Discounting and Divergence of Opinion

Transcription

1 Dscounng and Dvergence of Opnon Elyès Joun y Jean-Mchel Marn z Clolde Napp x{ December, 8 Absrac The objecve of hs paper s o adop a general equlbrum model and deermne he socally e cen dscoun rae when here are heerogeneous ancpaons abou he fuure of he economy as well as heerogeneous me preference raes. Among ohers we ackle he followng quesons. Is he socally e cen dscoun facor an arhmec average of he ndvdual subjecvely ancpaed dscoun facors as n he cerany equvalen approach of Wezman (998, )? As a sor of addonal rsk or uncerany, can belefs dsperson lead o lower dscoun raes? Is socally e cen, when dversy of opnon s aken no accoun, o reduce he dscoun rae per year for more dsan horzons? Does he socally e cen dscoun rae converge n he long run o he lowes ndvdual subjecvely ancpaed dscoun rae as n he homogeneous belefs seng? More generally, wha s he shape of he yeld curve?. Inroducon The concep of a dscoun rae s cenral o economc analyss, as allows e ecs occurrng a d eren fuure mes o be compared by converng each fuure dollar amoun no equvalen presen dollars. The problem of he deermnaon of a dscoun rae has acqured renewed relevance laely n order o analyse envronmenal projecs or acves he e ecs of whch wll be spread ou over hundreds of years, and he evaluaon of whch, hrough Coss and Bene s Analyss (CBA), s very sensve o he dscoun rae beng used. For nsance, concernng global clmae change, has been argued ha he srong conclusons of he Sern Revew were essenally drven by he low assumed dscoun rae (see, e.g. Nordhaus, 7 or Wezman, 7). As underlned by e.g. Nordhaus (7) or Wezman (7), here s an mporan dsncon beween he uly socal dscoun rae and he consumpon socal dscoun rae. The former refers o a pure me preference rae ha dscouns uly. I re ecs he level of mpaence The nancal suppor of he GIP ANR and of he Rsk Foundaon (Groupama Char) are graefully acknowledged by he auhors. We have bene ed from commens from parcpans a he 6h Toulouse Conference on Envronmen and Resources Economcs and a he FIME (EDF-Dauphne) semnar. y CEREMADE, Unversé Pars Dauphne, Pars, F Pars cedex 6. z INRIA FUTURS, Proje Selec, Unversé Pars-Sud x CNRS, UMR 788, F-756 Pars. { Unversé Pars Dauphne, DRM, F-756 Pars and Cres Promnen examples nclude: global clmae change, radoacve wase dsposal, loss of bodversy, hnnng of sraospherc ozone, groundwaer polluon, mnerals depleon, and many ohers.

2 or, for long me horzon projecs, he relave weghs of d eren people or generaons. The laer s he rae used o dscoun fuure consumpon. There are essenally hree deermnans of he level of hs dscoun rae. The rs deermnan s relaed o a psychologcal preference for he presen e ec and s represened by he uly dscoun rae. The more mpaen he ndvduals, he hgher he value of one un of consumpon oday relave o one un of consumpon omorrow, he hgher he dscoun rae. Bu here are oher reasons o dscoun fuure consumpon. The second deermnan s relaed o a wealh e ec whle he hrd deermnan s relaed o a precauonary savngs e ec. These wo e ecs are drecly mpaced by agens belefs abou he fuure of he economy. In hs paper we are neresed n he properes of he consumpon socal dscoun rae snce our am s o deermne he value oday (n presen dollars) of fuure dollars amoun n order o apply for CBA. A crcal feaure ha mus be aken no accoun s dvergence of opnon abou he fuure of he economy. Forecasng for he comng year s already a d cul ask. I s naural ha forecass for he nex cenury/mllennum are subjec o poenally enormous dvergence. I s doubful ha agens or economss currenly have a complee undersandng of he deermnans of long erm economc evoluons. The debae on he noon of susanable growh s an llusraon of he degree of possble dvergence of opnon abou fuure of socey. Some wll argue ha he e ecs of mprovemen n nformaon echnology have ye o be realzed and he world faces a perod of more rapd growh. On he conrary, hose who emphasse he e ecs of naural resource scarcy wll see lower growh raes n he fuure. Some even sugges a negave growh of he GNP per head n he fuure, due o he deeroraon of he envronmen, populaon growh and decreasng reurn o scales. Anoher crcal feaure ha mus be aken no accoun s he heerogeney of uly dscoun raes among agens. These raes may re ec d eren levels of mpaence. In a seng wh long-lved agens ha represen presen and fuure generaons, hese raes may also re ec dvergence of opnon abou he mporance graned o he welfare of fuure generaons relave o he presen. The mporan debae among economss (and also among phlosophers) on he noon of nergeneraonal equy s an llusraon of hs possble dvergence. Some wll argue ha nergeneraonal choces should be reaed as neremporal ndvdual choces leadng o wegh more presen welfare. Ohers wll argue ha fundamenal ehcs requre nergeneraonal neuraly and ha he only ehcal bass for placng less value on he welfare of fuure generaons s he uncerany abou wheher or no he world wll exs and wheher or no hese generaons wll be presen. In hs paper we explcly ake no accoun possble dsagreemen among agens abou he fuure of he economy and abou he reamen of fuure generaons. As underlned by Wezman (), hese and many more are fundamenally maers of judgmen or opnon, on whch fully nformed and fully raonal ndvduals mgh be expeced o d er. In such a framework, our am s o deermne he socally e cen dscoun rae and o analyse s properes; n parcular, we wan o deermne he analog of Ramsey Equaon n a farly In a deermnsc seng, he well-known Ramsey equaon gves he followng expresson for he dscoun rae R = + g;

