The Dynamics of Efficient Asset Trading with Heterogeneous Beliefs

Transcription

1 The Dynamcs of Effcen Asse Tradng wh Heerogeneous Belefs Pablo F. BEKER # Emlo ESPINO ## Deparmen of Economcs Unversy of Warwck Deparmen of Economcs Unversdad Torcuao D Tella Ths Draf: July 2, 2008 Absrac Ths paper analyzes he dynamc properes of porfolos ha susan dynamcally complee markes equlbra when agens have heerogeneous prors. We argue ha he convenonal wsdom ha belef heerogeney generaes connuous rade and sgn can ucuaons n ndvdual porfolos may be correc bu also needs some qual caons. We consder an n ne horzon sochasc endowmen economy where he acual process of he saes of naure consss n..d. draws. The economy s populaed by many Bayesan agens wh heerogeneous prors over he sochasc process of he saes of naure. Our approach hnges on sudyng porfolos ha suppor Pareo opmal allocaons. Snce hese allocaons are ypcally hsory dependen, we propose a mehodology o provde a complee recursve characerzaon when agens know ha he process of saes of naure s..d. bu dsagree abou he probably of he saes. We show ha even hough heerogeneous prors whn ha class can ndeed generae genune changes n he porfolos of any dynamcally complee markes equlbrum, hese changes vansh wh probably one f he suppor of every agen s pror belef conans he rue dsrbuon. Fnally, we provde examples n whch asse radng does no vansh because eher () no agen learns he rue condonal probably of he saes or () some agen does no know he rue process generang he daa s..d. Keywords: heerogeneous belefs, asse radng, dynamcally complee markes. We hank Rody Manuell and Juan Dubra for dealed commens. All he remanng errors are ours. # Correspondng Auhor: Unversy of Warwck, Deparmen of Economcs, Warwck, Covenry CV4 7AL, UK. E-mal: Pablo.Beker@warwck.ac.uk. ## Unversdad Torcuao D Tella, Deparmen of Economcs, Saenz Valene 00 (C428BIJ), Buenos Ares, Argenna. E-mal: eespno@ud.edu.

2 Inroducon A long-sandng ene n economcs s ha belef heerogeney plays a prme role n explanng he behavor of prces and quanes n nancal markes. In spe of he emphass ha economss gve o e cency, surprsngly, very lle s known abou he mplcaons of belef heerogeney on dynamcally complee markes. However, here are some noable excepons. Sandron [20] and Blume and Easley [4] provde an analyss of he asympoc properes of consumpon. Cogley and Sargen [7] focus on asse prces. Our paper, nsead, focuses on he e ec of belef heerogeney on asse radng. Before proceedng s useful o recall wha s known abou asse radng n a dynamcally complee markes equlbrum when agens have dencal belefs. Judd e al. [4] consdered a saonary Markovan economy where agens have homogeneous and degenerae belefs bu d eren audes owards rsk and show ha each nvesor s equlbrum holdngs of asses of any spec c maury s consan along me and across saes afer an nal radng sage. I follows ha d erences n rsk averson by self canno explan why nvesors change her porfolos over me. We consder an exchange economy where boh he endowmens as well as he asses reurns are..d. draws from a common probably dsrbuon. Invesors who are n nely lved do no know he one-perod-ahead condonal probably of he saes of naure and updae her prors n a Bayesan fashon as daa unfolds. We begn wh wo examples of dynamcally complee markes equlbrum ha llusrae ha he convenonal wsdom ha belef heerogeney causes sgn can rade may be correc bu also needs some qual caons. In example, agens know he rue process s..d. and hey only dsagree abou he probably of he saes of naure. In he long run, condonal probables, wealh and porfolos converge. In example 2, agens do no know he rue process s..d., condonal probables converge and ye wealh bounces back and forh beween hem n nely ofen so ha each of hem holds almos all he wealh n nely many mes. Ths second example shows ha even hough agens may learn, pror belef heerogeney may ndeed generae sgn can ucuaons n he wealh dsrbuon and he correspondng porfolos ha do no exhaus n he long run. We argue ha he d eren dynamcs n he wo examples re ec d erences n he lm behavor of he lkelhood rao of he agens prors. Ths paper lnks he evoluon of he wealh dsrbuon and he correspondng To avod any confuson, we use he followng ermnology. By a pror, we refer o he subjecve uncondonal probably dsrbuon over fuure saes of naure. In he parcular case where he pror can be characerzed by a vecor of parameers and a probably dsrbuon over hese parameers, we call he laer he agen s pror belef.

3 porfolos n any dynamcally complee markes equlbrum o he evoluon of he lkelhood rao. Ths s useful because he lkelhood rao s an exogenous varable and several properes of s lm behavor are well undersood from he sascs leraure (see Phllps and Ploberger [9] and references heren.) Our examples and 2 rase he queson of wha ype of belef heerogeney maers for asse radng. In order o answer hs queson, we rs carefully assess a class of prors sasfyng wo assumpons ha are ubquous n he leraure. Namely, every agen knows he lkelhood funcon generang he daa and here s a leas one agen who learns n he sense ha her one-perod-ahead condonal probably converges o he ruh. In our seup, hs s ensured by assumng ha every agen knows he daa s generaed by..d. draws from a common (unknown) dsrbuon and he suppor of her pror belefs conans he rue probably dsrbuon of he saes of naure as n example. We rs show ha even hough heerogeneous prors n ha class can ndeed generae changes n he porfolos of a dynamcally complee markes equlbrum, hese changes vansh wh probably one. Very mporanly, we fully characerze he dynamcs of porfolos and s correspondng lm. Aferwards, we show by means of wo addonal examples ha f one wans o argue ha heerogeney of prors can have endurng mplcaons on he volume of rade n a saonary envronmen hen one needs o relax one of he aforemenoned assumpons; ha s, eher () no agen learns he rue condonal probably of saes or () some agen does no know he lkelhood funcon generang he daa. Snce solvng drecly for he porfolos of a dynamcally complee markes equlbrum s no always possble, we follow an ndrec approach developed by Espno and Hnermaer [9]. Ths approach hnges on sudyng porfolos ha suppor Pareo opmal allocaons. The d culy s ha belef heerogeney makes opmal allocaons hsory dependen because opmaly requres he rao of margnal valuaons of consumpon of any wo agens -whch ncludes prors ha could be subjecvely held- o be consan along me. Consequenly, a any dae he rao of margnal ules a any fuure even mus be proporonal o he hsory dependen rao of he agens prors abou ha even,.e. he lkelhood rao of he agens prors. Ths rao represens he novel margn of heerogeney among agens consdered n hs paper, whch we call he B-margn of heerogeney. The evoluon of he B-margn deermnes he dynamcs of he opmal dsrbuon rule of consumpon and, consequenly, he evoluon of he wealh dsrbuon n any dynamcally complee markes equlbrum. The law of moon of hs margn s ypcally hsory dependen and, very mporanly, he curren sae and he curren B-margn are no enough o summarze he hsory. Under he assumpon ha every agen knows he daa s generaed by..d. draws from a common (unknown) dsrbuon bu have d eren belefs over 2

