The Market Selection Hypothesis

Transcription

1 The Market Selecton Hypothess Lawrence Blume and Davd Easley Department of Economcs Cornell Unversty 1 June 1999 The authors thank Alvaro Sandron for stmulatng conversaton, and The Natonal Scence Foundaton for research support under grant SES Research support for Blume from the John D. and Catherne T. MacArthur Foundaton s gratefully acknowledged.

2 1 1 Introducton It s conventonal to assume that traders n asset markets have ratonal expectatons about asset returns, and choose savngs rates and portfolos as f they maxmze expected utlty usng these belefs. As the hypothess that traders are expected utlty maxmzers places few restrctons on behavor n the absence of the ratonal expectatons hypothess, much attenton has been focused on the valdty of assumng ratonalty correctness of expectatons. Although traders would certanly prefer to hold accurate rather than naccurate belefs, an explanaton of how traders come to correctly forecast endogenous equlbrum rates of return s lackng. Varous approaches have been devsed to provde the mssng foundaton for the ratonal expectatons hypothess. One approach posts that correct belefs can be learned. In other words, ratonal expectatons are stable steady states of learnng dynamcs; see [2] and [5] for more dscusson. The learnng approach s not completely satsfactory. Achevng ratonal expectatons through learnng requres too much pror knowledge, to the pont where the requrements for learnng essentally assume the concluson. An alternatve approach based on evolutonary forces operatng through wealth dynamcs has arsen n recent years. Ths approach actually has a long tradton. Both Alchan [1] and Fredman [7] argued n the early 1950 s that evolutonary forces would eventually result n behavor consstent wth correct maxmzaton. Although ther argument s plausble, untl recently there was no careful analyss of the market dynamcs that would supposedly select for expected utlty maxmzers, and, wthn the class of expected utlty maxmzers, select for those wth ratonal expectatons. In [3] we analyzed an economy wth repeated markets for rsky, one perod assets. We showed that f savngs rates are equal across traders, then wealth dynamcs do not necessarly lead to traders actng as f they maxmze expected utlty usng ratonal expectatons. Expected utlty maxmzaton may fal to emerge because the market selects for traders whose nvestment portfolos generate hgher expected growth rates of ther share of wealth. Although our prmary focus was not on the lnk between portfolo rules and expectatons, we showed that a trader wth logarthmc utlty and correct expectatons maxmzes the expected growth rate of wealth share and so domnates the market. However, traders wth correct expectatons and non-logarthmc utlty need not maxmze the expected growth rate of wealth share and so can be drven out of the market even by traders wth ncorrect expectatons. Our analyss addressed the possble emergence of expected utlty maxmzaton. The queston of belef selecton among expected utlty maxmzers was examned by Sandron [9], who analyzed an economy wth nfntely lved rsky assets. He showed that f markets are dynamcally complete, and some assumptons are made on returns, then when savngs rates are endogenous, and all traders are expected utlty maxmzers wth a common dscount factor, only traders wth ratonal expectatons survve. Ths occurs because when markets are complete, traders can place bets on any dsagreement about the probablty of states and traders wth correct expectatons wll wn the bets. So n Sandron s world the market selects for those traders whose expectatons are correct. Ths analyss dffers from [3] n that savngs behavors are

3 2 endogenous, and the exogenous parameters are the prmtves nvolved n descrbng preferences, such as payoff functons and dscount factors. In ths paper we explore more completely when selecton occurs, when t does not occur and why. We return to the asset structure of [3], so frst we show that wth our assets, market completeness and some ancllary assumptons mply selecton for ratonal expectatons. Second, we show that dynamc completeness of markets s necessary to guarantee selecton for ratonal expectatons. In economes wth ncomplete markets, a trader who s overly optmstc about the return on some asset n some state can choose to save enough to more than overcome the poor asset allocaton decson that hs ncorrect expectatons create. Ths result s even more strkng than t seems, because when traders belefs are heterogeneous, some market ncompleteness s nevtable. For wth heterogeneous belefs, market completeness mples that traders can bet on any dfferences n belefs. Ths amounts to openng a new set of markets every tme a new trader wth dfferent belefs enters the economy. In the context of the model, the relevant state space contans the unon of the supports of each player s belefs. Thus addng a new trader can requre expandng the state space, and therefore addng new markets. We conclude that, wthn the evolutonary framework, the condtons requred to ensure market selecton for ratonal expectatons are too strong to be useful. In general there are no market mpedments to long-run heterogeneous belefs. 2 The Model Our model magnes a group of traders repeatedly buyng shares n exogenously suppled assets. The assets delver unts of a sngle consumpton good, corn. Traders are nfntely-lved, and all assets have a lfe of one perod. That s, each asset avalable today promses delvery of corn tomorrow. Asset returns are uncertan, and wll be modeled as functons of some stochastc process of states. An equlbrum s a stochastc process of prces for assets n terms of corn wth the property that the asset markets clear when traders are optmzng gven ther belefs about the future evoluton of the economy. 2.1 Notaton and Bascs Formally, we assume that tme s dscrete, begnnng at date 1. Furthermore, the state avalable at each date comes from a fnte set {1,..., S}. The set of all sequences of states s Σ, and Σ together wth ts product σ-feld s the measurable space on whch everythng wll be bult. Let p denote the true probablty meaasure on Σ. For many of the examples we construct, the stochastc process of states wll be assumed to be..d. In ths case p = p 1,..., p S wll refer to the sngle-perod probablty dstrbuton on states. In the next few paragraphs we ntroduce a number of random varables of the form x t σ. All such random varables are assumed to be date-t measurable; that s, ther value depends only on the realzaton of states through date t.

