Aggregation of Heterogeneous Beliefs

Transcription

1 Aggregaon of Heerogeneous Belefs Clolde Napp, Elyès Joun To ce hs verson: Clolde Napp, Elyès Joun. Aggregaon of Heerogeneous Belefs. Journal of Mahemacal Economcs, Elsever, 2006, 42 (6), pp <halshs > HAL Id: halshs hps://halshs.archves-ouveres.fr/halshs Submed on 6 Jun 2007 HAL s a mul-dscplnary open access archve for he depos and dssemnaon of scenfc research documens, wheher hey are publshed or no. The documens may come from eachng and research nsuons n France or abroad, or from publc or prvae research ceners. L archve ouvere plurdscplnare HAL, es desnée au dépô e à la dffuson de documens scenfques de nveau recherche, publés ou non, émanan des éablssemens d ensegnemen e de recherche franças ou érangers, des laboraores publcs ou prvés.

2 Aggregaon of heerogeneous belefs E. Joun CEREMADE-Unversé Pars -Dauphne and Insu unversare de France C. Napp CEREG-Unversé Pars-Dauphne and CREST March 23, 2006 Absrac Ths paper s a generalzaon of Calve e al. (2002) o a dynamc seng. We propose a mehod o aggregae heerogeneous ndvdual probably belefs, n dynamc and complee asse markes, no a sngle consensus probably belef. Ths consensus probably belef, f commonly shared by all nvesors, generaes he same equlbrum prces as well as he same ndvdual margnal valuaon as n he orgnal heerogeneous probably belefs seng. As n Calve e al. (2002), he consrucon sands on a cous adjusmen of he marke porfolo. The adjusmen process re- ecs he aggregaon bas due o he dversy of belefs. In hs seng, he consrucon of a represenave agen s shown o be also vald.. Inroducon In a recen paper, Calve, Grandmon and Lemare (2002) propose a way o consruc a represenave agen n a sac heerogeneous belefs seng. In he presen paper, we exend hs approach o an neremporal framework. The auhors hank Jean-Mchel Grandmon for helpful dscussons and manly for brngng hs problem o her aenon. The nancal suppor of he Europlace Insue of Fnance s graefully acknowledged.

3 The man purpose of Calve e al. (2002) s o ncorporae n he represenaon of he economy some degree of heerogeney n he nvesors belefs and o analyze how one could exend and modfy he radonal expeced uly maxmzng represenave agen approach n order o cover he case, whch appears o be emprcally more relevan, of heerogeneous belefs. Among he ssues he auhors nvesgae are: s possble o de ne a consensus probably ha would aggregae heerogeneous ndvdual subjecve belefs and could be used o explan (mmc) equlbrum prces? Is sll possble n such a conex o de ne a verson of an expeced uly maxmzng aggregae nvesor ha would represen an equlbrum of hs economy,.e. generae he same equlbrum asse prces and mmc equlbrum prcng of asses by ndvduals? These ssues are addressed n he smple framework of a sac exchange economy. Gven an observed equlbrum wh heerogeneous ndvdual subjecve probables, Calve e al. (2002) de ne an equvalen equlbrum, where all nvesors would share he sngle consensus probably by: ) he equvalen equlbrum generaes he same equlbrum prces and 2) every nvesor s margnal expeced uly valuaons of an asse reman he same n boh equlbra. They prove he exsence of such an equvalen equlbrum modulo a scalar adjusmen of he marke porfolo. Ths means ha n order o aggregae he ndvdual heerogeneous belefs no a sngle aggregae belef, he marke porfolo may have o be scalarly adjused, upward or downward, n he equvalen common probably equlbrum, a re econ of an aggregaon bas due o he dversy of belefs. Moreover, he auhors show ha he sandard consrucon of an expeced uly maxmzng aggregae nvesor, who s desgned so as o generae he observed equlbrum asse prces when endowed wh he marke porfolo, and o value hen asses a he margn as does every ndvdual nvesor n equlbrum, does carry over o he case of heerogeneous subjecve probables, provded ha ) hs aggregae nvesor s assgned he same consensus probably as prevously found and ha 2) he marke porfolo (aggregae consumpon) s scalarly adjused upwardly or downwardly as prevously. The am of he presen paper s o address he same ssues as Calve e al. (2002) bu n an neremporal, dscree or connous me, framework. We sar n Secon 2 by nroducng our dynamc and complee markes model. Gven an observed heerogeneous belefs Arrow-Debreu equlbrum, we hen de ne an equvalen equlbrum where all nvesors would share a common consensus belef by he same requremens as n Calve e al. (2002),.e. nvarance of he equlbrum prces and of he nvesors margnal valuaons. We prove (Subsec- 2

