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Optimal emission-extraction policy in a world of scarcity and irreversibility
Fabien Prieur, Mabel Tidball
To cite this version:
Fabien Prieur, Mabel Tidball. Optimal emission-extraction policy in a world of scarcity and irre- versibility. 18. Annual Conference EAERE, European Association of Environmental and Resource Economists (EAERE). INT.; Université Panthéon-Sorbonne (UP1). Paris, FRA. Ecole d’Economie de Paris (Paris School of Economics) (PSE)., Jun 2011, Rome, Italy. 25 p. �hal-02807037�
Optimal emission-extraction policy in a world of scarcity and irreversibility
PRELIMINARY DRAFT Abstract
This paper extends the classical exhaustible-resource/stock-pollution model with the irreversibility of pollution decay. Within this framework, we are wondering first how the potential irreversibility of pollution affects the extraction path. Our aim is also to emphasize the conditions under which the economy will optimally adopt a reversible policy. Once the situation has turned irreversible, we show that the pollution problem does affect the total amount of resource extracted. In particular, it may be optimal to leave a positive amount of resource in the ground forever. As far the optimal extraction/emission policy is concerned, three types of solutions may arise. We derive a simple condition that guarantees that it is optimal to stay in what is called the reversible region. When this condition does not hold, it is difficult to conclude whether the optimal policy is reversible or irreversible. Using a numerical example, one can find a situation where the optimal path is unique and identify the set of initial conditions associated with each possible policy. Due to the non-convexity introduced by the decay function, the occurrence of multiple optimal solutions cannot be ruled out. Indeed, we present another numerical example in which two optimality candidates – one being reversible, the other irreversible – simultaneously exist. The computation of the present values of both paths reveals that the reversible policy yields the highest value.
JEL classification: Q30, Q53, C61.
Keywords: non-renewable resource, irreversible pollution, optimal policy.
1 Introduction
In the design of optimal climate policy it should be taken into account that most emissions of CO2 result from burning fossil fuel, which originate from non-renewable resources. It is forcefully argued by e.g., D’Arge and Kogiku (1973) that "the ’pure’ mining problem must be coupled with the ’pure’ pollution problem". Although not really applicable to the climate change problem they also state that questions like these become relevant: which should we run out first, air to breathe of fossil fuels to pollute the air we breathe? An early prototype model can be found in Withagen (1994) where utility is derived from consumption of fossil fuel from a non-renewable resource and where the use of the fossil fuel also contributes to the accumulation of CO2. The accumulated stock of CO2 causes damage, represented by a convex damage function. One of the findings is that the extrac- tion path of the fossil fuel becomes flatter than in the absence of environmental damage.
In a similar model Sinclair (1994) and Ulph and Ulph (1994) derive the optimal carbon tax needed to implement model the first-best optimum. Still focusing on the exhaustible- resource/stock-pollution model, Tahvonen (1997) fully characterizes the properties of the optimal extraction/emission policy. He notably shows that, since extraction and pollution necessarily converge toward zero, in the long run, the pollution problem does not have an influence on the total amount of resource extracted over the planning period. In that sense, the pollution and the resource management problems are independent of each other.
A widely used alternative to capturing environmental damage is to impose a ceiling on the total accumulated stock of CO2. Examples of this approach are Chakravorty et al.
(2006, 2008). This is usually motivated in the following ways. First one could argue that a ceiling is a political reality. International negotiations are indeed aiming at keeping the temperature rise below 2 degrees C and it is widely accepted that in order to accomplish this the CO2 concentration should be no more than 450 ppmv. Second, and related to the first argument, taking a damage function rather than a ceiling, may lead to highly undesired outcomes because there is high uncertainty surrounding the effects of climate change and catastrophes may occur (see e.g., Tsur and Zemel, 2008). So, with a ceiling one models the relatively safe region and irreversibility. The focus of the present contribution is on irreversibility beyond a certain pollution level, but in a different sense. Usually the decay of pollution is modeled as linear, meaning that a constant percentage of the existing stock is diluted per unit of time. However, this approach has been criticized by many authors including Dasgupta (1982), Fiedler (1992), and Pethig (1993). Alternative specifications have been proposed by e.g., Forster (1975). They usually allow for inverted-U shaped decay with the important feature that there exists a critical threshold of pollution
above which the assimilation capacity of Nature becomes permanently exhausted, thereby implying an irreversible concentration of pollution. In this way a ceiling is introduced, not on the allowed stock of pollutants but on the stock of pollutants that allows for decay.
The decision maker is then faced with the problem whether it is optimal to stay below the ceiling and benefit from decay or going beyond it, because of higher consumption, and then stay in the irreversible region.
