Should low-carbon capital investment be allocated earlier to achieve carbon emission reduction?

Journal Pre-proofs Should low-carbon capital investment be allocated earlier to achieve carbon emission reduction? Jianxin Guo, Ying Fan PII: DOI: Reference:

S0048-9697(19)34940-X https://doi.org/10.1016/j.scitotenv.2019.134948 STOTEN 134948

To appear in:

Science of the Total Environment

Received Date: Accepted Date:

31 August 2019 10 October 2019

Please cite this article as: J. Guo, Y. Fan, Should low-carbon capital investment be allocated earlier to achieve carbon emission reduction?, Science of the Total Environment (2019), doi: https://doi.org/10.1016/j.scitotenv.2019.134948

This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

© 2019 Published by Elsevier B.V.

1

2

3

Should low-carbon capital investment be allocated earlier to achieve carbon emission reduction?

Abstract

22

Abatement effort can bring about different consequences towards emissions, which is mainly due to the mechanism within the abatement investments. In this paper, the study aims to study in more depth how differentiated evanescent and inertia abatement investments should be allocated to achieve carbon reduction, and also to show that the abatement path and costs differ sensibly between these two allocations. Inertia performance is the critical determinant to this distinction, however which can be influenced deeply by the overestimation of the economic growth. The work confirms the results in theory that the abatement trends differentiated between the evanescent and the inertia case, in which the former is more likely to be an early-move action and however the latter tends to suffer from the converted burden with the underassessment of the growth. Turning to the numerical method, the work gauges the magnitude of the differentiated abatement efforts as well as the burden shifting between them. These qualitative results also confirm our ratiocination, which further help to analyze some intricate disturbances beyond the power of theory. Also we can find that the total cost is increasing due to the introduction of the operating cost, and the accumulated capital of the inertia abatement is more smooth and weighs less. The peak of the accumulated capital is dropping by 38%. Moreover, the evanescent abatement effort is more attractive than the basic case, which is risen universally by 22%. The study derives some policy implications for the ranking to the implementation of the abatement efforts and for incentive instruments to be set up at the industrial level.

23

Keywords: abatement strategy; abatement capital; inertia effect; emission path.

24

1. Introduction

4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21

25 26 27 28 29 30 31 32 33 34

It has been acknowledged widely that the greenhouse gas (GHG) emission, especially carbon emissions caused by human use of fossil energy, is the vital reason that can account for the increase of temperature, which is claimed to be reduced primely. Climate scientists believe that if global temperatures risen by more than two degrees Celsius compared with the temperature in industrial revolution, the consequences could be catastrophic, for instance, increasing the likelihood of flooding, reducing crop yields and nutrient content in the tropics, and exposing more people to extreme heat waves. In order to achieve the goal of control temperature under two degrees Celsius, the parties in the Paris agreement have pledged to circumvent increasing greenhouse gas emissions as soon as possible. A key issue in carbon emission control is the adoption of more efficient low-carbon abatement capital. In turn, the Preprint submitted to Elsevier

November 23, 2019

35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64

incentive to invest in capital equipment embodying such technologies can be significantly influenced by the specific form of environmental regulation. In contrast to productive capital, the stock of abatement capital will either make it possible to keep emissions within the specified limits or will reduce the total amount paid for emissions. From this angle, abatement capital can be regarded as a defensive expenditure. In fact, this factors differentiates from productive capital by differences in installation, training costs, etc [1], which claims decision-makers to be face with different adjustment costs and abatement avenues when investing in abatement capital1 . Hence, this perspective could restore to understand the optimal decision of the carbon abatement path under the benefit-cost criterion. In fact, it has been studied widely how optimal abatement investment path should be designed to achieve the abatement target. One important source of influencing the abatement progress is the knowledge-based accumulation [2], through which the costs declined as the policy-maker gains experience utilizing a certain abatement effort 2 Thus, how to allocate these endowments in different period is relevant. The policy-maker faces a decision dilemma about the evaluation of the opportunity “cost” or the opportunity “benefit” of postponing investment given a decision environment characterized by uncertainties and ignorance on future climate policy, path-dependencies, and large up-front sunk costs3 . A conventional point is that investment decision often acts as an early-move initiative, which could lay the foundation for stronger and potentially more secure profit growth. In the work of Goulder [5], they used analytical models in which technological change results from profit-maximizing investments in research and development (R&D). They showed the significance of induced technological change for the attractiveness of CO2 abatement policies. Bramoulle [6] examined how learning by doing affects the allocation of abatement between heterogeneous technologies over time. The optimal policy balances current abatement costs against reductions in future costs and infant technologies may be preferred to mature technologies despite greater initial costs. Moreaux et al. [7] studied optimal CCS from point sources, taking into account damage incurred from the accumulation of carbon in the atmosphere and the exhaustibility of fossil fuel reserves. Guo et al. [8] described a novel dynamic abatement cost curve and investigate their interactions in improving the performance of the revised model. Fan et al. [9] used a learning curve model and a cost optimization model to explore the 1

Under environmental control conditions, production capital and emission reduction capital have a large positive correlation. It can be considered that the abatement capital is attached to the production assets, which is an upgrade to the existing production capital. For example, energy-saving technology assets are technical upgrades to production lines, and reduction activities are achieved by reducing energy consumption per unit of product. 2 This learning process takes place in the production stage after the product has been designed according to [2]. As the abatement capital is a special one, the cost variation can also be described in this way such as a common learning form [3] [4]. 3 Path dependence means that once a society chooses a system, due to factors such as economies of scale, learning effects, and adaptive expectations, the system will continue to strengthen itself in a given direction. It is worth noting that a large part of the difficulty in transforming the energy system at the national level comes from the path dependence characteristics. In the process of structural adjustment of the economic system, if macro regulation is required, it will inevitably generate large sunk costs for existing economies of scale investment.

2

65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107

total cost of CCS commercialization and the national and provincial CCS retrofit potential of coal-fired power plants in China. In general, these works describe such a fact that due to the early-move abatement action the increased accumulation today lowers costs at all future dates, which ensure the marginal cost and secure profit growth. This also means that a spot of upset abatement plan can distort the accumulation in a reinforced compound effect to bring or delay the abatement which should be. Another important source of influencing the abatement planning is the available abatement measures, which might be differentiated from abatement capital in installation, training costs, abating potential, and the time it takes to implement them and so forth. A representative abatement measure disparity is discussed here, which is known as the inertia effect. The important role of inertia in climate policy has already been acknowledged by Grubb [10]. He maintained the fact that socioeconomic systems were characterized by considerable inertia. Both of the macroeconomic level and the microeconomic level, a certain system has to pay cost to complete the change. This effect is more obvious at the systemic level owing long lived period such as energy systems. Investments in mines, power stations, oilfields and pipelines are often expected to earn returns over decades; buildings and transport infrastructure last even longer. The faster the change the greater the costs tend to be. In the work [11], they examined optimal CO2 policies, given long-term constraints on atmospheric concentrations. They found that the integrated assessment models so far applied under-represent inertia, and showed that higher adjustment costs made it optimal to spread the effort across generations and increased the costs of deferring abatement. In the work of Hourcade [12], they also attached a grate importance upon the function of the inertia effect. They confirmed that daily decisions in transport infrastructure, urban planning or land use could indeed trigger bifurcations towards GHGs intensive development paths, embedding high inertia and consequently likely to imply very high transition costs in the case of a further need for curbing GHG emissions. In the work of [13], Marechal summarized the contributions of evolutionary economics to the issue of climate change by pointing to both its departure from the perfect rationality hypothesis and its shift of focus towards a better understanding of economic dynamics. Karen et al. [14] treated the inertia of carbon emissions due to such mutually reinforcing physical, economic, and social constraints as carbon lock-in. They maintained that such a carbon lock-in was a special case of path dependency, which was common in the evolution of complex systems. In general, to a certain degree these works showed that early attention to the carbon content of new and replacement investments reduces the exposure of both the environmental and the economic systems to the risks of costly and unpleasant surprises. Based on the above review, we can come up with a very meaningful question, that is, how the differentiated abatement effort can influence the abatement reduction. To attain this aim, this study names two abatement efforts including the evanescent abatement which can generate the abatement effort instantly in the decision period, and the inertia abatement which will give rise to the abatement accumulation contributed to the lagging effort. The present investigation differs from the mentioned studies in two ways. One side, we derive analytical results to reveal the intrinsic reasons of allocation between the differentiated abatement efforts on optimal abatement paths, and we also give the answer to the question 3

