98 intermediate goods sector, 94 solving the, 98–100 steady state, 99–100, 115 transitional dynamics, 100–105 Daubanes, J., 51 Decay rate of pollution, 61–62 De Groot, H. L., 101 Delays, climate policy, 213–14, 222–23 consequences of, 218–21 Demand green paradox theories and, 225–26, 226–34, 246–47 in model of imperfect substitutability and, 65–66 responses to change in supply, 238–43 restrictions and imperfect climate policy, 243–46 -side climate policy, 21–23 size of green paradox and, 230–34 Di Maria, C., 216, 225, 229, 232, 243 Acid Rain Program and, 236, 237 on adverse response to anticipation of taxes, 62 on carbon cap, 215

291

on climate policy and coal, 237–38 on policies that generate green paradox, 227–28, 246 Direct effect and green paradox, 76–77 Dirty backstop, 33–35, 36, 207n2 Dynamic exhaustible fuel market model, 197–98 alternative setups, 202–205 interpretation of results, 198–200 main setup, 198 result main setup, 200–202 Dynamic system, DHSS model, 98–99, 113–15 Efficiency, fossil fuel use, 138–43, 264–65 Eichner, T., 151, 259 on adverse response to anticipation of taxes, 62 on demand elasticity, 231 on unilateral cap, 228, 229 Elasticity of substitution, 108–10 constant, 90–92 Emissions. See Carbon emissions Endogenizing reserves, 48–51 Endogenous future regime change, 195–96 Energy generation sector, DHSS model, 93–94, 109 Energy price index, 112 Equilibrium backstop technology effect on, 157, 262–63 derivation of, 37–39 extraction path, 23–26 first-best optimum and, 124–26 multiple pools, 260–62 nonexhaustion, 157–64 path phases, 69 supply theory and, 45 trade-offs between emissions and backstop transition, 269–72 uncertain innovation in clean energy technology and, 128–33 without carbon tax, 126–28 Ex ante green paradox, 122, 132–33 Ex ante long-run innovation green paradox, 146 Exhaustion time direct computation of, 77–79 effect of increase on substitutability on, 69–75 first-best optimum and, 124–26 general equilibrium model and, 164–67

292

Ex post strong green paradox, 133 Extraction costs, 6, 10–11, 157. See also General equilibrium model of resource extraction biofuel, 63 constant elasticity of substitution and, 90–91 Cournot path, 212, 222–23 demand-side climate policy and, 21–23 derivation of equilibrium for, 37–39 effects of carbon tax, 26–30 equilibrium extraction path and, 23–6 extraction speed and, 212–13 Hotelling rent, 129–30 impact of announcing climate policy on, 216–18 investment in capital stock and, 239–41 multiple pools of oil, 260–66 non-energy, 32–33, 39n6 pure intertemporal substitution effect and, 45 reserve-to-extraction rate and, 116–17 stock-dependent marginal, 63 strong green paradox and, 62–63, 174–75 taxes, 53n2 total quantity extracted and, 45–46 Extraction rates, 178–80 demand constraints and, 230–34, 247–48 investment in capital stock and, 239–41 policy stringency and, 278, 279 Factor markets, DHSS model, 97 Factors of production, necessary, 117–18n6 Farzin, Y. H., 62 Final good sector, DHSS model, 91–93 First-best optimum, green paradoxes at, 124–26 Fischer, C., 45, 46, 49, 259 Florax, R. J., 101 Flow budget constraint, 112–13 Foreign Policy, 258 Fossil fuels. See Nonrenewable fossil fuels Frame, D. J., 26 Gans, W., 238 Gasoline, 134–35, 238–39 Gately, D., 239, 268 Gaudet, G., 46, 49, 282n18

