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Ecological Modelling 220 (2009) 2536–2542

Contents lists available at ScienceDirect

Ecological Modelling

journa l homepage: www.e lsev ier .com/ locate /eco lmodel

eduction of CO2 emission by optimally tracking a pre-defined target

arco Antonio Leonel Caetano a,∗, Douglas Francisco Marcolino Gherardi b,ustavo de Paula Ribeiro a, Takashi Yoneyama c

INSPER, Instituto de Ensino e Pesquisa, São Paulo, BrazilDivisão de Sensoriamento Remoto, INPE, São José dos Campos, BrazilITA, Instituto Tecnológico de Aeronáutica, São José dos Campos, Brazil

r t i c l e i n f o

rticle history:eceived 18 February 2009eceived in revised form 20 May 2009ccepted 2 June 2009vailable online 1 July 2009

eywords:ontrolimulation

a b s t r a c t

The recent global financial crisis has highlighted the need for balanced and efficient investments in thereduction of the greenhouse effect caused by emissions of CO2 on a global scale. In a previous paper, theauthors proposed a mathematical model describing the dynamic relation of CO2 emission with investmentin reforestation and clean technology. An efficient allocation of resources to reduce the greenhouse effecthas also been proposed. Here, this model is used to provide estimates of the investments needed in landreforestation and in the adoption of clean technologies for an optimum emission and abatement of CO2,for the period of 1996–2014. The required investments are computed to minimize deviations with respectto the emission targets proposed in the Kyoto Protocol for European Countries. The emission target can be

athematical modellobal warmingarbon dioxidereenhouse gases

achieved by 2014 with investments in reforestation peaking in 2004, and a reduction of the expected GDPof 42%, relative to 2006. Investments in clean technology should increase between 2008 and 2010 withmaximum transfer figures around 70 million American dollars. Total (cumulative) costs are, however,relatively high depending on the price of carbon abatement and the rate at which the expected CO2

concentration in the atmosphere should be reduced. Results highlight the advantages for policy makersstmehat tr

to be able to manage inveon a portfolio of actions t

. Introduction

The Kyoto Protocol recommends the collective reduction ofreenhouse gases (carbon dioxide, methane, nitrous oxide, sulfurexafluoride, hydrofluorocarbons, and perfluorocarbons) for indus-rialized countries by 5.2%, averaged over the period of 2008–2012,aking as reference the year of 1990. However, some countries areeluctant to ratify the emission target due to the alleged negativempacts of reduction costs on economic growth. Meanwhile, manyndustrialized economies are suffering from an acute deteriorationrocess caused by the financial crisis (the sub-prime crisis) thatit the world economy after mid-2007. The assumptions that haveeen made to estimate the costs of reducing emission by the USA

mply high costs, and these estimates seem to be biased by wrongnterpretation of literature, confusion of short-term with long-runosts and the selection of inadequate parameters (Barker and Ekins,001). On the other hand, Ackerman and Stanton (2008) reported

∗ Corresponding author at: IBMEC Sao Paulo, Graduacao, Rua João Bicudo, 345splanada II, 12.242-530 São José dos Campos, SP, Brazil.

E-mail addresses: marcoALC1@isp.edu.br, malcaetano@uol.com.brM.A.L. Caetano), douglas@dsr.inpe.br (D.F.M. Gherardi), contato@guribeiro.com.brG.d.P. Ribeiro), takashi@ita.br (T. Yoneyama).

304-3800/$ – see front matter © 2009 Elsevier B.V. All rights reserved.oi:10.1016/j.ecolmodel.2009.06.003

nts in climate policy more efficiently, controlling optimum transfers basedacks a pre-defined CO2 concentration target.

© 2009 Elsevier B.V. All rights reserved.

that the total cost of global warming to the USA by 2100, in abusiness-as-usual (BAU) scenario, can be as high as 1.8% of theirgross domestic product (GDP, in dollars for 2008). Economic lossesare expected to come from impacts on areas such as hurricane dam-ages, real estate losses, energy-sector and water costs, the lattercontributing with the largest cost percentage.