It is well known that the climate system is extremely complex and that economists’
modelling of it is rather rudimentary. We do not aim at catching this complexity in a simple model. But it is clearly the case that representing decay as a constant fraction of the existing stock is far too simplistic and that with a large stock of CO2 the absorp- tion capacity of oceans and forests may be reduced considerably. This is what we want to capture by introducing the ceiling or irreversibility threshold. Indeed, experts of the second working group of the IPCC (2007) have identified positive climate feedbacks due to emissions of greenhouse gases (GHG). There is more andmore evidence that increasing emission levels and concentrations of GHG disturbs the regeneration capacity of natural ecosystems. Oceans, that form the most important carbon sink, display a buffering ca- pacity that begins saturating. At the same time, the assimilation capacity of terrestrial ecosystems (lands and forests form the other important carbon sink) will likely peak by mid-century and then decline to become a net source of carbon by the end of the present century. Therefore, the irreversible degradation of the assimilation capacity of Nature does not seem so distant from today. Irreversibility of decay has been studied before by Tahvo- nen and Withagen (1996) but they neglect exhaustibility. An important result obtained by Tahvonen and Withagen is the potential multiplicity of equilibria. The inverted-U shaped decay function introduces a non-convexity that may give rise to multiple paths satisfying the necessary conditions, starting from the same initial stock values.
The present paper adds to the contributions of Tahvonen and Withagen (1996) and Tahvonen (1997) by introducing the irreversibility of decay in the classical exhaustible- resource/stock-pollution model, without extraction costs and backstop technology. Our approach to modelling irreversibility is to assume that the decay rate is linear for levels of pollution lower than some critical level, after which decay is zero and remains zero.
Within this framework, the first question is to know how the potential irreversibility of pollution affects the extraction path. Our aim is also to emphasize the conditions under which the economy will optimally adopt an irreversible policy.
Once the situation has turned irreversible, we show that the pollution problem does affect the total amount of resource extracted. In particular, it may be optimal to leave a
positive amount of resource in the ground forever. As far the optimal extraction/emission policy is concerned, three types of solutions may arise. We derive a simple condition that guarantees that it is optimal to stay in what is called the reversible region. When this condition does not hold, it is difficult to conclude whether the optimal policy is reversible or irreversible. Using a numerical example, one can find a situation where the optimal path is unique and identify the set of initial conditions associated with each possible policy.
We also have a rudimentary non-convexity and it is one of the aims of the present paper to investigate the occurrence of multiplicity in our model. We present another numerical example in which two optimality candidates – one being reversible, the other irreversible – simultaneously exist. Computation of present values reveals that the reversible policy yields the highest value.
The paper is organized as follows. In section 2 we present the formal model. Section 3 characterizes the optimum for the cases of i/ an abundant natural resource and ii/ a scarce resource, which can serve as benchmarks. We also provide some economic intuition about when irreversibility should play a role. In section 4, we look at the case where pollution might become irreversible and discuss whether it is optimal to deplete the fossil fuel or not. Section 5 analyzes properties of the optimal extraction/emission policy and addresses the issue of multiplicity. Section 6 concludes. Throughout the text we illustrate our findings with a leading example for the sake of exposition.
2 Problem description
We consider a partial equilibrium representation of the global warming problem. Our carbon economy is described by the following set of assumptions. The economy produces one good with a technology that uses a natural resource,x(t) ≥ 0. There is no extraction cost. Emissions, y(t) ≥ 0, are a production by product. We assume that emissions are proportional to the amount of resource extracted and used at each date, with a one-for- one relationship. The resource is non-renewable (fossil fuels). It means that the stock of resource, x, follows the usual law:
x(0) = x0 and ˙x(t) = −y(t) ↔ x(t) = x0− Z t
0
y(u)du (1)
wherex0, the initial stock, is given.
Let U (y) be the utility derived from the production. In the same vein as Tahvonen and Withagen (1996):
Assumption 1. The utility function is such that: U (0) = 0, U00(y) < 0, U0(0) < ∞ and ∃!¯y/U0(¯y) = 0.
Remark. The utility function can be understood as a profit function: U (y) = py − c(y) with p a constant and exogenous price and c(y) a convex production cost.
Emissions contribute to the accumulation of a pollution stock, z(t). Pollution accu- mulation is not innocuous to ecosystems and, in particular, it affects their capacity to regenerate. We assume that pollution would turn irreversible if the stock were to reach a critical threshold ¯z > z0. This irreversibility threshold is known to the policy-maker.
Therefore, we do not consider any uncertainty surrounding ¯z. To account for irreversibility, the dynamics of the pollutant are defined piecewise:
z(t) =˙
( y(t) − αz(t) if z(t) < ¯z
y(t) else (2)
The natural regeneration or assimilation rate α is constant and positive as long as accumulated emissions are not too high that is, as long as the stock remains below the irreversibility threshold ¯z. Once the threshold is reached, a new stage occurs where the re- generation capacity is completely and permanently overwhelmed. Thus, pollution becomes irreversible.
The domain where z < ¯z is called hereafter the reversible region whereas whenever pollution is higher than or equal to ¯z, the economy lies in the irreversible region.
Pollution is damaging to the economy. For any levelz(t), denote the pollution damage asD(z(t)).
Assumption 2. The damage function is such that: D(0) = 0, D0(z) > 0, D00(z) > 0, D0(0) = 0 and limz→∞D0(z) = ∞.
Finally, note that all of the following analysis will be illustrated using a simple quadratic example:
U (y) = θy(¯y − y) D(z) = γz22
with θ, γ ≥ 0
(3)
The next section is devoted to the analysis of our benchmarks. We first recall the features of the optimal emission policy in an economy that would not be submitted to neither the irreversibility nor the exhaustibility issues. Then, the focus is on a second economy that would deal with the non-renewable character of the natural resource only.