3 general heerogeneous, me-varyng and sochasc 3 seng. Among ohers, we wsh o ackle he followng quesons. Is here general equlbrum foundaon for synheszng opnons by akng an average as n he cerany equvalen approach of Wezman (998, )? How do dscoun raes vary wh he degree of belefs and ases dvergence? Is socally e cen, when here s belefs heerogeney, o reduce he dscoun rae per year for more dsan horzons? More generally, wha s he shape of he yeld curve? The las quesons abou he shape of he yeld curve are of parcular neres. There s a wde agreemen ha dscounng a a consan posve rae for long me horzons s problemac, rrespecve of he parcular dscoun rae employed. Indeed, wh a consan rae, he coss and bene s accrung n he dsan fuure appear relavely unmporan n presen value erms. Hence decsons made oday on hs bass may expose us o caasrophc consequences n a dsan fuure. Wezman (998) summarses hs succncly when he saes : «To hnk abou he dsan fuure n erms of sandard dscounng s o have an uneasy nuve feelng ha somehng s wrong somewhere». A recenly proposed soluon o hs problem s o use a dscoun rae whch declnes over me. I s clear ha usng a declnng dscoun rae could make an mporan conrbuon owards he goal of susanable developmen. Bu wha formal jus caons exs for usng a declnng dscoun rae and wha s he opmal rajecory of he declne 4? In a deermnsc world, decreasng dscoun raes can arse as a resul of known changes n he growh rae, changes n rsk averson, ec. Addonal movaons emerge once uncerany s consdered. For example, Wezman (998, ) consders uncerany on he dscoun rae self. Sarng from he fac ha here s a huge dvergence of expecaons abou fuure dscoun raes among economc expers, Wezman (998, ) 5 nroduces a probably dsrbuon for he fuure dscoun rae and s behavour over me. The auhor adops a cerany equvalen analyss n order o deermne he dscoun raes for varyng horzons and obans decreasng dscoun raes. More generally, Goller (a), Dybvg e al.(996), Wezman (998, ) show n d eren conexs ha from oday s perspecve, he only relevan lmng scenaro s he one wh he lowes neres rae. In he presence of unceran growh, Goller (a, b) shows ha he shape of he yeld curve depends upon preferences for rsk and prudence, and hgher order momens of he uly funcon. Regardless of wheher s he dscoun rae or he growh rae ha s unceran, he naure of he dsrbuon of random growh s of parcular mporance; for nsance, decreasng dscoun raes are obaned wh Bayesan learnng n Wezman (4), and Goller (7) shows ha seral correlaon n he growh raes leads o downward slopng yeld curves when he represenave agen s pruden. Decreasng dscoun raes also emerge from he spec caon of a susanable welfare funcon à la Chchlnsky (996) and L and Löfgren (). Lasly, here s consderable emprcal and expermenal evdence o show ha ndvduals are frequenly hyperbolc dscouners (see, where denoes he rae of pure me preference, g s he per capa growh rae of consumpon and = s he elascy of margnal uly, or equvalenly he degree of relave rsk averson. 3 As n sandard models of he erm srucure of neres raes (see, e.g., Cox e al., 985, Ingersoll and Ross, 99, Vascek, 977, Cochrane, ). 4 See Groom e al. (5) for a survey. 5 Wezman () underakes a survey of over academc economss, and a so-called blue rbbon selecon of 5, as o her opnon on he consan rae of dscoun o use for Cos Bene Analyss. The responses were dsrbued wh a gamma dsrbuon wh mean 4% and sandard devaon of 3%. 3

4 e.g. Loewensen and Thaler, 989, Loewensen and Prelec, 99). In hs paper, we wan o examne f dvergence of belefs and heerogeney of me preference raes can be a jus caon for he use of declnng dscoun raes n a general equlbrum framework. More generally, we wan o analyse f he equlbrum yeld curve wh belefs and ases heerogeney has desrable properes. Our heorecal framework bulds upon Joun and Napp (7) and generalzes o ake no accoun heerogeneous me preference raes and n ne horzon. The agens hold heerogeneous expecaons abou he fuure of he economy. Our model encompasses dsagreemen abou he fuure growh rae or abou s dsrbuon and dynamcs (snce we allow for sochasc growh raes) or, for example, dsagreemen abou he possble mpac of economc acvy on he clmae 6, or dsagreemen abou he esmaon of he possble economc damages ha would be nduced by clmae change. In our model, he agens also d er n her rae of pure me preference. Ths may re ec d eren levels of mpaence as well as d eren concepons of nergeneraonal equy or dsagreemen abou he probably ha he world wll exs a a gven fuure dae (see he Sern Repor for he lnk beween hs probably and he pure me preference rae). Noe ha whle he am of Joun and Napp (7) was o provde an aggregaon procedure, he am of hs paper s o analyse he properes of equlbrum dscoun raes. As n Goller (a, b), Wezman (998,, 7) and Nordhaus (7), we focus on an exchange economy n order o analyse he radeo beween curren and fuure wealh, n parcular for CBA. Neverheless, our model can also shed some lgh on he analyss of he radeo beween curren producon and he envronmenal and economc welfare of fuure generaons; ndeed, our resuls perm o characerse he margnal cos we are wllng o nves n any echnology ha can reduce clmae change mpac n he fuure. We provde he followng answers o he quesons above. We rs oban ha he cerany equvalen approach of Wezman (998, ) and Renschmd (), ha consss n akng he arhmec average 7 of he ndvdually recommended dscoun facors s compable wh an equlbrum approach f we assume ha all uly funcons are logarhmc and ha all he agens have he same endowmen and he same me preference rae 8. More generally, for logarhmc uly funcons, he socally e cen dscoun facor s gven by a weghed average of he ndvdually recommended dscoun facors, he weghs beng deermned by he ndvdual me preference raes and he nal endowmens. For more general CRRA uly funcons wh relave rsk averson level =, he rgh concep of average o consder whn an equlbrum approach s no he arhmec average bu an average. Ths average s, as n he logarhmc case, a weghed average. Fnally, here s a bas nduced by agens heerogeney. Excep n very spec c sengs, s no possble o recover he socally e cen dscoun facor as an average of he ndvdual subjecvely ancpaed 9 dscoun facors. These 6 One agen may assume ha economc acvy wll lead o a C excess emperaure n 5 years wh a 95% con dence nerval of.5 C, whle anoher agen may assume ha economc acvy wll lead o a 3 C excess emperaure wh a 95% con dence nerval of.5 C. 7 See also Noce e al. (8) for an approach leadng o (possbly weghed) arhmec averages. 8 More precsely, su ces ha he produc of he endowmen and of he me preference rae be he same for all he agens. 9 For a gven agen ; he ndvdual subjecvely ancpaed prces, dscoun raes and dscoun facors are hose ha would preval f he economy was made of agen only. 4

