Global Carbon Finance

Transcription

1 Global Carbon Fnance Offce of Clmate Change Global Carbon Fnance A uanttatve modellng framework to explore scenaros of the Global Deal on Clmate Change Federco Gallo, Alon Carmel, Stephen Prchard, Jenne Rayson, Nat Martn, Robert Dxon and Alexander MacDowall Offce of Clmate Change, DECC. Aprl, 2009

2 Global Carbon Fnance Offce of Clmate Change Abstract. The Kyoto Protocol s set to expre n 202 and there s now urgency to reach a new nternatonal agreement to replace t f the world s to avod dangerous clmate change. Negotatons are under way, and an nternatonal framework to tackle global warmng s expected to be agreed at the end of 2009 n Copenhagen under the UNFCCC. To acheve ths, the world wll need excellent modellng capabltes to estmate the costs of mtgaton. Currently there are a number of numercal carbon fnance models, such as GLOCAF and FAIR. However, such models reure sgnfcant effort to develop and mantan, and most governments do not own ther own models. One problem s that, as we enter the fnal negotaton stages, governments wll need to rely and trust second hand estmates of mtgaton costs from a small number of models held n a few developed countres. Ths nformaton asymmetry may lead to lack of trust and ultmately adversely affect the negotatons. Ths paper attempts to address ths ssue. The purpose of ths paper s to help Governments and research nsttutons kckstart and accelerate the development of ther n-house modellng capabltes to analyse the shape of the Global Deal on Clmate Change. More concretely, the model offers the opportunty to readly obtan estmates of costs and fnancal flows, as well as other uanttes of nterest. The paper presents a set of formulae that could be readly used to set up a smple spreadsheet model to analyse the clmate change negotatons. Moreover, the paper also descrbes the underlyng modellng framework, whch can be used as a platform to desgn and analyse new scenaros and generally to develop more complex models of the global deal on clmate change. The valdty of the model results s tested aganst state-of-the-art numercal models (GLOCAF and FAIR), and s found to be n excellent agreement. Moreover, the mathematcal formulaton presented n ths paper was used to develop the UK s Global Carbon Fnance model, GLOCAF. Applcatons of ths model have so far been presented to the EU Commsson, Number 0 as well as to the UNFCCC conference n Bal. Furthermore, the model s also beng used to facltate techncal dscussons between the UK, Inda, Brazl, Mexco, Argentna and Colomba. It s hoped that ths model wll help to accelerate the process of reachng an nternatonal agreement on clmate change. 2

3 Global Carbon Fnance Offce of Clmate Change 3 Summary of key formulae For uck reference, the boxes below summarse the key euatons from the carbon fnance model. These are derved formally and fully explaned n the followng pages. Box : Key formulae (free trade) Eulbrum carbon prce: Q R p e Global Net costs [$bn]: Q R C C G Regonal Domestc abatement [GtCO 2 e]: Q R d / Net costs [$bn]: I e N C D p C Fnancal flows [$bn]: D p F e F Carbon flows [GtCO 2 e]: d C F Box 2: Related constants (free trade) Domestc credts [$bn]: e d p D Incremental costs [$bn]: / Q R C I Abatement targets [GtCO 2 e]: Q R /