4 on 2.) ha n a dynamc seng a scalar adjusmen of he marke porfolo s no su cen n order o ensure he exsence of an equvalen equlbrum. We nroduce a predcable adjusmen of he marke porfolo and prove he exsence (and uncy) of an equvalen equlbrum under hs condon (Subsecon 2.2). Ths means ha equlbrum prces and ndvdual margnal valuaons n he heerogeneous belefs seng are he same as n an oherwse smlar homogeneous belefs economy n whch aggregae endowmen s mod ed, upward or downward, n a predcable way. We also show ha hs homogeneous belefs equvalen equlbrum s naurally assocaed o a represenave agen endowed wh he adjused marke porfolo (Subsecon 2.3). We end Secon 2 by showng on spec c examples (HARA uly funcons) how he consensus probably and he predcable adjusmen process can be explcly obaned. We nd ha he consensus probably s drecly relaed o some weghed mean of ndvdual subjecve belefs. We also characerze he suaons where he adjusmen of he marke porfolo has o be made upward (resp. downward). In parcular, we nd ha he adjusmen process s nondecreasng and greaer han (resp. nonncreasng and smaller han ) hence leads o an equvalen equlbrum n whch aggregae endowmen s ncreased (resp. decreased) f he cauousness parameer s smaller han (resp. greaer han ). In a sandard seng whou belefs heerogeney, when here s more rsk nvolved and when he nvesor s cauous (.e. s cauousness parameer s smaller han ), can be shown ha he nvesor wll ncrease curren consumpon acng as f fuure wealh was decreased. Now n our conex, a possble nerpreaon of our resul consss n consderng he dsperson of belefs as a source of rsk, hereby leadng for he represenave agen o an upward or downward adjusmen of aggregae endowmen dependng on wheher he cauousness parameer s smaller or greaer han. Our resuls are conssen wh hose of Rubnsen (974) and Zapaero (998) where he only e ec of he heerogeneous belefs on he asse prces s relaed o a change of probably (no scalng e ec). Indeed, n hese papers, he agens are endowed wh logarhmc uly funcons and appears ha hese funcons are he only ones n he HARA class for whch he predcable adjusmen process s consan and equal o. Secon 3 essenally consss of remarks and exensons. We rs explore he mplcaons of he aggregaon procedure on he rsk sharng rule. In he sandard case of homogeneous belefs and sae ndependen ules, we know ha he equlbrum allocaons are comonoonc,.e. are ncreasng funcons of he same facor (he aggregae consumpon) and hs feaure s commonly called he 3

5 rsk sharng rule. We show ha our resuls can be renerpreed as generalzng he sandard rsk sharng rule resul o he case of heerogeneous belefs. Indeed, as n Calve e al. (2002), we show ha, n a dynamc framework, he equlbrum allocaons n an heerogeneous belefs seng can be dvded no wo pars. The rs par sas es he sandard rsk sharng rule and he second par can be nerpreed as a resdual rsk due o heerogeney of belefs and s monoone n ndvdual belefs devaons from he consensus probably. We also show (Subsecon 3.2) how he same ssues can be addressed n a connuous me seng. We prove n such a conex he exsence of an equvalen equlbrum sasfyng he same requremens as before, modulo an adjusmen of he marke porfolo, whch s a ne varaon process. In Subsecon 3.3, we analyze he second nvarance requremen of our equvalen equlbrum, o w, ha he nvesors margnal ules reman he same n boh equlbra, and more precsely s lnks wh he desrable propery for an equvalen equlbrum ha each nvesor s observed (or nal) demand be larger han (resp. equal o, less han) hs demand n he equvalen equlbrum f and only f he aaches a subjecve probably ha s larger han (resp. equal o, less han) he consensus probably. Fnally, we consder n Subsecon 3.4 alernave aggregaon procedures. In parcular, we prove he exsence of an equvalen equlbrum (of he second knd), n whch all agens would share he same common probably leavng unchanged he equlbrum prces, he margnal valuaons and he marke porfolo (bu wh possble ncome ransfers) modulo a predcable aggregaon bas, whch akes he form of a dscoun facor on he uly funcons. Ths means ha he equlbrum prces and margnal valuaons n an heerogeneous belefs model are he same as n an homogeneous belefs economy modulo he nroducon of dscouned uly funcons. All he proofs are n he Appendx. 2. Aggregaon of heerogeneous belefs n dscree me We consder a collecon of ndvdual nvesors ndexed by = ; :::; N. We x a ne me horzon T on whch we are gong o rea our problem and we consder a lered complee probably space ; F; (F ) T=0 ; P ; whch sas es he usual 4

6 condons. The se represens he se of all hsores and he algebra F can be hough of as represenng all (heorecally) observable evens up o and ncludng me : Each nvesor solves a sandard dynamc uly maxmzaon problem. He has a curren endowmen a dae denoed by e ; and h a Von Neumann Morgensern uly funcon for consumpon of he form E PT =0 M u (; c ), where M s a posve marngale process sasfyng M0 =,.e. he posve densy process of a probably measure Q equvalen o P, and corresponds o he subjecve belef of ndvdual. We make he followng assumpons. For all = 0; :::; T, u (; ) : R +! R[ f g s of class C on R +, srcly ncreasng and srcly concave. 2. For all, u (; ) sas es Inada condons,.e. he dervave of u (; ) denoed by u 0 (; ) s such ha u 0 (; 0 + ) = and u 0 (; ) = 0. We shall denoe by (u 0 ) (; ) he nverse funcon of u 0 (; ) ; whch s connuous and srcly decreasng. 3. The aggregae endowmen e P N = e sas es e e e unformly n (;!) for some posve consans e and e. We recall ha an Arrow-Debreu equlbrum relavely o he belefs (M ) and he endowmen processes (e ) s de ned by a posve, unformly bounded prce process q and a famly of opmal admssble consumpon plans y such ha markes clear,.e. y = y (q ; M ; e ) P N = y = P N = e e where " TX y (q; M; e) arg max E P M E P [ P u (c ) : T =0 q(y e )]0 =0 We sar from an Arrow-Debreu equlbrum q ; y relavely o he belefs (M ) and he endowmen processes e : Such an equlbrum, when exss, A lered complee probably space ; F; (F ) T=0 ; P s sad o sasfy he usual condons f () F 0 conans all he P -null ses of F and () F = \ u> F u ; all, 0 < ;.e., he lraon s rgh-connuous. # 5

