Absolute abundance and relative scarcity: Environmental policy with implementation lags

Ecological Economics 74 (2012) 104–119

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Analysis

Absolute abundance and relative scarcity: Environmental policy with implementation lags☆ Corrado Di Maria a, b, Sjak Smulders b, c, d, Edwin van der Werf b, e,⁎ a

Department of Economics, JG Smith Building, University of Birmingham, Edgbaston, Birmingham B15 2TT, United Kingdom CESifo, Munich, Germany Department of Economics Tilburg Sustainability Center, and CentER, Tilburg University, P.O. Box 90153, NL-5000 LE Tilburg, The Netherlands d Department of Economics, University of Calgary, Canada e Environmental Economics and Natural Resources Group, Wageningen University, P.O. Box 8130, 6700 EW, Wageningen, The Netherlands b c

a r t i c l e

i n f o

Article history: Received 25 June 2010 Received in revised form 28 November 2011 Accepted 2 December 2011 Available online 18 January 2012 JEL classification: Q31 Q41 Q54 Q58 Keywords: Non-renewable resources Implementation lags Announcement effects Scarcity Order of extraction Climate policy Clean Air Act Green Paradox

a b s t r a c t We study the effectiveness of environmental policy in a model with nonrenewable resources and an unavoidable implementation lag. We find that a time lag between the announcement and the implementation of an emissions quota induces an increase in emissions in the period between the policy's announcement and implementation. Since a binding constraint on emissions restricts energy use during the implementation phase, more of the resources must be extracted outside of it. We call this the abundance effect. In the case of multiple resources that differ in their pollution intensity, a second channel emerges: since cleaner sources are relatively more valuable when the policy is implemented, it is optimal to conserve them before the cap is enforced. This ordering effect tends to induce a switch to dirtier resources before the policy is implemented, compounding the increase in emissions via the abundance channel. Using the announcement lag in Title IV of the 1990 CAAA as a case study we are able to empirically show that the abundance effect and ordering effect are both statistically and economically significant. We discuss a number of alternative policy options to deal with these undesirable side effects of policy announcements. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Major regulatory efforts, be they labor market reforms, privatizations, energy market deregulation efforts or overhauls of tax systems, tend to be characterized by the existence of significant lags between the announcement of the policy and its implementation. These implementation lags exist, for example, because complex policy changes require the development of institutions and mechanisms to monitor, manage and enforce the new rules, or because in many cases giving firms and consumers time to adapt to the changes increases the political palatability of the measure, or for political economy considerations. 1 ☆ We would like to thank Michael Caputo, Reyer Gerlagh, Michael Hoel, Ian Lange, Emiliya Lazarova, Charles Mason, Heinz Welsch, Cees Withagen, and three anonymous referees for useful comments and suggestions. ⁎ Corresponding author. E-mail addresses: [email protected] (C. Di Maria), [email protected] (S. Smulders), [email protected] (E. van der Werf). 1 Classic references on the political economy of regulation are Stigler (1971) and Becker (1983), while Magat et al. (1986) provide an excellent early reference on the empirics of agency behavior. 0921-8009/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2011.12.003

Environmental policy initiatives are no exception to this rule. For example, Title IV of the United States Clean Air Act Amendments of 1990 was phased in over a period of 10 years (Schmalensee et al., 1998), and the European Union Emissions Trading Scheme was first announced in 2001, with a preliminary ‘pilot’ phase in 2005–2007, and the first commitment phase starting in 2008 (Ellerman et al., 2010). The fact that some of the largest environmental policy endeavors take the form of international environmental agreements that need to be ratified by a sufficient number of signatories to enter into force does nothing to shorten the length of the implementation lags. The Convention on Long-Range Transboundary Air Pollution was agreed upon in 1979 and entered into force in 1983; the 1987 Montreal Protocol took 2 years to enter into force, with a first commitment period starting in 1991; the Kyoto protocol, signed in 1997, entered into force in 2005, and its first commitment period started in 2008. While policies take time to implement, however, market participants respond quickly to anticipated future measures. Sometimes responses to announced policy go in the intended direction, e.g. when firms use the time allowed to develop and adopt cleaner technologies. In other cases, however, the agents' actions during

C. Di Maria et al. / Ecological Economics 74 (2012) 104–119

the implementation lag may have adverse environmental effects. Sterner (2004), for example, indicates that the 25% increase in the demand for chemicals containing Trichloroethylene (TCE) in 1999 in Norway might be attributed to pretax hoarding by firms who anticipated the imposition of the (very high) tax on TCE from 2000 onward. In the context of fisheries management, Loehle (2006) shows with a numerical example that implementation lags in quota enforcement may lead to the collapse of the fishery, even when the theoretically correct policy is announced. In their analysis of the effect of the US Endangered Species Act (ESA) on land development in Arizona, List et al. (2006) find evidence that listing a species under the ESA led to plots of land included in the proposed critical habitat map being developed up to 1 year earlier than comparable plots not classed as part of the critical habitat. Finally, there is abundant anecdotal evidence of hoarding behavior on the part of retailers and consumers ahead of announced incandescent light bulbs phase-outs in Australia, the EU and the US (Green, 2011). In this paper, we focus on the effects of implementation lags in the context of environmental policies aimed at regulating the use of polluting non-renewable resources, such as fossil fuels. We use a theoretical model to show that the owners of polluting non-renewable resources have an incentive to increase early extraction in anticipation of future measures that prevent the implementation of the resource owners' initial extraction plan. As an illustration of the existence and the size of the effects we identify in our theoretical model, we study the effects of the implementation lags embedded in Title IV of the 1990 US Clean Air Act Amendments (CAAA). Our empirical findings are consistent with our theoretical predictions. Our estimates suggest that, due to the existence of implementation lags, SO2 emissions in the US might have been 9% higher than would have otherwise have been the case. Both the theoretical and empirical results of our analysis suggest that implementation lags might have substantial effects on emissions. Hence, we also discuss the implications of our findings for the design of environmental policy, starting from a cost–benefit perspective and covering possible early-action policies suggested in the literature. We conclude with some specific implications for the design of environmental policy for non-renewable resources, when implementation lags are unavoidable. In our analytical model, utility is derived from the use of nonrenewable resources. Resource use, however, is associated with polluting emissions. We concentrate on a simple policy constraining the flow of emissions, which is however implemented with a lag. 2 We fully characterize the optimal extraction (and emission) path when policies are announced ahead of implementation. Our results show that the existence of an implementation lag affects emissions via two channels that lead to an instantaneous increase in emissions: an abundance effect, and an ordering effect. The former arises whenever resources are abundant, i.e. when the available stock of resources is large enough to make the constraint binding. The ordering effect, instead, emerges to different degrees, depending on how scarce cleaner resources are, relative to the overall stock. As the result of both effects, the fact that polluters anticipate the policy leads to an increase in emissions following the policy announcement, and to a sudden drop once the policy is finally enforced. We first discuss the case in which resources have the same pollution content and show that announcing mitigation policies induces an abundance effect: when extraction is constrained over some period of time, more of the resource is extracted at other points in time. Crucially, we show that, along any optimal path, some of this ‘extra’

2 In our analysis we take both the existence of the implementation lag and its length as given. A rich literature exists, however, that instead focuses on the determinants of regulatory delay in environmental policy (see e.g. Alberini and Austin, 1999; Ando, 1999; Metrick and Weitzman, 1996).

105

resource is consumed between the time of announcement and the policy's implementation. The associated increase in resource consumption in the interim phase may induce an increase in polluting emissions relative to the pre-announcement situation. This emissions increase due to the announcement of the policy has not been explicitly discussed in the literature until now. The second part of the paper discusses the case of multiple resources that differ in their pollution intensity. In this context, we show that the announcement may induce an additional ordering effect, which increases the (expected) pollution content of resource use. When the economy faces a binding constraint on emissions at some point in the future, it may be optimal to save (some of) the cleanest resource for this phase, and front-load the use of the dirtier resource ahead of enforcement. Thus polluting emissions may increase in the period between announcement and implementation. Importantly, the ordering effect compounds the abundance effect and further contributes to the increase in emissions. Given that the goal of the environmental policy is to reduce emissions, these announcement effects go directly against the spirit of the policy. Few papers have studied environmental policy in the presence of implementation lags. Kennedy (2002) and Parry and Toman (2002) focus on the effects of domestic action in the period between the announcement and the implementation of an internationally imposed binding cap on emissions. Both papers argue that policies aimed at emission reductions in this period may be costly and inefficient. In Kennedy's small-open-economy optimal-policy model, early domestic emission reductions have no effect on global damages. Hence, such efforts are welfare reducing, unless the associated co-benefits (e.g. reduced damages from local pollutants) are sufficiently large. In their exogenous policy framework, Parry and Toman find that early action may be undesirable if the abatement target is high, the environmental benefits are low, banking (or credits for emissions reductions) is not allowed and if there are no co-benefits (in the form of learning-bydoing, for example). Our analysis complements these contributions in that we show that, even when no additional measures are introduced between announcement and implementation, the mere existence of implementation lags in the presence of exhaustibility leads to increases in emissions, due to the optimizing behavior of resource owners. Stavins (2006) considers regulatory delay that arises when older regulated units are subject to less stringent standards than newer ones. This differential treatment not only delays regulation but also introduces an intratemporal distortion, which makes regulation more costly. Our work contributes to this discussion by focusing on intertemporal distortions, arising from uniform regulation at some point in time in the absence of regulation early on. Our model builds on the traditional optimal extraction framework developed by Hotelling (1931), and finds its place in the long literature that studies the problem of resource use in the presence of taxation (Sinclair, 1992; Tahvonen, 1997; Withagen, 1994). The present paper is also closely related to several recent contributions that discuss the optimal ordering of resource extraction in the context of climate change policy. Chakravorty et al. (2006) discuss a ceiling on the stock of pollution, and show that in the case of one polluting resource and a clean backstop technology, the two inputs might be used jointly during the constrained phase, even though they are perfect substitutes. Lafforgue et al. (2008) extend this analysis to the case where polluting emissions can be stockpiled in carbon sinks, and show that a sink without leakage can be treated as a second, non-polluting non-renewable resource. Finally, Chakravorty et al. (2008) study the case of two perfectly substitutable resources that differ in carbon content, and Smulders and Van der Werf (2008) study the case of imperfect substitutes without, however, discussing emissions levels. Our model is more general and more widely applicable to different types of polluting non-renewable resources. Moreover, none of these papers explicitly discuss emission profiles, nor do they focus on the effects of announcing policy in advance.