4 Global Carbon Fnance Offce of Clmate Change. Introducton Scentfc research and poltcal negotatons on clmate change have been gong on for decades. However n the past couple of years the debate has been gatherng momentum thanks to a number of factors. Supported by growng evdence, the scentfc communty has reached an almost unanmous consensus that clmate change s real and s happenng at a faster rate than prevously predcted. Also, awareness among the general populaton has grown consderably, wth the ssue now appearng regularly on the front pages of natonal newspapers, and the meda constantly presentng new scentfc evdence. A seres of events have also contrbuted to puttng clmate change at the top of the nternatonal poltcal agenda. In 2006 the Stern Revew provded the frst systematc economc assessment of the potental conseuences of nacton and the costs of mtgaton (Stern, 2006). The 2007 Noble Peace Prze gave further mpetus to the poltcal debate. Also busness s becomng an actve voce; for example, 50 of the bggest companes n the world demanded an nternatonal agreement on clmate change n November The UN conference n Bal n December has ntated negotaton talks between the world leaders and set up a Road Map to desgn the post-kyoto framework. The G8 summts n both Germany (2007) and Japan (2008) concluded that clmate change s a real threat and that the ndustralsed natons are commtted to cuttng greenhouse gas emssons sgnfcantly n the near to md-term. Now countres are enterng full negotaton mode, and s expected that by the end of 2009 a poltcal agreement should have been reached n Copenhagen. However, before countres commt to legally bndng targets, they wll need to understand the fnancal costs nvolved. Therefore, excellent modellng capabltes wll need to be avalable to nform decson-makers. The UK s GLOCAF model s one of the tools capable of uantfyng the costs and fnancal flows nvolved n many negotaton scenaros. An alternatve model s MNP s FAIR. One weakness of these models s that they are complex computer smulatons that reure sgnfcant resources to develop and mantan. Moreover, these models can be very data hungry, and to both obtan and update the relevant data can be both tme consumng and expensve. Because of ths, currently many developng countres do not have n-house carbon fnance models, and nstead often rely on data from foregn nsttutons to desgn ther negotaton strategy. Ths nformaton asymmetry may affect levels of trust and have a negatve mpact on the negotaton process. The purpose of ths paper s to address ths ssue by presentng a mathematcal formulaton that would allow Governments and research nsttutons that do not yet possess these capabltes to develop ther n-house model to descrbe and analyse scenaros of the Global Deal on Clmate Change. Startng from greenhouse gas emsson targets and margnal costs of abatement, ths paper develops a set of smple formulae that descrbe a global carbon fnance market. The formulae are derved n two dfferent ways: by ether assumng mnmsaton of abatement costs or by matchng demand and supply of the carbon 4

5 Global Carbon Fnance Offce of Clmate Change credts. Both methods are shown to be euvalent. The model outputs are uanttes of nterest such as nternatonal fnancal flows and net costs ncurred by ndvdual countres. It s hoped that ths mathematcal formulaton s more transparent and concse, n the sense that t does not reure complex numercal smulatons. Also, the data reurements are substantally reduced, thus constranng the reured nvestment n terms of tme and money. Secton 2 presents the assumptons underlyng the model: emsson targets and margnal abatement costs. Sectons 3, 4 and 5 present the results of the model. In partcular, Secton 3 dscusses the eulbrum carbon prce and the domestc abatement. Secton 4 looks nto the country specfc results, whle Secton 5 presents the global costs. Secton 6 concludes and suggests further research. A number of techncal appendces explan the formal dervatons of the model and llustrate some practcal applcatons that may be used to support the nternatonal negotatons as well as the desgn of the global carbon market. 2. Assumptons: global emssons target and margnal abatement costs It s of nterest to obtan an estmate of the costs that would be ncurred by countres as they reduce ther greenhouse gas emssons. If an nternatonal carbon market s set up, countres wll also be nterested n the resultng fnancal flows. In partcular, ndvdual countres wll need to be confdent n ther numbers as they jon the negotaton table. In order to obtan an estmate of these uanttes we need three peces of nformaton (Ellerman and Decaux, 998):. the greenhouse gas emssons targets, 2. the margnal costs of abatement, and 3. the mechansm by whch the abatement s acheved: here we consder an nternatonal carbon market wth ratonal actors. By ratonal actors we mean utlty-maxmsng economc actors as defned n mcroeconomc theory. Ths has some lmtatons, for example n the case of energy effcency where there may be barrers such as splt ncentves whereby the landlord does not nsulate the property because he wll not gan from energy effcency savngs, whle the tenants wll not beneft from the hgher prce of the property. 5

