workng paper Where to Go To wth CO2 Emssons A Comparson of Resource Sharng Models Professor Manfred Sargl Günter Wttmann, Master of Mathematcs Andreas Wolfstener, Master of Economcs save-the-clmate@onlne.ms (mal to) Abstract The COP 21 n Pars decded to start a ratchet up mechansm n 2018 that leads to progressve NDCs. If the connecton between the cumulatve CO2 emssons and global warmng, as descrbed n the IPCC Ffth Assessment Report, s taken serously, the queston arses whether the progressve NDCs submtted by the countres allocate ths remanng CO2 budget n a far and reasonable manner. Resource sharng models drectly address the allocaton of such a remanng global budget. Ths artcle wll therefore gve an overvew of the propertes of selected resource sharng models. We found out a new essental property of convergence models: convergence models show under certan condtons a specfc mplct weghtng of populaton dependng of the selected global pathway. Ths mplct weghtng of populaton s obtaned by allocaton a global remanng budget drectly to countres usng a weghted dstrbuton key of current emssons and current populaton such that the same natonal budgets are obtaned as n the convergence models. Aganst ths background the queston arses, whch mplct weghtng of populaton contan NDCs. The mplct weghtng of populaton could be an mportant addtonal key ndcator, when t comes to a ratonal dscourse on realstc and targeted NDCs. 9 September 2018
A Comparson of Resource Sharng Models page 2 of 21 Contents 1 Introducton and underlyng data... 3 2 Convergence models... 5 2.1 Contracton & Convergence Model... 5 2.2 The Regensburg Model... 6 2.3 Dfferent pathways for emergng and developng countres... 6 2.3.1 Common but Dfferentated Convergence Model... 6 2.3.2 Modfed Regensburg Model... 6 2.4 Comparson of the convergence models... 7 3 Smooth Pathway Model... 9 4 Emsson Probablty Model... 11 5 Comparson of the models... 12 6 Concluson... 16 7 Notes... 18 8 References... 19 9 Supplementary Materal... 21
A Comparson of Resource Sharng Models page 3 of 21 1 Introducton and underlyng data Accordng to the Pars Agreement, countres must regularly present ther ambtons (NDCs),. e. ther plans for CO2 emssons. Ths leads to the queston, what crtera ther ambtons are based on. The framework s gven by the IPCC, whch consders a cumulatve budget of 2,900 GtCO2 snce the begnnng of ndustralsaton to be compatble wth the 2 C lmt wth a probablty of over 66%. About 1,890 GtCO2 of ths budget were already emtted by the end of 2011 (cf. IPCC, 2013, p. 27; IPCC, 2014, p. 24). Hence the remanng cumulatve budget from 2011,. e. as of 2012, s approxmately 1,000 GtCO2 wth current global annual anthropogenc emssons of about 41 GtCO2 (cf. Le Quéré, et al., 2017). In ths paper we have concentrated on resource sharng models 1 currently dscussed, whch take nto account current emssons and populaton when t comes to determnng natonal emsson pathways. We selected these models, as they are sutable for ndcatng realstc NDCs, whch n sum fulfl the 2 C lmt. Current emssons reflect the present realty and populaton can map justce. Ths artcle wants to present dfferent and common features and ncrease the transparency of the models. In Chapter 2 we consder models wth a lmted convergence perod, at the end of whch global emssons are allocated to countres accordng to populaton only. The Smooth Pathway Model n Chapter 3 calculates natonal pathways startng from allocated remanng natonal budgets. The Emsson Probablty Model n Chapter 4 determnes country specfc emsson densty functons and caps the emssons of ndvduals. 2 By way of llustraton we show the results of the models for three pure type countres (see Chapter 2.4 and Chapter 5) wth the followng underlyng data: Fgure 1: Global emsson pathways. Source: Supplementary Tool 2.
A Comparson of Resource Sharng Models page 4 of 21 In the followng fgures we used the global pathway number 3 n Fgure 1. country world A B C n the base year 2019 ndustral emergng developng emssons n GtCO 2 21.0 19.5 1.2 41.7 populaton n bllons 1.2 4.0 1.7 6.9 per capta emssons n t 17.5 4.9 0.7 6.1 Table 1: Underlyng data for the three pure type countres. Detals ncludng the values of further parameters can be found n the Supplementary Tool 1. Our am s not to explore real countres but pure type countres of whch per capta emssons are typcal of an ndustral, an emergng and a developng country. In sum the pure type countres approxmately reflect the global data. In our calculatons we used a frozen populaton of the base year. For a mathematcal descrpton of the models wth proofs of ts propertes we refer to the Supplementary Text.