3 Optimal emission policy: benchmarks
3.1 Stock-pollution problem
To start with, let us consider the problem of controlling polluting emissions when the source of pollution is not limited (that is when ignoring the constraint corresponding to eq. 1). The optimal emission policy is a sequence {y} that solves:
max{y} W =
∞
Z
0
[U (y) − D(z)] exp−δtdt
subject to,
( z = y − αz˙
y ≥ 0 for all t and z(0) = z0 given with δ ∈ (0, 1) the discount rate.
From any textbook (see also the seminal contribution of Plourde, 1972), we know that the resulting dynamics,
( y =˙ U001(y)((α + δ)U0(y) − D0(z)) z = y − αz˙
lead the economy to a unique saddle point equilibrium (y∗, z∗), which satisfies (α + δ)U0(y∗) =D0(z∗) andy∗ =αz∗.
The optimal policy is this situation is depicted in Figure 1. It consists in appropriately choosing initial emissions so that (y(0), z(0)) is located on the stable branch of the saddle point. So, we see that when initial pollution is low, the extraction rate is high and decreases; pollution monotonically increases until the steady state is reached. For a high level of initial pollution, the economy starts with a low extraction rate, that is increasing towards the steady state. Pollution decreases.
What may be the impact of introducing the irreversibility?
Let us first assume that there exists a threshold ¯z at which pollution turns irreversible.
If the threshold is below the initial pollution stock, the system is irreversible from the beginning of the planning period. The optimal policy in that case is studied in section 4.1. The most interesting is the one whenz0 < ¯z: the economy is initially reversible. The important question is to know whether a central planner should optimally let the system reach the irreversible region. If the initial stock is higher than the steady state value, then nothing changes compared to what we had without the threshold. Let us therefore assume that z0 < ¯z < z∗, when moving along the stable branch, the threshold is hit. In
z y
˜ z
˙y = 0
˙z = 0
0
Figure 1: Phase diagram for the case of an unlimited stock of oil and no irreversibility threshold.
that case, the central planner has to incorporate the potential irreversibility of pollution in her planning program. It is the only situation when the irreversibility constraint may influence the optimal policy.
The next section introduces the exhaustibility of fossil fuel and reassess the extrac- tion/emission problem.
3.2 Exhaustible resource/stock pollution problem
Now, we assume that the consumption of a finite resource is the source of emissions.
These emissions still contribute to the accumulation of pollution. The problem faced by the policy maker can be written as:
max{y} W =
∞
Z
0
[U (y) − D(z)] exp−δtdt
subject to,
z = y − αz˙ x = −y˙
y ≥ 0 for all t and z(0) = z0, x(0) = x0 given and limt→∞x(t) = 0.
The features of the optimal policy has been shown by Tahvonen (1997). The hamiltonian isH = U (y)−D(z)−µq −λ(y −αz), where µ ≥ 0 is the shadow price of the non-renewable
(or the scarcity rent) andλ ≥ 0 is the shadow price of pollution, which can be interpreted as an emission tax. The necessary optimality conditions:
U0(y) − λ − µ 5 0, (U0(y) − λ − µ)y = 0, y = 0
˙λ = (δ + α)λ − D0(z) µ = δµ˙
z = y − αz˙ x = −y˙
limt→∞exp−δtλ(t) = 0
limt→∞exp−δtµ(t) = 0, limt→∞exp−δtµ(t)x(t) = 0
(4)
Since the resource stock is finite, extraction – and consequently pollution – must asymp- totically go to zero. It implies that the pollution problem has no influence on the total amount of resource extracted in the course of the planning period. The scarcity rent is strictly positive and increases at an exponential rate. In addition, whenever U0(0) < ∞ (the condition imposed in assumption 1), extraction decreases to zero and the resource stock is exhausted in finite time. Of course, the larger the initial fossil fuel stock, the later exhaustion (if any) takes place.
As for the properties of the optimal policy, we can no longer draw a nice phase diagram.
But, Tahvonen (1997) has shown the following:1
Lemma 1 i/ When z0 is low enough, the pollution stock trajectory is inverted U-shaped and extraction decreases monotonically to 0.
ii/ When z0 is high enough, pollution decreases monotonically to 0 and the extraction path is inverted U-shaped.
Figure 2 provides an illustration of the time paths of the pollution stock when exhaus- tion occurs in finite time. Again, the question is when irreversibility should play a role when added to the exhaustible resource/stock pollution problem?
In case where z0 is high enough and above the ceiling, or threshold, the economy is irreversible from the initial date. The analysis is postponed to section 4.1. Still with a high enough z0 but below the threshold, clearly the issue of irreversibility will never be raised since the pollution stock monotonically decreases and approaches 0 asymptotically.
It means that the only situation in which irreversibility can be a matter for the central planner is the one where the economy starts with a low enough pollution stock. In this situation, the economy has the incentive to extract a lot of resource at the beginning of the planning period. This optimal choice is thus accompanied by a phase during which
1The result holds for D000(z) = 0, a condition that is consistent with a quadratic damage function.
pollution ceiling
0 2 4 t 6 8 10
3.2 3.3 3.4 3.5 3.6 3.7 3.8
Figure 2: Time paths of z for the exhaustible resource/stock pollution problem.
the pollution stock increases. If the maximum level of pollution is below the threshold, again we do not have to care about irreversible pollution. However, if on the contrary, this maximum lies above ¯z, which should intuitively be the case with a high enough x0, then irreversibility matters. In other words, in this situation, one expects that the feature that the pollution problem is independent of the resource extraction problem will be lost.