6 Global Carbon Fnance Offce of Clmate Change 2. Greenhouse abatement targets The global level of abatement, Q (measured n GtCO 2 e), may be nternatonally agreed and should be nformed by the latest scence as to avod dangerous anthropogenc clmate change. Ths global target s then shared among countres accordng to a burden sharng scheme: each country s reured to accept a natonal abatement target (also measured n GtCO 2 e) so that the global target s met 2. The natonal targets should add up to the global target: Q () 2.2 Margnal costs of abatement The costs of abatement may be descrbed by Margnal Abatement Cost (MAC) curves. These may be dsaggregate at regonal (natonal) or sectoral level, examples of whch are the steel and cement ndustres. MAC curves may be obtaned n a number of ways; for example, from general eulbrum models, energy models or from an nventory of abatement optons, such as energy effcency or the ntroducton of green technologes (Kuk et al, 2008). In general the data may be approxmated by a functon p =f( ), where p s the margnal abatement cost (or euvalent carbon prce) and f( ) s an arbtrary functon of abatement,. The uantty p measured n US$/tCO2e. For a gven country, ths euaton represents the amount of carbon that wll be economcally vable to abate at a gven carbon prce p. GLOCAF uses emprcal data to specfy the euaton. It s found that the followng famly of functons can accurately approxmate the data (see Appendx II): p (2) The values of the parameters are obtaned from the data through standard statstcal estmaton e.g. least suare ft and are specfc for each MAC curve. (An ndvdual MAC curve could be global, regonal or sectoral, dependng on the underlyng data.) The parameter, on the other hand, s common to all curves and s also estmated emprcally. See Appendx II for further data consderatons. 2 In practse, n some cases the targets may be set eual to zero (e.g. developng countres partcpatng n CDM schemes). 6

7 Global Carbon Fnance Offce of Clmate Change Fgure : Dfferent models produce substantally dfferent estmates of margnal abatement cost curves. Source: Fscher and Morgenstern, Euatons and 2 are very smple but powerful. These models are subject to sgnfcant levels of uncertanty: for example, Fgure shows how estmates of margnal abatement costs vary sgnfcantly dependng on the approach used to generate them (Fscher and Morgenstern, 2005; Kuk et al, 2008). The formulaton of the present global carbon fnance model s such that allows us to use dfferent underlyng datasets and explore the senstvty to them. Moreover, the uncertanty can be ncorporated nto the model to obtan a probablstc verson: ths would allow us to obtan ranges rather than exact estmates of the nternatonal costs and fnancal flows. Although turnng the determnstc model nto a probablstc one s relatvely straghtforward, as major constrant s the data avalablty: currently there are no standard probablstc MAC curves, and producng them would reure a number of ad-hoc consderatons. Euatons and 2 also help us focus on the sources as well as the mplcatons of these underlyng uncertantes. 2.3 The carbon market Each country could be responsble for meetng ts entre abatement target domestcally. However, t s a well known economc fact that the total costs can be consderably reduced f countres are allowed to buy carbon allowances from countres where abatement s cheaper (Stern, 2007). Indeed, the Kyoto Protocol allows for such an nternatonal carbon market. Moreover, a number of such markets already exst: the EU s Emsson Tradng Scheme s currently the largest, but other are beng set up, for example n Australa and the Unted States. 7

8 Global Carbon Fnance Offce of Clmate Change In Appendx I, carbon tradng rules 3 are appled to the above margnal abatement costs and emsson abatement targets to derve explct formulae that descrbe the nternatonal carbon market; these nclude, for example, the global costs, regonal costs, fnancal flows and can be easly manpulated to deduce further uanttes of nterest. The formulae can be derved by two dfferent methods: both by applyng cost mnmsaton condtons or by matchng demand and supply of carbon credts. The followng sectons descrbe these uanttes n more detal. 3. Key results: eulbrum carbon prce (p e ) and domestc abatement ( d ) Ths secton presents the two key results of the carbon market model: the eulbrum carbon prce and the level of domestc abatement. These are specal uanttes because all the other model outputs can be derved from them through algebrac manpulaton. 3. The eulbrum carbon prce The exchange of carbon credts n the nternatonal market would gve rse, by matchng demand and supply, to an eulbrum carbon prce or market clearng prce, p e (measured n US$/tCO 2 e). Ths s the frst key result of the model, and s gven by: p e R Q (3) The constant R s gven by: R / (4) As can be seen, R depends only on the margnal abatement costs of ndvdual countres, as measured by the emprcal parameters and. Euaton 3 offers a number of mportant nsghts: 3 The carbon tradng rules are based on classcal economc theory, whch assumes perfectly ratonal behavour. 8

