Resource Sharing Models

Transcription

1 ESA / T. Reer Mahemacal Deals on Resource Sharng Models Auhors: Güner Wmann, Maser of Mahemacs Andreas Wolfsener, Maser of Economcs save-he-clmae@onlne.ms (mal o) Verson: 7 h December 216

2 Mahemacal Deals on Resource Sharng Models page 2 of 14 Conens 1 Inroducon Models wh convergence a a ceran pon of me Per Capa Model Conracon & Convergence Model LIMITS Model Common bu dfferenaed convergence Model (CDC) Regensburg Model Models wh convergence a nfny Smooh Pahway Model Emsson Probably Model The basc seps Excurson: Probably densy funcon and Lorenz Curve Specal case: gamma probably dsrbuon Ls of abbrevaons References... 14

3 Mahemacal Deals on Resource Sharng Models page 3 of 14 1 Inroducon Accordng o he Pars Agreemen, counres mus regularly presen her ambons,. e. her plans of CO2 emssons (NDCs). Ths leads quckly o he queson, on whch crera her ambons are based. Several models have been developed so far. Effor or burden sharng models can be undersood as an answer o he queson whch s he effor of each counry so ha he busness as usual pahway of each counry s changed n a way o oban n sum a pahway ha mees he remanng global budge. 1 We should lke o concenrae on resource sharng models who gve a drec answer o he queson how he global remanng budge s shared. In some models, he pon of me can be chosen when global emssons are arbued o counres accordng o populaon. In oher models hs pon s a nfny when global and all naonal emssons are zero. As a consequence, resource sharng models arbue all counres posve/negave emssons when he global emssons are posve/negave. On he oher sde effor sharng models can arbue n he same year some counres posve and some counres negave emssons. Our am s o faclae he comparson of he resource sharng models pung he spolgh on he mahemacal formulae and on an Excel ool (download: hp://downloads.save-he-clmae.nfo/ or comparng he resource sharng models for hree ypcal counres. 1 An elaboraon of he Ausralan Clmae Change Auhory provdes an nal overvew:

4 Mahemacal Deals on Resource Sharng Models page 4 of 14 2 Models wh convergence a a ceran pon of me All models wh convergence a a ceran pon of me sar wh a global pahway ha mees a remanng global budge correspondng o a ceran global warmng. The dea s o sar n each counry n a base year (BY) wh he acual emssons and o ransform hese emssons no emssons based on a per capa arbuon durng a convergence perod endng wh a convergence year (CY). 2.1 Per Capa Model Ths s he eases model. In he years before he convergence year each counry s arbued emssons accordng o he emssons n he base year. In he convergence year each counry s arbued emssons accordng o populaon. Ths leads o a rough ranson n he convergence year. E : = E BY E E, for BY + 1 < CY BY E {, for CY (1) 2.2 Conracon & Convergence Model 2 The Conracon & Convergence Model, he LIMITS Model and he Regensburg Formula are smlar. Each model connuously replaces he arbuon rao n he pas wh he arbuon populaon whn a convergence perod. As of he convergence year, only he arbuon populaon s appled. However, he underlyng formulae are dfferen n each model. The Conracon & Convergence formula E : = { ((1 C ) E 1 + C E P ) E 1, for BY + 1 < CY E, for CY was already propounded by he Global Commons Insue n he early 199s. C s any nondecreasng weghng funcon ha akes he value n he base year (BY) and he value 1 n he convergence year (CY). The mos popular varans for C are (2) exponenal: C = exp ( a (1 BY )) wh he parameer a > o be deermned. CY BY Noe ha CBY = exp(-a) s only approxmaely zero. For a > 4 populaon has he leas nfluence compared wh he oher varans. BY konvex quadrac: C = ( CY BY )2 ; populaon has more nfluence han n he exponenal varan wh a > 4. lnear: C = BY CY BY konkav quadrac: C = 1 (1 lnear varan. ; populaon has more nfluence han n konvex quadrac varan. BY CY BY )2 ; populaons has more nfluence han n he 2 (cf.meyer, n.d.)