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Chakravorty et al. (2008) study the immediate implementation of a ceiling on the stock of carbon dioxide in the atmosphere, which results in an immediate positive price on carbon emissions, even when the ceiling is initially not binding. The authors show that it may be optimal to initially use only the high-carbon input when it is abundant, and the cap is not initially binding, in order to reach the ceiling as quickly as possible. This effect differs from our ordering effect in that it is due to the social planner's incentives to take full advantage of the costless natural uptake of CO2 by natural sinks like forests and the oceans (this effect was first discussed by Wigley et al., 1996). We show that the initial ordering of resource extraction may be affected even in the absence of a positive emissions price, and that this result does not depend on the presence of natural uptake. Smulders and Van der Werf (2008), instead, show that the expectation of future policies might increase the demand for the relatively clean resource. In the current paper, due to the high degree of substitutability between resources, there is never an incentive to do so. The ordering effect that we describe below emerges as a consequence of the relative scarcity of different types of inputs rather than from their relative productivity. Finally, another strand of literature that our paper is related to, is the recent literature on delayed participation in international climate policy agreements. Both Bosetti et al. (2009) and Blanford et al. (2009) use forward-looking numerical models with non-renewable resources – WITCH and MERGE, respectively – to study the announced future entry of additional groups of countries into an existing climate coalition of OECD countries aiming at stabilizing radiative forcing at a given level. Neither set of authors focuses specifically on the time profile of emissions by late entrants during the interim phase, but Bosetti et al. (2009) discuss emissions ahead of entry for China. In their simulations, China's emissions are equal to the business-as-usual level for most of the interim phase, and become strictly smaller starting just before the policy's implementation. In this calibrated model (which has a much richer representation of energy technologies than our analytical model), the effects that tend to reduce emissions (innovation and technology diffusion/spillovers/adjustment costs/inertia, etc.) seem to outweigh the abundance effect. Unfortunately, neither Bosetti et al. (2009) nor Blanford et al. (2009) discuss incentives to increase emissions coming from resource abundance. The insights provided in the present paper suggest that in the absence of these perverse incentives, emissions by regions with delayed climate policy would have been even lower. The remainder of this paper proceeds as follows. After we introduce the model in Section 2, we study the optimal path of resource consumption of an unregulated (laissez-faire) economy. We then show in Section 3 that the announcement of emissions reduction policy leads to an abundance effect that induces an increase in resource use at the instant of announcement, irrespective of the number of resources available. In Section 4, we show under which conditions the two resources can be considered abundant, and when the cleaner one is relatively scarce. We then study (Section 5) how this affects the ordering of extraction after the policy announcement, and discuss how the abundance and the ordering effects affect the optimal emission paths. Section 6 takes a preliminary look at data on the effect of implementation lags for Title IV of the Clean Air Act Amendments of 1990, finding evidence in support of the effects discussed in the paper. We discuss implications of our results for the design of environmental policy in the presence of implementation lags in Section 7. Finally, we suggest possible extensions in Section 8. 2. The Model We study the optimal response by consumers, producers, and resource owners to an announced ceiling on the flow of the polluting emissions associated with the use of exhaustible resources. Emissions

can be thought of as CO2 emissions from the use of fossil fuels, but the model applies equally well to the analysis of any pollutant emitted as a consequence of the use of an exhaustible resource, e.g. SO2 and NOx from coal use, arsenic, sulfur and mercury leaching from mining operations, and so on. A high-emission nonrenewable resource, H, and a low-emission one, L, are perfect substitutes from the perspective of resource consumers. 3 The use of one unit of i ∈ {H, L}, entails εi units of emissions, Z, with εH > εL. The economy faces a constraint on the flow of emissions, Z≤Z
, which is enforced from some exogenously set and known point in the future, T ≥ 0. 4 Consumers derive utility from total resource consumption R. They maximize utility U(R), which is a C 2 function with U′ > 0 and U″ b 0, that satisfies the Inada conditions. Producers and resource owners maximize profits, taking prices and policies as given. Since we do not model market failures in our model (we study an exogenous ceiling on emissions, rather than optimal environmental policy), the decentralized economy can be represented by the social planner's solution in which utility is maximized subject to technological, resource and emission constraints. Hence, the model reads: max

∞

−ρt

∫ U ðRðt ÞÞe

fRH ðt Þ;RL ðt Þg∞ 0 0

dt

ð1:aÞ

s:t: Rðt Þ ¼ RH ðt Þ þ RL ðt Þ;

ð1:bÞ

S_ H ðt Þ ¼ −RH ðt Þ; RH ðt Þ≥0; SH ð0Þ ¼ SH0 ;

ð1:cÞ

S_ L ðt Þ ¼ −RL ðt Þ; RL ðt Þ≥0; SL ð0Þ ¼ SL0 ;

ð1:dÞ

Z ðt Þ≡εH RH ðt Þ þ εL RL ðt Þ≤ Z
;∀t≥T:

ð1:eÞ

Ri(t) denotes extraction of nonrenewable i at time t, and ρ is the rate of time preference. Eqs. (1.c) and (1.d) show that the stock Si of each nonrenewable declines with extraction. The initial endowment of each resource, Si0, i ∈ {H, L}, is given. Throughout the paper, both stocks and extraction flows of the resources are expressed in units of consumption. Environmental policy is described in Eq. (1.e): emissions (Z) that arise from resource use are constrained from time T onwards not to exceed Z
. The fact that the policy is announced at the beginning of the planning horizon but only becomes effective with a lag constitutes the key ingredient of our model. This entails the division of the planning horizon in two phases: a first period when the constraint is not yet enforced (the interim phase), and a second period when the constraint is enforced and (at least initially) binding (the enforcement phase). The problem in Eqs. (1.a)–(1.e) is therefore an infinite-horizon discounted two-stage optimal control problem, with a fixed switching time at t = T. In the first stage, the problem 3 The assumption of perfect substitutability we make here lends itself to two possible interpretations in the context of electricity production from non-renewable energy resources. First, it can be seen as implying that the consumer only cares for her electricity consumption, irrespective of it source. Second, it can be interpreted as implying that the two types of fuel are perfect substitutes in electricity production, at the plant level. For any given power plant this is usually not the case, at least in the short-run. One should notice however, that in coal-fired power plants, it is technically relatively easy to substitute high-sulfur coal with low-sulfur coal. In addition, oil-fired plants can be converted to natural gas fire plants in less than a year. Clearly, the longer the relevant period (interim phase), the more flexibility a power plant has. Most contributions in the literature assume perfect substitutability as well, e.g. Herfindahl (1967), Lewis (1982) and Chakravorty et al. (2008) for the case of multiple non-renewable resources, and tetTahvonen97, Chakravorty et al. (2006) and Lafforgue et al. (2008) for the case of one non-renewable resource and a clean ‘backstop’ energy source. 4 We opt for this type of modeling since it allows us to connect the model to the way climate policy is conducted in reality (“Kyoto forever”) as well as to US SO2 policy under the Acid Rain Program. As discussed in Section 7 below, this choice does not have dramatic impacts on our conclusions, in that we would obtain the same qualitative results under several alternative modeling choices. Several recent contributions adopt a strategy similar to ours, e.g. Ishikawa and Kiyono (2006), Leach (2009), Eichner and Pethig (2009, 2011).

C. Di Maria et al. / Ecological Economics 74 (2012) 104–119

faced by the agent is to optimally extract resources in order to maximize discounted utility over the interim phase, while leaving an optimal amount of resources for the second stage. Given the resource stock at the time of enforcement, in the second stage the agent will once more maximize discounted utility over the remaining horizon. The Lagrangians for the two stages of the problem are: 1

1

1

L ð·Þ ¼ U ðRðt ÞÞ− ∑ λi ðt ÞRi ðt Þ þ ∑ γ i ðt ÞRi ðt Þ; i∈fH;Lg

i∈fH;Lg

ð2Þ


 2 2 2 L ð·Þ ¼ U ðRðt ÞÞ− ∑ λi ðt ÞRi ðt Þ þ ∑ γ i ðt ÞRi ðt Þ þ τ ðt Þ Z
−Z ðt Þ ; i∈fH;Lg

i∈fH;Lg

ð3Þ where superscript 1 indicates the Lagrangian for the period t ∈ [0, T), while 2 indicates the Lagrangian for t ≥ T. The λi's are the co-state variables associated to Eqs. (1.c) and (1.d), the γi's the multipliers for the nonnegativity constraints on the extraction rates, and τ is the multiplier associated with the emission constraint. As the decision regarding the stock to bequeath to the second stage is made optimally, it is easy to show 5 that in this case the standard necessary conditions, 6 ′

j

j

U ðRðt ÞÞ ¼ λi ðt Þ−γ i ðt Þ þ εi τ ðt Þ≡pi ðt Þ;

ð4Þ

j j λ_ i ðt Þ ¼ ρλi ðt Þ;

ð5Þ

where i ∈ {H, L} and j ∈ {1, 2}, need to be complemented by the following matching conditions for the co-state variables in the two stages:   1 − 2 þ λi ðT Þ ¼ λi T :

ð6Þ

To simplify notation, in what follows we drop the superscripts to the λ's. The complementary slackness conditions for the constraints are,   τðt Þ≥0; Z
−Z ðt Þ≥0; τðt Þ Z
−Z ðt Þ ¼ 0; ∀t≥T; j

j

γ i ðt Þ≥0; Ri ðt Þ≥0; γi ðt ÞRi ðt Þ ¼ 0;

ð7Þ ð8Þ

and the transversality conditions are, −ρt

lim λi ðt ÞSi ðt Þe

t→∞

¼ 0:

ð9Þ

In the absence of (binding) regulation, environmental policy has no influence on the agent's choices. For this reason, we refer to this benchmark case as the laissez-faire economy. We discuss the laissez-faire economy in the remainder of this section. Since the two resources are perfectly substitutable as sources of utility, the two non-renewables are de facto identical as long as their emission content is irrelevant, and can be treated as one. For an economy that is never constrained, we can then define the total stock of available resources (in units of consumption) at time t as S(t) ≡ SH(t) + SL(t), and the initial total stock as S(0) ≡ S0. Along the optimal extraction path, marginal utility grows in parallel with the scarcity rent λ at rate ρ, as can be seen from Eqs. (4) and (5). At each point in time total extraction equals demand, and is simply given by R(t) = d(λ(t)) ≡ U′ − 1(λ(t)). Thus, extraction is continuous and declines along the optimal path. From Eqs. (5) and (9), it follows that it is not optimal to leave any of the resource in the ground, hence the stock will be exhausted over the infinite time horizon. The initial scarcity rent λ0 ≡ λ(0) thus solves 5

See Appendix A. In the interest of compactness, we have indicated the necessary conditions for the two stages as one. Note that τ(t) = 0 for all t b T. 6