The European Union (EU) has accepted a quantitative absolutereduction of 8% of its GHG emissions and the projected welfare costmay vary across countries between 0.6% and 5% of GDP (exclud-ing favorable terms of trade effects), assuming that no climatepolicies are implemented (Viguier et al., 2003). The projected emis-sion within the EU, without specific reduction efforts, will reach3.8 × 1012 kg of CO2 in 2010 and 4.1 × 1012 kg of CO2 in 2020, cor-responding to an increase of 14% above 1990 levels, instead of areduction of 8%. On a global scale, the stabilization at 450 ppmCO2-equivalent would require a reduction of present average globalper-capita emission of about 25% during the next three decades andby 50% by 2050 (Bolin and Kheshgi, 2001). Projections presented byEnkvist et al. (2007) suggest that the costs for the global economy

of achieving the 450 ppm CO2-equivalent scenario in 2030 couldlie between 500 billion euros, or 0.6% of that year’s projected GDP,and 1 trillion euros, or 1.4% of global GDP. More pessimistic projec-tions produced by Keohane and Goldmark (2008) argue that, evenextending the time horizon for GHG reduction programs until 2030,

al Modelling 220 (2009) 2536–2542 2537

aw9

cinptgpwa(aGiataUcrcsot

cnoamoTdabLtmtCeoa

Table 1Model parameters fitted for WesternEurope.

Parameters Values

r 0.15s 700h 0.0001u1 0.00012u2 0.0008� 0.035q 11r1 3.5E + 8

M.A.L. Caetano et al. / Ecologic

n adequate target might be unachievable, as the global emissionould have to be cut by more than half in just 20 years, and by over

5% by the end of the century.Under such conditions of high costs and uncertainties in

ost–benefit relations, it is crucial to determine the efficiency of anynvestment policy for reduction of GHG among the available alter-atives, conditional to the maximum economic development. Theotential costs of Kyoto and the burden sharing seem to be sensitiveo the implementation of domestic policies, and it has been sug-ested that inefficient policies in some sectors could lead to higherrojected costs (Viguier et al., 2003). Basic emission control policiesere discussed by Janssen (1997) using two levels of reductions

s indicated by the intergovernmental panel on climate changeIPCC): BAU and accelerated policies. The BAU policy resulted inn expected temperature increase of 4 ◦C, whereas the controlledHG in the accelerated policies scenario resulted in a temperature

ncrease limited to 1.3 ◦C. The idea of establishing targets has beenlso explored by Azar and Schneider (2002), whose estimates ofhe (present) value of future abatement costs for reaching 350, 450nd 550 ppm CO2-equivalent are US$ 18 trillion, US$ 5.2 trillion andS$ 1.9 trillion, respectively, in the period of 1990–2100. Abatementosts may also incorporate equity issues (Tol, 2001) such as deathsesulting from changes in heat stress, cold stress, malaria, tropicalyclones, and inequity aversion or the maximization of cooperativeolutions. Despite the difficulty in distinguishing between sourcesf inequity, international cooperation can reduce emission in con-rast with a no-cooperation scenario.

This work addresses the problem of analyzing the investmentsonsidering two decision variables, reforestation and clean tech-ology, one type of GHG (CO2), and measuring the performancef the adopted policy by computing the accumulated deviationsbout a target path based on the Kyoto Protocol. The mathematicalodel uses ordinary differential equations to relate the production

f CO2 with forest area and GDP, following Caetano et al. (2008).he parameters of the model were adjusted using widely publishedata, such as those available at UNEP (UNEP GRID ARENDAL, 2008),nd the estimates of the required joint investments are determinedy solving numerically an optimal tracking problem (Kirk, 1970;ewis, 1986). The work is organized as follows. Section 2 presentshe mathematical model describing the dynamics of CO2, the opti-

um control problem and the tracking procedure. For details ofhe optimum control problem the reader is referred to the paper of

aetano et al. (2008). In Section 3, the results are presented withmphasis on the future costs of CO2 abatement with investmentsn reforestation and on the development of clean technologies. It islso discussed the usefulness of the model as an aid to the selection

Fig. 1. Relationships among the state variables in the adopted model.

r2 1E + 9p 10˛1 0.15˛2 0.00005

of the most effective investments capable of tracking the maximumatmospheric concentration indicated on the Kyoto Protocol. Section4 summarizes de main conclusions.

