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Temporal flexibility of permit trading. What if
pollutants are correlated?
Sophie Legras, Georges Zaccour
To cite this version:
Sophie Legras, Georges Zaccour. Temporal flexibility of permit trading. What if pollutants are correlated?. Cahiers du Gerad, 2008, pp.1-23. �hal-02664133�
Temporal Flexibility of Permit Trading – What if Pollutants Are Correlated?
S. Legras G. Zaccour G–2008–80 December 2008
Les textes publi´es dans la s´erie des rapports de recherche HEC n’engagent que la responsabilit´e de leurs auteurs. La publication de ces rapports de recherche b´en´eficie d’une subvention du Fonds qu´eb´ecois de la recherche sur la nature et les technologies.
What if Pollutants Are Correlated?
Sophie Legras
Georges Zaccour
GERAD and Chair in Game Theory and Management HEC Montr´eal
Montr´eal (Qu´ebec) Canada, H3T 2A7 sophie.legras@toulouse.inra.fr
georges.zaccour@gerad.ca
December 2008
Les Cahiers du GERAD G–2008–80
multiple atmospheric trace constituents. The importance of adopting a comprehensive approach to global warming that would account for multiple interacting pollutants is in-creasingly recognized. In this paper we propose to extend the analysis of intertemporal permit trading to a framework encompassing multiple correlated pollutants. In doing so our aim is to assess the consequence of providing “when flexibility” to participants in a pollution market scheme with respect to the timing of use of their permits. In the same manner as local pollution hotspots have been identified as potential drawbacks of allowing intra-regional trade, temporal flexibility has the potential to induce a type of “temporal hotspot” of pollution that could undermine the efficiency of intertemporal trading schemes – especially when the stock of a particular pollution induces damage in itself but also im-pacts on the accumulation of another detrimental pollution stock. In a first step, we assess the impact of various types of correlations (technological and physical) on the so-cially optimal accumulation of regional and global pollutants. We illustrate that even in the case of a linear damage function the regional stocks may have ambiguous impacts on the global stock. In a second step, we show that it is possible for a global benevolent regulator to have recourse to a set of intertemporal trading schemes to induce individual agents to take socially optimal decisions over time. One requirement is to implement a set of time-dependant intertemporal trading rates. We also analyze the impact on pollutants accumulation of implementing non-optimal intertemporal trading rates.
Key Words: Intertemporal trade, Stock pollution, Multiple pollutants, Optimal con-trol.
R´esum´e
Il est maintenant reconnu que le changement climatique est imputable `a l’action cu-mul´ee de plusieurs gaz atmosph´eriques, ce qui rend n´ecessaire l’adoption d’une d´emarche globale capable de prendre en compte ces interactions. Nous proposons d’´etendre l’analyse des march´es de droits de pollution intertemporels `a un cadre comportant plusieurs pollu-tants corr´el´es. Notre objectif est d’´evaluer en quoi autoriser la flexibilit´e temporelle des ´echanges va modifier les d´ecisions d’´emission des agents. De la mˆeme mani`ere qu’une cons´equence potentielle de la flexibilit´e spatiale des ´echanges est l’apparition de “points chauds” locaux de pollution, la flexibilit´e temporelle a le potentiel de g´en´erer des “points chauds” temporels de pollution. Ces derniers pourraient menacer l’efficacit´e des sch´emas de permis `a polluer, d’autant plus s’il est pris en compte que plusieurs polluants intera-gissent. Dans un premier temps, nous ´evaluons l’impact de plusieurs types d’interaction (technologiques et physiques) sur le chemin d’accumulation de polluants globaux et r´egio-naux. Dans un deuxi`eme temps, nous montrons que le r´egulateur peut avoir recours `a un ensemble de march´es de pollution intertemporels pour induire les agents `a se conformer aux d´ecisions socialement optimales. Une condition est la mise en œuvre de taux d’´echanges temporels. Nous analysons aussi l’impact de la mise en place de taux d’´echanges temporels non optimaux.
Mots cl´es : ´echanges intertemporels, pollution de stock, polluants multiples, contrˆole optimal.
1
Introduction
It is now accepted that climate change is due to the cumulative and joint effect of multiple
atmospheric trace constituents.1 The importance of adopting a comprehensive approach to
global warming that would account for multiple interacting pollutants is increasingly
recog-nized (Reilly and Richards,1993;Reilly et al.,1999;Aaheim,1999;Manne and Richels,2000;
Schmieman et al.,2002;Caplan and Silva,2005;Yang,2006). In the literature, this argument is usually used to challenge the current policy approach to climate change, the recourse to Global Warming Potentials (GWPs). GWPs are indices defined, within the framework of the Kyoto Protocol (KP), to compare and aggregate the various greenhouse gases (GHGs) recognized by the KP; parties are then required to ensure that their “aggregate carbon
diox-ide equivalent emissions of the greenhouse gases [. . . ] do not exceed the assigned amounts”.2
Studies show that static and exogenously defined GWPs are unable to account for the
dy-namic interactions that develop between the GHGs (Reilly and Richards, 1993; Schmieman
et al.,2002; Moslener and Requate,2005).
If such analyzes tackle the multiple greenhouse gases issue, they do not account for the interactions between GHGs and other atmospheric trace constituents. Indeed, the GHGs listed in Annex A of the Kyoto Protocol are not the only human-induced contributors to the evolution of global temperature. Other gases have indirect effects on GHGs and other relevant
variables, through a complex set of feedbacks (Reilly et al., 1999). An increasing number of
studies analyze the complex links between GHGs and these other gases to which they can
be correlated in various ways: at the source, through joint emission or abatement (Caplan
and Silva,2005;Moslener and Requate, 2007), or at the receptor side, through joint damage (Moslener and Requate,2005) or interacting stocks (Yang, 2006).
To the best of our knowledge, there has been only very few analyzes of the main tool introduced
by the KP, that is CO2 emission permits market, in a multiple gas context. Emission permits
markets, through the flexibility they introduce with respect to where and when pollutants can be emitted, have the potential to greatly alter the patterns of GHGs emissions. Consequently, both “when” and “where” flexibility may have side-effects on GHGs-correlated pollutants.
Caplan and Silva(2005) analyze, in a static framework, a self-enforcing mechanism to ensure the efficient management of both a global pollutant (carbon) and numerous local pollutants (smog) by competitive regions through the use of markets for emission permits. However, most atmospheric constituents are stock pollution, rather than flow pollution. Consequently, the timing of emissions is important as it alters the accumulation, and damage, patterns, potentially leading to local or temporal pollution “hotspots” . Adopting a dynamic approach is thus essential to fully capture the interactions between GHGs and non-GHGs.
