The Dynamics of Efficient Asset Trading with Heterogeneous Beliefs

Transcription

1 The Dynamcs of Effcent Asset Tradng wth Heterogeneous Belefs Pablo F. BEKER # Emlo ESPINO ## Department of Economcs Unversty of Warwck Department of Economcs Unversdad Torcuato D Tella Ths Draft: February 9, 2009 Abstract Ths paper analyzes the dynamc propertes of portfolos that sustan dynamcally complete markets equlbra when agents have heterogeneous prors. We argue that the conventonal wsdom that belef heterogenety generates contnuous trade and sgn cant uctuatons n ndvdual portfolos may be correct but t also needs some qual catons. We consder an n nte horzon stochastc endowment economy where the actual process of the states of nature conssts n..d. draws. The economy s populated by many Bayesan agents wth heterogeneous prors over the stochastc process of the states of nature. Our approach hnges on studyng portfolos that support Pareto optmal allocatons. Snce these allocatons are typcally hstory dependent, we propose a methodology to provde a complete recursve characterzaton when agents know that the process of states of nature s..d. but dsagree about the probablty of the states. We show that even though heterogeneous prors wthn that class can ndeed generate genune changes n the portfolos of any dynamcally complete markets equlbrum, these changes vansh wth probablty one f the support of every agent s pror belef contans the true dstrbuton. Fnally, we provde examples n whch asset tradng does not vansh because ether () no agent learns the true condtonal probablty of the states or () some agent does not know the true process generatng the data s..d. Keywords: heterogeneous belefs, asset tradng, dynamcally complete markets. We thank Rody Manuell and Juan Dubra for detaled comments. All the remanng errors are ours. # Correspondng Author: Unversty of Warwck, Department of Economcs, Warwck, Coventry CV4 7AL, UK. E-mal: Pablo.Beker@warwck.ac.uk. ## Unversdad Torcuato D Tella, Department of Economcs, Saenz Valente 00 (C428BIJ), Buenos Ares, Argentna. E-mal: eespno@utdt.edu.

2 Introducton A long-standng tenet n economcs s that belef heterogenety plays a prme role n explanng the behavor of prces and quanttes n nancal markets. In spte of the emphass that economsts gve to e cency, surprsngly, very lttle s known about the mplcatons of belef heterogenety on dynamcally complete markets. However, there are some notable exceptons. Sandron [20] and Blume and Easley [4] provde an analyss of the asymptotc propertes of consumpton. Cogley and Sargent [7] focus on asset prces. Our paper, nstead, focuses on the e ect of belef heterogenety on asset tradng. Before proceedng t s useful to recall what s known about asset tradng n a dynamcally complete markets equlbrum when agents have dentcal belefs. Judd et al. [4] consdered a statonary Markovan economy where agents have homogeneous and degenerate belefs but d erent atttudes towards rsk and show that each nvestor s equlbrum holdngs of assets of any spec c maturty s constant along tme and across states after an ntal tradng stage. It follows that d erences n rsk averson by tself cannot explan why nvestors change ther portfolos over tme. We consder an exchange economy where both the endowments as well as the assets returns are..d. draws from a common probablty dstrbuton. Investors who are n ntely lved do not know the one-perod-ahead condtonal probablty of the states of nature and update ther prors n a Bayesan fashon as data unfolds. We begn wth two examples of dynamcally complete markets equlbrum that llustrate that the conventonal wsdom that belef heterogenety causes sgn cant trade may be correct but t also needs some qual catons. In example, agents know the true process s..d. and they only dsagree about the probablty of the states of nature. In the long run, condtonal probabltes, wealth and portfolos converge. In example 2, agents do not know the true process s..d., condtonal probabltes converge and yet wealth bounces back and forth between them n ntely often so that each of them holds almost all the wealth n ntely many tmes. Ths second example shows that even though agents may learn, pror belef heterogenety may ndeed generate sgn cant uctuatons n the wealth dstrbuton and the correspondng portfolos that do not exhaust n the long run. We argue that the d erent dynamcs n the two examples re ect d erences n the lmt behavor of the lkelhood rato of the agents prors. Ths paper lnks the evoluton of the wealth dstrbuton and the correspondng To avod any confuson, we use the followng termnology. By a pror, we refer to the subjectve uncondtonal probablty dstrbuton over future states of nature. In the partcular case where the pror can be characterzed by a vector of parameters and a probablty dstrbuton over these parameters, we call the latter the agent s pror belef.

3 portfolos n any dynamcally complete markets equlbrum to the evoluton of the lkelhood rato. Ths s useful because the lkelhood rato s an exogenous varable and several propertes of ts lmt behavor are well understood from the statstcs lterature (see Phllps and Ploberger [9] and references theren.) Our examples and 2 rase the queston of what type of belef heterogenety matters for asset tradng. In order to answer ths queston, we rst carefully assess a class of prors satsfyng two assumptons that are ubqutous n the lterature. Namely, every agent knows the lkelhood functon generatng the data and there s at least one agent who learns n the sense that her one-perod-ahead condtonal probablty converges to the truth. In our setup, ths s ensured by assumng that every agent knows the data s generated by..d. draws from a common (unknown) dstrbuton and the support of ther pror belefs contans the true probablty dstrbuton of the states of nature as n example. We rst show that even though heterogeneous prors n that class can ndeed generate changes n the portfolos of a dynamcally complete markets equlbrum, these changes vansh wth probablty one. Very mportantly, we fully characterze the dynamcs of portfolos and ts correspondng lmt. Afterwards, we show by means of two addtonal examples that f one wants to argue that heterogenety of prors can have endurng mplcatons on the volume of trade n a statonary envronment then one needs to relax one of the aforementoned assumptons; that s, ether () no agent learns the true condtonal probablty of states or () some agent does not know the lkelhood functon generatng the data. Snce solvng drectly for the portfolos of a dynamcally complete markets equlbrum s not always possble, we follow an ndrect approach developed by Espno and Hntermaer [9]. Ths approach hnges on studyng portfolos that support Pareto optmal allocatons. The d culty s that belef heterogenety makes optmal allocatons hstory dependent because optmalty requres the rato of margnal valuatons of consumpton of any two agents -whch ncludes prors that could be subjectvely held- to be constant along tme. Consequently, at any date the rato of margnal utltes at any future event must be proportonal to the hstory dependent rato of the agents prors about that event,.e. the lkelhood rato of the agents prors. Ths rato represents the novel margn of heterogenety among agents consdered n ths paper, whch we call the B-margn of heterogenety. The evoluton of the B-margn determnes the dynamcs of the optmal dstrbuton rule of consumpton and, consequently, the evoluton of the wealth dstrbuton n any dynamcally complete markets equlbrum. The law of moton of ths margn s typcally hstory dependent and, very mportantly, the current state and the current B-margn are not enough to summarze the hstory. Under the assumpton that every agent knows the data s generated by..d. draws from a common (unknown) dstrbuton but have d erent belefs over 2

