Optmal Expectaton Marku K. Brunnermeer rnceton Unverty Jonathan A. arker rnceton Unverty and NBER June 2003 Frt Draft: Augut 2002 Abtract Th paper ntroduce a tractable, tructural model of ubjectve belef. Forward-lookng agent care about expected future utlty flow, and hence have hgher current felcty f they beleve that better outcome are more lkely. On the other hand, expectaton that are baed toward optmm woren decon makng, leadng to poorer realzed outcome on average. Optmal expectaton balance thee force by maxmzng the lfetme well-beng of an agent. We apply our framework of optmal expectaton to three dfferent economc ettng. In a portfolo choce problem, agent overetmate the return on ther nvetment and may nvet n an aet wth negatve expected exce return f uffcently potvely kewed. In general equlbrum, agent pror belef are endogenouly heterogeneou, leadng to gamblng. Fnally, n a conumpton-avng problem wth tochatc ncome, agent are both overconfdent and overoptmtc, and conume more than mpled by ratonal belef early n lfe. Keyword: expectaton, heterogeneou belef, belef bae, conumpton, avng, portfolo choce, overconfdence, gamblng JEL Clafcaton: D, D8, E2, G, G2 For helpful comment, we thank Roland Bénabou, Andrew Capln, Lar Hanen, Davd Labon, Augutn Lander, Sendhl Mullanathan, and Larry Samuelon, a well a emnar partcpant at Unverty of Amterdam, UC Berkeley, Brkbeck, Unverty of Bonn, Carnege-Mellon, Unverty of Chcago, the Federal Reerve Board, Harvard Unverty, the Inttute for Advanced Study, London Bune School, London School of Economc, New York Unverty, Unverty of ennylvana, rnceton Unverty, Stanford Unverty, Tlburg Unverty, the Unverty of Wconn, Yale Unverty, and conference partcpant at the Summer Meetng of the Econometrc Socety, the NBER Behavoral Fnance Conference, and the Wetern Fnance Aocaton Annual Meetng. Both author acknowledge fnancal upport from the Natonal Scence Foundaton (SES 40-639 and SES 009-6076). arker alo thank the Sloan Foundaton and an NBER Agng and Health Economc Fellowhp through the Natonal Inttute on Agng (T32 AG0086). Department of Economc, Bendhem Center for Fnance, rnceton Unverty, rnceton, NJ 08544-02, e-mal: marku@prnceton.edu, http://www.prnceton.edu/ marku Department of Economc, Bendhem Center for Fnance, and Woodrow Wlon School, rnceton Unverty, rnceton, NJ 08544-03, e-mal: jparker@prnceton.edu, http://www.prnceton.edu/ jparker
Introducton Modern pychology vew human behavor a a complex nteracton of cogntve and emotonal repone to external tmul that ometme gve dyfunctonal outcome. Modern economc take a relatvely mple vew of human behavor a governed by unlmted cogntve ablty appled to a mall number of concrete goal and unencumbered by emoton. The central model of economc allow coherent analy of behavor and of economc polcy, but elmnate dyfunctonal outcome, and n partcular the poblty that houehold mght pertently err n attanng ther goal. One area n whch there ubtantal evdence that houehold do contently err n the aement of probablte. In partcular, agent often overetmate the probablty of good outcome, uch a ther ucce (Alpert and Raffa (982), Wenten (980), and Buehler, Grffn, and Ro (994)). We provde a tructural model of ubjectve belef n whch agent hold ncorrect but optmal belef. Thee optmal belef dffer from objectve belef n way that match many of the clam n the pychology lterature about rratonal behavor. Further, n the canoncal economc model that we tudy, thee belef lead to economc behavor that match oberved outcome that have puzzled the economc lterature baed on ratonal behavor and common pror. Our approach ha three man element. Frt, at any ntant, people care about current utlty flow and expected future utlty flow. We alo allow, although t not central to our man pont, that pat utlty flow alo nfluence current felcty. Whle t tandard that agent that care about expected future utlty plan for the future, forward-lookng agent have hgher current felcty f they are optmtc about the future. hrae lke antcpaton exceed realzaton are content wth th dea. Agent that care about expected future utlty flow are happer wth dtorted belef about the payoff of ther nvetment and/or the tochatc proce for future labor ncome. The econd crucal element of our model that uch optmm affect actual decon. Dtorted belef dtort acton. In th ene, agent are not chzophrenc. For example, an agent cannot derve utlty by optmtcally belevng that he wll be rch tomorrow, whle alo The common German phrae Vorfreude t de chönte Freude tranlate roughly to antcpaton exceed realzaton.