The analysis of emission credits trading in a dynamic setting usually focuses on the possibility, and consequences, of authorizing intertemporal trade. The question of temporal flexibility in emission markets, though, has received relatively few attention in the literature, given the considerable policy interest in such a mechanism. Indeed, the banking (and more rarely borrowing) of emission permits is explicitly authorized in existing schemes such as the US
Sulphur Dioxide trading program.3 Theoretical analyzes of intertemporal trading programs
for flow pollutants show that the banking/borrowing provision enables firms, or countries, to
achieve a cumulative pollution target at the lowest discounted cost (Rubin,1996; Cronshaw
1
Greenhouse gases, identified in Annex A of the Kyoto Protocol, specifically CO2, CH4, N2O, P F Cs,HF C
and SF6. 2
Kyoto Protocol, Article 3. Cited inMoslener and Requate(2005).
3
2 G–2008–80 Les Cahiers du GERAD
and Kruse,1996;Kling and Rubin,1997). This is due to the possibility given to firms to
op-timize emissions over both time and space. Rubin(1996) shows that, because of discounting,
the introduction of banking and borrowing induces firms to borrow in early periods. Conse-quently, even if the cumulative target can be reached cost-effectively, firms’ behavior when banking/borrowing is introduced may not be socially optimal because high initial emissions may result in higher social damage. The authors then analyze the design of an optimal
in-tertemporal trading rate, set at the discount rate if damage is linear in emissions.4 The issue
of the timing of emissions is exacerbated when stock pollutants are considered. Leiby and
Ru-bin(2001) provide a framework encompassing both flow and stock pollutants and non-linear
damage. They show that, for stock pollutants, the optimal intertemporal trading rate should account for the lack of coincidence between the timings of emissions and damage.
This paper seeks to fill a gap in the literature by analyzing intertemporal trade in emission permits markets when multiple pollutants are explicitly accounted for. More precisely, it investigates the consequences of allowing temporal flexibility of polluting emissions on the accumulation pattern of correlated stocks. The impact of the nature of the correlation (tech-nological, physical) and of the intertemporal rate design is addressed.
This paper is organized as follows. Section 2 reviews the literature on multiple pollutants
situations and details the model. Section3examines the socially optimal management of a set
of regional stock pollutants that impact of the accumulation of a global pollutant. Illustrations
for linear and quadratic specifications of the damage function are provided. Section4extends
previous analyzes of intertemporal permit trading (Leiby and Rubin, 2001) to a multiple
pollutants setting. Section5 concludes.
2
Modelling a Multiple Pollutants Situation
2.1 Related Literature
There is a growing literature on the joint emission of multiple pollutants. An early paper by Montgomery addresses the design of market for licences in the presence of multiple linearly
correlated pollutants (Montgomery, 1972); Beavis and Walker (1979) extend the analysis to
nonlinear interactions in watercourses. However, most papers dealing with multiple pollutants are set in the framework of atmospheric pollution and address one of the numerous links that exist between the various sources and receptors of pollution due to atmospheric trace
constituents (Schmieman et al.,2002).
More often analyzed are the gases listed under Annex A of the Kyoto Protocol for which GWPs have been defined. These indices are used to aggregate and compare the various greenhouse gases listed above with respect to their contribution to global warming; parties to the Kyoto Protocol are then required to ensure that their“aggregate carbon dioxide equivalent
emissions of the greenhouse gases [. . . ] do not exceed the assigned amounts”.5 Moslener and
Requate first address this issue with an optimization model where abatement technologies are correlated: pollutants are said complements if the cross derivative of the abatement cost
function is positive, and substitutes if the derivative is negative (Moslener and Requate,2007).
Case of substitute pollutants arise because of the considerable amount of GHG generating
energy needed to abate some pollutants (Moslener and Requate, 2007). The same authors
4
Hence this system only corrects one source of divergence between individual and social programs (the fact that firms discount the future), not the second one (that damage depend on the timing of emissions – which doesn’t arise if the damage function is linear).
5
consider, in a subsequent paper (Moslener and Requate, 2005), a model where abatement technologies are independent, but the damage function is not separable. In both cases, they show that the difference in decay rates between the various gases may lead to non-monotonicity of emissions and stock accumulation patterns. Consequently, they argue that static and exogenously defined GWPs are unable to account for the dynamic interactions between the
multiple GHGs . However, as pointed out by Reilly et al. (1999), failure to account for gases
not identified as GHGs, but that have an effect on global warming, such as CO, N Oxor SOx,
may also lead to large errors – potentially larger than concentrating on a CO2-only rather
than on a multi-GHGs strategy. A number of studies address global warming in combination with another, sometimes more localized, air pollution problem.
The links between the Kyoto Protocol and the Montreal Protocol (MP), the international framework for stratospheric ozone management, are very illustrative in this respect. The substances subject to regulation under the Montreal Protocol, the Ozone Depleting Substances (ODS), have two effects on global warming: they are powerful GHGs, with GWPs far above
CO2’s, and they also have a cooling effect by destroying stratospheric ozone which is itself
a powerful GHG. Furthermore, the MP promotes the use of HCFs, now listed in Annex A of the KP, as substitutes for some ODS. In other words, part of the problem in the KP has
been promoted as part of the solution in the MP (Oberthur, 2001). Accounting for these
links of different natures (chemical, physical, policy-induced) in future policy-making is a real challenge, to ensure that potential synergies are captured and potential drawbacks minimized. Authors have also concentrated on another important linkage, between climate change and acidification. Indeed, sulphur emissions are responsible for acid deposition and for increasing atmospheric levels of sulfate aerosols, that mask global warming by scattering global radiations (Yang, 2006; Posch et al., 1996). Posch et al. (1996) use coupled models of climate change
and acid rains to address the impact of various SO2 reduction policies. They show that a
50% reduction policy would have different impacts in Europe and Asia: the positive effect of reducing acid deposition in Asia balances the indirect negative effect on climate change; while the positive direct impact on acidification would greatly overweight the negative indirect
decreased cooling effect in Europe. Yang (2006) uses an optimization model to derive the
socially optimal management of a global stock of carbon that interacts with multiple stocks
of SO2. The cooling effect induced by the stock of SO2 could make a case for subsidizing
sulphur emissions (Yang,2006).