4 the unknown parameters, ths hstory dependence can be succnctly captured by the agents belefs (over the parameters). Ths assumpton allow us to use a strategy smlar to Lucas and Stokey s [6] to obtan a recursve characterzaton of the set of Pareto optmal allocatons n our stochastc framework. 2 The key nsght s that the planner does not need to know the partal hstory tself n order to contnue the date zero optmal plan from date t onwards. In fact, t su ces that he knows the state of nature, the agents pror belefs over probabltes and, very mportantly, the current B-margn,.e. the lkelhood ratos of the agents prors that summarze how the weght attached to each agent depends on hstory. We argue that the sequental formulaton of the planner s problem s equvalent to a recursve dynamc program where the planner, who takes a vector of welfare weghts as gven, allocates current feasble consumpton and assgns next perod attanable utlty levels among agents. The planner s optmal choce of contnuaton utltes nduces a law of moton for welfare weghts that s somorphc to the evoluton of the lkelhood rato of the agents prors. Afterwards, we use the planner s polcy functons to characterze recursvely nvestors nancal wealth n any dynamcally complete market equlbrum. Ths allows us to establsh that the nancal wealth dstrbuton (and the correspondng supportng portfolos) converges f and only f both the B-margn vanshes and the agents belefs over the parameters become homogeneous. When the agents know that the true process conssts n..d. draws from a common dstrbuton and the true dstrbuton s n the support of ther prors, the well-known consstency property of Bayesan learnng mples that the agents pror belef become homogeneous wth probablty one. To get a thorough understandng of the lmtng behavor of portfolos, therefore, what remans to be explaned s the asymptotc behavor of the B-margn. When the support of the agents pror belefs over the parameters s a countable set contanng the true probablty dstrbuton, the true probablty dstrbuton over paths s absolutely contnuous wth respect to the agents prors and, therefore, the convergence of lkelhood ratos follows from Sandron [20]. When the agents pror belefs have a postve and contnuous densty wth support contanng the true parameter, the hypothess n Sandron [2] are not sats ed and so we apply a result n Phllps and Ploberger [9] to show that the lkelhood rato of the agents prors stll converges wth probablty one. The mportant message here s that the heterogenety of prors by tself can generate changes n portfolos but these changes necessarly vansh because the B-margn vanshes. Furthermore, we show that portfolos converge to those of a ratonal expectatons equlbrum of an economy where the nvestors relatve wealth s determned by the 2 Lucas and Stokey [6] characterze recursvely optmal programs n a determnstc settng where recursve preferences nduce the dependence upon hstores. 3

5 denstes of ther pror belefs evaluated at the true parameter and the date zero welfare weght. 3 To conclude we analyze the exact role played by the aforementoned assumptons on prors and we argue that t s crtcal that they are coupled together. We do so by provdng two addtonal examples, each of whch relax one of these assumptons, n whch the B-margn does not vansh and consequently portfolos change n ntely often. In example 3, agents know the data s generated by..d. draws from a common dstrbuton but they do not have the true parameter n the support of ther pror belefs and so no agent learns. We assume that ther pror belefs are such that the assocated one-perod-ahead condtonal probabltes have dentcal entropy, a condton that ensures that the lkelhood rato of ther prors uctuates n ntely often between zero and n nty and, consequently, portfolos uctuate n ntely often. Fnally, example 4 underscores the mportance of assumng that every agent knows the process of states conssts n..d. draws for the portfolos to converge. To stretch the argument to the lmt, we consder an example n whch only one agent does not know the data s generated by..d. draws. Ths agent makes exact one-perodahead forecasts n ntely often but t also makes mstakes n ntely often though rarely. We show that the lkelhood rato of these agents prors fals to converge wth probablty one mplyng that the set of paths where the equlbrum portfolo converges has probablty zero. Ths paper s organzed as follows. In Secton 2 we revew the related lterature. In secton 3 we descrbe the model. In secton 4 we present a smple example that llustrate the man deas n ths paper. The recursve characterzaton of Pareto optmal allocatons s n secton 5. Secton 6 characterzes the asymptotc behavor of the agents nancal wealth and ther correspondng supportng portfolos. Fnally, sectons 7 and 8 dscuss when the agents portfolo converge and when t does not. Conclusons are n secton 9. Proofs are gathered n the Appendx. 2 Related Lterature Ths paper relates to two branches of the lterature on the e ect of belef heterogenety n asset markets: models amng to explan the dynamc consequences of belef heterogenety on nvestors behavor and models analyzng the market selecton hypothess. Harrson and Kreps [3] and Harrs and Ravv [2] who study the mplcatons of belef heterogenety on asset prces and tradng volume, respectvely, are 3 In partcular, even though agents learn the true probablty of states of nature, these lmtng portfolos need not concde wth those of an otherwse dentcal economy that starts wth homogeneous prors and zero nancal wealth. 4