7 can be characerzed by he rs order necessary condons for ndvdual opmaly and he marke clearng condon. These condons can be wren as follows 8 >< >: h M u 0 (; y ) = q ; PT E P =0 q y e = 0 = ; :::; N (2.) P N = y = e for some se of posve Lagrange mulplers ( ) : Our am s o nd an equvalen equlbrum n whch he heerogeneous subjecve belefs would be aggregaed no a common belef M,.e. all nvesors would share he common belef M; he process M beng lke he M s a posve marngale process sasfyng E [M T ] =,.e. he posve densy of a probably measure Q equvalen o P: We wan he equvalen equlbrum o generae he same equlbrum prce process q as n he orgnal equlbrum wh heerogeneous belefs, so ha every asse ges he same valuaon n boh equlbra. The rs order condons would hen lead o M u 0 ; y = M u 0 ; y where y sands for he new equlbrum allocaon and where s a gven posve mulpler. Takng he produc of all hese equaons n he parcular case where u 0 (; x) = exp (x=), we oban M =! =N NY =! =N NY M : = In ha case, appears ha M s a marngale only f all he M s are equal. There s herefore n general no soluon o he aggregaon problem. Calve e al. (2002) n a sac framework propose o perm a scalar adjusmen of he aggregae endowmen and o replace he nvarance prncple on he aggregae endowmen by an nvarance prncple on he ndvdual margnal valuaon of asses. 2.. Aggregaon of heerogeneous belefs wh a scalar adjusmen of he marke porfolo Followng he approach of Calve e al. (2002), we de ne he equvalen equlbrum by wo requremens. Frs, he equvalen equlbrum should generae 6

8 he same equlbrum prce process q as n he orgnal equlbrum wh heerogeneous belefs, so ha every asse ges he same valuaon n boh equlbra. Second, every nvesor should be nd eren a he margn beween nvesng one addonal un of ncome n he orgnal equlbrum wh heerogeneous belefs and n he equvalen equlbrum, so ha every asse ges he same margnal valuaon by each nvesor (n erms of hs margnal expeced uly) n boh equlbra. We show n Secon 3 ha hs requremen s essenally equvalen o a monooncy requremen beween changes n he ndvdual porfolos and he underlyng changes of ndvdual probables. Calve e al. (2002) prove n a sac seng ha he consrucon of an homogeneous belefs equvalen equlbrum desgned so as o mmc equlbrum prces and margnal asse valuaons by ndvdual nvesors, s possble n a dverse belefs framework. The proposed desgn may requre a scalar adjusmen of he marke porfolo. We show on he followng example ha hs resul does no exend o he dynamc seng,.e., prces and ndvdual margnal valuaons n an heerogeneous probably belefs seng canno necessarly be expressed as prces and ndvdual margnal valuaons n an homogeneous probably belefs economy wh a possble scalar adjusmen on aggregae endowmen. Le f! ;! 2 g, P = (=2; =2) ; M and M 2 be such ha M 2 0 =, M 2 (! ) = 2=3; M 2 (! 2 ) = 4=3: Le u () = u (; ) = u 2 (; ) be such ha u 0 (x) = x wh 2 [ ; 0[ : We ake e such ha M u 0 e does no depend upon and we x q M u 0 e : In hs seng, q ; e s an equlbrum relave o he belefs (M ) and he endowmen processes e. We look for M; y ; y 2 and r 2 R such ha for = ; 2; we have Mu 0 y = M u e 0 = ; 2 X y = r X e = I appears ha =2M (! )+=2M (! 2 ) =3 s gven by 2 M 0 (+(2=3) = ) + 2=3, (+(4=3) = ) whch s equal o one f and only f =. Consequenly, excep for he logarhmc uly funcon, all he oher uly funcons n our class lead o processes 7

9 M ha are no marngales. Hence, hey can no be nerpreed as he densy process of a gven probably Aggregaon of heerogeneous belefs no a consensus probably belef and a predcable adjusmen process of he marke porfolo We have jus seen n he prevous subsecon ha we canno n general aggregae he belefs M n such a way ha M s a marngale whle keepng he adjusmen of he marke porfolo a scalar. We shall nroduce a predcable adjusmen process. We shall nd erenly use he ermnology equlbrum relave o he belefs Q or equlbrum relave o he belefs M ; where as above M represens he densy process of Q wh respec o P. Proposon 2.. Consder an Arrow-Debreu equlbrum q ; y relave o he belefs (Q ) and he endowmen processes e wh P N = e e : There exss a unque equvalen probably measure Q; a unque posve and bounded predcable adjusmen process r, a unque famly of endowmen processes e wh P N = e = re and a unque famly of consumpon processes y such ha. q ; y s an equlbrum relave o he common belef Q and he endowmen processes e 2. Tradng volumes and ndvdual margnal valuaon reman he same before and afer he aggregaon procedure,.e. for all = ; :::; N y e = y e and u 0 ; y = u 0 ; y : Noce ha hs aggregaon procedure sas es he addonal homogeney requremen. We oban hrough hs aggregaon procedure ha equlbrum prces and ndvdual margnal valuaons n he heerogeneous belefs seng are he same as n an oherwse smlar homogeneous belefs economy n whch aggregae endowmen s adjused n a predcable way. A naural queson s hen o deermne wheher hs adjusmen of he marke porfolo s o be made upward or downward. We shall a he end of Secon 2 analyze, n he spec c case of HARA uly funcons, he suaons leadng o an upward (resp. downward) 8