A Comparson of Resource Sharng Models page 5 of 21 2 Convergence models All convergence models presented here start wth a global pathway that meets a remanng global budget usually correspondng to a certan degree of global warmng. Then the models break down the annual global emssons on country level, transformng the actual emssons n a base year (BY) nto emssons based on a per capta allocaton n a convergence year (CY) at the end of a lmted convergence perod. In the llustraton wth the three pure type countres, we have chosen 2050 as the convergence year and 2019 as the base year. The models n Chapters 2.1 and 2.2 gradually replace the allocaton key emssons n a base year wth the allocaton key populaton wthn the convergence perod. However, the underlyng formulae are dfferent n each model. In Chapter 2.3 we present enhancements of the prevous models that enable dfferent rules for some countres and n Chapter 2.4 we compare the models n Chapter 2. 2.1 Contracton & Convergence Model The Global Commons Insttute already propounded the followng Contracton & Convergence Model (C&C) n the early 1990s. Ths model defnes the emssons of country n the year t (E ) t recursvely (Meyer, No date) 3 : E : t = ((1 C t ) E t 1 + C E t P t ) E t 1 P t, for BY + 1 t < CY t, P t E { P t, for CY t t (1) where E t global emssons n the year t, P t global populaton n the year t and P t populaton of country n the year t. C t denotes the weghtng of the populaton when allocatng global emssons to countres. The Global Commons Insttute consdered two specfcatons of C t : exponental (C&C-exp): C t = exp ( a (1 t BY )) wth the parameter a > 0 to be CY BY determned. The hgher the value [a], the more the convergence happens towards the end of the convergence perod, and vce-versa. Choosng a = 4 gves an even balance. (Meyer,
A Comparson of Resource Sharng Models page 6 of 21 1998, p. 21) lnear (C&C-ln): C t = t BY CY BY. Some more specfcatons of C t are presented and dscussed n the Supplementary Text. 2.2 The Regensburg Model In the Regensburg Model (RM) the emssons of country n the year t (E ) t are gven by E : t = (1 C t ) E BY + C t E CY, for BY + 1 t < CY P t E { P t, for CY t t (2) where C t = E BY E t E BY E CY and E CY = E CY P P CY (cf. Sargl, et al., 2017). CY 2.3 Dfferent pathways for emergng and developng countres In ths chapter we present enhancements of the prevous models that enable more favourable rules for some (normally emergng) countres. These modfcatons nvolve greater efforts to be made by the other (normally ndustral) countres. 2.3.1 Common but Dfferentated Convergence Model The Common but Dfferentated Convergence Model (CDC) refnes C&C (cf. Höhne, et al., 2006). Ths approach [CDC] elmnates two concerns often voced n relaton to gradually convergng per-capta emssons: () advanced developng countres have ther commtment to reduce emssons delayed [ ] () CDC does not provde excess emsson allowances to the least developng countres. (Höhne, et al., 2006, p. 181) Ths s acheved by allocatng countres below a contnuously decreasng threshold emssons accordng to ther free decson recorded n a busness-as-usual scenaro. Thus the C&C model s only used for countres wth per capta emssons above ths threshold. 2.3.2 Modfed Regensburg Model The RM can also be combned wth the dea of CDC, where some countres are exempt from the emsson allocaton regme as long as ther per capta emssons are below the threshold.
A Comparson of Resource Sharng Models page 7 of 21 It s even possble to exempt some countres from the emsson allocaton regme throughout the convergence perod. Ths would be a way of allowng, for example, countres whch start sgnfcantly below the convergence level n the base year to get emssons accordng to straght pathways to the convergence level. Ths shortest way to the convergence level can be seen as a mnmum justce level for developng countres. In ths case global, n the descrpton of formula (2), must be read as of the countres under the emsson allocaton regme. 2.4 Comparson of the convergence models Fgure 2: Comparson of the convergence models emsson pathways (n the C&C-exp model we chose a = 4). Fgure 2 shows that the RM requrng the least reductons for country A (typcal ndustral country) corresponds to greater efforts for country B (typcal emergng country) and country C (typcal developng country). If the effects of changes n populaton are suppressed and f global emssons fall durng the convergence perod, then the RM n contrast to the other models allocates countres startng above the convergence level lower emssons from the frst year on n the convergence perod, rrespectve of how far and how long they have already been above the convergence level. As a consequence, most emergng countres have to reduce ther emssons from the frst year on n the convergence perod. countres startng below the convergence level hgher emssons n each year, but, n contrast to other models n Chapter 2, never greater emssons than the convergence level. The RM, therefore, smlarly to CDC, does not provde developng countres wth excess emsson allowances. For a gven global pathway the weght of populaton C t n C&C can be calculated n such a way that the resultng natonal pathways of C&C and the RM are the same f the populaton s frozen. Ths allows makng clear the dfferent weghtng of populaton n the RM and n C&C-exp.
