Discounting, Risk and Inequality: A General Approach

Transcription

1 Dcounng, Rk and Inequaly: A General Approach Marc Fleurbaey a Séphane Zuber b Ocober 2013 Abrac The common pracce con n ung a unque value of he dcoun rae for all publc nvemen. Endorng a ocal welfare approach o dcounng, we how how dfferen publc nvemen hould be dcouned dependng on: he rk on he reurn of he nvemen, he background rk on conumpon, he drbuon of gan and loe, and nequaly. We alo udy he lm value of he dcoun rae for very long erm nvemen, and he ype of nformaon ha needed abou long-erm cenaro n order o evaluae nvemen. Keyword: Socal dcounng, rk, nequaly. JEL Clafcaon number: D63. Th paper ha benefed from he commen of S. Pacala and he audence a a Prnceon workhop and a he TIGER Conference n Touloue. S. Zuber acknowledge he hopaly of he Prnceon Inue for Inernaonal and Regonal Sude Reearch Communy Communcang Uncerany: Scence, Inuon and Ehc n he Polc of Global Clmae Change. a Prnceon Unvery, Woodrow Wlon School and Cener for Human Value. E-mal: marc.fleurbaey@prnceon.edu. b Par School of Economc CNRS, France. E-mal: ephane.zuber@unv-par1.fr.

2 dcounng, rk and nequaly 1 1 Inroducon Invemen and polce havng long erm mpac are crucal for he developmen of he economy, and hey ofen arac much aenon n he publc debae. One promnen example are of coure mgaon polce amed a prevenng dramac fuure clmae change ha may hreaen he mere urvval of many pece, ncludng humanknd. The ue of clmae polcy ha recenly revealed ha here no agreemen among econom abou he approprae welfare framework for evaluang uch polce. The Sern revew on he economc of clmae change Sern 2006 ha been heaedly debaed on h ground Nordhau, 2007; Wezman, 2007; Dagupa, Alhough hey reach very dfferen concluon, all hee paper endore he ame bac welfare model, namely he Expeced Dcouned Ularan creron e δ Euc, 1 =0 where c, o mplfy, he conumpon of he repreenave agen of generaon. Th creron yeld he andard Ramey equaon Ramey, 1928: An nvemen from perod 0 o perod ha yeld a ure rae of reurn r worh dong, n he margn, f.e., f u c 0 < e δ Eu c e r, r > δ 1 Eu ln c u. 2 c 0 The debae around h welfare model ha been moly confned o a dcuon of he parameer nvolved n Eq. 2, n parcular he rae of pure me preference δ and he elacy of he margnal uly of conumpon. One can roughly denfy wo poon: he ehcal or precrpve approach ha recommend ha ehcal conderaon hould gude he choce of he parameer, and he decrpve approach argung ha he parameer hould be choen o mach oberved marke rae. In he end, he choce beween he wo approache an ehcal choce, n he ene ha a normave jufcaon of he welfare evaluaon framework called for. Anoher ue ha ha been much dcued he mpac of uncerany on he ocal dcoun rae. In parcular, ha been howed ha he ocal dcoun rae lkely o be lower when here a large uncerany on fuure growh Wezman, 1998; Goller, 2002; Goller and Wezman, Th knd of uncerany generally nduce he Wezman effec Wezman, 1998 ha he ocal dcoun rae hould decreae wh he me horzon. In he exreme, when he probably of havng bad cenaro yeldng nfne margnal

3 dcounng, rk and nequaly 2 ule of conumpon uffcenly hgh, we may oban Wezman dmal reul ha he ocal dcoun rae nfnely negave Wezman, Bu a dfferen knd of uncerany for rky projec ha on he rae of reurn of he nvemen. I generally yeld he oppoe Goller effec Goller, 2004 ha he ocal dcoun rae ncreae wh he me horzon. In general, boh knd of uncerane co-ex, and hey hould be jonly uded Goller, 2012, conan a chaper on he ue n he rerced framework of o he dcouned ularan approach. In addon o rk, equy anoher dmenon ha affec he ocal dcoun rae, even n he andard dcouned ularan framework. The uual echnque o deal wh equy conderaon n he leraure on clmae change polcy ha been o nroduce equy wegh pung greaer empha on he damage affecng he poor han o damage affecng he rch. Early reference nclude Azar and Serner 1996, Fankhauer, Tol and Pearce 1997 and Pearce 2003, and reach ambguou concluon regardng he mpac of equy conderaon on he co of carbon. Anhoff, Hepburn and Tol 2009 perhap he more recen and complee udy, and hey fnd ha equy conderaon can gnfcanly ncreae he ocal co of carbon. Thee approache do no drecly ncorporae equy conderaon n he dcoun rae. Goller 2012, Chap. 9 a ep n ha drecon, whch how ha equy conderaon yeld a lower dcoun rae n he long run. Th paper however conder effcen ocal dcoun rae, where he co and benef of he nvemen are effcenly hared whn generaon. We wan o conder a more general cae allowng neffcen harng of he co and benef. To um up, he leraure ha denfed ome of he lmaon of he bac welfare model ued o derve ocal dcounng. Several horcomng however reman. Fr, he analy almo alway confned o he andard Expeced Dcouned Ularan Model. The excepon are paper conderng non-expeced uly model allowng, for nance, ambguy averon Goller, 2002, 2012; Traeger, In he preen paper, we hall ck o he expeced uly framework ha eem normavely appealng from he ocal evaluaon pon of vew. Oher excepon are a few paper uggeng alernave o Ularanm Bommer and Zuber, 2008; Fleurbaey and Zuber, 2013b; Zuber and Ahem, Bu hey generally lm hemelve o ome apec of he rk. A econd horcomng of he leraure ha he dfferen ue background rk, uncerany of he reurn, nequaly are generally reaed eparaely. A hrd lmaon ha he dffcul ue raed by he uncerany of he number of fuure generaon are generally kpped, whle here are reaon o hnk hey are core ue. La, only he effec of he polce on fuure conumpon have been uded whle oher mpac lke he effec of he polcy on

4 dcounng, rk and nequaly 3 he probably of fuure caarophe are no condered. In h paper, we propoe a general welfare framework o analyze he ue of rk, equy, and varable populaon. Our analy doe no focu on a parcular welfare approach bu uncover common feaure of a wde e of approache. We allow boh for background rk and rk on he reurn of he polcy, examnng how her neracon affec ocal dcounng. The man reul of he paper are he followng. Fr, we provde a generalzed defnon of he ocal dcoun rae and oban general formula whch decompoe he man componen of he ocal dcoun rae relaed o growh, nequaly and uncerane n conumpon and nvemen reurn. Compared o andard reul, hee formula dplay mporan addonal covaran erm. Second, we how ha n he long run, he key deermnan of he dcoun rae are he uaon of he wor-off ndvdual n he wor cae cenaro, a well a he maxmum reurn on he nvemen. La, we how ha he dcoun rae a weghed average of he marke rae ndvdual own dcoun rae and he ocal dcoun rae beween dfferen generaon, mplyng ha marke rae relevan for hor erm nvemen, bu much le o n he long run. The dffcul ue raed by he clmae change problem alo pon o he lmaon of he ocal dcoun rae. When here a rk of an early exncon of he humany, clear ha he mpac of a polcy on fuure conumpon are no he only one o be condered, and perhap no he mo mporan one. The effec of he polcy on he propec of a caarophe are alo a key elemen, and we need ool o value change n he probably of caarophc even. Thee ue are dcued a he end of he paper. Our paper organzed a follow. Secon 2 nroduce a general eng and propoe a defnon of he ocal dcoun rae. Secon 3 dcue how hree apec of he rk, he background rk, he rk on reurn and he rk on he plannng horzon, affec he ocal dcoun rae. Secon 4 ackle he ue of nra-generaonal nequale n conumpon and drbuon of co and benef. Secon 5 derve an approxmaon formula for he ocal dcoun rae n he long run, howng ha he key fgure are he maxmum reurn of he nvemen and he maxmum ne reurn for a poor-o-poor nvemen. Secon 6 provde furher exenon. Fr conder an OLG economy where ndvdual lve for everal perod and how how ndvdual own dcoun rae ener he general formula for ocal dcounng. Second dcue he lmaon of he ocal dcoun rae, n parcular when polce affec he propec of fuure caarophe.