bang today conumpton-avng decon on ratonal belef about future ncome. How are thee force balanced? We aume that ubjectve belef maxmze the agent expected total well-beng, the expected dcounted um of felcty acro perod. Th thrd key element lead to a balance between the frt two the beneft of optmm and the cot of bang acton on dtorted expectaton. There are everal reaon why belef mght maxmze overall well-beng. Frt, n the long tradton of economc, we want to explore what optmal. Second, f nature (or parent) care about the happne of agent, t would chooe to endow them wth optmal expectaton. Th content wth chldren beng raed belevng they can do or be anythng, they are pecal, etc. Fnally, the centfc tet of the theory t performance, not t aumpton. So far, our approach help to explan heterogenou belef, gamblng, overconfdence, procratnaton, and ntertemporal preference reveral, all wthn one coherent and tractable model. We demontrate how thee utlty-ervng bae dtort belef and behavor. In general, belef are le ratonal when bae have lttle cot n realzed outcome and when bae have large beneft n term of expected future happne. Belef tend toward optmm tate wth greater utlty flow are perceved a more lkely. More pecfcally, we llutrate our theory of optmal expectaton ung three example. Frt, n a portfolo choce problem, agent overetmate the return of ther nvetment and underdverfy. Second, n general equlbrum, agent pror belef are endogenouly heterogeneou and agent gamble agant each other. The prce of the rky aet may dffer from that n an economy populated by agent wth ratonal belef. Thrd, n a conumpton-avng problem wth quadratc utlty and tochatc ncome, agent are overconfdent and overoptmtc; agent conume more than mpled by ratonal belef early n lfe. In addton, Brunnermeer and arker (2003) how n a dfferent economc ettng that agent wth optmal expectaton can exhbt ntertemporal preference reveral, a greater readne to accept commtment, regret, and a context effect n whch non-choen acton can affect utlty. Our model of belef dffer markedly from treatment of rk n economc. Whle early model n macroeconomc pecfy belef exogenouly a nave, myopc, or partally updated (e.g. Nerlove (958)), nce Muth (960, 96) andluca(976) nearly all reearch ha proceeded 2
under the ratonal expectaton aumpton that ubjectve and objectve belef concde. There are two man argument for th. Frt, the alternatve to ratonalty lack dcplne. But our model provde exactly uch dcplne for ubjectve belef by pecfyng an objectve for belef, that they maxmze lfetme well-beng. The econd argument that ratonal expectaton an optmal a f agent have the ncentve to hold ratonal belef (or act a f they do) becaue thee expectaton make the agent a well off a they can be. However, th ratonale for ratonal expectaton rele on an ncontency: agent care about the future but at the ame tme expectaton about the future do not affect current felcty. Our approach of optmal expectaton the outcome of an optmal a f argument that take nto account the fact that agent care n the preent about utlty flow that are expected n the future. Mot mcroeconomc model aume that agent hare common pror belef. Th Harany doctrne weaker than the aumpton of ratonal expectaton that all agent pror belef are equal to the objectve probablte governng equlbrum dynamc. But lke ratonal expectaton, the common pror aumpton qute retrctve and doe not allow agent to agree to dagree (Aumann (976)). Savage (954) provde axomatc foundaton for a more general theory n whch agent hold arbtrary pror belef, o agent can agree to dagree. But f belef can be arbtrary, theory provde lttle tructure or predctve power. Optmal expectaton provde dcplne to the tudy of ubjectve belef and heterogeneou pror. Framed n th way, optmal expectaton a theory of pror belef for Bayean ratonal agent. The key aumpton that agent derve current felcty from expectaton of future pleaure ha t root n the orgn of utltaranm. Detaled expoton on