Consequently, there are various ways in which atmospheric gases interact. On the source side, they can be jointly emitted from the same production activity. Transport activities are
typically responsible for emitting a number of pollutants, including CO, CO2, volatile
com-pounds, N Ox and SO2; energy production generates SO2 and N Ox. Another technological
correlation is due to abatement activities. As coined by Caplan and Silva, abating activities can be “coarse” as abating for smog, for instance, indirectly abates for carbon. As noted
ear-lier, Moslener and Requate (2007) consider both coarse abatement activities (complement in
their terminology) and substitute ones. Finally, pollutants can be correlated on the receptor side, through a complex and evolving set of feedbacks that are yet to be fully understood. In this paper, we propose a model that encompasses these three types of correlation between atmospheric pollutants; in this regard we extend previous works that focused on only one or
two aspects of the multiple pollutant setting (Yang, 2006; Caplan and Silva,2005; Moslener
4 G–2008–80 Les Cahiers du GERAD
2.2 The Model
We consider N countries, indexed by i, with a representative firm in each country. In the
process of producing a good xi firms jointly emit 2 types of pollution, that participate to
the accumulation of both a global pollutant stock G and a regional pollutant stock Rj with
j ∈ {1, . . . , n}. The n regional pollutants are of the same nature and only differ with respect to the location of accumulation. Define S the set of agents that contribute to the global
pollutant stock and Sj the set of those who are responsible for stock Rj; then it is assumed
that S =S
jSj.
Firms have two abatement technologies, one for each type of pollution: global, gi, and regional,
ri. Emissions by country i of the regional pollutant, ei,6 and of the global pollutant, Ei, are
defined as:
ei = f (xi, gi, ri), (1)
Ei= F (xi, gi, ri). (2)
We assume additive separability of functions F and f as well as the following conditions on
their first-order derivatives7: F
x, fx > 0, Fr, Fg, fr, fg < 08. The technological correlation is
captured by the terms Fr and fg: the former is the fraction of regional abatement that also
reduces global emissions, the latter illustrates how global abatement also reduces regional emissions.
The revenue function of agent i is πi(xi, gi, ri) and the pollution stocks dynamics are:
˙ G =X i∈S Ei− δG − n X j=1 h(Rj), (3) ˙ Rj = X i∈Sj ei− γjRj , ∀j (4)
where δ and γj are the natural decay rates, comprised between 0 and 1. Function h(·)
captures the impact of the n regional stocks on the global one. This function is assumed additively separable and the two main situations, positive and negative feedbacks from the regional stocks, are accounted for: h > 0 illustrates the cooling effect and h < 0 denotes regional stocks that increase the global stock. Both types of stocks are harmful to society
as described by the additively separable damage function: D(G, R1, . . . , Rj, . . . , Rn), with
DG> 0, DGG ≥ 0, DRi > 0, DRjRj ≥ 0. The discount rate is denoted by ρ.
Consequently, three types of correlations are potentially captured by this model:
• a technological correlation characterized by the joint emission of two types of pollutants,
assuming strict positivity of parameters fx and Fx. This sets the multiple pollutants
scene;
6
We do not discrimate between regional stocks – production and abatement technologies are assumed the same over all stocks.
7
denoted by subscripts.
8
Refer toMoslener and Requate(2007) for a general analysis of the optimal management of both substitute pollutants, Fr>0 and fg >0, and complement pollutants, Fr <0 and fg <0 – the case analyzed here.
• a correlation between abatement technologies, assuming that parameters fg and Fr are
non zero. We further assume that these parameters are strictly negative; in other words we only consider complement technological correlations as it is the most common case; • a physical correlation that intervenes after the production process, captured by function
h(·).
Note that there is a major difference between the technological and the physical correlations. Indeed, the former are de facto accounted for by the agents as a component of their pro-duction process while the latter are captured by the regulator, as part of his model of stock accumulation, but not necessarily by individual agents.
3
Socially Optimal Management of Multiple Correlated
Pol-lutants
In this section we analyze the socially optimal allocation of production and abatement efforts in the presence of correlated regional and global pollutants. First we provide a general analysis of the impact of various types of correlations – technological and physical – on emissions decisions over time. Then we illustrate the steady state levels and emissions paths under two specifications of the damage function, linear and quadratic. Finally we address how the socially optimal paths, derived in an infinite horizon framework, are used in the following section where the horizon of policy implementation is assumed finite.
3.1 Socially Optimal Emissions Decisions in a General Framework
The environmental regulator’s objective is to maximize social welfare, defined here as total profits less damage from the accumulation of the various pollutants, subject to the accumu-lation of both types of pollutants:
max {xi,gi,ri} Z ∞ 0 e−ρt " n X i=1 πi(xi, gi, ri) − D(G, . . . , Rj, . . .) # dt, (P1) s.t.: ˙G =X i∈S Ei− δG − n X j=1 h(Rj), ˙ Rj = X i∈Sj ei− γjRj , ∀j, xi ≥ 0 , gi≥ 0 , ri≥ 0.
The Hamiltonian and first order necessary conditions for this problem are described in
Ap-pendix 1. Reordered FOCs are presented and discussed below, with λso(resp. µso
j ) the shadow
cost of global pollutant (resp. regional pollutant Rj) accumulation, where superscripts stand
for social optimal decisions:
πix= −λsoFx− µsoj fx, (5)
6 G–2008–80 Les Cahiers du GERAD πir = −λsoFr− µsoj fr, (7) ˙ λso= (ρ + δ)λso+ D G, (8) ˙ µso j = (ρ + γj)µsoj + DRj+ λ so hRj. (9)
An analysis of the FOCs offers some first insights on how the different correlations are
ac-counted for by the social planner. Equation (5) states that countries should equate their
marginal revenue to the marginal cost of adding a unit of pollution to each stock – which is equal to the respective stocks’ shadow cost weighted by the relative contribution of production
to emissions. Equation (6) states that the marginal revenue from abating the global pollutant
should be equated to the associated benefits in terms of the reduction of pollution: lower accumulation of the global stock and lower accumulation of the local one due to the coarse
nature of abatement captured by the correlation term fg6= 0. The interpretation of equation
(7) is similar. The coarse nature of abatement induces higher abatement compared to the
uncorrelated pollutants case; the resulting increase in the value of the RHS of equations (6)
and (7) is counterbalanced by a higher level of abatement to increase the LHS. Furthermore,
the stronger the abatement correlation, the higher the induced level of abatement. Equations
(8) and (9) show the evolution of the shadow prices of pollution accumulation through time.
Equation (8) is the standard depiction of a unique stock case: the shadow price evolves at
the “environmental” discount rate ρ + δ accounting for a damage-related term. Equation (9)
is peculiar as it incorporates the correlation between the global and local stocks, through the
term attached to hRj. If hRj < 0, this terms acts as another term of damage as long as
λso < 0: if the local stocks enhance the accumulation of the global stock, then their
associ-ated shadow prices should account for this indirect damage term. In the case of negatively
correlated stocks, the beneficial cooling effect of SO2, for instance, is captured as if hRj > 0
and λso < 0 the correlation term enters the shadow cost dynamics as a benefit term.
Conse-quently, according to the sign of the physical correlation, the local stocks dynamics are altered to include the costs, or benefits, of their impact on the accumulation pattern of the global pollutant.