5 dcounng, rk and nequaly 4 2 A general framework and he defnon of he ocal dcoun rae 2.1 The framework We le N 0 denoe he e of non-negave neger, N he e of pove neger, R he e of real number, and R + he e of non-negave real number. For a e X and any n N, X n he n-fold Carean produc of X. We focu on evaluang drbuon of conumpon or ncome a he ndvdual level acro perod. An alernave c a collecon of conumpon level, one for each ndvdual alve n he alernave. The e of poenal ndvdual N, o ha alernave are elemen of C = N N\ N R +. We herefore conder a varable-populaon framework, n whch he ze of he populaon may vary from one alernave o anoher, dependng on he ube of ndvdual alve n he alernave. For any c C, we le Nc be he e of ndvdual alve n he alernave and nc = N c be he number of ndvdual n he alernave. We alo need o know o whch generaon he people alve n an alernave belong. To do o, we aume ha here ex a paron of N no ube N conanng he poenal ndvdual of generaon N 0. We wll rerc aenon o C = {c C N 0 Nc}, whch mean ha all he member of he curren generaon are preen n all he alernave we conder. For any c C and any N 0, we denoe N c = N Nc and n c = N c. Uncerany decrbed by m N \ {1} ae of he world. The e of ae of he world S = {1,, m}. A propec a vecor belongng o he e C = C m wh ypcal elemen c = c 1,, c m. For a propec c, c denoe he conumpon of ndvdual n ae of he world, whenever Nc. To eae he expoon, when here no ambguy, we ue he noaon N = Nc, n = nc, N = N c and n = n c for c C and S. Le P = { p 1,, p m [0, 1] m m =1 p = 1 } denoe he cloed m 1-mplex. A loery he combnaon of a probably vecor p = p 1,, p m P wh a propec c = c 1,, c m C. The e of loere denoed L = { p, c P C }. 2.2 The ocal evaluaon funcon A ocal evaluaon funcon F a funcon F : L R ued o ranked loere. For any p, c, p, c L, F p, c F p, c mean ha he loery p, c deemed ocally a lea a good a he loery p, c. We aume ha he ocal evaluaon funcon an

6 dcounng, rk and nequaly 5 expeced uly o ha here ex a funcon W : C R uch ha: F p, c = m p W c. 3 =1 Alhough we wll moly work wh h general funcon, wll be ueful o llurae our reul wh a few more pecfc example. When W c = N uc, 4 we oban he uual Toal Ularan creron. The creron ha been crczed for yeldng he Repugnan Concluon Parf, 1984 where, for any populaon wh excellen lve, here a populaon wh lve barely worh lvng ha beer, provded ha he laer nclude uffcenly many people. Several auhor Blackorby and Donaldon, 1984; Broome, 2004; Blackorby, Boer and Donaldon, 2005 have herefore propoed o ue he Crcal-Level Ularan creron W c = uc ū, 5 N where ū he crcal-level of uly. If ū = 0 1, we are back o he Toal Ularan cae, bu f ū > 0 only lve wh a hgh enough welfare level are worh addng o a populaon. One problem when ū > 0 ha we ge he Very Sadc Concluon Arrhenu, 2012 where, for any populaon wh errble lve no worh lvng, here a populaon wh good lve ha wore, provded ha he laer nclude uffcenly many people. Ye anoher form he Equally Drbued Equvalen Fleurbaey, 2010, whch can ake he form: W c = φ 1 1 n N φ uc, 6 where φ an ncreang and weakly concave funcon. When φ affne, we oban he Average Ularan Creron, whch doe no afy he Negave Expanon Prncple Blackorby, Boer and Donaldon, 2005 ha addng a lfe no worh lvng hould decreae he value of a populaon. Th rue of all Equally Drbued Equvalen evaluaon funcon whenever φ0 = 0. More generally, one may wan o conder he followng cla of generalzed addve ocal evaluaon funcon: W c = Ψ n N φ uc. 7 1 u = 0 known a he neural uly level uch ha a lfe wh hgher welfare worh lvng and a lfe wh lower welfare no.

7 dcounng, rk and nequaly 6 Alhough h la formulaon hghly general, he purpoe of h paper no o endore a pecfc ocal evaluaon funcon. Several dfferen form have been propoed n he leraure ee for nance: Ng, 1989; Bommer and Zuber, 2008; Ahem and Zuber, 2013, and hey all have normave drawback n erm of populaon ehc or ocal rk evaluaon. Our purpoe o how a common rucure of he dcoun rae for all hee dfferen normave approache. The key role of he ocal evaluaon funcon wll be o deermne he ocal margnal value of he conumpon by a pecfc ndvdual. 2.3 Defnng he ocal dcoun rae Followng he uual approach o ocal dcounng recalled n he nroducon, he compuaon for our general evaluaon funcon 3 goe a follow. Suppoe ha ndvdual n perod 0 and ae naure nve $ε for an ndvdual j lvng n perod and ae. Wha he reurn rae r on he nvemen ha mnmally requred o make he nvemen worhwhle? 2 Ung he evaluaon funcon 3, he welfare change nduced by he nvemen a he margn : W, c ε,, c j + e r ε, W c. When ε uffcenly mall, he nvemen make he ocey ndfferen whenever: W c c r W c = e c j 8 Equaon 8 mplcly defne he dcoun rae o be ued for h pecfc nvemen. Th dcoun rae concern nvemen nvolvng only wo pecfc ndvdual, wh a pecfc me dfference, n a pecfc ae of he world. We hall herefore name he ae-pecfc peron-o-peron dcoun rae. Denoe W = W c / c, he ocal prory of ndvdual conumpon n ae. We have he followng defnon. Defnon 1 For all p, c L, for all S, and for all N 0, j N, he ae-pecfc peron-o-peron dcoun rae from peron o peron j n ae, denoed δ,j, : δ,j = 1 ln W. 9 W j 2 In dcree me, for a rae r he reurn on $1 equal o 1 + r. In h paper we adop he alernave formula e r whch correc n connuou me becaue provde convenen formula. Th mean ha he rue dcree-me rae of reurn n our paper e r 1, no r e.g., r ha o go o for he rae of reurn o go o -100%.