4 3 For a gven path σ, σ t s the state at date t and σ t = σ 1,..., σ t s the partal hstory through date t of the evoluton of states. There are A assets. The number R a t σ s the amount of corn asset a, whch s traded at date t 1, pays off at date t n realzaton σ. At date t 1 the prce of ths asset s q a t 1σ. A prce sequence s a collecton of random varables q = {q t } t=1, where q t s the prce vector q 1 t,..., q A t. At each date t, 1 unt of each asset s avalable to the economy. Thus the total supply of corn at date t along path σ s R t σ = a Ra t σ. The economy contans I nfntely lved traders. Each trader has a dscount factor β < 1 and a sngle-perod payoff functon u : R R defned on unts of corn consumpton. A consumpton plan c for trader s a collecton of random varables {c t} t=1. For a gven consumpton plan c and path σ, the utlty of the consumpton plan s v c σ = t βt 1 u c t σ. Each payoff functon u s C 1, strctly ncreasng, strctly concave and satsfes an Inada condton at the orgn: lm c 0 u c = +. Each player also has belefs p, whch s a probablty measure on Σ. In the..d. case, p wll refer to a probablty measure on {1,..., S}. In every case, E refers to the expectatons operator for belefs p. At the begnnng of each date t on path σ, all the assets pay off. Total corn wealth s R t σ, and trader owns share r tσ. The wealth shares sum to 1. Thus trader s date t wealth on path σ s w tσ = r tσr t σ. At date 1 traders have an aggregate endowment R 1 of corn, and trader s share s r 1 > 0. These date 1 parameters are exogenously gven. 2.2 Demand and Equlbrum Each trader chooses a savngs plan and a portfolo plan to maxmze the expected value of v c σ gven hs belefs p. A savngs plan δ for trader s a collecton of [0, 1]-valued random varables {δt} t=1, whch record the fracton of current wealth whch s to be nvested n assets for tomorrow. The remanng fracton s eaten. A portfolo plan α for trader s a collecton of random varables {αt} t=1 such that each αt = α1t,..., αat s a vector representng the shares of savngs gong to each asset a. The coeffcents of ths vector sum to 1, but t may have negatve terms whch correspond to short sales. Thus trader s optmzaton problem s to solve the problem P : { max c,δ,α E t=1 βt 1 u c t σ } s.t. c tσ = 1 δtσ wtσ A wt+1σ α = atσδtσw tσ R a qt a σ t+1σ a=1 1 w 1σ = r 1R 1 A suffcent condton for the exstence of optmal plans s that feasble consumpton not grow to fast. See [8, Theorem 16.12, pg. 116]. Although ths dynamc programmng problem s not