To sum up, the various types of correlation under study affect the socially optimal production and abatement decisions in the following manner. The joint emissions feature reduces the marginal benefit from producing – as production is associated with two types of polluting emissions. The coarse abatement feature increases the marginal benefit from abating – as abating one pollutant indirectly abates the other pollutant and the agents are induced to account for the marginal benefits derived from the avoided accumulation of the correlated pollutant. The physical correlation impacts on the marginal benefit from avoided regional
pollutant accumulation; according to the sign of hRj, it either increases or decreases it.
Proposition 1. Current and future benefits and costs associated with production and abate-ment decisions are balanced in a socially optimal way as follows:
πik− ˙πik ρ + γj = FkDG+ fkDR ρ + γj + RSHk , k ∈ {x, g, r}.
Proof. See Appendix 2.
These relations state that current marginal benefits from producing (or abating) minus the present value of future changes of those marginal benefits should equate the present value of
the future marginal damage from accumulating both types of pollution (positive if production, negative if abatement) plus a corrective term characteristic of the multiple pollutants setting
detailed below. Proposition 1 extends the results of Leiby and Rubin (2001) to a multiple
pollutants context: both types of stocks are considered in assessing future damage and a term corrects for discounting when the stocks’ decay rates are different.
The corrective terms RHSx, RHSg and RHSr, superscripts denoting the marginal cost they
refer to, can be decomposed in three sub-terms: (a) an expression of the shadow cost of
global pollution accumulation, λk, expressed as function of the other marginal costs, (b) a
term correcting any discrepancy between environmental decay rates DRk and (c) a term
accounting for the physical correlation CORk:
RHSk= λk(DRk+ CORk) with k ∈ {x, g, r}.
(a) To derive explicit expressions of each marginal cost evolution, the shadow cost associated
with the global pollutant λso is expressed in each equation, relating to a particular marginal
cost, as a function of the other marginal costs. Those expressions of the shadow cost are: λx= πirfg− πigfr Fgfr− Frfg , λg= πixfr− πirfx Frfx− Fxfr and λr= πixfg− πigfx Fgfx− Fxfg .
The denominators of these expressions illustrate (environmental) efficiency discrepancies: for
instance, the stronger the abatement correlations, the higher λx in absolute terms. The
numerators illustrate the arbitrage between abating more and producing less; or between the various abatement strategies, in order to reduce the accumulation of the global pollutant.
(b) The equations presented in Proposition 1 also include terms that are discounted on the
basis of the regional pollutant stock decay rate. Corrections to discount global
pollutant-related terms with respect to the global decay rate are contained in terms DRk:
DRx= F x δ − γj ρ + γj , DRg = F g δ − γj ρ + γj and DRr= F r δ − γj ρ + γj .
In the absence of any type of correlation, except the joint emission, a correction is needed to assess the evolution over time of the marginal costs of production and global abatement. The extent, and sign, of the correction depends on the difference between the natural decay rates.
If δ > γj then the expressions relating to global pollution have been under-discounted: RHSx
needs to be reduced (note that the signs are opposite in the abatement cases, as Fg,Fr < 0).
(c) The terms that account for the physical correlations are the following:
CORx= Fx hRj ρ + γj , CORg= fg hRj ρ + γj and CORr= fr hRj ρ + γj .
The stronger the physical correlation, the higher this corrective term. As long as the shadow
costs are negative, the sign of hRj governs the sign of RHSg and RHSr and RSHx.
Conse-quently, if the regional pollutant worsens the accumulation of the global pollutant, then the marginal benefits from producing are reduced while the marginal benefits from abating are increased. The opposite is true when the regional pollutant contributes to lowering the detri-mental impacts of global pollution: the benefits from producing are, provocatively, increased, while the benefits from abating are lowered.
8 G–2008–80 Les Cahiers du GERAD
Table 1: Impact of the correlations (technological and physical) on various steady state levels. Linear damage function. *: − if h < 0, +/− if h > 0
λL
∞ µL∞ RL∞ GL∞
h 0 + + +/−
α3 0 0 − −
β2 0 0 +/ − ∗ +/ − ∗
3.2 Illustrations with Linear and Quadratic Damage Functions
Here we illustrate the impact of the various types of correlations on the steady state levels and accumulation paths of the regional and global pollutants. We consider the following emissions and revenue functions:
Ei(xi, gi, ri) = α1xi− α2gi− α3ri, (10) ei(xi, gi, ri) = β1xi− β2gi− β3ri, (11) πi(xi, gi, ri) = pxi− hc1 2x 2 i + c2 2g 2 i + c3 2r 2 i i . (12)
and two specifications of the damage function: DL(G, . . . , Rj, . . .) = d0G + n X j=1 djRj, (13) DQ(G, . . . , Rj, . . .) = 1 2d0G 2+ n X j=1 1 2djR 2 j. (14)
Analytical results are derived for both specifications (see Appendix 3). However, it is difficult to sign most analytical expressions in the quadratic case, especially when h > 0. Consequently, we provide here an analytical analysis of the linear case and graphical illustrations of the quadratic case.
Table 1 describes the impact of the various types of correlations on the steady-state values
for a linear damage function.
The evolution of the stocks and shadow costs over time are, when the regional and global stocks’ decay rates are different, but all regional rates are equal:
GL(t) = G∞+ (G0− G∞)e−δt+ n h(R0− R∞) γ − δ (e −γt− e−δt) , RL(t) = R∞+ (R0− R∞)e−γt , λL(t) = λ∞= −d0 ρ + δ , µL(t) = µ∞= −d + hλL ∞ ρ + γ .
Table 2: Parameters used for simulations
α1 = 0.5 β1 = 0.5 c1= 2 N = 10 n = 1 pt= 2
α2 = 0.05 β2 = 0.01 c2= 2 δ = 0.05 γ = 0.1 ρ = 0.06
α3 = 0.01 β3 = 0.05 c3= 2 d = 0.05 d0= 0.1
First note that both shadow costs are unaffected by the technological correlations. Indeed, in a linear setting, marginal damage are constant through time, and the shadow costs only depend on the damage parameters and the terms affecting discounting – the decay and discount rates. However, the regional stock’s steady state level is affected by the physical interaction; indeed the cost of accumulating R accounts for the indirect impact it has the accumulation of the global pollutant.
The ratio of regional abatement that also abates the global pollutant, α3, has an unambiguous
impact on pollution accumulation: a stronger correlation decreases both stocks. The situation
is more complex for the ratio of global abatement that abates the regional pollutant, β2.
Indeed, its impact depends on the sign of the physical correlation. If the stocks interact positively (h < 0, more regional pollution leads to more global pollution), then it has the
same impact as α3: a stronger correlation leads to less accumulation of both pollutants. If,
however, the regional pollutant affects the global pollutant negatively, then the impact of the technological correlation on both stocks is ambiguous. In fact, there is a value of h above
which the impact changes sign: a high β2 means that a given level of global abatement will
lead to a higher level of regional abatement, hence a lower regional stock. A lowering of the regional stock has different impacts on the global stock depending on the value of h. If h > 0 then two opposite effects on the global stock arise: a higher global abatement that decreases G, and a lower regional stock that doesn’t reduce G as much.
