aa r X i v : . [ q -f i n . E C ] J un Economics of limiting cumulative CO emissions Ashwin K Seshadri
Divecha Centre for Climate Change, Indian Institute of Science, Bangalore 560012, India. email:[email protected]
Abstract
Global warming from carbon dioxide (CO ) is known to depend on cumulative CO emis-sions. We introduce a model of global expenditures on limiting cumulative CO emissions, takinginto account effects of decarbonization and rising global income and making an approximationto the marginal abatement costs (MAC) of CO . Discounted mitigation expenditures are shownto be a convex function of cumulative CO emissions. We also consider minimum-expendituresolutions for meeting cumulative emissions goals, using a regularized variational method yield-ing an initial value problem in the integrated decarbonization rate. A quasi-stationary solutionto this problem can be obtained for a special case, yielding decarbonization rate that is pro-portional to annual CO2 emissions. Minimum-expenditure trajectories in scenarios where CO emissions decrease must begin with rapid decarbonization at rate decreasing with time. Dueto the shape of global MAC the fraction of global income spent on CO mitigation ("burden")generally increases with time, as cheaper avenues for mitigation are exhausted. Therefore failureto rapidly decarbonize early on reduces expenditures by a small fraction (on the order of 0.01%) of income in the present, but leads to much higher burden to future generations (on theorder of 1 % of income). Global warming from carbon-dioxide (CO ) is related to its cumulative emissions, i.e. emissionsintegrated across time, and independent of emissions pathway ( Allen et al. (2009);
Matthews et al.
Stocker et al. (2013);
Seshadri (2017)). Mitigation of global warming requires large-scaleand expensive efforts to decarbonize the world economy (
Manne and Richels (1993);
Grubb (1993);
Weyant (1993);
Nordhaus (1993a)). Estimated costs of mitigation vary across different studiesand depend on the assumptions that are made (
Grubb (1993);
Rogelj et al. (2015)). The SternReview of Climate Change (
Stern (2007)) estimated an average cost of reducing anthropogenicemissions of climate forcers to a 550 ppm CO equivalent stabilization level of about 1 percent ofGlobal Gross Domestic Product (GGDP), which is much smaller than damage costs of unmitigatedglobal warming ( Stern (2007)). In a recent study
Rogelj et al. (2015) illustrate using the integratedassessment model MESSAGE that for scenarios limiting global warming to less than 2 degrees Cthe cost of mitigation ranges from a small fraction of 1% to a few % of GGDP, and an importantfactor governing the costs is baseline energy demand (
Rogelj et al. (2015)).This paper introduces an analytical model for global expenditures on reducing CO emissionsfrom economic activity ("decarbonization"). We represent the marginal abatement cost (MAC)of reducing CO emissions as a function of emissions intensity of GGDP. If increase in GGDPleads to relative increase in CO emissions by a smaller factor, because global income elasticity ofCO emissions is less than 1, there will occur exogenous reductions in emissions intensity as theglobal economy grows; this will happen even in the absence of any deliberate mitigation effort, inthe "business as usual" scenario. Exogenous reductions in emissions intensity may be viewed asbeing the result of technological improvements with time, in models where the production functionis explicitly present ( Nordhaus (1993b)). Our model does not include the production function,but instead uses the income elasticity of global CO emissions as a constant parameter. Thispermits a simple treatment of the time-dependence of the MAC curve, even in the presence ofexogenous reductions in emissions intensity, but our discussion is limited by the assumption ofconstant elasticity.We integrate the resulting MAC curve to consider global expenditures on mitigation. Our modelconsiders expenditures for reducing emissions intensity (i.e. decarbonization) as well as thoseinvolved in scaling up decarbonization activity as the global economy expands. This can helpunderstand factors behind the scale of economic effort involved in global decarbonization. Be-cause of the centrality of cumulative CO emissions to the global warming problem ( Stocker et al.
Friedlingstein et al. (2014);
Raupach et al. (2014);
Rozenberg et al. (2015);
Peters (2016);
Pfeiffer et al. (2016)), we also consider how to minimize costs of limiting cumulative emissions overtime. This is examined through a constrained variational problem ( van Brunt (2004)), minimizinga functional describing discounted total mitigation expenditures while satisfying a constraint oncumulative emissions across a specified time-horizon. The policy variable is the decarbonizationrate, the latter being the rate with which emissions intensity is reduced. Minimizing the func-tional leads to the familiar Euler-Lagrange equation ( van Brunt (2004)). For our model of CO mitigation expenditures, the variational problem is degenerate and the Euler-Lagrange equationis algebraic, whereas we seek an initial condition problem in the integrated decarbonization rate.This is an ill-posed problem ( Tikhonov and Arsenin (1977)), because a solution satisfying the initialcondition on the integrated decarbonization rate does not exist for the algebraic Euler-Lagrangeequation that is obtained. One approach for dealing with ill-posed problems is through regulariza-tion (
Tikhonov and Arsenin (1977)), by adding an additional term to the quantity being minimizedin order to render it soluble. In our case we include an additional contribution to the functionalbeing minimized for rendering an initial value problem in the Euler-Lagrange equation.The aforementioned approach contrasts with the optimal control problem of choosing an emissionspathway to maximize discounted utility, taking into account costs of mitigation as well as damagecosts of global warming, which underlies integrated assessment models of global warming suchas DICE (
Nordhaus (1993a,b)). In such approaches the optimal mitigation pathway is such thatthe effect on utility of marginal increment to current consumption from increasing emissions isbalanced by the present value of the diminution in future consumption (
Nordhaus (1993a)). Withoutconsidering the production function or climate damages in the present paper we cannot modelthe above features, and instead assume that cumulative emissions goals are specified exogenously,following contemporary discussions in climate policy (
Meinshausen et al. (2009);
Peters (2016)).Section 2 introduces the models of global CO emissions and mitigation expenditures, and theresults derived from them. Section 3 considers the expenditure-minimizing pathways of decar-bonization, subject to a cumulative emissions constraint. The control variable is the integrateddecarbonization rate. As mentioned previously, the original problem is degenerate, and must be See, for e.g., van Brunt (2004) for a general discussion of such cases. emissions under business as usual We describe global CO emissions m ( t ) as the product of GGDP, denoted by g ( t ), and globalemissions intensity µ ( t ); with present values m , g and µ . In the absence of deliberate mitigation,under business as usual, growth in GGDP leads to increase in global CO emissions that is governedby the constant-elasticity model △ m/m = θ △ g/g , where θ is the global income elasticity of globalCO emissions. Considering small time-interval △ t the emissions intensity at t + △ t has formula m + △ mg + △ g = (cid:18) θ − △ gg (cid:19) µ (1)The change △ µ in emissions intensity is the above ratio minus µ ( t ), so its rate of change duringinterval △ t is △ µ △ t = − (1 − θ ) rµ (2)where r = ( △ g/ △ t ) /g is growth rate of GGDP during this period. Generally θ is smaller than one,leading to exogenous decrease of emissions intensity at rate σ = (1 − θ ) r . This effect is independentof any deliberate mitigation effort, occurring under business as usual. It is larger for higher growthrates of GGDP if θ <