A Comparson of Resource Sharng Models page 8 of 21 Fgure 3: Comparson of the weghtng of populaton n C&C-exp (a=4) and n C&C yeldng the same results as the RM. We also examned whether the results of the RM and the C&C model wth the classc exponental specfcaton of C&C (C&C-exp) are smlar. Choosng the parameter a = 8, we obtaned smlar results for the ndustral country A and the emergng country B. However, the results for the developng country C are dfferent and not plausble: a sharp ncrease of emssons at the end of the convergence perod,. e. shortly before the ntended decarbonsaton, s followed by a sharp drop n emssons. Fgure 4: Comparson of the RM and C&C-exp (a = 8) - emsson pathways We would lke to stress that n the RM C t s determned by the underlyng global pathway. In C&C, by contrast, C t, whch specfes how fast the per capta dstrbuton comes nto effect, can be chosen arbtrarly.
A Comparson of Resource Sharng Models page 9 of 21 3 Smooth Pathway Model Raupach, M. R. et al. suggest the followng weghted key for allocaton the remanng global budget to countres (cf. Raupach, et al., 2014)): RB = (C RB P BY + (1 C RB ) E BY ) RB (3) P BY E BY where RB RB C RB global remanng budget remanng budget of country weghtng of populaton A lot of crtera on how to obtan a remanng budget for each country (gross domestc product (per capta), cumulatve emssons per capta, populaton, emssons n the past, ) are possble. But there are two outstandng crtera: populaton n a base year (equty) and emssons n a base year (nerta). These two alternatves act as bounds to a range of blended optons, and demonstrate how natonal quotas [(remanng natonal budgets)] can be allotted usng any mx of the two alternatves, [ ]. (Peters, et al., 2015, p. 3). Raupach, M. R. et al. also showed how to transform the allocated remanng budget of a country nto a postve pathway (. e. a pathway whch has no negatve emssons), wth a smooth transton from the current pathway and wth near-zero emssons at nfnty. In the Smooth Pathway Model (SPM) the emssons of country n the year t (E t ) are gven by where E t = E BY+1 +E BY+1 e m (t BY) e m (t BY 1) (m ) 2 [(r m + (m ) 2 ) (t BY) + 2m +r ] (m ) 2 [(r m + (m ) 2 ) (t BY 1) + 2m + r ], E BY+1 emsson power,. e. the dervatve of emssons wth respect to tme or the emssons per unt of tme, of country at the end of the base year, r change rate of the emsson power of country at the end of the base year and m the mtgaton rate (or the decay parameter) of country.
A Comparson of Resource Sharng Models page 10 of 21 If r > 1/T, the mtgaton rate m s gven by where T = RB E BY+1 m = 1 + 1 + r T T, s the tme defned by the remanng budget of country and the emsson power of country at the end of the base year.
A Comparson of Resource Sharng Models page 11 of 21 4 Emsson Probablty Model The Emsson Probablty Model (EPM) from Chakravarty, S. et al. not only takes emssons and populaton nto consderaton, but also the ncome dstrbuton of a country (cf. Chakravarty, et al., 2009). EPM assumes that the emsson dstrbuton s a scaled ncome dstrbuton. Then EPM allocates a country the emssons of ts nhabtants whose emssons are below a cap and the cap for each nhabtant whose emssons are above the cap. The global cap s chosen each year such that the global emssons are met. Thus EPM, as well as the convergence models n Chapter 2, allocates a preset global pathway to all countres. In EPM the emssons of country n the year t are gven by CA t E t = P t ( z f (z; p ) dz + CA t f (z; p ) dz), CA t (4) where CA t f (z; p ) the cap n the year t and the estmated emsson probablty densty functon (PDF) of country wth parameters p.
