Should marginal abatement costs differ across sectors? The effect of low-carbon capital accumulation

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1 Should margnal abatement costs dffer across sectors? The effect of low-carbon captal accumulaton Adren Vogt-Schlb 1,, Guy Meuner 2, Stéphane Hallegatte 3 1 CIRED, Nogent-sur-Marne, France. 2 INRA UR133 ALISS, Ivry-Sur-Sene, France. 3 The World Bank, Sustanable Development Network, Washngton D.C., USA Abstract Clmate mtgaton s largely done through nvestments n low-carbon captal that wll have long-lastng effects on emssons. In a model that represents explctly low-carbon captal accumulaton, optmal margnal nvestment costs dffer across sectors. They are equal to the value of avoded carbon emssons over tme, mnus the value of the forgone opton to nvest later. It s therefore msleadng to assess the cost-effcency of nvestments n low-carbon captal by comparng levelzed abatement costs, measured as the rato of nvestment costs to dscounted abatement. The equmargnal prncple apples to an accountng value: the Margnal Implct Rental Cost of the Captal (MIRCC) used to abate. Two apparently opposte vews are reconcled. On the one hand, hgher efforts are justfed n sectors that wll take longer to decarbonze, such as transport and urban plannng; on the other hand, the MIRCC should be equal to the carbon prce at each pont n tme and n all sectors. Equalzng the MIRCC n each sector to the socal cost of carbon s a necessary condton to reach the optmal pathway, but t s not a suffcent condton. Decentralzed optmal nvestment decsons at the sector level requre not only the nformaton contaned n the carbon prce sgnal, but also knowledge of the date when the sector reaches ts full abatement potental. Keywords: clmate change mtgaton; carbon prce; path dependence; sectoral polces; optmal tmng; nerta; levelzed costs JEL classfcaton: L98, O21, O25, Q48, Q51, Q54, Q58 A verson of ths paper s avalable at 1. Introducton Many countres have set ambtous targets to reduce ther Greenhouse Gas (GHG) emssons. To do so, most of them rely on several polcy nstruments. The European Unon, for nstance, has mplemented an emsson tradng system, feed-n tarffs and portfolo standards n favor of renewable power, and Correspondng author Emal addresses: (Adren Vogt-Schlb ), (Guy Meuner), (Stéphane Hallegatte) World Bank Polcy Research Workng Paper 6154 (213) May 14, 213

2 dfferent energy effcency standards on new passenger vehcles, buldngs, home applances and ndustral motors. These sectoral polces are often desgned to spur nvestments that wll have long-lastng effects on emssons, but they have been crtczed because they result n dfferent Margnal Abatement Costs (MACs) n dfferent sectors. Exstng analytcal assessments conclude that dfferentatng MACs s a second-best polcy, justfed f multple market falures cannot be corrected ndependently (Lpsey and Lancaster, 1956). For nstance, f governments cannot use tarffs to dscrmnate mports from countres where no envronmental polces are appled, t s optmal to dfferentate the carbon tax between traded and non-traded sectors (Hoel, 1996). A government should tax emssons from households at a hgher rate than emssons from the producton sector, f labor supply exerts market power (Rchter and Schneder, 23). Also, many abatement optons nvolve new technologes wth ncreasng returns or learnng-bydong (LBD) and knowledge spllovers. Ths twn-market falure n the area of green nnovaton can be addressed optmally by combnng a carbon prce wth a subsdy on technologes subjected to LBD and an R&D subsdy (e.g., Jaffe et al., 25; Fscher and Newell, 28; Grmaud and Lafforgue, 28; Gerlagh et al., 29; Acemoglu et al., 212). But f the only avalable nstrument s a carbon tax, hgher carbon prces are justfed n sectors wth larger learnng effects (Rosendahl, 24). These studes often model a socal planner who takes an abatement cost functon as gven, and may choose any quantty of abatement at each tme step. Wthn ths framework, MACs can be easly computed as the dervatve of the cost functons; but the nerta nduced by slow captal accumulaton s neglected (Vogt-Schlb et al., 212). Only a few studes explctly model nerta or slow captal accumulaton; they conclude that hgher efforts are requred n the partcular sectors that wll take longer to decarbonze, such as transportaton nfrastructure (Lecocq et al., 1998; Jaccard and Rvers, 27; Vogt-Schlb and Hallegatte, 211); but they do not provde an analytcal defnton of abatement costs. In ths paper, we assess the optmal cost, tmng and sectoral dstrbuton of greenhouse gas emsson reductons when abatement s obtaned through nvestments n low-carbon captal. We use an ntertemporal optmzaton model wth three characterstcs. Frst, we do not use abatement cost functons; nstead, abatement s obtaned by accumulatng low-carbon captal that has a long-lastng effect on emssons. For nstance, replacng a coal power plant by renewable power reduces GHG emssons for several decades. Second, these nvestments have convex costs : accumulatng low-carbon captal faster s more expensve. For nstance, retrofttng the entre buldng stock would be more expensve f done over a shorter perod of tme. Thrd, abatement cannot exceed a gven maxmum potental n each sector. Ths potental s exhausted when all the emttng captal (e.g fossl fuel power plants) s replaced by non-emttng captal (e.g. renewables). We fnd that t s optmal to nvest more dollars per unt of low-carbon captal n sectors that wll take longer to decarbonze, as for nstance sectors wth greater baselne emssons. Indeed, wth maxmum abatement potentals, nvestng n low-carbon captal reduces both emssons and future nvestment opportuntes. The optmal nvestment costs can be expressed as the value of avoded carbon, mnus the value of the forgone opton to abate later n the 2