126

of how the differentiated abatement should be allocated and what amount they are. Another side, we consider two channels for the process of abatement accumulation in a unified framework, and analyze what is the difference when allocating the abatement investment in these two ways, what is the interplay performing when incorporating these two effects into the abatement allocation problem. Also, we consider some key factors which may bring a great influence towards the final results, which employ both analytical and numerical methods in an integrated complementary way to verify each other. This work helps to understand how different abatement means perform themselves in the long-period decision making process. We find that it is not always effective to introduce the abatement capital in the early period. The timing and amount of adopting the differentiated abatement depend on some key factors, which may bring about the change of intrinsic mechanism in the model results in the interplay between the abatement amount and the emission path. Combined with the differentiated abatement avenues, these external or internal parameters can lead to the final results performing in a counterintuitive way. This paper is organized as follows. In Section 2, we describe the constitutions of the model to be used and derive some analytical results. In Section 3, we illustrate and analyzes the results of the model, and moreover some special cases of the model are discussed as well. In Section 4, we present one frame of the extensive version of the model. In Section 5, we summarize our conclusions and render some policy implication.

127

2. Model

108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125

128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145

2.1. Abatement capital and abatement cost Optimal abatement trajectories are frequently calculated assuming that a social planner has to choose when to abate GHG emissions. The planner does so by spending on abatement expenditures that reduce emissions at one given point in time. Considering the abatement avenues are different from each other, it is necessary to distinguish the function of the abatement effort in detail. Generally, a given amount of abatement requires a nonnegative amount of time for its implementation [15]. Some abatement effort can be referred as “evanescent abatement”, which is assumed that the instant abatement efforts have an immediate effect. Such a case is not unusual, for instance, energy-saving bulb substitution, automobile exhaust tail processing equipment, CCS (carbon capture and storage) technique and so forth. Under the inter-temporal decision making problem, the direct consequence of using this measure is that the current emissions can be significantly reduced. Contrary to that, the abatement efforts will lag far behind though the related cost is incurred with the formulation of decisions. Moreover, once such a capital is founded, permanent or for a long time in the near future, decision-makers can benefit from this decision. For instance, replacing coal-fired plants with gas-fired plants, green building in the residential sector, etc. In fact, this inertia abatement has been widely used in the related work. Form the dynamic aspect, the inertia might offset some of the gains from the abatement benefit, if any. This study divides such an effort into these two categories, i.e. evanescent abatement effort and inertia abatement effort. Moreover, this study takes interest what is the difference between the allocation if carrying out them both. The expression of abatement cost function 4

can be easily derived from the so-called marginal abatement cost (MAC) especially in some applications, such as [16], [17] and [18]. In more economic theory frameworks, the abatement cost function can be directly expressed as c(·) = c(et , Θt ), which includes the abatement action et and potential parameters set Θt . Usually, Θt includes the parameters that can bring about great influence upon the change of the abatement function, for instance [19]. The explicit representation including logarithmic form power form [20], exponential form [21], and so forth. This expression to some degree can specify the abatement action in a micro-aspect related problem such as [22, 23]. This study uses the power form as that in [22, 23], namely: c(et ) = ct · eςt , b(et ) = bt · ktς , which denote the evanescent abatement effort and the inertia abatement effort, respectively. Here ς > 1; ct denotes the unit evanescent abatement effort; bt denotes the unit inertia abatement effort. The inertia abatement effort kt will bring about the abatement accumulation to form an abatement capital At , which evolves as following: A˙ t = −δAt + kt . 146

147 148 149 150 151 152 153 154 155 156 157 158

159 160 161 162 163 164 165

Here, the parameter δ denotes the rate of depreciation coefficient. 2.2. Dependent emission path and concentrated target The cumulative emission Et in many works such as [5] are exogenous irrespective of the abatement effort et . In this framework, the amount of emission is decided dependently at the period before, and thus the corresponding cost at one period depend on actions undertaken before. Consequently, the emission magnitude will exhibit path dependency. In detail, this study considers that the actual emission depends on the emissions in the previous period and the emission abatement proportion and natural emission growth rate gt in the current period, which is similar to that used in [20]. Is is assumed that if there are no exogenous emission reduction policies, the emissions from a country will keep rising. Natural growth rates of emissions derive from many factors, such as a countrys phase of economic development, energy resource endowments, and industrial structure. In this paper, the dependent emission path is expressed as: E˙ t = gt Et − (1 + gt )(At + et ), (2.1) in which gt is the natural emission growth rate with the same meaning as that in [20]. From (2.1) the abatement effort is composed of two parts, the evanescent abatement effort et and the inertia abatement effort At . A social planner can set a benevolent goal to control GHG emissions. Usually, this can be done by constraining cumulative emissions. According to the evolution of the equation (2.1), due to the dependent characteristic the policy maker can constrain the terminal state of Et to achieve the concentrated target, i.e. Ω(ET ) ≤ 0, in which Ω(·) is a function of ET .

5

166 167 168 169

2.3. Problem formulation Based on the equations above, the final goal of the planner is to choose the time paths of evanescent abatement effort and the inertia abatement effort to achieve the concentration target, which can be solved by the following model: Z T M in J(et ) = M in ρt [b(kt )ς + c(et )ς ]dt et ,kt

et ,kt

0

s.t. E˙ t = gEt − (1 + g)(At + et ); A˙ t = −δAt + kt ; Ω(ET ) ≤ 0. 170 171 172 173 174 175 176 177 178 179 180 181 182

183 184 185 186 187 188 189

This study formulates the problem in the form of a continuous form for the convenience of analyzing the theory property. In the result section, the discretized form will be used to implement the simulation result. It is conceivable that the evanescent abatement effort can generate a instant result contributed to the the accumulation process Et , nevertheless the inertia abatement effort may exhibit a lagging result if conducting the same effort. In this study, the cost function does not include the inertia abatement capital operating cost. In fact, the introduction of such a cost will complicate our problem to a great extent. This situation will be discussed in the model extension section. Besides, policy makers will use the terminal emission intensity constraint ET /T IPT ≤ E0 /T IP0 to stand for the general form Ω(Et ) ≤ 0 to make a further study. This type of constraint is usually adopted by governments and other public agencies at some point in time. Here, T IP is the total industry profit about the considered level, from regional or national level. Denoting Γ = T IPT · E0 /T IP0 , the terminal constraint becomes ET ≤ Γ. 2.4. Model property In this section, the study intends to analyze some properties of the model, through which the study can understand how the abatement efforts are allocated between the evanescent abatement and inertia abatement. Without loss of generality, the study assumes that the parameter ς = 2, which corresponds to many researches [20][23]. In order to solve the optimization problem (2.2), the optimal control theory [24] is applied and the current value Hamiltonian function is formed as:

H 190 191

192 193

(2.2)

= ρt [b(kt )2 + c(et )2 ] + λt (gEt − (1 + g)(At + et )) + µt (−δAt + kt ).