5 4 statonary, t nonetheless s solvable by backward nducton. Let P σ t denote the dynamc programmng problem begnnng at tme t after partal hstory σ t. Equlbrum n ths model s gven by the condton that asset markets clear. That s, for all paths σ, dates t and assets a, I =1 α tσδ tσw tσ q a t σ = 1 Ths condton can be rewrtten n terms of wealth shares. I =1 αtσδ tσr tσ R qt a t σ = 1 2 σ Total wealth s exogenously gven, and so 2 shows that prces are determned by the optmal savngs and portfolo plans of each agent, and the path of wealth shares. 3 Evoluton The evoluton of wealth shares determnes both the fate of ndvdual traders and the path of market aggregates such as equlbrum prces. Here we work out the mplcatons for both ndvdual selecton and the long-run behavor of market prce. 3.1 Selecton across Traders Characterstcs Let 1 s tσ denote the ndcator functon for state s at date t; that s, t equals 1 f σ t = s and 0 otherwse. The evoluton of trader s wealth share s gven by the equaton r t+1σ = = s { A s=1 a=1 s { A s=1 a=1 α atσδ tσw tσ q a t σ Rt+1σ a R t+1 σ } 1 s t σ αatσδ tσr tσ R qt a t σ Ra t+1σ σ R t+1 σ } 1 s t σ 3 In the frst equaton, δtw t s the amount of wealth trader saves. Fracton αat s nvested n asset a, and so the fracton represents the number of shares of asset a purchased by trader. Multplyng ths by Rt a gves the wealth next perod that trader accrues from hs holdngs of asset a. Summng over all the assets gves the gross return on hs nvestment, and dvdng by R t turns that nto a share. Usng the ndcator functon as an exponent and multplyng over all states pcks out the rght state realzaton for the path σ.

6 5 The equlbrum condton 2 determnes the equlbrum prce q a of asset a as a functon of the savngs, wealth shares and portfolo decsons of all traders. Substtutng from 2 nto 3 gves r t+1σ = s { A αatσδ tσr tσ j αj atσδ j t σr j t σ s=1 a=1 Rt+1σ a R t+1 σ } 1 s t σ 4 Ths equaton dentfes trader s share tomorrow on path σ as the average of hs share of nvestment n each asset, weghted by the share of total wealth tomorrow that asset generated. Our nterest s n the long-run fate of traders. Traders can vansh f ther wealth share goes to 0, or survve f t does not. Formally, Defnton 3.1. Let S Σ denote a measurable set of paths. Trader vanshes on S f for p-almost all σ S, lm t r tσ = 0. If trader does not vansh, he survves. The crtera for survval worked out n [3] apply n ths model too. These crtera are based on the noton of relatve entropy. Defnton 3.2. The relatve entropy of a probablty measure ρ on S wth respect to the probablty measure q on S s I q ρ = s q s logq s /ρ s. The relatve entropy s a way of measurng a dstance from ρ to q. Clearly ths functon s mnmzed at ρ = q, where t takes the value 0. In [3] we drectly analyzed the lmt behavor of 3 usng approprate strong laws of large numbers. A weakness of our approach there s the dffculty n workng backwards from behavors to prmtves such as preferences. Here we characterze the long run dstrbuton of wealth drectly n terms of preference prmtves, n a manner smlar to [9]. Our methods drectly extend the Euler equaton technques of [4] to the stochastc envronment of the model developed here. The man result n ths secton s a characterzaton of who vanshes and who survves based on dynamcally complete markets wth bounded asset returns. All assets pay off n the numerare good, and so the necessary and suffcent condton for dynamc market completeness s that each A S matrx of asset returns have rank S. Let R t σ denote the asset returns matrx at date t on path σ. Ths s the matrx whose as th element s Rt a σ where σ t = s. Ths matrx-valued random varable s measurable wth respect to state observatons through date t. The followng Theorem s the basc result whch enables the demonstraton of varous propostons about market selecton for traders wth heterogeneous preferences and belefs n dynamcally complete markets. Ths Theorem compares two traders, and j. Let Z t σ = log p j σ τ+1 σ τ /p σ τ+1 σ τ. Notce that the condtonal expectaton EZ t F t 1 wth respect to the probablty dstrbuton p s a dfference of relatve entropes wth respect to p σ t 1. Theorem 3.1. Consder 2 agents and j. Suppose that the followng condtons hold:

7 6. For all t and σ the asset returns matrx R t σ has rank S.. Every trader s utlty functon s bounded from below.. There are B > R > 0 such that for all assets a and dates t, B > R a t σ > R; v. t t 2 VarZ t F t 1 < ; v. lm sup t t 1 t τ=1 E{Z τ F t 1 } > logβ j /β. Then r j t σ 0 p-almost surely. Theorem 3.1 concerns the asymptotc behavor of traders consumpton and wealth paths n economes wth complete markets. The role of the varance condton v s to justfy the use of a law of large numbers. It s easy to state condtons mplyng that v holds almost surely. For nstance, fx ɛ > 0 and suppose that almost surely, each trader beleves at every date that each state tomorrow wll happen wth probablty at least ɛ. The weght of the Theorem s carred by condton v, whch s a condton on the dstances of belefs of the two traders from the truth. For example, f β = β j, condton v states that the relatve entropy of s belefs wth respect to the truth s smaller than that of j s belefs. The remanng two condtons are necessary to move from the concluson of small consumpton to small wealth share. Condton bounds gross returns. Ths condton mples that rates of return are bounded from below by B/R. Theorem 3.1 compares the experences of dfferent traders to determne who survves and who vanshes. The followng Corollares are obvous consequences of the Theorem. Corollary 3.1. Suppose that all traders have a common dscount factor and at least one trader has correct expectatons, and that condtons,, and v are satsfed. Then for any trader j, on the set of paths such that r j t 0 p-almost surely. lm sup t 1 t t Elog pσ t+1 σ t /p j σ t+1 σ t < 0 τ=1 Corollary 3.2. Suppose that all traders have the same belefs, but dfferent dscount factors. If condtons, and are satsfed, then only those traders wth the hghest dscount factor n the populaton survve, and all others vansh p-almost surely. The man pont of these Corollares and of Theorem 3.1 s that so long as the Inada condton s satsfed and the Euler equatons work, payoff functons have nothng to do wth survval. The market selects across dscount factors and belefs, but not across payoff functons. Unfortunately the hypotheses of the Theorem exclude mportant utlty functons such as the log functon, whch can n fact be handled by dstnct arguments, and CRRA utlty wth

8 7 relatve rsk averson coeffcents exceedng 1. The dffculty wth unbounded payoff functons comes n ensurng that the Euler condtons used n the proof are n fact necessary condtons for an optmal path. The proof of 3.1 s an extenson of the Euler equaton arguments of [4], appled to the optmzaton problem 1 under the assumpton of dynamcally complete markets. It s not a best posble theorem. For nstance, suppose that trader has correct belefs, such that all condtonal probabltes are unformly bounded away from 0. Suppose that trader j s belefs have the probablty of one state rapdly convergng to 0. Then trader j wll be nvestng ever less n ths state, whch keeps recurrng at some fxed rate, so hs wealth should go to 0 almost surely. But the varance condton v fals, and so the Strong Law argument of the proof wll not work. Proof of Theorem 3.1. Snce markets are complete, we can construct portfolos, one for each state s at date t, such that the payoff to holdng the entre supply of ths portfolo, whch s taken to be 1 share, s Wt+1σ s = a Ra t+1σ t, s n state s at date t, and 0 n all other states. These portfolos can be prced from the equlbrum asset prces. Let πt s denote the prce of the portfolo that pays off n state s at date t. In equlbrum each consumer s optmzng, and the soluton to 1 nduces a soluton to the optmzaton problem n whch shares of these portfolos are traded at the prces π. In ths soluton savngs are the same, and portfolo allocatons are such that the returns to savngs at each date-event par are dentcal n the two problems. At each date agents consumptons are bounded above by A B, and so u A B/1 β s an upper bound on the value of the problem. Snce ths bound s fnte, and snce utlty s bounded from below, Euler equatons are necessary frst-order condtons that optmal plans must satsfy. u c t σ = β p σ t+1 σ t u c t+1 σ W σ t+1 t σ σ Iteratng ths equaton through tme along the path, u c t+1 σ = β t πt σt+1 σ π1 σ2 σ p σ t+1 σ t p σ 2 σ 1 Now defne the random varable 1 t log u c t σ u j c j tσ = log β j + 1 β t π σt+1 t 1 W σ t+1 t σ W σ 2 1 σ u c 1 σ 5 t Z τ σ + 1 t log u c 1 σ u j c j 1σ 6 A SLLN [6, Corollary 4.5b, pg. 108] shows that on the set of paths V satsfyng condton v, t t 1 Z t converges almost surely. The usual Kronecker lemma argument now mples that on ths set t 1 τ=1 X t τ EX τ F t 1 converges to 0 almost surely. It follows that on the set V, 1 lm t t log u u j c t σ c j tσ log β j β 1 t τ=1 τ=1 t E log pj σ τ+1 σ τ p σ τ+1 σ τ F t 1 0 p a.s.