1. For example, GGDP growth at constant rate r = 4 . θ = 0 .
75 yields constant exogenous decarbonization rateof 1 % per year. However we caution that income elasticity of energy demand varies by countryand is smaller for developed nations (
Webster et al. (2008)). Therefore it is liable to change withtime as countries develop, and our assumption of constant elasticity is only an idealization. Absence of mitigation can also adversely impact economic growth. Quoting
Stern (2016), "So the business-as-usual baseline, against which costs of action are measured, conveys a profoundly misleading message to policymakersthat there is an alternative option in which fossil fuels are consumed in ever greater quantities without any negativeconsequences to growth itself." We write m + △ mg + △ g as µ θ △ gg △ gg , expand the denominator by its Taylor series in △ gg and approximate to first-degreein △ gg , obtaining equation (1). . σ = 0 . r and θ and in the absence of deliberatereductions, the emissions intensity at time t would be µ ( t ) = µ e − σt , where t = 0 denotes thepresent. Of course, if elasticity θ = 1 then there is no exogenous decarbonization in our model, anddecarbonization occurs only through deliberate mitigation. Increase in GGDP leads to increases in emissions as described above and corresponding expansion ofmitigation possibilities, thereby stretching horizontally the marginal abatement cost (MAC) curveas the global economy grows. In the absence of deliberate reductions, this effect is governed byexogenous decarbonization rate σ so the MAC curve expressed in terms of emissions intensity is C ( µ ( t )) = α (cid:16) µ ( t ) µ e − σt (cid:17) ν (3)where C ( µ ( t )) is cost of reducing emissions intensity by one unit, for each unit of GGDP and ν > µ ( t ) is equal to µ e − σt for all times then the MAC in our model remains constant at α , and thereby this modelneglects effects of learning. As deliberate mitigation proceeds thereby reducing ratio µ ( t ) /µ e − σt the marginal cost increases. The MAC C ( µ ( t )) is in units of billion $ / (Gton CO year -1 ). Thegraph represented by C ( µ ( t )) has the same scale as MAC curves described in $ / ton CO .Parameter α describes the present MAC, in billion $ / (Gton CO year -1 ). The present cost ofreducing emissions intensity by △ µ in a year is αg △ µ . The factor g △ µ has units of emissions(Gton CO year -1 ). For decreasing emissions intensity between time t and t + △ t by △ µ ( t ),the mitigation expenditure is C ( µ ( t )) g ( t ) △ µ ( t ). GGDP is in units of trillion $ / year, andglobal emissions intensity in Gtons CO / trillion $, so expenditure is in (billion $ / (Gton CO year -1 ))*(trillion $ / year)*(Gton CO / trillion $), or billions of dollars. A simple model for exogenous learning would make α a function of time. .3 Effect of mitigation at rate k ( t ) Deliberate reduction of emissions intensity, or mitigation, occurs at rate k ( t ), with the effect overtime-interval △ t being △ µ ( t ) = − k ( t ) µ ( t ) △ t . In conjunction with exogenous reductions arisingduring economic expansion in case θ <
1, the rate of change in emissions intensity is △ µ ( t ) / △ t = − ( k ( t ) + σ ) µ ( t ), which is integrated for µ ( t ) = µ e − ´ t k ( s ) ds e − σt . We write this in terms ofintegrated decarbonization rate K ( t ) = ´ t k ( s ) ds , so that µ ( t ) = µ e − K ( t ) e − σt . Decarbonizationin the form of this integrated rate is the policy variable in the optimization problem of Section 3. The expenditure on mitigation has two contributions: initial cost associated with reducing emissionsintensity, and scaling up mitigation to maintain reduced levels of emissions intensity as GGDPincreases.
Deliberate reductions in emissions intensity require expenditures at levels described by the MACcurve. Reducing emissions intensity by △ µ ( t ) between time t and t + △ t costs C ( µ ( t )) g ( t ) |△ µ ( t ) | .Only the deliberate reduction in emissions intensity △ µ ( t ) = − k ( t ) µ ( t ) △ t contributes to mitiga-tion expenditures, so this contribution between time t and t + △ t is α (cid:0) µ ( t ) / (cid:0) µ e − σt (cid:1)(cid:1) − ν g ( t ) k ( t ) µ ( t ) △ t ,with discounted sum across the specified time-period being E µ = αµ ν △ t X t e − δt e − νσt g ( t ) k ( t ) ( µ ( t )) − ν (4)where δ is the rate of time-discounting. Substituting for emissions intensity µ ( t ) = µ e − K ( t ) e − σt E µ = αµ △ t X t e − δt e − σt g ( t ) ˙ K ( t ) e ( ν − K ( t ) (5)6 .4.2 Expansion of mitigation with growth in GGDP The second contribution to expenditures is due to scaling up of mitigation for maintaining lowerlevels of emissions intensity as the global economy expands. Between times t and t + △ t theactivities involved in reducing emissions intensity from µ e − σt , the value it would have in theabsence of deliberate reductions, to µ ( t ) that it actually has must be expanded proportionally toincrease in GGDP during this period. For each unit of GGDP, total cost of reducing emissionsintensity from µ e − σt to µ ( t ) at time t is ˆ µ ( t ) µ e − σt α ( µ/ ( µ e − σt )) ν ( − dµ ) (6)Time t is fixed in the above equation so factor e − νσt can be taken outside the integral. Thiscontribution increases proportionally with change in GGDP, so expenditure between time t and t + △ t , on scaling up mitigation when GGDP increases from g ( t ) to g ( t ) + △ g ( t ), is αµ ν e − νσt ν − µ ( t ) ν − − µ ν − e − ( ν − σt ! △ g ( t ) (7)and substituting for µ ( t ) as before the total discounted expenditure becomes E g = αµ ν − X t e − δt e − σt (cid:16) e ( ν − K ( t ) − (cid:17) △ g ( t ) (8) The total discounted expenditure E µ + E g in continuous-time is E ( t ) = ˆ t e − δs P µ ( s ) ds + ˆ t e − δs P g ( s ) ds (9)where P µ ( t ) = βe − σt g ( t ) ˙ K ( t ) e ( ν − K ( t ) (10)7s annual expenditure from reducing emissions intensity, where β = αµ , and P g ( t ) = βν − e − σt ˙ g ( t ) (cid:16) e ( ν − K ( t ) − (cid:17) (11)is that from expansion of mitigation. Exogenous decarbonization has the effect of decreasing bothcontributions to future expenditure by e − σt . We therefore introduce the parameter ρ = σ + δ ,combining its effect with that of discounting in time.In scenarios with constant growth rate of GGDP and constant mitigation rate k the discountedexpenditure then becomes E ( t ) = βg k ˆ t e (( ν − k + r − ρ ) s ds + βg rν − ˆ t (cid:16) e (( ν − k + r − ρ ) s − e ( r − ρ ) s (cid:17) ds (12)integrating to E ( t ) = βg k e (( ν − k + r − ρ ) t − ν − k + r − ρ + βg rν − e (( ν − k + r − ρ ) t − ν − k + r − ρ − e ( r − ρ ) t − r − ρ ! (13)In case of large mitigation rates and long time-horizons e (( ν − k + r − ρ ) t − ν − k + r − ρ ≫ e ( r − ρ ) t − r − ρ , so that E ( t ) ∼ = βg (cid:18) k + rν − (cid:19) e (( ν − k + r − ρ ) t − ν − k + r − ρ (14)and the ratio of expenditures from expansion and reducing emissions intensity is approximately ν − rk .For short time-horizons, such that (( ν − k + r − ρ ) t ≪