10 adjusmen of he marke porfolo. We sar by showng n he nex secon ha hs homogeneous belefs equvalen equlbrum s naurally assocaed o a represenave agen endowed wh he adjused marke porfolo (Subsecon 2.3) Represenave agen As n he sandard case of homogeneous belefs, we wan o consruc an expeced uly maxmzng aggregae nvesor, represenng he economy n equlbrum. More precsely, we look for a sngle aggregae nvesor, endowed wh he marke porfolo, who, when maxmzng hs expeced uly under he consensus probably generaes he same equlbrum prces as n he orgnal equlbrum. The nex proposon esablshes he exsence of such a represenave agen, as long as he marke porfolo s beng adjused by he same predcable process r as before. As n he sandard case, for 2 R + N ; we nroduce he funcon u (; x) = max P N = x x NX = u (; x ) and we recall ha we denoe by 2 R+ N he posve Lagrange mulplers of he heerogeneous belefs nal equlbrum. Proposon 2.2. Consder an equlbrum q ; y relave o he belefs (Q ) and he endowmen processes e wh P N = e e : The aggregae nvesor de ned by he normalzed VNM uly u s an equlbrum represenave agen when endowed wh he common probably Q as n Proposon 2., and he adjused marke hporfolo re ; n he sense ha he porfolo re maxmzes hs expeced uly PT h E Q =0 u PT (; y ) under he marke budge consran E =0 q (y r e ) 0. The consrucon of he represenave agen s exacly he same as n he sandard seng. As a consequence, all classcal properes of he represenave agen uly funcon reman vald n our seng (see e.g. Huang-Lzenberger, 988). Among oher properes, f all ndvdual uly funcons are sae ndependen, hen he aggregae uly funcon s also sae ndependen, and f all ndvdual uly funcons exhb lnear rsk olerance,.e. are such ha hen he aggregae uly funcon s also such ha P = N = : 9 u 0 (;x) u 00 (;x) u 0 (;x) u 00 = (;x) + x; = + x where

11 Remark ha our aggregaon procedure apples o a framework where agens have common belefs bu possbly d eren sae dependen uly funcons of he followng separable form U (;!; x) = v (;!) u (; x) : In ha case 2, we oban a represenave agen uly funcon of he same form U (;!; x) = v (;!) u (; x), where he funcon u s obaned from he u s as n he sandard framework, and where v s an average of he v s. Our consrucon can be compared wh Cuoco-He (994) represenave agen consrucon n ncomplee marke models (or Basak-Cuoco (998) n models wh resrced marke parcpaon). The man d erence s ha n Cuoco and He s consrucon, he represenave agen s uly funcon appears as a sochasc weghed average of he ndvdual uly funcons, whereas n our consrucon, he weghs are deermnsc, he represenave agen s uly funcon s he same as n he sandard case, however he oal endowmen s sochascally adjused. In parcular, as underlned above, n our consrucon, he classcal properes of he represenave agen s uly funcon reman vald, whch s no he case n Cuoco and He s consrucon..besdes, our consrucon (or more precsely, Calve e al. s conrucon) seems o be more racable n order o compare he equlbrum characerscs n he sandard and n he heerogeneous belefs sengs Example: HARA uly funcons In hs subsecon we assume ha all he uly funcons are n he HARA class. u More precsely, we suppose ha u (; x) s such ha 0 (;x) = u "(;x) + x for all. Alhough some of hese funcons do no sasfy Inada condons, he consensus probably and he predcable adjusmen process, f hey exs, should sasfy he same rs order condons as before and any par (M; r) sasfyng hese condons solves our aggregaon problem. We oban n he nex proposon as n Calve e al. (2002) explc expressons for he agggregae consensus probably and he adjusmen process r n he case of HARA uly funcons. In parcular, we are able o deermne f he aggregaon bas conrbues o an ncrease or a decrease of aggregae endowmen. Proposon 2.3. Assume ha ndvdual VNM ules belong o he HARA u famly wh 0 (;x) = u "(;x) + x > 0. 2 Noe ha even f he v s are no marngales, our resuls sll apply. 0

12 . The represenave agen who suppors he equlbrum wh he common probably Q n Proposon 2.2 belongs o he same HARA famly wh u 0 (;x) = + x, where = P N u "(;x) = : 2. The densy M wh respec o P of he correspondng common probably Q and he adjusmen process r are gven when 6= 0, by M s a marngale r 0 = M 0 = + r e = " X N M = + e = # = M wh = P N = when = 0, by : M s a marngale r 0 = M 0 = NY M = M = (r exp = ) e 3. The adjusmen process r sas es r (;!) f < r (;!) f > r (;!) = f = 4. If P N = = 0, hen we oban a smple consrucon algorhm r 0 = M 0 = r = E M = r " N X M PN = (M ) = = # M