The purpose of the next sections is precisely to assess the conditions under which the optimal extraction/emission policy may feature irreversibility.
4 Optimal policy under irreversibility and exhaustibil- ity
Now we address the extraction/emission problem by explicitly accounting for the potential irreversibility of pollution, together with the scarcity of resources causing emissions. This is an optimal control problem that typically exhibits non convexities. Indeed, the law
of pollution is neither concave nor convex nor differentiable at z = ¯z. This may cause problems when there occurs a transition from pollution levels below the threshold to the critical level or beyond it. Hereafter, we proceed by following Tahvonen and Withagen (1996) who solve the problem in two stages. First, we consider that the economy starts, at some date, with a level of pollution equal to ¯z and assess the subproblem of maximizing discounted welfare from this date onwards. Then, the overall problem is to determine optimal emission path given anyz0 < ¯z.
Denote the instant when the threshold ¯z is reached (when the resource gets exhausted x = 0) by Tz (by Tx).
4.1 Second period problem
After the threshold has been reached, that is in the irreversible region, the problem faced by the policy-maker consists in solving:
max{y} WTz =
∞
Z
Tz
[U (y) − D(z)] exp−δtdt
subject to,
z = y˙ x = −y˙
y ≥ 0, z(Tz) = ¯z; x(Tz) = xTz ≥ 0 given and limt→∞x(t) = 0.
This problem is mathematically similar to the one studied in section 3.2 when we impose α = 0. Keeping the notations unchanged, the hamiltonian thus reads: H = U (y) − D(z) − (λ + µ)y. The set of necessary optimality include:
U0(y) − λ − µ 5 0, (U0(y) − λ − µ)y = 0, y = 0
˙λ = δλ − D0(z) µ = δµ˙
z = y˙ x = −y˙
(5)
According to the first necessary condition, when deciding the amount of emissions the planner faces the following simple trade-off: benefits from an additional unit of extraction (LHS) should be equal to the social cost that extraction entails (RHS). This social cost has two components: the damage from pollution (λ) and the increase in the scarcity rent (µ).
The transversality condition is finally given by:
t→∞lim exp−δtµ(t) = 0, lim
t→∞exp−δtλ(t) = 0 and lim
t→∞exp−δt(µx + λz) = 0 (6) Define by ˆz the sum of stock variables at the critical date Tz: ˆz = xTz + ¯z. Therefore, z is endogenous. This value yields the maximum level of pollution that can be releasedˆ by the economy during the second period.2
As it will become apparent in what follows, the behavior of the economy in the irre- versible region, provided that this region is reached, is fully determined by the relative size ofU0(0), D0(¯z)/δ and D0(ˆz)/δ, with D0(¯z)/δ < D0(ˆz)/δ. Actually, three distinct and mutually exclusive cases may occur.
Case 1 (no extraction, no exhaustion): Assume first that U0(0) ≤D0(¯z)/δ then, it cannot be optimal to increase the pollution stock over ¯z because the present value of pollution damage exceeds the benefit from the first unit of extraction (and emissions). It means that once the threshold is reached, all of the activities stop.
Case 2: Assume next that D0(¯z)/δ < U0(0), extraction is profitable at least at the beginning of the second period and there will be an interval of time with positive extraction.
Then, two sub-cases can be distinguished.
2.1 (extraction, no exhaustion): IfD0(¯z)/δ < U0(0) ≤D0(ˆz)/δ then extraction will come to an end asymptotically. It means the economy will optimally leave some resource in the ground forever.
2.2 (extraction, exhaustion): IfD0(ˆz)/δ < U0(0) then extraction will continue until the resource will be exhausted. All of the oil will be extracted and used in finite time.
Let us first analyze case 1(no extraction, no exhaustion). Assume thaty(t) = 0 ∀t ≥ Tz. It means that z(t) = ¯z and x = xTz > 0 for all t ≥ Tz.3 The dynamical system can be rewritten as:
µ = δµ˙
˙λ = δλ − D0(¯z) x = ˙z = 0˙ The general solutions, for the shadow prices, are:
λ(t) = (λ(Tz) − D0(¯z)
δ ) expδ(t−Tz)+D0(¯z)
δ and µ(t) = µ(Tz) expδ(t−Tz). (7)
2Combining the differential equations in z and x, one gets: z(t) = xTz+ ¯z − x(t) for all t ≥ T .
3The case where x(Tz) = 0, that is Tz = Tx, is a very special case. Discussion is relegated to the end of section 4.1.
In order for the transversality condition to hold, it must be true that µ(Tz) = 0 an λ(Tz) = D0δ(¯z), which implies that µ(t) = 0 an λ(t) = D0δ(¯z) for all t ≥ Tz.
It means that from Tz onwards, damages from extraction are so high (in comparison with the benefits they create) that the economy leaves the resource in the ground forever and does not value extraction anymore. Thus, we obtain that:
Proposition 1 When D0δ(¯z) ≥ U0(0), the economy directly stops all of the activities and for any t ≥ Tz:
y(t) = 0, x(t) = xTz > 0, z(t) = ¯z, λ(t) = D0(¯z)
δ , µ(t) = 0.