1, the expenditure from reducing emis-sions intensity increases linearly with time as E µ ( t ) ∼ = αm kt , being proportional to the decar-bonization rate. The second contribution from expansion is quadratic in time as E g ( t ) ∼ = αm krt ,and increases with the decarbonization rate and GGDP growth rate. Their ratio is E g ( t ) /E µ ( t ) = rt , and initially rt ≪ Strictly, the model of constant exogenous decarbonization rate σ assumes constant GGDP growth rate, so thisis the implicit assumption throughout the paper. We expand the corresponding exponentials in equation (13) by their Taylor series and consider the leading-orderterms. .5 Burden of mitigation expenditure as fraction of GGDP The global economy of the future is expected to be richer than the present, and therefore bet-ter poised to manage CO mitigation expenditures. However marginal abatement costs increasewith time as cheaper mitigation activities are exhausted and more expensive activities must beundertaken for continued decarbonization. Consider "burden", defined as the ratio of mitigationexpenditure in a year and corresponding GGDP; the numerator is the integrand in equation (9)without the time-discount factor. The burden is b ( t ) = βe − σt k ( t ) e ( ν − K ( t ) + βrν − e − σt (cid:16) e ( ν − K ( t ) − (cid:17) (15)using ˙ g/g = r . The two terms above arise from reducing emissions intensity and expansion respec-tively. Let us consider their respective contributions to ˙ b ( t ) for the case of constant decarbonizationrate k .From reducing emissions intensity this is˙ b ( t ) = βk (( ν − k − σ ) e − σt e ( ν − kt (16)so its sign depends on the sign of ( ν − k − σ . If the MAC rises steeply enough that ν > b ( t ) > σ so that( ν − k > σ . Increasing burden to future generations can result from a sharply rising MACcurve not being compensated adequately by exogenous reductions in emissions intensity. In case0 < ν < From expansion the contribution to ˙ b is simplified to˙ b ( t ) = βrν − n (( ν − k − σ ) e − σt e ( ν − kt − σe − σt o (17)and generally the last term is small so that if (( ν − k − σ ) / ( ν − > < ν < σ/k does this contribution decrease with time, In the approach ν → b ( t ) = βe − σt k ( t ), so this generallydecreases with time unless decarbonization rate is increasing. σ/k ≪ ν > − θ ) rk (18)since increasing MAC would not be compensated by effects of exogenous reductions in emissionsintensity.In the future if e ( ν − K ( t ) ≫ b ( t ) = βe − σt e ( ν − K ( t ) (cid:18) k ( t ) + rν − (cid:19) (19)so for approximately constant values of integrated decarbonization K ( t ) the burden from reducingemissions intensity is proportional to decarbonization rate k ( t ). Here we examine the approximate relation between discounted mitigation expenditures and cumu-lative CO emissions. Such a relation of course depends on the function K ( t ), and this subsectionconsiders only scenarios with constant decarbonization rate so K ( t ) = kt . We also fix the GGDPgrowth rate and autonomous decarbonization rate, so the graph between expenditures and cumu-lative emissions reflects only differences in the decarbonization rate.Cumulative CO emissions between the present at time t = 0 and the time-horizon at t = T is M ( T ) = ´ T m ( t ) dt . With m ( t ) = µ g ( t ) e − K ( t ) e − σt , and in case of constant decarbonization rate k this becomes M ( T ) = m (cid:16) − e − χT (cid:17) /χ , with χ = k + σ − r . The slope of the graph betweendiscounted expenditure E ( T ) and cumulative emissions M ( T ) equals derivative ∂E/∂M , which iswritten as ∂E/∂M = ( ∂E/∂k ) ( ∂k/∂χ ) ( ∂χ/∂M ). The value of ∂M/∂χ = m χ (cid:16) (1 + χT ) e − χT − (cid:17) .We examine separately the very different cases with short and long time-horizons.For short time-horizons the leading order terms are ∂E/∂k ∼ = αm T , and ∂M/∂χ ∼ = − m T . Then,10ince ∂k/∂χ = 1 with constant r and σ , we obtain ∂E/∂M ∼ = − α/T . This slope is constant forfixed time-horizon T . For T = 1 the slope of the graph is − α , which also represents the increasein expenditure (in billion $s) during one year from the present time that is needed for decreasingemissions by 1 Gton year -1 . This much should be obvious from the model of the MAC curve, butthere is also an analogous linear relation for short time-horizons of a few years. For long time-horizons T for which e − χT ≪ M ∼ = m /χ , and eliminating k and substitutinginto equation (14) one obtains the relation between expenditure and cumulative emissions E ∼ = βg e ( r − ρ ) T − r − ρ (cid:16) m M + rνν − − σ (cid:17) , where E is a convex function of M and has increasing slope forsmaller M . For ν > k due to the exponential term in k , so the effect is even larger. Convexity of the graph between E and M arises from two effects in the case of constant decarbonization rates: ∂M/∂χ for the caseof large T behaves as − m /χ , so larger χ in case of more rapid decarbonization has progressivelysmaller effect on limiting cumulative emissions; secondly, increasing decarbonization rate increases E nonlinearly for long time-horizons, due to the exponential term in k . These effects are moregeneral and therefore not limited to scenarios with constant decarbonization rates.In summary, for short time-horizons the graph of mitigation expenditures versus cumulative emis-sions is linear, whereas for longer time-horizons it is convex. CO is long-lived and policies tolimit cumulative emissions should take into account sufficiently long time-horizons. Under theseconditions the marginal cost, measured in terms of additional discounted expenditures needed toachieve more stringent mitigation goals, is increasing.Parameters used in the paper are listed in Table 1. The slope of the graph between expenditure and cumulative emissions becomes smaller as time-horizon T isincreased because sensitivity of expenditure to decarbonization rate grows linearly with T , whereas that of cumu-lative emissions has a quadratic relation with T . The benefits of higher decarbonization rates are sensitive to T ,because decarbonization takes time to influence cumulative emissions. Therefore for time horizons of a few years,decarbonization appears more attractive while considering the longer periods. m ( t ) CO emissions Gton CO year -1 M cumulative CO emissions goal PgC µ ( t ) CO emissions intensity Gton CO (trillion $) -1 g ( t ) Global gross domestic product (GGDP) trillion $ year -1 r annual GGDP growth rate year -1 k ( t ) decarbonization rate year -1 θ income elasticity of CO emissions dimensionless σ exogenous decarbonization rate year -1 α coefficient for MAC curve billion $ / (Gton CO year -1 ) ν exponent in MAC curve dimensionless E µ ( t ), E g ( t ), E ( t ) discounted expenditures until year t billion $ P µ ( t ), P g ( t ), P ( t ) expenditure in year t billion $ year -1 b ( t ) expenditure / GGDP in year t dimensionless K ( t ) integrated decarbonization rate dimensionless δ time-discount rate year -1 ρ σ + δ year -1 χ k + σ − r year -1 β αµ year12 Minimum-expenditure pathways for reducing emissions inten-sity of CO This section examines quasi-stationary pathways of decarbonization that minimize mitigation ex-penditure subject to constraint on cumulative CO emissions. Such pathways need not correspondto constant decarbonization rate. There is a constraint on the integrated decarbonization rate,which is K (0) = 0 at the present time, and we therefore seek the initial value problem in K ( t )whose solution minimizes mitigation expenditure.Cumulative CO emissions is written as M ( T ) = ´ T m (cid:16) t, K, ˙ K (cid:17) dt , where m (cid:16) t, K, ˙ K (cid:17) = µ g ( t ) e − K ( t ) e − σt is emissions. We wish to find K ( t ) that minimizes discounted mitigation expenditure E ( T ) = ´ T f (cid:16) t, K, ˙ K (cid:17) dt , where f (cid:16) t, K, ˙ K (cid:17) = βe − δt e − σt g ( t ) ˙ K ( t ) e ( ν − K ( t ) + βν − e − δt e − σt ˙ g ( t ) (cid:16) e ( ν − K ( t ) − (cid:17) .Consider choosing stationary pathway of integrated decarbonization K ( t ) in order to minimize E ( T ) subject to cumulative emissions constraint ˆ T m (cid:16) t, K, ˙ K (cid:17) dt = M (20)where M is the cumulative emissions goal. For such a pathway, the derivative of functional I (cid:16) K, ˙ K (cid:17) = ´ T f (cid:16) t, K, ˙ K (cid:17) dt + λ n ´ T m (cid:16) t, K, ˙ K (cid:17) dt − M o must be stationary with respect to K ( t ) for arbitrary perturbations δK satisfying the initial condition. This yields the familiar Euler-Lagrange (E-L) equation derived in Appendix 1 ∂f∂K + λ ∂m∂K = ddt (cid:18) ∂f∂ ˙ K + λ ∂m∂ ˙ K (cid:19) (21)simplifying to e νK ( t ) = λ µ ( δ + σ ) β e δt , where λ can be eliminated using the constraint on cumulativeemissions. In addition the solution must satisfy a "natural boundary condition" ∂f∂ ˙ K ( T )+ ∂m∂ ˙ K ( T ) = 0arising from the fact that the value of K ( T ) is not fixed by the specification of our problem(Appendix 1). However such satisfaction is not possible, and we seek a solution that is onlystationary with respect to perturbations that leave intact not only K (0), which follows from initialcondition K (0) = 0, but also K ( T ). Fixing K ( T ) is an artificial constraint on our problem that has Emissions do not depend explicitly on ˙ K , but we include this argument for consistency with the formulation ofthe rest of the functional that we seek to minimize. K ( t ).Although K ( t ) increases in time in the presence of a non-zero discount rate, the absence of a termin ˙ K in the E-L equation precludes imposing initial condition K (0) = 0. The E-L equation isalgebraic in our optimization problem because integrand f + λ m depends linearly on ˙ K , leadingto a degenerate case ( van Brunt (2004)) as shown in Appendix 1.In order to introduce a term in ˙ K in the E-L equation, we seek a second integral constraint involvinga different function h (cid:16) t, K, ˙ K (cid:17) dt that obeys ddt (cid:18) ∂h∂ ˙ K (cid:19) = ˙ Ke − γt (22)with γ >
0, for reasons that will become evident. Then ∂h∂ ˙ K = ´ t ˙ K ( s ) e − γs ds and integrating byparts ∂h∂ ˙ K = e − γt K ( t ) + γ ˆ t K ( s ) e − γs ds (23)using initial condition K (0) = 0. Furthermore, choosing γ large so that the first term can beneglected we obtain h (cid:16) t, K, ˙ K (cid:17) = γ ˙ K ( t ) ˆ t K ( s ) e − γs ds + h ( K ( t ) , t ) (24)We seek only one such function parameterized by γ satisfying equation (22), so make the simplestchoice and set h ( K ( t ) , t ) to zero. Then h (cid:16) t, K, ˙ K (cid:17) = γ ˙ K ( t ) ´ t K ( s ) e − γs ds , and we choose γ large so that ´ T h (cid:16) t, K, ˙ K (cid:17) dt can be made small. We therefore impose a further equality constraint on our original problem ˆ T h (cid:16) t, K, ˙ K (cid:17) dt = ε (25) To see how this is possible, consider the example of constant decarbonization rate where K ( t ) = ζt with ζ > h ( t ) = γζ ´ t se − γs ds . Integrating by parts this becomes − ζ te − γt + ζ γ (cid:0) − e − γt (cid:1) , which can be made smallby choosing γ to be sufficiently large. The terms involved in defining h ( t ) are positive so h ( t ) > h ( t ) < ζ /γ , so ´ T h ( t ) dt < ζ T /γ , which can be made small by choosing γ sufficiently large. ε ≪ λ n ´ T h (cid:16) t, K, ˙ K (cid:17) dt − ε o ,so the modified E-L equation becomes ∂f∂K + λ ∂m∂K + λ ∂h∂K = ddt (cid:18) ∂f∂ ˙ K + λ ∂m∂ ˙ K + λ ∂h∂ ˙ K (cid:19) (26)Here too there is an analogous natural boundary condition, but we cannot satisfy it and only seeka solution that is stationary with respect to the restricted class of perturbations discussed above.Recall that ddt (cid:16) ∂h∂ ˙ K (cid:17) = ˙ Ke − γt . Furthermore, ∂h∂K = γ ˙ K ( t ) ´ t e − γt dt = ˙ K (cid:0) − e − γt (cid:1) . Then the E-Lequation is − λ µ g ( t ) e − σt e − K ( t ) + λ ˙ K ( t ) (cid:16) − e − γt (cid:17) = − ( δ + σ ) βe − ( δ + σ ) t g ( t ) e ( ν − K ( t ) + λ ˙ K ( t ) e − γt (27)We simplify by neglecting e − γt compared to 1 because γt ≫
1, so the evolution equation for K becomes ˙ K ( t ) = λ µ λ e − σt g ( t ) e − K ( t ) − ( δ + σ ) βλ e − ( δ + σ ) t g ( t ) e ( ν − K ( t ) (28)with initial condition K (0) = 0. Multiplying by e K ( t ) on both sides and defining x ( t ) = e K ( t ) theevolution equation for x ( t ) is˙ x ( t ) = λ µ λ e − σt g ( t ) − ( δ + σ ) βλ e − ( δ + σ ) t g ( t ) ( x ( t )) ν (29)with x (0) = 1. The above equation is not of a standard type that can be solved exactly, and weresort to approximate solutions (Appendix 2). Integrated decarbonization rate K ( t ) must increaseto be economically relevant, but with an initial value problem in K ( t ) we face the risk that itactually decreases in the solution to the above equation. In case of either σ > δ > σ > K ( t ) is a decreasing function. Wecannot avoid this by regularizing the E-L equation for a 2-point boundary value problem. Thevalue of K (0) is known but not K ( T ), and the latter is not uniquely determined by the constrainton cumulative emissions. The cumulative emissions constraint of equation (20) discretized in time corresponds to a single equation inseveral unknown values of K at the various time-steps, so this does not yield a unique constraint on K ( T ). σ = 0, involving unit income elasticity of emissions, and in the absence of time-discounting. Forthis special case, x ( t ) = 1 + λ µ λ G ( t ) , where G ( t ) = ´ t g ( s ) ds is integrated GGDP, and hencethe quasi-stationary solution is K ( t ) = ln (cid:18) λ µ λ G ( t ) (cid:19) (30)which is increasing. Lagrange multipliers λ and λ are estimated by substituting equation (30)into integral constraints provided by equations (20) and (25). The E-L equation has reduced ourinfinite-dimensional problem of choosing function K ( t ) to the finite-dimensional one of estimating λ and λ satisfying these constraints.For the above solution, being quasi-stationary only if σ = 0, we have e − K ( t ) = 1 / (cid:16) λ µ λ G ( t ) (cid:17) ;and therefore in this case cumulative emissions at time t is M ( t ) = λ λ ln (cid:16) λ µ λ G ( t ) (cid:17) , or M ( t ) = λ λ K ( t ) in the quasi-stationary solution.Thus the quasi-stationary solution, stationary relative to a restricted class of perturbations in K ,exists for the special case of σ = 0 and δ = 0, and has cumulative emissions graph proportional tothe graph of K ( t ), and the emissions graph m ( t ) proportional that of decarbonization rate k ( t ),with k ( t ) = λ λ m ( t ). Scenarios requiring decreasing emissions, so that cumulative emissions even-tually becomes approximately constant, have K ( t ) increasing at diminishing rate in this solution;decarbonization rate k ( t ) is initially large and decreases with time. We estimate the global MAC curve (in 1990 US dollars) for CO by aggregating estimates presentedin Morris et al. (2008), which in turn are based on the MIT Emissions Prediction and PolicyAnalysis (EPPA) model (
Paltsev et al. (2005)). Figure 1a shows results from
Morris et al. (2008)for the year 2050. Estimation of the model of MAC is illustrated in Figure 1b, and we must know thereference emissions in the business as usual case, corresponding to the effect of µ e − σt . This is taken16rom reference-case emissions in 2050 documented for the EPPA model in Paltsev et al. (2005).Least squares regression yields estimates for α and µ (Table 2). The exponent ν is significantlylarger than one, and this relation affects the properties examined in Section 2. There is largeuncertainty in MACs ( Criqui et al. (1999);
Klepper and Peterson (2004);
Amann et al. (2009)) butit appears that this exponent in our model cannot be much smaller than 2, because that wouldlead to very slow increase of the MAC as decarbonization proceeds. Figure 1c shows the effect ofchanging ν in our model. In case ν is closer to 1 then the MAC even after 50 % reductions comparedto the reference case would be less than 50 $ /tonne in 2005 prices, which is substantially smallerthan studies suggest ( Ellerman and Decaux (1998);
Klepper and Peterson (2004);
Amann et al. (2009)).For income elasticity of CO emissions, we use constant value of θ = 0 .