9 Global Carbon Fnance Offce of Clmate Change. Frst of all, t shows that the eulbrum carbon prce depends only on the global emssons abatement target, Q; n other words t does not matter how ths target s allocated among natons. Ths s consstent wth economc theory It s also nterestng to compare Euaton 3 wth Euaton 2: ths allows us to see that the eulbrum carbon prce corresponds to the global margnal abatement cost of meetng the global target Q Another nsght s that, snce the ncremental cost s the ntegral of the margnal abatement cost (Euaton 3), t should follow that the global R ncremental cost s gven by: C C G Q. Ths s ndeed the case, and wll be confrmed below The domestc abatement Under free trade, the amount of abatement that s carred out n each country s ndependent of the country s abatement target: countres wll abate up to the pont where the margnal cost of abatement s eual to the carbon prce. Ths s because when abatement costs are hgher than the market carbon prce, t would be more convenent to buy credts from the market. On the other hand, f the natonal target s exceeded, the country wll be able to sell carbon credts at a hgher prce than they spent on abatement to create them. The second key result of the model s an explct formula for the level of domestc abatement, d (measured n GtCO 2 e), that s economcally optmal to abate n each country. Ths s done by rearrangng Euaton 2 n terms of, and replacng the carbon prce wth Euaton 3: 4 It s mportant to note that ths statement s vald only n the case of free trade. Restrctons to trade would result n hgher costs. 5 As a matter of fact, the parameter R can ether be obtaned from the through Euaton 4, or t can be estmated emprcally drectly from the global margnal abatement cost curve (Euaton 3). The values obtaned from the two approaches would dffer slghtly. Ths s because of the way the statstcal errors enter n the parameter estmatons: f the R s estmated drectly from the global cost curve, the assocated error wll result from the standard least suares estmaton method. On the other hand, s the R s obtaned from the above formula, ts value would ncorporate all the errors assocated wth the ndvdual. In some cases these errors may cancel out, however n other cases these may add up. Therefore attenton should be pad to these ssues durng the estmaton of the parameters. In general, f we are nterested n global costs, t may be more accurate to estmate R from the global cost curve. On the other hand, f we are nterested n regonal costs and flows, t would be necessary to obtan R from the regonal, as ths would ensure that total fnancal flows add up to zero. 9