107

∞

∫ d(λ0e ρt)dt = S0, so that λ0 is a function of S0, λ0 = Λ(S0). Hence, the 0 larger the initial (total) resource stock, the lower λ0, the higher initial extraction (see Eq. (4)). In the laissez-faire economy, optimal extraction profiles are purely determined by the initial scarcity of the resource, hence we denote the optimal aggregate unconstrained ~ ðt; S0 Þ ¼ dðΛ ðS0 Þeρt Þ. Since the two resources are extraction path by R identical as long as their emission content does not matter, the exact composition of extraction is undetermined, and so is the emission intensity of consumption. For the remainder of the paper, we assume that SH0, SL0 and Z
are such that the constraint is binding (i.e. τ(t) > 0) for a measurable interval of time from t = T onward. 3. The Abundance Effect Suppose the emission constraint (1.e) were introduced without announcement, that is let T = 0. Then, provided that the constraint is binding for some period, less of the resources can be extracted during this phase, compared to the laissez-faire extraction path. Hence, more of the resource is available after the constraint becomes slack (at time TH, say) than would otherwise have been the case. In order for both stocks to be exhausted over time, extraction during this period needs to be larger than under laissez-faire. When the policy is announced ahead of enforcement (i.e. T > 0), instead, the agents realize that the amount of resource that cannot be extracted during the constrained period, can be extracted not only after the constraint ceases to be binding, but also before the implementation, i.e. in the interim phase. The resource aggregate thus becomes de facto more abundant already during the interim phase. Consequently, it is optimal to increase the resource consumption in the (0, T) interval. This is, in a nutshell, the essence of the abundance effect: Proposition 1. The Abundance Effect Suppose an emission constraint is announced at t = 0 and becomes binding at t = T > 0. Then, resource use jumps up at the instant of announcement. Proof. Let tU be any t ∈ [0, T) ∩ [TH, ∞), i.e. any date at which the emission constraint does not bind. Let tC be any t ∈ [T, TH), i.e any date at which the emission constraint binds. Let W be the integral in (1.a). In a constrained optimum, reallocating extraction from some tC to some tU is always feasible but should not lead to an increase in welfare. That is, for small dR = – dR(tC) = dR(tU) > 0, dW/dR = e –ptU U′(R(tU))– e–ptC U′ (R(tC))≤ 0. Equations (4)-(5) imply e–ptU U′ (R(tU)) = c and e–ptU U′ ∽ (R(tU))= ∽ c , where c and ∽ c are constants. Then, combining this with the inequality above and (4)-(5), we find e–ptC U′ (R(tC)) ≥ c and ∽ e –ptC U′ (R (tC)) = c∽. We compare laissez-faire equilibrium and ∽ constrained equilibrium, taking into account U″ b 0. If R(tU) = R (tU) ∽ ∽ then c = c and either R(tC) b R(tC ) so that cumulative extraction is ∽ less than in laissez-faire and (9) with (5) is violated, or R(tC) = R(tC ) ∽ and the constraint never (strictly) binds. If R(tU) b R(tU) then c >∽c and ∽ R(tC) b R (tC) so that cumulative extraction is less than in laissezfaire and (9) with (5) is violated. Hence, in an equilibrium in which the ∽ ∽ constraint ever binds, R(tU) > R(tU) so that, if T > 0, R(0) > R(0). □ From the necessary conditions, we know that the optimal path entails constant discounted marginal utility across all phases during which the economy is unconstrained — see Eqs. (4) and (5). Hence the amount of the resource that comes available outside the constrained period due to the abundance effect must be extracted partly before and partly after the constrained phase. An interesting aspect of the abundance effect is that at the instant of implementation (t = T), extraction (and hence utility) jumps down relative to the laissez-faire situation. This might be surprising since one might expect that risk-averse consumers would smooth their

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SH

consumption over time, knowing in advance of the binding constraint. Consumption is smoothed wherever possible. Only at time T consumption discretely drops and marginal utility jumps up. However, this cannot be avoided since any reallocation of consumption towards the period just after T (with relatively high marginal utility) would result in violation of the emission constraint. Fig. 1 compares the extraction paths in the laissez-faire economy and in the economy with an announced constraint, for the special case in which all resources have the same emission intensity, or – equivalently – in which there is only one resource (note that the result in Proposition 1 holds for any number of resources, possibly with different pollution intensities). The level of extraction in the constrained economy is higher than under laissez-faire both before implementation at time T, and after t = TX, the instant after which laissez-faire emissions drop below Z
. From the proposition, it immediately follows that

IV III Sˇ H ( S L0 ) B S hL

Sm H

4. Absolute Abundance and Relative Scarcity For the constraint to be binding, it must be the case that initially so much of the resource is available that along a laissez-faire extraction R (t ) R (0) R (0)

R¯

T

TX TH

t

Fig. 1. Extraction paths if all resources have the same emission intensity for the laissezfaire economy — dashed line, and for the economy with an announced emissions constraint — solid line.

B

B

A I

Corollary 1. Keeping emissions per unit of resource consumption constant, the announcement of environmental policy leads to an increase in emissions during the interim phase. Announcing environmental policy leads to a perverse increase in emissions at the instant of announcement, if the emissions intensity of resource consumption does not fall. Note that the point made in the corollary can be strengthened: as Proposition 1 states that resource use increases ahead of implementation, so does the level of emissions, unless the emission intensity of resource consumption falls proportionally more than the increase in the level of resource consumption. In a setting with multiple resources, this emissions intensity is determined by the order, or composition, of extraction. In the following sections we investigate the case of two resources that differ in their emission content to study whether our ceteris paribus assumption of constant emissions intensity holds in the case of multiple resources. We show that in addition to the abundance effect, there exists an ordering effect of the announcement on the pollution intensity of consumption. As the optimal extraction paths are sensitive to initial conditions, however, it is first necessary to present a taxonomy of initial conditions in terms of the relative scarcity of the two resources.

II

A

O

Sm L

C

S hH

D

SL

Fig. 2. A taxonomy of initial stocks.

path, more would be emitted than allowed by the constraint. All resource endowments such that a given constraint will be binding for some period of time lead to the emergence of the abundance effect. Hence, the abundance effect is directly linked to the initial absolute abundance of resources. Identifying all endowments such that a laissez-faire extraction path leaves the economy unconstrained, we obtain a contrario those that lead to an abundance effect. 4.1. Absolute Abundance without Implementation Lags To begin with, assume that the cap Z
is immediately binding, i.e. let T = 0. In this case, it is possible to identify the largest stock of each type of resource i ∈ {H , L} such that, if it were the only resource in the economy, its extraction would occur along a path that is never constrained by the ceiling on emissions, Z
. Such a stock, which we denote by Sih, must  be such that the associated optimal ex~ 0; Sh ¼ Z
; i.e. the constraint is exactly traction path satisfies εi R i binding at t = 0, and never again. We call these the maximal Hotelling   ~ 0; Sh ¼ Z
=εi ≡R
i is the maximum amount of i stocks. Notice that R i

that can be extracted when the emission constraint is binding. For fu
ture use,
 it is useful to denote the price associated with R i by
i . From the properties of U and the fact that εH > εL, it follows p
i ≡U ′ R that p
H > p
L . Finally, since the emission coefficient determines the amount of resource that can be used at the ceiling, the cleaner the resource, the larger the maximal Hotelling stock, i.e. SLh > SHh. These definitions are useful to identify initial endowments that lead to an unconstrained path when T = 0. We refer to Fig. 2, which has stocks of the low-emission resource on the abscissa, and stocks of the high-emission resource on the ordinate, both expressed in consumption-equivalents rather than in physical units.7 We begin by indicating the maximal Hotelling stocks in the figure. When stocks are
H , the price is p
, and the path at point A = (0, SHh), extraction equals R H of extraction is unconstrained from the instant when stocks are at A onwards. Next, focus on the iso-consumption line through C = (SLh, 0), which denotes all vectors such that SH(t) + SL(t) = SLh. The economy is unconstrained at point C by definition of maximal Hotelling stock. However, moving up along this line from point C, the emission content of the vector of stocks increases. As a consequence, if initial demand (equal to 7

Given this convention, lines with negative unit slope are iso-consumption lines.

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L ) were satisfied using only the high-emission source, emissions R would exceed the cap. To avoid excessive emissions, enough of the low-emission resource must be extracted at each point in time in combination with the high-emission resource. This requires a minimum amount of the low-emission resource to be available initially. Along the iso-consumption line through point C (i.e. when the total stock equals SLh), the minimum amount of low-emission resources needed along an unconstrained path is SLm (formally defined in Appendix B), so that whenever initial stocks are on line segment BC, extraction is unconstrained. Starting from an iso-consumption line with smaller total stock than SLh, unconstrained extraction is lower and correspondingly lower amounts of low-emission resources are needed to make the unconstrained extraction path compatible with the emissions constraint. For any given total stock (i.e. along a particular isoconsumption line), we can identify the minimum L-stock and the corresponding maximum H-stock. For S0 ∈ (SHh, SLh) this is represented by AB. In Appendix B we show that along the trajectory AB, emissions exactly equal Z
, and the resource (user) price increases from pL to p
H , as more and more RH is used. Thus, when the vector of stocks is within the OABC area, the emission constraint is never binding, and we can state the following:

unit of emissions. When the stock of L is relatively small, i.e. not sufficient to allow exclusive use of L during the entire enforcement phase, the cleaner resource becomes scarce relative to H due to the introduction of the cap. So the cap changes the relative scarcity rent of the two inputs when too little of L is available, irrespective of the existence of an implementation lag. In the presence of implementation lags (i.e. when T > 0), the agents prefer to conserve the clean input and use it during the enforcement phase, provided that the stock of L is small enough. Indeed, in this case it becomes attractive to use only the dirty input during the interim period, so that all of the clean one is still available when enforcement starts. When enough of the clean input is available from the start, however, no such incentive emerges in the interim phase. If this intuition is correct, the introduction of an announced, binding constraint should have different effects on the relative scarcity of the two resources, depending on their initial endowment. As a first step, we show that the constraint never makes the highemission resource scarcer than the low-emission one:

Lemma 1. When stocks are in area OABC of Fig. 2, the economy is unconstrained along an optimal extraction path.

Proof. First, notice that if the cap is binding at time t, then τ(t) ≥ 0. Now suppose that λH > λL. From the necessary conditions Eq. (4) we must have: (i) if τ = 0, only L is used; or (ii) if τ > 0, then pH > pL, and again only L is used. Hence, H is never used, which, given Eq. (5), violates Eq. (9). Hence we must have λH(t) ≤ λL(t) ∀ t. □

□

Proof. See Appendix B. 4.2. Absolute Abundance with Implementation Lags

When the policy is announced ahead of implementation, i.e. when T > 0, the possibility arises that during the interim phase some of the resources can be extracted without violating the constraint. Thus, in the presence of an announced constraint, the set of endowment vectors consistent with unconstrained paths is larger than the area identified above. The key issue is whether, for a given level of the constraint Z
, the vector of resource endowments lies within OABC at time T. Let the vectors S0 = {SL(0), SH(0)} identify initial levels of the stock of resources. Recalling from Section 2 that the optimal unconstrained extraction path associated with an initial total stock of resources ~ ðt; S0 Þ, we can define the set of all initial endowments equal to S0 is R leading to an unconstrained path as:

Lemma 3. If the emissions constraint is binding for some period of time, then λH(t) ≤ λL(t) ∀ t along any optimal extraction path.