4 he unknown parameers, hs hsory dependence can be succncly capured by he agens belefs (over he parameers). Ths assumpon allow us o use a sraegy smlar o Lucas and Sokey s [6] o oban a recursve characerzaon of he se of Pareo opmal allocaons n our sochasc framework. 2 The key nsgh s ha he planner does no need o know he paral hsory self n order o connue he dae zero opmal plan from dae onwards. In fac, su ces ha he knows he sae of naure, he agens pror belefs over probables and, very mporanly, he curren B-margn,.e. he lkelhood raos of he agens prors ha summarze how he wegh aached o each agen depends on hsory. We argue ha he sequenal formulaon of he planner s problem s equvalen o a recursve dynamc program where he planner, who akes a vecor of welfare weghs as gven, allocaes curren feasble consumpon and assgns nex perod aanable uly levels among agens. The planner s opmal choce of connuaon ules nduces a law of moon for welfare weghs ha s somorphc o he evoluon of he lkelhood rao of he agens prors. Aferwards, we use he planner s polcy funcons o characerze recursvely nvesors nancal wealh n any dynamcally complee marke equlbrum. Ths allows us o esablsh ha he nancal wealh dsrbuon (and he correspondng supporng porfolos) converges f and only f boh he B-margn vanshes and he agens belefs over he parameers become homogeneous. When he agens know ha he rue process consss n..d. draws from a common dsrbuon and he rue dsrbuon s n he suppor of her prors, he well-known conssency propery of Bayesan learnng mples ha he agens pror belef become homogeneous wh probably one. To ge a horough undersandng of he lmng behavor of porfolos, herefore, wha remans o be explaned s he asympoc behavor of he B-margn. When he suppor of he agens pror belefs over he parameers s a counable se conanng he rue probably dsrbuon, he rue probably dsrbuon over pahs s absoluely connuous wh respec o he agens prors and, herefore, he convergence of lkelhood raos follows from Sandron [20]. When he agens pror belefs have a posve and connuous densy wh suppor conanng he rue parameer, he hypohess n Sandron [2] are no sas ed and so we apply a resul n Phllps and Ploberger [9] o show ha he lkelhood rao of he agens prors sll converges wh probably one. The mporan message here s ha he heerogeney of prors by self can generae changes n porfolos bu hese changes necessarly vansh because he B-margn vanshes. Furhermore, we show ha porfolos converge o hose of a raonal expecaons equlbrum of an economy where he nvesors relave wealh s deermned by he 2 Lucas and Sokey [6] characerze recursvely opmal programs n a deermnsc seng where recursve preferences nduce he dependence upon hsores. 3

5 denses of her pror belefs evaluaed a he rue parameer and he dae zero welfare wegh. 3 To conclude we analyze he exac role played by he aforemenoned assumpons on prors and we argue ha s crcal ha hey are coupled ogeher. We do so by provdng wo addonal examples, each of whch relax one of hese assumpons, n whch he B-margn does no vansh and consequenly porfolos change n nely ofen. In example 3, agens know he daa s generaed by..d. draws from a common dsrbuon bu hey do no have he rue parameer n he suppor of her pror belefs and so no agen learns. We assume ha her pror belefs are such ha he assocaed one-perod-ahead condonal probables have dencal enropy, a condon ha ensures ha he lkelhood rao of her prors ucuaes n nely ofen beween zero and n ny and, consequenly, porfolos ucuae n nely ofen. Fnally, example 4 underscores he mporance of assumng ha every agen knows he process of saes consss n..d. draws for he porfolos o converge. To srech he argumen o he lm, we consder an example n whch only one agen does no know he daa s generaed by..d. draws. Ths agen makes exac one-perodahead forecass n nely ofen bu also makes msakes n nely ofen hough rarely. We show ha he lkelhood rao of hese agens prors fals o converge wh probably one mplyng ha he se of pahs where he equlbrum porfolo converges has probably zero. Ths paper s organzed as follows. In Secon 2 we revew he relaed leraure. In secon 3 we descrbe he model. In secon 4 we presen a smple example ha llusrae he man deas n hs paper. The recursve characerzaon of Pareo opmal allocaons s n secon 5. Secon 6 characerzes he asympoc behavor of he agens nancal wealh and her correspondng supporng porfolos. Fnally, secons 7 and 8 dscuss when he agens porfolo converge and when does no. Conclusons are n secon 9. Proofs are gahered n he Appendx. 2 Relaed Leraure Ths paper relaes o wo branches of he leraure on he e ec of belef heerogeney n asse markes: models amng o explan he dynamc consequences of belef heerogeney on nvesors behavor and models analyzng he marke selecon hypohess. Harrson and Kreps [3] and Harrs and Ravv [2] who sudy he mplcaons of belef heerogeney on asse prces and radng volume, respecvely, are 3 In parcular, even hough agens learn he rue probably of saes of naure, hese lmng porfolos need no concde wh hose of an oherwse dencal economy ha sars wh homogeneous prors and zero nancal wealh. 4

6 he leadng arcles of he rs branch. These rs-generaon papers consder paral equlbrum models where a ne number of rsk-neural nvesors rade one un of a rsky asse subjec o shor-sale consrans. Invesors do no know he value of some payo relevan parameer bu hey observe a publc sgnal and have heerogeneous bu degenerae pror belefs abou he relaonshp beween he sgnal and he unknown parameer. Snce hey are rsk neural and have heerogeneous belef, hey have d eren margnal valuaons and so rade occurs f and only f agens "swch sdes" regardng her valuaon of he asse. In addon, Harrson and Kreps [3] show ha an speculave premum mgh arse, n he sense ha he asse prce mgh be srcly greaer han every rader s fundamenal valuaon. Snce each nvesor s absoluely convnced her model s he correc one, her dsagreemen does no vansh as he daa unfold. The possbly ha agens learn s addressed by Morrs [7] who exends Harrson and Kreps [3] model o consder agens ha have heerogeneous and non-degenerae pror belefs over he probably dsrbuon of dvdends. He characerzes he se of pror belefs for whch he speculave premum s posve. He assumes he rue process s..d., nvesors know hs fac bu hey have heerogeneous pror belefs abou he dsrbuon of hese draws wh suppor conanng he rue dsrbuon. Snce hey are Bayesan, hey evenually learn he rue dsrbuon. Consequenly, rsk neuraly mples he prce converges and he speculave premum vanshes. We underscore ha asse radng does no vansh because here s always a perod n he fuure when he asse changes hands once agan. Morrs [7] asympoc resuls, however, are a drec consequence of he assumpon ha agens are rsk-neural. Indeed, under rsk-neuraly he neremporal margnal raes of subsuon are ndependen of he equlbrum allocaon and, herefore, hey are lnear n he agens one-perod-ahead condonal probables. Ths has wo drec mplcaons. On he one hand, when he ndvduals one-perod-ahead condonal probables swch sdes perpeually, so do her neremporal margnal raes of subsuon and, herefore, new ncenves for a change n he ownershp of he asse arse n nely ofen. On oher hand, asse prces hemselves are parameerzed by he one-perod-ahead condonal probables and, hus, hey converge ogeher. In hs paper, we argue ha hese forces do no operae n a seng where agens are rsk-averse and allocaons are Pareo opmal. More precsely, Pareo opmaly mples ha he agen s neremporal margnal raes of subsuon mus be equalzed and, unlke n Morrs [7] where hey swch perssenly, every rader s valuaon of any fuure ncome sream always concde. Consequenly, here s never a speculave premum n spe of belef heerogeney. Our analyss makes evden ha he speculave premum s no necessarly drven by belef heerogeney bu, more mporanly, by he 5