2. Materials and methods

2.1. The mathematical model

Several mathematical models are available in the literature todescribe the dynamics of the GHG (see, for instance, Nordhaus,1991a,b, 1993, 2006). In this work, the model presented in Caetanoet al. (2008) is used. It consists of a system of three coupledordinary differential equations involving the concentration ofatmospheric carbon dioxide x(t), forest area z(t) and gross domesticproduct—GDP, y(t). In what follows, the “dot” notation representsthe derivative of a variable with respect to time t.⎧⎪⎪⎨⎪⎪⎩

x = rx(

1 − x

s

)− ˛1z + (˛2 − u2)y

z = u1y − hz

y = � y

(1)

The decision variables are u1 and u2, representing the sharesof GDP in reforestation and in the adoption of clean technology,respectively. The model parameters (constants) are r, s, h, ˛1, ˛2 and� . The relationships among the variables in mathematical modelcan be visualized in Fig. 1 (Caetano et al., 2008).

2.2. Model parameters

The model parameters (Table 1) were adjusted to fit the coniferforest data for the Western Europe region available at UNEP (UNEP,2007) and Earth trends environmental information (2007) for thetracking problem. Results presented in Fig. 2 represent the actualdata and the numerical simulations using the model defined in (1),show good agreement using numerical values of parameters fromCaetano et al. (2008). The initial conditions for the Western Europein 1960 (UNEP, 2007) are x(0) = 398 million tons of CO2, as for the,z(0) = 43 million m3 in 1960 and y(0) = 2787 billion internationaldollars in 1960, GDP value for 1995 (The World Economy, 2007).

2.3. The tracking problem

Let the planned reduction of the GHG be a reference curvedenoted ref(t). The idea is to keep x(t) close reference ref(t), by penal-izing deviations from the target trajectory and the intensity of the

control variables, i.e., minimizing the quadratic cost function:

J[u1(·), u2(·)] = p(x(tf ) − ref (tf ))2 +∫ tf

0

[q(x(t) − ref (t))2 + r1u21(t)

+ r2u22(t)] dt (2)

2538 M.A.L. Caetano et al. / Ecological Modelling 220 (2009) 2536–2542

a and

svpa

ecd

r

msot

H

Fig. 2. Comparison between the actual dat

ubject to the state Eq. (1). The final time tf is fixed and the stateariables x, z and y are assumed free at the final time. The constants, q, r1 and r2 are positive constants. The initial conditions are thectual values observed in 1996:

x(0) = x0(actual)

z(0) = z0(actual)

y(0) = y0(actual)

x(tf ) = free

z(tf ) = free

y(tf ) = free

tf = fixed

(3)

The reference trajectory ref(t) for x(t), i.e., the target curve for CO2mission, should reach a established value at tf. In this work, twoases have been studied as targets, namely, linear and exponentialecay rate, or:

ef (t) ={ −�t + � (linear)

� exp(−ıt) (exp onential)(4)

The optimal control problem is to find U(·) = [u1(·) u2(·)] thatinimizes J(u1,u2) given by Eq. (2), subject to the dynamic con-

traint (1) and to the boundary conditions (3). In order to solve theptimal control problem, it is necessary to find the expressions forhe co-state variables using the Hamiltonian function:

= �x

(rx

(1 − x

s

)− ˛1z + (˛2 − u2)y

)+ �z (u1y − hz) + �y�y

+(

q(x − ref )2 + r1u21 + r2u2

2

)(5)

the simulation using the estimated model.