5 Mahemacal Deals on Resource Sharng Models page 5 of 14 The Global Commons Insue consdered only he lnear und he exponenal form for C. 2.3 LIMITS Model LIMITS, a research projec funded by he EU, uses he followng formula for he emssons of he counry n he year : 3 E : = ((1 C ) E BY + C E P ) E BY, for BY + 1 < CY E {, for CY where C s a non decreasng weghng funcon ha akes he value n he base year and he value 1 n he convergence year. Alhough all of he varans for C menoned n 2.2 are concevable, LIMITS only consdered he lnear form. 2.4 Common bu dfferenaed convergence Model (CDC) 4 The common bu dfferenaed convergence aproach refnes he Conracon & Convergence Model. Ths approach [CDC] elmnaes wo concerns ofen voced n relaon o gradually convergng per-capa emssons: () advanced developng counres have her commmen o reduce emssons delayed [ ] () CDC does no provde excess emsson allowances o he leas developng counres. (Höhne, e al., 26, p. 181) Ths s acheved by arbung conres below a connuously decreasng hreshold emssons accordng o her free descon noed n a busness as usual (bau) scenaro. Thus he Conracon & Convergence Model s only appled for counres wh per capa emssons above hs hreshold. In deal: Frs we defne a hreshold TH n he year ha decreases f he global emssons decrease: TH E PT, where PT s a gven percenage, e. g..95. If he average emssons of counry n he year n a busness as usual scenaro are below or equal o he hreshold,. e. E _bau TH, he counry s arbued emssons accordng o he busness as usual scenaro and we se E E _bau. Oherwse,. e. f he average emssons of counry n he year n he busness as usual scenaro are above he hreshold ( E _bau > TH ), a counry s arbued emssons accordng o he conracon and convergence formula and we se where E ((1 C ) E 1 EPoTH + C 1 oth) E oth, C s a non-decreasng weghng funcon, (3) 3 LIMITS uses he formula o deermne emssons pahways for dfferen regons of he world (cf. Tavon, e al., 213). 4 (Cf. Höhne, e al., 26). Unforunaely hs source doesn conan any formulae. So he formulae we presen are our nerpreaon of he descrbon of he CDC Model.

6 Mahemacal Deals on Resource Sharng Models page 6 of 14 E oth are he remanng emssons n he year for he counres over he hreshold n he year,. e. E oth = E E f E _bau P TH oth E 1 are he emssons n he year -1 of he counres over he hreshold n he year,. e. oth = E 1. E 1 f E _bau P > TH oth s he populaon n he year of he counres over he hreshold n he year,. e. P oth =, f E _bau > TH Remark: Obvously he equaon E oth = E, f E _bau > TH holds, bu hs equaon can be used o defne E oth, because E s defned wh he help of E oth. 2.5 Regensburg Model 5 In he Regensburg Formula he emssons of he counry n he year are gven by E : = { (1 C ) E BY E, for CY + C E CY, for BY + 1 < CY (4) where C = E BY E E BY E CY and E CY = E CY P P CY. CY Ths formula should only be used when global emssons are decreasng o ensure ncreasng C. For oher represenaons of hs formula or for a proof ha E = E (cf. Wmann & Wolfsener, curren verson). 5 (Cf. Sargl, e al., 216).

7 Mahemacal Deals on Resource Sharng Models page 7 of 14 The Regensburg Model smlar o he CDC Model does no provde excess emsson allowances o he leas developng counres (cf. chaper 2.4). Snce he Regensburg Model s he mos favourabel convergence model for ndusral counres we know, naonal emsson pahways calculaed wh he Regensburg Formula descrbe a knd of floor of ambon for hese counres. Indusral counres are n dffculy o explan her NDC f falls below hs floor. The Regensburg Model can also be combned wh he dea of he CDC Modell, ha some counres are exemp from he emsson arbuon regme as long as her emssons are below a hreshold. In hs case global n he descrbon of he formula above mus be read as of he counres above he hreshold.

8 Mahemacal Deals on Resource Sharng Models page 8 of 14 3 Models wh convergence a nfny 3.1 Smooh Pahway Model Raupach e al. 6 showed how o ransform an allocaed remanng budge of he counry (RB ) no a pahway wh a smooh ranson from he curren pahway and wh near-zero emssons n he fuure. Thus, n conras o he oher resource sharng models, he global pahway s obaned by summng up he pahways of all counres. he Smooh Pahway Model can accoun for planned negave emssons of some counres n he fuure. Ths model gves no hn how o oban a remanng budge for each counry. Of course, he global remanng budge can be arbued o counres accordng o populaon or emssons n he pas. In he Smooh Pahway Model for he emsson of he counry a he me z he followng funcon s used where E (z) = E BY (1 + (r + m )(z BY))e m (z BY), (5) E BJ are he emssons of counry a he begnnng of he base year (E (BY) = E BY ), r s he rae of change of emssons of counry a he begnnng of he base year ( E (BY) E (BY) = r ) m s he mgaon rae (or he decay parameer) of counry. The mgaon rae m s deermned such ha he allocaed remanng budge of counry (RB ) s me: E (z) dz = RB Thus, f r > 1/T, he mgaon rae m s gven by BY m = r T T, where T = RB E s he emsson me defned by he remanng budge of counry and he emssons of counry a he begnnng of he base year. Oherwse, here s no soluon for he m- BY gaon rae m. Snce we are more neresed n he emssons of counry n he year (E ) han n he emssons a a pon of me z, we negrae equaon (5) and oban: 6 (Cf. Raupach, e al., 214). Free supplemenary nformaon conanng mahemacal deals on he properes of he formula n equaon (5) can be rereved from hp://