Thus, the scarcity rent of the dirtier resource never exceeds the rent of the cleaner one. However, the optimal extraction path during the unconstrained phases is very different depending on whether λH is equal to, or strictly smaller than λL: in the latter case, only the cheaper, dirtier resource will be extracted during these phases. We now show how initial endowments determine the optimal extraction paths in the presence of announced emissions reduction policy. For ease of reference, we draw these paths in Fig. 3, for initial stocks that sum up to the same amount of reserves (in terms of consumptionequivalents). The proofs behind this Figure and the discussion below can be found in Appendix C.

   ~ ðt; S Þ∀t∩ ∑ ε R t ′ ≤Z
∀t ′ ≥T : H≡ S0 ¼ fSL ð0Þ; SH ð0Þg : ∑ Ri ðt Þ ¼ R 0 i i i

i

ð10Þ

SH

The following result relates the definition above to the graphical representation in Fig. 2:

P

IV

N

Lemma 2. The set H is represented by the area OA′B′D in Fig. 2. Proof. See Appendix B.

III

□

Sˇ H ( S L0 ) B

Thus, only initial endowments outside the set H can be considered abundant, in the sense that any path starting from such endowments will be constrained by the policy. Proposition 2. Any path beginning from a vector of initial stocks S0 ∉H will be subject to the abundance effect.

S hL

II

A Sm H

M B

B

A I

Proof. It immediately follows from the definition in Eq. (10) and Lemma 2. □

L

4.3. Relative Scarcity and Optimal Extraction Paths O

When the emissions constraint is binding, agents prefer to use the low-emission resource, as it allows more resource consumption per

Sm L

S hH

C

D

Fig. 3. Initial endowments and optimal extraction paths.

SL

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4.3.1. Equal Scarcity Rents Scarcity rents are equal whenever welfare cannot be improved by (marginally) changing initial endowments, replacing one unit of one resource by one unit of the other. This indifference situation arises if the initial stock of L is large enough, while that of H is not too large. Intuitively, this is the case when enough of L is available to guarantee exclusive use when the cap is binding, while the initial stock of H is sufficiently small to be

exhausted outside of  the constrained phase. Define zone I as SI ≡ S0 : S0 ∉H∩SH0 bSm H . This elucidates that initial endowments in this zone are abundant in absolute terms as the total stock is outside H, while at the same time the initial stock of the high-emission resource is small enough not to represent a constraint to extraction, i.e. SH0 b SHm (see Appendix C). Thus, L is not scarce relative to H, and hence λH = λL. Along paths starting in zone I, from point L in Fig. 3 say, the economy is constrained from time T until the instant when the path enters area OABC. During the constrained phase, only the low-emission input is used in order to maximize consumption (remember that
L > R
H ). Since the low-emission input is abundant, there is no R need to economize its use during the interim phase. As λH = λL, the composition of extraction – both during the interim phase and after the constraint ceases to be binding – is a matter of indifference. Thus, the path drawn in the picture is only one of the infinitely many possible ones. Note that, in the case where T = 0, the cap does not affect the relative scarcity of the resources, and zone I is accordingly extended until CB: when initial stocks are in this extended zone I, the scarcity rents and hence the order of extraction are not affected by the policy. Besides being used after TH, H can of course be used also before the cap is enforced. How much of it can be extracted along the optimal path ahead of the constrained phase, however, depends on how much of L is initially available, as discussed in Appendix C. For each level of SL0, we can identify an upper bound on SH0 (ŠH(SL0) in m Fig. 2), such that n SH can be reached beforetime T. Weican o now define m zone II as SII ≡ S0 : S0 ∉H∩SL0 > Sm . When initial L ∩SH0 ∈ SH ; Š H ðSL0 Þ stocks are in this zone, the dirty resource is relatively more abundant than in zone I and restricts the composition of extraction in the interim phase, as described above. At the same time, the initial stock of H is small enough so that there is no need to use it exclusively ahead of enforcement: mixed use is still optimal, and λL(t) = λH(t). Thus, ŠH(SL0) represents the border between areas where the clean resource is abundant (zones I and II) and those where it is relatively scarce (zones III and IV). When S0 ∈ SII, the optimal extraction paths are similar to zone I: initially the composition of extraction is indeterminate, provided that enough H is extracted to reach S I at t ≤ T. During the constrained phase only the low-emission resource is used; finally, the composition of extraction is again indeterminate once OABC is reached and the constraint ceases to be binding. In Fig. 3, we draw one of the many possible paths starting from M.

4.3.2. Different Scarcity Rents Initial endowments in zones III (defined as SIII ≡fS0 : S0 ∉H∩SL0 >

m Sm ∩S H0 > Š H ðSL0 Þg) and IV (where SIV ≡ S0 : S0 ∉H∩SL0 bSL ) are charL acterized by a relatively scarce clean resource. Along paths starting from these zones, scarcity rents are no longer equal, and we have λH b λL. This implies, from Eq. (4), that whenever the constraint is not binding (i.e. as long as τ = 0) only the high-emission resource is used. For simplicity, we begin with paths originating from zone IV (point P in Fig. 3). In this case, the low-emission input is very scarce, in the sense that its endowment is too small to warrant exclusive use during the constrained phase. Still, since it is the input that gives the highest amount of consumption per unit of emissions, it is optimal to conserve as much as possible of it during the interim phase, hence only the high-emission resource is used initially. At T (or immediately in the case without announcement, i.e. when T = 0) both inputs are

used jointly until L is exhausted. For the rest of the constrained
H . Finally, the traphase the dirty input is used at its maximum rate R jectory of stocks enters OABC at point A, the constraint ceases to bind, and extraction of H proceeds to exhaustion. When S0 ∈ SIII, the extraction paths are similar to the ones discussed above: trajectories from initial stocks in zone III eventually enter zone IV and follow the path described above. Prior to entering zone IV, however, the optimal path is already constrained and only the clean resource is extracted at the cap (hence, when T = 0, only L is used until zone IV is entered). In the interim phase it is optimal to save L and extract only H. Paths of this type are illustrated by the path starting from point N in Fig. 3. 5. The Ordering Effect In Section 3 we have shown that announcement of a constraint on the flow of emissions shifts extraction to the period before implementation if the total initial stock of resources is abundant enough to make the emissions constraint binding. In the previous section we have shown how the initial composition of the resource stock affects the composition and order of extraction: if initially the low-emission resource is scarce relative to the total stock, the announcement tends to shift extraction to the high-emission resource in the interim period. In this case the announced constraint exerts what we will call an ordering effect. 5.1. The Extreme Ordering Effect Depending on initial endowments, the ordering effect occurs with different intensity. When initial stocks are in zone III or IV, total reserves are large and mostly consist of high-emission resources. The large total stock provides the incentive to consume a lot during the interim phase, while the relative scarcity of the low-emission stock makes it optimal to maintain these high resource consumption levels economizing on the use of the low-emission resource. We thus observe the ordering effect in its extreme form: Proposition 3. The Extreme Ordering Effect Suppose an emission constraint is announced at t = 0 and becomes binding at t = T > 0. Then, whenever S0 ∈SIII ∪SIV, it is optimal to extract only the high-emission resource ∀ t ∈ [0, T). Proof. As shown in Appendix C, paths initiated in zones III and IV are characterized by λH b λL. It follows from Eqs. (4) and (8) that whenever τ = 0, RH > 0, and RL = 0. □ The extreme ordering effect compounds the abundance effect in countering the intended outcome of the regulation in the interim phase: not only does resource use increase, it also becomes biased towards the most polluting resource. 5.2. The Weak Ordering Effect Proposition 3 shows that for initial endowments in zone III or IV, the announcement induces an extreme ordering effect, as optimality requires that only the high-emission input be extracted in the interim phase. On the other hand, Section 4.3.1 shows that for initial endowments in zone I, enough of the clean resource is available to allow its exclusive use in the interim period, and yet conserve enough for the constrained period. Therefore, just as in the laissez-faire economy, the ordering and composition of resource extraction are undetermined in the interim phase. Between these two extremes, we find paths that start from zone II. Based on our discussion in Section 4.3.1, we know that for such paths, although the scarcity rents are equalized, L is nevertheless scarce enough to dictate a precise requirement for extraction: optimality

C. Di Maria et al. / Ecological Economics 74 (2012) 104–119

requires the path to enter zone I by time t = T. This restriction imposes a lower bound on the amount of H that needs to be extracted in the interim phase, and de facto introduces an ordering effect, albeit in a weaker form than the one discussed in Proposition 3. Since L is not scarce enough to univocally determine the extraction path (as λH = λL, see Section 4.3.1), however, such weak ordering effect can only be analyzed in terms of expected, rather than actual, emission intensity. To see what the weak ordering effect entails, we now refer to Fig. 4. Consider any point Q in zone II. As discussed in Section 2, under laissez-faire the exact composition of extraction at any point in time is indeterminate. Hence, the actual extraction path from Q for t ∈ [0, T) can evolve in any direction within the quadrant to the SouthWest of the initial point. Assuming that any of these paths is equally likely, the ‘expected’ extraction path is given by the bisector of the SW quadrant (i.e. the dashed 45° vector to the SW from Q). With announced emissions reduction policy, instead, the optimal extraction path originating from Q must enter zone I before the enforcement of the cap. This means that at least an amount of H equal to the vertical distance between Q and the horizontal SHm line in Fig. 4 – the segment Qy – has to be extracted in the interim phase. Given that the total amount of resources that can be consumed during the interim phase is given by wx (the vertical distance between point w, where the iso-consumption line through Q crosses the ŠH locus, and the SHm line), at most xy of the clean input can be extracted before time T. Although it is impossible to be precise as to which path will actually be followed, the composition of extraction → will range from 100% H, along→the vector Q z , to the path with the lowest emission content, along Qx . The expected extraction path in this dz , as depicted in the case is given by the bisector of the angle xQ Figure. Notice that the range of possible optimal trajectories from Q after the announcement is reduced relative to laissez-faire; moreover, the trajectories that are not feasible are the ones with the lowest emission content. Hence, we can conclude that the weak ordering effect induced by the announcement raises the expected emission intensity of extraction during the interim phase. Proposition 4. The Weak Ordering Effect Suppose an emission constraint is announced at t = 0 and becomes binding at t = T > 0. Then, whenever S0 ∈SII, the expected emission intensity of extraction in the interim phase increases, relative to laissez-faire.