75 yielding exogenousdecarbonization rate of σ = 1 % per year in a 4 % GGDP growth scenario that is close to therecent historical value in real dollars ( DeLong (1998)). As a result the estimate of σ correspondsto the 2015 value in the DICE model ( Nordhaus and Sztorc (2013)). In DICE the autonomousdecarbonization rate is decreasing at 0 .
1% at each time-step of five years (
Nordhaus and Sztorc (2013)). If were were to consider scenarios with decreasing GGDP growth rate, a similar decreasein exogenous decarbonization rate would occur in our emissions model.For future GGDP growth estimates, considering that our time-horizon is 100 years from the present,most IAMs show a gradual decrease in GGDP growth reflecting demographic transitions duringthe present century (e.g.
Paltsev et al. (2005)). Nevertheless we consider annual growth rates (inreal dollars) ranging between 0 .
012 - 0 . Krakauer (2014)). Generally there is a tendency even amongexperts to underestimate uncertainty (
Morgan and Henrion (1990)), and we included a wide rangeof long-term growth rates in our calculations. Estimates of parameters of the model, includingpresent emissions, GGDP, and emissions intensity, are listed in Table 2.We study cumulative emissions goals during the next 100 years of 300, 600, 900 and 1200 PgC. Global warming is approximately proportional to cumulative CO emissions, with the ratio esti- The same caution applies to long term MACs but we do not consider that effect here. Only the cumulative emissions are specified in PgC, because carbon accounting is usually performed in theseunits. However the economic model is carried out in units of Gton CO . . − . Allen and Stocker (2014)). Assuming a symmetricdistribution of this "transient climate response to cumulative carbon emissions" with mean value of1 .
65 K per 1000 PgC yields mean global warming contribution of CO of approximately 0 .
5, 1 . .
5, and 2 . .
5, 2 .
0, 2 .
5, and 3 . Meinshausen et al. (2009);
Peters (2016)).18able 2: Parameter estimates for model of expenditures. Monetary units refer to 1990 USD.Parameter Value Unit α . ± . year -1 ) ν . ± .
46 dimensionless θ .
75 dimensionless m
36 Gton CO year -1 µ .
46 Gton CO (trillion $) -1 g . -1 β . × − year 19 missions reduction (GtCO ) C o s t ( $ /t on C O ) µ / µ e - σ t M A C , C ( µ ) b illi on $ / ( G t on C O y ea r - ) µ / µ e - σ t M A C , C ( µ ) b illi on $ / ( G t on C O y ea r - ) ν = 0.5 ν = 1.0 ν = 2.0 ν = 2.5 (a)(c) (b) α = 10.4 ± -1 ), ν = 2.4 ± Figure 1: Estimate of MAC (in 1990 USD) and effect of changing exponent ν in our model of MAC:(a) crosses show global MAC versus quantity of emissions reductions from Morris et al. (2008) forthe year 2050, after conversion from 2005 to 1990 dollars; (b) results of panel (a) presented inthe form of the model in equation (3), where µ e − σt is taken to correspond to a 60% increase in2050 CO emissions compared to the present in the reference scenario, roughly based on resultsdocumented for the EPPA integrated assessment model ( Paltsev et al. (2005)). The power law isestimated from a least squares fit using the logarithm of the ordinate, yielding α = 10 . year -1 ) and ν = 2 .
4. In both panels, curves correspond to estimate of equation (3);(c) effect of changing ν in the model, keeping α constant. Values of ν much smaller than 2 lead tovery slow increase of MAC (see text). Figure 2 shows expenditure-minimizing pathways in case global income elasticity of emissions isunity, so there is no exogenous decarbonization. Recall that these are "quasi-stationary", i.e. sta-tionary relative to a restricted type of perturbations in the variable K ( t ) that keep the endpoints asfixed. The quasi-stationary pathway of annual CO emissions is approximately invariant of GGDPgrowth rate. The time-dependence of emissions arises from term e rt − ´ t k ( s ) ds , and larger r is beingcompensated by larger k . More rapid economic growth requires faster decarbonization for limiting20 ear de c a r bon i z a t i on r a t e k ( t ) ( y ea r - ) -2 -1 r=0.012r=0.024r=0.036 year C O e m i ss i on s ( P g C y ea r - ) Figure 2: Quasi-stationary decarbonization rates k ( t ) in case θ = 1 and δ = 0, for cumulativeemissions goals during the next 100 years of 300, 600, 900 and 1200 PgC in the different colors, anddifferent GGDP growth rates r : (a) decarbonization rate k ( t ), on a logarithmic scale; (b) annualCO emissions. Where exogenous decarbonization is absent, i.e. σ = 0, the quasi-stationaryemissions trajectory is approximately invariant of GGDP growth rate. The decarbonization rateis proportional to emissions as k ( t ) = λ λ m ( t ) and scenarios involving emissions reduction havedecarbonization rate decreasing with time. Solid, dashed, and dotted curves indicate GGDP growthrate r of 0 . . .
036 respectively.cumulative emissions to the same levels.As noted in Section 3, the quasi-stationary solution has k ( t ) ∝ m ( t ) so the decarbonization rateis constant only in scenarios involving constant emissions, and increasing in time in scenariosinvolving increasing emissions. For scenarios requiring decreasing emissions, the decarbonizationrate is initially larger and decreases with time. Large annual decarbonization rates of more than10 percent are initially involved in the quasi-stationary solution for limiting cumulative emissionsto 300 PgC or less. Henceforth we will only describe cases with income elasticity of emissions θ = 0 .