As already mentioned, the physical correlation affects only the regional shadow costs, as it introduces an extra cost (or benefit) of accumulating the regional pollutant because of its indirect impact on the accumulation of the global stock. h as an unambiguous impact on the regional stock: a stronger correlation in absolute value increases the regional stock, which is not surprising. Finally, the impact of the physical correlation on the global stock depends on its sign, and is not straightforward.
Figures1–3illustrate the socially optimal accumulation paths of the global and regional stocks,
showing the impact of the various types of correlations. Table2describes the parameters that
have been used for the simulations, unless specified otherwise.
A first observation derived from Figures 1–3is that the quadratic specification of the damage
function accentuates the impact of the correlations on the stock accumulation paths. The impact of h is as expected: increasing (resp. decreasing) the global stock when h < 0 (resp.
h > 0). A negative h also induces a non monotonic regional pollutant accumulation path. β2
has nearly no impact on G, but greatly affects R – even causing negative values of R. In the
same manner, α3 has a greater impact on the global pollutant.
3.3 Transposition to a Finite Policy Horizon
Having characterized the socially optimal paths of pollution emissions, we are now interested in analyzing the ability of emission trading schemes to induce individual agents to take decisions regarding production and abatement that ensure these paths are followed. To be more in line with current policy initiatives regarding atmospheric pollution management, we consider that
10 G–2008–80 Les Cahiers du GERAD 0 25 0 50 100 50 75 25 75 t 75 50 3 25 t 0 100 1 0 2 2 t 75 25 5 50 1 4 0 100 0 6 3 t 1.5 1.0 0 100 0.5 50 25 0.0 75
Figure 1: Global – Regional pollutant accumulation path – impact of h. dotted line: h = 0, black solid line: h = 0.08, grey solid line: h = −0.08. Top: linear damage. Bottom: quadratic damage.
trading schemes are implemented over a finite time horizon. Hence the regulator’s objective in the following section is to have individual agents follow the optimal emissions paths until a determined time T .
4
Temporal Flexibility of Emission Trading in a Multiple
Pol-lutants Setting
Our objective in this section is to assess how various ITRs affect individual agents’ decision regarding production and abatement efforts. It has been established that the allocation of permits governs the level of the price, while the ITR drives its evolution over time. In this
paper we focus on the second point; in this respect we followLeiby and Rubin(2001) in that
we do not formally justify the recourse to ITR, as policy instruments, compared to an optimal allocation of emission permits at each point of time. Indeed, we consider that the current interest in intertemporal trading is sufficient to justify this type of analysis.
20 60 t 25 30 40 70 50 10 0 100 50 0 75 3.0 1.5 1.0 75 3.5 2.0 2.5 0.5 100 50 0 0.0 t 25 t 75 0 25 50 1 0 3 4 100 2 0.2 100 25 t 0.3 −0.1 0.1 75 50 0 0.0
Figure 2: Global – Regional pollutant accumulation path – impact of β2. dotted line: β2= 0,
black solid line: β2= 0.5, grey solid line: β2 = 0.25. Top: linear damage. Bottom: quadratic
damage.
4.1 Market Description
Suppose that the regulator seeks to have a cumulative emissions target respected for each type
of pollution, global ¯E and regional ¯ej. The first step towards the implementation of a market
for pollution permits is to endow each agent i with an allocation ¯Eit of global pollutant and
¯
eit of the regional pollutant at each point of time, so that:
¯ E = Z T 0 X i∈S ¯ Eitdt and ¯ej = Z T 0 X i∈Sj ¯ eitdt. (15)
The enforcement of these endowments translates into an individual compliance constraints stating that each agent should not emit more of the global pollutant than what his endowment authorizes him to. This constraint is relaxed when spatial flexibility is introduced and
exchanges authorized. Then global and regional permits, respectively Yi and yi, can be sold
(Yi, yi < 0) or purchased (Yi, yi > 0) at the competitively determined market prices pG, pRi,
assuming agents are price-takers. Relaxing the individual constraint to allow for temporal
flexibility leads to the definition of individual “permit banks” Bit, bit. Individual agents are
then allowed to emit either more or less of the pollutant in one period than what would be permitted in a purely static market as they can either bank permits, for later use, or borrow
12 G–2008–80 Les Cahiers du GERAD 20 60 t 25 30 40 70 50 10 0 100 50 0 75 3.0 1.5 1.0 75 3.5 2.0 2.5 0.5 100 50 0 0.0 t 25 4 75 0 50 100 0 25 t 2 3 1 2.0 75 0.0 50 100 0 25 t 1.0 1.5 0.5
Figure 3: Global – Regional pollutant accumulation path – impact of α3. dotted line: α3 = 0,
black solid line: α3= 0.5, grey solid line: α3 = 0.25. Top: linear damage. Bottom: quadratic
damage.
permits for immediate use. Hence, depending on the agent being a net borrower or a net
banker of permits, Bit can be positive or negative. Permit accumulation in the bank is then
a state variable for each agent. It has been established, in a single stock framework, that
ITRs are necessary to induce socially optimal decisions over time (Rubin,1996). ITRs act as
interest rates on the individual banks: they reward the banking and penalize the borrowing of emission credits. For instance, the restoration rule, as stated in the Marrakesh Accords that govern trading rules under the KP, imposes that for each unit of permits short in one period,
1.3 units should be subtracted from next period’s endowment (Godal and Klaassen, 2006).
Furthermore, both banks are required to clear at T . Consequently, the banks’ dynamics are
described as follows, when intertemporal trading rates τG, τRare introduced:
˙
Bi = ¯Ei− F (xi, ri, gi) + Yi+ τGBi , with B(0) = 0 and B(T ) = 0,
˙
bi= ¯ei− f (xi, ri, gi) + yi+ τRbi , with b(0) = 0 and b(T ) = 0.
In the next section we analyze how ITRs affect individual agent’s decision making process, hence the accumulation pattern of global and regional pollutants over time.