5 spec c sengs are rs, he already menoned logarhmc uly seng, and second, he seng wh deermnsc heerogeney n pure me preference raes and no belefs dsperson (whch ncludes he seng wh raonal agens and deermnsc me preference raes of Goller and Zeckhauser, 5, or Lengwller, 5). In a general seng, here s an aggregaon bas and we show ha can be que sgn can. The bas can be owards hgher or lower dscoun raes dependng on he relave poson of wh respec o. Ths resul s conssen wh he nerpreaon of belefs and me preference heerogeney as an addonal source of rsk or uncerany n he fuure, leadng agens o value more or less fuure consumpon (wh respec o presen consumpon) dependng on he relave level of prudence and rsk averson. We examne he mpac of belefs and me preference heerogeney on he expresson of he dscoun rae as well as on he relaonshp beween he dscoun rae and he me horzon (he possble shapes of he yeld curve). Increased belefs dsperson leads o a decrease of dscoun raes when s greaer han one. We show ha aggregae pessmsm as well as aggregae paence reduce he socally e cen dscoun rae. Snce hese aggregae levels are gven by sochasc, me-varyng (rsk-olerance) weghed averages of he ndvdual levels of pessmsm and paence, possble correlaon e ecs are nduced. Ths leads n he medum erm o a rch class of possble shapes for he yeld curve. In parcular, our model can observed behavor of he yeld curve n nancal markes where he "long erm rae" (3 years) s usually hgher han he shor erm rae. More precsely, our model s compable wh yeld curves ha are ncreasng n he shor erm and n he medum erm (ha corresponds o he nancal markes long erm) and ha are decreasng n he long erm. We show ha he yeld curve s always decreasng n he (very) long run. Indeed, he bas due o belefs dsperson vanshes n he long run and he asympoc dscoun rae s gven by he lowes ndvdually ancpaed asympoc dscoun rae among all agens. Noe ha hs lowes ndvdually ancpaed asympoc dscoun rae does no necessarly correspond o he rae of he mos paen agen as n he homogeneous belefs seng. For example, n he case of homogeneous me preference raes, he asympoc dscoun rae s gven by he dscoun rae of he mos pessmsc agen whch would conss o focus on he wors case scenaro among he agens (f each agen s assocaed o some scenaros). More generally, boh he dsrbuons of me preference raes and of pessmsm are necessary o deermne he asympoc dscoun rae. In he seng of a CBA, hs leads o dscoun long erm coss and bene s a he lowes ancpaed rae nducng a bas owards he opmal polcy of he agen who values he mos fuure consumpon n he long erm. In fac, he agen who values he mos fuure consumpon (eher because she s very pessmsc abou he fuure or because she s very paen or any combnaon of hese wo possbles) makes he marke for long erm bonds and herefore mposes her prce. Ths provdes us wh a gudelne for long erm CBA. Ths s especally useful snce, whle he observed rsk free rae provdes a useful ool for CBA n he shor erm, nancal markes are no very helpful when bene s and coss of he se of curren poenal acons are expeced o las n he medum and/or n he long run. Noe ha we rereve a he consumpon dscoun rae level resuls ha are already known a he uly dscoun rae level. Ths ransfer of properes from uly o consumpon dscoun raes s mmedae when agens share he same belefs. For example, greenhouse gas ha one ems oday yelds very long erm coss lke global warmng. Lqud 5

6 The paper s organzed as follows. Secon presens he heorecal framework. Secon 3 deals wh socally e cen dscoun facors and dscoun raes and n parcular, analyses her lnk wh her ndvdual subjecvely ancpaed counerpars. Secon 4 s devoed o long erm consderaons. In Secon 5, we analyse more n deal he shape of he yeld curve n spec c sengs. The concluson summarses he man conrbuons of he paper. Proofs of aggregaon resuls and oher exensons of Joun-Napp (7) are n Appendx A. Appendx B consss of he proofs of all oher resuls.. The heorecal framework We consder a connuous-me Arrow-Debreu economy wh an n ne horzon, n whch rsk averse agens ry o maxmze he expeced uly of fuure consumpon. A lered probably space (; F; (F ) ; P ) s gven. Each agen ndexed by = ; :::N; has a curren endowmen a me denoed by h e and a Von Neuman-Morgensern uly funcon for fuure consumpon R of he form E Q exp R (s;!)ds u (c (!)) d ; whch means ha agens may d er n her subjecve belefs (represened by he probably measure Q ); n her pure me preference rae process ; and n her endowmen process e : The uly funcon u can represen he ndvdual s uly funcon bu may also ncorporae he preferences of hs descendans. As n Joun-Napp (7), agens have d eren expecaons abou he fuure of he economy. In a purely nancal framework, such belefs heerogeney mgh resul, for nsance, from d eren subjecve probables of occurrence of a boom or of a krach. In a macroeconomc seng, belefs heerogeney mgh resul from heerogeneous growh rae forecass. More generally, belefs heerogeney mgh also resul from d erences of opnon abou he lkelhood of a caasrophe, abou he probably dsrbuon of he mpac of human acvy on he clmae or on he possble damages resulng from clmae change. In our model, agens may also dsagree abou he pure me preference rae. Ths may re ec d eren levels of preference for he presen. Wh long-lved agens ha represen presen and fuure generaons, hs re ecs d eren concepons of nergeneraonal equy. For racably reasons and n order o focus on he mpac of belefs and me preference rae heerogeney, we resrc our analyss o homogeneous uly funcons of he power ype,.e. we suppose ha agens share he same CRRA uly funcon for consumpon, of he form u (x) = x = : As far as belefs heerogeney s concerned, he unque assumpon we essenally make s he equvalence of he probably measures Q. In oher words, we assume ha he agens have he same se of possble evens (.e. evens wh a posve subjecve probably 3 ). Leng M denoe he posve densy process of Q wh respec o P and leng D exp R (s;!)ds denoe he ndvdual pure me preference dscoun facor a me ; he uly funcon of agen can equvalenly be wren n he form E R M (!) D (!) u (c (!)) d : We le e P N = e denoe he aggregae endowmen process. We make he assumpon nancal nsrumens wh such large duraons do no exs. For he sake of comparson, US reasury bonds have me horzons ha do no exceed 3 years. Our approach can be exended o he case wh HARA uly funcons of he form u (x) = ( + x) 3 Noe ha hs s a naural assumpon for he exsence of an equlbrum. Oherwse some agens wll consder as possble some evens ha are consdered as mpossble by ohers and opmal demand n he assocaed Arrow- Debreu asse wll be (posvely or negavely) n ne.. 6