Proof. From the first necessary condition, it must hold that: U0(0) −λ(Tz) ≤ 0, this term being constant. This is equivalent to D0δ(¯z) ≥ U0(0).
In sum, it is optimal to leave oil in the ground because of the pressure exerted by pollution damages, which are irreversible in the second period.
Now, we turn to the analysis of case 2 (extraction with or without exhaustion). Assume that y ≥ 0 ∀t ≥ Tz. Differentiating the first optimality condition and combining this expression with the other optimality conditions, one immediately obtains the dynamical system in {y, x, z}:
y = −σ(y)˙
δ − DU00(y)(z)
y z = y˙
x = −y˙
(8)
with σ(y) = −yUU000(y)(y) the elasticity of intertemporal substitution parameter (which is pos- itive since it cannot be optimal to set extraction to a level whereU0(y) < 0).
For any feasible sequence {y, x, z}∞Tz, the shadow price of the pollution stock is defined as follows:4
λ(t) =
λ(Tz) exp−δTz− Z t
Tz
D0(z(u)) exp−δudu
expδt (9)
Assume that a long run equilibrium exists. The corresponding values of z and x are respectively denoted byz∞ > ¯z and x∞. In addition, for now on, assume thatx∞ > 0. The transversality condition implies thatµ(t) = 0 and λ(t) = R∞
t D0(z(u)) exp−δudu expδtfor allt ≥ Tz. As in case 1, the shadow price of the resource is nil in the irreversible region.
Moreover the general solution for λ(t) echoes the definition of this shadow price which
4The general solution to the differential equation in λ is obtained by using the variation of constant method. The expression of the resource rent is unchanged and given by equation (7).
represents the present value of total damages from extraction. Next, we can establish an existence result:
Proposition 2 In the regime of positive extraction, the necessary condition for the exis- tence of a long run equilibrium is:
U0(0)> D0(¯z)
δ (10)
i/ this solution is a saddle point characterized by:
{y∞, x∞, z∞, λ∞} =0, x∞, D0−1(δU0(0)), U0(0)
ii/ Along the unique optimal path leading to this solution, {y}∞Tz is monotonically decreasing whereas {z}∞Tz is monotonically increasing.
Proof. i/ When evaluating (8) at a steady state, one has y∞ = 0 and z∞ solution to D0(z∞) =δU0(0). Since z∞> ¯z and D0(.) strictly increasing, it is necessary that U0(0) >
D0(¯z)
δ . The stability property directly derives from the observation that the dynamics of z andλ (or y) are independent of x. Writing the jacobian of the dynamical system in (z, λ), one finds two eigenvalues with opposite sign. Point ii/ is a consequence of the saddle point property.
The necessary condition balances the marginal benefit of extracting the last drop of oil in the ground with the total discounted cost of doing so. Now, it is clear that if U0(0)> D0δ(¯z) it is worthwhile to extract the resource at the beginning of this second period when pollution has turned irreversible. This is true until the long run when pollution reaches the level z∞ such that D0(z∞) = δU0(0). It could be that this level is never reached in particular if the initial stock of resource is too low. In such a case, the system will experience a new stage where emissions are set to zero. Note that if x∞ > 0 then it may be optimal to leave resource in the ground because damage related to extraction becomes high enough during this second stage with irreversible pollution.
It is worth mentioning that, in contrast with the simple problem stated in section 3.2, the irreversible pollution problem has now an impact on the total amount of resource extracted during the planning period. More precisely, as far as the equilibrium value of the resource stock x∞ is concerned, we can show that:
Corollary 1 x∞> 0 if and only if U0(0) < D0δ(ˆz).
When U0(0) < D0δ(ˆz), the economy will not find profitable to extract all of the stock of oil. It means that the long run value of the pollution stock will be lower than the maximum attainable value ˆz. The stock of resource that will be left in the ground in the long run is thus equal tox∞ = ˆz − z∞. In addition, recalling that ˆz = xTz+ ¯z, it is worth noting that what is stated as a standard result in the literature on the optimal management of non- renewable resources – we mean asymptotic exhaustion – corresponds in this framework to a knife-edge situation. Asymptotic exhaustion withx∞ = 0 is possible only when xTz, which is endogenous in our problem, is exactly equal to the difference between z∞ and ¯z (which are exogenously given by the fundamentals of the economy). That is why we have chosen not to pay attention to the casex∞= 0.
The case just analyzed corresponds to 2.1 with extraction and no exhaustion. It remains to examine what happens when U0(0) > D0δ(ˆz) which implies that x∞ < 0: the economy cannot reach the long run equilibrium anymore. In this situation, the stock of resource will be exhausted in finite time (case 2.2). The economy will behave according to case 2.2 when ∃Tx ≥ Tz such that x(t) = 0, y(t) = 0 ∀t ≥ Tx, z = z(Tx) ≥ ¯z. This case includes the very particular situation where irreversibility and exhaustion occur at the same date that is, Tz =Tx. The expression ofλ is now: λ(t) = D0(z(Tδ x)), where z(Tx) is endogenous.