Index

General equilibrium model of resource extraction, 151–52, 167–68 exhaustion of the resource and, 164–67 nonexhaustion of the resource and, 157–64 two periods, 152–157, 169n4 Gerlagh, R., 151, 174, 227, 259 on adverse response to anticipation of taxes, 62 on cumulative extraction, 45, 53n4, 259 on weak green paradox, 22, 89, 122 Global warming. See Climate change Gordon, R. L., 45 Grafton, Q., 63, 151, 227, 232 on biofuel subsidies, 59, 259 on cumulative extraction, 45 Graphical apparatus, DHSS model, 102–105 Gray, L. C., 44 Green paradox. See also Baseline scenario (BS) backstop technologies and, 87–89, 109–10 baseline scenario, 178–97 climate policy and, 14–16, 173–74 defined, 43, 59, 225 definitions, 174 demand for resources and theories of, 226–34, 246–47 demand-side climate policy and, 22–23 dynamic exhaustible fuel market model and, 197–205 empirical evidence of, 234–38 ex ante, 122, 132–33 ex ante long-run innovation, 146 existence of real, 227–30 ex post, 133 facilitating, 178 at first-best optimum, 124–26 green policies and, 211–13 Hotelling rule and, 4, 23, 43–44 imperfect substitutability and, 11 increase in substitutability and, 75–79 innovation, 122–24, 146–48 and its determinants, 1–9 literature on, 13–14 long-term, 122–23 other sources of emissions and strong, 133–37 perverse consequences of policy and, 121–24 size and demand for resources, 230–34

Index

strong, 3, 13, 62, 122, 132–33, 174 as substitution between resource units extracted at different dates, 44 timing, announcement effects, and time consistency, 12–13 unilateral cap and, 228–29 van der Ploeg and Withagen on, 124–28 very strong, 174 weak, 3, 22, 30, 174, 259 under wide variety of circumstances, 59–60 without carbon tax, 126–28 Green welfare, 122, 125 when innovation is impossible, 131–32 Groth, C., 61 Habermacher, F., 181, 182, 186, 194, 195, 198 Hamilton–Jacobi–Bellman equation, 95–96 Hansen, J. E., 267 Harstad, B., 21, 37 Heal, G., 89, 194, 214, 229 Helm, D., 2 Herfindahl, O. C., 45 Hilaire, J., 5 Hoel, M., 21, 63, 151, 228, 259 on adverse response to anticipation of taxes, 62 on backstop technology innovation, 59, 165 on cumulative extraction, 45 on increasing extraction cost effect, 175 on relationship between carbon taxes and carbon extraction, 22 on resource exhaustion, 134 on social cost of carbon, 276 Hotelling rent, 129–30, 134–35 Hotelling rule, 4, 23, 43–44, 45, 257 supply of fossil fuels and, 67 synthetic supply model, 46–48 Households, DHSS model, 97–98 flow budget constraint, 112–13 Huntingford, C., 26 Imperfect climate policy and demandside restrictions, 243–46 Imperfect substitutability, 64–69. See also Dasgupta–Heal–Solow–Stiglitz (DHSS) model backstop technology and, 105–10

293

Indirect effect and green paradox, 76–77 Innovation aggregate resource extraction increases with, 162–64 in carbon abatement, 137–38 clean energy, 122–24, 128–33, 258–60 in efficiency of fossil fuel energy use, 138–43 green paradox, 122–24, 146–48 isoclines, 100–101, 102–105, 116 rate, 95, 108 substitutability and, 11–12 uncertainty in clean energy technology, 128–33 Interaction effect, 137–38 Interest rate and exhaustion, 158–59, 163, 164–65, 208n15 Intergovernmental Panel on Climate Change (IPCC), 255, 256 Intermediate goods sector, DHSS model, 94 International Energy Agency (IEA), 198, 267 Intertemporal leakage, 256–58 rate comparison, 272–78 Intertemporal substitution effect, 46, 52 Investment in capital stock, 239–41 Isoclines, 100–101, 116 graphical apparatus, 102–105 reserve to extraction rate, 116–17 Jevons, W. S., 139, 149n11 Jones, C. D., 26 Jus, D., 229 Kellogg, R., 241 Kennedy, P., 216 Kharecha, P. A., 267 Kilian, L., 268 Kirchgässner, G., 181, 182, 186, 194, 195 Koch, K.-J., 100 Koetse, M. J., 101 Kompas, T., 45, 63, 151, 227, 232 on biofuel subsidies, 59, 259 Kverndokk, S., 62, 134 Kyoto Protocol, 175, 177, 196–97, 213–14, 229, 255 Lange, I., 225, 237–38, 243, 246 Lasserre, P., 46, 49, 51 Laxminarayan, R., 46, 49 Leach, A., 259