Here, λ is the shadow price associated with the state variable E(t), and satisfies the adjoint equation: ∂ −λ˙ t = =λg. (2.3) ∂Et

H

µ is the shadow price associated with the inertia capital state variable A(t), and satisfies the adjoint equation: −µ˙ t =

H

∂ = −(1 + g)λ − µδ, µT = 0. ∂At 6

(2.4)

194

Combing the equations (2.3) and (2.4), we can obtain that: λt =se−gt , s > 0; µ˙ t = (1 + g)se−gt + µδ.

195

Solving the differential equation above, we can obtain that: µt = reδt −

196 197

198

(1 + g)s −gt e . (δ + g)

Here, s and r are two parameters to be confirmed. Using the bound condition µT = 0, we obtain that: (δ + g)e(δ+g)T s= r. (2.5) (1 + g) Define β = e(δ+g)T , we obtain that µt = reδt − βre−gt .

199

Besides, the necessary condition of the problem about the control variable is:

H

(2.6)

H

(2.7)

∂ λt (1 + g) se−gt (1 + g) se(τ −g)t (1 + g) = 2ρt cet − λt (1 + g) = 0 ⇔ et = = = . ∂et 2ρt c 2ρt c 2c ∂ µ reδt − βre−gt re(τ +δ)t − βre(τ −g)t = 2ρt bkt + µ = 0 ⇔ kt = − t = − = − . ∂kt 2ρ b 2ρt b 2b 200 201

202 203

Based upon the induction above, the necessary condition of the optimal solution to (2.2) can be easily obtained as follows: Proposition 2.1 Considering the problem (2.2), the necessary condition of the decision variables satisfy that: et =

204 205 206 207

208 209 210 211

se(τ −g)t (1 + g) re(τ +δ)t − βre(τ −g)t ; kt = − . 2c 2b

Here, the parameter s and r are to be determined, which satisfy the equation (2.5). Moreover, the evanescent abatement is increasing if and only if τ −g > 0; however the inertia abatement must has one inflection point, before which the path is increasing and after which the path is decreasing into zero. We find that the evanescent abatement and the inertia abatement strategy are different from each other. The evanescent abatement is keeping increasing, nevertheless the inertia abatement performs the opposite exhibition. Moreover, under some circumstance its strategy will be concave. No matter what condition occurs, the inertia abatement will be 7

212 213 214 215

216 217

218

219 220 221

222 223 224 225 226 227 228 229 230 231

232 233 234 235

236 237 238 239 240 241 242

243 244 245

246 247 248

diminishing into zero. The intrinsic reason will be discussed in the final section. Apart from the inertia abatement strategy, we take more interesting about the inertia capital accumulation performance. Before we rendering the main result about the state At , the following Lemma is required though its simplicity. Lemma 2.2 The function with the shape of Aeξt − Beςt = K owns precisely one root, in which A, B, K, ξ, ς are parameters and satisfying: ς − ξ > 0, A − B − K > 0. Further, we have the following result: Proposition 2.3 Considering the problem (2.2) with the emission intensity target, then in the time period [0, T ) inertia abatement capital accumulation abatement At owning no more than one inflection point. Moreover, if there exists, it is a concave point. How to allocate the evanescent and inertia abatement is the center work in this paper. Compared to the result in Proposition(2.1), this result reveals that the cumulative abatement does not always keep being monotonous. It is meaningful to find out if investments are really postponed in response to this type of abatement. In some case, as we can view in the numerical section, the peak of the path does occur. A banal situation is the system is accompany with large δ, at the near of the duration the contribution from the direct inertia abatement can hardly compensate for the consumption in the abatement capital. Broadly, which of the two effects dominate should be reconsidered as the other parameters performance. Parallel to this conclusion, emission path Et have similar characteristics however with slight difference. Proposition 2.4 Considering the problem (2.2) with the emission intensity target, then in the time period [0, T ) inertia abatement capital accumulation abatement Et owning no more than two inflection points. Moreover, if there exists just one such a point, it must be a concave point. In fact, we can obtain all the parameters which are to be defined in the model. However, searching exact explicit solution can only lead us into the process of complicated parameter discussion and redundant scope analysis, which is far away from the central issue we intend to discuss. Fortunately, some parameters such as the bound condition and the initial value are decouple with the remain ones, and hence we can obtain some further results. These analysis are rendering important criterion for the policy-maker to estimate the effectiveness of the carbon reduction aim. The following results will reveal relevant results. Proposition 2.5 Considering the problem (2.2) with the emission intensity target, then the optimal index J is increasing with respect to the initial emission value and decreasing with ∂J respect to the bound condition, i.e. ∂E > 0 and ∂J < 0. ∂Γ 0 So far, we have discussed some basic properties of the model, especially the peak and inflection points of emission reduction and emission path. These basic conclusions are crucial for decision makers to judge the timing and trend of emission reduction investment strategy. 8

249 250 251 252 253

The perturbation analysis of the final emission reduction cost also reflects an intuitive result of this process. It is worth noting that these paths and features will be further analyzed in the model extension section. In addition, in the following part, we will use numerical methods to further verify the validity of the theory and the issues that beyond the power of it.

285

2.5. Parameters Values This paper uses the following values to carry out illustrative experiments, which help to draw more general conclusions. Consider a case in China’s iron and steel sector, in which the abatement technologies are used to implement the abatement task. For illustrative purpose, assuming that the abatement tools containing only two contrasted measures, evanescent abatement and inertia abatement. Evanescent abatement has a cheaper abatement cost than inertia abatement cost. Such an evanescent abatement can represent for instance the measure of switching energy sources in buildings, and the inertia could represent the retrofitting of these buildings. Use the case of general abatement costs ς = 2, which ensure that the abatement is strictly convex in eit . In the absence of reliable data, assuming that it takes years to target the final constraint at year 2030. The discretized form will be used to reflect the practical problem. The unit evanescent abatement is approximated as 50 yuan/t and the unit inertia abatement is 200 yuan/t, which are set as the parameters values for b = 200 and c = 50. Besides, we have to describe some parameters values which will be used in the simulation. These values are not meant to represent accurately concrete sectors of the economy. The choice of the discount rate will have some effects on the results, such as the discussion in [6] [25]. In this paper, we adopt the standard approach assuming a constant and positive discount rate δ = 0.1 to value the stream of future benefits and costs. We assume that the considered industry baseline emissions Eind which occupies 10% of the contemporaneous China GHG emissions. The initial baseline emission E0 of the considered entity occupies 1% of the industry constant baseline emission. We assume that the growing rate is aligned with the GDP performance. The growing rate of the GDP from 2015–2020 is approximately 7.2%, from 2021–2030 is approximately 4.5%, respectively. Thus, the average growing rate for the considered entity g can be set between 2% − 6%. Besides, the initial emission E0 is assumed to be 500 ten thousand tons, and the final constraint ET is assumed to be 550 ten thousand tons. It is difficult to find the appropriate value of A0 , we must make some assumption. As we know, this parameter can influence the result to a great extent. Obviously, the larger it is, the less the inertia abatement task will undertake. This is mainly due to the accumulation effect. Besides, the discounter factor r reflecting the time value of the currency can be set between 1%–20%, and moreover we can calculate the discounter value δ = e−r . The whole parameters used in the simulation are shown in the Table.1.