3 same sector. Snce the latter term s dfferent n each sector, t leads to dfferent nvestment costs and levels. There are multple possble defntons of margnal abatement costs n a model where abatement s obtaned through low-carbon captal accumulaton. Here, we defne the Margnal Levelzed Abatement Cost (MLAC) as the margnal cost of low-carbon captal (compared to the cost of conventonal captal) dvded by the dscounted emssons that t abates. Ths metrc has been smply labeled as Margnal Abatement Cost (MAC) by scholars and government agences, suggestng that t should be equal to the prce of carbon. We fnd that MLACs should not be equal across sectors, and should not be equal to the carbon prce. Instead, the equmargnal prncple apples to an accountng value: the Margnal Implct Rental Cost of the Captal (MIRCC) used to abate. MIRCCs generalze the concept of mplct rental cost of captal proposed by Jorgenson (1967) to the case of endogenous nvestment costs. On the optmal pathway, MIRCCs expressed n dollars per ton are equal to the current carbon prce and are thus equal n all sectors. If the abatement cost s defned as the mplct rental cost of the captal used to abate, a necessary condton for optmalty s that MACs equal the carbon prce. It s not a suffcent condton, as many suboptmal nvestment pathways also satsfy ths constrant. Optmal nvestment decsons to decarbonze a sector requre combnng the nformaton contaned n the carbon prce sgnal wth knowledge of the date when the sector reaches ts full abatement potental. The rest of the paper s organzed as follows. We present our model n secton 2. In secton 3, we solve t n a partcular settng where nvestments n low-carbon captal have a permanent mpact on emssons. In secton 4, we solve the model n the general case where low-carbon captal deprecates at a non-negatve rate. In secton 5 we defne the mplct margnal rental cost of captal and compute t along the optmal pathway. Secton 6 concludes. 2. A model of low-carbon captal accumulaton to cope wth a carbon budget We model a socal planner (or any equvalent decentralzed procedure) that chooses when and how (.e., n whch sector) to nvest n low-carbon captal n order to meet a clmate target at the mnmum dscounted cost Low-carbon captal accumulaton The economy s parttoned n a set of sectors ndexed by. For smplcty, we assume that abatement n each sector does not nteract wth the others. 1 Wthout loss of generalty, the stock of low-carbon captal n each sector starts at zero, and at each tme step t, the socal planner chooses a non-negatve amount of physcal nvestment x,t n abatng captal a,t, whch deprecates at rate δ (dotted varables represent temporal dervatves): a, = (1) ȧ,t = x,t δ a,t (2) 1 Ths s not completely realstc, as abatement realzed n the power sector may actually reduce the cost to mplement abatement n other sectors usng electrc-powered captal. In order to keep thngs smple, we let ths ssue to further research. 3

4 For smplcty, abatng captal s drectly measured n terms of avoded emssons (Tab. 1). 2 Investments n low-carbon captal cost c (x ), where the functons c are postve, ncreasng, dfferentable and convex. The cost convexty bears on the nvestments flow, to capture ncreasng opportunty costs to use scarce resources (sklled workers and approprate captal) to buld and deploy low-carbon captal. For nstance, x,t could stand for the pace measured n buldngs per year at whch old buldngs are beng retroftted at date t (the abatement a,t would then be proportonal to the share of retroftted buldngs n the stock). Retrofttng buldngs at a gven pace requres to pay a gven number of scarce sklled workers. If workers are hred n the mert order and pad at the margnal productvty, the margnal prce of retrofttng buldngs c (x ) s a growng functon of the pace x. In each sector, a sectoral potental ā represents the maxmum amount of GHG emssons (n GtCO 2 per year) that can be abated n ths sector: a,t ā For nstance, f each vehcle s replaced by a zero-emsson vehcle, all the abatement potental n the prvate moblty sector has been realzed. 3 The sectoral potental may be roughly approxmated by sectoral emssons n the baselne, but they may also be smaller (f some fatal emssons occur n the sector) or could even be hgher (f negatve emssons are possble). We make the smplfyng assumpton that the potentals ā and the abatement cost functons c are constant over tme and let the cases of evolvng potentals and nduced techncal change to further research Carbon budget The clmate polcy s modeled as a so-called carbon budget for emssons above a safe level (Fg. 1). We assume that the envronment s able to absorb a constant flow of GHG emssons E s. Above E s, emssons are dangerous. The objectve s to mantan cumulatve dangerous emssons below a gven celng B. Cummulatve emssons have been found to be a good proxy for clmate change (Allen et al., 29; Matthews et al., 29). 4 For smplcty, we assume that all dangerous emssons are abatable, and, wthout loss of generalty, that dong so requres to use all the sectoral potentals. Denotng E d the emssons above E s, ths reads E d = ā. In other words, (ā a,t ) stands for the hgh carbon captal that has not been replaced by low carbon captal yet n sector, 2 Investments x,t are therefore measured as addtonal abatng capacty (n tco 2 /yr) per unt of tme,.e. n (tco 2 /yr)/yr. 3 Ths modelng approach may remnd the lterature on the optmal extracton rates of non renewable resources. Our maxmal abatement potentals are smlar to the dfferent deposts n Kemp and Long (198), or the stocks of dfferent fossl fuels n the more recent lterature (e.g, Chakravorty et al., 28; Smulders and Van Der Werf, 28; van der Ploeg and Wthagen, 212). Our results are also smlar, as these authors fnd n partcular that dfferent reservors or dfferent types of non-renewable resources should not necessarly be extracted at the same margnal cost. 4 Our conclusons are robust to other representatons of clmate polcy objectves such as an exogenous carbon prce, e.g. a Pgouvan tax; or a more complex clmate model. 4