294

Leakage intertemporal, 256–58, 272–78 spatial, 255–56 Lijesen, M. G., 238 Linear quadratic case, 71–75 Liski, M., 62 Lomborg, B., 258 Long, N. V., 44, 45, 63, 90, 227, 232 on biofuel subsidies, 59, 259 on effects of taxation on resource extraction, 22 on interest rates, 60 on intertemporal adjustment, 151 Long-term green paradox, 122–23 Lowe, J. A., 26 Low Emission Zones (LEZ), 242 Marginal social cost of emissions, 135–36, 148n3 Meier, V., 229 Meinshausen, M., 26 Meinshausen, N., 26 Michielsen, T., 5, 89 Monopoly, 79–80 Moreaux, M., 259 Multiple pools, 260–66 Murphy, D. P., 268 National Ambient Air Quality standards, 242 Natural gas, 238–39, 243–44, 248–49n2 Necessary factors of production, 117–18n6 Non-energy extraction costs, 32–33, 39n6 Nonexhaustion of resource, 157–164 Nonrenewable fossil fuels, 12, 69. See also Biofuels choke price for, 66–67, 74, 140–41 demand-side climate policies, 21–23 dynamic exhaustible fuel market model, 197–205 effect of increase on substitutability on exhaustion time of, 69–75 endogenizing reserves and, 48–51 energy efficiency, 138–43, 264–65 extraction costs, 6, 10–11, 90–91 high-cost versus low-cost reserves, 37 innovation in efficiency of use of, 138–43 literature review, 61–63

Index

model of imperfect substitutability between renewable energy and, 64–69 monopolies, 79–80 multiple pools, 260–66 perfect substitutes, 60 reduced-form demand function for, 60, 67–69, 73–74 spatial leakages, 255–256 substitutability and innovation, 11–12, 87–110 supply theory and, 44–45 synthetic two-period model of, 46–48 Nordhaus, W., 3, 136, 276 “Not in my backyard” (NIMBY), 241 Obama, B., 121 Oil climate policies in model with multiple pools of, 260–66 current emissions levels, 134–35 dirty backstop for, 33–35, 36 global reserves estimates, 266–67 sands, 267–68, 278 Parameterizing substitutability, 71–75 Partial equilibrium analysis, 90 Pearce, D., 173 Perfect substitution, 60, 89–90 backstop technology and, 105–10 Pethig, R., 151, 259 on adverse response to anticipation of taxes, 62 on demand elasticity, 231 on unilateral cap, 228, 229 Polborn, S., 174 Policy, climate, 14–16, 173–74 consequences of delaying, 218–21 Cournot path, 212, 222–23 delays, 213–23 demand-side, 21–23, 243–46 gradual greening of, 43 green paradox and, 211–13, 249n3 imperfect, 243–46 implications of innovation green paradox on, 122–24 literature review, 214–16 in model with multiple pools of oil, 260–66 monopolist behavior model, 216–18 reactions in anticipation of future, 121–24 stringency levels, 278, 279

Index

supply-side, 30 time between announcement and implementation of, 229 US Acid Rain Program, 229, 234–38 Pollution decay and carbon tax, 61–62 new plants and regulations on, 241 Positive feedback effects, 144–46 Present-value emissions targets, 275–77, 278 Prices global resource, 243–45 index, energy, 112 spatial leakage and, 255–56 supply effects on changes in, 46–48, 51, 53n6, 238–43 Pure intertemporal substitution effect, 45 Reduced-form demand function, 60, 67–69, 73–74 effect of increase in substitutability on exhaustion time and, 69–75 Regular oils and dirty backstop, 33–35, 36 Renewable resources backstop technology and, 87–88, 87–110 blend mandate, 265–66 equilibria, 12 model of imperfect substitutability between fossil fuels and, 64–69 substitutability and innovation, 11–12 supply, 65–67 varieties of, 88 Reserves climate policies in model with multiple pools of, 260–66 endogenizing, 48–51 estimates of global, 266–67 fixed, 46–48 high-cost versus low-cost, 37 total quantity extracted, 45–46 Reserve-to-extraction rate, 116–17 Roeger, W., 101 Rogner, H.-H., 198 Romer, P. M., 95 Rose, S., 276 Salant, S., 45, 259 Schou, P., 61 Sequestration and carbon capture, 266 Sinclair, P., 21, 228 Sinclair, P. J. N., 61 Sinn, H.-W., 1, 3, 151, 226, 227, 258

