Calculations for the Greenhouse Development Rights Calculator

Transcription

1 Calculatons for the Greenhouse Development Rghts Calculator Erc Kemp-Benedct September 4, 8 Stockholm Envronment Insttute Workng Paper Abstract The Greenhouse Development Rghts (GDRs) on-lne calculator s a web-based tool for examnng the mplcatons of dfferent assumptons under the GDRs framework. The calculator reports a number of ndcators that requre techncal documentaton.

2 Copyrght 8 by the Stockholm Envronment Insttute Ths publcaton may be reproduced n whole or n part and n any form for educatonal or non-proft purposes, wthout specal permsson from the copyrght holder(s) provded acknowledgement of the source s made. No use of ths publcaton may be made for resale or other commercal purpose, wthout the wrtten permsson of the copyrght holder(s). For more nformaton about ths document, contact Erc Kemp-Benedct at erckb@se-us.org. Stockholm Envronment Insttute - US 11 Curts Avenue Somervlle, MA , USA and 1

3 Introducton The Greenhouse Development Rghts (GDRs) Framework (Baer et al., 7) s a general framework for burden-sharng for clmate change that takes both responsblty and fnancal capacty nto account. The framework takes the ndvdual as the unt of analyss, whch means that at a mnmum nstantaneous ncome dstrbutons wthn countres are a necessary nput nto calculatons that make use of the framework. The avalablty, snce the md-199s, of a reasonably complete nternatonal database on ncome dstrbuton (Dennger and Squre, 1996; UNU-WIDER, 8) makes ths calculaton possble. Whle ncome moblty the trajectory of ndvduals over tme wthn an ncome dstrbuton would also be useful, there are nsuffcent data on ncome moblty to support an nternatonal calculaton. An on-lne calculator s beng developed that s consstent wth the GDRs framework and that s used by the developers of the GDRs framework n order to explore the mplcatons of the framework at natonal and ndvdual level. 1 Ths workng paper descrbes some of the calculatons that are mplemented n the calculator. Lognormal Income Dstrbutons Followng Kemp-Benedct (1) and Lopez and Servén (6), ncome s assumed to be dstrbuted lognormally wthn countres. The lognormal dstrbuton has two parameters: the mean ncome y and the standard devaton of the log of ncome,. The standard devaton of the log of ncome s a measure of how equally or unequally dstrbuted ncome s wthn the country, and can be related to the well-known Gn coeffcent G usng the followng formula, In ths expresson, N -1 s the nverse normal functon. 1G = N 1. (1) It s convenent when workng wth the dstrbuton wthn an ndvdual country to transform ncome y nto a new varable z, gven by In terms of z, the lognormal becomes smply a normal dstrbuton, z= 1 ln y/ y. () y~lognormaly, z~n,1. (3) Partal Moments of the Lognormal Income Dstrbuton The GDRs framework proposes that ncome below a certan lower ncome threshold y l s exempt, n that t does not contrbute to an ndvdual s capacty to pay. Furthermore, n the practcal mplementaton of the framework wthn the calculator, t s assumed that emssons wthn countres vary wth ncome wth an elastcty that s the same for all countres, and emssons assocated wth ncomes below y l are exempt, n that they do not contrbute to an ndvdual s responsblty. 1 The calculator s under development. It s avalable on a password-protected web ste, and at present access s very lmted. Please contact the author of ths workng paper to nqure about accessng the calculator.