7 ha e and M sasfy he followng sochasc d erenal equaons ( de = e d + e dw e = dm = M dw where W denoes a sandard undmensonal ((F ) ; P ) Brownan moon and where ; and sand respecvely for (;!); (;!) and (;!); whch means ha hey mgh depend upon me and saes of he world. There s no Markovan assumpon; he coe cens ; and for = ; :::; N may depend on he enre pas hsory of he economy 4. Le us recall he followng noaon. For a gven process ( ) ; he Doléans-Dade exponenal E () denoes he quany exp( R R sdw s sds): Wh hs noaon we have, for all ; M = E : Accordng o Grsanov Theorem, for agen, aggregae endowmen s an Iô process wh d uson parameer (;!) and wh subjecve drf parameer (;!) (;!)+(;!) (;!); whch means ha n our model, agens d er n her expeced nsananeous growh rae of aggregae endowmen and agree abou s volaly 5. Noe ha even f he agens agree on he nsananeous level of rsk (;!), he fac ha (; :) s sochasc perms dsagreemen among agens abou he level of rsk. For nsance, agens for whch he sochasc process exhbs posve (resp. negave) seral dependence wll wegh more (resp. less) exreme evens (see Goller, 7). Even n he exreme case where here s no me dependence, where s consan and where s a dscree random varable (ndependen of and of W ), he varance of he logarhmc reurn beween dae and dae T s equal o T + V ar T allowng for dvergence of opnon among agens abou he level of rsk. Noe ha he process as well as are (F ) adaped sochasc processes, whch may be updaed connuously accordng o he avalable nformaon. In parcular, he ndvdual belefs we consder mgh resul from Bayesan updang as n e.g. Deemple and Murhy (994) and Zapaero (998) or from adapave learnng as n Brock and Durlauf (). For nsance, f he saes of he world are assocaed h o d eren possble levels of clmae change (d eren R T levels of excess emperaure), E (s;!)ds F wll represen he average logarhmc reurn beween daes and T ha s ancpaed by agen a dae based on dae scen c knowledge abou clmae change and s possble mpac on he economc growh. In our general framework, an Arrow-Debreu equlbrum sde ned, as usual, by a posve prce process q and a famly of opmal consumpon plans y such ha markes =;:::;N clear,.e. ( y = y q ; M ; D ; e P N = y = e where y (q; M; D; e) = arg max E[ R q(y e )d] E R M D u (c ) d : 4 We only assume ha for all T; R T j j d < ; R T R T d < ; and d <, almos surely: R 5 More precsely, leng W W ds, we oban hrough Grsanov Theorem ha W s a Brownan moon under Q and de = + e d + e dw : 7

8 3. Dscoun facors and dscoun raes As n Joun-Napp (7), we sar from an Arrow-Debreu equlbrum q ; y : Noe =;:::;N ha snce our uly funcons sasfy Inada condons, all equlbra are neror, hence here exs posve Lagrange mulplers ( ) =;:::;N such ha for all, he equaly M Du y = q holds for all : We le (= ) P N j= (= j) : We x a dae > ; a whch a cos or bene s ncurred. The socally e cen dscoun facor A beween dae and dae s he prce of a zero coupon bond maurng a me and s gven by A E [q ] : The (average) socally e cen dscoun rae R s de ned as R log A : Analogously, he dscoun facor A ; and he dscoun rae R ; ha would preval f he economy was made of agen only are gven by R ; log A; log E M D u (e ) : These are he dscoun facor and dscoun rae ha agen ancpaes. The am of hs secon s wofold. Frs, o analyse he lnk beween he socally e cen dscoun rae R and he ndvdual subjecvely ancpaed dscoun raes R ; : Second, o analyse he expresson of he socally e cen dscoun rae R and, n parcular, how depars from he sandard seng. We rs consder o wha exen ndvdual subjecvely ancpaed socally e cen dscoun facors can be averaged no a consensus dscoun facor. Our queson can be rephrased as follows: can he socally e cen dscoun facor A be represened as an average of he ndvdual A ;? Proposon 3... If here s no belefs heerogeney and f me-preference raes are deermnsc,.e., f (s;!) (s;!) and (s;!) (s) ; hen he socally e cen dscoun facor s an dscoun facors, more precsely. In he general seng, If = ; hen A = P N = A ; : Oherwse, we have and average of he ndvdual subjecvely ancpaed socally e cen " N = X # A = A ; : = " N = X # A A ; for < = " N = X # A A ; for > ; = wh equaly holdng only when he dvergence n ndvdual characerscs N M D s deermnsc,.e. f N =N j s deermnsc for all (; j). Proposon 3. means rs ha he rgh concep of average o consder for dscoun facors (n he case of power uly funcons) s an average, whch s an arhmec average only n he 8

9 case of logarhmc uly funcons. Moreover, hs average s a weghed average, he weghs beng gven by he parameers. These weghs are deermnsc and hey wll be analysed more n deal n Secon 5. Fnally, excep n very spec c sengs, s no possble o recover he socally e cen dscoun facor as an average of he ndvdual subjecvely ancpaed dscoun facors. These spec c sengs are rs, he seng wh deermnsc heerogeney n pure me preference raes and no belefs dsperson 6 (whch ncludes he seng wh raonal agens and deermnsc me preference raes of Goller and Zeckhauser, 5), and second, he seng wh logarhmc uly funcons. In a general seng, here s an aggregaon bas. The prce a dae of a zero-coupon bond maurng a dae s lower (resp. hgher) han he (weghed ) average of he subjecvely ancpaed prces for < (resp. > ): Ths means ha wh belefs dsperson and/or sochasc me preference heerogeney, ndvduals value less (resp. more) one un of consumpon a dae when < (resp. > ): As far as he magnude of hese bases are concerned, we shall see n Secon 5 ha he d erence beween he socally e cen dscoun rae and he rae assocaed o he of he ndvdual subjecvely ancpaed dscoun raes can be sgn can. Le us elaborae on why hese bases (wh respec o he average average) are n oppose drecons dependng on he poson of wh respec o : The nerpreaon of as a degree of relave rsk olerance s no enlghenng for our purpose. I seems more meanngful o observe ha he condon s equvalen, n our seng, o he condon ha prudence s larger han wce absolue rsk averson. Indeed, hs las condon 7 appears as crucal n neremporal choces analyss; Goller and Kmball (996) show ha, n a sandard porfolo problem, he opporuny o nves n a rsky asse rases (resp. reduces) he aggregae savng f and only f absolue prudence s larger (resp. smaller) han wce absolue rsk averson: Moreover, Goller () sudes he problem of he opmal use of a good whose consumpon can produce damages n he fuure and shows ha scen c progress provdng nformaon on he dsrbuon of he nensy of damages nduces earler prevenon e or only f prudence s larger han wce rsk averson. Hence, a possble nerpreaon of he cenral role of = s he followng. Inerpre belefs heerogeney and me preference heerogeney n a sochasc seng as addonal rsk or as less nformaon or more uncerany abou he fuure. Accordng o Goller and Kmball (996) or Goller (), hs should lead agens o value more fuure consumpon n he case > and less fuure consumpon n he case <, whch s essenally he resul of Proposon 3:: Noce ha Proposon 3: mples ha he approach ha consss n consderng an arhmec average of he ndvdual subjecvely ancpaed dscoun raes s compable wh a general equlbrum approach as far as agens are endowed wh logarhmc uly funcons. We now analyse more precsely he expresson of he socally e cen dscoun rae and, n parcular, we compare wh he sandard seng. For hs purpose, le us rs recall some resuls abou he rsk free rae. In he sandard seng wh raonal belefs and homogeneous 6 Noe ha no belefs dsperson does no mean ha ndvduals are raonal; hey can all share he same subjecve belef. Analogously, all me preference raes need no be deermnsc bu hey need o be wren n he form (;!) = (;!)a () where s a common erm and where a s a deermnsc process. 7 Ths condon has been horoughly suded by Goller () and has appeared n d eren conexs (Drèze and Modglan, 97, Caroll and Kmball, 996, Snclar-Desgagné and Gabel, 997, Donne and Fombaron, 996). 9