III w Q

S hL Q B x

A

y

y z

I

z

O

Sm L

5.3. Announced Policy and Emissions In Section 3 we have shown that, irrespective of the number and emission-content of resources, resource use jumps up at the instant at which a future binding constraint on emissions is announced. This leads to an increase in emissions if the emission content of consumption does not fall too much according to Corollary 1. We can now combine these results with the ones derived above on the ordering – and hence the emission content – of resource extraction. First, for initial stocks in zone I, we know that the abundance effect operates (Proposition 2), but there is no ordering effect at work. Although we cannot say anything specific in terms of actual extraction paths, there is no reason to expect that the announcement would affect the ordering of extraction compared to the laissez-faire economy. Thus, we expect that emissions increase at the instant of announcement. When initial stocks are in zone II, the abundance effect is compounded by the weak ordering effect of Proposition 4, and emissions are expected to increase in the interim phase also in this case. Since the strength of the ordering effect increases with the proximity to the boundary between zone II and III (Š(SL0) in Fig. 4), our degree of confidence in this results also increases for endowments closer to zone III. Finally, for initial stocks in zones III and IV, any ambiguity is resolved. For such initial endowments, Propositions 1 and 3 both imply that emissions increase in the interim phase. Not only does the level of extraction increase due to the abundance effect, but all of this extraction is made up of the most polluting resource due to the extreme ordering effect. Going strongly against the spirit of the regulation, emissions increase in the interim phase due to the policy announcement. We summarize this discussion in the following proposition:

2. ∀ S0 ∈ SII, resource use increases due to the abundance effect, and emission-intensity of consumption is expected to increase, leading to an increase in expected emissions; 3. ∀ S0 ∈ SIII ∪ SIV, resource use increases due to the abundance effect, and the emission-intensity of consumption increases, leading to an increase in emissions.

II

A Sm H

Finally, it is worth pointing out that, while the weak ordering effect operates everywhere in zone II, its impact is weaker, the lower the emission content of initial stocks. Along any iso-extraction line, initial endowments closer to zone I will be subject to a weaker ordering effect. This is illustrated in Fig. 4 by point Q′. For initial stocks in Q′, the range of possible paths is much larger than for point Q, since only a small amount of H needs to be extracted in the interim phase.

1. ∀ S ∈ S , resource use increases due to the abundance effect, and the 0 I emission-intensity of consumption is expected not to be affected, leading to an increase in expected emissions;

Sˇ H ( S L0 ) B

□

Proposition 5. Suppose an emission constraint is announced at t = 0 and becomes binding at t = T > 0. Then, during the interim phase,

SH IV

Proof. In the text above.

111

C

D

6. Announcement Effects of the 1990 Clean Air Act Amendments

SL

Fig. 4. Feasible paths during the interim phase for initial stocks in zone II.

As discussed previously, our theoretical results apply to any delay between the announcement and the implementations of policies affecting the future use of non-renewable resources. As an empirical illustration of the relevance of the theoretical results derived above, we investigate the effects of the announcement of the introduction of a cap on and an emission trading scheme for sulfur dioxide (SO2)

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emissions, contained in Title IV of the US Clean Air Act Amendments (CAAA) of 1990. 8 Title IV of the 1990 CAAA represents a good case study from our point of view for three main reasons: Firstly, because it is aimed at controlling polluting emissions (of sulfur dioxide, SO2, a precursor of acid rains) associated with the use of a non-renewable resource (coal in this case); Second, because the polluting resource occurs in different grades in North America, which differ in terms of their sulfur intensity; Finally, because between the passing into law of the 1990 CAAA and its enforcement, there was a lag of at least five years. 9 In the parlance of our theoretical model, low-sulfur coal would represent the cleaner resource, L, whereas high-sulfur coal plays the role of H in this context. The abundance effect described in Proposition 1 occurs when the amount of pollution embedded in non-renewable resource deposits is sufficiently large for the environmental policy to be binding (cf. Proposition 2). Since less resources can be used during the period in which the cap is binding, more will be used in the period between announcement and implementation. The fact that SO2 allowance prices have been positive testifies of the regulation stringency. The ordering effect of Proposition 3 occurs when the initial stock of high-pollution resources is so large, that exclusive use of the cleaner resource during the period in which the cap is binding, results in part of the stocks of the dirtier resource remaining unexploited. This induces owners of those stocks to supply more of their resource at each point in time outside the constrained phase, including the interim phase. Data from the United States Energy Information Administration (US EIA, 1993) suggest that the ratio of coal reserves with a sulfur content above 1.2 pounds per million Btu (lb/MBtu) to reserves with sulfur content lower than this threshold is roughly 0.82.10 In terms of Fig. 2, this suggests that the initial stocks are located slightly below a ray with unity slope, outside the hatched area. The large amounts of coal available in the US suggest that initial stocks are likely to fall in zone II or III, but we cannot rule out zone I a priori. In the first case the weak ordering effect would occur, whereas in the second our model predicts the emergence of a strong ordering effect. In both cases, however, we expect our empirical analysis to show both an increase in total energy input, and in the sulfur intensity of coal used. In the case in which initial stocks are in zone I, however, we expect only an abundance effect to arise, implying an increase in total energy input in response to the signing into law of the 1990 CAAA. To gauge the existence of the abundance effect, we use data on the heat content of coal used by US energy generators to produce electricity. For the ordering effect, instead, we analyze the sulfur content per unit of energy of the coal used. We use data collected by the US Federal Energy Regulatory Commission, FERC, via form FERC-423 ‘Monthly Cost and Quality of Fuels for Electric Plants’. We focus on deliveries of coal to utilities, and use the information on fuel quantity and quality (in terms of calorific and sulfur content) to construct a yearly dataset that covers coal deliveries to 552 plants (231 companies) across the United States between 1972 and 2000. In terms of our data, the existence of an abundance effect would imply that energy inputs to plants increased in the period 1990–1999 due to the announcement of the trading system in 1990. Thus, the first of our regressions explains the first difference of energy 8 There exists a large literature that studies the consequences of the Acid Rain Program of the 1990 CAAA, but nowhere have the consequences of the implementation lag been discussed. Schmalensee et al. (1998) present an overview of the literature on the Acid Rain Program and summarize the main findings. Ellerman et al. (2000) expand on this discussion. 9 Since the majority of boilers became subject to the regulation only in 2000, in our empirical investigation we consider the interim phase to run between 1990 and 1999. 10 Our calculation is based on Table C1 of US EIA (1993). We chose the 1.2 lb/MBtu cut-off point because the permit allocation from 2000 onwards was based on this number. The amount of reserves is expressed in thermal power (Btu).

Fig. 5. The abundance effect: predicted heat input (TBtu) over time (solid), and counterfactual (dashed).

inputs (in trillion Btus, or TBtu) using plant-level fixed effects, a set of year dummies to control for year specific fixed-effect Dyear, and an ‘announcement’ dummy (δA), which takes on the value 1 for every year of the interim phase (1990–1999), and zero elsewhere. 11 Our estimates for the relative coefficients are as follows (t-statistics in parenthesis): ΔEnergy Inputt ¼ −3:14 þ 1:89 δA þ Dyear : ð−2:90Þ

ð11Þ

ð1:55Þ

All coefficients are statistically significant and have the expected sign. The positive sign of the announcement dummy δA suggests that the signing into law of Title IV of the 1990 CAAA in 1990 induced an increase in energy input by coal-fired power plants between 1990 and 2000, the year of implementation (relative to the hypothetical case of an unexpected cap on SO2 emissions in 2000). Fig. 5 plots emissions calculated using our estimated coefficients and counterfactual emissions in the absence of the announcement. Using the results from this first counterfactual experiment, we can calculate that the 1990 CAAA announcement produced an increase in primary energy input in coal-fired power plants of about 5% over the decade 1990–1999. The ordering effect implies substitution towards coal with higher sulfur content, and hence an increase in the sulfur content of the coal burned by power plants. It is important to note, however, that the choice between different types of coal is affected also by the costs of transporting coal from different basins. Following railroad deregulation in the US, rail rates for hauling coal fell by nearly twothirds from the early 1980s to 2000. This increased the appeal of lower-sulfur coal from the Powder River Basin (PRB), located in Wyoming and Montana, and favored its penetration into the Midwest markets (Ellerman and Montero, 1998). To control for this effect, we use rail cost data provided by the US EIA (US EIA, 2004), and construct an average rate per ton-mile for the years 1981–2000, as a proxy for plant level transportation costs. 12 To verify the existence of an ordering effect, we thus regress the sulfur intensity of coal inputs (in lb/MBtu), on an indicator of rail prices, plant-level and year fixed effects, and a dummy for the effect of the announcement in period 1990–1999, δA. We estimate the following parameters (t-statistics in parenthesis): Sulfur Intensityt ¼ 0:57 þ 0:03 Rail Pricet þ 0:04 δA þ Dyear : ð9:74Þ

ð10:92Þ

ð2:11Þ

ð12Þ

11 As the data for heat input may suffer from unit root, we decided to take first differences. 12 Data are unfortunately not available for earlier years.

C. Di Maria et al. / Ecological Economics 74 (2012) 104–119

Once again, all coefficients are statistically significant and have the expected sign. The positive coefficient for the rail price variable indicates that sulfur intensity decreased alongside rail transportation rates, as rail deregulation tipped the financials in favor of PRB coal inputs. The coefficient for the announcement dummy is positive and significant at the 5% level, suggesting an increasing in sulfur intensity after the announcement and before implementation. Based on our estimates in Eq. (12), we can compute the sulfur intensity for the average plant over 1990–1999 and the counterfactual intensity assuming that no announcement had been made. The sulfur intensity of coal used was roughly 4% higher, on average, over the entire period as a consequence of the increased attractiveness of ‘dirty’ coal, brought about by the announcement of Title IV of the CAAA in 1990. Fig. 6 graphically describes the results of this exercise. The combined impact of the abundance and ordering effect computed above implies that – relative to the hypothetical case of an unexpected cap on SO2 emissions in 2000 – the announcement of the regulation to come induced an increase in the amount of sulfur emitted between 1990 and 1999 by roughly 9%, a significant amount by any standard. Although further empirical analysis is needed, these preliminary results suggest that significant abundance and ordering effects may have occurred in response to the signing into law of the 1990 Clean Air Act Amendments. 7. Policy Implications Our theoretical model shows that the delayed implementation of a restriction on emissions from a non-renewable resource (e.g. SO2- or CO2-emissions) results in an increase of emissions in the period between announcement and implementation, through an abundance effect and an ordering effect. We found evidence that both effects have played a role in SO2 regulation in the US, and that the effects are non-negligible. The question now is whether this effect entails a relevant welfare cost, and if so, to what extent and in what way we should take these effects into account when designing policy. 7.1. Cost–Benefit Analysis Our model is a model of physical exhaustibility of the resource, in the sense that over the planning horizon it is optimal to completely exhaust the resource. It follows from this that the total (cumulative) amount of polluting emissions is given, together with the initial resource stocks. As a consequence, the role of policy in our model is only to change the timing of emissions. In particular, we have shown that implementation lags have the effect of concentrating polluting emissions in the early stages of the planning horizon, i.e. they tend to induce a ‘frontloading’ of emissions. As compared to a

Fig. 6. The ordering effect: predicted sulfur intensity (lb/MBtu) over time (solid), and counterfactual (dashed).