8 dcounng, rk and nequaly 7 Now, realc nvemen do no nvolve only wo ndvdual n a pecfc ae of he world. In addon, he reurn on he nvemen are rarely ceran, bu may vary dependng on he ae of he world. Le r denoe he rae of reurn n ae. The expeced rae of reurn hen defned a he oluon r o e r = m p e r, 10 =1 and he relave reurn n ae defned a θ = er e r. 11 Conder an $ε nvemen ha all ndvdual from he curren perod 0 make ogeher, and whch hared beween hem ung he ae-ndependen harng rule σ N 0, uch ha N 0 σ = 1. The aggregae reurn hared n perod and ae by ndvdual j N ung he harng rule σ j j N, uch ha j N σj = 1. Indvdual j n ae and perod herefore receve σ j θ e r ε. If we conder an nvemen wh expeced aggregae reurn r, ung he evaluaon funcon 3 and aumng ha ε mall enough, he welfare change nduced by he nvemen df = m p =1 N 0 σ W ε + m p σθ j e r W j ε. 12 We wan o defne a rk-and-equy adjued ocal dcoun rae δ, uable for dcounng he expeced fuure aggregae reurn of he nvemen. Followng he mehodology ued for ae-pecfc peron-o-peron dcoun rae, h rk-and-equy adjued ocal dcoun rae δ he rae r uch ha he welfare change n Eq. 12 df = 0. Th yeld he followng defnon. Defnon 2 The perod rk-and-equy adjued ocal dcoun rae δ for a projec wh relave reurn θ S and harng rule σ N 0 =1 j N m δ = 1 ln =1 p N 0 σ W m =1 p j N σj θ W j and σ j j N, S :. 13 The fac ha he dcoun rae depend on characerc of each parcular nvemen o be evaluaed hare σ, σ j, relave reurn θ hould no lead o he confuon ha h formula compue an nernal rae of reurn of he nvemen,.e., he dcoun rae ha would render h nvemen a maer of ocal ndfference. Invemen wh he ame σ, σ, j θ parameer bu dfferen expeced aggregae reurn r hould be evaluaed wh he ame ocal dcoun rae gven by 13.

9 dcounng, rk and nequaly 8 We can expre he rk-and-equy adjued ocal dcoun rae a a generalzed mean of he ae-dependen peron-o-peron dcoun rae. To do o, le w = W/ m =1 p W denoe he ocal prory of ndvdual of generaon 0 n ae relave o her expeced prory. Propoon 1 For any p, c L, he perod rk-and-equy adjued ocal dcoun rae δ for a projec wh relave reurn θ {1,,m} and harng rule σ N 0 and σ j j N, S gven by he formula: δ = 1 ln Proof. From Eq. 13, we have: N 0 σ m p wθ =1 m δ = 1 ln =1 p N 0 σ W m =1 p j N σj θw j = 1 ln N 0 σ = 1 ln N 0 σ = 1 ln N 0 σ j N m =1 pw m =1 p =1 j N σj θ W j m p θ j N σ j m p wθ =1 j N σe j δ,j e δ,j W m =1 p W σe j δ,j he nex o la ep ung he fac ha, by Defnon 1, W j = e δ,j W. If w no unform acro ae of he world, he formula n Prop. 1 no ju a generalzed mean of he δ,j. When all δ,j are dencal = δ acro ndvdual and ae, he formula become δ = δ + 1 ln N 0 σ m 1 p wθ, o ha he dcoun rae lower han δ f he nvemen ha greaer reurn n he ae n whch nveor or, equvalenly, benefcare, nce a conan δ,j =1, mean ha her prore are perfecly correlaed have greaer prory, a meaured by w. In he nex econ we provde a more ranparen analy of h formula. In all he above formula, a key varable he ocal prory of ndvdual conumpon. The normave choce of funcon W wll be crucal n deermnng h value.

10 dcounng, rk and nequaly 9 For nance, n he well-known Toal Ularan cae defned n Eq. 4, h value gven by he margnal uly of conumpon W = u c. Th alo he cae for he Crcal-Level Ularan creron 5, o ha he crcal level doe no affec ocal dcounng n ha cae. 3 In boh cae, w = u c / m =1 u c, o ha w = 1 when here no uncerany abou fr perod conumpon, whch furher mplfe Eq. 14. Th mple deny beween he margnal uly of conumpon and ocal prory no longer he cae for he more general EDE creron 6. Indeed, hen, W = u c φ uc n φ EDEc 1, 15 where EDEc = φ 1 1 n N φ uc. Th expreon how ha he ocal prory of an ndvdual may depend on a lea hree facor n addon o he margnal uly of conumpon: a nequaly averon repreened by an addonal equy wegh o nure welfare raher han conumpon equaly; b populaon ze; c global welfare n he whole populaon ncludng all preen and fuure people. 3 Rk on he reurn, background rk and populaon rk The leraure udyng he mpac of rk on he ocal dcoun rae generally dnguhe wo form of he rk. One he background rk ha affec he baelne cenaro under conderaon. The oher he rk on he reurn of he nvemen. A hrd mpac ha been uded by a maller rand of he leraure: he rk on he exence of fuure generaon. I brefly ackled n he Sern repor Sern, 2006, Appendx o Chaper 1, followng he emnal conrbuon by Dagupa and Heal Bommer and Zuber 2008, Ahem and Zuber 2013, and Fleurbaey and Zuber 2013b are recen conrbuon addreng h knd of rk. In h econ, we how how he hree apec of rk ener he formula for he ocal dcoun rae n our more general framework. More precely, we eparae hree erm. A fr erm he probably of he exence of fuure generaon. A econd erm he weghed ocal prory weghed by he harng rule, whch repreen he background rk: unfavorable ae of he world have a hgher ocal prory. The hrd erm nvolve he reurn on he nvemen. To decrbe how hee hree erm ener he formula for he ocal dcoun rae, we need o nroduce ome addonal noaon. For wo varable x and y, expeced value and covarance are denoed Ex = m =1 p x and covx, y = m =1 p x Exy Ey. 3 The crcal level may however affec he choce of he opmal polcy, a hown n Mllner 2013.

11 dcounng, rk and nequaly 10 For any j N, h probably of exence p j = :j N p, and he expeced value and covarance condonal on h exence are E j x = :j N p p j x and cov j x, y = :j N p p j x E j xy E j y. Propoon 2 For any p, c L, he perod rk-and-equy adjued ocal dcoun rae δ for a projec wh relave reurn θ S and harng rule σ N 0 gven by he formula: δ = 1 ln N 0 σ j EjσjW j p j N EW E j θ Proof. From Eq. 13, we have: m δ = 1 ln =1 p N 0 σ W m =1 p j N σj θw j = 1 ln m p σθ j W j N 0 σ = 1 ln N 0 σ = 1 ln N 0 σ = 1 ln N 0 σ = 1 ln N 0 σ 1 EW j N =1 1 EW j N 1 + cov j θ E j θ, σ j W j E j σ j W j 1 p θσ W j j :j N EjσjW j EW p j E j θ j N p j E j θ j N j EjσjW j E j θ EW j N p :j N 1 p p j EjσjW j EW E j θ θ σw j E j θ j E j σ j W j σ j W j E j θ E j σ j W j 1 + cov j θ E j θ, where he la ep nvoke E j xy = E j xe j y + cov j x, y. and σ j j N, S σ j W j E j σ j W j 16 1 I poble o dnguh hree erm n formula 16. Fr he probably of he exence of he fuure ndvdual j, p j. Second, he relave prory of ndvdual j weghed by her hare of he benef, Ej σ j W j. Th erm mplcly capure he background rk, EW whch wll deermne wheher fuure people are more or le well-off. I may alo nclude he rk on populaon, for he relave prory of ndvdual j may depend on populaon ze, a exemplfed n Equaon 15. E j θ 1 + cov j θ E j θ, σ j W j E j σ j W j, Thrd, a erm nvolvng he rk on he reurn,, whch elf decompoed no a erm of aocaon beween j exence and he relave reurn, E j θ, and a erm of aocaon beween j ocal prory and he relave reurn.