The two cases discussed above encompass all of the possible policies the economy can adopt once it has reached the irreversible region. In particular, we have identified a regime under which the system features asymptotic conservation of the resource. This is a striking result with respect to the related literature (see in particular Tahvonen, 1997) that predicts that, in the absence of irreversibility, the economy will use the resource until exhaustion (in finite time or asymptotic).
Let us investigate what happens before irreversibility, when the economy can rely on natural assimilation to control pollution damage.
4.2 First period problem
Starting with a level of pollution below the irreversibility threshold (z0 < ¯z), the intertem- poral extraction/emission problem entails finding a policy {y(t)} that solves:
max{y} WTz =
Tz
Z
0
[U (y) − D(z)] exp−δtdt
subject to,
z = y − αz˙ x = −y˙
y ≥ 0, z(0) = z0, x(0) = x0 given and x(Tz) = 0 z(t) < ¯z ∀t < Tz 5 ∞
z(Tz) = ¯z if Tz < ∞; limt→∞z(t) < ¯z else
This problem is very close to the one studied by Tahvonen (1997) and discussed in section 3.2. The set of necessary conditions is thus similar to the system (4). In particular,
U0(y) − λ − µ 5 0, (U0(y) − λ − µ)y = 0, y = 0
˙λ = (δ + α)λ − D0(z) µ = δµ˙
z = y − αz˙ x = −y˙
(11)
However, the problem has a significant difference since the planning horizonTz becomes a choice variable. The transversality condition is modified in the following way. Whenever Tz is finite, one has to replace the transversality condition in (4) by a new condition related to the choice of the optimal Tz. This transversality condition refers to the existence of an optimal finite Tz to reach ¯z. Define W = WTz +WTz then it must hold that
∂W
∂Tz
/Tz =Tz∗ = 0, (12)
if it is not optimal to reach the threshold in finite time, then lim sup
Tz→∞
∂W
∂Tz
≥ 0
and the usual transversality conditions hold whenTz = ∞ (see Tahvonen and Withagen, 1996). When there exists a finiteTz the new transversality condition, which simply states that the hamiltonian is continuous at the date when the threshold is reached, requires the control variabley to be continuous at Tz.
We have learned about the general features of the solution from Tahvonen (1997)’s analysis. However, for the remainder of our study, it is useful to examine in more details the kind of dynamics that govern the economy in the reversible region. Three different regimes can be observed.
First, the focus is on the regime with positive extraction, y ≥ 0, that may lead to the exhaustion of the non-renewable resource or to the crossing of the irreversibility
threshold. From the system of necessary conditions (11), one obtains µ(t) = µ0expδt. As for the shadow price of pollution, using the variation of constants method: λ(t) =
λ0+Rt
0 D0(z(u)) exp−(α+δ)udu
exp(α+δ)t. Differentiating the first eq. in (11) and com- bining the resulting expression with the other conditions, one gets the dynamics under positive extraction:
y =˙ U001(y)((α + δ)U0(y) − αµ − D0(z)) z = y − αz˙
x = −y˙
We already know that the economy cannot end up in the regime with positive extrac- tion. It means that there is no steady state in the reversible region. Two situations may then occur. Either there exists a date Tx < Tz = ∞ such that x(Tx) = 0: the economy always remains reversible. Or, there exists Tz < Tx 5 ∞: the economy reaches the ir- reversible region in finite time and then behaves according to the solution of the second period problem (see section 4.1).
Let us now examine what is going on when the economy switches to the second possible regime with no extraction and exhaustion. This occurs once all of the oil available in the ground has been extracted and used in the production process. The analysis of the dynamics in the case where x(t) = 0 ∀t ≥ Tx is as follows. Direct calculations yield z(t) = z(Tx) exp−α(t−Tx) with z(Tx) an unknown. The resource rent is unchanged and the shadow priceλ is given by: λ(t) = R∞
t D0(z(u)) exp−(α+δ)udu exp(α+δ)t. In sum, once the economy enters this second regime then economic activity ceases and Nature progressively assimilates the pollution stock. The system ends up in a state with no extraction, no more resource and no pollution.
There is finally a last regime that must be considered. It corresponds to periods of time during which the economy will stop extracting for a while and leave a positive stock in the ground. This is likely when the marginal damage due to resource extraction exceeds the marginal benefits. Stopping temporarily extraction is a means to let Nature regenerate itself. Denote the instant when extraction ceases as T2 = 0, then it must be true that x(t) = x(T2) > 0 and y(t) = 0 for all t ∈ [T2, T3], with T3 5 ∞ the instant when the economy may switch regime and extract again. The following proposition states that this regime can only be observed at the beginning of the planning period (which means that T2 = 0 if it exists).
Proposition 3 Assume that pollution is reversible along the optimal path. Then, i/ the economy will not leave resource in the ground in the long run and ii/ there is no period of
time during which the economy stops extraction and leaves a positive amount of resource unexploited.
Proof. i/ From the transversality condition, on has µ(t) = 0 for all t ≥ T2 and λ∞ = 0.