10 me preference rae, he rsk free rae s gven by r f (sdd) = + + (3.) where all parameers ; ; hence r f may depend upon and!: Ths s an exenson of Ramsey Equaon o a sochasc seng, whch llusraes he paence e ec, he wealh e ec as well as he precauonary savngs e ec on he rsk free rae. In our seng wh heerogeneous me preference raes and belefs, we oban (see Proposon A- n Appendx A) ha he rsk free rae s gven by r f = D + + M + + B (3.) wh D P N = ; M P N =, and B ( ) V ar () where = y e represens agen rsk olerance and where V ar () represens he varance of he s across he agens when agen s endowed wh a wegh : In oher words, he homogeneous me preference rae of he Ramsey formula s replaced by he average me preference rae D and he objecve growh rae s replaced by he average subjecve growh rae + M. Furhermore, here s an addonal erm B ha s drecly relaed o belefs dsperson and whose mpac s owards an ncrease or a decrease of he rsk free rae dependng on he poson of wh respec o. Ths s a sraghforward generalzaon of Joun-Napp (7) ha akes no accoun heerogeneous me preference raes. Noe ha he precauonary savngs erm s he same n Equaons (3.) and (3.) snce, as prevously underlned, n our model all agens necessarly agree on he volaly level (as a consequence of he equvalence of he subjecve probably measures Q ): Noce also ha even when all parameers, ; and are consans, he rsk olerance weghed averages D and M as well as he varance erm B are me-varyng and sochasc, hence he rsk free rae s also me-varyng and sochasc (whch s no he case n he sandard seng). Comparng Equaons (3.) and (3.), s easy o see ha here are essenally hree possble ways hrough whch belefs and me preference raes heerogeney may lead o lower rsk free raes; rs, a negave correlaon beween mpaence and rsk olerance or a low level of mpaence, second, a posve correlaon beween pessmsm and rsk olerance or a hgh level of pessmsm, hrd, a negave belefs dsperson e ec ha corresponds o he case >. The followng proposon enables us o show ha analogous resuls are obaned for he socally e cen dscoun rae. We adop he same noaons as n Equaon (3:) : Proposon 3.. The socally e cen dscoun rae R s gven by wh B = exp R = = R B(s)ds ; D = exp log E D M B u (e ) (3.3) Z log EQ exp rs f ds (3.4) R D(s)ds, M = E ( M ) and dq dp = E M Noce rs hrough Equaon (3.4) ha when r f s deermnsc (or when r f and dq dp are

11 ndependen), all ha we have jus sad abou he mpac of belefs and me preference raes heerogeney on r f s rue of R : In parcular, aggregae paence, aggregae pessmsm, as well as ncreased belefs dsperson when > nduce a socally e cen dscoun rae ha s lower han n he sandard seng. More generally, he comparson of Equaon (3.3) wh he expresson of he socally e cen dscoun rae n he sandard seng, whch s gven by R (sdd) = log E [exp ( ) u (e )], perms o exhb hree deermnans of he mpac of belefs and me preference raes heerogeney on he dscoun rae: rs, he consensus me preference facor D; second, he belef (densy) M; and hrd, he aggregaon bas B. The analyss of he e ec of hese hree facors on he dscoun rae (beween dae and ) s less sraghforward han for he (nsananeaous) rsk free rae a a gven dae. However, he man conclusons reman vald. Indeed, excess paence a he aggregae level, n he form of an average me preference rae R D(s)ds ha s lower han he sandard me preference rae R (s)ds leads o a hgher D hence, ceers parbus, o a lower dscoun rae R : Belefs dsperson for leads o a nonposve parameer B and o a facor B ha s greaer han, hence, ceers parbus, o a dscoun rae ha s lower han n he sandard seng. Analogously, ncreased belefs dsperson n he form of a hgher V ar () leads o an ncrease n B for hence, ceers parbus, o a decrease n he dscoun rae. As far as aggregae pessmsm s concerned, nuvely, a pessmsc belef ncreases he expeced value of a decreasng funcon of he oal endowmen e ; hence should lead o a lower dscoun rae R: The case wh deermnsc parameers and llusraes hs nuon. Indeed, s hen easy o oban (see Appendx A, Proposon A-3) ha, f he consensus belef s neural or pessmsc,.e., when M ; hen log E M u (e ) log E u (e ) ; hence he e ec of pessmsm only s oward a lower dscoun rae. In fac, we oban ha he mpac of belefs heerogeney s owards a lower (resp. hgher) socally e cen dscoun rae f he consensus belef s neural or pessmsc ( M ) and when (resp. he consensus belef s neural or opmsc,.e. M, and when ). Ths resul s n he spr of he ndngs of Dumas e al. (8) ha obans, n a spec c senmen framework, ha whenever rsk averson s [an neger] greaer han, an ncrease n he varance of senmen reduces he expeced values of all he fuure sochasc dscoun facors. 4. Long erm consderaons We now urn o (very) long erm consderaons. In parcular, s socally e cen, when dversy of opnon s aken no accoun, o reduce he dscoun rae per year for far dsan horzons? We oban he followng resul. Proposon 4.. We suppose ha for all, he ndvdual asympoc dscoun rae R ; lm! R; exss 8. The asympoc socally e cen dscoun rae exss and s gven by he lowes 8 These lms can be replaced by lms along sequences,.e. R ; = lm n! R n; for some sequence n such ha lm n = : In hs case, he asympoc socally e cen dscoun rae would be de ned along he same sequences.