13 and he predcable adjusmen process r sas es r s nondecreasng f < r s nonncreasng f > Noce ha he consensus belef s always gven by some weghed mean of he ndvdual heerogeneous belefs, adjused by a process dependng on r. The mean s eher a geomerc mean (n he case of exponenal uly funcons) or a power-" mean (n he case of power uly funcons). The weghs are gven by he ndvdual rsk olerances T +y : The process r s drecly relaed o +e he dsperson of he belefs M. In he spec c case of power uly funcons wh = 0;.e. f for all = ; :::; N; u 0 (; x) = (x) =, hen " N # = X M = (r ) = M : = h PN The process r measures he defaul of marngaly of he mean = (M ) = : We know by he prevous proposon ha r s nondecreasng (resp. nonncreasng) f < (resp. > ): The nerpreaon s he followng. In he sandard seng, when here s more rsk nvolved, dependng on wheher he nvesor s cauous or no, ha s o say dependng on wheher he cauousness parameer s smaller or greaer han, can be shown ha he nvesor wll ncrease or reduce curren consumpon acng as f fuure wealh was decreased or ncreased. For nsance, a cauous nvesor (cauousness parameer smaller han ) ncreases curren consumpon acng as f fuure wealh was ncreased. Now n our conex wh heerogeneous belefs, a possble nerpreaon consss n consderng he dsperson of belefs as a source of rsk, hereby leadng for he represenave agen o an upward or downward adjusmen of aggregae endowmen dependng on wheher he cauousness parameer s smaller or greaer han. In parcular f all agens have he same logarhmc uly funcons,.e. f for all = ; :::; N; u 0 (; x) = x, hen r and M = NX M = 2

14 where = P N = y = 0 P N. In hs case, we nd ha here s no adjusmen = y 0 e ec on he marke porfolo, and he consensus belef s gven by a weghed arhmec mean of he ndvdual heerogeneous belefs. Ths s he resul of Rubnsen (976). 3. Remarks and exensons 3.. Rsk sharng rule As underlned by Rubnsen (976), one poenal use of he aggregaon procedure s o relae he heerogeney of ndvdual demands o he heerogeney of ndvdual belefs. I s well known ha n an homogeneous belefs seng all he ndvdual allocaons are comonoonc. Ths propery s called he rsk sharng rule. Le us see n our seng he mplcaons of he aggregaon procedure n erms of rsk sharng rule. In he equvalen equlbrum q ; Q; re ; e, he belefs are homogeneous (represened by he probably measure Q) and all he ndvdual allocaons y are hen comonoonc. In he nal equlbrum wh heerogeneous belefs, he ndvdual allocaons a me denoed by y are gven by y = y + y y = y + ' M ; M M ; M 0 when M M, and ' M ; M 0 when M M. Furhermore ' M ; M = (u 0 ) M u 0 M y where for all, he y s are comonoonc and ' y s monoone wh respec o M : The heerogeney of he belefs nduces hen a dsoron of he rsk sharng rule and, for each agen, hs dsoron s monoone n ndvdual belefs devaons from he aggregae probably. The funcon ' can be explcly compued for some spec c classes of uly funcons. For nsance, n he exponenal case, we ge ha y y = log M log M 3.2. Aggregaon of heerogeneous belefs n a Connuous Tme Seng Le us now consder a connuous me framework. We x a ne me horzon T on whch we are gong o rea our problem, we le T [0; T ] ; and we con- 3

15 sder a lered probably space ; F; (F ) T=0 ; P ; where (F ) 2T denoes he P -augmenaon of he naural lraon generaed by a Brownan moon W on (; F; P ). We assume ha F T = F. The oal endowmen of he economy s descrbed by a sochasc process e sasfyng he followng sochasc d erenal equaon de = e d + e dw and e 0 s gven. As prevously, we assume ha each ndvdual s subjecve belef M s gven by he posve densy process of a probably measure Q, whch s equvalen o he orgnal probably measure P: Such a posve densy process can be represened as he soluon of he followng sochasc d erenal equaon dm = M dw ; M 0 = : We recall ha an Arrow-Debreu equlbrum relave o he belefs (M ) and he endowmen processes (e ) s de ned by a posve, unformly bounded prce process q and a famly of opmal admssble consumpon plans y such ha markes clear,.e. y = y (q ; M ; e ) P N = y = P N = e e where Z T y (q; M; e) arg max E P M u (c )d : E P [ R T 0 q(y e )d]0 We sar from an Arrow-Debreu equlbrum 0 q ; y relave o he belefs (M ) and he endowmen processes e and we assume ha he equlbrum allocaons y sasfy he followng equaons dy = y d + y dw : The R valued processes f ; 2 Tg ; f ; 2 Tg ; f ; 2 Tg ; ; 2 T ; ; 2 T are he coe cens of he model and are aken o be progressvely measurable wh respec o (F ) 2T and bounded unformly n (;!) n T : Furhermore, we clearly have P N = y = e P N = y = e : 4

16 Recall ha our condons on he densy M and on he equvalen equlbrum lead o nd an adjusmen process r, a densy M and an allocaon scheme y such ha NX y = re and M u 0 = ; y = Mu 0 ; y for all : We have seen ha we can no mpose ha r be a scalar f we wan M o be he densy process of a probably measure equvalen o he nal probably measure P,.e. a posve marngale wh M 0 =. In he prevous secons, we mposed ha r be a posve predcable process and he unqueness of such an r con rms ha hs condon s a naural condon. In he presen seng, we are dealng wh connuous processes and here s no d erence beween predcable processes and adaped processes. The predcably condon becomes hen oo weak. Indeed, su ces o ake any M; o choose he y s accordngly,.e. such ha M u 0 ; y = Mu 0 ; y and nally o ake r = P N = y =e : The rgh condon n he curren seng seems o be a zero-d uson coe cen for r;.e. dr = r d for some process ( ) : In he nex, we assume ha he uly funcons are C ;3 : If we mpose ha he y s and M be soluons of sochasc d erenal equaons of he form dy = a y d + b y dw (3.) dm = M dw and f we d erenae hese condons we oban he followng sysem of equaons u cc (; y ) u ccc (; y ) + u cc (; y ) M = a u cc (; y ) + 2 b2 u ccc (; y ) + b u cc (; y ) M u c (; y ) + u cc (; y ) M = u c (; y ) + b u cc (; y ) M (3.2) P a = r P P + P P y b = r 5