6 the leadng artcles of the rst branch. These rst-generaton papers consder partal equlbrum models where a nte number of rsk-neutral nvestors trade one unt of a rsky asset subject to short-sale constrants. Investors do not know the value of some payo relevant parameter but they observe a publc sgnal and have heterogeneous but degenerate pror belefs about the relatonshp between the sgnal and the unknown parameter. Snce they are rsk neutral and have heterogeneous belef, they have d erent margnal valuatons and so trade occurs f and only f agents "swtch sdes" regardng ther valuaton of the asset. In addton, Harrson and Kreps [3] show that an speculatve premum mght arse, n the sense that the asset prce mght be strctly greater than every trader s fundamental valuaton. Snce each nvestor s absolutely convnced her model s the correct one, ther dsagreement does not vansh as the data unfold. The possblty that agents learn s addressed by Morrs [7] who extends Harrson and Kreps [3] model to consder agents that have heterogeneous and non-degenerate pror belefs over the probablty dstrbuton of dvdends. He characterzes the set of pror belefs for whch the speculatve premum s postve. He assumes the true process s..d., nvestors know ths fact but they have heterogeneous pror belefs about the dstrbuton of these draws wth support contanng the true dstrbuton. Snce they are Bayesan, they eventually learn the true dstrbuton. Consequently, rsk neutralty mples the prce converges and the speculatve premum vanshes. We underscore that asset tradng does not vansh because there s always a perod n the future when the asset changes hands once agan. Morrs [7] asymptotc results, however, are a drect consequence of the assumpton that agents are rsk-neutral. Indeed, under rsk-neutralty the ntertemporal margnal rates of substtuton are ndependent of the equlbrum allocaton and, therefore, they are lnear n the agents one-perod-ahead condtonal probabltes. Ths has two drect mplcatons. On the one hand, when the ndvduals one-perod-ahead condtonal probabltes swtch sdes perpetually, so do ther ntertemporal margnal rates of substtuton and, therefore, new ncentves for a change n the ownershp of the asset arse n ntely often. On other hand, asset prces themselves are parameterzed by the one-perod-ahead condtonal probabltes and, thus, they converge together. In ths paper, we argue that these forces do not operate n a settng where agents are rsk-averse and allocatons are Pareto optmal. More precsely, Pareto optmalty mples that the agent s ntertemporal margnal rates of substtuton must be equalzed and, unlke n Morrs [7] where they swtch persstently, every trader s valuaton of any future ncome stream always concde. Consequently, there s never a speculatve premum n spte of belef heterogenety. Our analyss makes evdent that the speculatve premum s not necessarly drven by belef heterogenety but, more mportantly, by the 5

7 d erences n the agents ntertemporal margnal valuatons due to the exstence of short-sale constrants. 4 In our settng, portfolos mght stll change persstently but these changes depend purely on the asymptotc behavor of the e cent allocaton. Furthermore, as we emphaszed above, the convergence of the one-perod-ahead condtonal probabltes by tself does not guarantee the convergence of allocatons, asset prces and portfolos. Belef heterogenety may have fundamental mplcatons on the behavor of asset markets even n the absence of the aforementoned captal market mperfectons. In the context of the Lucas [5] tree model, Cogley and Sargent [6] and [7] focus on the e ects of learnng and pror belef heterogenety, respectvely, on asset prces under the assumpton that agents know the true lkelhood functon. In [6], they consder an economy wth a rsk-neutral representatve agent wth a pessmstc but non-degenerate pror belef over the growth rate of dvdends. Even though learnng eventually erases pessmsm, pessmsm contrbutes a volatle multplcatve component to the stochastc dscount factor that an econometrcan assumng correct prors would attrbute to mplausble degrees of rsk averson. 5 Cogley and Sargent [7] analyze the robustness of that ndng by consderng an economy wth complete markets wth some agents who know the true probablty dstrbuton (.e., they add belef heterogenety). For a plausble calbraton of ther model, they show that unless the agents wth correct belefs own a large fracton of the ntal wealth, t takes a long tme for the e ect of pessmsm to be erased. Ther work s close n sprt to ours n that they use a general equlbrum model wthout any addtonal market mperfecton. Snce they are prncpally nterested n studyng the market prces of rsk, however, they are slent about the mplcatons of belef heterogenety for tradng volume. Consequently, the asset tradng mplcatons stemmng purely from d erences n prors are stll an open queston. The second branch of the lterature related to our paper analyses the market selecton hypothess and s exempl ed by the work of Sandron [20] and Blume and Easley [4]. Sandron [20] shows that, controllng for dscount factors, f the true dstrbuton s absolute contnuous wth respect to some trader s pror then she survves and any other trader survves f and only f the true dstrbuton s absolute contnuous wth respect to her pror as well. 6 He also consders some cases n whch the true dstrbuton s not absolute contnuous wth respect to any agent s pror. He shows that 4 Indeed, n any economy where tradng constrants are occasonally bndng for d erent agents, the agent who prces the asset changes and thus the speculatve premum can arse naturally. 5 Ther model can generate substantal and declnng values for the market prces of rsk and the equty premum and, addtonally, can predct hgh and declnng Sharpe ratos and forecastable excess stock returns. 6 An agent s sad to survve f her consumpton does not converge to zero. 6