antcpatory utlty can be found n the work of Bentham, Hume, Böhm-Barwerk and other early economt. More recently, the temporal element of the utlty concept have re-emerged n reearch at the juncture of pychology and economc (Loewenten (987), Kahneman, Wakker, and Sarn (997), Kahneman (2000)), and have been ncorporated formally nto economc model n the form of belef-dependent utlty by Geanakoplo, earce, and Stacchett (989), Capln and Leahy (200), and Yarv (200). In partcular, Capln and Leahy (2000) how that compettve equlbra are genercally ntertemporally ub-optmal and o open the door for belef dtorton to ncreae well-beng. 3
Several paper n economc tudy related model n whch forward-lookng agent dtort belef. In partcular, Akerlof and Dcken (982) model agent a choong belef to mnmze ther dcomfort from fear of bad outcome. In a two-perod model, agent wth ratonal belef chooe an ndutry to work n, undertandng that n the econd perod they wll dtort ther belef about the hazard of ther work and perhap not nvet n afety technology. Lander (2000) tude a two-perod game n whch agent chooe a pror before recevng a gnal and ubequently takng an acton baed on ther updated belef. Unlke our approach, belef dynamc are not Bayean; common to our approach, agent tend to ave le and be optmtc about portfolo return. Fnally, Bénabou and Trole (2002) and Harbaugh (2002) analyze belef bae a reultng from conflctng multple elve that play ntra-peronal game contraned by elfreputaton. Whle our approach not drectly related to uch ettng, model of ntra-peronal game, bounded ratonalty, and ncomplete memory ugget mechanm for how houehold acheve optmal expectaton n the face of pobly contradctory data. ychologcal theore provde many channel through whch the human mnd able to hold belef ncontent wth the ratonal proceng of objectve data. Frt, ndvdual forget and remember event baed on aocatvene, rehearal, and alence. 2 To the extent that thee charactertc of memory ba the agent to remember better outcome and to perceve them a more lkely n the future, bae n belef are optmtc a n our optmal expectaton framework. Second, mot human behavor not baed on concou cognton but automatc, proceedonlynthelmbcytemandnotthecortex(barghandchartrand(999)). If automatc proceng optmtc, then the agent may naturally approach problem wth optmtc bae. However, the agent may alo chooe to apply cognton to dcplne belef bae when the take are large, a n our optmal expectaton framework. The tructure of the paper a follow. In Secton 2, we ntroduce the general optmal expectaton framework. Subequently, we ue the optmal expectaton framework to tudy behavor n three dfferent canoncal economc ettng. Secton 3 tude a two-perod two-aet portfolo choce problem and how that agent hold belef that are baed toward the belef that ther nvetment wll pay off well. Secton 4 how that n a two-agent economy of th type 2 See for example Mullanathan (2002). 4
wth no aggregate rk, optmal expectaton are heterogeneou and agent gamble agant one another. Secton 5 analyze a clacal conumpton-avng problem of an agent wth quadratc utlty recevng tochatc labor ncome over tme and how that the agent baed toward optmm and overconfdent, and o ave le than a ratonal agent. Secton 6 conclude. An appendx contan proof of all propoton. 2 The optmal expectaton framework To olve for optmal belef and the reultng acton of agent, we proceed n two tep. Frt, wedecrbetheproblemoftheagentgvenanarbtraryetofbelef. Atanypontntme, agent maxmze felcty, the preent dcounted value of expected flow utlte. Second, optmal expectaton are the et of belef that maxmze lfetme well-beng n the ntal perod. Lfetme well-beng the expected dcounted um of the agent felcte at each pont n tme, and o a functon of the agent belef and the acton thee belef nduce. 