75, so exogenousdecarbonization occurs at a rate increasing with GGDP growth rate. Figure 3 shows pathwayssatisfying equation (30) with the cumulative emissions constraint taking into account exogenousdecarbonization at rate σ . Limiting cumulative emissions below 600 PgC requires near-term de-carbonization rates of at least a few percent. Emissions pathways are generally not invariant of21GDP growth rate in the presence of autonomous decarbonization, and larger early emissions inhigh growth scenarios is compensated by reduced emissions later on. When autonomous decar-bonization is present, it is not necessary that decreasing emissions pathways be accompanied byinitially high decarbonization rates that subsequently decrease. However even in this case stringentmitigation scenarios generally involve initially larger decarbonization rates.Despite decreasing decarbonization rates in such scenarios, mitigation expenditures are rising hereas in all mitigation scenarios, as Figure 4 shows. Expenditures increase from a few to tens ofbillion dollars in the present time to a few orders of magnitude more as the MAC rises and GGDPincreases. The carbon price needed to achieve these reductions in CO is equal to the MAC, withformula αe νK ( t ) following equation (3). Figure 4a shows the carbon price (in constant 1990 USdollars) increasing to 100 $ / ton CO in a few decades in the 300 PgC scenario, and much morerapidly in case GGDP growth can expected to be rapid, because decarbonization must occur morequickly. However the average rate of growth of GGDP during the next 100 years will not be knownuntil the end of this period.In our model of mitigation cost, there are two contributions: from reducing emissions intensity,and from subsequent expansion of mitigation as the global economy grows. Recall from Section 2.4that the first contribution initially grows linearly in time, whereas the second grows quadratically.The first contribution is initially much larger, but the second contribution grows in importancewith time (Figure 4d). Its role increases with time because, as the economy grows, decarbonizationefforts have to be scaled up in proportion to the level of decarbonization at that time present theglobal economy, as measured by e ( ν − K ( t ) −
1. The ratio of the two contributions is P g ( t ) /P µ ( t ) = ν − rk ( t ) (cid:16) − e − ( ν − K ( t ) (cid:17) . For large t this approaches ν − rk ( t ) . Since ν ∼ = 2 in our model, thisalways remains smaller than one in scenarios where the decarbonization rate must be larger thanthe GGDP growth rate.One lesson from Figures 2-4 is the importance of assumptions about future global economic growthto estimates of the scale of effort involved in decarbonization. Faster growth requires furtherdecarbonization, entailing not only higher marginal costs, but also these must be scaled up to alarger extent with greater economic expansion. Uncertainty in future economic growth therebypresents substantial uncertainty for estimates of the scale of mitigation required.22 ear de c a r bon i z a t i on r a t e k ( t ) ( y ea r - ) -2 -1 r=0.012r=0.024r=0.036 year C O e m i ss i on s ( P g C y ea r - ) Figure 3: Quasi-stationary decarbonization rates for limiting cumulative emissions during the next100 years to 300, 600, 900 and 1200 PgC in the different colors, where effects on emissions ofexogenous decarbonization arising from income elasticity of emissions θ = 0 .
75 have been included:(a) decarbonization rate k ( t ); (b) annual CO emissions. Solid, dashed, and dotted curves indicateGGDP growth rate r of 0 . . .
036 respectively.
Figure 5 plots the mitigation expenditure as fraction of GGDP ("burden") for the cases shownin the previous figures. This generally increases with time because the condition of equation(18) is met, and the effect of increasing MAC is not mitigated by exogenous decarbonization.For short times the second contribution from expansion is small and the burden approximates to b ( t ) ∼ = βe ( ν − K ( t ) − σt k ( t ), and with β = 4 . × − years in the present model this becomes a verysmall fraction of GGDP even for large decarbonization rates. At the current time there are manylow-cost options for decarbonization and the mitigation burden therefore can be very small.The burden after a long time is b ( t ) ∼ = βe ( ν − K ( t ) − σt (cid:16) k ( t ) + rν − (cid:17) . The exponent ( ν − K ( t ) − σt increases substantially as mitigation proceeds (Figure 5a) and the burden approaches 1 % of GGDPor more in scenarios with stringent mitigation goals, especially in the presence of rapid GGDPgrowth (Figure 5b). The latter is generally consistent with results of Rogelj et al. (2015) showingthe large influence of baseline energy demand on discounted mitigation costs.One effect is that delaying decarbonization can impose much larger burdens on future generationsif they seek to meet the same stringent cumulative emissions goal. Figure 6 considers emissionsscenarios leading to cumulative emissions of 300 PgC. We compare, in the presence of exogenous23 ear c a r bon p r i c e ( $ /t on C O ) r=0.012r=0.024r=0.036 year annua l e x pend i t u r e P ( t ) ( b illi on $ y ea r - ) year P µ ( t ) ( b illi on $ y ea r - ) year r a t i o P g ( t ) / P µ ( t ) -2 -1 (a) (b) (d)(c) Figure 4: Carbon price and CO mitigation expenditures for the scenarios in Figure 3. Colorsindicate cumulative emissions goals of 300, 600, 900 and 1200 PgC during the next 100 years. Solid,dashed, and dotted curves indicate GGDP growth rate r of 0 . . .
036 respectively.Prices and expenditures are in constant 1990 USD: (a) carbon price, which is equal to the marginalabatement cost; (b) total expenditure in each year P ( t ); (c) expenditure from reducing emissionsintensity P µ ( t ); (d) ratio P g ( t ) /P µ ( t ) of expenditures from expansion and reducing emissionsintensity. Keeping cumulative emissions below 600 PgC typically requires carbon price of 50 USD (in1990 dollars) or higher within the next two decades. Carbon price and expenditures are much higherin case of higher GGDP growth. Mitigation expenditures are increasing even as decarbonizationrate is decreasing, as the MAC rises. At the present time expenditures are dominated by those onreducing emissions intensity, but the contribution from expansion plays an increasing role.24 ear ( ν - ) K ( t )- σ t -2 -1 r=0.012r=0.024r=0.036 year m i t i ga t i on bu r den b ( t ) -4 -3 -2 (a) (b) Figure 5: Mitigation expenditure as a fraction of GGDP ("burden") in the scenarios of Figure3. Colors indicate cumulative emissions goals of 300, 600, 900 and 1200 PgC for the next 100years. Solid, dashed, and dotted curves indicate GGDP growth rate r of 0 . . . ν − K ( t ) − σt appearing in the equation; (b) mitigation burden.decarbonization, the aforementioned quasi-stationary scenario with one having constant decar-bonization rate. The former serves as archetype for rapid and early mitigation. A small fraction( ≤ . e ( ν − K ( t ) − σt ismuch larger. The larger future decarbonization rates occurring in the scenario with late mitigationare therefore amplified by a much greater amount than present savings. Let us consider the overall costs of mitigation, by evaluating f = ´ T e − δs P ( s ) ds/ ´ T e − δs g ( s ) ds ,the ratio between discounted expenditures and GGDP for the quasi-stationary solution. Figure7 shows that this is a convex function of cumulative emissions goal, as Section 2.6 argued. Thescale is logarithmic, and mitigation cost increases rapidly for stringent cumulative emissions goals as25 ear C O e m i ss i on s ( P g C y ea r - ) year m i t i ga t i on bu r den b ( t ) -4 -3 -2 -1 r=0.012r=0.024r=0.036(a) (b) Figure 6: Effect of delay in decarbonization on future burden of meeting the same cumulativeemissions target of 300 PgC: (a) quasi-stationary emissions pathway with early mitigation and apathway with constant decarbonization rate; (b) mitigation burden. Solid, dashed, and dottedcurves indicate GGDP growth rate of 0 . . .