5 Emssons GtCO 2 /yr a (t) E d B E s t tme Fgure 1: An llustraton of the clmate constrant. The cumulatve emssons above E s are capped to a carbon budget B (ths requres that the long-run emssons tend to E s). Dangerous emssons E d are measured from E s. as measured n emssons. The carbon budget reads: (ā a,t ) dt B (3) 2.3. The socal planner s program The full socal planner s program reads: mn e rt c (x,t ) dt (4) x,t subject to ȧ,t = x,t δ a,t (ν,t ) a,t ā (λ,t ) (ā a,t ) dt B (µ) The Greek letters n parentheses are the respectve Lagrangan multplers (notatons are summarzed n Tab. 1). The socal cost of carbon (SCC) µ does not depend on nor t, as a ton of GHG saved n any sector at any pont n tme t contrbutes equally to meet the carbon budget. In every sector, the optmal tmng of nvestments n low-carbon captal s partly drven by the current prce of carbon µe rt, whch follows an Hotellng s rule by growng at the dscount rate r. The multplers λ,t are the sector-specfc socal costs of the sectoral potentals. They are null before the potentals ā are reached (slackness condton). The costate varable ν,t may be nterpreted as the present value of nvestments n low-carbon captal n sector at tme t. 3. Margnal nvestment costs (MICs) wth nfntely-lved captal In a frst step, we solve the model n the smple case where δ =. Ths case helps to understand the mechansms at sake. However, wth ths assumpton, margnal abatement costs cannot be defned: one sngle dollar nvested produces an nfnte amount of abatement (f a MAC was to be defned, t would be null). Ths ssue s dscussed further n the followng sectons. 5

6 Name Descrpton Unt c Cost of nvestment n sector $/yr ā Sectoral potental n sector tco 2 /yr δ Deprecaton rate of abatng captal n sector yr 1 r Dscount rate yr 1 a,t Current abatement n sector tco 2 /yr x,t Current nvestment n abatng actvtes n sector (tco 2 /yr)/yr ν,t Costate varable (present socal value of green nvestments) $/(tco 2 /yr) λ,t Socal cost of the sectoral potental $/tco 2 µ Socal cost of carbon (SCC) (present value) $/tco 2 µe rt Current carbon prce $/tco 2 c Margnal nvestment cost (MIC) n sector $/(tco 2 /yr) l,t Margnal levelzed abatement cost (MLAC) n sector $/tco 2 p,t Margnal mplct rental cost of captal (MIRCC) n sector $/tco 2 Table 1: Notatons (ordered by parameters, varables, multplers and margnal costs). Defnton 1. We call Margnal Investment Cost (MIC) the cost of the last unt of nvestment n low-carbon captal c (x,t ). MICs measure the economc efforts beng orented towards buldng and deployng low-carbon captal n a gven sector at a gven pont n tme. Whle one unt of nvestment at tme t n two dfferent sectors produces two smlar goods a unt of low-carbon captal that wll save GHG from t onwards they should not necessarly be valued equally. Proposton 1. In the case where low-carbon captal s nfntely-lved (δ = ), optmal MICs are not equal across sectors. Optmal MICs equal the value of the carbon they allow to save less the value of the forgone opton to abate later n the same sector. Equvalently, sectors should nvest up to the pace at whch MICs are equal to the total socal cost of emssons avoded before the sectoral potental s reached. Proof. Wth nfntely-lved captal (δ = ), the generalzed Lagrangan reads: L(x,t, a,t, λ, ν, µ) = e rt c (x,t ) dt + λ,t (a,t ā ) dt ( ) + µ (ā a,t ) dt B ν,t a,t dt ν,t x,t dt 6

7 Fgure 2: Left: Optmal margnal nvestment costs (MIC) n low-carbon captal n a case wth two sectors ( {1, 2}) wth nfntely-lved captal (δ = ). The dates T denote the endogenous date when all the emttng captal n sector has been replaced by low-carbon captal; after ths date, addtonal nvestment would brng no beneft. Optmal MICs dffer across sectors, because of the socal costs of the sectoral potentals (λ,t). Rght: correspondng optmal abatement pathways. The frst-order condtons are: 5 (, t), L = ν,t = λ,t µ a,t (5) (, t), L = e rt c (x x,t) = ν,t,t (6) The optmal MIC can be wrtten as: 6 c (x,t) = e rt (µ λ,θ ) dθ (7) The complementary slackness condton states that the postve socal cost of the sectoral potental λ,t s null when the sectoral potental ā s not bndng: t (, t), λ,t, and λ,t (ā a,t ) = (8) Each nvestment made n a sector brngs closer the endogenous date, denoted T, at whch all the producton of ths sector wll come from low-carbon captal (Fg. 3). After ths date T, the opton to abate global GHG emssons thanks to nvestments n low-carbon captal n sector s removed. The value of ths opton s the ntegral from t to of the socal cost of the sectoral potental λ,θ ; t s subtracted from the ntegral from t to of µ (the value of abatement) to obtan the value of nvestments n low-carbon captal. Usng (7) and (8) allows to express the optmal margnal nvestment costs 5 The same condtons can be wrtten usng a Hamltonan. 6 We ntegrated ν,t as gven by (5) between t and ; used the relaton lm t ν,t = (justfed latter); and replaced ν,t n (6). 7