10 Global Carbon Fnance Offce of Clmate Change d / R (5) Q Ths corresponds to the level of abatement that each country wll undertake domestcally under free trade. In other words, under a completely free market each country wll reduce emssons domestcally untl the domestc margnal abatement cost matches the global carbon prce. Each country wll then be able to buy or sell carbon credts n the nternatonal carbon market to meet ts agreed abatement target. Countres wth hgh domestc margnal abatement costs wll be able to purchase carbon credts at lower costs on the market, whle countres wth cheap abatement costs wll be able to abate more than ther target and then sell the credts at a hgher prce, thus makng a proft. Euaton 5 shows explctly how the level of domestc abatement s ndependent of the country s abatement target (.e. the abatement target,, does not appear n the formula). On the other hand, the domestc abatement s a fxed fracton of the global abatement target. Ths proporton wll be larger n those countres where the abatement costs are smaller (.e. smaller ). In other words, countres wth cheaper abatement costs wll abate more domestcally, as we would expect. The next two sectons present a number of regonal and global uanttes of nterest: these are obtaned through algebrac manpulatons based on the key formulae presented above. 4. Regonal costs and flows Negotatng a burden sharng agreement s a phenomenal task due to the large number of partes nvolved, ther dfferent geopoltcal weghts and the wdely dvergng natonal nterests and crcumstances. Moreover, dfferent levels of economc development and hstorc emssons, as well as ethcal ssues add to the complexty of the problem. Before any country commts to a mandatory emssons target, t wll need to assess and accept the responsbltes that come wth t; for example the rsk of havng to pay a fne f the target s not met. Therefore, countres wll need estmates of the ncremental costs and fnancal flows nvolved. Ths and the next sectons descrbe how these uanttes can be derved. In essence, Euatons 3, 4 and 5 descrbe the global carbon fnance model: by manpulatng them t s possble to defne and calculate smple formulae for country specfc uanttes such as ncremental costs or fnancal flows. Ths secton derves formulae for the most freuently used fnancal uanttes. It s mportant to note, however, that the lst below s by no means exhaustve, and more uanttes can be defned f necessary. 0

11 Global Carbon Fnance Offce of Clmate Change 4. Carbon flows As an ntal example, defne the regonal carbon flows, F C (measured n GtCO 2 e) as the dfference between the country s abatement target,, and the level of domestc abatement, d. F C d / R (6) Q Here postve flows ndcate that the country s buyer of carbon credts, and vce versa. 4.2 Fnancal flows As dscussed above, carbon credts wll be traded at the eulbrum carbon prce, p e. Therefore, t s straghtforward to obtan the fnancal flows, F F (measured n US$bn), by multplyng the carbon credt flows by the eulbrum carbon prce: F F p e F C R / (7) Q R Q We can obtan a more ntutve formula for the fnancal flows by defnng the d domestc credts as: D p. Ths s the market value of the credts generated e through domestc abatement. We are now n a poston to rewrte the fnancal flows as: F F p e D (7a) In words, ths s sayng that the fnancal flows are gven by the dfference n carbon credts that would be needed to meet the natonal target and the credts generated domestcally. The dfference between the two would be ether sold to or bought from the market.

12 Global Carbon Fnance Offce of Clmate Change 4.3 Incremental regonal costs Each country wll ncur ncremental costs, C I (measured n US$bn), due to the amount of carbon abated domestcally. Ths can be obtaned by takng the ntegral of the margnal abatement curve up to the level of domestc abatement, whch s gven by Euaton 5: C I R Q (8) Euaton 8 shows that the ncremental costs for each country are ndependent of the abatement targets. Snce t was already shown that the level of domestc abatement, Euaton 5, s also ndependent of the abatement target, ths s not surprsng. 4.4 Net regonal costs The net costs of each country, C N (measured n US$bn), wll be the sum of the fnancal flows and ncremental costs of the country. They are gven by: C N R / Q R / Q R Q (9) Euaton 9 can be rewrtten n a more ntutve form by substtutng n the formulae for the domestc credts, D, ncremental costs (Euaton 8): C N p e D C I (9a) In words, Euaton 9a s sayng that the net costs are gven by the ncremental costs and the fnancal flows. The above formula shows that the net costs depend on the natonal abatement target, as well as on the natonal and global margnal abatement costs and global abatement target. 4.5 Net regonal costs n the absence of carbon tradng For comparson, t s useful to determne the net costs, K, that each country would experence n the absence of an nternatonal carbon market: 2