4.2 Individual Decisions in the Presence of Temporal Flexibility in Pollu-tion Permits Markets
We consider a situation where both temporal and spatial flexibility are introduced, so that an agent has several ways of meeting his pollution standard: by producing and abating so that emissions amount directly to his allocated standard, by emitting less than what he is allowed to generate permits to be sold to other participants or deposited in his bank, or finally by emitting more than his standard and compensating with permits purchased from other agents or borrowed from his individual bank. Consequently, an agent i seeks to maximize his utility, accounting for his own pollution banks’ dynamics:
max xi,gi,ri,Yi,yi Z T 0 e−ρt[πi(xi, gi, ri) − pGYi− pRyi] dt, (P2) ˙ Bi= ¯Ei− F (xi, ri, gi) + Yi+ τGBi , B(0) = 0 and B(T ) = 0, (16) ˙ bi = ¯ei− f (xi, ri, gi) + yi+ τRbi , b(0) = 0 , b(T ) = 0, (17) −Yim≤ Yi ≤ YiM , Yim≥ 0 and YiM ≥ 0 (18) −yim≤ yi ≤ yiM, , yim≥ 0 and yiM ≥ 0 (19) xi≥ 0 , gi ≥ 0 , ri ≥ 0. (20)
Refer to Rubin (1996) for a demonstration of the existence of such a market equilibrium.9
The subsequent analyzes assume that agents have an internal solution, so that the constraints
described in equations (18)-(20) do not bind. This implies, in particular, that they neither
buy nor sell at the maximum rate.
Proposition 2. Agents participating in both a global pollution and regional pollution trading schemes perform the following balancing of costs and benefits:
πik= FkpG+ fkpR , k ∈ {x, g, r}. (21)
Proof. Appendix 4 presents the derivation of the FOCs that lead to these results.
These relations state that current marginal benefits from producing (or abating) should equal the sum of market prices weighted by the relative contribution of the decision variable (produc-tion or abatement) to emissions. Consequently, assuming sta(produc-tionarity of emission func(produc-tions, the evolution of marginal benefits over time depends exclusively on the evolution of the market prices:
˙πik= Fkp˙G+ fkp˙R, k ∈ {x, g, r}. (22)
Compared to a single stock setting (and hence a single market context), Proposition 2 high-lights the impact of the various technological correlations on how individual agents account for pollution permits markets. The coarse abatement feature ensures that both markets affect
9
Rubin (1996) discusses the necessity to assume an interior solution to agent’s problem as the objective function is linear in the market-related controls Yiand yi. The recourse to explicit bounds on permit purchase
poses technical difficulties that are not relevant to the economic problem at hand. Hence the more simple strategy to assume the existence of an interior solution that places implicit bounds on permit purchase/sale.
14 G–2008–80 Les Cahiers du GERAD
both abatement decision variables as long as Fr6= 0 and fg6= 0. A higher global market price
has a direct effect on global abatement and an indirect effect on regional abatement, and the stronger the correlation, the stronger the weight of the global price on regional pollutant deci-sions (the same applies for the regional market price). Note also that the physical correlation doesn’t appear explicitly in the above-equations. It is the regulator’s task to design the ITRs in such a way as the physical correlation is properly accounted for by the agents.
Proposition 3. To ensure the participation of the agents in the market at a non bounded rate, the permit prices should follow a modified Hotelling rule when full temporal and spatial flexibility are implemented:
˙pm
pm
= ρ − τm , m ∈ {G, R} (23)
Proof. From the FOCs (Appendix 4) we know that firms will buy and sell permits so that:
pG− θiG = 0 , with ˙θiG= (ρ − τG)θiG, (24)
pR− θiR= 0 , with ˙θiR= (ρ − τR)θiG. (25)
Equation (23) follows. See also Rubin (1996); Kling and Rubin (1997); Leiby and Rubin
(2001) for this result in the single stock setting.
Indeed, agents treat each type of permit as a non renewable resource over the entire time horizon, and apply the usual economic principle of maximization of extraction rent over time. However, temporal flexibility alters this rule, as the rate of change of the market price is equal to the difference between the discount rate and the ITR. Consequently a positive intertemporal trading rate decreases the rate of growth of the associated market price.
The multi-pollutant context does not modify the Hotelling rule identified previously in
in-tertemporal markets for unique pollutant stocks (Rubin,1996;Kling and Rubin,1997;Leiby
and Rubin,2001), as a market has been considered for each type of pollution.
The design of the trading rate is of prime importance to assess the level of individual decisions regarding production and both types of abatement, as it impacts on the market price path. Next, we analyze various design strategies for the ITRs and assess their impact on emission decisions by individual agents. After examining the properties of hypothetical optimal rates, that would ensure optimality of individual agents regarding production and both abatement
activities, we followFeng and Zhao (2006) by analyzing the more plausible cases of a trading
rate equal to the financial discount rate, or the case of a 1-to-1 trading, which amounts in our context to setting a rate equal to zero.
4.3 Socially Optimal ITRs
Designing optimal ITRs consists in ensuring that the evolution of the market prices over time induces the agents to align their abatement and production paths with the socially optimal
ones described in Section 3.10
10
The level of price then depends on the amount of permits distributed – allocation strategies are not discussed here.
Proposition 4. Optimal intertemporal trading rates are characterized by: τG∗ = ρ − ˙λ so λso and τ ∗ R= ρ − ˙µso µso.
The socially optimal trading rates are set equal to the discount rate minus the optimal rate of change of the respective pollutant stock shadow costs of accumulation.
Proof. Compare equations (5)–(7) and (22) to derive that socially optimal and individual
decisions are compatible under the following condition: θG= λso and θR= µsoi ⇒ ˙θG= ˙λso and ˙θR= ˙µsoi .
Then use equation (23) to obtain the above result.
The regulator can implement a set of ITRs that induce the agents to manage optimally each
of the stocks. As in the single stock case (Leiby and Rubin,2001), the optimal ITRs are set
equal to the discount rate minus the rate of change of the shadow costs associated to the respective pollutant stocks; they ensure that agents’ production and abatement decisions are optimal along the time horizon. However, in this multiple pollutants setting, the optimal
shadow costs are linked, as addressed in Section3. Consequently, so are the trading rates. To
illustrate this, rewrite the ITRs as follows, assuming strict negativity of the socially optimal shadow costs: τG∗ = −δ + DG −λso (26) τR∗ = −γi+ DR −µso − h −λso −µso (27)
Because of the structure of our model itself, in other words that stock G is affected by stocks
Ri, and not the reverse, the correlated feature of the multiple pollutants context we study
appears in equation (27) that defines the optimal regional ITR.
Equation (26) describes the optimal global ITR, which is like in the single pollutant case
equal to the ratio of current marginal damage from stock accumulation to discounted value of
future stock related damage less the decay rate11 (Leiby and Rubin,2001). High discounted
future damage tend to decrease the optimal trading rate – translating into less abatement moved in early periods. Symmetrically, a high current damage from pollution increases the trading rate, inducing more abatement and less production in early periods.
The ITR associated with regional pollution permits depends on both market prices, or both shadow costs, when the correlation term h is non zero. It is equal to the decay rate (regional) plus the ratio between current and discounted future damage, due to the emissions of regional pollutant; plus, contrary to the global case, a third term equal to the physical correlation parameter times the ratio between global and regional shadow costs. The sign of the latter expression depends on the nature of the correlation, assuming both shadow costs are negative: the stronger the positive (resp. negative) correlation, and the higher the cost of accumulating the global pollutant, the smaller (higher) the regional ITR.