9 8 Let W denote the set of states satsfyng v, that the term n large brackets has a negatve lm sup. On V W, u c t σ /u j c j tσ converges to 0 p-almost surely. Ths can happen ether because c t becomes large or c j t becomes small. Because aggregate corn stocks are bounded unformly from above, c j t must be convergng to 0. Now need to show that r j t σ converges to 0. Suppose not. Then we can fnd a state s and a subsequence of dates wth the followng propertes: w j t k σ 2ɛ, ɛ c j t k σ 0, and α js t k σ 1/S, where α js t s the fracton of savngs nvested by j at date t n the portfolo that pays off n state s. Let σs t denote the partal hstory of length t + 1 whch agrees wth σ through date t and ends wth state s at date t+1. Let v j w, σ t denote the value functon for the dynamc program begnnng at node σ t n the date-event tree wth wealth w. It s necessary to ndex the value functon by the entre partal hstory because the dynamc program s not n general statonary. Standard arguments show that despte the non-statonarty, the value functon s concave, and the margnal value of wealth equals the margnal utlty of consumpton n the current perod. Consequently the Euler equaton mples u jc j t k σ t k = βr s σ t k v w j t k σ t k, σ t k s p j σ t k s σ t k Snce trader j nvests at least ɛ/a n portfolo s, and snce by assumpton t pays off at most B, we have the followng nequalty: u jc j t k σ t k β B ɛa v w j t k σ t k, σ t k s Snce consumpton s convergng to 0, the left hand sde of the nequalty converges to +, and therefore so does the rght hand sde. Furthermore, w j t k σ t k ɛr/ba takng account of the lower bound on the rate of return of the portfolo gven by the assumed upper and lower bounds on gross returns. Snce the value functons are ncreasng n wealth, t follows that v ɛr/ba, σ t k s dverges. But ths cannot happen. The value functon at any node s bounded from below by v0 = u0/1 β. Then, usng the subgradent nequalty for concave functons, whch s a contradcton. v0 = v 0, σ t k s v ɛr/ba, σ t k s ɛr BA v ɛr/ba, σ t k s 3.2 Long-Run Asset Prcng PUT THE ASSET-PRICING FORMULA HERE. 4 Selecton Theorem 3.1 shows that the market selects for ratonal expectatons under some assumptons. The most mportant of these assumptons are:

10 9 1. Traders dsagreement about the future s lmted to dfferng probabltes on a common state space. In partcular, all traders know the returns on asset holdngs and equlbrum prces n each future state. Dsagreement about realzatons of these varables can be modeled as dsagreement about the probablty dstrbuton on an enrched state space, but ths then calls nto queston the next mportant assumpton. 2. Markets are dynamcally complete. Ths assumpton allows traders to place bets on all possble dsagreements and t s crucal for the analyss. It s an mportant lmtaton partcularly n lght of the frst assumpton. 3. Returns on assets are unformly bounded from above and away from zero. Ths prohbts, for example, the case of assets wth dvdends that grow at a non-zero, constant rate. The followng three example economes show that each of these assumptons s necessary for selecton for ratonal expectatons. In each of these economes there s one trader wth logarthmc utlty and correct expectatons and one trader wth CRRA utlty and ncorrect expectatons. We show that the logarthmc trader vanshes and that equlbrum prces do not converge to ther ratonal expectatons equlbrum values. We endow the trader wth correct expectatons wth logarthmc utlty for two reasons. Frst, t s easy to compute the optmal polcy for ths utlty functon. Second, and more mportant, we know from our prevous work [1] that f all traders save at the same rate a trader wth logarthmc utlty and correct expectatons wll survve and any trader whose portfolo s not asymptotcally equvalent to the log traders portfolo wll vansh. Thus our examples gve selecton for ratonal expectatons the best possble chance of success. ADD BELIEFS EXAMPLE HERE Example I: Ths frst example llustrates the effects of traders dsagreeng about what wll happen n the exogenously gven states. There s one state and one asset wth a gross return of 1 at each date. There are two traders. Trader 1 has logarthmc utlty, u 1 c 1 t = log c 1 t, and trader 2 s utlty functon exhbts constant relatve rsk averson CRRA, u 2 c 2 t = γ 1 c 2 t γ, for γ < 1, γ 0. Both traders have dscount factor β = 1/2. Trader 1 knows the return process and trader 2 beleves at each date t that the gross return on the asset wll be Rt+1 e = /3t /3 t 1 The return trader 2 expects s greater than one at each date and converges to 9/4 as t. Actutal total wealth at each date s 1 and traders begn wth equal shares of ths wealth.