113

situation in which resource extraction does not anticipate the implementation, extraction and emissions are higher before implementation but smaller after the constraint ceases to be binding. To understand whether this intertemporal shift makes implementation lags very costly, one has to consider when and to what extent emissions result into damages, and at which rate future damages are discounted. In our model, a marginal increase of short-run emissions unequivocally leads to an increase in the present value of damages whenever the emergence of damages from pollution coincides with the release of emissions (i.e. pollution matters as a flow rather than a stock) and marginal damages do not decrease with emissions (i.e. constant or convex damages). This is likely to be the case for flow pollutants such as short-lived toxic compounds like 1,3 Butadiene and Benzene, and easily-dispersed pollutants such as SO2, and particulate matter, which is the target of most air-quality regulation. In this case, indeed, a shift in emissions from the future to the present also entails a shift in discounted damages. Since extraction and emissions are decreasing over time in our model, marginal damages stay constant or decline, and decline when discounted. Hence the shift increases the total value of damages. 13,14 This can lead to substantial cost: in the simplest case of constant marginal damages, if the policy binds for 35 years and the discount rate is 2%, a pollutant is at least twice as costly when emitted in the interim period rather than after the implementation period. 15 When damages from emissions persist over time – which is the case for heavy metals such as lead and cadmium, for arsenic, and for other pollutants such as greenhouse gases – frontloading emissions might have much bigger effects. In this case, the shift in emissions not only affects damages in the interim period, but also in the (early periods of the) implementation phase since concentration levels of the pollutant are higher. 16 In general, shifting emissions to the interim period entails a cost whenever the net present value of the social cost of emissions is larger before the implementation period than after the constraint ceases to bind. Hence, the only case in which the announcement effect is not very costly is if, between the interim and the postimplementation phase, marginal damages grow at a rate that is larger than the discount rate. For stock pollutants that have an economywide impact, marginal damages could grow as fast as GDP (as in e.g. Eyckmans, 1999) but even in this case implementation lags are costly since GDP growth is typically smaller than the interest rate. If the policy binds for 70 years, the discount rate is 2%, and GDP grows at 1%, again a pollutant is at least twice as costly when emitted in the interim period rather than after the implementation period. 17 Despite the environmental costs arising from immediate emission increases due to implementation lags, there is no reason to expect that such costs would outweigh the gains from well-designed policy. A complete analysis of the desirability of implementation lags would require a full-blown assessment of costs and benefits, which is clearly beyond the scope of the present paper. Such an analysis ought to include sufficient detail about the effects of (the timing of) emissions 13 The same logic applies when damages accrue with a lag after the release of the pollutants but still coincide with emissions. 14 This might not be the case in more general contexts. For example, our reasoning in the text would not go through in the presence of increasing emissions, provided that the damage function is sufficiently convex. 15 Indeed, in this case the anticipation of policy shifts emissions at least 35 years and the cost difference is at least (1 + 0.02)35 ≈ 2. 16 The anticipation of emission reduction policy increases emissions during the interim phase (see Fig. 1), which, given the inertia in the stock, lead to higher concentration, at least for some time during the implementation phase. 17 The back of the envelope calculation is, in this case, ((1 + 0.02)/(1 + 0.01))70 ≈ 2. These numbers are very conservative estimates for the climate change case, since longer implementation periods are likely to be needed, and the calculation ignores the possibility that the probability of a run-away climate problem increases with emissions.

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on damages and welfare. Moreover, it should probably recognize the fact that the upward flexibility of emissions is somehow restricted because of limited substitution between pollution-intensive and clean sectors, and because of the capacity constraints (or adjustment costs) of power plants and other polluting firms. A key implication of our analysis is, however, that in the presence of non-renewable resources and implementation lags, the abundance and ordering effects should be an essential component of such a cost–benefit endeavor.

for emissions in excess of the cap after the policy is implemented. If we denote the instant at which the banking period ends by TB, and assume no permit trading is allowed among firms, the following constraint for firms would replace Eq. (5): n o T T ∫ B Z ðt Þdt ¼ ðT B −T ÞZ
þ ∫ max 0; Z~ ðt Þ−Z ðt Þ dt: T

0

Although minimizing implementation lags seems desirable, these lags are inherent in the policy process, for the reasons stated in the introduction. This leaves two alternative responses, temporary ‘early action’ policies before the implementation of the policies, and alternative policy design after implementation. Before the implementation of the policy, existing regulation and taxes that were designed for other purposes could be modified to address the new pollution. For example, an increase in value added tax (VAT) rates for emission-intensive goods acts as an emission tax, but is likely to require time to implement as well. Pre-existing output-related taxes, such as VAT, regulate pollution in a less direct and hence more costly way than targeted emission reduction policies. They come at a cost, but might bridge the period until the pollution regulation is in place. Early-action policy options that are more directly linked to incentives for polluters may be available. In particular, it is worth revisiting the results that Parry and Toman (2002) derived for the case without non-renewables, in light of our discussion of abundance and ordering effects. In our setting, early action policies not only serve the purpose of avoiding damages and reaping co-benefits until the full policy is in place, as in Parry and Toman (2002); they can also be targeted to mitigate the abundance and ordering effect. Hence, early action measures – as far as available and politically feasible – become more desirable when considering non-renewable resources. The question, then, is how to design such policies. The aim is to generate incentives for firms to reduce emissions, before the actual emission policy comes into force. Early action policies thus should imply an abatement subsidy high enough to make firms reduce their emissions. This could be achieved through early reduction credits (ERCs) and emissions banking provisions, for example. Parry and Toman (2002) show that these measures may induce emissions reductions and decrease overall abatement costs, respectively. However, the abundance and ordering effect seriously lower the effectiveness of these policies. First, consider ERCs that subsidize firms for reducing emissions below business-as-usual (BAU) emissions. 18 This implies that the subsidies the firm receives amount to:

For a short banking period, i.e. when TB is small, however, firms would never find it profitable to reduce emissions below BAU levels. In the absence of banking, firms have an incentive to increase emissions. Hence, if the period over which the permits can be redeemed (TB − T) is short, the gains from relaxing the emissions constraint over this period are likely to be too small to outweigh the cost of reducing emissions below BAU levels during the interim phase. Even for a long banking period, banking is never attractive. This is easiest to see in the case with one resource, shown in Fig. 1. Banking allows firms to smooth out extraction between time 0 and TB. They will choose the intertemporally efficient path along which the marginal value of extraction grows at rate ρ (i.e. a Hotelling path) over the entire period between 0 and TB. Hence, the effect of banking is to allow for an extended period of declining emissions (until TB > T). After the emission constraint ceases to bind (after TH), extraction must be on a similar path. The possibility to arbitrage between unconstrained phases requires that extraction before T and after TH must follow the same Hotelling path. Hence, if emissions are initially below BAU, so that firms can bank credits, they must be below BAU levels after TH as well. In this case, however, cumulative extraction is lower than the initial stock of resources and firms are left with unsold resources, which is suboptimal. Hence, banking cannot arise in equilibrium. Combining the possibility to bank ERCs and permit trading among firms introduces problems of its own (cf. Kim and Lyon, 2011; Michaelowa and Rolfe, 2001). Trade in permits between polluting firms creates ‘leakage’ problems, in analogy to situations in which some sectors are regulated and others are not. During the grace period, firms can only be rewarded for early reductions but cannot be punished for releasing more emissions. It is then possible that some firms would reduce their emissions and accumulate permits, while others increase extraction enough to (more than) offset such early efforts. In addition, early action policies of the type discussed here suffer from ‘additionality’ problems similar to those encountered with Clean Development Mechanism (CDM) and Reducing Emissions from Deforestation and Forest Degradation (REDD) schemes. In both cases it is difficult to establish a credible BAU for emissions/deforestation. In our non-renewables case, firms can choose at what point in time to extract in the absence of regulation and will be indifferent between extracting earlier or later along the equilibrium path. Hence, it is extremely difficult to establish a baseline emission path for individual firms for the interim phase.

n o T ∫ σ ðt Þmax 0; Z~ ðt Þ−Z ðt Þ dt;

7.3. Alternative Policies after Implementation

7.2. Early Action Policies

0

where Z~ is BAU emissions and where the subsidy σ(t) might be related to the permit price after T, as in Parry and Toman (2002). Much of the present paper has been devoted to showing that, in the absence of these subsidies, firms benefit from increasing emissions. As a consequence, the level of the subsidy must be substantial to outweigh the gains from emissions expansions — this instrument may therefore be quite costly from the point of view of the policy maker. Second, consider the case in which firms can reduce emissions relative to the BAU emissions level, and ‘bank’ these reductions to allow

18 In our case, BAU emissions coincide with the laissez-faire (unregulated) emissions, but this does not need to be the case in the presence of pre-existing regulation.

The second possible response to the problems stemming from implementation lags is to redesign the announced policies themselves. The policy that is implemented with a lag should be designed in such a way that the incentive to anticipate the policy with increased emissions is minimized. In practice, regulation has taken the form of caps on emission flows, and we considered this type of policy in our theoretical model. 19 Possible alternatives to this type 19 In our discussion above we have opted for a type of policy where the cap is fixed over time (‘Kyoto forever’). An obvious alternative is to model a cap that gets tighter over time (‘cap-and-squeeze’). With a tighter constraint over time, more of the resource needs to be extracted in the interim phase, reinforcing the abundance effect. Moreover, our second effect, the ordering effect, also becomes stronger. As the cap gets tighter, the low-emission source becomes more valuable, and hence scarcer. Thus, the scarcity rents of the two resources diverge.