13 Global Carbon Fnance Offce of Clmate Change K (0) Ths s obtaned by replacng the term wthn brackets n Euaton 8 by. 5. Global costs and flows The formulae presented n the prevous secton descrbe costs and flows that apply to ndvdual regons. The secton presents a number of global varables. 5. Net global costs The net global costs of abatement are gven by the sum of all the country specfc / ncremental costs (Euaton 8): C C R C G C Q. Ths s measured n US$bn. Smplfyng and substtutng for R, t follows that the global cost of abatement s gven by the smple formula below: C C G R Q () Ths s a smple yet very powerful formula. Once the value of the parameters R and has been determned emprcally, t becomes straghtforward to obtan the global costs of meetng the global abatement target, Q. For example, we can draw any emsson pathway and, gven a BAU trajectory, automatcally get an estmate of the assocated costs by calculatng the global abatement target, Q. Moreover, Euaton allows us to gan more nsght nto the problem. For example, t shows that the ncremental costs ncrease nonlnearly wth abatement. Ths means that a small ncrease n the level of abatement can lead to a large ncrease n the resultng costs. 5.2 Net global costs n the absence of trade For comparson, the global costs (agan measured n US$bn) n the absence of tradng may be calculated n the same way. Ths results n the followng formula: 3

14 Global Carbon Fnance Offce of Clmate Change K G K (2) It s nterestng to note that n ths case the costs are a functon of the abatement target, as ths determnes the amount of abatement done domestcally. 6. Conclusons and suggestons for applcatons and further research We developed a new mathematcal model to descrbe a global carbon tradng market. The model s transparent, and offers new nsghts. For example, we showed that t may be wse to have more open dscussons on the underlyng data, specfcally:. busness as usual projectons, 2. global abatement targets and 3. margnal abatement costs before determnng natonal targets or the characterstcs of the tradng system. The model results were compared wth state-of-the-art numercal models, showng that the model s accurate. Perhaps the most powerful property of ths model s that t produces smple formulae nstead of numercal outputs. These formulae could then be used n other applcatons. 4

15 Global Carbon Fnance Offce of Clmate Change Acknowledgements We would lke to thank Alban Ktous and Patrck Cru from the Unversty of Grenoble for the ENERDATA POLES data underlyng the energy MAC curves, Mchel den Elzen from MNP for the non-co 2 data, Jayant Sathaye from the Internatonal Energy Studes Group Lawrence Berkeley Natonal Laboratory Unversty of Calforna, Berkeley, for the GCOMAP land use data, and Mchael Oberstener from IIASA for the DIMA land use data. References Ellerman AD and Decaux A (998) Analyss of Post-Kyoto CO 2 emssons tradng usng Margnal Abatement Curves. MIT Jont Program on the Scence and Polcy of Global Change. Fscher C and Mongenstern RD (2005) Carbon abatement costs. Why the wde range of estmates? Dscussons paper. Resources for the future. Gallo F, Contucc P, Coutts A and Gallo IA (2008) Tacklng clmate change through energy effcency: mathematcal models to offer evdence-based recommendatons for polcy makers. Quantty and Qualty. Sprnger. Kuk O, Brander L and Tol RSJ (2008) Margnal abatement costs of carbon doxde emssons: a meta-analyss. ESRI Workng paper No Larsky M (2003) The Economc cost of reducng emssons of greenhouse gases: a survey of economc models. Washngton DC: Congressonal Budget Offce. Stern N (2007) The economcs of clmate change. The Stern revew. Cambrdge Unversty Press. Weyant JP and Hll JN (999) Introducton and Overvew. Energy Journal Specal Issue: v-xv. 5

16 Global Carbon Fnance Offce of Clmate Change Appendx I Dervaton of formulae Ths secton formally derves the formulae presented n the man text. The dervatons presented here assume free trade, but these can be generalsed to nclude polcy restrcton to the trade of carbon credts, for example constrants on supplementarty. Assume the exstence of an nternatonal carbon market, as well as country specfc abatement targets and abatement costs. Classcal economc theory suggests that countres wll act n order to mnmse the costs ncurred by ther economes. They can choose how much of ther target to meet through domestc reductons and how much by buyng carbon credts from the nternatonal market: economc theory suggests that demand should match supply. Based on ths, ths secton presents two alternatve dervatons of the carbon tradng formulae:. one by matchng demand and supply and 2. the other by mnmsng costs. The two dervatons produce the same results: ths confrms ther valdty. Matchng demand and supply Ths Secton provdes a smple proof of the eulbrum prce formula by matchng supply and demand. The total abatement target (see Euaton ) s: Q (IV.) where are the regonal targets 6. The margnal abatement cost curve s gven by (see Euaton 2): p (IV.2) / p whch can be rearranged as:. The carbon credt flows are defned as: d (IV.3) 6 The abatement targets, measured n GtCO 2 e, are generally a user nput. However, t s possble to treat them as a dependent varable. Ths would allow us to reverse engneer a soluton: for example t s possble to set a target n terms of fnancal flows or ncremental costs and obtan the correspondng GHG abatement target. 6