8 Fgure 3: From a gven nvestment pathway (x,t) leadng to the abatement pathway (a,t), one supplementary unt of nvestment at tme θ, as n ( x,t), has two effects: t saves more GHG (a,t), but t also brngs closer the date when the maxmum sectoral potental s reached (T T ). Both effects shape optmal margnal nvestments costs (7,9). as a functon of T and µ: 7 c (x,t) = { µe rt (T t) f t < T f t T (9) Optmal MICs equal the total socal cost of the carbon expressed n current value (µe rt ) that wll be saved thanks to the abatement before the sectoral potental s reached the tme span (T t). The followng lemma fulflls the proof. Lemma 1. When the abatng captal s nfntely-lved (δ = ), for any cost of carbon µ, the decarbonzng date T s an ncreasng functon of the sectoral potental ā. Proof. See Appendx A. Snce potentals ā dffer across sectors, the dates T also dffer across sectors, and optmal MICs are not equal. The general shape of the optmal MICs s dsplayed n Fg. 2. Vogt-Schlb et al. (212) provde some numercal smulatons calbrated wth IPCC (27) estmates of costs and abatng potentals of seven sectors of the economy. 4. Margnal levelzed abatement costs (MLACs) In ths secton, we solve for the optmal margnal nvestment costs n the general case when the deprecaton rate s postve (δ > ). Then, we show that the levelzed abatement cost s not equal across sectors along the optmal path, and n partcular s not equal to the carbon prce. 7 t < T, a,t < ā = λ,t = (8); for t T the abatement a (t) s constant, equal to ā, thus x (t) = ȧ,t s null, and, usng (6) t T, ν,t =, = ν,t = = λ,t = µ (5). Ths last equalty means that once the sectoral potental s bndng, the assocated shadow cost equals the value of the carbon that t prevents to abate. 8

9 Optmal margnal nvestment costs Proposton 2. Along the optmal path, abatement n each sector ncreases untl t reaches the sectoral potental ā at a date denoted T ; before ths date, margnal nvestment costs can be expressed as a functon of ā, T, the deprecaton rate of the low-carbon captal δ, and the socal cost of carbon µ: t T, c (x rt 1 e δ(t t),t) = µe +e (r+δ)(t t) c (δ ā ) (1) δ } {{ } K Proof. See Appendx B. Equaton 1 states that at each tme step t, each sector should nvest n lowcarbon captal up to the pace at whch margnal nvestment costs (Left-hand sde term) are equal to margnal benefts (RHS term). In the margnal benefts, ( the current carbon prce µe rt appears multpled by 1 a postve tme span: ) δ 1 e δ (T t). Ths term s the equvalent of (T t) n the case of nfntely-lved abatng captal (9). We nterpret K as the margnal beneft of buldng new low-carbon captal. The longer t takes for a sector to reach ts potental (.e. the further T ), the more expensve should be the last unt of nvestments drected toward low-carbon captal accumulaton. K reflects a complex trade-off: nvestng soon allows the planner to beneft from the persstence of abatng efforts over tme, and prevents nvestng too much n the long-term; but t brngs closer the date T, removng the opton to nvest later, when the dscount factor s hgher. Ths results n a bell-shaped dstrbuton of mtgaton costs over tme: n the short term, the effect of dscountng may domnate 8 and the effort may grow exponentally; n the long term, the effect of the lmted potental domnates and new captal accumulaton decreases to zero (Fg. 4). Margnal benefts also have another component, that tends to the margnal cost of mantanng abatement at ts maxmal level: e (δ+r)(t t) c (ā ). As the frst term n (1) tends to, the optmal MIC tends to the cost of mantanng low-carbon captal at ts maxmal level: c (x,t) t T c (δ ā ) (11) After T, the abatement s constant (a,t = ā ) and the optmal MIC n sector s smply constant at c (δ ā ). Margnal levelzed abatement costs Our model does not feature an abatement cost functon that can be dfferentated to compute the margnal abatement costs (MACs). In ths secton, we compute the levelzed abatement cost of the low-carbon captal. Ths metrc s sometmes labeled margnal abatement costs by some scholars and nsttutons. We fnd that margnal levelzed abatement costs should not be equal to the carbon prce, and should not be equal across sectors. 8 f ether T or δ s suffcently large 9