9 Mahemacal Deals on Resource Sharng Models page 9 of 14 E = 1 +E BY E (z) dz = E BY e m ( BY 1) 3.2 Emsson Probably Model e m ( BY) (m ) 2 [(r m + (m ) 2 ) ( BY) + 2m +r ] (m ) 2 [(r m + (m ) 2 ) ( BY 1) + 2m + r ]. From a mahemacal pon of vew, hs model s challengng because does no only ake emssons and populaons no consderaon bu also ncome probables The basc seps Chakravary e al. 7 descrbed hree seps how o oban and cu an emsson probably densy funcon sarng wh he pons of a Lorenz curve. Le (x j, y j ) be pons of he Lorenz curve L of counry,. e. y j = L (x j ), and Z be random varables represenng he ncome of a person n he counry. In a frs sep, he parameers p of an assumed ncome probably densy funcon (PDF) f (z; p ) for each counry s esmaed by fng he Lorenz curves L (z; p ) wh a leas square f: mn p { (L (x j ; p ) y j ) 2 j }. In a second sep ncome random varables Z are scaled o oban emsson random varables Z : Z = s Z wh he scalng facor s average emssons n counry average ncome n counry of counry. In a hrd sep n each year a cap CA s deermned such ha he emsson n all counres yeld he agreed upon global emssons n he year (E ): CA P ( z f (z; p ) dz + The emssons of he counry n he year are hen gven by CA E = P ( z f (z; p ) dz + CA f (z; p ) dz) = E. CA CA f (z; p ) dz). CA Global negave emssons: Normally, s assumed ha each person earns a posve ncome. Tha s why he scalng n he second sep s possble. However, when global emssons are negave an oher ransformaon mus be found ha convers an ncome PDF, whch s zero for nega- 7 (Cf. Chakravarya, e al., 29).

10 Mahemacal Deals on Resource Sharng Models page 1 of 14 ve ncomes, no an emsson PDF ha addresses negave emssons. Such ransformaons are concevable, bu hey are no ndspuable Excurson: Probably densy funcon and Lorenz Curve For some readers a summary on how o oban a Lorenz Curve from a probably funcon mgh be useful. Le f be an ncome probably densy funcon. Then f akes for he ncome z he probably value f(z), he cumulave ncome share x s gven by he cumulave dsrbuon funcon (CDF) F, z. e. he probably for an ncome equal o z or less s x = F(z) = f() d and a paramerc represenaon of he Lorenz curve L s gven by z L (z) = ( y = x = F(z) z f() d) (6) f() d f() d: average ncome of he persons wh an ncome equal o z or less f() d: average ncome of he populaon For a CDF F wh he nverse funcon F 1 he Lorenz curve L s drecly gven by y = L(x) = F 1 (x) f() d. (7) f() d Subsung = F 1 ( ) yelds d d = (F 1 ) ( ) = curve can be wren as y = L(x) = x F 1 1 F 1 1 = 1 F (F 1 ( )) f(f 1 ( )) and he Lorenz ( )d. (8) ( )d Scalng Theorem: The Lorenz curve s ndependen of he scalng of he z-axs. Proof: Wh a scalng facor s he scaled PDF f for a PDF f s gven by For he CDF F we oban F (z ) = z f ( ) d f (z ) = s f(sz ). z = s f(s ) d = Thus F 1, he nverse funcon of he CDF F, s gven by F 1 = 1 s F 1. sz f()d = F(sz Wh he help of he represenaon (8) of he Lorenz curve we see ha, he Lorenz curve from he PDF f and he PDF f are he same. ).