A Comparson of Resource Sharng Models page 12 of 21 5 Comparson of the models The followng two fgures show the emsson pathways and the per-capta emssons of the pure type countres resultng from the presented models for a comparson: Fgure 5: Comparson of Resource Sharng Models emsson pathways Fgure 6: Comparson of Resource Sharng Models t per capta emssons
A Comparson of Resource Sharng Models page 13 of 21 Fgure 5 shows that, f SPM-n s not taken nto account, RM s clearly the most favourable resource sharng model for ndustral countres. For a detaled comparson of the models n Chapter 2 we refer to the results n Chapter 2.4. The followng key fgures of the models can be reproduced wth the help of the Supplementary Tool 1. Approxmaton of C&C, RM and EPM wth SPM In the model wth three pure type countres and the underlyng global pathway (number 3 n Fgure 1) we approxmated the pathways of C&C-exp (a = 4), RM and EPM wth SPM, mnmsng the sum of the squared relatve devatons n each year. By ths means we obtaned a weghtng of the populaton (see Formula (3)) of 50% for C&C-exp (a = 4), 16% for RM (see Fgure 7) and 24% for EPM. Fgure 7: Comparson of SPM (weghtng of populaton = 16%) and RM - emsson pathways Consderaton of populaton The dea of the models n Chapter 2 s an annual ncrease of the nfluence of the allocaton key populaton, whereas the SPM can only consder populaton once when determnng natonal budgets. EPM does not take nto account populaton as an allocaton key, but calculates a global cap for per capta emssons. Convergence Convergence models lead to postve dentcal per capta emssons n a gven convergence year. The SPM leads to zero emssons at nfnty n each country. Therefore, from a mathematcal pont of vew, SPM leads to per capta emssons that are convergng to zero. EPM only leads to convergng per capta emssons when global emssons are tendng to zero. Role of the global pathway In contrast to the other resource sharng models, SPM does not share a global pathway, but a remanng global budget. Natonal postve pathways can then be calculated and the global pathway 4 s obtaned by summng up natonal pathways. The global pathway s therefore an output
A Comparson of Resource Sharng Models page 14 of 21 value of SPM, whereas n the other models the global pathway s an nput value. Due to the SPM formula, the resultng global pathway at the begnnng has relatvely fast declnng emssons. Ths should be taken nto account when comparng natonal reference values of the dfferent models. Dependence of remanng natonal budgets on the global pathway In the convergence models n Chapter 2 and EPM (Chapter 4) the remanng natonal budgets also depend on the choce of the global pathway. Here the prncple holds that global pathways that stpulate hgh reductons only at a late stage are more favourable - from the perspectve that the reducton of emssons carres dsadvantages for a country - for ndustral countres than global pathways that stpulate hgh reductons at an early stage. Ths property holds all the more for the RM, snce ts weghtng of populaton also depends on the global pathway. In order to llustrate the mpact of the choce of the global pathway, we consdered sgnfcant dfferent postve global pathways (see Fgure 1) meetng the global cumulatve budget of 2,900 Gt CO2. We then calculated wth the dfferent global pathways the remanng natonal budgets that result from the RM, C&C-exp, C&C-ln and EPM. We also calculated the remanng natonal budgets drectly wth a weghtng of the allocaton keys populaton and emssons n a base year usng Formula (3). We then mnmsed over the weghtng of populaton the sum over each country of the squared relatve devatons of the two remanng natonal budgets (best approxmaton of the remanng natonal budgets of the RM, C&C and EPM wth a drect blended allocaton of populaton and emssons n a base year). Ths led to the results n Table 2. Number of the global pathway n Fgure 1 2 3 1 RM pure type country model and real world 3% 15% 35% C&C-exp pure type country model and real world 39% 43% 51% C&C-ln pure type country model and real world 68% 69% 73% EPM pure type country model 11% 18% 30% Table 2: Weghtng of populaton resultng from dfferent global pathways If the populaton s frozen and f there are no global negatve emssons n the perod under consderaton the results of the convergence models (RM, C&C-exp and C&C-ln) do not depend on whether a pure type country model or the real world s used. The weghtng of populaton s hence a charactersng key fgure of the convergence model that depends only on the global pathway. Ths fndng leads to the dea that such a key fgure should also exst for NDCs. It should be possble to deduct a natonal emsson pathway and then the remanng natonal budget. Usng ths remanng natonal budget as well as the emssons and the populaton n a base year the mplct weghtng of populaton can be calculated.