10 Fgure 4: Left: rato of margnal nvestments to abated GHG (MLACs) along the optmal trajectory n a case wth two sectors ( {1, 2}). In a frst phase, the optmal tmng of sectoral abatement comes from a trade-off between () nvestng later n order to reduce present costs thanks to the dscountng, () nvestng sooner to beneft from the persstent effect of the abatng efforts over tme, and () smooth nvestment over tme, as nvestment costs are convex. Ths results n a bell shape. After the dates T when the potentals ā have been reached, margnal abatement costs are constant to δ c (δ ā ). Rght: correspondng optmal abatement paths. Defnton 2. We call Margnal Levelzed Abatement Cost (MLAC) and denote l,t the rato of margnal nvestment costs to dscounted abatements. MLACs can be expressed as: Proof. See Appendx D. x,t, l,t = (r + δ ) c (x,t ) (12) MLACs may be nterpreted as MICs annualzed usng r+δ as the dscount rate. Ths s the approprate dscount rate because, takng a carbon prce as gven, one unt of nvestment n low-carbon captal generates a flow of real revenue that decreases at the rate r + δ. Practtoners may use MLACs when comparng dfferent technologes. 9 Let us take an llustratve example: buldng electrc vehcles (EV) to replace conventonal cars. Let us say that the socal cost of the last EV bult at tme t, compared to the cost of a classc car, s 7 $/EV. Ths fgure may nclude, n addton to the hgher upfront cost of the EV, the lower dscounted operaton and mantenance costs complete costs computed ths way are sometmes called levelzed costs. If cars are drven 13 km per year and electrc cars emt 11 gco 2 /km less than a comparable nternal combuston engne vehcle, each EV allows to save 1.43 tco 2 /yr. The MIC n ths case would be 4 9 $/(tco 2 /yr). If electrc cars deprecate at a constant rate such that ther average lfetme s 1 years (1/δ = 1 yr), then r + δ = 15%/yr and the MLAC s 734 $/tco We defned margnal levelzed costs. The gray lterature smply uses levelzed costs; they equal margnal levelzed costs f nvestment costs are lnear (Appendx E). 1 The MIC was computed as 7 $/(1.43 tco 2 /yr) = $/(tco 2 /yr); and the MLAC as.15 yr $/(tco 2 /yr)= 734 $/tco 2. 1

11 Proposton 3. Optmal MLACs are not equal to the carbon prce. Proof. Combnng the expresson of optmal MICs from Prop. 2 and n the expresson of MLACs from Def. 2 gves the expresson of optmal MLACs: t T, l,t = (r + δ ) c (x,t) ( = µe rt r 1 e δ(t t)) + e (r+δ)(t t) (r + δ ) c (δ ā ) (13) Corollary 1. In general, optmal MLACs are dfferent n dfferent sectors. Proof. We use a proof by contradcton. Let two sectors be such that they exhbt the same nvestment cost functon, the same deprecaton rate, but dfferent abatng potentals: x >, c 1(x) = c 2(x), δ 1 = δ 2, ā 1 ā 2 Suppose that the two sectors take the same tme to decarbonze (.e. T 1 = T 2 ). Optmal MICs would then be equal n both sectors (1). Ths would lead to equal nvestments, hence equal abatement, n both sectors at any tme (1,2), and n partcular to a 1 (T 1 ) = a 2 (T 2 ). By assumpton, ths last equalty s not possble, as: a 1 (T 1 ) = ā 1 ā 2 = a 2 (T 2 ) In concluson, dfferent potentals ā have to lead to dfferent optmal decarbonzng dates T, and therefore to dfferent optmal MLACs l,t. A smlar reasonng can be done concernng two sectors wth the same nvestment cost functons, same potentals, but dfferent deprecaton rates; or two sectors that dffer only by ther nvestment cost functons. Ths fndng does not necessarly mply that mtgatng clmate change requres other sectoral polces than those targeted at nternalzng learnng spllovers. Well-tred arguments plead n favor of usng few nstruments (Tnbergen, 1956; Laffont, 1999). In our case, a unque carbon prce may nduce dfferent MLACs n dfferent sectors. 5. Margnal mplct rental cost of captal (MIRCC) The result from the prevous secton may seem to contradct the equmargnal prncple: two smlar goods, abatement n two dfferent sectors, appear to have dfferent prces. In fact, nvestment n low-carbon captal produce dfferent goods n dfferent sectors because they have two effects: avodng GHG emssons and removng an opton to nvest later n the same sector (secton 3). Here we consder an nvestment strategy that ncreases abatement n a sector at one date whle keepng the rest of the abatement trajectory unchanged. It consequently leaves unchanged any opportunty to nvest later n the same sector. From an exstng nvestment pathway (x,t ) leadng to an abatement pathway (a,t ), the socal planner may ncrease nvestment by one unt at tme θ and 11

12 Fgure 5: From a gven nvestment pathway (x,t) leadng to the abatement pathway (a,t), savng one more unt of GHG at a date θ wthout changng the rest of the abatement pathway, as n (ã,t), requres to nvest one more unt at θ and (1 δ dθ) less at θ + dθ, as ( x,t) does. mmedately reduce nvestment by 1 δ dθ at the next perod θ + dθ. The resultng nvestment schedule ( x,t ) leads to an abatement pathway (ã,t ) that abates one supplementary unt of GHG between θ and θ + dθ (Fg. 5). Movng from (x,t ) to ( x,t ) costs: P = 1 [ c (x,θ ) (1 δ ] dθ) dθ (1 + r dθ) c (x,θ+ dθ ) (14) For margnal tme lapses, ths tends to : P dθ (r + δ ) c (x,θ ) dc (x,θ ) dθ (15) Defnton 3. We call margnal mplct rental cost of captal (MIRCC) n sector at a date t, denoted p,t the followng value: p,t = (r + δ ) c (x,t ) dc (x,t ) dt (16) Ths defnton extends the concept of the mplct rental cost of captal to the case where nvestment costs are an endogenous functons of the nvestment pace. 11 It corresponds to the market rental prce of low-carbon captal n a compettve equlbrum, and ensures that there are no proftable trade-offs between: () lendng at a rate r; and () nvestng at tme t n one unt of captal at cost c (x,t ), rentng ths unt durng a small tme lapse dt, and resellng 1 δdt unts at the prce c (x,t+dt) at the next tme perod. Proposton 4. In each sector, before the date T, the optmal mplct margnal rental cost of captal equals the current carbon prce:, t T, p,t = (r + δ ) c (x,t) dc (x,t ) dt = µe rt (17) 11 We defned margnal rental costs. Ths dffers from the proposal by Jorgenson (1963, p. 143), where nvestment costs are lnear, and no dstncton needs to be done between average and margnal costs (Appendx E). 12