12 dcounng, rk and nequaly 11 If one abrac from he ue of he exence of fuure generaon, formula 16 relaed o exng formula n he cae where here boh a background rk and a rk on he reurn. For nance, wh a Dcouned Ularan evaluaon funcon wh only one ndvdual per generaon, Goller 2012, Chap. 12 propoe o evaluae fuure cah flow by fr compung a cerany equvalen of he rky cah flow, and hen dcounng ung a dcoun rae for rk-free nvemen. To follow h noaon, wh whch he creron e δ u c, =0 and he rky cah flow a random varable B, he fuure cah flow hould be evaluaed ung he formula e r F, where r = δ 1 ln Eu c and F = EB u c Eu c. Eu c 0 If one would lke o dcoun he expeced reurn nead, defnng θ = Eu c 0 Eθ u c Eu c B EB and followng Goller lne of argumen, one could alo ue he formula e r EB, where r = δ 1 ln Eu c 1 ln. Realzng ha u c he ame a W n he Ularan pecal cae, clear ha formula 16 exend Goller formula. The dfference herefore ha our formula ued o drecly dcoun expeced cah flow, raher han he cerany-equvalen of he cah-flow. I poble o follow h roue o defne a rk-adjued peron-o-peron dcoun rae. Defnon 3 For any p, c L, and for any N 0, j N, he rk-adjued peron-operon dcoun rae from peron o peron j, denoed δ,j, : δ,j = 1 ln p j 1 ln E j W j E W 1 ln 1 + cov j θ, W j E j W j. 17 When here no uncerany abou he compoon of he populaon and he harng rule n perod, he rk-and-equy adjued ocal dcoun rae can be wren a a generalzed mean of he rk-adjued peron o peron dcoun rae: Corollary 1 Aume ha p, c L uch ha, for all S, N, N = N a and σ j = σ j a for all j N a. The perod rk-and-equy adjued ocal dcoun rae δ gven by he formula: δ = 1 ln N 0 σ σae j δ,j j N a 1 18 Th formula generalze equaon 21 and 24 n Fleurbaey and Zuber 2013a. One key facor n Equaon 16 he covarance beween he relave reurn and he relave prory of fuure people n dfferen ae of he world. A pove covarance wll

13 dcounng, rk and nequaly 12 decreae he dcoun rae, and herefore ncreae he value of fuure co and benef. To underand wha gn we can expec for h erm, we now dcu a mple example. Example 1: Reurn are proporonal o aggregae conumpon. Aume ha reurn are proporonal o aggregae ncome, whch we model a reurn beng proporonal o conumpon. Aume ha ocey face a mple bnomal rk. Wh probably p, he growh rae g forever, o ha c = e g c 0. Wh probably 1 p, he growh rae g for ever, o ha c = e g c 0. Gven ha he reurn proporonal o oal conumpon θ = c Th mean ha wh probably p, θ e = g, and wh probably 1 p, θ = pe g +1 pe g e g. pe g +1 pe g Throughou he example, we aume ha here are N dencal ndvdual n each generaon and no uncerany on he plannng horzon all generaon ex unl perod T >. We conder wo ocal evaluaon funcon: he Toal Ularan creron 4 and he Equally Drbued Equvalen creron 6. The Ularan cae: Aume ha u a power funcon uc = c 1 η /1 η. Then, for j N : W j = c j η and E j W j E W o ha, when uffcenly large, = pe ηg + 1 pe ηg 1 ln E j W j E W ηg, whch mlar o Wezman 2009: when he probably of very bad oucome uffcenly large, he dcoun rae end o mnu nfny. We can alo compue E θ W j EW j = = pe η 1g + 1 pe η 1g p 2 e η 1g + 1 p 2 e η 1g + p1 pe η+1g + p1 pe η+1g pe 2η 1g + 1 p p 2 e 2ηg + 1 p 2 e 2g e 2g + p1 p + p1 pe 2η+1g o ha, when η > 1 and when uffcenly large, 1 ln 1 + cov j θ W, j = 1 E j W j E ln The rk on he reurn ncreae he ocal dcoun rae. θ W j E j W j 2g. The Equally Drbued Equvalen cae: Conder he Equally Drbued Equvalen creron 6, and aume ha u and φ are power funcon,.e., uc = c 1 η /1 η, where Ec.

14 dcounng, rk and nequaly 13 η > 1 and φv = v 1+γ /1 + γ. 4 Denoe κ = η 11 + γ. Then we have, for j N : where c τ = c k for k N τ. Hence, T γ/1+γ W j = c j 1+κ T + 1 1/1+γ c τ κ, 1 e κgt +1 γ/1+γ E j W j = pe 1+κg 1 e κg +1 pe 1+κg e κgt +1 γ/1+γ 1 e κg 1 E W 1 e κgt +1 γ/1+γ p +1 p e κgt +1 γ/1+γ 1 1 e κg e κg 1 γκ = e 1+κg p+1 pe21+κg 1+γ gt γκ. p+1 pe 1+γ gt Aumng ha T > τ=0 21+κ1+γ κγ, he econd erm n he above equaon end o 1 when g +. For large value of g, one herefore oban 1 ln E j W j E W 1 + κg, whch rkngly dfferen from he Ularan cae. The reaon for h dfferen reul ha ndvdual n perod have a hgher W j n he favorable cenaro when growh pove han n he bad cenaro. One compue ha he rao of W j n he good cenaro over W j n he bad cenaro equal o e whch greaer han one when T > 21+κ1+γ κγ γκ 21+κg+ 1+γ gt,. Th becaue ocal prory doe no depend only on margnal uly bu alo on he relave rankng of ndvdual n her ae of he world. When growh pove, generaon relavely le well-off when here are many fuure generaon. On he conrary, generaon relavely beer-off when all fuure generaon wll be poorer n he bad cenaro. Ung he ame mplfcaon a above, wh probably p and wh probably 1 p. Therefore E θ W j EW j = W j 1 = E j W j γκ 21+κg p+1 pe 1+γ gt γκ W j e = 21+κg 1+γ gt E j W j γκ 21+κg p+1 pe 1+γ gt γκ 2κg p+1 pe 1+γ gt γκ γκ < 0 p 2 +p1 pe g 21+κg +p1 pe 1+γ gt +1 p 2 2κg e 1+γ gt 4 The cae γ < 1 can be reaed mlarly wh φv = v 1 γ /1 γ.