036 respectively. The mitigation burdenincreases with time, and increases more rapidly in scenarios where mitigation is postponed. A smallfraction ( ≤ . Rogelj et al. (2015). The graph shows anapproximately linear relationship on a logarithmic scale, so the relation between costs f and thecumulative emission goal M roughly obeys power law f ( M ) = f ( M /M ) − n , with M = 1000PgC being a reference goal and leading to cost f .Figure 8a shows that higher economic growth increases the reference cost f substantially, andthis is more sensitive to assumptions about long-term economic growth than to the exponent in theMAC curve. The power n describes sensitivity △ f/f −△ M /M of relative changes in f to relative changesin the cumulative emissions goal. Figure 8b shows higher sensitivity to the cumulative emissionsgoal in case the MAC curve rises sharply and GGDP rises slowly. Rapid economic growth wouldmake meeting cumulative emissions goals more expensive regardless of how stringent they might26 umulative CO emissions (PgC)
300 400 500 600 700 800 900 1000 r a t i o f -4 -3 -2 -1 r=0.012; δ =0r=0.012; δ =0.03r=0.024; δ =0r=0.024; δ =0.03r=0.036; δ =0r=0.036; δ =0.03 Figure 7: Ratio f between discounted expenditures on mitigation and discounted GGDP, graphedversus cumulative CO emissions. Each curve shows results for a different combination of GGDPgrowth rate and discount rate. For medium economic growth scenarios the expenditures are limitedto 1% of discounted GGDP, but can be a few percent in high growth scenarios. The ratio ofexpenditures and GGDP increases with time (Figures 5 and 6), and therefore f is smaller at higherdiscount rates. The relationship is roughly linear on a logarithmic scale, indicating a power law.be, whereas sharply rising MAC makes cost more sensitive to cumulative emissions. For medianvalues the power in the above model n ∼ = 2 .
2, so halving the future cumulative emissions goal (forexample from a 2 C scenario to a 1 . . We have estimated the scale of expenditures involved in reducing cumulative CO emissions byreducing emissions intensity of GGDP ("decarbonization"). Our model assumes constant globalincome elasticity of CO emissions, which leads to exogenous decarbonization, independent of mit-igation policy, occurring in the business as usual scenario. This occurs at rate increasing with theGGDP growth rate; and in case of constant GGDP growth rate, the effects are analogous to dis-counting in time. We also neglect decreases in marginal abatement costs with time, but expect thatreal costs would have to decrease by a large degree to undermine our main conclusions. Mitigationexpenditure has two contributions, arising from reducing emissions intensity and expanding effortas the global economy grows. The first contribution is initially dominant for the scenarios we have27 . . . . . . . . . n MAC exponent ν GG D P g r o w t h r a t e r - . - . - . - . - - - . - . - . - . log f MAC exponent ν GG D P g r o w t h r a t e r Figure 8: Parameters of power law between cumulative emissions goal and mitigation cost, asa function of GGDP growth rate and exponent of MAC curve. Mitigation cost is measured by f , the discounted expenditure as fraction of discounted GGDP, and we fit the model: f ( M ) = f ( M /M ) − n , with f being the cost of limiting M to a reference value of M = 1000 PgC: (a)logarithm of reference cost f , which is larger with higher growth rate and the MAC exponent, butis more sensitive to the former; (b) exponent n is higher with larger MAC exponent and smallereconomic growth. Mitigation cost is more sensitive to cumulative emissions goal if MAC risessharply and economic growth is slow. Mitigation cost can vary more than an order of magnitudedepending on assumptions about the MAC and economic growth.examined but the second contribution plays a growing role.The MAC is estimated to rise steeply enough that the CO mitigation burden, defined as expen-diture as a fraction of GGDP, is expected to increase following equation (18). Future generationswould have to expend larger fractions of GGDP on decarbonization, even in scenarios where thedecarbonization rate is decreasing, as cheaper mitigation options become exhausted. Rapid globaleconomic growth increases the mitigation burden on future generations, and despite being wealthierin scenarios with higher-growth they would have to spend larger fractions of GGDP as it becomesnecessary to ascend the MAC curve more rapidly.Discounted mitigation expenditures are convex functions of cumulative CO emissions. Over longtime-horizons, larger rates of decarbonization have diminishing benefits for reducing cumulativeCO emissions, whereas mitigation expenditures increase more rapidly. Global warming is approxi-mately proportional to cumulative emissions ( Matthews et al. (2009);
MacDougall and Friedlingstein (2015);
MacDougall (2016);
Tokarska et al. (2016)), and the cost of emissions reduction is a convexfunction of the level of global warming and expenditures increase more steeply if the policy goalinvolves a smaller degree of global warming. Nonconvexities could still arise in climate change28conomics due to non-convex damages from, for example, threshold effects (
Fisher and Hanemann (1993);
Lempert et al. (1996);
Keller et al. (2004)).We measure overall costs by the discounted mitigation expenditures as a fraction of discountedGGDP, and there is an approximate power law relation between costs and cumulative emissionsbelow 1000 PgC, with exponent substantially larger than one. This exponent depends mainlyon the exponent in the MAC. Implications are best understood through an example. Considertwo alternate mitigation trajectories where future cumulative emissions differ by a factor of two,for example with global warming of 2.0 K versus 1.5 K where contributions from other forcersare assumed to remain unchanged from present-day values. With the exponent in the powerlaw being generally larger than 2, the more stringent trajectory would involve CO mitigationcosts that are atleast 4 times larger. However such an account neglects the benefits of earlyinvestments in low-carbon technologies and knowledge spillovers to other sectors ( Aghion et al. (2014);
Dechezleprêtre et al. (2014)).The optimization problem examined here seeks the pathway for global decarbonization that min-imizes discounted mitigation expenditures, while meeting an exogenous constraint on cumulativeCO emissions. Solving it required us to "regularize" the variational problem because the origi-nal problem led to an algebraic Euler-Lagrange equation whose solution did not satisfy the initialcondition on the integral of decarbonization rate. Regularization yields a soluble Euler-Lagrangeequation in the form of an initial value problem in the integrated decarbonization rate. However thesolution carries economic meaning only for the special case of absence of exogenous decarboniza-tion and time-discounting. The general problem of minimizing discounted mitigation expendituresin the presence of exogenous decarbonization is important, but beyond our present scope. Fur-thermore the solution is only "quasi-stationary", with respect to a restricted class of perturbationsleaving intact the integrated decarbonization rate at the end of the time-horizon as well.For the soluble case noted above, the optimal solution has decarbonization rate proportional to emis-sions. An attraction of this solution is that it does not depend on the parameters of the MAC model.However, policy cannot be chosen without making long-term economic growth forecasts, illustratinganother difficulty of choosing climate policy under uncertainty ( Roughgarden and Schneider (1999);
Pindyck (2013)). The difficulty of forecasting economic growth makes policy-choice in a cumulative29O framework uncertain. For example the levels of price or quantity instruments are likely todetermine the position on the MAC reached, but their efficacy cannot be determined except ex-postonce average growth rate of GGDP during the long time-horizon of interest becomes known.The discounted mitigation expenditure is not only a function of cumulative emissions, but dependsalso on the decarbonization pathway. Expenditures are higher in scenarios requiring larger decar-bonization rates in the future during periods of much higher MAC. We illustrate using the quasi-stationary solution for limiting cumulative emissions to 300 PgC over the next hundred years. Although decarbonization starts rapidly at an initially higher rate, the cost of mitigation amountsto a small burden at the present time ( < .