13 Proof. In Appendx C we show that ths relaton s a consequence of the frst order condtons. Equaton 17 also gves a suffcent condton for the margnal levelzed abatement costs (MLACs from Def. 2) to be equal across sectors to the carbon prce: ths happens when margnal nvestment costs are constant along the optmal path: dc (x,t )/dt =. When ths condton s satsfed, MLACs can be used labeled as MACs, and should be equal across sectors to the carbon prce. However, f nvestment costs are convex functons of the nvestment pace, margnal nvestment costs change n tme and MLACs dffer from MIRCCs (9,1). Proposton 5. Equalzng MIRCC to the socal cost of carbon (SCC) s not a suffcent condton to reach the optmal nvestment pathway. Proof. Equatons (16) and (17) defne a dfferental equaton that c (x,t ) satsfes when the IRRC are equalzed to the SCC. Ths dfferental equaton has an nfnty of solutons (those lsted by equaton B.6 n the appendx). In other words, many dfferent nvestment pathways lead to equalze MIRCC and the SCC. Only one of these pathways leads to the optmal outcome; t can be selected usng the fact that at T, abatement n each sector has to reach ts maxmum potental (the boundary condton used from B.7). The cost-effcency of nvestments s therefore more complex to assess when nvestment costs are endogenous than when they are exogenous. In the latter case, as Jorgenson (1967, p. 145) emphaszed: It s very mportant to note that the condtons determnng the values [of nvestment n captal] to be chosen by the frm [...] depend only on prces, the rate of nterest, and the rate of change of the prce of captal goods for the current perod. 12 In other words, when nvestment costs are exogenous, current prce sgnals contan all the nformaton that prvate agents need to take socally-optmal decsons. In contrast, n our case wth endogenous nvestment costs and maxmum abatng potentals the nformaton contaned n prces should be complemented wth the correct expectaton of the date T when the sector s entrely decarbonzed. 6. Dscusson and concluson We used three metrcs to assess the socal value of nvestments n low-carbon captal made to decarbonze the sectors of an economy: the margnal nvestment costs (MIC), the margnal levelzed abatement cost (MLAC), and the margnal mplct rental cost of captal (MIRCC). We fnd that along the optmal path, the margnal nvestment costs and the margnal levelzed abatement costs dffer from the carbon prce, and dffer across sectors. Ths may brng strong polcy mplcatons, as agences use levelzed abatement costs labeled as margnal abatement cost (MAC), and exstng sectoral polces are often crtczed because they set dfferent MACs (or dfferent carbon prces) n dfferent sectors. Our results show that levelzed abatement 12 In our model, these correspond respectvely to the current prce of carbon µe rt, the dscount rate r, and the endogenous current change of MIC dc (x,t )/dt. 13

14 costs should be equal across sectors only f the costs of nvestments n low-captal do not depend on the date or the pace at whch nvestments are mplemented. In the optmal pathway, the margnal mplct rental cost of captal (MIRCC) equals the current carbon prce n every sector that has not fnshed ts decarbonzng process. In other words, f abatement costs are defned as the mplct rental cost of the low-carbon captal requred to abate, MACs should be equal across sectors. Ths fndng requred to extend the concept of mplct rental cost of captal to the case of endogenous nvestment costs at our best knowledge t had only been used wth exogenous nvestment costs. A theoretcal contrbuton s to show that when nvestment costs are endogenous and the captal has a maxmum producton, equalzng the MIRCC to the current prce of the output (the carbon prce n ths applcaton) s not a suffcent condton to reach the Pareto optmum. In other words, current prces do not contan all the nformaton requred to decentralze the socal optmum; they must be combned wth knowledge of the date when the captal reaches ts maxmum producton. The bottom lne s that two apparently opposte vews are reconcled: on the one hand, hgher efforts are actually justfed n the specfc sectors that wll take longer to decarbonze, such as urban plannng and the transportaton system; on the other hand, the equmargnal prncple remans vald, but apples to an accountng value: the mplct rental cost of the low-carbon captal used to abate. Our analyss does not ncorporate any uncertanty, mperfect foresght or ncomplete or asymmetrc nformaton. We also dsregarded nduced techncal change, known to mpact the optmal tmng and cost of GHG abatement; and growng abatng potentals, a key factor n developng countres. A program for further research s to nvestgate the combned effect of these factors n the framework of low-carbon captal accumulaton. Acknowledgments We thank Alan Ayong Le Kama, Maranne Fay, Mchael Hanemann, Jean Charles Hourcade, Chrstophe de Gouvello, Lonel Ragot, Jule Rozenberg, semnar partcpants at Cred, Unversté Pars Ouest, Pars Sorbonne, Prnceton Unversty s Woodrow Wlson School, and at the Jont World Bank Internatonal Monetary Fund Semnar on Envronment and Energy Topcs, and the audence at the Conference on Prcng Clmate Rsk held at Center for Envronmental Economcs and Sustanablty Polcy of Arzona State Unversty for useful comments and suggestons. The remanng errors are the authors responsblty. We are grateful to Patrce Dumas for techncal support. We acknowledge fnancal support from the Sustanable Moblty Insttute (Renault and ParsTech). The vews expressed n ths paper are the sole responsblty of the authors. They do not necessarly reflect the vews of the World Bank, ts executve drectors, or the countres they represent. References Acemoglu, D., Aghon, P., Bursztyn, L., Hemous, D., 212. The envronment and drected techncal change. Amercan Economc Revew 12 (1),