8 the entropy of prors determnes survval and, therefore, an agent who persstently makes wrong predctons vanshes n the presence of a learner. Absolute contnuty s a strong restrcton on prors that s not sats ed, for nstance, f the true process s..d., the agent knows ths fact but her pror belefs over the probablty of the states of nature have contnuous and postve densty. 7 Ths s precsely the case that Blume and Easley [4] consder and they prove that among Bayesan learners who have the truth n the support of ther prors, only those wth the lowest dmensonal support can have postve consumpton n the long run. Techncally speakng, Blume and Easley s noton of convergence s n probablty and they establsh ther asymptotc result for almost all parameters n the support of the agent s pror belef. Although we do not focus on survval, one sde contrbuton of ths paper s to make Blume and Easley s results more robust because we show that every Bayesan agent wth a pror belef wth the lowest dmensonal support actually survves wth probablty one (not just n probablty), not only for almost every parameter n the support of her pror belef but actually for all parameters n the support of her pror belef. 8 Our treatment of prors s very general n that we consder a famly that ncludes prors for whch the one-perod-ahead condtonal probablty converges to the truth regardless of whether the agents prors merge wth the truth or whether traders know the true process conssts n..d. draws. In addton, t ncludes cases n whch some agents have the truth n the support of ther prors whle some other agent do not learn the true one-perod-ahead condtonal probablty and yet the latter survves as n our example 4. To the best of our knowledge ths s the rst example of ts knd n the lterature. Our results characterzng the portfolos that support a Pareto optmal allocaton are a novel contrbuton to the lterature snce nether Sandron [20] nor Blume and Easley [4] analyze portfolo dynamcs. Indeed, the mappng between consumpton and ts supportng portfolo s only smple when agents have degenerate homogeneous prors as n Judd et al. [4]. Ths s most evdent when one consder the case where agents have homogeneous but non-degenerate pror belefs. In ths case, the dstrbuton of consumpton s tme ndependent whle the supportng portfolos are not because the state prces change as agents learn. We also contrbute to the analyss of the asymptotc behavor of portfolos snce t s not evdent that Sandron s [20] and Blume and Easley s [4] results on the lmt behavor of consumpton mply that () portfolos must converge when lkelhood ratos do and, very mportantly, 7 In that case, snce the entropy of every agent s pror s the same, one cannot apply Sandron s results relatng survval wth the entropy of prors ether. 8 Ths dstncton s economcally relevant because both n Blume and Easley s [4] settng as well as n ours the data (and agents ultmate fate) may be produced by a probablty measure wth parameters that may le n a zero measure set of the agents support. 7

9 () when portfolos converge, what the lmtng portfolos are. The recursve characterzaton of the nancal wealth dstrbuton that we obtan allows to answer these two questons. Frst, snce t provdes a contnuous mappng between the portfolos supportng a PO allocaton and the nvestors lkelhood ratos, t makes evdent that they converge together. Second, t makes possble to sngle out the PO allocaton that can be decentralzed as a compettve equlbrum wthout transfers by means of the applcaton of a recursve verson of the Negsh s approach. Ths allocaton s parametrzed by ts correspondng welfare weght that depends upon date 0 pror belefs, ndvdual endowments and aggregate resources. Fnally, and very mportantly, t allows to conclude that the lmtng wealth dstrbuton s pnned down by the denstes of ther pror belefs evaluated at the true parameter and the correspondng date 0 welfare weghts. 3 The Model We consder an n nte horzon pure exchange economy wth one good. In ths secton we establsh the basc notaton and descrbe the man assumptons. 3. The Envronment Tme s dscrete and ndexed by t = 0; ; 2; :::. The set of possble states of nature at date t s S t f; :::; Kg. The state of nature at date zero s known and denoted by s 0 2 f; :::; Kg. We de ne the set of partal hstores up to date t as S t = fs 0 g t k= S k wth typcal element s t = (s 0 ; :::; s t ). S fs 0 g ( k= S k) s the set of n nte sequences of the states of nature and s = (s 0 ; s ; s 2 ; ), called a path, s a typcal element. For every partal hstory s t, t 0, a cylnder wth base on s t s the set C(s t ) fs 2 S : s = (s t ; s t+ ; )g of all paths whose t + ntal elements concde wth s t. Let F t be the -algebra that conssts of all nte unons of the sets C(s t ). The - algebras F t de ne a ltraton on S denoted ff t g t=0 where F 0 ::: F t ::: F where F 0 f;; S g s the trval S algebra F t. t=0 algebra and F s the -algebra generated by the For any probablty measure : F! [0; ] on (S ; F), s t : F! [0; ] denotes ts posteror dstrbuton after observng s t. 9 Let t (s) be the probablty of the nte hstory s t,.e. the F t measurable functon de ned by t (s) (C(s t )) for all t and 0. Let t be the F t measurable functon de ned by t (s) t(s) t (s). That s, gven the partal hstory s t up to date t, t s the one-perod-ahead 9 Formally, s t (A) (A s t) (C(s t )) for every A 2 F, where A s t s 2 S : s = s t ; s 0 ; s 0 2 A. 8

10 condtonal probablty of the states at date t and s t denotes ts realzaton at s t = after the partal hstory s t. Fnally, for any random varable x : S! <, E (x) denotes ts mathematcal expectaton wth respect to : Let K be the K dmensonal unt smplex n < K, B K be ts Borel sets and P( K ) be the set of probablty measures on K ; B K. Consder a set of probablty measures on (S ; F) parameterzed by 2 K, wth typcal element, wth the addtonal property that the mappng 7! (B) s B K measurable for each B 2 F. Ths set ncludes the subset of probablty measures on (S ; F) unquely nduced by..d. draws from a common dstrbuton : 2 K! [0; ], where () > 0 for all 2 f; :::; Kg, wth typcal element P. We make the followng assumpton. A.0 The true stochastc process of states of nature s P for some >> 0. We assume the true process of states of nature s..d. to ease the exposton. However, all our results hold true for any tme-homogeneous Markov process. 3.2 The Economy There s a sngle pershable consumpton good every perod. The economy s populated by I (types of) n ntely-lved agents where 2 I = f; :::; Ig denotes an agent s name. A consumpton plan s a sequence of functons fc t g t=0 such that c t : S! R + s F t measurable for all t and sup (t;s) c t (s) < : The agent s consumpton set, denoted by C, s the set of all consumpton plans Preferences We assume that agents preferences satsfy Savage s [2] axoms and, therefore, they have a subjectve expected utlty representaton. Ths representaton provdes a pror P over paths and, as t s well-known, t also mples that agents are Bayesans (.e., they update ther pror usng Bayes rule as nformaton arrves). But, most mportantly, t does not otherwse restrct agent s prors n any partcular way. 0 We denote by P the probablty measure on (S ; F) representng agent s pror and we make the standard assumptons that the utlty functon s tme separable and the dscount factor s the same for all agents. preferences are represented by U P (c ) = E P! X t u (c ;t ) ; t=0 That s, for every c 2 C her 0 See Blume and Easley [3] for a complete dscusson on the mplcatons of Savage s axoms. 9