11
16 G–2008–80 Les Cahiers du GERAD
Corollary 1. When the damage function is linear, both optimal intertemporal trading rates are equal to the financial discount rate ρ.
Proof. When the damage function is linear, the shadow costs are constant over time ˙λso =
˙µso
i = 0, as seen in Section3.
When the damage function is linear, the marginal cost of each additional unit of pollution accumulating is constant over time. Consequently the trading rate only corrects for discount-ing. The physical correlation is internalized by the regional market price, optimally set as the opposite of the regional stock shadow cost:
pR=
hλL
∞+ d
ρ + γi
.
Consequently there is a direct link between the sign, and intensity, of the correlation between the stocks and the marginal costs incurred by the agents. A positive h, denoting a benefi-cial impact of the regional stock on the global stock, decreases the market price – regional accumulation bearing some benefits, the associated market price is intuitively set lower.
4.4 Exogenous (Constant) ITRs
Now we consider ITRs more likely to be considered for implementation by a policymaker. The first case is the most straightforward: no trading rate, so that permits are treated equally in all periods and there is no cost of borrowing – and no reward for banking. In the second case the regulator is assumed to use the same interest rate for the permit bank and for the monetary system.
These exogenous ITRs both induce simplistic patterns of emissions: when the ITR is equal to the monetary interest rate, market prices are constant over time, while with a zero ITR
market prices to grow over time at the discount rate. Indeed, remember equation (23):
˙
pm
pm
= ρ − τm.
The following proposition examines to which extent these exogenous ITRs affect polluting emissions compared to the socially optimal case.
Proposition 5. The extent of the difference in emission levels between the social optimum and the ones induced by the choice of exogenous ITRs is captured by the following terms, when
functions are specified as in equations (10)-(12) and (14):
∆E(t) = K1µso(t) + p0R + K2λso(t) + p0G if τm= ρ ∆e(t) = K3µso(t) + p0R + K1λso(t) + p0G ∆E(t) = K1µso(t) + p0Reρt + K2λso(t) + p0Geρt if τm = 0 ∆e(t) = K3µso(t) + p0Reρt + K1λso(t) + p0G with: K1= X i αiβi ci , K2 = X i α2i ci , K3 = X k βi2 ci
and µso(t) and λso(t) as defined in
Proof. Recall the definition of global and regional emissions: E(t) = α1xit+ α2git+ α3rit,
e(t) = β1xit+ β2git+ β3rit.
Decision variables xit, git and rit can be expressed as functions of the socially optimal shadow
costs of pollution accumulation (see Section 3) or of the market price (see Proposition 2). Proposition 5 follows.
With a trading rate equal to the monetary interest rate (taken as the discount rate), prices are constant over time, and so are the emissions. Consequently, the evolution of the emission discrepancy over time is attributable to the socially optimal pattern of polluting emissions. With a trading rate equal to zero, the emission discrepancy evolves over time with both the socially optimal emission paths and the price growth.
Note that the physical correlation impacts on the optimal shadow cost dynamics; however
the technological correlations appear in constants kj, j ∈ {1 . . . 3}, and as such affect all the
terms of the above equations.
Figure 4 illustrates ∆E(t) and ∆e(t), when h = 0, for various levels of prices. We see that
the choice of exogenous ITR is important as the difference between optimal and individual decisions do not follow the same pattern according to the value of the ITR.
Corollary 2. When the damage function is linear:
• with τG= τR= ρ, ∆E(t) and ∆e(t) are constant over time,
• with τG= τR= 0, ∆E(t) and ∆e(t) evolve over time at the discount rate,
and in both cases: ∂∆E(t) ∂h = d0 k1 (ρ + γi) > 0 and ∂∆e(t) ∂h = d0 k3 (ρ + γi) > 0
Under the assumption of a linear damage function, the expressions of the shadow costs are simplified, allowing us to derive a few more insights on the impact of the various correlations on the emission patterns. The physical correlation has a non ambiguous impact on the difference: a strong positive correlation increases the discrepancy, while a negative one decreases it. Indeed, the physical correlation only has an impact on the shadow cost of regional pollution accumulation, which itself impacts (directly) on regional emissions and (indirectly) on global emissions. And the sign of the impact is opposite to the sign of h: a negative h induces less regional pollution, a positive h induces more regional emissions (compared to cases with no correlation).
5
Conclusion
In this paper we proposed to extend the analysis of intertemporal permit trading to a frame-work encompassing multiple correlated pollutants. In doing so our aim was to assess the consequence of providing “when flexibility” to participants in a pollution market scheme with respect to the timing of use of their permits. In the same manner as local pollution hotspots have been identified as potential drawbacks of allowing intra-regional trade, temporal flexibil-ity has the potential to induce a type of “temporal hotspot” of pollution that could undermine
18 G–2008–80 Les Cahiers du GERAD 0.0 5 3 0.5 1 7 1.25 1.0 0.75 0.25 10 8 6 4 2 −0.25 0 t −0.5 9 0.0 5 3 0.5 1 7 1.25 1.0 0.75 0.25 10 8 6 4 2 −0.25 0 t −0.5 9 0.0 5 3 0.5 1 7 1.25 1.0 0.75 0.25 10 8 6 4 2 −0.25 0 t −0.5 9 0.0 5 3 0.5 1 7 1.25 1.0 0.75 0.25 10 8 6 4 2 −0.25 0 t −0.5 9
Figure 4: Discrepancy between global (left) and regional (right) emissions – social optimum
minus constant ITR (up: τ = ρ, down: τ = 0) for various levels of prices: p∗(0) (dotted line),
p∗(T ) (grey line),p∗(0)/2 (thick black line), 2p∗(T ) (black line) where p∗ is the optimal price.
(h=0)
the efficiency of intertemporal trading schemes – especially when the stock of a particular pol-lution induces damage in itself but also impacts on the accumulation of another detrimental pollution stock.
In a first step, we assessed the impact of various types of correlations (technological and physical) on the socially optimal accumulation of regional and global pollutants. The multiple pollutants context has already been shown to give rise to non-monotonicity of emission paths (Moslener and Requate,2005,2007) – the physical correlation as described in the present paper complicates even more the derivation of analytical results. We have illustrated that even in the case of a linear damage function the physical correlation may have ambiguous impacts on the global stock. The complete analysis of the social optimal, including the identification of constraints on parameters’ value to ensure monotonicity of the emission paths, is outside the scope of this paper but constitutes an interesting extension that should be considered for future research. Consequently, the remainder of the analyzes presented in this paper were made assuming that the socially optimal emission and accumulation paths respected reasonable conditions of existence (positivity, for instance).