295

on adverse response to anticipation of taxes, 62 on demand for resources, 230 on effect of rising carbon tax, 59 on future carbon taxes, 175, 177 on gradual greening of economic policies, 43–44 green paradox term, 22, 225 on optimal carbon tax, 173, 174 Smulders, J., 97, 102, 105, 111 Smulders, S., 62, 215, 229, 232 Social costs of carbon (SCC), 3 Solar power, 88 Solow, R. M., 89 Spatial leakage, 255–56 Stähler, F., 60, 151 “Standing on shoulders effect,” 95 Steady state, DHSS model, 99–100, 115 transitional dynamics and, 100–105 Steger, T. M., 100 Stern Report, 3 Stiglitz, J., 89, 123, 129 Stochastic introduction of future scheme, 194–95 Stock effect, 46, 51 Strand, J., 62, 258 Stranded assets, 9 Stringency, policy, 278, 279 Strong green paradox, 3, 13, 122, 174 backstop technology and, 62 ex ante, 132–33 ex post, 133 other sources of emissions and, 133–37 Subsidies, biofuel, 63, 82n5–6, 259 Substitutability. See also Dasgupta–Heal– Solow–Stiglitz (DHSS) model backstop technology and, 105–10 exhaustion time and, 77–79 imperfect, 64–69 increase effect on exhaustion time, 69–75 innovation and, 11–12 parameterizing, 71–75 perfect, 60, 89–90 sufficient conditions for green paradox outcome caused by increase in, 75–79 Supply, 51–53 demand-side responses to changes in, 238–43 emissions from the extraction process and, 30–33 high-cost versus low-cost reserves and, 37

296

Supply (cont.) price change and, 46–48, 51, 53n6, 238–43 renewable energy, 65–67 -side climate policies, 30 synthetic two-period model of nonrenewable resource supply and, 46–48 Sweeney, J. L., 44, 53n4 Synthetic fuel, 35 Synthetic two-period model of nonrenewable resource supply, 46–48 Tahvonen, O., 62 Targets, emission cumulative, 272–75, 283n19 present-value, 275–77, 278 Taxes. See Carbon taxes Terms-of-trade channel, 244–45 Tol, R. S. J., 276 Tol, S. J., 3 Transitional dynamics, 100–105 Trimborn, T., 100 Ulph, A., 61 Ulph, D., 61 Unilateral cap and green paradox, 228–29 United Nations Climate Change Conference (UNCCC), 213–14 United Nations Framework Convention on Climate Change (UNFCCC), 2, 255 US Acid Rain Program, 229, 234–38, 241 US Clean Air Act of 1995, 4 US Energy Information Administration, 102, 239, 266 van der Meijden, G., 97, 102, 105, 111 van der Ploeg, F., 59, 148n3, 174, 195, 207n2, 227 on adverse intertemporal adjustment, 151 on cumulative extraction, 45, 53n4, 259 on green paradox, 124–28 on innovation green paradox, 123 on optimal tax, 173 on stock-dependent marginal extraction costs, 63 van der Werf, E., 62, 215, 216, 225 on climate policy and coal, 237–38 on policies that generate green paradox, 227–29, 232, 243, 246 Velez-Lopez, D., 238 Very strong green paradox, 174

Index

Waldhoff, S., 276 Weak green “orthodox,” 11, 89, 109–10 Weak green paradox, 3, 22, 174, 259 backstop technology and, 62 carbon tax and, 30 positive demand elasticity and, 231 Welfare, green, 122, 125 when innovation is impossible, 131–32 Welfare analysis limitations, 278 Welsch, H., 151 Wind power, 88 Winter, R. A., 123, 129, 144 Withagen, C., 59, 148n3, 174, 207n2, 227 on adverse intertemporal adjustment, 151 on cumulative extraction, 45, 53n4, 259 on green paradox, 124–28 on innovation green paradox, 123 on stock-dependent marginal extraction costs, 63 Wolfram, C. D., 239