A Comparson of Resource Sharng Models page 15 of 21 Further propertes of the models under consderaton In SPM wth a low weghtng of the populaton, developng countres have to reduce ther emssons relatvely quckly, but emssons reman at a hgher level longer than n the RM (see Fgure 7). If the effects of changes n populaton are suppressed and f global emssons fall, then the EPM s the only model that also contnually reduces the emssons n the developng countres after a base year. SPM, CDC and EPM n one way or another take nto consderaton the change rate of emssons from the base year. Usually ths leads to a soft transton from the emssons n the base year. SPM always leads to postve natonal pathways. Hence, the resultng global pathway s also always postve. SPM can therefore map nether global nor natonal negatve emssons n a year. Natonal pathways under the EPM wll always declne. Natonal pathways under the SPM wll fall after they have reached a maxmum. In contrast natonal pathways, partcularly of developng countres n C&C, show a change of drecton wthn the convergence perod. The pathways of developng countres n the RM usually show a clear knk at the end of the convergence year, f the global pathway declnes rapdly after the convergence perod (see Fgure 7).
A Comparson of Resource Sharng Models page 16 of 21 6 Concluson The drect comparson of resource sharng models usng three pure type countres has revealed sgnfcant dfferences between the models that are relevant when dscussng reference values for the NDCs n the ratchet up mechansm of Pars, whch wll start wth a facltatve dalogue n 2018. If SPM-n s not taken nto account, RM s the most favourable resource sharng model for ndustral countres. Therefore natonal emsson pathways calculated wth the RM descrbe a floor of ambton for ndustral countres f they accept takng nto account equal per capta rghts as an am. They wll otherwse have dffculty explanng ther NDCs f they fall below ths floor. As an example the RM means for the EU that t has to reduce emssons n comparson to 1990 rather n a scale of 50% than only 40% proposed so far (cf. Sargl, et al., 2017, p. 672). In an overall assessment of the characterstcs of the models, SPM has the advantage that the queston of clmate justce s addressed explctly and does not depend on a global pathway. Furthermore SPM leads to smooth pathways untl nfnty. Other models mght gve hnts of the range of the weghtng of populaton when t comes to determnng remanng natonal budgets as an nput value n SPM. If any other model s used to justfy or to assess NDCs, the underlyng global pathway must be dsclosed, because ts results depend on the selected global pathway. The queston of the convergence of the per capta emssons as a property of a model becomes less mportant the sooner global emssons must be zero or even negatve. The later global emssons begn to declne, the sooner emsson neutralty must be reached. Ths s due to the budget property of CO2. Although n SPM t s not necessary to determne a global pathway when allocatng the global remanng budget to countres, t would be useful for the world communty to agree on a sutable global pathway, after havng agreed on a remanng global CO2 budget compatble wth the 2 C lmt. Ths would enable an effectve control of devatons. The global pathway would not be carved n stone, but could and should be constantly adapted to the latest developments and scentfc status. Consderng the ratchet up mechansm of Pars the focus should be more on the remanng natonal cumulatve budgets seen as a far and reasonable share of the remanng global cumulatve budget. We have shown: resource sharng models gve useful help when t comes to determnng these budgets.
A Comparson of Resource Sharng Models page 17 of 21 It would make sense to calculate the weghtng of the allocaton keys current populaton and current emssons leadng to the same remanng natonal budgets that result from NDCs. Ths makes clear, whch mplct weghtng of the allocaton key current populaton s consdered as legtmate. Thus, ths weghtng of populaton could contrbute to a more ratonal dscourse of the core queston: Who gets respectvely takes how much of the remanng global cumulatve budget?