But, it implies that the first necessary condition U0(0) − λ(t) ≤ 0 is not satisfied at the limit. ii/ Assume that there exists an interval [T2, T3], with T2, T3 < ∞ such that y(t) > 0 just before T2 and just after T3 and y(t) = 0 for all t ∈ [T2, T3]. Since y(t) is continuous, there exists two dates t2 < T2 and t3 > T3 such that y(t2) = y(t3) and arbitrary small. Denote the corresponding stocks as x(t2), x(t3) and z(t2), z(t3). Then, we have z(t2) > z(t3) since during the interval of time [T2, T3], Nature assimilates part of the stock of pollution. Choose t2 close enough to T2, then t3 is close enough to T3
which means that at these two dates, the economy owns approximately the same stock of resource. Since the stock of accumulated pollution is larger at t2 than at t3, the shadow price of pollution must also be larger att2 whereas the economy approximately faces the same scarcity rents, µ(t2) and µ(t3). But, from the first necessary condition, extraction rates,y(t2) and y(t3), cannot be the same. Thus, the contradiction.
So, it may only be the case that during an interval of time following the beginning of the planning period, the economy stops extraction and leaves a positive stock in the ground.
This is likely when the initial marginal damage due to resource extraction exceeds the marginal benefits. For instance, this may be observed when the initial stock of pollution is high. As soon as extraction ceases, the pollution stock – and the environmental damage – decrease until a dateT3 at which it is optimal to extract again. But, note that it cannot be optimal to keep a positive amount of resource forever.
5 Candidates for optimality
To address the question raised by the paper, the problem basically consists in determining whether the resource will be exhausted before or after the threshold is reached that is, Tx < Tz(= ∞) or Tz < Tx. To simplify the discussion, we do not consider the case where the economy starts, at datet = 0, in the regime without extraction.
A first result can be established.
Proposition 4 Suppose U0(0) < D0δ(¯z). i/ Then z(t) < ¯z for all t ≥ 0. The optimal extraction policy is always reversible: Tz = ∞. ii/ There exists Tx such that y(t) = 0 for all t ≥ Tx.
Proof.
Assume that U0(0) < D0δ(¯z) and that there exists Tz < ∞ such that z(Tz) = ¯z. Then, from proposition 1,y(t) = 0 for all t ≥ Tz. In particular, y(Tz+) = 0. Moreover, from the transversality condition (12), y is continuous at Tz. Now, two cases are possible: either Tz < Tx = ∞ or Tz = Tx. When Tz < Tx, x(Tz) > 0 implies, from proposition 3, that y(Tz) > 0. In particular, y(Tz−) > 0. But then, the control cannot be continuous at Tz, which provides the contradiction. WhenTz =Tx, one hasy(t) = 0 and λ(t) = D0δ(¯z) for all t ≥ Tz (see proposition 1). Before Tz =Tx, from (11), the following holds: U0(y) = λ + µ.
The continuity of the costateλ and of the control y at Tz implies that: λ(Tz−) = D0δ(¯z) and y(Tz−) = 0. Combining these relations, one obtains: µ0 =
U0(0) − D0δ(¯z)
exp−δTz, which is negative under the assumptionU0(0) < D0δ(¯z), thus the contradiction.
Thus, the economy will always remain in the reversible region. It is worth mentioning that this feature holds whatever the initial conditions (z0, x0). It means that whenU0(0)<
D0(¯z)
δ , (marginal) damages related to irreversible pollution are so high that the economy never finds optimal to reach this point and consequently will exhaust the resource before facing irreversibility (...) In addition, the lower the discount rate that is, the more patient the economy, the more likely is this situation.
As noticed by Tahvonen and Withagen (1996), this condition is only sufficient to obtain a reversible extraction policy. Thus, reversible solutions may also be optimal when the reverse holds. But then the choice between reversible and irreversible trajectories becomes less simple.
What may happen in the opposite case has already been discussed in the preceding sections. WhenU0(0) ≥ D0δ(¯z), reversible and irreversible emission policies are candidate to optimality. Features of reversible optimal paths have already been emphasized in section 3.2. Figure 3 depicts the time paths of y, x and z in this case. The extraction rate and emissions are monotonically decreasing. Exhaustion occurs in finite time which in turn implies that pollution (after an initial stage of growth) starts to decline before reaching the threshold and will asymptotically converge toward 0.
When Tz exists such that the irreversibility threshold is hit, if D0(¯z)/δ < U0(0) ≤ D0(ˆz)/δ then extraction will come to an end asymptotically. The optimal trajectory can be decomposed into two stages (see figure 4, right). Both stages are characterized by positive extraction, the pollution being first reversible then irreversible. The optimal path features resource conservation. It means that, in this case, the economy will optimally leave some resource in the ground, forever. Finally, Figure 4 left illustrates the situation
stock resource emission pollution threshold
0 10 20 t 30 40 50
0 1 2 3
Figure 3: Optimal reversible extraction/emission policy.
when U0(0) > D0(ˆz)/δ that is, the resource will be exhausted in finite time. 5 The economy first experiences a stage of positive extraction with reversible pollution. Then, the irreversible threshold is hit in finite time. After the threshold has been hit, a second stage takes place where the economy continues to extract the resource. This stage lasts untilTx < ∞ and finally, the economy is left with exhaustion and irreversible pollution.
In order to gain insight into the circumstances under which one or the other – reversible or irreversible – policy will be the optimal one, a numerical analysis is performed. To start with, by making use of the following set of baseline parameters: {δ, α, γ, θ, ¯y, ¯z} = {0.05; 0.005; 0.1; 1; 10; 3.8}, we can illustrate how the optimal policy, that prevails, depends on the set of initial condition (z0, x0).