286

3. Basic results and discussions

254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284

287 288

In this section, we intend to render some basic results about the abatement path and emission. The parameters used are the same as those in Table.1. 9

Table 1: Parameters used in the basic case

Parameters Description Value Source T Time duration 2018–2030 Given τ Discount rate 0.05 Given ρ Discount value 0.95 ρ = e−τ c evanescent abatement coefficient 50 Calibration b inertia abatement coefficient 200 Calibration g Growing rate for the emission 0.02 Calibration E0 The initial emission 500 Given ET The terminal emission 550 Given δ Capital depreciation 0.1 Assumed

Figure 1: Basic results

289 290 291 292 293 294 295 296 297 298 299 300

3.1. Abatement and emission path The related comparison results are illustrated in Fig.1. In this group, figures are showing the evanescent abatement path, the inertia abatement path, the inertia capital accumulation and the actual emission path. Viewed from the pictures, we can find that abatements are allocated between evanescent abatement and inertia abatement. These two allocations are different from each other apparently. The evanescent abatement is allocated increasingly from the initial period 0.95 unit to the end of the decision period 1.35 unit. As we know, there is no lagging effect from the evanescent abatement effort, and thus it only has to compare the total benefit conserved from the evanescent abatement effort and the marginal implementation of the evanescent abatement cost. Besides, the result also verify the the Proposition (2.1), since it is easy to check that the parameters in the basic case meet the condition in the Proposition and thus the evanescent abatement is keeping increasing. Contrary to the evanescent abatement path, the inertia path is distinctly different. Viewed from the second picture in the Fig.1, we find that the inertia abatement is keeping decreasing from 1.5 unit till zero at the end of the period. In fact, at the beginning of the decision period, the inertia abatement experiences a transient increasing though it is unconspicuous. It is easy to verify this path by list the time series date of the related inertia abatement decision. Another side, we calculate that β(τ − g) < τ + δ, and thus this affirmation holds itself according to (2.1) as well. The performance of the inertia abatement is aligning with its accumulation in capital stock At . Also, it is easy to check the terminal value about the function f (t) = −δue−δt + (

301

βr r − δp)e(τ −g)t + (δq − )e(τ +δ)t , 2b 2b

which is defined in Proposition (2.3). 10

Figure 2: Singular case: large emission parameter

302 303 304 305 306 307 308 309 310 311 312

313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331

The actual emission is illustrated in the last picture in Fig.1. It can be confirmed that this path is increasing and has two inflections, which highlights that the emission has experienced an exponential phase, a stationary phase and then a growth phase again. A plausible reason is due to the interplay between the evanescent abatement effort and the inertia abatement effort. In fact, at the early stage of the decision, the inertia abatement effort is accumulated not enough, which is in according with the evanescent abatement effort to give rise to the boosting of the emission. Straight after that in the middle stage, the inertia abatement has expand to a great extent to restrict the growing rate of the actual emission. Finally, as the time period is drawing the inertia abatement is diminishing as well as the inertia accumulation, which undertake the synergistic effect with the evanescent abatement effort. This extinction effect releases the actual emission once again. 3.2. Singular solution As confirmed in the Proposition (2.1), (2.3), and (2.4), the relationship among some key parameters in the model function significantly upon the path. In this subsection, we intend to scrutinize some exotic solution and depict the strategy under it. To begin with, we set the emission growing rate to be 0.06, which is larger than τ . The related results are illustrated in the Fig.2. In the light of Proposition (2.1), the evanescent abatement effort is decreasing till the end of the duration. Apart from the theory aspect, we intend to render a rational excuse intuitively. As the emission rate is growing to some great extent, say larger than τ , the ex post abatement is less attractive which can transfer the economic burden to the lagging implementation with favorable time discounting. Thus, it is more favorable to adopt the abatement strategy early to circumvent the initial blooming growing of the emission. If not, the abatement burden will be reinforced in the near future, which invalidate the same amount abatement effort comparatively. Secondly, viewed from the inertia strategy kt and At , we can see that these path are similar to those in the basic case only with larger magnitude. This is mainly due to the fact that the evanescent abatement and the inertia abatement are “de-coupled” from each other. These two decision are fundamental, whose performance is related to the other only in the dimension of the magnitude. This explanation can be verified from theoretical aspect in the expression in Proposition (2.1), i.e. et =

332 333 334 335 336

se(τ −g)t (1 + g) re(τ +δ)t − βre(τ −g)t ; kt = − . 2c 2b

Last, we can find that the emission path is more twisted and presenting a characteristic of peak. This performance is not contradicted to Proposition (2.4), which claims there at most owning two inflection points on the path of Et . Contrary to the results in the Fig.1, the emission experiences a path with increasing way, declining way and increasing way. Noticing that the evanescent abatement is decreasing always, the emission reaching a peak 11

Figure 3: Sensitivity analysis with δ 337 338 339

340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361

in advance specifies that at the end of the duration the abatement is mainly contributed by the remaining inertia capital. We can conclude that earlier substantial investment in emission reduction is the optimal strategy to cope with the large growing emission rate. 3.3. Discussion over key parameters We specify two groups of parameter in the model, the first δ and the second E0 (Γ). In the Fig.3, we set the parameter δ = 0.25 compared with the basic case δ = 0.1. As we know that the larger the δ, the more depreciation level it will bring to the capital accumulation upon the inertia abatement effort. Compared with the result in Fig.1, we can find that the inertia abatement path is more concave. The larger depreciation level δ entails a long period investment accumulation, which makes the inertia abatement more obvious. Corresponding the capital accumulation declining rapidly, this is mainly due to the fact that the large depreciation and the declining abatement effort give rise to dash towards it. Viewed from the evanescent abatement path, we can find that the magnitude is risen from 1 unit to 1.85 unit in 2020, and the other contemporary period is also increased. Since the depreciation level is large, the burden has been transferred into the evanescent abatement. Adjustment is reconsidered before the optimal allocation, and a considerable amount from the inertia abatement is attached upon the evanescent abatement Viewing the the parameters set in Table.2, we intend to investigate the disturbance from the central case E0 and Γ. The results are also presented in the Table.2. It is easy to find that the total abatement cost is strictly increasing with E0 and decreasing with the larger Γ. As we understand intuitively, a larger value of ET means benevolent policy and the little value of E0 means stringent policy. Moreover, from the abatement efforts et and kt we can see that collaborative effort between them are “equal”. No matter what cases occur about E0 and ET , the abatement alleviation or the burden appeared to be allocated equally between them. Table 2: Sensitivity comparison