15 dcounng, rk and nequaly 14 when T > 21+κ1+γ κγ, o ha 1 ln 1 + cov j θ W, j = 1 EW j E ln θ W j EW j < 0 The reaon why he covarance beween ocal prory and he rk on he reurn pove n h cae agan he greaer prory of generaon n he good cenaro, for he reaon gven above. The background rk on conumpon and he rk on he reurn are well known and uded n he leraure on ocal dcounng and n he relaed fnance leraure on valung unceran projec. Th no he cae of he rk on populaon ze whch ha only receved lmed aenon ye. The man conrbuon addreng he ue lm hemelve o nocng ha he hazard rae of he rk can be ued o pnpon he value of a parameer known a he rae of pure me preference Dagupa and Heal, 1979; Sern, A clear from Eq. 16, he relave probably of he exence of fuure generaon ndeed one apec of he rk on he populaon ze ha would ener he dcounng formula for any ocal evaluaon funcon. Bu no he only way populaon ze maer f one goe beyond he andard Toal Ularan cae, a demonraed by he followng example. Example 2: The rk on populaon ze. Aume ha here no rk on conumpon or he reurn of he nvemen. We conder a cae n whch people of a gven generaon are all equal and conume c. The only rk on he exence of he fuure generaon. Each perod, wh probably p he world urvve o he nex perod, and wh probably 1 p he human pece and any pece relevan for welfare dappear. 5 We alo conder ha poenal populaon.e. aben he exncon rk grow a a gven exponenal rae n. Hence denong n = N, we have n +1 = 1 + nn. Wh probably 1 pp, he populaon ze herefore exacly τ τ=1 n0 1 + n = n 0 1+n +1 1 n = 1 + n n0 1+n n n0 n1+n. Generaon ex wh probably p. Conder fr he Toal Ularan creron 4. Then, for any j N : E j W j = u c and, ung formula 17 and 18 we oban: δ = ln p 1 ln u c u c 0 where only he hazard rae of he exncon rk ener he dcounng formula. 5 Noe ha we conder a counably nfne number of ae of he world. All our formula can be exended o ha cae.,

16 dcounng, rk and nequaly 15 If we now urn o he Average Ularan creron.e. creron 6 wh φ an affne funcon, we oban ha for any j N : E j W j = u c 1 pp τ 1 + n τ n0 1 + n n τ= n 0 1 n1 + n τ = u c 1 + n 1 pp τ 1 + n τ n0 1 + n n τ=0 Ung formula 17 and 18 we oban: δ,j = ln p + ln1 + n 1 ln u c u c 0 1 ln τ=0 pτ 1+n τ n0 1+n n n 0 1 n1 + n τ+. n 0 1 n1+n τ+ τ=0 pτ 1+n τ n0 1+n 1 n0 n n1+n τ Neglecng he la erm, whch lkely o be neglgble for a large, eem ha adopng an Average Ularan vew, raher han a Toal Ularan vew, mple adjung he dcoun rae for populaon growh. In fac, compared o he Toal Ularan formula, we have o add up he populaon growh rae n he Average Ularan formula, whch can gnfcanly aler he dcoun rae. The example how ha mporan choce concernng populaon ehc.e., he creron ued o ae populaon of dfferen ze may deeply aler he ocal dcoun rae and herefore polcy recommendaon. Th no a debae ha can reman unaddreed.. 4 Inequaly The drbuon of co and benef one of he key ue n he heory of co-benef analy. The manream approach o he problem con n aumng an effcen drbuon of he co and benef or an mplc redrbuon compenang hoe reaed unfarly, o ha equy conderaon can be dpened wh. If one doe no wan o aume away he equy ue, he andard echnque n cobenef analy con n nroducng equy wegh. Thee are wegh on he co and benef dependng on ndvdual relave welfare: he le well-off receve hgher wegh. The echnque ha been ued n he cae of clmae change o adju he ocal co of carbon for equy conderaon. We propoe a echnque o ncorporae equy conderaon drecly n he dcoun rae. We fr hghlgh he mporance of he covarance beween he ndvdual relave ocal prory and her hare of co and benef. To do o, we need o nroduce ome addonal noaon. For j N, w j = W j / 1 n k N W k he relave ocal prory of j wh repec o ndvdual n he ame generaon and ae of he world. We alo defne Covpopx, y = 1 n j N x j 1 n k N xk y j 1 n k N yk.

17 dcounng, rk and nequaly 16 Propoon 3 For any p, c L, he perod rk-and-equy adjued ocal dcoun rae δ for a projec wh relave reurn θ S and harng rule σ N 0 and σ j j N, S gven by he formula: m δ = 1 ln =1 p 1 n 0 N 0 W [ 1 + n 0 Covpop 0 σ 0, w 0 ] m =1 p θ 1 [1 n j N W j + n Covpop σ, w ] 19 Proof. From Eq. 13, we have: m δ = 1 ln =1 p N 0 σ W m =1 p = 1 ln m =1 p j N σj θw j 1 n 0 N 0 W N 0 σ w m 1 =1 pθ n j N W j j N σj w j m = 1 ln =1 p 1 n 0 N 0 W [ 1 + n 0 Covpop 0 σ 0, w 0 ] m =1 p θ 1 [1 n j N W j + n Covpop σ, w ], where he la ep ue he fac ha b k = k a k b + k a k ā b k b, leng ā and b denoe he average value. Noe ha N 0 σ = j N σj = 1 and 1 n N 0 0 w = 1 n j N wj = 1. Formula 19 nroduce he erm Cov 0 pop σ 0, w 0 and Cov pop σ, w o ake no accoun he equy n he drbuon of co and benef. When co are born by he poor oday, h wll herefore end o ncreae he dcoun rae: he nvemen le valuable becaue nvolve ncreang nequaly oday. Bu f he benef are receved by he poor omorrow, h wll end o decreae he dcoun rae: he nvemen more valuable becaue wll reduce nequaly omorrow. One apec of equy no hghlghed n formula 19: he level of nequaly oday and omorrow. I mplcly aken no accoun n he average ocal prory of people n generaon and ae. For mo ocal evaluaon funcon, he expreon 1 n j N W j ncreang wh nequaly. For nance, n he Ularan cae, f u c = 1 1 η c1 η and η > 0, one oban u c = c η, an expreon ha decreae when a progreve ranfer made beween wo agen. Therefore formula 19 how ha ncreang nequaly n he fuure end o lower he dcoun rae. Inuvely, a more unequal fuure deerve more nvemen from he preen generaon, becaue, oher hng equal, more people wll be n grea need of he benef of he nvemen.

18 dcounng, rk and nequaly 17 The rao 1 n j N j W j / 1 n 0 N 0 W no mply a meaure of he evoluon of nequaly becaue alo ncorporae conumpon growh whch decreae prory when pove. Separang he wo apec poble wh many ocal evaluaon funcon, a hown by he followng example. Example 3: The role of nrageneraonal nequaly. Conder he cae of a generalzed addve ocal evaluaon 7, wh power funcon u c = 1 1 η c1 η, for 0 < η < 1, and φ u = 1 1 γ u1 γ, for γ > 0. 6 Denoe υ = η + 1 ηγ and le c 0 and c denoe he average conumpon n generaon 0 and, repecvely. In ha cae, W j = c j υ Ψ 1 υ nc, o ha, The erm 1 n W j = c υ j N Eq = 1 n j N 1 n j N c 1 υ N c j υ c Ψ n c j c 1/υ υ c 1 υ 1 υ N a meaure of equaly, whch equal o 1 when he uaon perfecly equal and end o 0 when one ndvdual conume everyhng. In he mple uaon where here no uncerany, one oban 1 δ = 1 ln n 0 N 0 W [ 1 + n 0 Covpop 0 σ 0, w 0] [1 j N c W j + n Covpop σ, w ] 1 n c c = υ 1 Eq ln c 0 + υ n ln Eq 0 1 ln Covpop σ, w 1 + n 0 Covpop 0 σ 0, w 0 In h formula, he dcoun rae can herefore be decompoed n hree erm, whch all repreen one apec of equy. The fr erm relaed o he growh of average conumpon and accoun for nergeneraonal equy. The econd erm relaed o he change n equaly whn generaon, and herefore accoun for nrageneraonal equy. The la erm meaure he evoluon of he covarance beween relave welfare and he hare n co and benef; accoun for he equy n he drbuon of co and benef. 6 The ame reaonng would hold for u c = 1 1 η c1 η, for η > 1, and φ u = γ > 0., 1 1+γ1 η u1+γ, for