11 where 2 (0; ) and u : R +! R + s contnuously d erentable, strctly ncreasng, strctly concave and = + for all. One partcular famly of prors s that where the agent beleves that the true (x) process of states of nature belongs to a parametrc famly of probablty measures,, but the agent does not know the parameter 2 K. That s, the probablty of every event A 2 F s where 2 P( K Z (A) = K (A) (d), () ) s the pror belef over the unknown parameters. The hypothess of ratonalty can be further strengthened to requre that the agent s a Bayesan who knows that the process generatng the data s..d. but does not know the true probablty of the states of nature. We state ths assumpton as A:. A. = P for every 2 K. We want to emphasze that A: says that even though agents agree that the states of nature are generated by..d. draws from a common dstrbuton, they mght stll dsagree about tself. The followng assumpton mposes more structure on the subjectve dstrbuton of and t wll be dscussed further below. A.2 has densty f wth respect to Lebesgue that s contnuous at wth f ( ) > 0. Another nterestng spec caton of pror belefs s a pont mass probablty measure on de ned as : F! [0; ] where f 2 B (B) 0 otherwse. When prors belong to the class represented by (), Bayes rule mples that pror belefs evolve accordng to ;s t (d) = (s t s t ) ;s t (d) R K (s t js t ) ;s t (d), (2) where ;0 2 P( K ) s gven at date 0 and (s t s t ) C s t C s t. It s well-known that Bayesan learnng s consstent for any pror satsfyng A:. However, ths property apples to more general spec catons of prors (for nstance, those satsfyng (), see Schwartz [23, Theorems 3.2 and 3.3]), and snce our example 4 n Secton 8.2 does not satsfy A. but t does satsfy (), we state the consstency result n the followng Lemma to make precse ts scope. The celebrated De Fnett theorem states that ths s equvalent to the pror beng exchangeable. 0

12 Lemma Suppose that for ;0 almost all 2 K the probablty measures on (S ; F) are mutually sngular. Then ;s t t=0 converges weakly to for almost all s 2 S, for ;0 almost all 2 K : Remark : It s ubqutous n the learnng lterature related to asset prcng to assume both that () every agent knows the lkelhood functon generatng the data and () some agent learns the true condtonal probablty of the states. The latter s guaranteed n our setup by strengthenng A: to requre that the true parameter,, s n the support of some agent s pror. The case where ths holds for every agent s consdered n sectons 5, 6 and 7. Secton 8 deals wth the cases n whch ether () or () does not hold Endowments Agent s endowment at date t s a tme-homogeneous functon of the current state of nature, that s y (s t ) > 0 for all s t 2 f; :::; Kg and the aggregate endowment s y(s t ) P I = y (s t ) y <. An allocaton fc g I = 2 CI s feasble f c 2 C for all and P I = c ;t(s) y(s t ) for all s 2 S. Let Y denote the set of feasble allocatons. 4 Heterogeneous Prors and Portfolos: Examples The man purpose of ths secton s to llustrate our man results usng smple examples of dynamcally complete markets equlbra. In Secton 3 we assumed that the range of utlty functons was < +. Ths lower bound on utlty wll be used n the proofs of Theorems 3 and 4. We have ver ed the conclusons of those Theorems drectly for all of the examples n ths secton and n secton 8. Suppose there are two states, A:0 holds wth () = 2, two agents, u(c) = ln c and y () = y() > 0 for all 2 f; 2g where + 2 =. Agents can trade a full set of Arrow securtes n zero net supply. Arrow securty 0 pays unt of the consumpton good f s t+ = 0 and 0 otherwse. The prce of Arrow securty 0 2 f; 2g and agent s holdngs at date t after partal hstory s t are denoted by m 0 t (s) and a0 ;t (s), respectvely. We assume that agents have no endowment of Arrow securtes,.e. they have zero nancal wealth at date 0. In Appendx A we show that equlbrum consumpton and portfolos are P c ;t (s) = + j;t (s) j P ;t (s) y t (s); P + j;t (s) j P ;t (s) a 0 ;t (s) = y(0 )! p j ( 0 js t ), 0 2 f; 2g, (3) p ( 0 js t ) where P ;t (s) = P (C(s t )) and p ( 0 s t ) = P (C(s t ; 0 ))=P (C(s t )). Observe that ndvdual portfolos at date t are completely determned by the lkelhood rato at

13 t +, P j;t+ P ;t+. Portfolos converge f and only f the lkelhood rato converges. Thus, changes n portfolos are purely determned by the heterogenety of prors. The relevant margn of heterogenety descrbed by lkelhood ratos changes as tme and uncertanty unfold. Consequently, (3) suggests that the conventonal wsdom that changes n portfolos are fundamentally drven by heterogenety n prors s correct as long as ths margn of heterogenety perssts. Bayesan updatng, however, mposes a strong structure on the lmt behavor of belefs, n the sense that agents typcally end up agreeng on the one-perod-ahead condtonal probablty. What s pendng to explan s the lmt behavor of lkelhood ratos when one-perod-ahead condtonal probabltes converge. Benchmark Case: Homogeneous Prors Agents have dentcal one-perod-ahead condtonal probabltes of state after observng partal hstory s t, p j s t. Then, the lkelhood rato P j;t(s) P ;t (s) = for all t and s. Consequently, and thus portfolos are xed forever. a 0 ;t (s) = 0 for all t, s and 0, In every equlbrum, agents consume ther endowment every perod and, then, consumpton and Arrow Securtes prces are smple random varables wth support dependng only on the aggregate endowment. More precsely, c ;t (s) = y(s t ) m 0 t (s) = 2 y(s t ) y( 0 ) : From ths result and as a drect consequence of the convergence of the one-perodahead condtonal probabltes, one mght hastly make the followng conjectures: Conjecture I: Portfolos converge to a xed vector whle consumpton and Arrow securty prces converge to some smple random varable dependng only on the aggregate endowment. Conjecture II: Lmtng portfolos, consumpton and Arrow securty prces are those of an otherwse dentcal economy where agents begn wth homogeneous prors and zero nancal wealth. Example shows that Conjecture II mght fal even f Conjecture I holds. Example : Heterogeneous Prors I The agents one-perod-ahead condtonal probabltes of state are gven by p j s t = n s t + t + 2 and p 2 j s t = n2 s t + 2 ; t + 4 2