15 Allen, M. R., Frame, D. J., Huntngford, C., Jones, C. D., Lowe, J. A., Menshausen, M., Menshausen, N., 29. Warmng caused by cumulatve carbon emssons towards the trllonth tonne. Nature 458 (7242), Chakravorty, U., Moreaux, M., Tdball, M., 28. Orderng the extracton of pollutng nonrenewable resources. Amercan Economc Revew 98 (3), Fscher, C., Newell, R. G., 28. Envronmental and technology polces for clmate mtgaton. Journal of Envronmental Economcs and Management 55 (2), Gerlagh, R., Kverndokk, S., Rosendahl, K. E., 29. Optmal tmng of clmate change polcy: Interacton between carbon taxes and nnovaton externaltes. Envronmental and Resource Economcs 43 (3), Grmaud, A., Lafforgue, G., 28. Clmate change mtgaton polces : Are R&D subsdes preferable to a carbon tax? Revue d économe poltque 118 (6), Hoel, M., Should a carbon tax be dfferentated across sectors? Journal of Publc Economcs 59 (1), IPCC, 27. Summary for polcymakers. In: Clmate change 27: Mtgaton. Contrbuton of workng group III to the fourth assessment report of the ntergovernmental panel on clmate change. Cambrdge Unversty Press, Cambrdge, UK and New York, USA. Jaccard, M., Rvers, N., 27. Heterogeneous captal stocks and the optmal tmng for CO2 abatement. Resource and Energy Economcs 29 (1), Jaffe, A. B., Newell, R. G., Stavns, R. N., 25. A tale of two market falures: Technology and envronmental polcy. Ecologcal Economcs 54 (2-3), Jorgenson, D., The theory of nvestment behavor. In: Determnants of nvestment behavor. NBER. Kemp, M. C., Long, N. V., 198. On two folk theorems concernng the extracton of exhaustble resources. Econometrca 48 (3), Laffont, J.-J., Poltcal economy, nformaton, and ncentves. European Economc Revew 43, Lecocq, F., Hourcade, J., Ha-Duong, M., Decson makng under uncertanty and nerta constrants: sectoral mplcatons of the when flexblty. Energy Economcs 2 (5-6), Lpsey, R. G., Lancaster, K., The general theory of second best. The Revew of Economc Studes 24 (1), Matthews, H. D., Gllett, N. P., Stott, P. A., Zckfeld, K., 29. The proportonalty of global warmng to cumulatve carbon emssons. Nature 459 (7248),

16 Rchter, W. F., Schneder, K., 23. Energy taxaton: Reasons for dscrmnatng n favor of the producton sector. European Economc Revew 47 (3), Rosendahl, K. E., Nov. 24. Cost-effectve envronmental polcy: mplcatons of nduced technologcal change. Journal of Envronmental Economcs and Management 48 (3), Smulders, S., Van Der Werf, E., 28. Clmate polcy and the optmal extracton of hgh- and low-carbon fossl fuels. Canadan Journal of Economcs/Revue canadenne d économque 41 (4), Tnbergen, J., Economc polcy: prncples and desgn. North Holland Pub. Co., Amsterdam. van der Ploeg, F., Wthagen, C., 212. Too much coal, too lttle ol. Journal of Publc Economcs 96 (1 2), Vogt-Schlb, A., Hallegatte, S., 211. When startng wth the most expensve opton makes sense: Use and msuse of margnal abatement cost curves. Polcy Research Workng Paper 583, World Bank, Washngton DC, USA. Vogt-Schlb, A., Meuner, G., Hallegatte, S., 212. How nerta and lmted potentals affect the tmng of sectoral abatements n optmal clmate polcy. Polcy Research Workng Paper 6154, World Bank, Washngton DC, USA. Appendx A. Proof of lemma 1 Proof. As c s strctly growng, t s nvertble. Let χ be the nverse of c ; applyng χ to (9) gves: { χ (e rt (T t)µ) f t < T x,t = (A.1) f t T The relaton between the sectoral potental (ā ), the MICs (through χ ), the SCC (µ) and the tme t takes to acheve the sectoral potental T reads: ā = a (T ) Let us defne f χ = such that: T χ ( e rt (T t)µ ) dt f χ (t) = = df χ dt (t) = t t χ ( e rθ (t θ)µ ) dθ e rθ χ ( e rθ (t θ)µ ) dθ Let us show that f χ s nvertble: χ > as the nverse of c >, thus dfχ dt > and therefore f χ s strctly growng. Fnally: ā T = f χ 1 (ā ) s an ncreasng functon 16