(E0 , Γ) Total cost (500,550) 3327.88 (510,550) 4276.73 (490,550) 2497.88 (500,540) 4050.95 (500,560) 2675.84

362 363 364

Total et (mt) 14.66 16.62 12.70 16.17 13.14

Total kt (mt) 13.67 15.50 11.85 15.09 12.26

Total abatement(mt) 28.34 32.12 24.55 31.26 25.41

3.4. Validation test In this section, we consider other similar path planning models and compare them to what we have proposed. In the test model, the baseline case is the abatement effort without 12

365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387

differentiating the abatement ways. The accumulated model which based upon the learningby-doing effect will be added on each model to further investigate the results. The four groups for the validation test are denoted as base (the proposed model), non-differentiation (with no inertia effect), base considering learning-by-doing effect, and non-differentiation considering learning-by-doing effect, respectively. In the base case, the inertia effect is not considered, that is, δ = 1. Moreover, the learning effect is considered only in the first abatement effort with the rate α = 0.1, which means the abatement cost in problem(2.2) becomes bt kt2 A−α for the first technology. t All the results are illustrated in Fig.4. We can see that the learning effect does not change the trend of the inflection point and the emission path of the investment, whether or not the inertial effect is considered. The learning effect is only a quantitative adjustment to the share of emissions reductions in the short term. However, it can be clearly found that the inertial model and the non-inertial model are still very different in the investment path. Mis-modeling and analysis of emission reduction measures with time-delay characteristics will not only misjudge the investment path, but also misjudge the actual carbon emissions in the future, and underestimate the ability of carbon assets, resulting in economic losses. In addition, it can be found that in the two types of models that do not consider the inertial effect, the LBD effect is the opposite of the two types of emission reduction routes, that is, the emission reduction measures with learning effects. There are relatively many tasks. This also verifies the conclusions of some previous work on the other hand, because the early capital accumulation caused by the learning effect at this time will have a great effect on the future cost reduction, and the investment level as the long-term should be much higher than expected.

Figure 4: Results of validation test

388

389 390 391 392 393 394 395 396 397

4. Model Extension In this subsection, we intend to investigate the extension form for the proposed model. In the practice, the introduction of the inertia abatement entails the necessity of considering the operating cost, which is mainly due to the exist of the capital accumulation At in a period. This feature is reasonable and can be found in many literatures named as operating maintenance cost [26, 27]. Especially when the inertial effect is relatively obvious, the economy caused by the investment in emission reduction technology is large, and the operating cost needs to be considered. Here, we denote that the marginal unit operating cost to be m and the total operating cost is convex with respect to the capital accumulation. Thus, the extension model can be depicted as following:

13

Z M in J(et ) = M in et ,kt

et ,kt

T

ρt [b(kt )2 + c(et )2 + m(At )2 ]dt

0

s.t. E˙ t = gEt − (1 + g)(At + et ); A˙ t = −δAt + kt ; Ω(Et ) ≤ 0. 398 399

Similar to the basic case, we can derive the optimal path from the necessary condition. The difference here is the Hamilton function becomes

H 400

401 402

403 404 405 406

407 408 409

410 411 412 413 414

415 416 417

418 419 420

(4.1)

= ρt [b(kt )2 + c(et )2 + m(At )2 ] + λ(gEt − (1 + g)(At + et )) + µ(−δAt + kt ).

Thus, we have the following result without giving any proof. Proposition 4.1 Considering the problem (4.1) with the emission intensity target, then in the time period [0, T ) the necessary condition of the system become:  t 2ρ cet − λt (1 + g) = 0;    t   2ρ bkt + µt = 0; −λ˙ t = λt g;   −µ˙ = −(1 + g)λt − µt δ + 2ρt mAt ;    ˙ t At = −δAt + kt . We can find that the evolution of the accompanying variable µ is more complicated due to the introduction of the operating cost m(At )2 , which will finally influence the inertia abatement kt . In fact, this change can twist the inertia abatement into three pieces, which is illustrated in the following result. Proposition 4.2 Considering the problem (4.1) with the emission intensity target, then in the time period [0, T ), the inertia abatement path kt owning no more than two inflection points. From the ratiocination in the Proposition (2.3) (4.2), we can find that the expression of kt play a crucial role to confirm the assertion. The linear combination of the exponential function eαt owns a special characteristic, since both the derivative and the integral of that is still sharing the same form. This property entails the convenient analysis of the existence for the equation roots, which also helps to parsing the analytic properties of the path. Remark 4.3 Likewise, we can find that the expression of At is the same as the basic case or (5.1). We can apply the same procedure to scrutinize the proposition of At , so does Et . Here, we do not render further discussion. Besides, we illustrate the model results with considering the operating cost. All the parameters in the model are the same as the basic assumption in Table.1 except for m. The unit operating cost set for m is assumed to be 20. The corresponding results are illustrated 14

Figure 5: Results analysis with m = 20

428

in the Fig.5. Viewed from the pictures, we can find that the kt is of a great difference. As the Proposition (4.2) claims, there are two inflection points on the abatement path. The optimal inertia abatement effort should be allocated decreasing drastically, then smoothly and finally rapidly. Also we can find that the total cost is increasing due to the introduction of the operating cost, and the accumulated capital of the inertia abatement is more smooth and weighs less. The peak of the accumulated capital is dropping by 38% (from 8 unit to 5 unit). Moreover, the evanescent abatement effort is more attractive than the basic case, which is risen universally by 22% (from 1.8–2.6 unit to 2.2–3 unit).

429

5. Policy implication and Conclusion

421 422 423 424 425 426 427

430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458

Traditionally, it is convinced that low-carbon capital investment should be allocated earlier to achieve a reduction in CO2 emissions. This aspect is supported by the view that the accumulation of the abatement capital is conducive to the formation of technology market, which in turn could further reduce the price of the abatement technology and the related abatement cost. However, the case maybe much more complicated being faced with differentiated abatement investments. In this paper, we deem that it is reasonable for the policy-maker to initiate the abatement effort considering both from the external factors and internal factors. The external factors such as the economic growing rate, the policy restraint can shed light upon the magnitude of the abatement effort considerably. However, a more important factors are the internal factors such as the the characteristic of the abatement effort, the special abatement cost and the depreciation rate of the abatement capital. To scrutinize the depth of the specific influence of these reasons, we differentiate the abatement effort into evanescent abatement and the inertia abatement. Firstly, we find that under a certain case, these two abatement paths show a great disparity, or conversely under some circumstance. The evanescent abatement is more sensitive to the market environment. In a prosperous market, it strengths the view that abatement capital should be allocated as early as possible, while in a slow-growing market, the optimal choice is more inclined to make on the contrary. However, this distinction is not applicable for the inertia abatement. Due to the bundle of the capital accumulation and the depreciation rate, the inertia abatement is under a prior execution. In the early stage, the accelerated accumulation of the abatement capital maximizes the reduction benefit due to the inertial effect. In the second place, the investment path for inertia emissions reduction is different at an early stage if considering the inner factor. We discuss this situation in the extension of the model by introducing the operating cost, which is claimed to be proportionable to the quadratic function of the capital. It depicts the picture that this abatement is not always plunged smoothly through a large-scale at the initial stage. A drastic declining is occurring if the operation cost, say the operating cost, is considerable. A plausible reason to specify this difference is that the operating cost is comparable in a high level. The initial burden of the operating cost makes it possible to wait and see that the inertia abatement capital has 15