14 where n s t stands for the number of tmes state 2 f; 2g has been realzed at the partal hstory s t. Snce we assume A:0 holds wth () = 2, the Strong Law of Large Numbers mples that p j s t! 2 (P a:s:) as t!, for every agent 2 f; 2g. Therefore, both agents learn the true one-perod-ahead condtonal probablty. By the Kolmogorov s Extenson Theorem (Shryaev [22, Theorem 3, p. 63]), there exsts a unque P on (S ; F) assocated to the agent s one-perod-ahead condtonal probablty. Moreover, P sats es A: and A:2 and agents pror belefs over have denstes f () = and f 2 () = 6 ( ) on (0; ), respectvely. 2 The lkelhood rato s P ;t (s) P 2;t (s) = R 0 P t (s) d R 0 P t (s) 6 ( ) d = 6 [n (s t )+] [n 2 (s t )+] [t+2] [n (s t )+2] [n 2 (s t )+2] [t+4] = 6 (t+3) (t+2) (n (s t )+) (n 2 (s t )+) ; where stands for the Gamma functon. 3 The Strong Law of Large Numbers can be appled once agan to show that P ;t (s) P 2;t (s)! 2 3 = f 2 P a:s: f 2 2 It follows from (3) that portfolos converge to a xed vector, that s a 0 ;t (s)! 3 y(0 ) + 2! ; 0 2 f; 2g P a:s: 2 Although securty prces, asset holdngs and consumpton all converge, we want to underscore that only prces converge to those of an otherwse dentcal economy wth homogeneous prors. Indeed, c ;t (s)! m 0 t (s)! 2 y(s t) y( 0 ) ; and thus Conjecture I holds but Conjecture II does not. y(s t ) < y(s t ); The reason s that n the economy that starts wth homogenous pror belefs the agents nancal wealth s zero whle n the lmt economy pror belefs are homogeneous but the agents nancal wealth s not zero. In ths example lmt asset prces are dentcal to those of an otherwse dentcal economy that starts wth homogenous pror belefs because logarthmc preferences make ntertemporal margnal rates of substtuton, and thus asset prces, ndependent of the wealth dstrbuton. In general, however, asset prces 2 That s, agent s pror belefs over follow a Beta dstrbuton B (; ) on (0; ), as n Morrs [7]. 3 Recall that f n s an nteger, then (n) = (n )! 3

15 do depend on the wealth dstrbuton. In Secton 6 we fully characterze the lmt wealth dstrbuton and argue that t depends crtcally on date 0 prors. The followng example shows that Conjecture I mght be false as well. Example 2: Heterogeneous Prors II The agents one-perod-ahead-condtonal probabltes of state are gven by p p j s t = p and p 2 j s t = e =t p : + e =t + e =t.e., agents beleve that the states of nature are ndependent draws from tme-varyng dstrbutons. Observe that one-perod-ahead condtonal probabltes converge to 2 for both agents,.e. agents learn, and have the same entropy. That s, E P (log p ;t+ j F t ) = E P (log p 2;t+ j F t ) : The rato of one-perod-ahead condtonal probabltes atndate t after partal hstory s t p s a random varable, ;t p 2;t, that takes values n ep =t ; p e =t o. The logarthm of the lkelhood rato can be wrtten as the sum of condtonal mean zero random varables as follows log P;t (s) P 2;t (s) ty p ;k (s) = log p 2;k (s) k= tx p = sk = (s) log e =t + ( sk = (s)) log p k= e =t tx = 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. Consequently, the log-lkelhood rato s the sum of unformly bounded random varables wth zero condtonal mean. Addtonally, snce the sum of condtonal varances of x k dverges wth probablty, t follows by Freedman [, Proposton 4.5 (a)] that and, therefore, sup t tx k= x k (s) = + and nf t tx x k (s) = P a:s: k= lm nf P ;t(s) P 2;t (s) = 0 and lm sup P ;t(s) P 2;t (s) = + P a:s: 4

16 Ths behavor of the lkelhood rato mples that ndvdual portfolos uctuate n ntely often. In partcular, lm nf a 0 ;t (s) = y( 0 ) and lm sup a 0 ;t (s) = ( ) y( 0 ): Snce each agent s debt attans ts so-called natural debt lmt n ntely often, ndvdual portfolos are hghly volatle. Consequently, Conjecture I does not hold n ths example and, a pror, ths s rather surprsng snce every agent learns the true oneperod-ahead-condtonal probablty. The fact that the one-perod-ahead-condtonal probabltes converge certanly means that trade n each perod becomes eventually very small. However, snce the lkelhood rato of agents belefs fals to converge, ths small trade compounds over large perods of tme and so (n a su cently long span of tme) there are wde uctuatons n the dstrbuton of wealth. Why does Conjecture I hold n example whle t fals n example 2? The man d erence s that prors satsfy A: n example but not n example 2. It turns out that when A: holds for every agent, the lkelhood ratos always converge and, thus, Conjecture I holds n general. However, to generalze these lessons to the settng descrbed n secton 3 one faces two d cultes that we avod n the examples by carefully choosng preferences, ndvdual endowments and prors. Frst, equlbrum portfolos are typcally hstory dependent n a more general setup. Closed-form solutons for asset demands as n (3) are useful to tackle ths d culty but they are a partcular feature derved from logarthmc preferences and constant ndvdual endowment shares. Second, lkelhood ratos are typcally complcated objects whch makes the analyss of ther behavor a nonstandard task. Closed-form representaton for the lkelhood rato, as n the examples above, smpl es the analyss of ts asymptotc propertes but t s a consequence of the partcular famly of prors that we choose. The rest of the paper tackles the d cultes to extend the lessons from the examples to the more general setup descrbed n secton 3. Here we o er an outlne. We begn wth a recursve characterzaton of e cent allocatons and ther correspondng supportng portfolos under the assumpton that A. holds. In secton 5, we show that the evoluton of any Pareto optmal allocaton s drven solely by the evoluton of the lkelhood ratos of the agents prors and the agents belefs over the unknown parameters, as n the examples. In secton 6, we prove that the agents nancal wealth converges f and only f both the lkelhood rato as well as ther belefs (over the unknown parameters) converge. Afterwards, we tackle the d cultes assocated wth the lack of closed form for the lkelhood ratos. In secton 7, we consder a broad class of prors satsfyng A.. We apply recent results n probablty theory 5