Now, the co-state variables �x, �y and �z satisfy the following ordi-nary differential equations:

�x = −∂H

∂x= −�xr + 2r�xx

s− 2q(x − ref )

�z = −∂H

∂z= �x˛1 + �zh

�y = −∂H

∂y= −�x(˛2 − u2) − �zu1 − �y�

(6)

As the terminal conditions for the state variables (3) are free, thefinal conditions for the co-state variables are:

�x(tf ) = 2p(x(tf ) − ref (tf ))

�z(tf ) = 0

�y(tf ) = 0

(7)

The Pontriaguin’s maximum principle (Pontriaguin et al., 1983;Kirk, 1970; Lewis, 1986) requires the Hamiltonian H in Eq. (5) to beminimized along optimal trajectories. Therefore, the optimality ofthe control variables u1 and u2 are required to satisfy:

∂H

∂u1

∣∣∣∣u∗

1

= 0 ⇔ �zy + 2bu∗1 = 0 ⇔ u∗

1 = −�zy

2b(8)

and

∂H

∂u2

∣∣∣∣u∗

= 0 ⇔ − �xy + 2cu∗2 = 0 ⇔ u∗

2 = �xy

2c(9)

2

where the (*) denotes optimal values of u1 and u2.The numerical solution was obtained by solving the TPBVP using

the “Collocation Method” on Matlab 6.5 with rotine bvp4c.m. Thereare, however, other alternative approaches to solve a TPBVP, suchas the Multiple Shooting Method (Bulirsh and Stoer, 1980; Bulirshet al., 1991; Oberle, 1988; Pesh, 1980).

M.A.L. Caetano et al. / Ecological Modelling 220 (2009) 2536–2542 2539

Table 2Total cost to reduce CO2 emission for Western Europe obtained by simulation.

US$/ton CO2/year Total cost (18 years) inUS trillion dollars

q-Values US$/tonCO2/month

12 27 148 32 472 34 696 40 8

3

tivtmoiapuo2muaee(s(vbc

sttaa

r

o

r

pm4r1cwaf1aan

m

Fig. 3. Projection of the state variables for the period 1996–2014, if the proposedoptimal policy had been adopted in 1996.

132 48 11144 51 12156 53 13

. Results and discussion

A reasonable climate policy strategy should consider balancinghe investments for the reduction of CO2 in the atmosphere againstts effectiveness and the resulting welfare costs. This is a differentiew if compared to the typical cost–benefit assessments where theotal cost of abatement is compared with the perceived costs of cli-

ate change itself. The focus on benefits resulted from investmentsn climate policy may distract the decision maker from the most

mportant issue that is the reduction of CO2 concentration in thetmosphere to a level that will not enhance greenhouse effect. Asointed out by Hof et al. (2008), some cost–benefit analyses tend tonderestimate the optimal abatement effort, including proposalsf optimal concentrations of CO2-equivalent of up to 850 ppm in200. There has been much debate about the ability of cost–benefitodels to identify meaningful targets for climate policy, fueled by

ncertainties around marginal damage estimates (Yohe, 2003; Tolnd Yohe, 2007; Hof et al., 2008). On the other hand, abatementfforts based on incentives aimed at pre-defined targets of CO2mission can be an efficient means to tackle the costs of KyotoSchaeffer et al., 2008). This approach also contrasts with othertrategies based on integrated assessment model of climate changeTol, 1999; Elzen and Lucas, 2005; Hof et al., 2008). Our model pro-ides an evaluation of the possible investments by indicating theest available options and how efficiently the chosen alternativesan lead to a pre-defined abatement target.