459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501

been accumulated to a certain degree, and at this time such a operating cost is unavoidable. In general, in both case the investment paths have to go through a period of smooth decline and accelerated decline. Last, some key factors play an important role in the transition allocation between these two differentiated abatement. We still distinct and discuss the reasons between internal factors and external factors. Considering the market environment (mild or thriving), the allocation between the evanescent abatement and the inertia abatement are obvious. For instance, the evanescent abatement will behave conversely as discussed in the first view and moreover it suffers much from the total additional emission from the mild market to the thriving market. Accordingly, the contribution to emissions reduction has been reduced from the inertia abatement, which performs a much more smooth strategy path. However, being as the important external factors, the policy strict seems to influence these two avenues equally. Besides, the depreciated parameter also has important effects on the path and peak of emission reduction. In addition, through the cross-validation test in the results, we can find that even after considering the learning effect model, the proposed model is very robust. In other words, the learning effect does not essentially change the inflection point and topological nature of the emission reduction path in our proposed model. The results of the traditional learning model show similarities in the past when embedded in our model. To a certain extent, it is an extension of the previous work on the technical means of emission reduction, and hence the theory of learning effect is also adapted to analyze the path of the proposed model. The results have potentially important policy implications and can provide a rationale for supporting abatement investment. The optimal abatement efforts should be differentiated across abatement avenues. Both of the internal abatement factor and the external abatement factor play a duel role in the presence of introducing the abatement effort. The market environment, being as an important external factor, should be paid to considerable attention. The nature of the market has important implications not only for the evanescent abatement effort but also influences the peak for the optimal path of the inertia path. Thus, it is significant for the policy-maker to distinguish the current market attribute and choose the right time to introduce mitigation techniques. Especially, given the long-term horizons of environmental problems, the strict on the emission deserves more attention in policy design, which can to a certain extent be interfered with different market characteristics to stimulate the investment plans. Also for the industry decision-maker, identifying market characteristics is conducive to making more reasonable investment timing for emission reduction measures. Besides, the policy-maker should also consider the internal factor such as the capital depreciation rate, the operation cost and so forth. In fact, the operation cost directed to the introduction of the abatement capital is another important source of the strategy change. With these internal factors, marginal benefit in the inertia abatement investment occurs, and the evanescent abatement investment acts a by-product of its consequence. For the policy-maker, it is meaningful to make a distinction among abatement different investment path. This, however, may not be as clear cut as it actual seem in the practice. In the mild decreasing phase investment aggregates numerous purposive efforts and investments 16

515

intended to improve the overall abatement efficiency. In addition, it seems to be much advisable to adopt the less depreciated abatement capital to take a full use of the abatement ability. This tip is mainly from the designer’s consideration, and we conjecture that many of the core principles emerging from our implication would not change in the extension version. Based upon the discussion above, the related policies for low-carbon abatement technology promotion should seriously be considered both from the mechanism of the avenues or from the external environment. In the practice, these policies not only can speed up the new technology to its mature state, but also can result in a certain spillover effect to the current technologies in its development process. Besides, the optimal policy should be differentiated upon abatement investment effort to make a design according to their own characteristics. Although the main contribution of this article is theoretical, its main results are instructive for practical industrial applications. Due to the lack of specific investigation and data collection processes, the data used in the simulation here is still rough, and future work will focus on how to apply the model to more specific case in industry.

516

Appendix

502 503 504 505 506 507 508 509 510 511 512 513 514

517 518 519 520

Proof of Proposition 2.1 In the light of (2.6) and (2.7), we obtain the explicit expressions regarding et and kt . One side, since s > 0, and e(τ −g)t is increasing if and only if τ −g > 0, thus the conclusion being held. Another side, considering (2.7) if
r k˙ t = β(τ − g)e(τ −g)t − (δ + τ )e(δ+τ )t = 0, 2b we obtain

τ −g = e(δ+g)t . δ+τ Thus, if β(τ − g) > τ + δ, there must exist a unique root t∗ lying in the interval (0, T ) to meet k˙ t∗ = 0. Besides, the abatement is decreasing when t > t∗ and increasing when t < t∗ . Hence, we can confirm that the inertia abatement is either decreasing or concave. Noticing that µT = 0, it means that kT = 0 according to (2.7). Proof of Lemma 2.2 Define h(t) = Aeξt − Beςt − K, and thus β

521 522 523 524 525 526

h0 (t) = Aξeξt − Bςeςt > 0 ⇔ Aξeξt (1 − 527 528 529

Bς (ς−ξ)t Aξ e ) > 0 ⇔ e(ς−ξ)t < . Aξ Bς

View that there is a Fermat point t∗ , which is also a concave point on h(t). Since h(0) > 0, we obtain that there is just one root lying in (0, +∞). Proof of Proposition 2.3 Consider the equation A˙ t = −δAt + kt .

17

530

In the light of (2.7), At can be solved as: At = ue−δt +

531

rβe(τ −g)t re(τ +δ)t − = ue−δt + pe(τ −g)t − qe(τ +δ)t . 2b(τ − g + δ) 2b(τ + 2δ)

Denote p=

532 533 534

535

rβ r ,q = , 2b(τ − g + δ) 2b(τ + 2δ)

and u is a parameter to be determined by the initial condition. Noticing the initial condition A0 = 0, we obtain that u = q − p. Thus, βr r A˙ t = −δue−δt + ( − δp)e(τ −g)t + (δq − )e(τ +δ)t . 2b 2b Define βr r f (t) = −δue−δt + ( − δp)e(τ −g)t + (δq − )e(τ +δ)t , 2b 2b and it is easy to check that f (0) = −δu + (

536 537 538 539 540 541 542

βr r (β − 1)r − δp) + (δq − ) = > 0. 2b 2b 2b

Considering the fact that τ + δ > τ − g and Lemma (2.2), we confirm that function f (t) owing no more than one root. If the condition that f (T ) < 0 is holding, then there is just one root lying in (0, T ). Combining these results we arrive at the conclusion that At is either keeping increasing or at most decreasing from one certain time. Proof of Proposition 2.4 We consider the equation E˙ t − gEt = −(1 + g)(At + et ).