7 d erences n he agens neremporal margnal valuaons due o he exsence of shor-sale consrans. 4 In our seng, porfolos mgh sll change perssenly bu hese changes depend purely on he asympoc behavor of he e cen allocaon. Furhermore, as we emphaszed above, he convergence of he one-perod-ahead condonal probables by self does no guaranee he convergence of allocaons, asse prces and porfolos. Belef heerogeney may have fundamenal mplcaons on he behavor of asse markes even n he absence of he aforemenoned capal marke mperfecons. In he conex of he Lucas [5] ree model, Cogley and Sargen [6] and [7] focus on he e ecs of learnng and pror belef heerogeney, respecvely, on asse prces under he assumpon ha agens know he rue lkelhood funcon. In [6], hey consder an economy wh a rsk-neural represenave agen wh a pessmsc bu non-degenerae pror belef over he growh rae of dvdends. Even hough learnng evenually erases pessmsm, pessmsm conrbues a volale mulplcave componen o he sochasc dscoun facor ha an economercan assumng correc prors would arbue o mplausble degrees of rsk averson. 5 Cogley and Sargen [7] analyze he robusness of ha ndng by consderng an economy wh complee markes wh some agens who know he rue probably dsrbuon (.e., hey add belef heerogeney). For a plausble calbraon of her model, hey show ha unless he agens wh correc belefs own a large fracon of he nal wealh, akes a long me for he e ec of pessmsm o be erased. Ther work s close n spr o ours n ha hey use a general equlbrum model whou any addonal marke mperfecon. Snce hey are prncpally neresed n sudyng he marke prces of rsk, however, hey are slen abou he mplcaons of belef heerogeney for radng volume. Consequenly, he asse radng mplcaons semmng purely from d erences n prors are sll an open queson. The second branch of he leraure relaed o our paper analyses he marke selecon hypohess and s exempl ed by he work of Sandron [20] and Blume and Easley [4]. Sandron [20] shows ha, conrollng for dscoun facors, f he rue dsrbuon s absolue connuous wh respec o some rader s pror hen she survves and any oher rader survves f and only f he rue dsrbuon s absolue connuous wh respec o her pror as well. 6 He also consders some cases n whch he rue dsrbuon s no absolue connuous wh respec o any agen s pror. He shows ha 4 Indeed, n any economy where radng consrans are occasonally bndng for d eren agens, he agen who prces he asse changes and hus he speculave premum can arse naurally. 5 Ther model can generae subsanal and declnng values for he marke prces of rsk and he equy premum and, addonally, can predc hgh and declnng Sharpe raos and forecasable excess sock reurns. 6 An agen s sad o survve f her consumpon does no converge o zero. 6

8 he enropy of prors deermnes survval and, herefore, an agen who perssenly makes wrong predcons vanshes n he presence of a learner. Absolue connuy s a srong resrcon on prors ha s no sas ed, for nsance, f he rue process s..d., he agen knows hs fac bu her pror belefs over he probably of he saes of naure have connuous and posve densy. 7 Ths s precsely he case ha Blume and Easley [4] consder and hey prove ha among Bayesan learners who have he ruh n he suppor of her prors, only hose wh he lowes dmensonal suppor can have posve consumpon n he long run. Techncally speakng, Blume and Easley s noon of convergence s n probably and hey esablsh her asympoc resul for almos all parameers n he suppor of he agen s pror belef. Alhough we do no focus on survval, one sde conrbuon of hs paper s o make Blume and Easley s resuls more robus because we show ha every Bayesan agen wh a pror belef wh he lowes dmensonal suppor acually survves wh probably one (no jus n probably), no only for almos every parameer n he suppor of her pror belef bu acually for all parameers n he suppor of her pror belef. 8 Our reamen of prors s very general n ha we consder a famly ha ncludes prors for whch he one-perod-ahead condonal probably converges o he ruh regardless of wheher he agens prors merge wh he ruh or wheher raders know he rue process consss n..d. draws. In addon, ncludes cases n whch some agens have he ruh n he suppor of her prors whle some oher agen do no learn he rue one-perod-ahead condonal probably and ye he laer survves as n our example 4. To he bes of our knowledge hs s he rs example of s knd n he leraure. Our resuls characerzng he porfolos ha suppor a Pareo opmal allocaon are a novel conrbuon o he leraure snce neher Sandron [20] nor Blume and Easley [4] analyze porfolo dynamcs. Indeed, he mappng beween consumpon and s supporng porfolo s only smple when agens have degenerae homogeneous prors as n Judd e al. [4]. Ths s mos evden when one consder he case where agens have homogeneous bu non-degenerae pror belefs. In hs case, he dsrbuon of consumpon s me ndependen whle he supporng porfolos are no because he sae prces change as agens learn. We also conrbue o he analyss of he asympoc behavor of porfolos snce s no evden ha Sandron s [20] and Blume and Easley s [4] resuls on he lm behavor of consumpon mply ha () porfolos mus converge when lkelhood raos do and, very mporanly, 7 In ha case, snce he enropy of every agen s pror s he same, one canno apply Sandron s resuls relang survval wh he enropy of prors eher. 8 Ths dsncon s economcally relevan because boh n Blume and Easley s [4] seng as well as n ours he daa (and agens ulmae fae) may be produced by a probably measure wh parameers ha may le n a zero measure se of he agens suppor. 7