19 dcounng, rk and nequaly 18 5 Long-erm dcounng and he nformaon needed o evaluae clmae polce The formula ha we have preened above requre a lo of nformaon abou he drbuon of co and benef, he drbuon of wealh whn generaon and he probably of he dfferen ae of he world. A key queon whch nformaon mu be known wh good accuracy o evaluae very long-erm mpac of polce. The reul of h econ how ha, forunaely, only a mall fracon of he nformaon relevan n he very long erm. The followng Propoon ae ha he mo mporan nformaon, for he compuaon of he ocal dcoun rae, he maxmum poble reurn of he nvemen, and he maxmum poble reurn ne of a pecfc peron-o-peron dcoun. We can dpene wh he drbuon hare of mpac and of conrbuon and wh he probable of he dfferen ae. Propoon 4 Conder p, c L uch ha for all N, for all S, for all j N, σ j σ 0 for ome σ 0 > 0. When, he perod rk-and-equy adjued ocal dcoun rae δ afe he formula: Proof. See Appendx A. δ = max S r mn max N 0 S,j N r δ,j + O1/ 20 In 20, he expreon O1/ mean ha here A > 0 uch ha max δ r mn max r δ,j < A/ S N 0 S,j N for hgh enough. Therefore h approxmaon reul compable wh he fac ha he drbuon of r and δ,j may vary wh me and even dverge. Obvouly, when goe o nfny, he dcoun rae converge no ju o he formula bu o he lm of he formula f here one. For nance, f he growh rae of conumpon end o zero n he very long run, he peron-o-peron dcoun rae δ,j all, j, and he lm dcoun rae zero. end o zero for In order o underand whch ndvdual N 0 and j N are relevan n he compuaon of mn N 0 max S max j N r δ,j, fr oberve ha for a gven and, he greae r δ,j obaned for j havng he lowe δ,j,.e., he greae W j. Tha wll be he mo dadvanaged ndvdual n ae. Now, aume ha for every gven and j, he maxmum value of δ,j obaned for he ame ndvdual,.e., for every he ame ha he greae W. Alhough n general

20 dcounng, rk and nequaly 19 he ndvdual wh he greae W mgh depend on, que naural, when here no rk on generaon 0, ha no uch dependence occur. For nance, when he ocal evaluaon funcon ake he generalzed addve form 7, one mply ha δ,j = 1 ln u c φ uc, u c j φ uc j o ha he ndvdual wh he greae W he ndvdual wh he lowe conumpon. When he ame ndvdual ha he greae W for all and obvouly for all j a well, one ha mn max N 0 S,j N r δ,j = max S r max mn δ,j N 0 j N To compue he long-erm dcoun rae, one can hen proceed a follow. For any gven S, focu on he wor-off-o-wor-off dcoun rae, and compue a ne reurn on he nvemen ung h dcoun rae he ne reurn beng he dfference beween he reurn and he dcoun rae. Then pck he ae S n whch h ne reurn maxmal. The dfference beween he maxmum reurn and he maxmum ne reurn he ocal dcoun rae. In cae he hghe ne reurn r δ j obaned n he ae where r greae, he fnal formula mply max mn δ,j N 0 j N. Th o when hgh reurn occur n ae where he wor-off of he fuure generaon lve n he deepe povery, a uaon ha may be he cae for clmae change. If clmae change affec growh and he clmae damage are hgher n hgh-emperaure cenaro, an nvemen o reduce clmae damage may be more profable n bad ae where fuure generaon are poor. Bu f greae ne reurn r δ,j obaned for a low r, becaue he reurn are correlaed wh he well-beng of j, hen he dcoun rae greaer han max N 0 mn j N δ,j for any ae. Th mean ha nvemen ha pay when he fuure wor-off are epecally badly-off hould be evaluaed wh a lower dcoun rae han nvemen payng when he fuure wor-off are le badly-off. A an exreme example, conder a cae n whch here no nequaly whn generaon and n every ae he dcoun rae an ncreang funcon of he rae of reurn, wh a coeffcen greaer han one a n a Ramey formula wh elacy of uly greaer han one, and a growh rae equal o he rae of reurn on nvemen. Then he maxmum of r δ,j obaned for he lowe r, and he dcoun rae hen equal o max S r mn S r +mn S δ,j, where and j are any repreenave agen of her generaon.

21 dcounng, rk and nequaly 20 Formula 20 alo provde reul whch are remncen of Goller 2004 and Wezman 1998 analye of he long-run dcoun rae, n pecal cae. When he rae of reurn on he nvemen, n every ae of he world, equal o he dcoun rae e.g., becaue of a marke equlbrum condon, a n Goller and Wezman 2010, or when here no nequaly whn generaon, he rae of reurn he ame a he growh rae, and uly logarhmc, hen he econd erm vanhe and he dcoun rae on expeced reurn, n he lm, he maxmum poble rae of reurn, a n Goller In conra, f here no uncerany abou he rae of reurn, hen r dappear from he formula, and n abence of nequaly whn generaon, mplfe no mn S δ,j, where and j are any repreenave agen of her generaon. Th mlar o Wezman 1998 perpecve on he ue. 7 Alhough Eq. 20 hed lgh on wha maer n he very long run, doe no mply ha he approxmaon formula can be appled whou precauon. In parcular, he approxmaon for he dcoun rae may be reaonable only n he very long run, a he convergence of he bracke o zero may be low n ome cae. The error on he dcoun rae made by ung he approxmaon formula bounded by A/, for A = max ln mn σ + ln mn p, σ /σ 0 N 0 S mn S p w N 0, whch vare lke 1/ ln 10 k 2.3k/, where k may be a hgh a 12 n parcular, σ 0 may be of he order of magnude of a bllonh, and he lowe p of he order of a percenage pon. One hen need o go beyond = 3000 n order o make he error go below one percenage pon. The large uncerany abou he mpac of our curren acon on uch remoe me may render he approxmaon uele, a he relevan conequence for decon-makng may be n he medum-erm, before he approxmaon correc. On he oher hand, he followng example how ha one can fnd cae for whch he convergence happen whn a few hundred year. Example 4: The long run dcoun rae. Aume ha we ue a Toal Ularan ocal welfare funcon, and ha u a power funcon uc = c 1. 7 Goller and Wezman argumen were no baed on a ocal welfare funcon. Goller 2004 noed ha, when, he expeced fuure ne value of an nvemen drven by he greae rae of reurn, wherea Wezman 1998 oberved ha he expeced preen value of an nvemen drven by he lowe dcoun rae when here uncerany abou growh. Neher he expeced fuure value nor he expeced preen value are generally he relevan creron n our ocal welfare approach.