In a second step, we showed that it is possible for a global benevolent regulator to have recourse to a set of intertemporal trading schemes to induce individual agents to take socially optimal
decisions over time. One requirement is to implement a set of time-dependant intertemporal trading rates. The regional trading rate has been shown to depend explicitly on the physical correlation parameter – illustrating the asymmetric position of the stocks we study, as the impact is unidirectional. We also investigated the case of constant intertemporal trading rates, due to the difficulties of implementation that time-dependant trading rates may cause. A trading rate equal to the monetary interest rate renders permit prices constant over time – inducing constant emission over time. A trading rate equal to zero translates into permit prices rising over time at the discount rate. In both cases, stock related damage are ignored, which induces non optimal decisions. The existence of the physical correlation either improves or worsens the situation, depending on the type of correlation between the stocks.
Appendix
Appendix 1: Hamiltonian and FOCs in the Social Optimum Case
The Hamiltonian writes as follows:
Hso = n X i=1 πi(xi, gi, ri) − D(G, . . . , Rj, . . .) + X i∈Sj µsoj [f (xi, gi, ri) − γjRj] + λso X i∈S [F (xi, gi, ri)] − δG − X jinS j h(Rj) .
Assuming an interior solution, the first-order conditions for this problem are:
Hxso= πix+ λsoFx+ µsoj fx= 0, (28) Hgso= πig+ λso+ µsoj fg= 0, (29) Hrso= πig+ λsoFr+ µsoj = 0, (30) −HGso= ˙λso− ρλso= − [−D G− δλso] , (31) −HRsoj = ˙µso j − ρµsoj = −−DR− γjµsoj − λsohRj . (32)
with the transversality conditions: lim t→∞λ so(t)G(t) = 0 (33) lim t→∞µ so j (t)Rj(t) = 0, ∀j. (34)
Appendix 2: Proof of Proposition 1
Differentiating and rearranging (28)–(32), we obtain:
(28) ⇒ µsoj = −λ soF x− πx fx ⇒ ˙µso j = − ˙λsoF x− ˙πx fx (31) ⇒ − ˙λ soF x− ˙πx fx = (ρ + γj)µj+ DR+ λsohRj
20 G–2008–80 Les Cahiers du GERAD ⇒ −Fx fx ((ρ + δ)λso+ DG) − ˙ πx fx = (ρ + γj)( −λsoF x− πx fx ) + DR+ λhRj ⇒ πix− ˙πix ρ + γj = FxDG+ fxDR ρ + γj + RSHx
Appendix 3: Social Optimum Steady State and Paths Derivation
The Modified Hamiltonian Dynamic System (for n symmetric agents) writes as follows, with
Φ = (G, λso, R j, µsoj ): ˙Φ = J · Φ + C with: JL= −δ nK1 −nh nK2 0 ρ + δ 0 0 0 K2 −γj K3 0 h 0 ρ + γj and CL= K4 d0 K5 d JQ = −δ nK1 −nh nK2 d0 ρ + δ 0 0 0 K2 −γj K3 0 h d1 ρ + γj and CQ= K4 0 K5 0 K1 = X i α2 i ci , K2 = X i αiβi ci , K3 = X i β2 i ci , K4 = n α1p c1 , K5 = β1p c1
The steady state values are obtained by setting ˙Φ = 0. The paths then follow:
Φ(t) =X
j
kjvjievjt+ Φ∞ (35)
where vj are the negative eigenvalues of J, vji the associated eigenvectors and kj constants
derived by setting t = 0. Linear Damage Function
G∞= −hn δ R ∞+nK2 δ µ ∞+nK1 δ λ ∞+K4 δ (36) R∞= K3 γi µ∞+K2 γj λ∞+K5 γj (37) λ∞= −d0 ρ + δ (38) µ∞= − −hλ∞+ d ρ + γj (39) c1= G0− G∞ − c2 (40) c2= R0− R∞ v2R (41)
v1 = −δ v2 = −γ
v1G = 1 v2G = 1
v1λ= 0 v2λ= 0
v1R= 0 v2R = γ−δhn
v1µ= 0 v2µ = 0
Quadratic Damage Function
G∞= − δ + ρ d0 λQ∞ (42) R∞= − γ + ρ d1 µQ∞− h d1 λQ∞ (43) λ∞= d0np (γ + ρ)(β1h − α1γ) + d1(β1K2− α1K3) DEN (44) µ∞= −p hd0n(β1h − α1γ) + d1d0n(β1K1− α1K2) + β1d1δ(ρ + δ) DEN (45) c1 = v2R(G∞− G0) + v2G(R0− R∞) v2Gv1R− v1Gv2R (46) c2 = v1R(G0− G∞) + v1G(R∞ − R0) v2Gv1R− v1Gv2R (47) v1 = 12ρ −12 q M +pT (h) v2= 12ρ −12 q M −pT (h) v1G = (ρ+δ−vdO[K1))(ρ+γ−v2d1+h(γ+v1))(γ+v1)] 1) v2G= (ρ+δ−v2))(ρ+γ−v2))(γ+v2) dO[K2d1+h(γ+v2)] v1λ= −(γ+vK2d11+h(γ+v)(ρ+γ−v1)]1) v2λ= −(γ+v2)(ρ+γ−v2) K2d1+h(γ+v2)] v1R= hKK32−Kd1+h(γ+v2(ρ+γ−v1)]1) v2R= hkr−kg(ρ+γ−v2) K2d1+h(γ+v2)] v1µ= 1 v2µ= 1 DEN = c1d0nK1γ(ρ + γ) + d1(K1K3− K22) − hK2(ρ + 2γ) + h2K3) + c1δ(δ + ρ) [γ(ρ + γ) + K3d1] M = ρ2+ 2ρ(γ + δ) + 2(δ2+ γ2) + 2d0nkK1+ 2K3d1 T (h) = h2(−4d0nK)+ h(4d0K2n(ρ + 2γ)) + ∆ Appendix 4
The Hamiltonian associated to the problem of an individual agent subject to markets for global and regional pollutants writes:
H = πi(xi, gi, ri) − pGYi− pRyi+ θiG
¯
Ei− F (xi, ri, gi) + Yi+ τGBi
22 G–2008–80 Les Cahiers du GERAD
First-order conditions follow: ∂H ∂k = πik− Fkθ G i − fkθiR= 0 for k ∈ {xi, gi, ri}, (49) ∂H ∂Yi = −pG+ θGi = 0 ⇒ pG= θGi , (50) ∂H ∂yi = −pR+ θRi = 0 ⇒ pR= θiR, (51) ˙ θiG− ρθiG = − ∂H ∂Bi = −τGθiG ⇒ ˙θiG= (ρ − τG)θiG, (52) ˙ θiR− ρθiR= − ∂H ∂bi = −τRθiR⇒ ˙θiR= (ρ − τR)θiR, (53) lim t→∞Bi(t)θiG = 0 and limt→∞bi(t)θiR= 0. (54)
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