The initial condition for the state variables in our model corre-pond to the actual data for 1996, starting with x(0) = 659 millionons of CO2, z(0) = 68 million m3 and y(0) = 8306 billion interna-ional dollars. The final target for Western Europe CO2 emission,ccording to the Kyoto Protocol is 590 million tons, this can bechieved numerically by the linear reduction scheme:

ef1(t) = −69t + 659 (10)

r the exponential reduction scheme:

ef2(t) = 659 e−0.11t (11)

The cost estimates for the abatement of CO2 emission in theresent work are given in million US$ and were integrated inonthly time steps (Table 2). Assuming the present estimate as US$

0/ton CO2/year in Western Europe, the adopted values for q, r1 and2 are 3.33/(10−12 million ton CO2)2, 3.5 × 108/(million US$)2 and× 109/(million US$)2, respectively. Tracking the proposed optimal

limate policy would lead to an increase of 47% in the forested areaithin an estimated GDP increase of 81%, relative to 1996 (Fig. 3). InBAU scenario for investments in climate policy, the expected GDP

or 2014 is 18.5 trillion dollars or an increase of 122.9%, relative to996. The GDP projection in a BAU scenario was calculated without

ny control by the investments considered in the proposed model,nd offer a maximum GDP increase figure if Western Europe wasot investing in the GHG reduction.

The model indicates that for an efficient use of resources, invest-ents in carbon capture (reforestation) should be higher during the

2540 M.A.L. Caetano et al. / Ecological Modelling 220 (2009) 2536–2542

Ft1

imrrliaMrhteMfhatieta

i

ig. 4. Investments in reforestation (top) and in incentives for clean technology (bot-om) for the period (1996–2014). Supposing optimal control had been adopted in996.

nitial years (Fig. 4), with maximum annual transfers of US$ 275illion (2000) for a GDP of US$ 10 trillion. The implementation of

eforestation programs tends to have low short-term investmenteturns and may demand public subsidies. Competition for localabor and substitution of crop lands to forest may also imposemplementation constraints and food security concerns in somereas. However, future estimates of carbon prices (Sohngen andendelsohn, 2003; Rokityanskiy et al., 2007) are expected to sky-

ocket after 2060, way above the projected abatement costs (asigh as US$ 156/ton CO2/year), suggesting that early implemen-ation of abatement initiatives is likely to increase its pay-off as thenergy abatement cost tend to show little variation (Sohngen andendelsohn, 2003). It is also important to highlight that alternative

orestland management, such as avoided deforestation or extendedarvest rotation, has not been explicitly modeled since investmentsre calculated based on a finite land surface limitation (Eq. (1)). So,he low short-term investment returns and the limited long-termmpact on CO2 reduction resulted from reforestation (see Caetanot al., 2008) clearly point to the urgent need of a concerted effort

o conserve and manage the unique ecological services provided byll major world biomes.

Investments in clean technology such as energy efficiencymprovements, bio-energy and renewable follow a symmetric pat-

Fig. 5. Some scenarios showing the strong dependence of the incurred total costin tracking the planned CO2 profile with respect to the investment rate and theexponential decay rate.

tern compared to reforestation (Fig. 4), progressively increasing butpeaking before 2014. As the emission target is approached bothinvestments in reforestation and clean technology tend to decrease,showing that the optimum tracking offers a pragmatic backgroundfor value judgment. It is largely accepted that coordinated invest-ments in a cooperative setting can reduce costs of investments inCO2 abatement (Tol, 2001). This might be the case, for instance, forthe development and dissemination of hydrogen powered vehiclesthat demand concerted policy intervention rather than a responseto market forces (Dougherty et al., 2009). The estimation of annualand cumulative costs to achieve a pre-defined target is more prag-matic than the cost–benefit approach with constant marginal gainsfor each dollar invested in climate policy, and the exponentialincrease of the costs for GHG concentration stabilization (Schaefferet al., 2008).