11 10 In ths economy there s no asset decson; traders smply choose savngs rates at each date. For trader the optmal savngs rate rate at date t s characterzed by the Euler equaton w u wt1 δt = βu t δt R Rt+11 δt+1 e t+1, 8 q 1 t where R t+1 s the return trader expects at date t + 1 and δ e t+1 s the savngs rate that trader plans at date t to employ at date t + 1. Usng the traders utlty funcatons 8 smplfes to γ 1 R 1 δt γ 1 = β δt1 δt+1 e γ t+1 9 From 9 we see that a trader s savngs plan depends only on hs expectatons about rates of return R t+1 and hs preference parameter γ. qt 1 The soluton of Equaton 9 for the logarthmc trader γ = 0 s straghtforward. He saves at rate β = 1/2 at each date. The soluton for the CRRA trader depends on the expected return process 7 and the prce process. Consder the prce process q 1 t q t = /3t /3 t 1 If trader 2 antcpates that prces wll follow ths process then Equaton 8 mples that hs savngs rate wll be 3/4 at each date. At date 1 ths yelds aggregate savngs, and thus aggregate nvestment n the asset, of 1/21/2 + 3/4 = 5/8 whch s the value of q 1 1 gven by Equaton 10. Then at date 2, trader 2 wll have share 3/5 of the one unt gross return on the asset. When he saves 3/4 of ths wealth and trader 1 saves 1/2 of hs wealth the market clearng prce wll be q 2 from Equaton 10. Contnung n ths way we see that Equaton 10 gves an equlbrum prce process. Because trader 2 always saves at a hgher rate than does trader 1 hs wealth share, and hs actual wealth, converges to 1. So trader 1 vanshes. The equlbrum evoluton of wealth share for trader 2 s gven by r 2 t = 1 + 2/3 t 1 1. In ths economy the prce of the asset converges to 3/4, the savngs rate of the rratonal trader 2. It s easy to check that f trader 2 had correctly antcpated the gross returns on the asset, then n equlbrum both traders would have saved at rate 1/2 at each date, there would have been no evoluton of wealth shares, and the prce of the asset would have been ts ratonal expectatons equlbrum value of 1/2 at each date. Thus n ths economy a trader wth ncorrect belefs drves out a trader wth ratonal expectatons, because hs overly optmstc belefs cause hm to save at a hgher rate. In consequence market behavor does not converge to a ratonal expectatons equlbrum. q 1 t

12 11 At frst glance ths example s hard to compare wth the models of Theorem 3.1 because there seems to be no room n the models of the Theorem for the knd of dsagreement we have posted here. But n fact we can recapture the conventonal understandng of states that all agents know what happens n each state by expandng the state space and approprately modfyng belefs. Example II: Another way of descrbng the economy n Example I s to expand the state space so that at each date there s a state n whch trader 1 s belefs hold, and a state n whch trader 2 s belefs hold. Each player beleves that only hs state wll happen, and trader 1 s correct n those belefs. Formally, suppose that there are two states, S = {s 1, s 2 }, whch are..d. wth probablty p = 1, 0. Trader 1 s belefs are correct, p 1 = p, and trader 2 s belefs are p 2 = 0, 1. There s one asset wth gross returns matrx R t σ = 1, R e t, for all σ. The analyss of ths economy s exactly as n I. Trader 2, who has ncorrect belefs, survves and trader 1, who has correct belefs, vanshes. Notce that n ths economy, markets are ncomplete. There are always two possble states tomorrow, but only one asset. One unsatsfactory feature of ths example s that traders dsagree so completely. If markets were complete, equlbrum would fal to exst because of ther extreme dsagreement. Each trader would want to nfntely short the state he beleves to be mpossble. Furthermore, condton v fals. But ths orthogonalty of belefs does not drve the example. The next Example shows that the economy can easly be modfed so that the traders have overlappng belefs. Example III: Consder agan Example II, but now suppose that the belefs of trader 2 are p 2 = 1 ɛ, ɛ for ɛ > 0. Change the return on the asset n state 2 to R e t+1 = /3 ɛ 1 t 1 1/ /3 t 1 The return trader 2 expects to receve n state 2 s greater than one at each date and converges to 1 + 5/2ɛ 2 as t. Actutal total wealth at each date s 1 as state 1 always occurs. The logarthmc utlty trader contnues to save 1/2 of hs wealth at each date regardless of hs expectatons. Calculaton shows that the CRRA utlty trader contnues to save 3/4 of hs wealth at each date. Thus as before the prces n Equaton 10 clear the market at each date. Trader 2 saves at a hgher rate than does trader 1, so trader 1 vanshes. Ths occurs regardless of trader 1 s expectatons. So even f trader 1 has correct expectatons he vanshes.