C. Di Maria et al. / Ecological Economics 74 (2012) 104–119

of policy might be constraints on emission concentrations, cap-andtrade systems, and banking provisions. When damages arise from concentrations of pollutants, these policies might be superior to the regulation of emission flows, in terms of providing better incentives for polluters to internalize the damages at minimum costs. We argue, however, that none of these alternative policies will eliminate the increase in emissions in the interim period that arise from the abundance and ordering effect. Whenever the pollution damage is a stock-externality (as in the case of climate change, since it is the stock of greenhouse gases in the atmosphere that causes global warming) environmental policy in the form of a ceiling on the stock of emission concentrations (i.e. a concentration target) has better potential to internalize damages. However, the abundance and ordering effects will arise unabated. To analyze this policy in our model, another state variable has to be introduced, and a corresponding co-state variable (the shadow price of emissions, to be interpreted as the optimal emissions tax). Provided that the ceiling becomes binding at some future time, a positive emissions tax emerges at time T irrespective of whether the ceiling is binding at that point in time. The presence of a grace period where no policy can be implemented means that between time 0 and T there is still an interim phase with unpriced emissions. Following the same steps as in Appendix A and the proof of Proposition 1, one can show that an abundance effect occurs given that the ceiling is binding for a positive period of time. Moreover, since the cleaner resource provides more consumption per unit of emissions, such resource becomes relatively scarcer under environmental policy if its initial stock is ‘too small’ in the sense discussed in Section 5. Hence, with a stock constraint or concentrations target, the abundance and ordering effects will occur, if the target is binding for some period of time. Similarly, a policy announcing a cap-and-trade system with banking and borrowing (i.e. banking and borrowing after the implementation date T only) might reduce compliance costs for firms, but is not eliminating the abundance and ordering effects. Consider the case in which cumulative emissions over a certain period (T, TB) are capped. Within this enforcement phase, emissions follow a Hotelling path as banking allows agents to efficiently spread extraction over time. The cumulative extraction over the constrained period, however, must be lower than under laissez-faire. Once again, more of the resource must be extracted outside of the constrained phase, both before implementation and after the end of the enforcement phase. Thus, the abundance effect remains unaffected. A special case is the case in which all cumulative emissions are capped from T onwards, i.e. TB goes to infinity, while the cap is binding. In this case we find an extreme abundance effect in its simplest form: the difference between capped cumulative emissions and the laissez-faire stock of emissions gets all emitted before the implementation of the policy. Even with cap and trade, the low-emission resource becomes more valuable for large initial stocks of the high-emission resource, which affects the relative scarcity rent along the lines of our analysis. Hence the ordering effect would also emerge. If quantity policies (emission quotas) cannot eliminate the abundance effect, would price policies work? It seems that a tax policy can be designed so as to eliminate the scarcity and abundance effects. (Sinclair, 1992) shows that emissions from non-renewable resources can be back-loaded by implementing an emissions tax that declines over time. This provides firms with the incentives to postpone extracting until taxes are low. Sinclair did not include an implementation lag. In our setting one would expect that if the carbon tax implemented at time T declines sufficiently fast, and perhaps becomes a subsidy, it can provide the incentive to not extract too much before T and thus remove the abundance and ordering effect. Notice, however, that this discussion relies on the expectation of low taxes in future. Moreover, emission subsidies (rather than taxes) might introduce additional problems of their own in terms of perverse entry effects.

115

These issues, however, are beyond the scope of the present paper, and additional research on these aspects is needed. 20

8. Conclusions In this paper we have focused on the possibility that implementation lags, a common feature of most large-scale policy changes, have effects that go against the spirit of the proposed regulation. In particular, we have shown that, in the specific case in which the policy being implemented consists in regulating the use of exhaustible resources, like in the case of climate policy or acid rain policy, resource use increases as a result of the implementation lag. When an emissions reduction policy is announced, an abundance effect occurs, which increases the demand for polluting resources at the time of announcement: under an implementation lag, resource use exceeds the laissez-faire's level until the constraint becomes binding. As a consequence, if the emission intensity of extraction does not fall enough during this phase, emissions will be higher than in the absence of regulation. Whether the emission content of extraction changes during the interim phase, depends on its composition. Our analysis shows that whenever the low-emission input is relatively scarce, only the highemission input is used during the interim phase. This is what we call the extreme ordering effect. When the low-emission input is (relatively) less scarce, a weak ordering effect occurs, leading to an increase in the expected emission intensity of aggregate consumption during the interim phase. Finally, when the low-emission input is not relatively scarce, no ordering effect materializes, and the expected emissions intensity is the same as under laissez-faire. In all cases, the pollution intensity does not fall. We have provided some empirical evidence that highlights that the pattern of use of high- vs. low-sulfur coal in the US in the 1990s suggests that the abundance and ordering effects may have occurred after the announcement of Title IV of the 1990 CAAA. Our estimates are consistent with power generators having increased their sulfur emissions by roughly 9% as a consequence of the policy implementation lag. Bringing together our theoretical and empirical results, we have concluded that the costs arising from implementation lags might be substantial in a number of cases, and should be included in cost–benefit analysis leading to the design of environmental policies. We have discussed several policy options available to policy makers, both in terms of ‘early-action’ policies, and in terms of policies designed to maximize the overall net benefit of environmental regulation. While tentative, our conclusions point to the fact that early actions might not be as beneficial in the presence of exhaustibility as otherwise claimed, and that quantity instruments might not be able to offset the abundance and ordering effects. Our results in this paper are derived using a rather stylized model. In what follows, we argue that allowing for different extensions, while making the model more complicated and less tractable, does not change the paper's main insights. A feature commonly found in the literature on exhaustible resources, and one that we have abstracted from in the present work, is the presence of extraction costs. Consider the benchmark case with constant marginal extraction costs, and assume that extraction costs are lower for the dirty input than for the cleaner input. Without environmental policy, the ordering of extraction will simply follow the least-cost-first result of Herfindahl (1967), and the dirty resource gets used first. In response to the policy announcement, the abundance effect still arises as the emission constraint makes

20 The interested reader is referred to a companion paper, Di Maria et al. (2010), that focuses on the optimal design of taxes under implementation lags.

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resources more abundant, independently of extraction costs. In this simple setting, however, there will be no additional ordering effect unless the high-emissions source is exhausted in the interim of the regulation. Thus, the average pollution intensity in the interim period is not likely to change unless the stock of the dirty resource is very small. When one thinks of large policy efforts such as the acid rains programs or climate policies, the dirty resource would be (highsulfur) coal. Indeed, the large stocks of coal available in reality, and the short time-span of the interim phase (5 to 20 years, say) devoid this possibility of realism, and the extension of our model with constant extraction costs is unlikely to reverse our main conclusions. The inclusion of a backstop technology will not affect the main conclusions of the paper either. If the price of the backstop is high enough to cause a switch away from the polluting resources after the emission ceiling ceases to be binding (which is, for example, assumed by Chakravorty et al., 2008), our results do not change at all. If the backstop price is lower, any changes to the extraction paths described in this paper, are to be found during the constrained phase: if a switch to the backstop occurred before time T, emissions would be zero from that instant onward, and the emission constraint would never be binding. With a switch to the backstop after time T, the abundance effect, which only requires that the cap be binding for some time, would certainly emerge; the ordering effect may also occur, as the conservation of the low-emission resource remains attractive in the presence of a more costly alternative. This brief discussion shows that, while many avenues remain open for possible extensions, the main insights of this paper are robust to many modifications of the model set-up.

Appendix A. Continuity of the Scarcity Rents Let us proceed backwards. The second stage problem can be simply written as: max

∞

−ρt

∫ U ðRðt ÞÞe

fRH ðt Þ;RL ðt Þg∞ T T

ðA:1Þ

dt

s:t:Rðt Þ ¼ RH ðt Þ þ RL ðt Þ;

ðA:2Þ

S_ j ðt Þ ¼ −Rj ðt Þ; Rj ðt Þ≥0; Sj ðT Þ given;

ðA:3Þ

Z ðt Þ≡ε H RH ðt Þ þ εL RL ðt Þ≤ Z
:

ðA:4Þ

Let VT(S(T)) be the value-function corresponding to the initial stock S(T) = {SL(T), SH(T)}: ∞

−ρðt−T Þ

V T ðSðT ÞÞ ¼ ∫ U ðR ðt ÞÞe T

dt;

where Rðt Þ ¼ fRL ðt Þ; RH ðt Þg. We then have (Leonard and Long, 1992, Eq. (4.80)): ∂V T ð⋅Þ 2 ¼ λj ðT Þ; ∂Sj ðT Þ

ðA:5Þ

S_ j ðt Þ ¼ −Rj ðt Þ; Rj ðt Þ≥0; Sj ð0Þ given:

ðA:8Þ

In this case, the transversality conditions read (Leonard and Long, 1992, Theorem 7.2.1): 1

λj ðT Þ ¼

∂V T ð⋅Þ : ∂Sj ðT Þ

ðA:9Þ

Using Eq. (A.5) together with (A.9), and ruling out discontinuities in the stocks, yields Eq. (6). □ Appendix B. Initial Stocks and the Abundance Effect Appendix B.1. Proof of Lemma 1 If, for given initial total resource stock S, we choose the Hotelling path for total extraction, R, and find it feasible to choose the composition of extraction so that ZbZ
, then it must also be optimal to follow such path, since the non-constrained path cannot be worse than any constrained one. Hence, we only need to prove that the unconstrained (Hotelling) solution is feasible in the constrained economy if we start in area OABC. For the unconstrained problem (1.a)–(1.d), we denote the solution of total extraction at time t when the total stock at time 0 equals ~ ðt; S0 Þ. Since the problem is time autonomous, we can also S0 by R ~ ðθ; ΣÞ for the solution of total extraction a period of length θ write R after S = Σ was reached, i.e. along an unconstrained path and for
 ~ ðθ; ΣÞ if S(t′) = Σ. Finally, we define Sih, any t′, we have R t ′ þ θ ¼ R   ~ i = L, H, as the solution of R 0; Shi ¼ Z
=εi , i.e. at the time the total stock equals Sih, total unconstrained extraction equals Z
=εi . 1. If S(t) b SHh (which is represented by a point to the left of the iso

 ~ t ′ ; Sðt Þ bZ
=εH and Z t ′ b Z
for extraction line through A), then R all t′ ≥ t along the Hotelling path. Hence the Hotelling path is feasible in the constrained economy. 2. If S(t) > SLh (which is represented by a point to the right of the iso~ ðθ; Sðt ÞÞ > Z
=εL so Z ðt þ θÞ > Z
for extraction line through C), then R θ > 0 sufficiently small along the Hotelling path. Hence the Hotelling path is not feasible in the constrained economy. 3. If SLh ≥ S(t) ≥ SHh (which is represented by any point between or on ~ ðθ; Sðt ÞÞ≤ the iso-extraction lines through A and C), then Z
=εH ≤R Z
=εL for θ > 0 sufficiently small. For any R≤Z
=εL , we have Z ¼ εL RL þ


  εH ðR−RL Þ≤Z
if RL ≥max 0; εH R−Z
=ðεH −εL Þ ≡RzL ðRÞ. Hence, RLz(R) is the minimum amount of L needed to ensure Z≤Z
when total extraction equals R, and puts a lower bound on the extraction of L in the constrained economy. Since RL ≤ R, we note that RLz(R) is defined only for R≤Z
=εL . Hence, by construction, to make h an unconstrained extraction  path, with  S(t) ≤ SL , compatible with ~ ð0; Sðt ÞÞ of L at any time t. Therefore Z ðt Þ≤Z
, we need at least RzL R the minimum amount of L needed over time to make an unconstrained path that starts with S(t) ≤ SLh feasible in the constrained economy equals   ∞ z ~ ðθ; Sðt ÞÞ dθ: ∫ RL R 0

where λj2 is the costate variable associated with the state Sj in the second stage problem. We can now write the first stage problem explicitly including the value-function VT(S(T)) as a scrap-value function, i.e. max

T

−ρt

∫ U ðRðt ÞÞe

fRH ðt Þ;RL ðt ÞgT0 0

s:t:Rðt Þ ¼ RH ðt Þ þ RL ðt Þ;