Under the cap and trade scheme used for GHG emission con-trol, uncertainties about compliance costs have caused countriesto withdraw from negotiations. However, if emission targets canbe made more flexible and incorporate optimum choices of invest-ments with minimum impact on the GDP, then climate agreementsmay become more attractive and efficient (Jotzo and Pezzey, 2007).The projected total carbon abatement costs are highly sensitiveto costs of carbon sequestration, different choices of weightsappearing in the cost function leads to different numerical results,although similar in qualitative terms (Fig. 5). The compoundedeffects on the costs of the CO2 abatement and the decay rate showlarger annual variances and higher cumulative costs for high (0.11)decay rate and lower variances and low costs for low (0.08) rate(Fig. 6). However, extremely low decay rates (0.01, not shown),meaning a careless climate policy, are as expensive as the moststringent policy. This means that the cost of a balanced climatepolicy tend to be cheaper, and less susceptible to uncertainties inabatement costs, than an indifferent or severe policy, which can beup to two orders of magnitude more expensive depending on theabatement cost. The high decay rate causes the tracking procedureto force initial investment to be the highest, characterizing a strin-gent pursuit of the target, therefore, leading to higher cumulative

costs. So, the longer the Western Europe countries (or any othercountry) delay their actions towards emission control, the moreexpensive it will become for them to achieve the atmospheric CO2concentration proposed by the Kyoto Protocol.

M.A.L. Caetano et al. / Ecological Mod

FeS(

dqaderlt9wtprCb

4

igo

ig. 6. Simulation of cumulative costs in tracking the planned CO2 profile with anxponential decay rate of 0.08 (top) and 0.11 (bottom), and different abatement costs.olid line, q = 1; circle, q = 4; square, q = 6; star, q = 8; diamond, q = 11; dashed line, q = 13see Table 2 for abatement costs/ton/year).

Extremely low abatement prices causes the model to pre-ict negative investments, clearly indicating that minimizing theuadratic cost function reduces penalization on the control vari-bles and the time necessary to reach the CO2 target. Consideringifferent values for the cost of keeping close to the target of CO2mission, i.e., US$ 48, 20 and 12 per ton of CO2 per year, theespective average annual costs are 5 (total costs of US$ 90 tril-ion/18 years), 4.7 and 1.5 trillion US$ in 2014. In terms of GDP, forhe optimistic case of US$ 12/ton CO2, the annual cost represents.3% of GDP of 2014. The simulations presented here agree veryell with results published elsewhere suggesting that achieving

he Kyoto Protocol target should involve massive investments, andoint out for sensible strategies for the allocation of resources. Theesults presented here have also shown that the maximization ofO2 reduction with the minimum possible impact on the GDP canecome part of a feasible and responsible climate policy.

. Conclusions

This work presents a method that shows how to balance thenvestments on a portfolio necessary to meet the CO2 emission tar-et recommended by the Kyoto Protocol. The main idea is to useptimal tracking control to force the concentration of CO2 to follow

elling 220 (2009) 2536–2542 2541

a pre-defined curve by selecting the best allocation of investmentsin reforestation and in the adoption of clean technology.

Although the recommended investments in reforestation and inthe adoption of clean technology vary significantly depending onthe choices of the weights used in the cost function, the proposedmethod can be useful in terms of gaining insight into alternativepolicies outcomes. The main qualities of the model are:

• allow a balanced evaluation of possible climate policies based onthe available portfolio of actions and investments for the reduc-tion of CO2 in the atmosphere, measured against its effectivenessand the resulting welfare costs;

• inform the expected cumulative (total) costs along a spectrum ofdifferent climate policies, from the careless to the severe imple-mentation scenarios;

• the qualitative results are invariant over a large range of parame-ter values;

• the use of optimal control theory offers an alternative modelingapproach to the typical cost–benefit assessments where the totalcost of abatement is compared with the perceived costs of climatechange itself.

The model was assumed to be time-invariant, so that economicalcrises, such as the sub-prime crash that initiated in 2007, may leadto the necessity of estimating new parameters. Moreover, the casestudy was applied to Western Europe as a whole and it is a factthat not all countries follow the same policy in terms of emissionof GHG.

Finally, because of the several simplifications, the proposedmethod may not be suitable for actual computation of the exactamounts to be invested in order to avoid the greenhouse effect.However, the method provides a first order approximation thatcould be used by decision makers concerned with the proposal ofsensitive climate policies.

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