17 When the margnal cost functon c s gven, χ and therefore f χ are also gven. Therefore, T can always be found from ā. The larger the potental, the longer t takes for the optmal strategy to acheve t. Appendx B. Proof of proposton 2 Lagrangan The Lagrangan assocated wth (4) reads: L(x, a, ȧ, λ, ν, µ) = e rt c (x,t ) dt + λ,t (a,t ā ) dt ( ) + µ (ā a,t ) dt B + ν,t (ȧ,t x,t + δ a,t ) dt In the last term, ȧ,t can be removed thanks to an ntegraton by parts: t ν,t (ȧ,t x,t + δ a,t ) dt = = ( ν,t ȧ,t dt + ( constant ) ν,t (δ a,t x,t ) dt ν,t a,t dt + The transformed Lagrangan does not depend on ȧ,t : L(x,t, a,t, λ, ν, µ) = (B.1) ) ν,t (δ a,t x,t ) dt e rt c (x,t ) dt + λ,t (a,t ā ) dt ( ) + µ (ā a,t ) dt B ν,t a,t dt + ν,t (δ a,t x,t ) dt (B.2) Frst order condtons The frst order condtons read: 13 L (, t), = ν,t δ ν,t = λ,t µ a,t (, t), L x,t = e rt c (x,t ) = ν,t (B.3) (B.4) 13 The same condtons may be wrtten usng a Hamltonan. 17

18 Slackness condton For each sector there s a date T such that (slackness condton): Before T, (B.3) smplfes: t < T, a,t < ā & λ,t = (B.5) t T, a,t = ā & λ,t t T, ν (t) δ ν,t = µ = ν,t = V e δt + µ δ Where V s a constant that wll be determned later. The MICs are the same quanttes expressed n current value (B.4): [ t T, c (x,t ) = e rt V e δt + µ ] (B.6) δ ] Any V chosen such that [V e δt + µδ remans postve defnes an nvestment pathway that satsfes the frst order condtons. The optmal nvestment pathways also satsfes the followng boundary condtons. Boundary condtons At the date T, a,t s constant and the nvestment x,t s used to counterbalance the deprecaton of abatng captal. Ths allows to compute V : x (T ) = δ ā = e rt c (δ ā ) = V e δ T + µ (from eq. B.6) δ ( = V = e δt e rt c (δ ā ) µ ) δ (B.7) Optmal margnal nvestment costs (MICs) Usng ths expresson n (B.6) gves: t T, c (x,t ) = e rt [ e δ T ( e rt c (δ ā ) µ δ ) e δt + µ δ ] Smplfyng ths expresson allows to express the optmal margnal nvestment costs n each sector as a functon of δ, ā, µ and T : c (x rt 1 e δ(t t),t) = µe + e (δ+r)(t t) c (δ ā ) (B.8) δ After T, the MICs n sector are smply constant to c (δ ā ). 18

19 Appendx C. Proof of proposton 4 The frst order condtons can be rearranged. Startng from (B.4): c (x,t ) = e rt ν,t = dc (x,t ) dt (B.4) = e rt ( ν,t + r ν,t ) (C.1) = e rt (δ ν,t µ + r ν,t ) (from B.3 and B.5) (C.2) = (r + δ)c (x,t ) µe rt (from B.4) (C.3) = µe rt = (r + δ ) c (x,t ) dc (x,t ) dt (C.4) Substtutng n the defnton of the mplct margnal rental cost of captal p,t (16) leads to p,t = µe rt. The solutons of (C.4), where the varable s c (x,t ), are those lsted n (B.6). Appendx D. Proof of the expresson of l,t n Def. 2 Let h be a margnal physcal nvestment n low-carbon captal made at tme t n sector (expressed n tco 2 /yr per year). It generates an nfntesmal abatement flux that starts at h at tme t and decreases exponentally at rate δ. For any after that, leadng to the total dscounted abatement A (expressed n tco 2 ): A = θ=t = h r + δ e r (θ t) h e δ(θ t) dθ (D.1) (D.2) Ths addtonal nvestment h brngs current nvestment from x,t to (x,t + h). The addtonal cost C (expressed n $) that t brngs reads: C = c (x,t + h) c (x,t ) = h h c (x,t ) (D.3) The MLAC l,t s the dvson of the addtonal cost by the addtonal abatement t allows: l,t = C A l,t = (r + δ ) c (x,t ) (D.4) (D.5) Appendx E. Levelzed costs and mplct rental cost when nvestment costs are exogenous and lnear Let I t be the amount of nvestments made at exogenous untary cost Q t to accumulate captal K t that deprecates at rate δ: K t = I t δ K t (E.1) 19

20 Let F (K t ) be a classcal producton functon (where the prce of output s normalzed to 1). Jorgenson (1967) defnes current recepts R t as the actual cash flow: R t = F (K t ) Q t I t (E.2) he fnds that the soluton of the maxmzaton program max I t e rt R t dt (E.3) does not equalze the margnal productvty of captal to the nvestment costs Q t : F K (K t ) = (r + δ) Q t Q t (E.4) He defnes the mplct rental cost of captal C t, as the accountng value: C t = (r + δ) Q t Q t (E.5) such that the soluton of the maxmzaton program s to equalze the margnal productvty of captal and the rental cost of captal: F K (K t ) = C t (E.6) He shows that ths s consstent wth maxmzng dscounted economc profts, where the current proft s gven by the accountng rule: Π t = F (K t ) C t K t (E.7) In ths case, the (untary) levelzed cost of captal L t s gven by: L t = (r + δ) Q t (E.8) And the levelzed cost of captal matches the optmal rental cost of captal f and only f nvestment costs are constant: Q t = F K (K t ) = L t (E.9) 2

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