22 dcounng, rk and nequaly 21 Conder a ocey compoed of wo group of people: he rch and he poor. populaon conan, he wo group have he ame ze n all perod, and co and benef are hared on a per capa ba. In perod 0, he rch conume fve me a much a he poor. The ocey face he followng rk. Wh probably 0.9 a good ae of naure, he conumpon of he rch grow a a 1.5% rae, and he conumpon of he poor grow a a 1.3% rae. Wh probably 0.1 a bad ae, he conumpon ay conan for boh he rch and he poor. We conder hree knd of nvemen. A fr knd of nvemen Invemen 1 only yeld a reurn n he bad ae of he world. A econd knd of nvemen Invemen 2 yeld a hgher reurn n he bad ae, bu ll ha a pove reurn n he good ae: he dfference beween he wo rae of reurn 1%. A la knd of nvemen Invemen 3 yeld he ame reurn n he wo ae. Tme mn,j, δ,j max,j, δ,j δ δ1 δ 2 δ The Table 1: Long-run convergence of he ocal dcoun rae Table 1 repor he maxmum and mnmum peron-o-peron dcoun rae, he approxmae formula δ = max S r mn N 0 max S,j N r δ,j, and he correc dcoun rae δ k, k = 1, 2, 3, for hee dfferen knd of nvemen for dfferen me horzon. In all cae preened above, he error of Approxmaon 20 le han 0.5 percenage pon afer 500 year. The approxmaon almo mmedaely correc for Invemen 1, where only he bad ae of he world maer. For he oher wo nvemen, he error around 1 percenage pon for a 100 year horzon, and 2 percenage pon for a 100 year horzon. So, whle he make can be ubanal for horer erm mpac, lmed for mpac a reaonably long horzon. Approxmaon 20 relevan for he compuaon of he ocal dcoun rae. Bu he mporan erm for he compuaon of he ne preen value of an nvemen e r δ, for whch h approxmaon reul no very ueful, a error on he dcoun rae ge compounded wh. Le u herefore examne how one can approxmae e r δ for large

23 dcounng, rk and nequaly 22 value of. To do o, we nroduce he followng condon. Condon C For p, c L, defne he followng e: { } J, = j N r δ,j = max r δ,j j N, { S = S max r δ,j j N = max S,j N I = { N 0 max S,j N r δ,j = mn r δ,j }, max N 0 S,j N r δ,j }. A loery p, c L afe Condon C f here ex α > 0 uch ha: mn / I max S,j N max S,j N r δ,j mn r δ,j max j N r δ,j max N 0 S,j N max / S,j N max j / J, r δ,j > α for all N, r δ,j > α for all N, N 0, r δ,j > α for all N, N 0, S. Condon C enure ha hey are uffcen dvergence beween he ne dcoun rae ung dfferen peron-o-peron ae-pecfc dcoun rae. If were no he cae, he ne reurn would be he ame n all ae of he world whaever he drbuon of he mpac, and we wan o leave ade h degenerae cae. Recall he followng andard defnon: a funcon f o 1/ f for all A > 0, f < A/ for hgh enough. Followng mehod mlar o he one ued o prove Propoon 4, one oban he followng reul on he preen value of he reurn on he nvemen. Propoon 5 Conder a loery p, c L afyng Condon C. Aume alo ha here ex σ 0 > 0 uch ha σ j σ 0 for all N, all S, all j N. Then, he preen value of he reurn a afe he formula: 1 e r δ = σ p w σ j I S Proof. See Appendx B. j J 1 mn + o1/ e N 0 max S,j N r δ,j. Propoon 5 how ha, when one look a he preen value of he nvemen, a role for hare and probable reored n he approxmaon for he long-run value. Bu he Propoon alo confrm he mporance of he expreon mn max N 0 S,j N r δ,j. I hould be clear ha h analy lead u very far from he uual pracce of akng dcoun rae baed on average uaon and ordnary marke condon.

24 dcounng, rk and nequaly 23 6 Exenon 6.1 Overlappng generaon Unl now, we have condered ha ndvdual lved for only one perod, or alernavely ha he uly of an ndvdual n dfferen perod condered eparaely n he ocal evaluaon. Of coure, ndvdual lve for everal perod, and arguably make ene normavely ha he ocal evaluaon ake no accoun lfeme uly Broome Conderng people lvng for everal perod mple conderng an overlappng generaon framework. I wll alo renroduce and clarfy he role of ndvdual conumpon dcoun rae n he ocal evaluaon. In h econ, we lghly aler he framework o nroduce overlappng generaon. 8 We aume ha ndvdual lve for A perod. 9 Hence, alernave are now elemen of C = N N\ N RA +. For any c C, a before, we le Nc be he e of ndvdual alve n he alernave and nc = N c be he number of ndvdual n he alernave. For any c C and Nc, c = c 1,, c A he conumpon ream of ndvdual n alernave c. For any a {1,, A}, c a herefore he conumpon of ndvdual a age a. We aume ha here ex a paron of N no ube N conanng he poenal ndvdual of generaon N 0 {1 A,, 1}. The generaon of an ndvdual he fr perod of h exence. 10 Hence, ndvdual N lve n perod,, +A 1 when he ex, and we le G = denoe h generaon name. For τ = G,, G + A 1, we alo le a τ = τ + 1 G denoe he age of ndvdual n perod τ. We rerc aenon o C = {c C 0 τ=1 A N τ Nc}. The currenly exng people are preen n all alernave. We alo need o know whch ndvdual lve n any gven perod n a gven alernave. We herefore change he noaon N c, and denoe N c = τ= A+1 N Nc. N c herefore he e of all ndvdual lvng n perod n alernave c. Accordngly, we denoe n c = N c he number of all ndvdual lvng n perod n alernave c. All 8 Our reul herefore complemen hoe by Dagupa and Makn preened n Dagupa 2012, Secon 6. In h man reul, Dagupa 2012 conder a dynay of ucceve non overlappng generaon, where he curren generaon ue a ularan creron. The non overlappng rucure may nduce cycle n he dcoun rae, whle he undcouned ularan creron mple no pure me dcounng. framework more general n he ene ha we do no comm o he Ularan ehc whle we do no exclude. Our overlappng rucure would n general no generae cycle n he ocal dcoun rae. 9 Exenon o he cae where ndvdual have dfferen lengh of lfe raghforward bu much more nvolved noaonally. 10 Tha why we have o nclude generaon {1 A,, 1} who ll lve n perod 0. Our