12 subjecvely ancpaed dscoun rae.e. R lm! R = nf R ; ; = ; :::; N : Le us remark ha n a model where each ndvdual corresponds o an exper who consuls a subjecve model n order o recommend dscoun raes, he exsence of R ; for all ; means ha each exper s able o propose an asympoc dscoun rae. Proposon 4. shows rs ha he bas, due o belefs heerogeney, ha we have exhbed n he prevous secon vanshes n he long run. The socally e cen dscoun rae behaves asympocally as he dscoun rae assocaed o an average of he ndvdual subjecvely ancpaed dscoun facors. Indeed, under he condons of he proposon, he socally e cen dscoun rae R converges o he lowes ndvdual subjecvely ancpaed dscoun rae as does he rae assocaed o any of he consdered averages of he ancpaed dscoun facors: The fac ha he dsperson erm vanshes n he long erm may seem counernuve accordng o Equaons (3.) and (3.3). Indeed, he bas beween R and an average of he R s s represened (up o a consan) by he varance of he s and s no clear wheher or no hs varance erm s neglgble n he long run. In parcular, he unweghed varance of he s s exogeneously gven n our model and does no necessarly converge o zero. The fac ha he weghed varance vanshes s hen drecly relaed o he dynamcs of he sochasc weghs. Furhermore, we emphasse ha he bas vanshes only asympocally, and we shall see n he nex secon ha we may have o consder very far horzons (hundreds of years) before observng hs asympoc behavor. Proposon 4. proves foremos ha, from oday s perspecve, among he possble subjecvely ancpaed asympoc behavors R ; ; he only relevan asympoc behavor s he one wh he lowes dscoun rae. In oher words, n a seng wh heerogeneous agens, only he agen wh he lowes ancpaed dscoun rae maers n he long run. In fac, he agen who values he mos fuure consumpon (eher because she s very pessmsc abou he fuure or because she s very paen or any combnaon of hese wo possbles) makes he marke for long erm bonds and herefore mposes her prce. Mahemacally, from oday s perspecves, all he oher agens become neglgble n he long run, because her wegh has been reduced by he power of compound dscounng a a hgher expeced dscoun rae. Asympocally, he value of he socally e cen dscoun facor s hen gven by he dscoun facor ha would preval n an economy made of he agen wh he lowes rae only. In he case of homogeneous belefs and heerogeneous me preference raes, hs mples ha he asympoc dscoun rae s gven by he rae assocaed wh he lowes rae of mpaence. In parcular, he resul of Goller-Zeckhauser (5) on he asympoc dscoun rae wh heerogeneous me preference raes 9 remans vald n a sochasc seng. In he case wh homogeneous me preference raes and heerogeneous belefs, Proposon 4. mples ha he asympoc socally e cen dscoun rae s gven by he rae of he mos pessmsc agen. More generally, boh he dsrbuons of me preference raes and of pessmsm are necessary o deermne he asympoc dscoun rae (whch s he lowes ancpaed dscoun rae). 9 See also Blanchard and Fscher (989) and L and Löfgren ().

13 In he seng of a CBA, hs leads o dscoun long erm coss and bene s a he lowes ancpaed rae nducng a bas owards he opmal polcy of he agen who values he mos fuure consumpon n he long erm. How does Proposon 4. relae o he prevous resuls of, e.g., Dybvg e al.(996), Goller (b), Wezman (998, ), who show - n a sandard sochasc seng wh a represenave agen wh raonal belefs and a gven me preference rae parameer - ha he long erm dscoun rae s assocaed wh he scenaro wh he lowes possble rae (wors case scenaro). Proposon 4. brngs abou an addonal elemen n favor of he choce of low dscoun raes n he long run. Indeed, he resul abou he wors scenaro mples ha for each agen denoed by ; he asympoc rae R ; corresponds o hs wors scenaro (over possble saes of he world). Proposon 4. mples hen ha he asympoc behavor of he socally e cen dscoun rae s gven by he lowes (among agens) of he lowes possble rae (among possble scenaros or saes of he world). As an llusraon, consder he smple case n whch he nsananeous rae of growh of aggregae endowmen denoed by s a dscree random varable, ndependen of and of W; akng values ; :::; J wh probably p ; :::; p J and suppose ha ; ; are consans. We easly ge ha for all = ; :::N; he asympoc subjecvely ancpaed dscoun rae s gven by he rae assocaed o he wors possble scenaro,.e., R ; = + nf j=;:::;j j + + : Accordng o Proposon 4., he socally e cen dscoun rae s hen gven by he lowes possble rae among he agens,.e. R = nf =;:::;N + + nf j=;:::;j j + : In order o deermne he whole shape of he yeld curve, we mus deermne explc formulas for he socally e cen dscoun raes. As seen n Proposon 3., we need o analyse how he ndvdual rsk olerances evolve over me, snce hey are key feaures n our analyss. Besdes, we have seen ha he covarance beween ndvdual rsk olerances and ndvdual belefs and ase parameers play an mporan par. Ths means ha s necessary o consder spec c sengs n order o analyse he shape of he yeld curve. 5. Spec c sengs and he shape of he yeld curve We consder he seng wh consan parameers. In such a seng, we know ha n he sandard model wh raonal belefs and homogeneous me preference raes, he yeld curve s a and he socally e cen dscoun rae s gven for all by R = + + : The am of hs secon s o analyse he mpac of belefs and me preference raes heerogeney on he shape of he yeld curve. More precsely, we assume ha he aggregae endowmen process parameers and are consan as well as he ndvdual belefs and me preference parameers and. We suppose ha for all ; he relave level of endowmen of agen sas es e w : Moreover, we assume ha, for all ; + + = w e for some consan > (5.) whch s a necessary condon for he ndvdual opmzaon problems o be well de ned. Noe ha assumng consan s may seem ncompable wh learnng. However, we consder he case wh consan parameers as an approxmaon of he suaon where all he parameers 3