13 12 In ths economy traders dsagree about the probablty dstrbuton on S, but because markets are ncomplete, ther ablty to bet on ths dsagreement s lmted. Ths dsagreement about states leads to trader 2 beng overly optmstc and therefore savng at a hgher rate than does trader 1. The state space n ths economy was constructed n order to ncorporate dfferng expectatons about returns on assets. It s apparent from 9 that t s only expectatons over rates of return that matter. So a smlar constructon could be used to ncorporate dfferng expectatons about future asset prces. An economy has dynamcally complete markets only f the set of assets s rch enough to allow wealth transfers from any partal hstory to any other partal hstory. Example III llustrates the fact the number of markets ths requres s controlled by the amount of subjectve dsagreement among traders. Consequently, we fnd the assumptons of dynamcally complete markets and heterogeneous belefs to be mplausble when taken together. In the precedng examples the source of adverse selecton effects came from the effect of dfferng belefs and market ncompleteness on savngs behavor. Blume and Easley [3] forced agents to have dentcal savngs behavor and studed the effects of selecton on portfolo choces. The next example shows how, even when nvestors have dentcal savngs behavor, portfolo effects can cause ncorrect belefs to survve and even prosper. Example IV: Consder an economy wth two assets and 3 states. Asset 1 pays off 1 unt of corn n state 1 and 0 n the other two states. Asset 2 pays off 0 n state 1, but 1 unt n each of states 2 and 3. There are two traders wth log payoff functons and common dscount factor β. The state probabltes and belefs are descrbed n the followng table: states s 1 s 2 s 3 truth 1/2 1/2 ɛ ɛ trader 1 1/2 1/2 ɛ ɛ trader 2 1/2 ɛ 1/2 ɛ The parameter ɛ > 0 s very small. Trader 1 has ratonal expectatons, whle trader 2 does not. Both traders wll save at the same rate β. Savngs must be allocated between assets 1 and 2. At any asset prce at every date, asset 2 wll have the same dstrbuton of returns for each trader. Consequently they wll hold the same portfolo at each date. and so the dstrbuton of wealth never changes despte the confguraton of traders belefs. To push ths pont farther, consder the followng confguraton of belefs where δ > 0 s

14 13 very small: states s 1 s 2 s 3 truth 1/2 1/2 ɛ ɛ trader 1 1 δ/2 1/2 ɛ1 + δ ɛ1 + δ trader 2 1/2 ɛ 1/2 ɛ Trader 1 has slghtly ncorrect belefs whle trader 2 has grossly ncorrect belefs. But trader 2 s belefs lead hm to make the same decsons that a trader knowng the truth would make, whle trader 1 wll do somethng else. Consequently trader 1 wll vansh and trader 2 wll domnate the market. The next example addresses the boundedness assumptons on asset returns. It demonstrates that f these do not hold even wth complete markets, market selecton for ratonal expectatons may fal. Example V: Now consder an economy n whch markets are dynamcally complete but returns are ether unbounded above or are not bounded away from zero. There are two states and two assets. Asset a paysoff R a t σ at date t f σ t = a and nothng n the other state. States are..d. and the probablty of state 1 s p 1 = 1 ɛ for 1/2 ɛ > 0. Trader 1 has logarthmc utlty and correct belefs. Trader 2 has CRRA utlty, u 2 c 2 t = γ 1 c 2 t γ, for γ < 1, γ 0, and belefs p 2 = 1/2, 1/2. The traders have common dscount factor β. Intally the traders have equal wealth shares of total wealth 1. It s apparent from the Euler equaton that the logarthmc utlty trader saves fracton β of hs wealth and nvests fracton 1 ɛ of hs savngs n asset 1 at each date regardless of current or future prces. The optmal polcy for the CRRA utlty trader s more complex. Suppose that prces are such that the rate of return on each asset, n the state n whch t paysoff, s x at each date. That s, for each asset a R a t+1σ q a t σ at each date on all paths σ. We wll construct returns and prces so that ths conjecture s satsfed n equlbrum. Wth ths constant rate of return the Euler equaton mples that the optmal polcy for the CRRA utlty trader s to save fracton βx γ 1 1 γ and nvest 1/2 of hs savngs n asset 1 at each date. = x Wth the polces above the equlbrum prcng equatons are qt 1 σ = Rtσ βx s γ 1 1 γ r 2 t σ1/2 + β1 ɛrt 1 σ qt 2 σ = Rtσ βx s γ 1 1 γ r 2 t σ1/2 + βɛrt 1 σ 11 12