4 More specfcally, ndvdual per capta emssons are assumed to vary n the followng way wth ncome, y= A y, (4) where y s emssons per capta for ndvduals n a narrow range of ncomes around y for country, and A s a country-specfc constant. The elastcty γ s the same for all countres. Chakravarty et al. (n preparaton) report evdence that ths assumpton stands up reasonably well to emprcal test, and that the data are consstent wth γ between about.7 and 1.. In order to mplement the framework, t s necessary to evaluate ntegrals of the followng form, whch are referred to n ths workng paper as partal moments, M y ; y l,, y dy y y l f y ; y,, (5) y l where f y ; y, s the (lognormal) ncome dstrbuton. Usng the convenent change of varables gven n Equaton (), for a lognormal dstrbuton the partal moments can be shown to be equal to M y ; y l,, y=y e 1 1 N z l y l 1 N z l, (6) where N s the cumulatve normal dstrbuton. Ths follows from the followng equaton, y l dy y f y ;,y = y e / dz e z 1 z l Capacty and Responsblty 1 e z = y e 1 1 N z l. (7) In prncple t s possble to have more than one ncome threshold and have dfferent fractons of ncome contrbute toward capacty to dfferent degrees at dfferent ncome levels. In some versons of the calculator ths s mplemented usng two dfferent thresholds, a lower threshold y l, and an upper threshold y u. Income s exempt below the lower threshold, whle above the upper threshold, 1% of ncome contrbutes to capacty. Between the two thresholds, a fracton φ of each margnal ncrement of ncome contrbutes toward capacty. The defntons of capacty and responsblty wll be gven for ths general case. Indvdual capacty and responsblty per capta Indvdual capacty per capta n country, c (y), s gven by y y l c y y y={ l y l y y u. (8) y u y l y y u y y u Responsblty s based on emssons above the threshold, where ndvdual emssons per capta vary as n Equaton (4). The annual contrbuton to ndvdual per capta responsblty r ann s then gven by 3

5 y y l r ann A y={ y y l y l y y u. (9) A [ y u y l y y u ] y y u The dffculty wth Equaton (9) s that the coeffcent A s not known. What s known s the total natonal annual emssons E, gven by As a result, E = dy A y f y ; y, = A y e 1. (1) and r ann y={ A = E y e 1, (11) y y l E y e 1 y y l y l y y u. (1) E y e 1 [ y u y l y y u ] y y u Natonal capacty and responsblty Natonal capacty C s gven by ntegratng per capta capacty [Equaton (8)] over the ncome dstrbuton. Usng the formula for partal moments gven n Equaton (6), and rearrangng, ths can be shown to be equal to C =y [1 1 N y u N y l ] 1 y u 1 N y u y l 1 N y l. (13) Note that n the case φ = 1, ths collapses to the expresson for a sngle threshold, =1 C =y [1 N y l ] y l 1 N y l. (14) Natonal annual responsblty R ann s found by ntegratng per capta annual responsblty [Equaton (1)] over the ncome dstrbuton. Ths can be shown to equal R ann = E [1 1 N y u N y l ] 1{ E e 1 y u y 1 N y u As wth capacty, ths collapses to the one-threshold case when φ = 1. y l y } 1 N y. (15) l In any gven year Y, natonal responsblty R s calculated as cumulatve annual responsblty from a startng year (currently, 199). Addng a tme ndex t to annual responsblty, ths can be wrtten 4

6 In subsequent formulas, the tme ndex wll be suppressed. The Responsblty-Capacty Indcator Y R,Y = R ann,t. (16) t =199 The Responsblty-Capacty Indcator (RCI) s the key ndcator for the GDRs framework. It s used to allocate burden-sharng. Natonal RCI, as currently mplemented, s a weghted sum of the natonal share of global responsblty and capacty. That s, RCI =a R N =1 R 1 a C =1 N C, (17) where N s the number of countres. Natonal RCI sums to one across countres, and so natonal RCI s the natonal share of total RCI. For an ndvdual wthn country, RCI per capta, rc (y), s calculated as rc y=a R N R =1 r ann y ann R 1 a c y N. (18) =1 C Ths formulaton assumes that cumulatve responsblty s allocated accordng to the current annual allocaton n any gven year. Ths s not deal, snce n fact ndvduals wll have ther own ncome trajectores that can be qute dvergent, dependng on the degree of ncome moblty wthn a country. However, gven the very lmted data on ncome moblty t s not feasble to try to capture ths. Burden Sharng as a Tax RCI s used to allocate burden sharng, whch can be thought of as a tax. The total tax T s gven as a fracton τ of gross world output, GWP, Natonal tax T s then gven by natonal RCI multpled by T, T =GWP. (19) T =RCI T, () whch sums to the total tax because natonal RCI sums to one across countres. Indvdual per capta tax t (y) s gven by t y=t rc y. () The Suts Index The Suts ndex (Suts, 1977) s a commonly-used ndcator of the degree of progressvty or regressvty of a tax or tax system. It s calculated by fndng the cumulatve share of the total tax across ndvduals and plottng aganst cumulatve ncome, n a way that s analogous to the calculaton of the Gn coeffcent from the Lorenz curve. In the GDRs calculator the Suts ndex s an optonal ndcator that can be calculated for the world as a whole. 5