14 are sochasc and where learnng s regularly compensaed by new shocks on he drf : For all, he ndvdual subjecvely ancpaed dscoun rae R ; s me and sae ndependen and gven by R = : We recall ha even n hs seng he consensus characerscs M and D as well as he aggregaon bas B are me-varyng, sochasc processes. 5.. Logarhmc uly funcons As we have underlned n Secon 3, he case of logarhmc uly funcons s very spec c. Indeed, n ha case, he socally e cen dscoun facor can be expressed as a weghed arhmec average of he ndvdual subjecvely ancpaed ones, more precsely A = P A ; (see Proposon 3.) and here s no aggregaon bas B n he expresson of he socally e cen dscoun rae (see Equaon 3.3). Noe ha Condon (5:) s equvalen n he logarhmc seng o he condon ha all me preference raes be posve. We oban he followng resul. Proposon 5.. In he case of logarhmc uly agens wh posve me preference raes, we have. The wegh of each ndvdual subjecvely ancpaed dscoun facor n he socally e cen dscoun facor s gven by = P w N ; = ; :::; N: j= w j j. The socally e cen dscoun rae sas es " N R = log X exp + # for all > ; = R = + + ; R = + nf =;:::N + : = The yeld curve R s downward slopng: We recall ha n he sandard seng wh logarhmc uly funcons, he yeld curve s a and for all endowmen dsrbuon, he raonal dscoun rae R s gven by R = for all : When belefs and me preference raes are heerogeneous, Proposon 5. shows ha he yeld curve (.e. he socally e cen dscoun facor as a funcon of me) s always downward slopng. The behavor of he socally e cen dscoun facor as a funcon of belefs dsperson s more complex snce depends on he correlaon beween ndvdual characerscs (w ; ; ). In order o focus on he mpac of belefs dsperson, Fgure and Fgure represen he yeld curve n a agens seng wh no pessmsm/opmsm on average,.e. + = : Fgure represens he yeld curve n he parcular seng wh w = w and =. In hs case, he dscoun facor s an equally weghed arhmec average of he ndvdual subjecvely ancpaed dscoun facors, as n Wezman (998, ). The shor erm rae s he raonal rae and he long erm rae s he pessmsc rae. Moreover, for all ; he dscoun rae R decreases wh belefs dsperson. 4

15 Inser Fgure The seng wh possbly d eren endowmen levels (bu sll wh = ) s llusraed n Fgure. The consensus dscoun facor s an endowmen-weghed arhmec average of he ndvdual subjecve dscoun facors. The yeld curve s sll downward slopng. The shor erm rae s an endowmen weghed average of he ndvdually ancpaed shor erm raes, whch d ers from he raonal rae f w 6= w. If here s a posve correlaon beween opmsm and nal endowmen, hen he shor-erm dscoun rae s hgher han n he raonal seng and an ncrease n belefs heerogeney ncreases he shor-erm rae. The long erm rae s sll gven by he pessmsc rae and an ncrease n belefs dsperson always decreases he long erm rae. More generally, an ncrease n belefs dsperson lowers he yeld curve when here s a negave correlaon beween opmsm and nal endowmen and roae clockwse when here s a posve correlaon beween opmsm and nal endowmen. Fnally, an ncrease n he nal relave wealh of he opmsc agen nduces a hgher shor-erm dscoun rae and a greaer spread beween he shor-erm dscoun rae R and he long-erm dscoun rae R ; he spread beng always posve. Inser Fgure Table sums up he possble resuls wh wo agens who are on average raonal,.e. + = : The resuls are smlar wh N agens ha are on average raonal. If here s a (pessmsc or opmsc) bas on average,.e. f N P N = = 6= ; hen here s an addonal opmsm (when s posve) or pessmsm (when s negave) e ec on he dscoun rae. We can consder sengs wh a connuous dsrbuon on he ndvdual belefs as a lm of a seng wh a large number of agens: If we suppose ha nal endowmen s equally dsrbued ; hen we oban as an easy exenson of he dscree seng ha he dscoun facor s an average of he ndvdual subjecvely ancpaed dscoun facors,.e. A = E A ; : Assumng a Gamma dsrbuon on he ndvdual subjecvely ancpaed dscoun raes (resp. a normal dsrbuon on ndvdual belefs), we rereve he expresson of he dscoun rae n Wezman () (resp. n Renschmd, ). I s neresng o noce ha whle he socally e cen dscoun rae always converges n he long run o he lowes ndvdual subjecvely ancpaed dscoun rae, he fuure shor erm raes are sochasc and may reman hgher han he raonal rae and may even converge n he long run o he hghes ancpaed rsk free rae. Consder for nsance he seng wh = ; + = and w = w. I s hen easy o oban ha for all ; () and () have he same dsrbuon, whch means ha none of he agens wns and he shor erm rae remans equal on average o he raonal rae. Wha happens now f, for nsance, one agen s pessmsc (or opmsc) wh M = E () and he oher s raonal wh M. In hs case, he weghs are he same as n he prevous seng snce hey do no depend upon ndvdual belefs. However, agen s wrong whle agen s rgh and can be shown More precsely, we need ha E h M = E M : 5

16 ha ()!! and ()!! ; a.s. Hence, by Equaon (3.), he fuure rsk free rae converges n he long run o he raonal rae. More generally, when agen s more wrong han agen, fuure shor erm raes converge o he shor erm rae ha would preval n he economy made of agen alone, whle he socally e cen dscoun rae converges o he lowes ancpaed dscoun rae. 5.. Power uly funcons We now consder he case of power uly funcons. As we have underlned prevously, here are essenally wo d eren sengs, < or > ; for whch he mpac of belefs dsperson s oppose The case < Le us sar by consderng he spec c case = =: In hs case, we recall ha n he sandard seng he yeld curve s a and, for all, R = 3 : We consder wo agens, who are raonal on average. Noe ha Condon (5:) s equvalen n hs seng o he condon ha + nf + >. Proposon 5.. Consder he case of power uly funcons wh = =. Suppose ha = >, = ; w = w, = = and ha > :. The rao s gven by = s +. The dscoun rae s a decreasng funcon of and s gven by R = 3 ln ( ) exp + ( ) exp + exp ; R = 3 + ( ) ; R = 3 : Accordng o he rs pon of he proposon, he relave wegh of he opmsc agen s greaer han he relave wegh of he pessmsc agen,.e., > : here s an opmsc bas a he aggregae level: Ths resul s vald n he general seng wh < (see Appendx A, Proposon A-4). Ths mples n parcular ha an ncrease n belefs dsperson leads o an ncrease of he shor erm rae. Noce ha n he long erm an ncrease n belefs dsperson leads o a decrease of he socally e cen dscoun rae snce corresponds o a more pessmsc belef for he more pessmsc agen. The man resul we oban s he fac ha he yeld curve s decreasng. We have already seen ha he bas beween he dscoun rae R and an average of he R s s represened (up o a consan) by he belefs dsperson erm 4 V ar (). In he case of logarhmc uly funcons; here s no belefs dsperson erm and, as already seen, he socally e cen dscoun rae R See Yan (8) for relaed ssues. 6