15 14 where Rtσ s s the gross return on the asset that paysoff at date t n state σ. Combnng these two equaton systems provdes the evoluton of gross returns necessary to verfy the conjectured constant rate of return on assets. Rt+1σ 1 = xrtσ βx s γ 1 1 γ r 2 t σ + β1 ɛrt 1 σ 13 Rt+1σ 2 = xrtσ βx s γ 1 1 γ r 2 t σ + βɛrt 1 σ. 14 Fnally, wealth share evoluton s βx γ 1 1 γ βx γ 1 1 γ r 2 t σ + β1 ɛ1 r rt+1σ 2 t 2 σ r2 t σ f σ t = 1 = βx γ 1 1 γ βx γ 1 1 γ r 2 t σ + βɛ1 rt 2 σ r2 t σ f σ t = 2 15 Now for any specfcaton of β, γ and x Equatons 11 through 15 descrbe the evoluton of the economy. We frst consder a specfcaton that yelds unbounded returns on the assets. Suppose that β = 1/4, γ = 1/2 and x = 10. Ths yelds an optmal savngs rate for trader 2 of 5/8. Trader 2 nvests one-half of ths, or 5/16, n each asset. Trader 1 s optmal savngs rate s β = 1/4. So no matter how trader 1 dvdes hs savngs between assets that s no matter what ɛ he purchases less of each asset than does trader 2. So trader 1 s wealth share converges to 0; he vanshes. In ths economy relatve asset prces converge to one as trader 2 nvests one-half of hs savngs n each asset. These lmt prces are ndependent of the actual probablty dstrbuton on states and are not ratonal expectatons equlbrum prces for the economy consstng of only the CRRA trader unless ɛ = 1/2. We next consder a specfcaton n whch the gross returns of assets converge to 0. For any 0 < β < 1 and 0 γ < 1 let x = β γ γ. As before, trader 1 saves at rate β and nvests fracton 1 ɛ of hs savngs n asset 1. Calculaton shows that the savngs rate of trader 2, βx γ 1 1 γ, tmes the share that he nvests n each asset, 1/2, s greater than the savngs rate β, of trader 1. So trader 2 wll purchase more of each asset than wll trader 1 and thus trader 1 wll vansh. In ths economy the growth rate of the return on asset 1 at date t s R 1 t+1σ R 1 t σ 3 1 γ γ 1 + 1/2r 2 t σ 3 1 γ 2 If γ < 0, then regardless of the wealth share evoluton, gross returns on asset 1 are convergng to 0. A smlar result holds for asset 2. Thus n ths economy the trader wth correct expectatons vanshes and the relatve prces of the assets converge to one for all possble state probabltes.

16 15 In each of the economes above selecton for traders wth ncorrect belefs s drven by savngs rates. In these economes dscount factors, belefs and preferences nteract to determne savngs rates and portfolos. When traders have dentcal savngs rates, markets select for portfolo rules that maxmze the condtonal expected growth rate of wealth share as n [3]. In general, ths does not lead to selecton for correct belefs, but t does f some trader has logarthmc utlty. But when savngs rates are endogenous, even logarthmc utlty traders wth correct belefs can be drven out of the market by expected utlty maxmzng traders wth ncorrect belefs. Ths occurs because traders wth ncorrect belefs may choose to save so much that the savngs rate effect overcomes the consequences of less ft portfolos. References [1] Armen Alchan. Uncertanty, evoluton and economc theory. Journal of Poltcal Economy, 58: , [2] Davd Easley Blume, Lawrence and Margaret Bray. Introducton to the stablty of ratonal expectatons. Journal of Economc Theory, 262:313 17, August [3] Lawrence Blume and Davd Easley. Evoluton and market behavor. Journal of Economc Theory, 581:9 40, October [4] Lawrence Blume and Davd Easley. Optmalty and natural selecton n markets. unpublshed, Cornell Unversty, December [5] Lawrence Blume and Davd Easley. Ratonal expectatons and ratonal learnng. In Mukul Majumdar, edtor, Organzatons wth Incomplete Informaton: Essays n Economc Analyss, pages Cambrdge Unversty Press, Cambrdge, UK, [6] Davd A. Freedman. Tal probabltes for martngales. Annals of Probablty, 31: , [7] Mlton Fredman. Essays n Postve Economcs. Unversty of Chcago Press, Chcago, [8] Karl Hnderer. Foundatons of Non-Statonary Dynamc Programmng wth Dscrete Tme Parameter. Lecture Notes n Operatons Research and Mathematcal Systems. Sprnger- Verlag, Berln, [9] Alvaro Sandron. Do markets favor agents able to make accurate predctons. unpublshed, Northwestern Unversty, March 1999.