543

(5.1)

In the light of (2.6), we can solve Et as: E˙ t − gEt = −(1Z+ g)(At + et );

Et = vegt + egt −(1 + g)(At + et )e−gt dt Z se(τ −g)t (1 + g) −gt gt gt = ve − e (1 + g)(ue−δt + pe(τ −g)t − qe(τ +δ)t + )e dt 2c  −(δ+g)t  ue pe(τ −2g)t qe(τ +δ−g)t se(τ −2g)t (1 + g) gt gt = ve − (1 + g)e + − + −(δ + g) τ − 2g τ +δ−g 2c(τ − 2g) = vegt +

(1 + g)ue−δt (1 + g)qe(τ +δ)t (1 + g)(2cp + s(1 + g))e(τ −g)t + − , δ+g τ +δ−g 2c(τ − 2g)

18

544

in which v is a parameter to be determined. Thus, δ(1 + g)ue−δt (τ + δ)(1 + g)qe(τ +δ)t (τ − g)(1 + g)(2cp + s(1 + g))e(τ −g)t E˙ t = vgegt − + − . δ+g τ +δ−g 2c(τ − 2g) Compared with the case in At , this situation is more complex. We consider the case of τ − 2g > 0 or τ − 2g < 0, and the sign of v. To avoid redundant discussion, supposing that vgegt −

545

δ(1 + g)ue−δt (τ + δ)(1 + g)qe(τ +δ)t (τ − g)(1 + g)(2cp + s(1 + g))e(τ −g)t + − = 0, δ+g τ +δ−g 2c(τ − 2g)

we consider the case of τ − 2g > 0 and v > 0. This is equal to that: (τ + δ)(1 + g)qe(τ +δ)t (τ − g)(1 + g)(2cp + s(1 + g))e(τ −g)t δ(1 + g)ue−δt + vgegt = + . τ +δ−g 2c(τ − 2g) δ+g

546 547 548 549 550 551 552

In most case, τ +δ −g > 0 4 and hence it is easy to check that the left side is an increasing convex function. An other side, it is easy to check that the right side is a convex function, however which has no more than one inflection point. This indicates that the equation above has no more than two roots. Also, it is similar to check the other cases of the parameters. The rest verification of the Proposition is trivial. Proof of Proposition 2.5 In the proof of (2.4), we can obtain that E = vegt +

(1 + g)ue−δt (1 + g)qe(τ +δ)t (1 + g)(2cp + s(1 + g))e(τ −g)t + − . δ+g τ +δ−g 2c(τ − 2g)

Considering the initial value of E0 , we get   (1 + g)u (1 + g)q (1 + g) [2cp + s(1 + g)] v = E0 − + − . δ+g τ +δ−g 2c(τ − 2g)

553

4

(5.2)

(5.3)

The strict parameter range assumption is not specified here. In fact, according to the actual meaning of the parameter, this inequality is always true. The range of the specific value of the parameter is shown in the numerical section.

19

554

555

These two equation together with (2.5) and p, q can form the following algebra equation: h i  (1+g)u (1+g)q (1+g)[2cp+s(1+g)]  v = E0 − δ+g + τ +δ−g − ;  2c(τ −2g)     u = q − p;     rβ    p = 2b(τ − g + δ) ; r  q= ;   2b(τ + 2δ)    (δ + g)β    s= r;   (1 + g)   β = e(δ+g)T . Putting all the results into the following equation, we can obtain the value of parameter r: Γ = vegT +

556 557 558

(1 + g)ue−δT (1 + g)qe(τ +δ)T (1 + g)(2cp + s(1 + g))e(τ −g)T + − . δ+g τ +δ−g 2c(τ − 2g)

It is easy to find that the formulated algebraic equations are linear with respect to r, and moreover p, q, s and u are homogeneous linear with respect to r. Thus, the equation (5.4) is equal to the following equation:

G (r) = e

gT

559

560 561

(5.4)

E0 − Γ > 0,

G (r) is a homogeneous linear function with respect to r and moreover. ∂G that r > 0, thus > 0. in which

∂r Beside, from Proposition (2.1) we have

se(τ −g)t (1 + g) ; 2c re(τ +δ)t − βre(τ −g)t kt = − . 2b et =

20

(5.5) Noticing

562

563

564

565

Thus, we can calculate the index as following: Z T J= ρt [b(kt )2 + c(et )2 ]dt  (τ +δ)t 2  (τ −g)t 2 Z T0 re − βre(τ −g)t se (1 + g) t = ρ [b +c ]dt 2b 2c  Z0 T  c(re(τ +δ)t − βre(τ −g)t )2 + b(se(τ −g)t (1 + g))2 = ρt dt 4bc  Z0 T  2 2(τ +δ)t − 2re(τ +δ)t βre(τ −g)t + β 2 r2 e2(τ −g)t ) + bs2 e2(τ −g)t (1 + g)2 t c(r e = ρ dt 4bc 0  2 2(τ +δ)t  Z T − 2cβr2 e2τ t + [cβ 2 r2 + bs2 (1 + g)2 ] e2(τ −g)t −τ t cr e = e dt 4bc  Z0 T  2 cr (τ +2δ)t 2cβr2 τ t [cβ 2 r2 + bs2 (1 + g)2 ] (τ −2g)t = e − e + e dt 4bc 4bc 4bc 0 cr2 2cβr2 τ t T [cβ 2 r2 + bs2 (1 + g)2 ] (τ −2g)t T = e(τ +2δ)t |T0 − e |0 + e |0 4bc(τ + 2δ) 4bcτ 4bc(τ − 2g) cr2 2cβr2 τ T [cβ 2 r2 + bs2 (1 + g)2 ] (τ −2g)T cr2 = e(τ +2δ)T − e + e − 4bc(τ + 2δ) 4bcτ 4bc(τ − 2g) 4bc(τ + 2δ) 2cβr2 [cβ 2 r2 + bs2 (1 + g)2 ] + − 4bcτ 4bc(τ − 2g)   e(τ +2δ)T 2β τ T [cβ 2 + b(δ + g)2 β 2 ] (τ −2g)T 1 2β [cβ 2 + b(δ + g)2 β 2 ] 2 = − e + e − + − r . 4b(τ + 2δ) 4bτ 4bc(τ − 2g) 4b(τ + 2δ) 4bτ 4bc(τ − 2g) Since the index J must be positive, thus J is a quadratic function about r i.e. J = ∂2 and > 0. Hence, considering (5.5) we have: ∂r2  −1 2 ∂J ∂J ∂r ∂ ∂r ∂2 ∂r ∂ gT ∂ = = =2 2 = 2e > 0. ∂E0 ∂r ∂E0 ∂r ∂E0 ∂r ∂E0 ∂r2 ∂r

F

F

F

F

F

F

F (r)

G

Similarly, we can have

F  ∂G 

∂J ∂J ∂r ∂ ∂r ∂2 ∂r ∂2 = = =2 2 = −2 2 ∂Γ ∂r ∂E0 ∂r ∂E0 ∂r ∂E0 ∂r

−1

∂r

< 0.

566 567 568

Proof of Proposition 4.2 Similar to the proof in Proposition(2.3), we can obtain that: −λ˙ t =

569

H

∂ µt =λt g ⇔ λt =se−gt , s > 0; kt = − t . ∂Et 2ρ b

Hence we only have to cope with the following second order linear inhomogeneous equations: 21

(

570

571 572

573 574

575

576

577

µ˙ t = δµt − 2ρt mAt + (1 + g)se−gt ; µt A˙ t = − t − δAt . 2ρ b

We rearrange it into the following form:   t       δ −2ρ m d µt µt (1 + g)se−gt   1 = + . − t −δ At 0 dt At 2ρ b According to the differential equations theory, it can be easily solved. Specifically, define the coefficient matrix   δ −2ρt m , 1 S= − t −δ 2ρ b p p whose characteristic roots are x1 = δ 2 + mb , x2 = − δ 2 + mb . Hence, the basic solution to this equations is       z1 z3 µt x1 t x2 t = C1 e + C2 e . 1 −δ)z1 2 −δ)z3 At − (x2ρ − (x2ρ tm tm Here, z1 and z3 are two free chosen parameters. Choosing z1 = z3 = 1, we have that:   µt = C1 ex1 t + C2 ex2 t ; (δ − x1 ) (5.6) x2 t (δ − x2 ) + C e .  At = C1 ex1 t 2 t t 2ρ m 2ρ m Using the method of variation of constant i.e. C1 = C1 (t), C2 = C2 (t), we have   C˙ 1 ex1 t + C˙ 2 ex2 t = (1 + g)se−gt ; (δ − x2 ) (δ − x1 )  C˙ 1 ex1 t + C˙ 2 ex2 t = 0. t 2ρ m 2ρt m Thus, according to Cramer’s Rule [28], we have  x2 t (1 + g)se−gt e    (δ−x ) 2 x t  2  0 e (δ − x2 )(1 + g)se(x2 −g)t t  2ρ m ˙  = C =  1  ex1 t ex2 t  x1 − x2   (δ−x ) (δ−x )  1 2 x t x t 1 2 e e ;        C˙ 2 =      