17 to prove that the lkelhood ratos converge wth probablty one, as n example. Fnally, n secton 8 we explan the exact role played by the assumptons that every agent knows the lkelhood functon generatng the data and that some agent learns and we argue that t s crtcal that they are coupled together. We do so by provdng two addtonal examples, each of whch relax one of these assumptons, n whch the lkelhood rato does not converge and consequently portfolos change n ntely often as n example 2. 5 A Recursve Approach to Pareto Optmalty In ths secton, we provde a recursve characterzaton of the set of Pareto optmal allocatons provdng a verson of the Prncple of Optmalty for economes wth heterogeneous pror belefs. Throughout ths secton we assume that A:0 and A: hold. It s well known that under A:, Bayes rule mples that pror belefs evolve accordng to ;s t (d) = (s t ) ;s t (d) R (s K t ) ;s t (d), (4) where ;0 2 P( K ) s gven at date 0. Lemma 2 Suppose agent s pror sats es A:. Then, for every B 2 F Z P ;s t (B) = Ps t (B) ;st (d) : (5) K 5. Pareto Optmal Allocatons A feasble allocaton fc gi = s Pareto optmal (PO) f there s no alternatve feasble allocaton fbc g I = such that U P (bc ) > U P (c ) for all 2 I. It s well known that the set of PO allocatons can be characterzed as the soluton to the followng planner s problem. Gven 0, s 0 and welfare weghts 2 R I +, de ne v (s 0 ; 0 ; ) sup fc g I = 2Y IX E P =! X t u (c ;t ). (6) Unlke the case where agents have homogeneous belefs, the recursve characterzaton of PO allocatons n our economy s rather trcky because belef heterogenety makes optmal allocatons hstory dependent. Ths can be seen from the followng t 6

18 (necessary and su cent) rst order condtons to the planner s problem: P ;t (s) j P j;t (c ;t j (c j;t (s)) = for all, j 2 I, for all t and all s, j;t IX c ;t (s) = y(s t ). (8) =. Snce j (c ;0 (c j;0 j;0, the planner dstrbutes consumpton among agents to make the rato of margnal valuatons of any two agents -whch, we recall, nclude prors that could be subjectvely held- to be constant along tme. Consequently, under the optmal dstrbuton rule of consumpton, the rato of margnal (c ;t (c j;t j;t, must be proportonal to the lkelhood rato of the agents prors, P j;t (s) /P ;t (s). Ths rato represents the novel margn of heterogenety among agents consdered n ths paper, whch we call the B-margn of heterogenety. The B-margn s purely drven by heterogenety n prors and ts evoluton determnes the dynamcs of the optmal dstrbuton rule of consumpton. Indeed, when all agents have the same prors the B-margn remans constant along tme and the optmal dstrbuton rule of consumpton s both tme and hstory ndependent. Consequently, ndvdual consumpton depends only upon the current shock s t (because t determnes aggregate output) and the date 0 vector of welfare weghts. When agents have heterogeneous prors, nstead, the B-margn s hstory dependent and so s the optmal dstrbuton rule of consumpton. Now we argue that ths hstory dependence can be handled wth a properly chosen set of state varables. Note that snce condton (7) holds f and only (c ;t (s)) ;t (s = R K t+ ) ::: (s t+k ) ;s t j (c j;t (s K t+ ) ::: (s t+k ) j;s t (d) (c ;t+k ;t+k j (c j;t+k j;t+k then the planner does not need to know the partal hstory tself n order to contnue the date 0 optmal plan from date t onwards. Indeed, t s su cent that he knows the rato of margnal utltes that the orgnal plan nduces at date (c ;t (c j;t j;t s t (d), snce ;s t (d) =, the state of nature at date t, s t, and the posteror belefs, (s t) ;s t (d) R K (s t) ;s t (d). Moreover, snce the rato of margnal utltes at date t equals the lkelhood rato weghted by the date zero welfare weghts, j P j;t (s) P ;t (s), the d cultes stemmng from the optmal plan hstory dependence can be handled by usng ( P ;t (s); :::; I P I;t (s); s t ) as state varables summarzng the hstory and the state of nature at date t, s t, descrbng aggregate resources. From the dscusson above, we conclude that a PO allocaton cannot be fully characterzed usng only the agents belefs over the unknown parameters (that s, 7