17 decreases wh and converges o he more "pessmsc" ndvdual subjecvely ancpaed rae. In he case = ; we have M = p = M ln N ; : In parcular, hs mples ha, for large, " s large wh respec o or s large wh respec o " wh a probably near. Loosely speakng, here are wo knds of saes of he world, hose for whch vanshes for large and hose for whch vanshes for large : The belefs dsperson erm 4 V ar () = vanshes hen asympocally. The socally e cen dscoun rae curve s hen globally decreasng and converges, as n he logarhmc case, o he mos pessmsc rae. Everyhng works hen as f we had wo scenaros one wh he opmsc rae and one wh he pessmsc rae and he asympoc socally e cen dscoun rae s assocaed o he wors scenaro as n Goller (a, b) and Wezman (998, ). Ths reasonng s vald for general <. Indeed, we hen have M p = M ln N ; ; and as n he case = = he belefs dsperson erm ( )V ar () vanshes asympocally. In Fgure 3 we represen he socally e cen yeld curve as well as he raes assocaed o he average and o he arhmec average of he ndvdual dscoun facors. All hese curves converge asympocally o he rae assocaed o he mos pessmsc belef bu he "" one s a much beer approxmaon of he yeld curve han he "arhmec" one. dsance beween he yeld curve and he "" curve measures he mpac of he bas due o belefs dsperson. The varance erm ncreases he shor rae bu s mpac decreases wh and vanshes asympocally. However hs mpac can reman non neglgble for cenures. The Inser Fgure 3 We have assumed so far ha boh agens are endowed wh he same nal endowmen. If we relax hs assumpon, we sll oban decreasng yeld curves convergng o he mos pessmsc rae. However, when he more opmsc agen has a larger nal endowmen, she has a greaer wegh n he average formula and he mpac of her opmsm lass longer. The yeld curve has hen a hgher sarng pon a = and s nal slope s smaller. When he more pessmsc agen has a larger nal endowmen, she has a larger wegh, he sarng pon of he yeld curve s lower and he convergence o s asympoc rae s more rapd The case > As already underlned, he dscoun rae exhbs boh an average belef/me preference e ec ha s measured by P and by P and a belefs dsperson e ec ha s measured by he varance erm. As n he case < ; he average e ec nduces a decrease of R when ncreases (snce he assocaed rae converges asympocally o he lowes rae) and he varance erm decreases and vanshes asympocally. However, n he case > ; he belefs dsperson erm ( )V ar () s negave. Ths leads hen o wo oppose e ecs when ncreases, he average e ec nducng a decrease of R and he dsperson e ec nducng an ncrease of R. Dependng on he relave sze of hese e ecs, we may oban decreasng curves as n he case < as well as ncreasng hen decreasng curves as n Fgure 4. The case > In order o evaluae he negrals and expecaons ha are nvolved n he formulas we used he adapve quadraure mplemened n he R funcon negrae o approxmae he arge negral. Ths funcon s based on QUADPACK rounes dqags and dqag (R. Pessens and E. dedoncker-kapenga,983) avalable from Nelb. 7

18 leads hen o a rcher famly of possble shapes and s compable wh he fac ha long-erm raes n bonds markes (.e. = 3) are usually hgher han shor-erm raes. In he case of an nally ncreasng yeld curve, hs nal shape resuls from belefs dsperson and s no rereved when we approxmae he yeld curve by he raes assocaed o he average or he arhmec average of he ndvdual dscoun facors as can be seen n Fgure 4. Inser Fgure 4 6. Concluson When publc nvesmen projecs enal coss and bene s n he very long run, a queson arses abou he selecon of he relevan dscoun rae o use for he Coss and Bene s Analyss. Indeed, nancal markes do no provde a gudelne n hs case. In hs paper we provde an equlbrum analyss of he e ec of expecaons heerogeney and pure me preference raes heerogeney on he socally e cen dscoun rae n a (que general) sochasc seng. Frs, we show ha he cerany equvalen approach of Wezman (998, ) ha consss n akng he (unweghed) arhmec average of he ndvdually recommended dscoun facors s compable wh an equlbrum approach f we assume ha all uly funcons are logarhmc and ha all agens have he same nal endowmen and he same me preference rae 3. However, for more general endowmen and me preference rae dsrbuon, he average s a weghed average. Furhermore, for more general uly funcons, he rgh concep of average o consder whn an equlbrum approach s no he arhmec average bu an average. Fnally, here s a bas nduced by belefs and pure me preference raes dsperson. Belefs and me preference rae heerogeney mpac he socally e cen dscoun facor as would an addonal source of rsk. Increased belefs and/or me preference raes dsperson leads o lower socally e cen dscoun raes when s larger han one. In he long run, we show ha he asympoc socally e cen dscoun rae s he lowes ndvdual subjecvely ancpaed dscoun rae whch depends on boh dsrbuon of paence and pessmsm across agens. Ths s anoher elemen n favor of decreasng dscoun raes n he long run. In he shor and n he medum erm, belefs heerogeney leads o a rcher class of possble shapes for he yeld curve. In parcular, we may oban ncreasng yeld curves for he rs 5 years as s ofen he case n nancal markes. Appendx A Aggregaon of ndvdual belefs and me-preferences The am of hs Appendx s o exend he resuls of Joun and Napp (7) o our seng wh an n ne horzon and heerogeneous me preference raes. We deal wh aggregaon ssues n he spr of Varan (985, 989), Abel (989), Calve e al. (), Shefrn (5). Proposon A- We le N denoe he ndvdual compose characersc M D : 3 In fac, su ces ha he produc of he me preference rae and of he nal endowmen be he same for all he agens. 8

19 . The ndvdual characerscs N can be aggregaed no a consensus characersc N such ha q = N u (e ) wh " N # = X N = N : =. The consensus characersc N can be wren n he form N = BDM where M s a Proof consensus probably belef, D s a pure me consensus dscoun facor and B s an aggregaon bas, relaed o belefs dsperson. More precsely, he marngale process M and he ne varaon processes D and B sasfy dm = M M dw ; dd = D D d; db = B B d wh M = D = B = and M = ; D = = = " B = N ( ) X = M # = ( ) V ar (). Snce q s an neror equlbrum prce process, we know ha here exs Lagrange mulplers ( ) such ha for all and for all ; Snce P N = y = e ; we ge. We can wre ha dy e N u y = q : (6.) " N # = X q = N u (e ) wh N = N : (6.) = = a ()d+b () dw for processes (a ) and (b ) such ha P N = a () = and P N = b () = e : Analogously, we nroduce he processes N and N such ha dn = N () N d + N () N dw. We apply Iô s Lemma o boh sdes of Equaon (6:). Idenfyng he d uson pars and he drf pars and afer smple compuaons, we oban N = M = = " N = N ( ) X = M # : = I s easy o check hen ha N s of he form N = BDM: 9