−ρT

dt þ e

V T ðSðT ÞÞ

ðA:6Þ

This integral equals zero if S(t) b SHh, since RLz = 0 for RbZ
=εL ¼   ~ 0; Sh , but this case was already discussed above; otherwise R H the integral equals:   t z ~ z ðθ; Sðt ÞÞ dθ≡SL ðSðt ÞÞ; ∫ H RL R 0

ðA:7Þ

where tH is the time it takes to reduce the stock from S(t) to SHh ~ ðt ; Sðt ÞÞ ¼ Z
=ε . along an unconstrained path, i.e. tH solves R H

H

C. Di Maria et al. / Ecological Economics 74 (2012) 104–119

Thus SLz(S) is the minimum amount of L needed to make the unconstrained extraction path feasible in the constrained economy, when starting from a total stock S. The corresponding maximum amount of H is then SHz(S) = S − SLz(S). Now it follows immediately that the minimum L and maximum H that are needed to ensure a Hotelling path with total resources S ∈ [SHh, SLh] can be realized in the constrained economy are given by the pairs {SLz(S), SHz(S)}. These pairs are represented by segment AB in the figure, of which the slope     0 0 ~ ð0; SÞ−Z
= εH R ~ ð0; SÞ−Z
, can be expressed as: Sz ðSÞ=Sz ðSÞ ¼ εL R H

L

so that the slopes of AB in points A and B are given by, respectively,
L and pðt Þ ¼ p
; in point ~ ðt; S0 Þ ¼ R infinity and 0. Hence in point B, R L ~ ðt; S0 Þ ¼ R
H and pðt Þ ¼ p
. Note that in the main text we denote A, R H SLm as the minimum amount of L needed to make the Hotelling path that starts at S = SLh feasible in the constrained economy, i.e. SLm ≡ SLz(SLh). Appendix B.2. Proof of Lemma 2 We are looking for the largest initial stocks, such that if an Hotelling path is followed from t = 0, the pollution constraint is not violated from T onwards. By Lemma 1, this implies that stocks at t = T must be on ABC. To allow unconstrained extraction between t = 0 and t = T, the initial total stock must equal S(T) + δ such that S(T) is on ABC and δ is cumulative extraction following Hotelling between t = 0 and t = T. Let δ solve ~ ðt; S þ δÞdt and call the solution δ(S, T). For any point (SL, SH) δ ¼ ∫T R 0

on ABC, δ(SL + SH, T) represents cumulative extraction along a Hotelling path over a period of length T that ends with total stock SL + SH. The maximal initial stocks are then represented by A′B′D such that the vertical distance between any point (SL, SH) on ABC and A′B′D equals δ(SL + SH, T). Note that the vertical distance between BC and B′D equals δ(SLh, T) ≡ δL and the vertical distance between A and A′ equals δ(SHh, T) ≡ δH. The vertical distance between AB and A′B′ increases the more we move to the right since points on AB further to the right are associated with higher S, and δ increases in S. Appendix C. Optimal Extraction Paths Lemma C.1. Along any optimal constrained path,  λL(t)=λH(t)m∀t∈[0, ∞) −ρ½TþðSL0 −SL Þ=R
L −t  T
dt . if and only if SL0 >SLm, and SH0 bŠ H ðSL0 Þ≡Sm þ ∫ d p e L H

117

Second, whenever τ > 0, switching from exclusive use of H to mixed use, and from mixed use to exclusive use of L cannot be optimal:
H. A • When τ > 0, exclusive use of H use implies Rðt Þ ¼ RH ðt Þ ¼ R switch to mixed use implies either lower pollution (contradicts τ > 0), or higher extraction (violating continuity) or both;
L . A switch • When τ > 0, exclusive use of L use implies Rðt Þ ¼ RL ðt Þ ¼ R to mixed use implies either higher pollution (violating the pollution constraint) or lower extraction (violating continuity) or both. Third, we note that it cannot be optimal to use L immediately before the economy becomes unconstrained since any use of L at
H , while when τ(t) = 0 only H is used, so the cap implies Rðt Þ > R
that Rðt Þ≤R H and R(t) has to be continuous. It follows that, when constrained, the economy uses both resources simultaneously until L is depleted, followed by exclusive use of H, and possibly preceded by exclusive use of L. When both resources are used simultaneously, Z ¼ Z
and p(t) = λH(t) + εHτ(t) = ˆ ˆ λL(t) + εLτ(t), so that λL ¼ λH ¼ τ^ ¼ p^ ¼ ρ. Simultaneous use and a binding cap requires that, given a path R(t) of total extraction, RL(t) = RLz(R(t)). A price growing at rate ρ implies that R(t) coincides ~ ðt; σ Þ for some σ. Thus, simulwith an unconstrained path, i.e. Rðt Þ ¼ R taneous use can at most last for the logðp
L =p
H Þ=ρ periods, during which the price grows from p
L to p
H and cumulative extraction is    ~ t; Sh dt = SLm. If SL0 > SLm, simultaneous use must be ∫logðp
L =p
H Þ=ρ RzL R L 0

L for SL0 −Sm =R
L periods) to preceded by exclusive use of L (at R L allow full exploitation of L. We can now calculate the minimum initial amount of H, given SL0, needed to make the path derived above feasible. The minimum amount is used when the period of exclusive use of H in the constrained period is minimized, which requires τ approaching 0 at the time L is depleted, which, in turn, requires τ approaching 0 when simultaneous use starts. But then, the use of H is, by construction, equal to Š H ðSL0 Þ, since this was calculated for exclusive H use between 0 and T and a zero tax after exclusive use of L. Hence, we have proven that if λL > λH and SL0 > SLm, we must have SH0 > Š H ðSL0 Þ. Thus, if S0 ∉H, SL0 > SLm and SH0 bŠ H ðSL0 Þ, we cannot have λL > λH, and, by lemma 3, we must have equal λ's. Hence, we have proven that if we are in zone I or II, we must have equal scarcity rents. □

0

Proof. Only if: Suppose λL(t) = λH(t). Since S0 ∉H, there exists T ¼ ½T; T H Þ such that τ(t) > 0 ∀t∈T, and τ(t) = 0 elsewhere. Since λL(t) =
L and RH(t) = 0 ∀t∈T, see Eq. (4). From this and λH(t), then RL ðt Þ ¼ R Lemma 1, it follows that SL(TH) ∈ [SLm, SLh], and SH(TH) ∈ [0, SHm]. Since TH > T, SL0 > SLm. As SL(TH) ∈ [SLm, SLh], it follows that ∀ SL(0) = SL0, at most SL0 − SLm of L can be extracted during the constrained phase. Thus, this phase lasts

at most SL0 −Sm L =R L periods. Since extraction must be continuous at TH, it follows that λH ðT H Þ ¼ λL ðT H Þ ¼ p
L . From Eqs. (4) and (5), it follows that the maximum amount of H that can be extracted during the interim phase is:   m T −ρ Tþ S −S =R
−t δ~ H ðSL0 Þ ¼ ∫ d p
L e ½ ð L0 L Þ L  dt; 0

m

where we have used T H ¼ T þ SL0R
−SL . Since SH(TH) ∈ [0, SHm], and RH(t) = L ~ 0 ∀t∈T, it follows that SH0 bŠ H ðSL0 Þ≡Sm H þ δ H ðSL0 Þ. If: Suppose λL > λH. Assume that both SL0 and SH0 are positive. We first note that then, whenever τ = 0, we cannot have simultaneous use, see Eq. (4), and that we cannot have a switch from exclusive use of one resource to exclusive use of the other resource (since it would imply a jump in price and hence in extraction, so that the Hamiltonian becomes discontinuous). Hence, RL = 0 whenever τ = 0, and all L must be depleted when t∈T to satisfy the transversality condition (9).

Corollary C.1. Any optimal constrained path outside of zones I and II, entails λL(t) > λH(t) ∀ t ∈ [0, ∞). We now have all the necessary elements to prove that the optimal paths look indeed like they are drawn in Fig. 3. Appendix C.1. Endowments in Zone I For initial endowments in the interior of zone I, λH(t) = λL(t), and optimal paths enter zone OABC crossing the iso-extraction line BC, or hit the abscissa, having exhausted H prior to enforcement. Both in the interim phase, and after the constraint becomes slack, extraction is a matter of indifference. When the constraint binds, however, τ(t) > 0 and, from Eq. (4), λL(t) + εLτ(t) b λH(t) + εHτ(t), thus implying that only L is used in the constrained phase. Hence, the ordering of extraction is indeterminate ∀ t ∈ [0, T) ∪ [TH, ∞), while only L is used in the constrained phase. The path of the price p(t) over time is presented in Figure C.1, where the solid line represents the actual path of the price of consumption. Appendix C.2. Endowments in Zone II The path is identical to the previous case, with the notable exception that now the trajectory must enter zone I before the enforcement

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C. Di Maria et al. / Ecological Economics 74 (2012) 104–119

p(t )

p(t )

λ i (0)e ρ t ε i τ (t )

p¯ H p¯ L p¯ L

λ L (0) λ H (0)

λ i (0)

L only

indeterminate

T

t

TH

T

H only

indeterminate

Tl

L at cap joint at cap

t

TH

Tlh H at cap

H only

Figure C.1. Price paths for endowments in zones I and II.

Figure C.3. Price paths for endowments in zone III.

phase begins at time T to ensure that the path enters zone OABC crossing the iso-extraction line BC, or hit the abscissa, having exhausted H prior to enforcement. This is required since in the proof of Lemma C.1 we show that, we must have SL(TH) ∈ [SLm, SLh], and SH(TH) ∈ [0, SHm], where TH is the time the constraint becomes slack. In terms of the evolution of the price over time, the graph is qualitatively identical to the one drawn for endowments in zone I (see Figure C.1).

point A – and the rest of the path is a pure Hotelling one to exhaustion. The price path corresponding to this extraction path is presented in Figure C.2.

Appendix C.3. Endowments in Zone IV Paths starting from zone IV, λH(t) b λL(t). Thus, it is optimal to extract only H ∀ t ∈ [0, T) ∪ [TH, ∞). Since SL0 b SLm, the phase of simultaneous extraction cannot be preceded by exclusive use of L (see proof of Lemma C.1). Hence, the constrained period begins with an initial phase of joint extraction at the cap along a trajectory parallel to AB, until L is exhausted. The path is parallel to AB, since path AB represents extraction at the cap with simultaneous use and the price, p, growing at rate ρ; these characteristics both apply to an unconstrained path with maximum H stock (i.e. along AB) and to a constrained path with simultaneous use and unequal λ's (see proof of Lemma C.1). L is exhausted as the price reaches p
H , at time Tlh say. Following this, only H is extracted at the cap until the cap ceases to bind – at p(t )

λ i (0)e ρ t ε i τ (t )

p¯ H

ps

λ L (0) λ H (0) T

H only joint at cap

t

TH

Tl h H at cap

H only

Figure C.2. Price paths for endowments in zone IV.

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