9 () when porfolos converge, wha he lmng porfolos are. The recursve characerzaon of he nancal wealh dsrbuon ha we oban allows o answer hese wo quesons. Frs, snce provdes a connuous mappng beween he porfolos supporng a PO allocaon and he nvesors lkelhood raos, makes evden ha hey converge ogeher. Second, makes possble o sngle ou he PO allocaon ha can be decenralzed as a compeve equlbrum whou ransfers by means of he applcaon of a recursve verson of he Negsh s approach. Ths allocaon s paramerzed by s correspondng welfare wegh ha depends upon dae 0 pror belefs, ndvdual endowmens and aggregae resources. Fnally, and very mporanly, allows o conclude ha he lmng wealh dsrbuon s pnned down by he denses of her pror belefs evaluaed a he rue parameer and he correspondng dae 0 welfare weghs. 3 The Model We consder an n ne horzon pure exchange economy wh one good. In hs secon we esablsh he basc noaon and descrbe he man assumpons. 3. The Envronmen Tme s dscree and ndexed by = 0; ; 2; :::. The se of possble saes of naure a dae s S f; :::; Kg. The sae of naure a dae zero s known and denoed by s 0 2 f; :::; Kg. We de ne he se of paral hsores up o dae as S = fs 0 g k= S k wh ypcal elemen s = (s 0 ; :::; s ). S fs 0 g ( k= S k) s he se of n ne sequences of he saes of naure and s = (s 0 ; s ; s 2 ; ), called a pah, s a ypcal elemen. For every paral hsory s, 0, a cylnder wh base on s s he se C(s ) fs 2 S : s = (s ; s + ; )g of all pahs whose nal elemens concde wh s. Le F be he -algebra ha consss of all ne unons of he ses C(s ). The - algebras F de ne a lraon on S denoed ff g =0 where F 0 ::: F ::: F where F 0 f;; S g s he rval S algebra F. =0 algebra and F s he -algebra generaed by he For any probably measure : F! [0; ] on (S ; F), s : F! [0; ] denoes s poseror dsrbuon afer observng s. 9 Le (s) be he probably of he ne hsory s,.e. he F measurable funcon de ned by (s) (C(s )) for all and 0. Le be he F measurable funcon de ned by (s) (s) (s). Tha s, gven he paral hsory s up o dae, s he one-perod-ahead 9 Formally, s (A) (A s ) (C(s )) for every A 2 F, where A s s 2 S : s = s ; s 0 ; s 0 2 A. 8

10 condonal probably of he saes a dae and s denoes s realzaon a s = afer he paral hsory s. Fnally, for any random varable x : S! <, E (x) denoes s mahemacal expecaon wh respec o : Le K be he K dmensonal un smplex n < K, B K be s Borel ses and P( K ) be he se of probably measures on K ; B K. Consder a se of probably measures on (S ; F) parameerzed by 2 K, wh ypcal elemen, wh he addonal propery ha he mappng 7! (B) s B K measurable for each B 2 F. Ths se ncludes he subse of probably measures on (S ; F) unquely nduced by..d. draws from a common dsrbuon : 2 K! [0; ], where () > 0 for all 2 f; :::; Kg, wh ypcal elemen P. We make he followng assumpon. A.0 The rue sochasc process of saes of naure s P for some >> The Economy There s a sngle pershable consumpon good every perod. The economy s populaed by I (ypes of) n nely-lved agens where 2 I = f; :::; Ig denoes an agen s name. A consumpon plan s a sequence of funcons fc g =0 such ha c : S! R + s F measurable for all and sup (;s) c (s) < : The agen s consumpon se, denoed by C, s he se of all consumpon plans Preferences We assume ha agens preferences sasfy Savage s [2] axoms and, herefore, hey have a subjecve expeced uly represenaon. Ths represenaon provdes a pror P over pahs and, as s well-known, also mples ha agens are Bayesans (.e., hey updae her pror usng Bayes rule as nformaon arrves). Bu, mos mporanly, does no oherwse resrc agen s prors n any parcular way. 0 We denoe by P he probably measure on (S ; F) represenng agen s pror and we make he sandard assumpons ha he uly funcon s me separable and he dscoun facor s he same for all agens. preferences are represened by U P (c ) = E P! X u (c ; ) ; =0 Tha s, for every c 2 C her where 2 (0; ) and u : R +! R + s connuously d erenable, srcly srcly concave and lm (x) = + for all. 0 See Blume and Easley [3] for a complee dscusson on he mplcaons of Savage s axoms. 9

11 One parcular famly of prors s ha where he agen beleves ha he rue process of saes of naure belongs o a paramerc famly of probably measures,, bu he agen does no know he parameer 2 K. Tha s, he probably of every even A 2 F s Z (A) = K (A) (d), () where 2 P( K ) s he pror belef over he unknown parameers. The hypohess of raonaly can be furher srenghened o requre ha he agen s a Bayesan who knows ha he process generang he daa s..d. bu does no know he rue probably of he saes of naure. We sae hs assumpon as A:. A. = P for every 2 K. We wan o emphasze ha A: says ha even hough agens agree ha he saes of naure are generaed by..d. draws from a common dsrbuon, hey mgh sll dsagree abou self. The followng assumpon mposes more srucure on he subjecve dsrbuon of and wll be dscussed furher below. A.2 has densy f wh respec o Lebesgue ha s connuous a wh f ( ) > 0. Anoher neresng spec caon of pror belefs s a pon mass probably measure on de ned as : F! [0; ] where f 2 B (B) 0 oherwse. When prors belong o he class represened by (), Bayes rule mples ha pror belefs evolve accordng o ;s (d) = (s s ) ;s (d) R K (s js ) ;s (d), (2) where ;0 2 P( K ) s gven a dae 0 and (s s ) C s C s. I s well-known ha Bayesan learnng s conssen for any pror sasfyng A:. However, hs propery apples o more general spec caons of prors (for nsance, hose sasfyng (), see Schwarz [23, Theorems 3.2 and 3.3]), and snce our example 4 n Secon 8.2 does no sasfy A. bu does sasfy (), we sae he conssency resul n he followng Lemma o make precse s scope. The celebraed De Fne heorem saes ha hs s equvalen o he pror beng exchangeable. 0

12 Lemma Suppose ha for ;0 almos all 2 K he probably measures on (S ; F) are muually sngular. Then ;s =0 converges weakly o for almos all s 2 S, for ;0 almos all 2 K : Remark : I s ubquous n he learnng leraure relaed o asse prcng o assume boh ha () every agen knows he lkelhood funcon generang he daa and () some agen learns he rue condonal probably of he saes. The laer s guaraneed n our seup by srenghenng A: o requre ha he rue parameer,, s n he suppor of some agen s pror. The case where hs holds for every agen s consdered n secons 5, 6 and 7. Secon 8 deals wh he cases n whch eher () or () does no hold Endowmens Agen s endowmen a dae s y (s ) > 0 for all s 2 f; :::; Kg and he aggregae endowmen s y(s ) P I = y (s ) y <. An allocaon fc g I = 2 CI s feasble f c 2 C for all and P I = c ;(s) y(s ) for all s 2 S. Le Y denoe he se of feasble allocaons. 4 Heerogeneous Prors and Porfolos: Examples The man purpose of hs secon s o llusrae our man resuls usng smple examples of dynamcally complee markes equlbra. Suppose here are wo saes, A:0 holds wh () = 2, wo agens, u(c) = ln c and y () = y() > 0 for all 2 f; 2g where + 2 =. Agens can rade a full se of Arrow secures n zero ne supply. Arrow secury 0 pays un of he consumpon good f s + = 0 and 0 oherwse. The prce of Arrow secury 0 2 f; 2g and agen s holdngs a dae afer paral hsory s are denoed by m 0 (s) and a0 ; (s), respecvely. We assume ha agens have no endowmen of Arrow secures,.e. hey have zero nancal wealh a dae 0. are In Appendx A we show ha for any 0 equlbrum consumpon and porfolos c ; (s) = + j P j; (s) a 0 ; (s) = y(0 ) P ; (s) y (s); + j P j; (s) P ; (s)! p j ( 0 js ), 0 2 f; 2g, (3) p ( 0 js ) where P ; (s) = P (C(s )) and p ( 0 s ) = P (C(s ; 0 ))=P (C(s )). Observe ha ndvdual porfolos a dae are compleely deermned by he lkelhood rao a