Figure 5 represents, in the (z0, x0) plan, three regions corresponding to the three pos- sible optimal trajectories that can arise in our model. These regions are separated by two
5When U0(0) = D0δ(¯z), we obtain the limit case where extraction ceases immediately after the threshold has been reached.
stock resource emission pollution threshold
0 1 2 3 4t 5 6 7 8
0 1 2 3 4
stock resource emission pollution threshold
0 10 t 20 30
1 2 3 4 5 6
Figure 4: Optimal irreversible policy. Left: with exhaustion. Right: without.
frontiers. The first frontier (red curve) is the limit case when z(Tx) = ¯z that is, when the instant when the resource is exhausted exactly coincides with the instant when the irre- versibility threshold is reached (thus Tx = Tz). Any pair (z0, x0) located on this frontier defines a unique date when this limit case occurs. For any pair (z0, x0) below the frontier, we are in the reversible scenario: the resource is exhausted before the threshold is reached.
We can observe that this is likely to occur not only when bothz0 and x0 are low but also whenz0 is close to ¯z, provided the initial stock of resource is very low.
The second frontier (green curve) is the other limit case when Tz finite exists and Tx = ∞ with x∞ = 0. Thus, in this case, the system features asymptotic exhaustion of fossil fuels.6Again, any pair (z0, x0) located on this second frontier defines a unique date when this limit case occurs. In addition, any combination (z0, x0), that lies between the two frontiers, corresponds to the optimal path with three stages: first extraction, reversible pollution (untilTz < ∞), then extraction, irreversible pollution (until Tx < ∞) and finally, exhaustion and irreversible pollution. Now it turns out that we can say a bit
6We can easily check that this frontier is defined by the condition we emphasize in the paper: D0δ(ˆz) = U0(0).
Figure 5: Sets of initial conditions associated with reversible vs. irreversible optimal policy.
more about the conditions under which the situation will turn irreversible. It was already straightforward that the higher x0 and the closer z0 to ¯z, the more likely this situation.
Here, we observe that the irreversible path can also emerge from a very low initial pollution stock, provided the stock of available resource is high enough etc.
The last region – located above the green curve – finally represents the set of initial conditions for which the optimal trajectory can be decomposed into two stages. Both stages are characterized by positive extraction, the pollution being first reversible then irreversible; some positive amount of resource being optimally not consumed.
It is worth mentioning that, for the example and the values considered, the optimal policy is uniquely defined. Now, the important question is to know whether there may exist multiple candidates for optimality. In other words, can reversible and irreversible simultaneously exist? Actually, the analysis has been developed in a framework exhibiting non-convexity, so we cannot ruled out multiplicity.
Figure 6 represents a situation where two paths satisfies the necessary conditions, start- ing from the same initial conditions. This example depicted corresponds to the particular case where δU0(0) = D0(¯z). In that case, from proposition 1, we know that once the threshold is reached the economy directly stops extraction (see the figure in the right).
Regarding the optimal reversible policy (left), it turns out that pollution first decreases then reaches a maximum at a level close to but strictly below ¯z and then decreases.
The policy maker, in our carbon economy, has thus two alternatives. On the one hand, He/She may adopt an extraction policy that is reversible since it will not lead the economy
stock resource emission pollution threshold
0 10 20 t 30 40 50
0 1 2 3 4 5
stock resource emission pollution threshold
0 5 10t 15 20
0 1 2 3 4 5
Figure 6: Multiplicity of optimality candidates. Left: reversible. Right: irreversible.
to the pollution threshold. But, at the same time, this policy causes the full exhaustion of the non-renewable resource. On the other, He/She may choose an extraction path that will optimally be accompanied by the crossing of the threshold. This in turn may force the economy to stop extraction and leave some positive amount of fossil fuels unexploited.
However, the computation of the present values of both paths reveals that the reversible policy yields the highest value.
6 Conclusion
This paper introduces the irreversibility of pollution decay in the classical exhaustible- resource/stock-pollution model. Within this framework, we wondering first how the po- tential irreversibility of pollution affects the extraction path. Our aim is also to emphasize the conditions under which the economy will optimally adopt an irreversible policy.
Once the situation has turned irreversible, we show that the pollution problem does affect the total amount of resource extracted. In particular, it may be optimal to leave a positive amount of resource in the ground forever. As far the optimal extraction/emission
policy is concerned, three types of solutions may arise. We derive a simple condition that guarantees that it is optimal to stay in what is called the reversible region. When this condition does not hold, it is difficult to conclude whether the optimal policy is reversible or irreversible. Using a numerical example, one can find a situation where the optimal path is unique and identify the set of initial conditions associated with each possible policy. Due to the non-convexity introduced by the decay function, the present paper also investigates whether the occurrence of multiple optimal solutions is possible. We present another numerical example in which two candidates for optimality – one being reversible, the other irreversible – simultaneously exist, the reversible policy yielding the highest present value.
Two main extensions of the present paper naturally come into mind. We shall first in- troduce a backstop technology in order to examine how the optimal timing of the backstop adoption and how the optimal combination of technologies are affected by irreversibility.
The second extension pertains to the observation that a lot of uncertainty surrounds both the extent of fossil fuels reserves present in the ground and the concentration of GHG that will initiate irreversible phenomena.
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