17 wh he nal condons r 0 = y 0 = y0 M 0 = Proposon 3.. There exss a unque soluon ((a ) ; (b ) ; ; ) o he sysem of equaons 3.2. Ths soluon depends on y ; (M ) ; y ; M ; ( ) ; ; ; r : If s regular enough n order o ensure ha he sochasc d erenal equaons adm a soluon, hen we have dy = a y d + b y dw ; y 0 = y 0 dm = M dw ; M 0 = dr = r d; r 0 = M u 0 NX y = re = ; y = Mu 0 ; y As n he dscree me framework, s possble, n he HARA uly seng, o characerze more precsely hese soluons. Proposon 3.2. If we suppose ha each uly funcon s such ha + x; he densy M and he adjusmen process r are gven by u 0 (;x) u 00 (;x) =. when 6= 0, M s a marngale r 0 = M 0 = = " + r e X N M = + e = # = M wh = P j j : 6

18 2. when = 0, by M s a marngale r 0 = M 0 = NY M = M = (r exp = ) e 3. If P N = = 0 and 6= 0, hen we oban a smple consrucon algorhm r 0 = M 0 = " X N M = r and he adjusmen process r sas es M s a marngale = # M r s nondecreasng f < r s nonncreasng f > : In parcular, as n he dscree me seng, we nd ha belefs heerogeney leads o an equvalen equlbrum wh ncreased aggregae endowmen f and only f he nvesor s cauous,.e. f he cauousness parameer s smaller han one Comonooncy propery and he second requremen A desrable propery of he equvalen equlbrum s ha each nvesor s observed (or nal) demand be larger han (resp. equal o, less han) hs demand n he equvalen equlbrum f and only f he aaches a subjecve probably ha s larger han (resp. equal o, less han) he aggregae common probably,.e. and y y y f and only f M M y f and only f M M ha we shall refer o as he comonooncy propery (a dae ). We wan n hs subsecon o analyze he lnks beween hs comonooncy propery and he 7

19 second nvarance requremen of our aggregaon procedure,.e. he nvarance of ndvdual margnal valuaons. I s easy o see ha, due o our second nvarance requremen, he comonooncy propery s auomacally sas ed by our aggregaon procedure. Indeed, as seen prevously, we have for all = ; :::; N and for all = 0; :::; T; M u 0 ; y = M u 0 ; y ; and, snce u 0 (; ) s decreasng, he comonooncy propery follows mmedaely a any dae = 0; :::; T. Conversely, le us sudy o whch exen he comonooncy propery as well as he rs requremen mply our second requremen. By he rs requremen, we mus have for all = ; :::; N M = u 0 M u 0 ; y ; y where s as above he Lagrange mulpler n he nal equlbrum, and s he Lagrange mulpler n he equvalen equlbrum. The second requremen s hen equvalen o he condon ha = for all = ; :::; N. Imposng drecly he comonooncy propery; nsead of he second requremen, leads o ; M M M M M : (3.3) M M M If we requre he comonooncy propery o be sas ed from dae 0; hen s mmedae ha mples our second requremen: ndeed, snce M0 = M 0 =, we ge by (3.3) a = 0, ha = for all = ; :::; N. If we only requre he comonooncy propery from dae =, hen s mmedae ha he followng condon (C) mples he second requremen. Condon (C) : For all 2 f; :::; Ng, for all " 2 R +, here exss some 2 f; :::; T g for whch M P M 2 ] "; ] P 2 [; + "[ > 0: M M Noce ha condon (C) can be nerpreed as a closeness condon beween M and he M s. 8

20 We nroduce he followng condon. Condon (C2) : For all 2 f; :::; Ng, here exss some 2 f; :::; T g for whch (a) The vecor M ; :::; M N ; e adms a posve densy wh respec o he Lebesgue measure on (R + ) N [e; e]. (b) The random varable M can be wren n he form M = g (M ; :::; M n ; e ) for some funcon g of class such ha g 6= 0: Noce ha condon (C2) (b) s sas ed f g s concave and nonnegave. We oban he followng resul: Proposon If he aggregaon procedure sas es he rs requremen and he comonooncy propery a all daes = 0; :::; T, hen he equvalen equlbrum sas es he second requremen. 2. If he aggregaon procedure sas es he rs requremen, he comonooncy propery a daes = ; :::; T as well as (C2), hen he equvalen equlbrum sas es he second requremen Oher possble aggregaon procedures We have consdered so far exensons of Calve e al. (2002). More precsely, knowng ha a rue aggregaon of ndvdual belefs leavng all equlbrum characerscs nvaran s mpossble, we have proposed aggregaon procedures ha auhorze an adjusmen of he marke porfolo. Anoher possble aggregaon procedure consss n leavng he marke porfolo nvaran and n auhorzng he nroducon of a dscoun facor on he uly funcons. We show n Joun-Napp (2003), n connuous me, ha gven an equlbrum prce process q relave o he belefs (M ); and he endowmen processes (e ) wh P N = e = e ; here exss a posve marngale process M wh M 0 = ; and a ne varaon posve process B exp R (s)ds such ha, wh he noaons of Secon 2, 0 M B u 0 (; e ) = q : The adjusmen process B measures hen he aggregaon bas nduced by he heerogeney of ndvdual belefs and leads o a (possbly negave) dscoun of uly from fuure consumpon hrough he dscoun rae ( ). I s shown n Joun-Napp (2003) ha he consensus belef M s gven by some weghed average of he ndvdual belefs, he weghs beng gven by he ndvdual rsk olerances, and ha he process B (or ) s drecly relaed o he weghed varance (wh he same weghs) of he ndvdual belefs. 9