13 +, P j;+ P ;+. Porfolos converge f and only f he lkelhood rao converges. Thus, changes n porfolos are purely deermned by he heerogeney of prors. The relevan margn of heerogeney descrbed by lkelhood raos changes as me and uncerany unfold. Consequenly, (3) suggess ha he convenonal wsdom ha changes n porfolos are fundamenally drven by heerogeney n prors s correc as long as hs margn of heerogeney persss. Bayesan updang, however, mposes a srong srucure on he lm behavor of belefs, n he sense ha agens ypcally end up agreeng on he one-perod-ahead condonal probably. Wha s pendng o explan s he lm behavor of lkelhood raos when one-perod-ahead condonal probables converge. Benchmark Case: Homogeneous Prors Agens have dencal one-perod-ahead condonal probables of sae afer observng paral hsory s, p j s. Then, he lkelhood rao P j;(s) P ; (s) = for all and s. Consequenly, and hus porfolos are xed forever. a 0 ; (s) = 0 for all, s and 0, In every equlbrum, agens consume her endowmen every perod and, hen, consumpon and Arrow Secures prces are smple random varables wh suppor dependng only on he aggregae endowmen. More precsely, c ; (s) = y(s ) m 0 (s) = 2 y(s ) y( 0 ) : From hs resul and as a drec consequence of he convergence of he one-perodahead condonal probables, one mgh hasly make he followng conjecures: Conjecure I: Porfolos converge o a xed vecor whle consumpon and Arrow secury prces converge o some smple random varable dependng only on he aggregae endowmen. Conjecure II: Lmng porfolos, consumpon and Arrow secury prces are hose of an oherwse dencal economy where agens begn wh homogeneous prors and zero nancal wealh. Example shows ha Conjecure II mgh fal even f Conjecure I holds. Example : Heerogeneous Prors I The agens one-perod-ahead condonal probables of sae are gven by p j s = n s and p 2 j s = n2 s + 2 ; + 4 2

14 where n s sands for he number of mes sae 2 f; 2g has been realzed a he paral hsory s. Snce we assume A:0 holds wh () = 2, he Srong Law of Large Numbers mples ha p j s! 2 (P a:s:) as!, for every agen 2 f; 2g. Therefore, boh agens learn he rue one-perod-ahead condonal probably. By he Kolmogorov s Exenson Theorem (Shryaev [22, Theorem 3, p. 63]), here exss a unque P on (S ; F) assocaed o he agen s one-perod-ahead condonal probably. Moreover, P sas es A: and A 2 and agens pror belefs over have denses f () = and f 2 () = 6 ( ) on (0; ), respecvely. 2 The lkelhood rao s P ; (s) P 2; (s) = R 0 P (s) d R 0 P (s) 6 ( ) d = 6 [n (s )+] [n 2 (s )+] [+2] [n (s )+2] [n 2 (s )+2] [+4] = 6 (+3) (+2) (n (s )+) (n 2 (s )+) ; where sands for he Gamma funcon. 3 The Srong Law of Large Numbers can be appled once agan o show ha P ; (s) P 2; (s)! 2 3 = f 2 P a:s: f 2 2 I follows from (3) ha porfolos converge o a xed vecor, ha s a 0 ; (s)! 3 y(0 ) + 2! ; 0 2 f; 2g P a:s: 2 Alhough secury prces, asse holdngs and consumpon all converge, we wan o underscore ha only prces converge o hose of an oherwse dencal economy wh homogeneous prors. Indeed, c ; (s)! m 0 (s)! 2 y(s ) y( 0 ) ; and hus Conjecure I holds bu Conjecure II does no. y(s ) < y(s ); The reason s ha n he economy ha sars wh homogenous pror belefs he agens nancal wealh s zero whle n he lm economy pror belefs are homogeneous bu he agens nancal wealh s no zero. In hs example lm asse prces are dencal o hose of an oherwse dencal economy ha sars wh homogenous pror belefs because logarhmc preferences make neremporal margnal raes of subsuon, and hus asse prces, ndependen of he wealh dsrbuon. In general, however, asse prces 2 Tha s, agen s pror belefs over follow a Bea dsrbuon B (; ) on (0; ), as n Morrs [7]. 3 Recall ha f n s an neger, hen (n) = (n )! 3

15 do depend on he wealh dsrbuon. In Secon 6 we fully characerze he lm wealh dsrbuon and argue ha depends crcally on dae 0 prors. The followng example shows ha Conjecure I mgh be false as well. Example 2: Heerogeneous Prors II The agens one-perod-ahead-condonal probables of sae are gven by p p j s = p and p 2 j s = e = p : + e = + e =.e., agens beleve ha he saes of naure are ndependen draws from me-varyng dsrbuons. Observe ha one-perod-ahead condonal probables converge o 2 for boh agens,.e. agens learn, and have he same enropy. Tha s, E P (log p ;+ j F ) = E P (log p 2;+ j F ) : The rao of one-perod-ahead condonal probables andae afer paral hsory s p s a random varable, ; p 2;, ha akes values n ep = ; p e = o. The logarhm of he lkelhood rao can be wren as he sum of condonal mean zero random varables as follows log P; (s) P 2; (s) Y p ;k (s) = log p 2;k (s) k= X p = sk = (s) log e = + ( sk = (s)) log p k= e = X = x k (s) k= where x k (s) 2 f p =k; p =kg, E P (x k j F k ) (s) = 0 and V ar P (x k j F k ) (s) = E P x 2 k Fk (s) = =k. Consequenly, he log-lkelhood rao s he sum of unformly bounded random varables wh zero condonal mean. Addonally, snce he sum of condonal varances of x k dverges wh probably, follows by Freedman [, Proposon 4.5 (a)] ha and, herefore, sup X k= x k (s) = + and nf X x k (s) = P a:s: k= lm nf P ;(s) P 2; (s) = 0 and lm sup P ;(s) P 2; (s) = + P a:s: 4