17 Global Carbon Fnance Offce of Clmate Change 7 where d s the level of domestc abatement, whch s one of the uanttes that we want to determne. At eulbrum the sum of the flows must be zero: 0 (IV.4) In other words, supply matches demand. Substtutng Euatons IV. and IV.2 nto IV.4, we obtan: p / 0 (IV.5) Economc theory says that at market eulbrum demand wll match supply and an eulbrum prce wll be establshed. We can replace all the p wth an eulbrum prce p e common to all regons. Ths allows us to extract p e from the summaton term: e p / / (IV.6) Now we can solve for the eulbrum prce (see Euaton 4): e Q p / / (IV.7) Defne R / (IV.8) So, as shown n Euaton 3, the eulbrum carbon prce s gven by: Q R p e (IV.9) Ths allows us to fnd the actual domestc abatement under free trade (see Euaton 5): Q R d / (IV.0)

18 Global Carbon Fnance Offce of Clmate Change Mnmsng costs Ths subsecton shows that the same results can be obtaned by assumng that countres trade n order to mnmse ther abatement costs. As before, the margnal abatement costs are gven by the functon p (Euaton IV.2). We can calculate the abatement costs ncurred by ndvdual regons under no tradng (.e. the ncremental costs): c d d d 0 (IV.) If we sum over all countres (assume there are N countres), we can obtan the global costs: N C c (IV.2) We now want to mnmse global costs subject to the constrant mposed by the global abatement target, Q: Mn : C d s. t.: N d Q (IV.2) Where d are the amounts to be abated domestcally by ndvdual countres under free trade: countres wll abate untl the margnal cost n ther MAC curve euals the eulbrum carbon prce. The Lagrangan n ths case s gven by: L N N d d Q (IV.3) The eulbrum condtons are gven by: L d 0 N L d Q 0 (IV.4) 8

19 Global Carbon Fnance Offce of Clmate Change 9 The second order dervatve s postve, confrmng that ths s ndeed a mnmum (.e. mnmum cost): L (IV.5) The solutons to the system IV.4 s gven by: / Q (IV.6) Where corresponds to the eulbrum carbon prce p e. Snce Soluton IV.6 s the same as solutons IV.7 and IV.0, we can conclude that the carbon tradng model obeys both costs mnmsaton and matches supply and demand. Moreover, snce the solutons were obtaned from two ndependent methods, we can nfer that they are correct.