7 In order to calculate the Suts ndex, the global ncome dstrbuton f(y) s needed. Ths s gven by N f y= =1 p f y ; y,, (1) where p s country s share of global populaton. Note that f(y) s not lognormal, although each of the country dstrbutons s lognormal. Usng the global ncome dstrbuton, the cumulatve share of global ncome up to ncome y*, x(y*), s gven by x y * = 1 y * dy y f y = 1 N y y =1 where y s global mean ncome, equal to lm y * x y*. The cumulatve share of total tax T(y*) up to ncome y* s gven by T y * = 1 N T =1 y * p dy y f y ; y,, () y * p dy t y f y ; y,, (3) where t (y) s ndvdual tax per capta, gven by Equaton (). The set of ponts (T(y*), x(y*)) traces out a curve T(x) as y* goes from to nfnty that s analogous to the Lorenz curve. In terms of the curve T(x), the Suts ndex s defned as (Suts, 1977) Because x s known as a functon of ncome, x(y*), ths can be re-wrtten 1 S=1 dx T x. (4) S=1 dy * dx dy * T y* (5) by a change of varables. It s more convenent to work wth the dervatve of T(y*) than wth the dervatve of x(y*). Ths can be accomplshed by ntegratng Equaton (5) by parts, to gve S=1 x y * T y * y * = y * = but snce x(y*) and T(y*) go from to 1, ths becomes smply dy * x y * dt y* dy *, (6) S= dy * x y * dt y* 1. (7) dy * Usng Equaton () and the formula for partal moments [Equaton (6)], x(y*) can be shown to be equal to x y * = 1 N y =1 p y N z *, (8) where z * s the value for z correspondng to y* usng Equaton (). The other factor n the ntegral n Equaton (7) s dt(y*)/dy*, whch can be calculated drectly from Equaton (3) as 6

8 dt y * = 1 N dy * T =1 p t y * f y * ; y,. (9) Wth Equaton (8) and (9), the Suts ndex can be calculated by numercally evaluatng the ntegral n Equaton (7). 7

9 References Baer, P., Athanasou, T., & Kartha, S. (7). The Rght to Development n a Clmate Constraned World: The Greenhouse Development Rghts Framework. Berln: Henrch Böll Foundaton. Chakravarty, S., Chkkatur, A., de Connck, H., Pacala, S., Socolow, R., & Tavon, M. (n preparaton). Clmate Polcy Based on Indvdual Emssons. Dennger, K., & Squre, L. (1996). A new data set measurng ncome nequalty. World Bank Economc Revew, 1(3), Kemp-Benedct, E. (1). Income Dstrbuton and Poverty: Methods for Usng Avalable Data n Global Analyss (No. 4). Boston, Mass.: Stockholm Envronment Insttute. Lopez, H., & Servén, L. (6). A Normal Relatonshp? Poverty, Growth, and Inequalty (No. WPS 3814). Washngton, D.C.: World Bank. Suts, D. B. (1977). Measurement of Tax Progressvty. Amercan Economc Revew, 67(4), Unted Natons Unversty, & World Insttute for Development Economcs Research. (8). UNU-WIDER World Income Inequalty Database, Verson.c. 8