19 s t ) and s t as state varables as n the sngle agent settng (see, Easley and Kefer [8]). In a multple agent settng, nstead, the planner needs to dstrbute consumpton and because of ths one needs to ntroduce ( P ;t (s); :::; I P I;t (s)) as an addtonal state varable, whch can be nterpreted as the date t welfare weghts, ;t (s) = P ;t (s). These weghts evolve accordng to the law of moton Z ;t (s) = ;t (s) (s t ) ;s t (d) where ;0 (s) =. (9) K In Secton 5.2 below we present a formal exposton of ths result. 5.2 Recursve Characterzaton of PO Allocatons Gven that n an envronment wth heterogeneous belefs and learnng PO allocatons are typcally hstory dependent, standard recursve methods cannot be appled. We tackle ths ssue by adaptng the method developed by Lucas and Stokey [6]. In Appendx B we show that v s the unque soluton of the functonal equaton X < v(; ; ) = max (c;w 0 ( 0 )) : u (c ) + X Z = ( 0 ) 0 (; ) (d) w( 0 0 ) 0 K ; ; (0) subject to 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 the recursve dynamc program de ned by (0) - (3), the current state,, captures the mpact of changes n aggregate output whle (; ) summarzes and solates the dependence upon hstory ntroduced by the evolvng B-margn of heterogenety. The planner takes as gven (; ; ) and allocates current consumpton and contnuaton utlty levels among agents. That s, nstead of allocatng consumpton from tomorrow on, the planner assgns to each agent the utlty level assocated wth the correspondng contnuaton sequence of consumpton. Indeed, the optmzaton problem de ned n condton (2) characterzes the set of contnuaton utlty levels 4 In sectons 5.2 and 6, we abuse notaton and let c to be a non-negatve vector and c ts th component. 8

20 attanable at ( 0 ; 0 (; )) (see Lemma 4 n Appendx B). 5 The weghts 0 ( 0 ) that attan the mnmum n (2) wll then be the new weghts used n selectng tomorrow s allocaton. The (normalzed) law of moton for the welfare weghts, 0 (; ; )(0 ), follows from the rst order condtons wth respect to the contnuaton utlty levels for each ndvdual and s gven by 0 (; ; )( 0 ) R ( 0 ) 0 P h h (; ) (d) R ( 0 (4) ) 0 h (; ) (d): Observe that the normalzaton s harmless snce optmal polcy functons are homogeneous of degree zero wth respect to. It follows by standard arguments that the correspondng consumpton polcy functon, c (; ), s the unque soluton to c (; ) + h (c (; )) = y(): for each 2 h denotes the nverse functon h. Gven (s 0 ; 0 ; 0 ), we say the polcy functons (c; 0 ) coupled wth 0 generates an allocaton bc f bc ;t (s) = c (s t ; t (s)), t+ (s) = 0 (s t ; t (s); s t )(s t+ ), s t = 0 (s t ; s t ), for all and all t 0 and s 2 S where 0 (s) = 0 and s = 0. The followng Theorem shows that there s a one-to-one mappng between the set of PO allocatons and the allocatons generated by the optmal polcy functons solvng (0) - (3). Theorem 3 (The Prncple of Optmalty) An allocaton (c )I = s PO gven (; ; ) f and only f t s generated by the polcy functons solvng (0) - (3). 5 To understand condton (2) notce that the utlty possblty set,.e. the set of expected lfetme utlty levels that are attanable by mean of feasble allocatons, s convex, compact and contans ts correspondng fronter. The fronter of a convex set can always be parametrzed by supportng hyperplanes. Moreover, under our assumptons, the correspondng parameters can be restrcted to le n the unt smplex and, therefore, they can be nterpreted as welfare weghts. Thus, a utlty level vector w s n the utlty possblty set f and only f for every welfare weght the hyperplane parametrzed by and passng through w, w, les below the hyperplane generated by the utlty levels attaned by the PO allocaton correspondng to that welfare weght, attanng the value v(; ; ). Ths s why we must have w v(; ; ) for all or, equvalently, mn e [v(; e; ) ew] 0. See Appendx B for techncal detals. 9

21 Informally, ths result can be grasped as follows. The characterzaton of the soluton to the sequental formulaton of the planner s problem hnts that once the planner knows both the lkelhood rato weghted by the date zero welfare weghts and the belefs at date t, he can contnue the optmal plan from date t onwards. It s key to understand that the consumpton plan from date t+ onwards can be summarzed by ts assocated utlty levels whch n turn can be summarzed by a vector of welfare weghts. Theorem 3 shows that the date zero optmal plan s consstent n the sense that the contnuaton plan s ndeed the soluton from date t onwards Dscusson: An Alternatve Approach There s an alternatve approach to state the dynamc program de ned by (0) - (3): nstead of parametrzng allocatons wth welfare weghts, the planner chooses current feasble consumpton and contnuaton utltes for both agents n order to maxmze the utlty of agent subject to two restrctons: () the utlty of agent 2 s above some prespec ed level (the so-called promse keepng constrant) and () contnuaton utlty levels le n next perod utlty fronter. Very mportantly, ths last condton mples that the correspondng value functon de nes the constrant set. 6 Snce both n our approach as well as n the alternatve one the correspondng value functon de nes the constrant set, nether of the two dynamc programs s standard n the sense that t s not obvous that any of the correspondng operators sats es one of Blackwell s su cent condtons, namely, dscountng. Indeed, for any functon v that de nes the constrant set there mght be some a > 0 such that v + a enlarges the feasble set of choces of contnuaton utltes wth respect to v. The key to show dscountng n our approach s to restrct the set of functons to be homogeneous of degree wth respect to the state varables,.e. the welfare weghts, (a property that s sats ed by v, see Lemma 3 n Appendx B). 7 Snce v + a s an a ne lnear transformaton of v, the choce of current consumpton s the same for v and v + a. In addton, homogenety of degree of the value functon wth respect to the welfare weghts mples that w 0 s the optmal choce for the constrant set de ned by v f and only f w 0 + a s the optmal choce for the constrant set de ned by v + a. Ths explans why homogenety of degree of the value functon wth respect to the welfare weghts s key to show that dscountng holds n our settng. 8 6 Snce we parametrzed the utlty levels wth ther assocated welfare weghts, our approach amounts to replacng the promse keepng constrant by usng the assocated Lagrange multplers as state varables. 7 Lucas and Stokey [6] do not make ths restrcton and so t s unclear whether dscountng holds n ther approach. 8 There s no obvous condton equvalent to homogenety n the alternatve approach descrbed above and then one needs to nd the soluton d erently. One plausble strategy would be to follow 20