21 4. Appendx Proof of Proposon 2. Snce q ; y s an equlbrum relave o he belefs (Q ) ; and he endowmen processes e wh P N = e = e, we know ha for all = 0; :::; T; NX = y = NX e = e = and ha here exs posve Lagrange mulplers ( ) such ha for all = ; ::::; N, for all = 0; :::; T, M u 0 ; y = q : h dq dp where M E j F : For a gven bounded and nonnegave predcable process r, we consder he maxmzaon problem (P r ) : max NX = U y under he consran NX y re ; h PT where U (c) = E =0 u (; c ). If we denoe by y ;(r) he soluon of he maxmzaon program (P r ), hen T he posve process ; y ;(r) s ndependen from and we denoe u 0 M (r) =0 by p (r) : Le M (r) q ; where by convenon, M (r) = 0 on p (r) =. We hen p (r) have for all, for all = 0; :::; T ; y = M Noce ha for all, he random varables y ;(r) u 0 u 0 ; y ;(r) r only hrough r and can herefore be denoed by y ;(r) = : (4.) (resp. M (r) ) depend on he process (resp. M (r) ). We wan o nd a posve and bounded predcable process r such ha M (r) 0 = and M (r) s a marngale. Le us consruc such a process by nducon on. For = 0, we ake r 0 = ; and we ge y ;(r 0) 0 = y0 and M (r 0) ha we have obaned he posve and bounded predcable process (r s ) 20 0 =. Le us suppose s=0 ; hence

22 h he posve process M s (r) and le us consruc r such ha E M (r) j F = s=0 h M (r ). We nroduce he funcon : r 7! E M (r) j F M (r ) and we wan o prove ha here exss an F -measurable posve and bounded random varable r such ha (r ) = 0 (almos surely). We rs show ha here exss a posve consan R such ha any nonnegave and bounded F measurable random varable a sas es (a) > 0 on fa > Rg. Noce ha snce u 0 ; y ;(a) P N = (u0 ) (; x). We have = q M (a), we have q M (a) = h (ae ) wh h (x) = 2 (a) = E 4 M u 0 ; y h (ae ) j F 3 5 M (r ) hence M (a) fa>rg > E u 0 (; e) h (er) fa>rg j F > M u 0 (; e) M (r ) h (er) fa>rg M (r ) fa>rg so ha he exsence of R such ha (a) > 0 on fa > Rg s gven by he exsence M of a posve consan A such ha nf (r ) < A. Now, M M (r ) u 0 ; y = M ; y ;(r ) and nf M (r ) M u 0 < sup u 0 ; e N u 0 ( ; R e) where R denoes he upper bound for r. Le us consder he se of nonnegave and bounded F measurable random varables r such ha (r ) 0: The se conans 0 and we have jus proved ha any r n s bounded by R: Furhermore, s easy o check ha, for all F nonnegave and bounded measurable random varables r and r 2 ; and any 2

23 even B 2 F, (r B + r 2 B c) = (r ) B + (r 2 ) B c and he monooncy of M (r) wh respec o r leads o (r _ r 2 ) = (r ) _ (r 2 ) : Le r ess sup r2 r : The famly beng dreced upward, we know ha here exss r k n such ha r = lm k! % r k. We prove now ha (r ) = 0. I s mmedae ha r s nonnegave, bounded and predcable. Besdes, s easy o see ha, when k! ; hen M (r k) % M (r ) ; and snce for all k, r k 0, we ge by he Beppo-Lev Theorem ha (r ) 0. I re- 2, mans o show ha for all ("; b) 2 R+ we have P (";b ) = 0, where ";b f (r ) < "; E [q ] bg 2 F : We nroduce br = r + ";b for 2 R +. We ge (br) = (r ) + [ (r + ) (r )] ";b. We shall prove ha for some posve, j (r + ) (r )j < " on ";b. Ths would lead o (br) 0, hence P ( ";b ) = 0 and (r ) = 0. Now, j (r + ) (r )j ";b = E q h ((r + ) e ) h (r e ) ";b b sup j (x) (y)j ";b jx yje x;y2[0;re] for h and hen j (r + ) (r )j < " on ";b for small enough. Takng r = r complees he consrucon by nducon of he process r. As far as unqueness s concerned, s rs easy o see (as n he proof of Proposon??) ha for any posve and bounded process r, here exss a unque M; y ; e = M (r) ; y ;(r) ; e y + y sasfyng requremens. and 2. of he Theorem. The unqueness of r s mmedae by monooncy of. Proof of Proposon 2.2 Smlar o he proof of he exsence of a represenave agen n he homogeneous belefs seng. Proof of Proposon 2.3 ) We know ha M u 0 ; y = M u 0 (; r e ) = () (2) M u 0 ; y We ge from he second equaly ha u 0 (; r e ) = u 0 ; y so ha, for > 0; u 0 ; y = [ u 0 (; r e )]. Snce u 0 (; x) = ( + x) =, hs leads o + y ;(r) = [ u 0 (; r e )], hence u 0 (; r e ) = 22 +re P N = : From he rs