16 Ths behavor of he lkelhood rao mples ha ndvdual porfolos ucuae n nely ofen. In parcular, lm nf a 0 ; (s) = y( 0 ) and lm sup a 0 ; (s) = ( ) y( 0 ): Snce each agen s deb aans s so-called naural deb lm n nely ofen, ndvdual porfolos are hghly volale. Consequenly, Conjecure I does no hold n hs example and, a pror, hs s raher surprsng snce every agen learns he rue oneperod-ahead-condonal probably. The fac ha he one-perod-ahead-condonal probables converge ceranly means ha rade n each perod becomes evenually very small. However, snce he lkelhood rao of agens belefs fals o converge, hs small rade compounds over large perods of me and so (n a su cenly long span of me) here are wde ucuaons n he dsrbuon of wealh. Why does Conjecure I hold n example whle fals n example 2? The man d erence s ha prors sasfy A: n example bu no n example 2. I urns ou ha when A: holds for every agen, he lkelhood raos always converge and, hus, Conjecure I holds n general. However, o generalze hese lessons o he seng descrbed n secon 3 one faces wo d cules ha we avod n he examples by carefully choosng preferences, ndvdual endowmens and prors. Frs, equlbrum porfolos are ypcally hsory dependen n a more general seup. Closed-form soluons for asse demands as n (3) are useful o ackle hs d culy bu hey are a parcular feaure derved from logarhmc preferences and consan ndvdual endowmen shares. Second, lkelhood raos are ypcally complcaed objecs whch makes he analyss of her behavor a nonsandard ask. Closed-form represenaon for he lkelhood rao, as n he examples above, smpl es he analyss of s asympoc properes bu s a consequence of he parcular famly of prors ha we choose. The res of he paper ackles he d cules o exend he lessons from he examples o he more general seup descrbed n secon 3. Here we o er an oulne. We begn wh a recursve characerzaon of e cen allocaons and her correspondng supporng porfolos under he assumpon ha A. holds. In secon 5, we show ha he evoluon of any Pareo opmal allocaon s drven solely by he evoluon of he lkelhood raos of he agens prors and he agens belefs over he unknown parameers, as n he examples. In secon 6, we prove ha he agens nancal wealh converges f and only f boh he lkelhood rao as well as her belefs (over he unknown parameers) converge. Aferwards, we ackle he d cules assocaed wh he lack of closed form for he lkelhood raos. In secon 7, we consder a broad class of prors sasfyng A.. We apply recen resuls n probably heory 5

17 o prove ha he lkelhood raos converge wh probably one, as n example. Fnally, n secon 8 we explan he exac role played by he assumpons ha every agen knows he lkelhood funcon generang he daa and ha some agen learns and we argue ha s crcal ha hey are coupled ogeher. We do so by provdng wo addonal examples, each of whch relax one of hese assumpons, n whch he lkelhood rao does no converge and consequenly porfolos change n nely ofen as n example 2. 5 A Recursve Approach o Pareo Opmaly In hs secon, we provde a recursve characerzaon of he se of Pareo opmal allocaons provdng a verson of he Prncple of Opmaly for economes wh heerogeneous pror belefs. Throughou hs secon we assume ha A:0 and A: hold. I s well known ha under A:, Bayes rule mples ha pror belefs evolve accordng o ;s (d) = (s ) ;s (d) R (s K ) ;s (d), (4) where ;0 2 P( K ) s gven a dae 0. Lemma 2 Suppose agen s pror sas es A:. Then, for every B 2 F Z P ;s (B) = Ps (B) ;s (d) : (5) K 5. Pareo Opmal Allocaons A feasble allocaon fc gi = s Pareo opmal (PO) f here s no alernave feasble allocaon fbc g I = such ha U P (bc ) > U P (c ) for all 2 I. I s well known ha he se of PO allocaons can be characerzed as he soluon o he followng planner s problem. Gven 0, s 0 and welfare weghs 2 R I +, de ne v (s 0 ; 0 ; ) sup fc g I = 2Y IX E P =! X u (c ; ). (6) Unlke he case where agens have homogeneous belefs, he recursve characerzaon of PO allocaons n our economy s raher rcky because belef heerogeney makes opmal allocaons hsory dependen. Ths can be seen from he followng 6

18 (necessary and su cen) rs order condons o he planner s problem: P ; (s) j P j; (c ; j (c j; (s)) = for all, j 2 I, for all and all s, j; IX c ; (s) = y(s ). (8) =. Snce j (c ;0 (c j;0 j;0, he planner dsrbues consumpon among agens o make he rao of margnal valuaons of any wo agens -whch, we recall, nclude prors ha could be subjecvely held- o be consan along me. Consequenly, under he opmal dsrbuon rule of consumpon, he rao of margnal (c ; (c j; j;, mus be proporonal o he lkelhood rao of he agens prors, P j; (s) /P ; (s). Ths rao represens he novel margn of heerogeney among agens consdered n hs paper, whch we call he B-margn of heerogeney. The B-margn s purely drven by heerogeney n prors and s evoluon deermnes he dynamcs of he opmal dsrbuon rule of consumpon. Indeed, when all agens have he same prors he B-margn remans consan along me and he opmal dsrbuon rule of consumpon s boh me and hsory ndependen. Consequenly, ndvdual consumpon depends only upon he curren shock s (because deermnes aggregae oupu) and he dae 0 vecor of welfare weghs. When agens have heerogeneous prors, nsead, he B-margn s hsory dependen and so s he opmal dsrbuon rule of consumpon. Now we argue ha hs hsory dependence can be handled wh a properly chosen se of sae varables. Noe ha snce condon (7) holds f and only (c ; (s)) ; (s = R K + ) ::: (s +k ) ;s j (c j; (s K + ) ::: (s +k ) j;s (d) (c ;+k ;+k j (c j;+k j;+k hen he planner does no need o know he paral hsory self n order o connue he dae 0 opmal plan from dae onwards. Indeed, s su cen ha he knows he rao of margnal ules ha he orgnal plan nduces a (c ; (c j; j; s (d), snce ;s (d) =, he sae of naure a dae, s, and he poseror belefs, (s ) ;s (d) R K (s ) ;s (d). Moreover, snce he rao of margnal ules a dae equals he lkelhood rao weghed by he dae zero welfare weghs, j P j; (s) P ; (s), he d cules semmng from he opmal plan hsory dependence can be handled by usng ( P ; (s); :::; I P I; (s); s ) as sae varables summarzng he hsory and he sae of naure a dae, s, descrbng aggregae resources. From he dscusson above, we conclude ha a PO allocaon canno be fully characerzed usng only he agens belefs over he unknown parameers (ha s, 7