25 dcounng, rk and nequaly 24 oher noaon reman he ame. We alo have o modfy he pecfcaon of ocal welfare. We aume ha for each poenal ndvdual N here ex a uly funcon u whch ued n he ocal evaluaon o ae h welfare gven a pecfc conumpon ream c. Therefore, we add he rercon afed by all pecfc crera nroduced n Secon 2.2 ha, for any c C: W c = F n u c. N Denong u a = u c / c a, F = F/ u and W W = u a F = ua u 1 = W c / c a, we oban: u 1 F. The rao u a /u 1 he margnal rae of ubuon beween conumpon a age a and conumpon a age 1 for ndvdual. Hence d a = ln u 1 /u a /a 1 approxmavely he conumpon dcoun rae a age a for ndvdual along a conumpon pah c. We are now able o rewre he ae-pecfc peron-o-peron dcoun rae from peron lvng n perod 0 o peron j lvng n perod n ae, ha we denoe δ,j. 11 I mply = 1 ln W 0 = 1 ln δ,j = djaj = djaj W j e da0 a 0 1 u 1 F e djaj a j 1 u j1 F j a j 1 d a0 a a0 a j a j 1 d a0 a Gj G ln u 1 F 0 u j F j + +a0 a j D,j. 21 Namng D,j 1 = Gj G ln u 1 F 0 he ae-pecfc peron-o-peron brh dcoun u j F j rae from peron o peron j, we ee ha h erm promnen n he long-run. Indeed, f conumpon dcoun rae d a are bounded for all relevan alernave, he fr erm n Eq. 21 end o 0 when goe o nfny. Smlarly, he erm +a0 a j goe o one. Th mean ha, n he long-run, he age of he peron nvolved n he nvemen and her conumpon dcoun rae do no play much role. brh dcoun rae wll maer. Only ae-pecfc peron-o-peron On he oher hand, for horer erm nvemen, we need o ake no accoun ndvdual conumpon dcoun rae. A mple example of coure when = j, ha when an ndvdual make an nvemen for h on ake. In ha cae, he econd erm n Eq. 21 dappear by defnon, D, = 0. Only he ndvdual conumpon dcoun rae maer 11 Becaue ndvdual lve for more han one perod, we now need o pecfy he perod of he fuure conumpon.

26 dcounng, rk and nequaly 25 for h knd of nvemen and he ocey doe no nerfere. The ndvdual conumpon dcoun rae ypcally he knd of nvemen we oberve n fnancal marke. The nformaon on marke rae herefore very relevan o he ocal dcoun rae for hor erm nvemen. Bu hey do no provde much gudance for long erm nvemen. Fnally, remark ha all our reul exend o he OLG cae, n parcular Propoon 4 and 5. I uffce o replace he value of δ,j 21. by he expreon for δ,j dplayed n Eq. 6.2 Concluon: Beyond he dcoun rae Sandard co-benef analy uually conder he mpac of polcy on conumpon. Conumpon can be nerpreed n a raher comprehenve way, ncludng non-marke good and publc good uch a bodvery and ecoyem ervce alhough h mple he dffcul ak of agnng a value o hee good. One omnou apec of clmae change, however, ha may hreaen lvelhood on earh, and herefore he mere exence of many fuure generaon no only human, bu alo for oher pece. I may herefore be he cae ha polcy affec no only conumpon, bu alo he propec ha we face concernng he fuure. We may be able o change he probably of fuure caarophc even. The echnque ued o evaluae change n probable, when here a rk on he exence or longevy of an ndvdual, con n compung he value of a acal lfe VSL. We can exend he mehodology n he cae of rk on he exence of fuure generaon he dea wa uggeed, bu ued n a very dfferen way by Wezman, Suppoe ha he goal of a polcy o hf probably from ae o ae by a mall amoun δ. Wha co ε can be mpoed on ndvdual who lvng now, for he ake of mplemenng h polcy? 12 The varaon n ocal welfare equal o m df = δ W c W c p Wε, and equal o zero when ε δ = W c W c m =1 p W. 22 The rgh-hand-de of he above expreon defne a concep mlar o VSL, ha we can name he ocal value of rk reducon. I deermne how much he ocey ready o pay for a mall reducon of he rk repreened by cenaro realzng. 12 Lke before, he co of he polcy could be hared beween dfferen ndvdual of he curren generaon, nvolvng addonal equy conderaon. Here we leave hee complcaon ade for he ake of mplcy. =1

27 dcounng, rk and nequaly 26 I urn ou ha he rgh-hand de of Eq. 22 can be reformulaed ung he perono-peron dcoun rae, when he wo ae and are no oo dfferen, and n parcular when he ame populaon N = N W c W c m =1 p W lve n he wo ae. In ha cae, j N W j c j c j m =1 p W = c j c j = w W m =1 pw j N j W W j N e δ,j c j c j In h cae, he ocal value of rk reducon proporonal o he dcouned um of he conumpon gan for all people n all generaon. Th expreon nuve: equal o he ocal wllngne o make pay for a 100% probably hf from o. In he cae of clmae change hough, here are reaon o hnk ha he margnal analy ued n he approxmaon wll no hold. Fr, we may wan o hf probably from a caarophc cenaro n whch people are all deprved o a very dfferen cenaro n whch people are able o enjoy much beer lve. Second, we would lke o hf probably from a caarophc cenaro n whch few generaon ex o a cenaro n whch many more generaon ex. Th nvolve comparng populaon of dfferen ze, and herefore makng hard ehcal decon concernng he value of addonal lve. Conder for nance he crcal level ularan creron 5. Ung h creron, and aumng ha N N, we oban:. W c W c m =1 p W = j N uc j uc j u c + j N j N \N uc j ū u c e δ,j c j c j + j N \N uc j ū u c. The new expreon crucally depend on he crcal level ū. If we e hgh andard for he fuure, may be he cae ha he econd erm n he expreon negave, becaue we are no able o boh ncreae he urvval of humany and keep hgh andard of lvng. If on he conrary ū raher low, he ocal value of caarophc rk reducon may be very hgh, uggeng an addonal and perhap more powerful reaon why clmae polce are ocally valuable. In any cae, he reaonng ugge ha we mu go beyond radonal co-benef analy valung fuure aggregae conumpon gan from polcy ung a ocal dcoun rae.

28 dcounng, rk and nequaly 27 Polce devoed o mgang he rk of clmae change ypcally modfy he probable of varou cenaro and change he level of conumpon for varou ubgroup of he populaon n he cenaro. Hence, ryng o conver he co and benef of uch a polcy no moneary amoun and comparng he correpondng oal rae of reurn o a ngle macroeconomc benchmark n he form of a dcoun rae may be a be que roundabou and a wor very mleadng. Reference Anhoff, D., Hepburn, C., Tol, R.S.J Equy Weghng and he Margnal Damage Co of Clmae Change, Ecologcal Economc, 68, Arrhenu, G Populaon Ehc The Challenge of Fuure Generaon, forhcomng. Ahem, G.B., Zuber, S Ecapng he Repugnan Concluon: Rank-Dcouned Ularanm wh Varable Populaon, forhcomng n Theorecal Economc. Azar, C., Serner, T Dcounng and Drbuonal Conderaon n he Conex of Global Warmng, Ecologcal Economc, 19, Bommer, A., Zuber, S Can preference for caarophe avodance reconcle ocal dcounng wh nergeneraonal equy?, Socal Choce and Welfare, 31, Blackorby, C., Donaldon, D Socal crera for evaluang populaon change, Journal of Publc Economc, 25, Blackorby, C., Boer, W., Donaldon, D Populaon Iue n Socal Choce Theory, Welfare Economc, and Ehc, Cambrdge: Cambrdge Unvery Pre. Broome, J Weghng Lve, Oxford: Oxford Unvery Pre. Dagupa, P.A Dcounng clmae change, Journal of Rk and Uncerany, 37, Dagupa, P.A Tme and he generaon, n Hahn, R.W., Ulph, A. Ed. Clmae Change and Common Sene: Eay n Honour of Thoma Schellng, Oxford: Oxford Unvery Pre, Dagupa, P.A., Heal, G Economc heory and exhauble reource, Cambrdge Unvery Pre, Cambrdge. Fankhauer, S., Tol, R.S.J., Pearce, D.W The Aggregaon of Clmae Change Damage: a Welfare Theorec Approach, Envronmenal and Reource Economc, 10, Fleurbaey, M Aeng rky ocal uaon, Journal of Polcal Economy, 118,