24 h h equaly, we ge ha u 0 ; y = M u 0 M (; r e ) = + y, hence + e = M u 0 (; r e ) P N, +re = M and M = = h PN +e = (M ) =. If = 0; we have u 0 (; x) = exp and adopng he same approach, usng he second equaly u 0 (; r e ) = exp equaly, we ge ha exp y x re M = exp M exp P N = ln : From he rs re exp P N = ln ; and hen M = Q N = (M ) = exp (r )e : +re 2) Snce M = = h PN +e = (M ) =, we have 2" N # = 3 X E M r< < E 4 M 5 r<: = Now, for 0 < <, we ge by Mnkovsk s nequaly E hpn = (M ) = h PN = M = +r ; hence E M r< < M e = r<; and + r e + e! r< < r<: +e Ths leads o fr < g fr < g : Snce r 0 =, we have r. The case > can be reaed smlarly. For = ; s clear ha r = and M = P N = M s he soluon of our equaons. h Q For = 0, E M r< E (M ) = r< Q N = M = and hen r< exp (r )e r< and as prevously fr < g fr < g. 3) In he case P N = h PN = 0; s mmedae ha M = r = (M ) h PN hpn = hence E M < r E = (M ) =. Now, for >, we ge he fol- hpn lowng Mnkovsk-lke nequaly E = (M ) = = M = ; h hence M r = PN = M = = ; and M r r M or equvalenly r r. The case > can be reaed smlarly. 23

25 Proof of Proposon 3. The second equaon n he sysem 3.2 gves us u c (; y ) + u cc (; y ) M u c (; y ) = u cc (; y )M u cc (; y ) + b and we deduce from he fourh equaon n he same sysem ha = P N = uc(;y )+ ucc(;y ) M u cc(;y )M P N u c(;y ) = u cc(;y ) P r N = : Ths gves and he b s. The a s (resp. ) are hen obaned from he rs (resp. he hrd) equaon n he sysem 3.2. Proof of Proposon 3:2 The relaons beween M; r and he M s are obaned exacly as n he dscree me framework. In he parcular case where = 0; we have " N # X M = r M (4.2) = and f we d erenae boh sdes, we oban 2 r ( ) NX = 2 M X N + r M = 2 ( ) 2 M r = NX M = M = or equvalenly NX = = 2 = ( ) 4 2! 2 NX X N = = where for all, r (M ) M and sasfy P N = = (by 4.2): 24

26 PN 2 Snce = P N = 2 s always nonposve, and ( ) have oppose sgns and r s nondecreasng (resp. nonncreasng) for < (resp. > ). Proof of Proposon 3.3 I s easy o see ha under condon (C2), he applcaon h : (x ; :::; x N ; y) 7! x x ; :::x ; g (x ; :::; x N ; y) ; x +; :::; x N ; y s a C -d eomorphsm from R+ N ]e; e[ o an open subse U of R N + ]e; e[ : Then he random vecor M ; :::; M ; M M ; M + ; :::; M N ; e adms a densy wh respec o he Lebesgue measure on (R + ) N [e; e], whch s posve on U. Furhermore, snce E M = E [M ] =, s mpossble o have g (x ; :::; x N ; y) < x for all (x ; :::; x N ; y) ; or g (x ; :::; x N ; y) > x for all (x ; :::; x N ; y). Snce Im U s conneced, Pr oj (Im U) s also conneced, where Pr oj denoes he projecon on he -h coordnae, hence here exss (bx ; :::; bx N ; by)such ha g (bx ; :::; bx N ; by) = bx. Snce U s open, here exss a neghborhood of (bx ; :::; bx ; ; bx + ; :::; bx N ; by) n U, whch leads o M P M 2 ] "; ] P 2 [; + "[ > 0: M M References [] Basak, S. and D. Cuoco, 998. An equlbrum model wh resrced sock marke parcpaon. Revew of Fnancal Sudes,, [2] Calve, L., Grandmon, J.-M., and I. Lemare, Aggregaon of Heerogenous Belefs and Asse Prcng n Complee Fnancal Markes. Workng Paper. [3] Cuoco, D. and H. He, 994. Dynamc equlbrum n n ne-dmensonal economes wh ncomplee nancal markes. Mmeo, Unversy of Pennsylvana. [4] Joun, E., and C. Napp, Consensus Consumer and Ineremporal Asse Prcng wh Heerogeneous Belefs. Workng Paper. 25

27 [5] Rubnsen, M., 974. An aggregaon heorem for secures markes. Journal of Fnancal Economcs,, [6] Rubnsen, M., 976. The Srong Case for he Generalzed Logarhmc Uly Model as he Premer Model of Fnancal Markes. Journal of Fnance, 3, [7] Zapaero, F., 998. E ecs of Fnancal Innovaons on Marke volaly when Belefs are Heerogeneous. Journal of Economcs, Dynamcs and Conrol, 22,