19 s ) and s as sae varables as n he sngle agen seng (see, Easley and Kefer [8]). In a mulple agen seng, nsead, he planner needs o dsrbue consumpon and because of hs one needs o nroduce ( P ; (s); :::; I P I; (s)) as an addonal sae varable, whch can be nerpreed as he dae welfare weghs, ; (s) = P ; (s). These weghs evolve accordng o he law of moon Z ; (s) = ; (s) (s ) ;s (d) where ;0 (s) =. (9) K In Secon 5.2 below we presen a formal exposon of hs resul. 5.2 Recursve Characerzaon of PO Allocaons Gven ha n an envronmen wh heerogeneous belefs and learnng PO allocaons are ypcally hsory dependen, sandard recursve mehods canno be appled. We ackle hs ssue by adapng he mehod developed by Lucas and Sokey [6]. In Appendx B we show ha v s he unque soluon of he funconal equaon X < v(; ; ) = max (c;w 0 ( 0 )) : u (c ) + X Z = ( 0 ) 0 (; ) (d) w( 0 0 ) 0 K ; ; (0) subjec o 2I where IX c = y() for all ; c 0; w 0 ( 0 ) 0 for all 0, () = 0 ( 0 ) arg mn e2 I " v( 0 ; e; 0 (; )) # IX e w( 0 0 ) 0 for all 0, (2) = R 0 B (; ) (B) = () (d) R () K (d) for any B 2 B(K ). (3) In he recursve dynamc program de ned by (0) - (3), he curren sae,, capures he mpac of changes n aggregae oupu whle (; ) summarzes and solaes he dependence upon hsory nroduced by he evolvng B-margn of heerogeney. The planner akes as gven (; ; ) and allocaes curren consumpon and connuaon uly levels among agens. Tha s, nsead of allocang consumpon from omorrow on, he planner assgns o each agen he uly level assocaed wh he correspondng connuaon sequence of consumpon. Indeed, he opmzaon problem de ned n condon (2) characerzes he se of connuaon uly levels 4 In secons 5.2 and 6, we abuse noaon and le c o be a non-negave vecor and c s h componen. 8

20 aanable a ( 0 ; 0 (; )) (see Lemma 4 n Appendx B). 5 The weghs 0 ( 0 ) ha aan he mnmum n (2) wll hen be he new weghs used n selecng omorrow s allocaon. The (normalzed) law of moon for he welfare weghs, 0 (; ; )(0 ), follows from he rs order condons wh respec o he connuaon uly levels for each ndvdual and s gven by 0 (; ; )( 0 ) R ( 0 ) 0 P h h (; ) (d) R ( 0 (4) ) 0 h (; ) (d): Observe ha he normalzaon s harmless snce opmal polcy funcons are homogeneous of degree zero wh respec o. I follows by sandard argumens ha he correspondng consumpon polcy funcon, c (; ), s he unque soluon o c (; ) + h (c (; )) = y(): for each 2 h denoes he nverse h. Gven (s 0 ; 0 ; 0 ), we say he polcy funcons (c; 0 ) coupled wh 0 generaes an allocaon bc f bc ; (s) = c (s ; (s)), + (s) = 0 (s ; (s); s )(s + ), s = 0 (s ; s ), for all and all 0 and s 2 S where 0 (s) = 0 and s = 0. The followng Theorem shows ha here s a one-o-one mappng beween he se of PO allocaons and he allocaons generaed by he opmal polcy funcons solvng (0) - (3). Theorem 3 (The Prncple of Opmaly) An allocaon (c )I = s PO gven (; ; ) f and only f s generaed by he polcy funcons solvng (0) - (3). 5 To undersand condon (2) noce ha he uly possbly se,.e. he se of expeced lfeme uly levels ha are aanable by mean of feasble allocaons, s convex, compac and conans s correspondng froner. The froner of a convex se can always be paramerzed by supporng hyperplanes. Moreover, under our assumpons, he correspondng parameers can be resrced o le n he un smplex and, herefore, hey can be nerpreed as welfare weghs. Thus, a uly level vecor w s n he uly possbly se f and only f for every welfare wegh he hyperplane paramerzed by and passng hrough w, w, les below he hyperplane generaed by he uly levels aaned by he PO allocaon correspondng o ha welfare wegh, aanng he value v(; ; ). Ths s why we mus have w v(; ; ) for all or, equvalenly, mn e [v(; e; ) ew] 0. See Appendx B for echncal deals. 9

21 Informally, hs resul can be grasped as follows. The characerzaon of he soluon o he sequenal formulaon of he planner s problem hns ha once he planner knows boh he lkelhood rao weghed by he dae zero welfare weghs and he belefs a dae, he can connue he opmal plan from dae onwards. I s key o undersand ha he consumpon plan from dae + onwards can be summarzed by s assocaed uly levels whch n urn can be summarzed by a vecor of welfare weghs. Theorem 3 shows ha he dae zero opmal plan s conssen n he sense ha he connuaon plan s ndeed he soluon from dae onwards Dscusson: An Alernave Approach There s an alernave approach o sae he dynamc program de ned by (0) - (3): nsead of paramerzng allocaons wh welfare weghs, he planner chooses curren feasble consumpon and connuaon ules for boh agens n order o maxmze he uly of agen subjec o wo resrcons: () he uly of agen 2 s above some prespec ed level (he so-called promse keepng consran) and () connuaon uly levels le n nex perod uly froner. Very mporanly, hs las condon mples ha he correspondng value funcon de nes he consran se. 6 Snce boh n our approach as well as n he alernave one he correspondng value funcon de nes he consran se, neher of he wo dynamc programs s sandard n he sense ha s no obvous ha any of he correspondng operaors sas es one of Blackwell s su cen condons, namely, dscounng. Indeed, for any funcon v ha de nes he consran se here mgh be some a > 0 such ha v + a enlarges he feasble se of choces of connuaon ules wh respec o v. The key o show dscounng n our approach s o resrc he se of funcons o be homogeneous of degree wh respec o he sae varables,.e. he welfare weghs, (a propery ha s sas ed by v, see Lemma 3 n Appendx B). 7 Snce v + a s an a ne lnear ransformaon of v, he choce of curren consumpon s he same for v and v + a. In addon, homogeney of degree of he value funcon wh respec o he welfare weghs mples ha w 0 s he opmal choce for he consran se de ned by v f and only f w 0 + a s he opmal choce for he consran se de ned by v + a. Ths explans why homogeney of degree of he value funcon wh respec o he welfare weghs s key o show ha dscounng holds n our seng. 8 6 Snce we paramerzed he uly levels wh her assocaed welfare weghs, our approach amouns o replacng he promse keepng consran by usng he assocaed Lagrange mulplers as sae varables. 7 Lucas and Sokey [6] do no make hs resrcon and so s unclear wheher dscounng holds n her approach. 8 There s no obvous condon equvalen o homogeney n he alernave approach descrbed above and hen one needs o nd he soluon d erenly. One plausble sraegy would be o follow 20