20 Carbon prce ($/tc) Global Carbon Fnance Offce of Clmate Change Appendx II Data consderatons It was explaned earler emprcal data s needed to estmate the MAC functons (Euatons 2). Here we use POLES data for fossl fuel emssons, GCOMAP data for deforestaton and MNP data for non-co 2 greenhouse gases (Please refer to the acknowledgements sectons for more detals on the sources of the underlyng data). Together, these amount to about 98% of greenhouse gas emssons. These sets of data, can be parametersed by =2.5 (ndvdual regons would have ndependent ). Fgure shows the ft of the functon to the data. The dagram shows global margnal abatement costs for a number of dfferent years. It can be seen that the ft s accurate and very satsfactory. The functon s generally accurate to wthn 5%, and often better than ths. Although the examples shown below apply to global abatement costs only, Euaton 2 s very general, and can be specfed to specfc sectors as well: for example, the model presented n ths paper could be used to descrbe a global carbon market that ncludes the power sector only. Consderng that dfferent approaches result n margnal abatement cost estmates that vary by a factor of fve or more (Fscher and Morgenstern, 2005), the approxmaton error ntroduced by Euaton 2 can be consdered to be acceptable. Moreover, the purpose of ths model s not to make exact predctons of future costs and carbon prces: the purpose s to have an nternally consstent model that allows us to compare, n a systematc way, dfferent scenaros of the global carbon market. Global Margnal Abatement Costs (Lnear) Data Model Abatement (GtC) Fgure 2: The dagram shows the ft of the model to the margnal abatement cost data. Although Euaton 2 s a very basc formula, the graph shows that t can accurately reproduce the data. In order not to dsclose senstve data, t was decded not to nclude the cost values. 20

21 Global Carbon Fnance Offce of Clmate Change It s mportant to note that more accurate fts are possble. Here, however, we have decded to use a very smple functon. There are a number of advantages to ths. Frst, t s convenent and concse to specfy each margnal abatement cost curve by a sngle parameter,.e.. Second ths partcular functon allows us to obtan explct formulae for all the economc uanttes of nterest, for example the eulbrum carbon prce or the ncremental costs ncurred by countres. On the other hand, for example, a functon of the form p=+ 2 would lead to mplct euatons 7. Ths s not an mprovement: a small gan n precson comes at a sgnfcant loss n flexblty as the solutons would need to be obtaned numercally. It s mportant to stress the fact that the man purpose of ths model s to retan the essence of the problem whle gettng rd of as much unnecessary complexty as possble. If we are nterested n more numercal precson, we can run the full numercal model (GLOCAF). On the other hand, ths analytcal verson of the model retans all the propertes of the numercal model but gans n flexblty, whle producng results that are very accurate. Moreover, the key strength of the model s n the nsghts t offers nto the problem. II. Numercal valdaton Ths secton compares the results from the analytcal model wth those obtaned from two state-of-the-art numercal models: GLOCAF and FAIR. Fgure 3 and Fgure 4 show that the results from the smple analytc model are n agreement wth the numercal models, and that the analytc model s therefore valdated. The scenaros presented correspond to free trade. In ths example, the emssons pathway leads to a stablsaton concentraton of 550ppm and the burden s shared accordng to Contracton and Convergence. 7 An example of mplct euaton s x=e x. An exact soluton to ths euaton does not exst, and the results need to be approxmated numercally. In practse ths complcates the mplementaton of the model: for example t would not be possble to set up a smple spreadsheet, and a numercal teraton would need to be programmed. 2

22 Global Carbon Fnance Offce of Clmate Change Fgure 3: The dagram compares the eulbrum carbon prces from the present analytc model wth those from GLOCAF and FAIR. In order not to dsclose senstve results, t was decded not to present the actual values of the carbon prces. Fgure 4: The dagram compares the global costs as a functon of global GDP from the present analytc model wth those from GLOCAF and FAIR. In order not to dsclose senstve results, t was decded not to present the actual values of the net costs. 22

23 Global Carbon Fnance Offce of Clmate Change The accuracy of the smple model s partcularly evdent when we compare the results produced by FAIR based on two dfferent sets of margnal abatement costs, POLES and TIMER. Fgure 5 compares two dentcal scenaros produced by FAIR. The only dfference s gven by the underlyng margnal abatement costs for energy CO 2 : one uses POLES and the other uses TIMER data. Clearly the results are sgnfcantly dfferent. In other words, snce currently avalable margnal abatement cost curves from dfferent sources can lead to dfferences n outputs by a factor of fve or more, a small dfference between analytc model and the numercal models s neglgble. Fgure 5: Two dentcal scenaros run by FAIR usng POLES and TIMER MAC curves, respectvely. In order not to dsclose senstve results, t was decded not to present the actual values of the carbon prces. 23