1% of GGDP), but rising to 1% or more at the end ofthe 100-year period. For trajectories meeting this cumulative emissions constraint, those with lowermitigation burdens in the present involve much higher burdens in the future, and small savings inthe present translate to much larger future costs. These results are consistent with the work of
Vogt-Schilb and Hallegatte (2014) who suggest that near-term policy should take into considerationlonger-term targets as well, otherwise costs of meeting the latter would be higher.Cumulative carbon accounting appears to make higher mitigation burdens to future generations in-evitable, but present choices can mitigate some of this. While future expenditures can be discountedat a goods-discounting rate based on the opportunity cost of capital (
Nordhaus (1993b)), discount-ing of future burdens can only be based on a positive value of the pure rate of time-preference thatreflects lower weights being ceded to welfare of future generations.
Acknowledgments
This work has been supported by Divecha Centre for Climate Change, Indian Institute of Science.Thanks to several colleagues for helpful suggestions. Although not being quasi-stationary in the presence of exogenous decarbonization, it provides an archetype of apathway involving early mitigation. ppendix 1: Derivation of quasi-stationary solution and degeneratecase For stationarity of I (cid:16) K, ˙ K (cid:17) = ´ T f (cid:16) t, K, ˙ K (cid:17) dt + λ n ´ T m (cid:16) t, K, ˙ K (cid:17) dt − M o , we require thefirst-variation δI in the integral due to small changes δK and δ ˙ K to vanish. The variation δI = I (cid:16) K + δK, ˙ K + δ ˙ K (cid:17) − I (cid:16) K, ˙ K (cid:17) is δI = ˆ T (cid:26) ∂f∂K δK + ∂f∂ ˙ K δ ˙ K + λ (cid:18) ∂m∂K δK + ∂m∂ ˙ K δ ˙ K (cid:19)(cid:27) dt = 0 (31)and, integrating by parts ´ T ∂f∂ ˙ K δ ˙ Kdt = ∂f∂ ˙ K ( T ) δK ( T ) − ´ T ddt (cid:16) ∂f∂ ˙ K (cid:17) δKdt , using δK (0) = 0 andthere is a corresponding equation involving m (cid:16) t, K, ˙ K (cid:17) . This yields δI = (cid:26) ∂f∂ ˙ K ( T ) + ∂m∂ ˙ K ( T ) (cid:27) δK ( T ) + ˆ T (cid:26) ∂f∂K − ddt (cid:18) ∂f∂ ˙ K (cid:19) + λ (cid:18) ∂m∂K − ddt (cid:18) ∂m∂ ˙ K (cid:19)(cid:19)(cid:27) δKdt = 0(32)for arbitrary changes δK . A subset of arbitrary changes δK involves those for which δK ( T ) = 0and for this ∂f∂K − ddt (cid:16) ∂f∂ ˙ K (cid:17) + λ (cid:16) ∂m∂K − ddt (cid:16) ∂m∂ ˙ K (cid:17)(cid:17) must vanish. This yields the Euler-Lagrange (E-L)equation (21), which must also be satisfied when one of the endpoints, in this case at t = T , isnot fixed by the specification of the problem. In addition the problem must satisfy the "naturalboundary condition" ∂f∂ ˙ K ( T ) + ∂m∂ ˙ K ( T ) = 0 in order to be stationary for arbitrary changes δK .Readers may refer to van Brunt (2004) for general discussion.However, is not possible to find a solution to our problem that satisfies the aforementioned naturalboundary condition. Therefore we consider only changes involving δK ( T ) = 0, so that for thesechanges equation (32) is equivalent to the E-L equation. The meaning of this is that, once theE-L equation with initial condition K (0) = 0 is solved for K ( t ), this solution is only stationarywith respect to changes that preserve both K (0) and K ( T ). In this sense the solution is "quasi-stationary", i.e. only relative to a restricted set of changes δK .Degeneracy arises in the following manner. In our problem, f (cid:16) t, K, ˙ K (cid:17) has form ˙ Kf ( t, K ) + f ( t, K ) so that ddt (cid:16) ∂f∂ ˙ K (cid:17) = ˙ K ∂f ∂K + ∂f ∂t and ∂f∂K = ˙ K ∂f ∂K + ∂f ∂K , so that ∂f∂K − ddt (cid:16) ∂f∂ ˙ K (cid:17) = ∂f ∂K − ∂f ∂t ,without any terms involving ˙ K . The term ∂m∂K − ddt (cid:16) ∂m∂ ˙ K (cid:17) does not produce any terms involving˙ K since m = m ( K, t ), and we are left with an algebraic E-L equation. Such degenerate cases31rise when dependence on ˙ K is linear ( van Brunt (2004)). To generate an initial value problemin K , regularization is necessary. Therefore we introduce another contribution to the functional,involving h (cid:16) t, K, ˙ K (cid:17) , in Section 3. Appendix 2: Solution to Euler-Lagrange equation for σ > Consider equation (29) for δ = 0 but σ > x ( t ) = λ µ λ n ( t ) − σβλ n ( t ) ( x ( t )) ν (33)where n ( t ) = e − σt g ( t ), and expanding x ( t ) ∼ = x ( t ) + σx ( t ) in small parameter σ ≪ x ( t ) + σ ˙ x ( t ) ∼ = λ µ λ n ( t ) − σβλ n ( t ) ( x ( t )) ν (cid:18) σ x ( t ) x ( t ) (cid:19) ν (34)with terms constant in σ equating to ˙ x ( t ) = λ µ λ n ( t ) with x (0) = 1 and those of first-degree in σ yielding ˙ x ( t ) = − βλ n ( t ) ( x ( t )) ν (35)with x (0) = 0. The 0 th - degree equation is integrated for x ( t ) = 1 + λ µ λ N ( t ), where N ( t ) = ´ t n ( s ) ds .Then x ( t ) = − ´ t βλ n ( s ) (cid:16) λ µ λ N ( s ) (cid:17) ν ds , which simplifies to x ( t ) = − β ( ν +1) λ µ (cid:26)(cid:16) λ µ λ N ( t ) (cid:17) ν +1 − (cid:27) .With λ µ λ N ( t ) ≪ x ( t ) = − βλ N ( t ) and using β = αµ x ( t ) ∼ = 1 + µ λ ( λ − σα ) N ( t ) (36)As it turns out, λ is small compared to σα so that λ − σα < x ( t ) is decreasingin time for σ > σ > δ > x ( t ) and hence K ( t ) is decreasingin time. Therefore we solve for the quasi-stationary pathway following equation (30), but with32 present in the emissions model and affecting how the cumulative emissions goal is met. Oncethis pathway is estimated, it is applied to our model of mitigation expenditures having in generalnonzero σ and δ . References
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