22 6 Deermnans of he Fnancal Wealh Dsrbuon In hs secon we sudy he deermnans of he nancal wealh dsrbuon ha suppors a dynamcally complee markes equlbrum allocaon. Frs, we characerze ndvdual nancal wealh recursvely as a me nvaran funcon of he saes (; ; ). Laer, we employ a properly adaped recursve verson of he Negsh s approach o pn down he PO allocaon ha can be decenralzed as a compeve equlbrum whou ransfers. Gven (; ; ), we consruc ndvdual consumpon usng c (; ) and de ne he sae prce by 9 Z M(; ; )( 0 ) = ( 0 ) 0 (; ) (c ( 0 ; 0 (;;)( 0 (c (;))/@c, (6) The funconal equaon ha deermnes agen s nancal wealh s A (; ; ) = c (; ) y () + X 0 M(; ; )( 0 ) A ( 0 ; 0 ; 0 ), (7) where 0 (; ) and 0 (; ; )( 0 ) are gven by (3) and (4), respecvely. Noe ha (7) compues recursvely he presen dscouned value of agen s excess demand a he PO allocaon. In Theorem 4, we show ha A s well-de ned. Furhermore, we apply Negsh s approach o show ha here exs a welfare wegh such ha A s zero for every : Theorem 4 Suppose A.0 and A. hold. Then, here s a unque connuous funcon A solvng (7). Moreover, for each (s 0 ; 0 ) here exss 0 = (s 0 ; 0 ) 2 R I + such ha A (s 0 ; 0 ; 0 ) = 0 for all. Remark 2: Observe ha f agens have boh homogeneous prors and dogmac belefs (.e., ;0 = for all and for some 2 K ), follows mmedaely ha ;+ (s) = 0 and ;s = for all s and all 0 and for each agen. Therefore, c ; (s) = c (s ; 0 ) for all s and all 0. Very mporanly,he soluon o (7) reduces o a vecor n R K and, hus, A ( 0 ; ) 2 R K such ha A (s 0 ; 0 ; ) = 0 for all. he semnal dea poneered by Abreu, Pearce and Sacche [] and consruc an alernave operaor ha eraes drecly on he uly possbly correspondence. Then, he value funcon (and he correspondng polcy funcons) could be recovered from he froner of he xed pon of ha operaor on uly correspondences. 9 The choce of agen o de ne M s whou loss of generaly snce Pareo opmaly mples ha he neremporal margnal raes of subsuon are equalzed across agens. 2

23 6. The Fxed Equlbrum Porfolo Propery We say ha he xed equlbrum porfolo (FEP hereafer) propery holds f here exss fa (); :::; a (K)g 2 < K such ha a () = A (; ; ) for all (; ; ) and all. If he FEP propery holds, any porfolo ha decenralzes a PO allocaon wh a xed se of non-redundan asses s kep consan over me and across saes. We say ha he B-margn of heerogeney vanshes on a pah s f (s) converge on s. Addonally, he FEP propery holds asympocally on s f ndvdual nancal wealh, A ; (s) = A (; (s); s ), converges on s for every and 2 S. Judd e al. [4] show ha he FEP propery s always sas ed afer a onceand-for all nal rebalancng when agens have homogeneous prors (he B-margn of heerogeney s consan) and degenerae belefs. Spec cally, Remark 6 mples ha a () = A (; 0 ; ) for all snce n her seng ;0 = for some 2 K and for all, herefore, he agens nancal wealh s a vecor n R K n any dynamcally complee markes equlbrum. When agens have homogeneous bu non-degenerae pror belefs, he welfare weghs are consan along me, (s) = 0, and he dsrbuon of consumpon s gven by c () = c (; 0 ) for each and so remans unchanged as me and uncerany unfold. However, he wealh dsrbuon, A ; (s) = A (; 0 ; s ), s sll hsory dependen because he agens learnng process make sae prces hsory dependen. Consequenly, he FEP propery does no necessarly hold. To ge a horough undersandng of why he wealh dsrbuon changes even hough he dsrbuon of consumpon does no, consder a wo-agen and wo-sae economy where y () = f = and 0 oherwse. Boh agens pror belef s ha sae s very lkely (.e. 0 concenrae mos of s mass around () = ) whle he rue probably of each sae s, say, () = =2 for 2 f; 2g. Thus, agen s rcher han agen 2 and hs mples ha agen s dae-zero welfare wegh, ;0 (s 0 ; 0 ), s relavely larger han ;0 (s 0 ; ) and she consumes accordngly forever. For su cenly large, s ges close o and he presen dscouned value of agen s endowmen wll no be enough o a ord he xed consumpon bundle c (; 0 ). Consequenly, she mus have accumulaed su cen nancal wealh o make he consumpon paramerzed by 0 (s 0 ; 0 ) a ordable (ha s, A ; (s) = A (; 0 ; s ) > 0 for all > T ). The remarkable feaure s ha he evoluon of (homogeneous) belefs has an mpac only on porfolos bu no on consumpon. Ths makes evden ha Judd e al s [4] resuls need boh homogeneous as well as degenerae belefs. In our seng, nsead, porfolos ypcally change as me and uncerany unfolds because he changes n he B-margn of heerogeney a ecs he dynamcs of he wealh dsrbuon hrough he evoluon of he welfare weghs. Therefore, he FEP 22

24 propery does no hold n a dynamcally complee markes equlbrum when prors are heerogenous mplyng ha he resul n Judd e al. [4] s no robus o he nroducon of hs margn of heerogeney. However, snce agens observe he same daa, have he rue model n he suppor of her prors and updae her prors n a Bayesan fashon, a much deeper queson s wheher hs radng acvy fades ou as he B-margn of heerogeney vanshes. Our recursve approach perms o sudy hs ssue drecly. The followng proposon, a drec consequence of he connuy of A, underscores ha f a PO allocaon can be decenralzed hrough a sequence of compeve markes, he assocaed wealh dsrbuon converges o a xed vecor for each whenever he B-margn of heerogeney s exhaused. Consequenly, asse radng reduces o he mnmum. Proposon 5 Suppose A.0 and A. hold. If he B-margn of heerogeney vanshes on a pah s, hen he FEP propery holds asympocally on s (P 7 Lmng Welfare Weghs a:s:) In hs secon, we analyze how he B-margn of heerogeney evolves over me when A: holds and every agen has he rue parameer n he suppor of her prors. From condon (4) and Theorem 3, he rao of welfare weghs s R ; (s) j; (s) = (s R K ) ;s (d) ; (s) (s K ) j;s (d) j; (s) = ;0 P ; (s) j;0 P j; (s), (8) and, herefore, he asympoc behavor of (s) depends on he lm behavor of he lkelhood raos P ;(s) P j; (s). Here, we show ha when agens agree ha he daa s generaed by..d. draws from a common dsrbuon (A: holds), lkelhood raos converge and so do welfare weghs. However, we need o dsngush he case where he suppor of he agens pror belefs s counable from ha when s uncounable. When he suppor s counable, he rue probably dsrbuon s always absoluely connuous wh respec o he agens prors and, herefore, he convergence of lkelhood raos follows from Sandron [20]. The assumpon of counable suppor, however, seems oo srong snce rules ou, for nsance, he case of pror belefs ha sasfy assumpon A:2. When boh A: and A:2 hold, he probably dsrbuon ha generaes he daa s never absoluely connuous wh respec o he agens prors and so Sandron s resul does no apply. 20 Noneheless, we show ha lkelhood raos converge applyng a recen resul by Phllps and Ploberger [9]. 20 Blume and Easley [4] also emphasze hs pon. 23