A Comparson of Resource Sharng Models page 18 of 21 7 Notes 1 There are two broad approaches to sharng emssons reducton efforts: sharng the global emssons budget - resource-sharng sharng the emssons reductons requred to meet that budget - effort-sharng [or burden sharng ] [ ]. In some ways, the two approaches are smlar - sharng the remanng budget mplctly sets a mtgaton task and vce versa. From a practcal perspectve, resource-sharng approaches are more straghtforward, as they requre only an estmate of the global emssons budget and equtable prncples. In contrast, effort-sharng also requres an estmate of global emssons n the absence of clmate change acton; that s, a BAU trajectory. As more countres take more acton, ths trajectory becomes ncreasngly abstract and dffcult to estmate. (Australan Government Clmate Change Athorty, No date) 2 Du Pont, Y. R. et al. dstngush fve allocaton approaches: (1) capablty, (2) equalty, (3) responsblty, (4) equal cumulatve per capta and (5) staged approaches (du Ponte, et al., 2017, p. 40). Usng ths classfcaton, the models n Chapter 2 belong to category (2). The classfcaton of the Smooth Pathway Model n Chapter 3 depends on the determnaton of the remanng natonal budgets. The Emsson Probablty Model n Chapter 4 belongs rather to category (1) wth equalty from category (2) playng a role, too. 3 LIMITS, a research project funded by the EU, defnes the emssons of country n the year t (E ) t explctly (cf. Tavon, et al., 2013): E : t = { ((1 C t ) E BY + C E t P t ) E BY P t, for BY + 1 t < CY t P t P t E t, for CY t 4 The global pathway n SPM-n meets the global cumulatve budget of 2,900 GtCO2 up to and ncludng the year 2100; the global pathway n SPM-eq even meets a cumulatve budget of 2,820 GtCO2 up to and ncludng the year 2100.
A Comparson of Resource Sharng Models page 19 of 21 8 References Australan Government Clmate Change Athorty, No date. Appendx C Sharng the global emssons budget. [Onlne] Avalable at: http://www.clmatechangeauthorty.gov.au/appendx-c-sharng-global-emssonsbudget [Accessed 25 02 2017]. Chakravarty, S. et al., 2009. Sharng global CO2 emson reductons among one bllon hgh emtters. PNAS, 106(29), p. 11884 11888. du Ponte, Y. R. et al., 2017. Equtable mtgaton to acheve the Pars Agreement goals. Nature Clmate Change, Jannuary, Volume 7, pp. 38-40. Höhne, N., den Elzen, M. & Wess, M., 2006. Common bud dfferentated convergence (CDC): a new conceptual approacht to long-term clmate polcy. Clmate Polcy, Volume 6, pp. 181-199. IPCC, 2013. Clmate Change 2013: The Physcal Scence Bass. Contrbuton of Workng Group I to the Ffth Assessment Report of the Intergovernmental Panel on Clmate Change. [Onlne] Avalable at: www.clmatechange2013.org [Accessed 23 12 2017]. IPCC, 2014. Clmate Change 2014: Synthess Report to the Ffth Assessment Report of the Intergovernmental Panel on Clmate Change. [Onlne] Avalable at: http://ar5-syr.pcc.ch/ [Accessed 23 12 2017]. Le Quéré, C. et al., 2017. Global Carbon Budget 2017. [Onlne] Avalable at: www.earth-syst-sc-data-dscuss.net/essd-2017-123/ [Accessed 14 01 2018]. Meyer, A., 1998. The Kyoto Protocol and the Emergence of Contracton and Convergence as a Framework for an Internatonal Poltcal Soluton to Greenhouse Gas Emssons Abatement. Hedelberg: Physca-Verlag. Meyer, A., No date. Contracton & Convergence. [Onlne] Avalable at: http://www.gc.org.uk [Accessed 7 12 2016].
A Comparson of Resource Sharng Models page 20 of 21 Peters, G. P., Andrew, R. M., Solomon, S. & Fredlngsten, P., 2015. Measurng a far and ambtous clmate agreement usng cumulatve emssons. Envronmental Research Letters, 15 October, Volume 10. Raupach, M. R. et al., 2014. Sharng a quota on cumulatve carbon emssons. Nature Clmate Change, Volume 4, pp. 873-879. Sargl, M., Wolfstener, A. & Wttmann, G., 2017. The Regensburg Model: reference values for the (I)NDCs based on convergng per capta emssons. Clmate Polcy, 17(5), p. 664 677. Tavon, M. et al., 2013. The dstrbuton of the major economes' effort n the Durban platform scenaros. Clmate Change Economcs, 4(4), pp. 1340009-1 - 1340009-25.
A Comparson of Resource Sharng Models page 21 of 21 9 Supplementary Materal Supplementary Materal can be downloaded from www.save-the-clmate.nfo (menu: "Downloads"). Supplementary Text: Resource Sharng Models - A mathematcal descrpton Supplementary Tool 1 (Excel tool): Comparng Resource Sharng Models Supplementary Tool 2 (Excel tool): Smooth Global Pathways (Tool_global_paths.xlsm)