11 Mahemacal Deals on Resource Sharng Models page 11 of Specal case: gamma probably dsrbuon In general, he evaluaon of he negrals n equaon (7) or (8) can cause rouble. However f Z s a gamma dsrbued random varable all hs work can be done by a spreadshee programme, such as EXCEL. Le Z be a gamma dsrbued random varable. Then he PDF g s gven by wh parameers a, b > and Γ(a) = The CDF s denoed by for z < g(z; a, b) = { 1 b a Γ(a) za 1 e z b for z z a 1 e z dz. z G(z; a, b) = g(; a, b) d = z 1 b a Γ(a) a 1 e Snce Γ(a + 1) = a Γ(a), he equaon g(; a, b) = ab g(; a + 1, b) holds. Thus he expeced value (or mean) of Z s gven by and E[Z] = b d g(; a, b) d = ab g(; a + 1, b)d = ab usng he represenaon (7) he Lorenz curve s gven by Scalng L(x) = G 1 (x;a,b) G g(; a, b) d ab 1 (x;a,b) g(; a + 1, b) d = g(, a, b) d ab = G(G 1 (x; a, b); a + 1, b). Wh a scalng facor s we easly fnd g (z ; a, b) = s g(sz ; a, b) = g(z ; a, b s ) Ths equaon shows ha he scalng of a gamma dsrbuon wh parameers a, b leads o anoher gamma dsrbuon wh parameers a, b s. Snce he Lorenz curve does no depend on scalng, he Lorenz curve mus be ndependen of he parameer b.

12 Mahemacal Deals on Resource Sharng Models page 12 of 14 4 Ls of abbrevaons BY C CA CY E BY base year weghng of per capa emssons n he year cap n he year convergence year global emssons n he base year E BY emssons of counry n he base year E BY emssons of counry a he begnnng of he base year (E (BY) = E BY ) E CY global emssons n he convergence year E CY emssons of counry n he convergence year E E E _bau E oth oth E 1 E (z) E[Z] f f F F F 1 F 1 f (z; p ) f (z; p ) global emssons n he year emssons of counry n he year emssons of counry n he year n a busness as usual scenaro remanng global emssons n he year for he counres over he hreshold n he year global emssons n he year 1 for he counres over he hreshold n he year emssons of counry a he me z expeced value (of mean) of he random varable Z ncome probably densy funcon scaled pdf, emsson probably densy funcon cumulave dsrbuon funcon,. e. he probably for an ncome equal o z or less z s F(z) = f() d cumulave dsrbuon funcon, he correspondng PDF s f nverse funcon of he cumulave dsrbuon funcon F nverse funcon of he cumulave dsrbuon funcon F assumed ncome probably densy funcon (PDF) of counry wh parameers p o be esmaed esmaed emsson probably densy funcon (PDF) of counry wh parameers p g(z; a, b) PDF of a gamma dsrbued random varable wh parameers a, b > G(z; a, b) CDF of a gamma dsrbued random varable wh parameers a, b > m L L P CY mgaon rae (or he decay parameer) of counry explc represenaon of he Lorenz curve paramerc represenaon of he Lorenz curve global populaon n he convergence year P CY populaon of counry n he convergence year

13 Mahemacal Deals on Resource Sharng Models page 13 of 14 oth PT r RB s s T TH global populaon n he year populaon of counry n he year populaon n he year of he counres over he hreshold n he year percenage rae of change of emssons of counry a he begnnng of he base year ( E (BY) E (BY) = r ) remanng budge of counry scalng facor average emssons n counry scalng facor of counry ( ) average ncome n counry emsson me defned by he remanng budge of counry and he emssons of counry a he begnnng of he base year (T = RB hreshold n he year E ) BY (x j, y j ) pons of he Lorenz curve L of counry,. e. y j = L (x j ) Z Z Z esmaed random varables represenng he ncome of a person n he counry random varables represenng he ncome of a person n he counry esmaed random varables represenng he emsssons of a person n he counry

14 Mahemacal Deals on Resource Sharng Models page 14 of 14 5 References Chakravarya, S. e al., 29. Sharng global CO2 emson reducons among one bllon hgh emers. PNAS, 21 July, 16(29), p Höhne, N., den Elzen, M. & Wess, M., 26. Common bud dfferenaed convergence (CDC): a new concepual approach o long-erm clmae polcy. Clmae Polcy, Volume 6, pp Meyer, A., n.d. Conracon & Convergence. [Onlne] Avalable a: hp:// [Accessed ]. Raupach, M. R. e al., 214. Sharng a quoa on cumulave carbon emssons. Naure Clmae Change, puplshed onlne: 21 Sepember, pp Sargl, M., Wolfener, A. & Wmann, G., 216. The Regensburg Model: reference values for he (I)NDCs based on convergng per capa emssons. Clmae Polcy, publshed onlne: 14 June, pp Tavon, M. e al., 213. The dsrbuon of he major economes' effor n he Durban plaform scenaros. Clmae Change Economcs, Volume 4(4). Wmann, G. & Wolfsener, A., curren verson. Mahemacal Deals on he Regensburg Formula. [Onlne] Avalable a: or hp://downloads.save-he-clmae.nfo [Accessed 2 December 216].