2ρt m x1 t

2ρt m e (1 + g)se−gt 1) ex1 t (δ−x 0 2ρt m x1 t x2 t e e (δ−x2 ) x1 t (δ−x1 ) x t 2 e e 2ρt m 2ρt m

22

=

−(δ − x1 )(1 + g)se(x1 −g)t . x1 − x2

578 579 580

In the light of the solutions above, we can obtain the solutions in the equations (5.6). Compared to the basic case, i.e. (5.1) and kt , we can find that the expression of At are the same and the kt are different. In fact, we can solve the equations by calculating the coefficients of C1 and C2 in the light of (5.6) and the bound condition µT = 0, A0 = 0. To circumvent the redundant discussion over the specific solution of the expression kt , we take more interest in the inflection point on the abatement path. According to the expression of (5.6), we can calculate kt as kt = a0 e(τ +x1 )t + b0 e(τ +x2 )t + c0 e(τ −g)t . Compared to the expression in the Proposition(1), we can find that there is an additional part in kt , which can contribute to the more complicated performance. Actually, in this time the second derivative of kt can be expressed as: k¨t = a0 e(τ +x1 )t + b0 e(τ +x2 )t + c0 e(τ −g)t .

581 582

Here, we still use the a0 , b0 , c0 to stand for the parameters in the expression. Let k¨t = 0 and consider that x1 + x2 = 0, the equation is equal to the following equation: a0 e2x1 t + c0 e(x1 +τ −g)t = −b0 .

584

Similar to the proof in Lemma (2.2) and Proposition (2.3), it is easy to confirm that the equation above has no more than two roots. Thus, the condition holds in the assertion.

585

Acknowledgement

583

587

Support from the National Natural Science Foundation of China under grant No. 71690245, 71801212.

588

References

586

589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604

[1] Xepapadeas A P. Environmental policy, adjustment costs, and behavior of the firm [J]. Journal of Environmental Economics & Management, 1992, 23(3):258-275. [2] Arrow K, The economic implication of learning-by-doing, Review of Economic Studies, 1962, 29: 155C173. [3] Nachtigall D, Rbbelke D. The green paradox and learning-by-doing in the renewable energy sector[J]. Resource and Energy Economics, 2016, 43: 74-92. [4] Dato P. Energy transition under irreversibility: a two-sector approach[J]. Environmental and resource economics, 2017, 68(3): 797-820. [5] Goulder LH , Schneider SH. Induced technological change and the attractiveness of CO2 , abatement policies[J]. Resource & Energy Economics, 1999, 21(3C4):211-253. [6] Yann Bramoull, Lars J. Olson. Allocation of pollution abatement under learning by doing[J]. Journal of Public Economics, 2005, 89(9):1935-1960. [7] Moreaux M, Withagen C. Optimal abatement of carbon emission flows[J]. Journal of Environmental Economics and Management, 2015, 74: 55-70. [8] Guo J X, Zhu L, Fan Y. Emission path planning based on dynamic abatement cost curve[J]. European Journal of Operational Research, 2016, 255(3):996-1013.

23

605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 632 633 634 635 636 637 638 639 640 641 642 643 644

[9] Fan J L, Xu M, Li F, et al. Carbon capture and storage (CCS) retrofit potential of coal-fired power plants in China: The technology lock-in and cost optimization perspective[J]. Applied energy, 2018, 229: 326-334. [10] Grubb M. The economics of changing course. Implications of adaptability and inertia for optimal climate policy[J]. Fuel & Energy Abstracts, 2006, 36(4-5):417-431. [11] Haduong M, Grubb M, Hourcade J C, et al. Influence of socioeconomic inertia and uncertainty on optimal CO2-emission abatement[J]. Nature, 1997, 390(6657): 270-273. [12] Hourcade J C, Chapuis T. No-regret potentials and technical innovation[J]. Energy Policy, 1995, 23(4):433-445. [13] Marechal, K, Lazaric, N. Overcoming inertia: insights from evolutionary economics into improved energy and climate policies.[J]. Climate Policy, 2010, 10(1):103-119. [14] Seto K C, Davis S J, Mitchell R B, et al. Carbon Lock-In: Types, Causes, and Policy Implications[J]. Annual Review of Environment & Resources, 2016, 41(1). [15] Vogt-Schilb A, Hallegatte S. Marginal abatement cost curves and the optimal timing of mitigation measures[J]. Energy Policy, 2014, 66: 645-653. [16] Wang K, Che L, Ma C, et al. The shadow price of CO2 emissions in China’s iron and steel industry[J]. Science of the Total Environment, 2017, 598: 272-281. [17] Kesicki F, Ekins P. Marginal abatement cost curves: a call for caution[J]. Climate Policy, 2012, 12(2): 219-236. [18] Baker E, Clarke L, Shittu E. Technical change and the marginal cost of abatement[J]. Energy Economics, 2008, 30(6): 2799-2816. [19] Lecuyer O, Quirion P. Can uncertainty justify overlapping policy instruments to mitigate emissions?[J]. Ecological Economics, 2013, 93(3):177-191. [20] Guo J X, Fan Y. Optimal abatement technology adoption based upon learning-by-doing with spillover effect[J]. Journal of Cleaner Production, 2017, 143. [21] Fan Y, The cost, Path and policy research of greenhouse gas emissions, Science Press, 2011, 112-151. [22] Rosendahl KE, Cost-effective environmental policy: implications of induced technological change, Journal of Environmental Economics and Management, 2004, 48(3): 1099-1121. [23] Park H. Real option analysis for effects of emission permit banking on investment under abatement cost uncertainty ?[J]. Economic Modelling, 2012, 29(4):1314-1321. [24] Becerra V M. Optimal control theory: Applications to management science and economics [Book Review][J]. IEEE Control Systems, 2003, 23(5):86-87. [25] Six M, Wirl F. Optimal pollution management when discount rates are endogenous[J]. Resource and Energy Economics, 2015, 42:53-70. [26] Heuberger C F, Rubin E S, Staffell I, et al. Power capacity expansion planning considering endogenous technology cost learning[J]. Applied Energy, 2017: 831-845. [27] Akashi O, Hanaoka T, Matsuoka Y, et al. A projection for global CO2 emissions from the industrial sector through 2030 based on activity level and technology changes[J]. Energy, 2011, 36(4): 1855-1867. [28] Anton, Howard. Elementary linear algebra : with supplemental applications / 10th ed. international student version[M]. Wiley, 2011.

24