2. Optmzaton gven belef Conder a wde and canoncal cla of optmzaton problem. In each perod from to T, agent take ther belef a gven and chooe a vector of control varable, c t, and the mpled evoluton of a vector of tate varable, x t, to maxmze ther happne. We conder frt a world where the uncertanty can be decrbed by a fnte number of tate, S. 3 Let π t t denote the true probablty that tate t S realzed after tate htory t S t. Wedepartfromthe canoncal model n that agent are endowed wth ubjectve probablte that may not concde wth objectve probablte. Condtonal and uncondtonal ubjectve probablte are denoted by ˆπ t t and ˆπ ( t) repectvely, and atfy the bac properte of probablte (precely pecfed ubequently). The felcty of an agent at tme t depend on current utlty flow and ubjectve expected future utlty flow, V (x t ; t, {ˆπ}) T t =Êt τ=0 βτ u (x t+τ,c t+τ ), () 3 Appendx A defne optmal expectaton for the tuaton wth a contnuou tate pace. 5
where Êt the ubjectve expectaton operator, 0 <β<, andu (x t+τ, ) the flow utlty functon whch ncreang and concave n c t+τ. Crucally, felcty depend on the complete et of ubjectve condtonal belef, denoted {ˆπ}. If β were zero, there would be no forwardlookng behavor. Whle t not crucal for our analy, we allow the poblty that felcty addtonally depend on pat utlty flow, whch we capture by the functon M x t, c t = t τ= δτ u (x t τ,c t τ ) where 0 δ β and x t and c t denote the htore of x and c repectvely. In the pecal cae of δ =0the agent ha no experenced utlty and current felcty collape to equaton (). The agent problem tandard: at each tme t, the agent chooe control varable to maxmze felcty ubject to the evoluton of the tate varable, the ntal level of x t and termnal condton on x T +.Formally, ubject to V (x t ; t, {ˆπ}) =max {c t } X β τ u (x t+τ,c t+τ ) Ê t T t τ=0 x t+ = g (x t,c t, t+ ) (2) h (x T + ) 0 (3) where the agent take x t a gven, h ( ) gve the endpont condton, and g ( ) gve the evoluton of the tate varable and contnuou and dfferentable n x and c. Denotng the optmal choce of the control a {c ( t, {ˆπ})}, the nduced path of the tate varable a {x ( t, {ˆπ})}, and the correpondng htore a c ( t, {ˆπ}) and x ( t, {ˆπ}), wecan wrte V and M n a recurve formulaton a: V (x t ; t, {ˆπ}) = max u (x t,c t )+β X ˆπ ( t+ t) V (g (x t,c t, t+ ); t+, t, {ˆπ}) (4) c t M x t, c t+ S ubject to (2) and (3) t = δu x t,c t + δm x M = 0, V T + =0 t 2, t 2 c Where not ambguou, we collape notaton a c t for c ( t, {ˆπ}), V t for felcty along the optmal path, V (x t ; t, {ˆπ}). 6 for V (x t ; t, {ˆπ}) and V t
So far we have focued on the optmzaton problem of a ngle agent. In a compettve economy, each agent face th maxmzaton problem takng a gven h belef and the tochatc proce of payoff-relevant aggregate varable, and market clear. Belef may dffer acro agent. Specfcally, x t nclude the payoff-relevant varable that the agent take a gven, and o reflect the acton of all other agent n the economy. The equlbrum choce of control varable for each agent mple an equlbrum allocaton {x T,c T },where{c T } wthout a upercrpt denote all agent control varable along any poble path of the event tree. In um, the problem reman tandard, wth the excepton that agent pror belef may be heterogeneou. 2.2 Optmal belef Subjectve belef are a complete et of condtonal probablte for each branch after any htory of the event tree, ˆπ t t ª. We requre that ubjectve probablte atfy four properte. Aumpton (Retrcton on probablte) () ˆπ t S t t = () ˆπ t t 0 () ˆπ ( 0 t) =ˆπ 0 0 t ˆπ 0 0 t t t 2 ˆπ ( 0 ) (v) ˆπ t t =0f π t t =0. Aumpton () mply that probablte um to one. Aumpton () () mply that the law of terated expectaton hold for ubjectve probablte. Aumpton (v) mple that n order to beleve that omethng poble, t mut be poble. You cannot beleve you wll wn the lottery unle you buy a lottery tcket. M t We further conder the cla of problem for whch a oluton ext and for whch V t are le than nfnte. Whle the condton to enure th are tandard, we requre that thee properte hold for all poble ubjectve belef. and Aumpton 2 (Condton on agent problem) () E [V (x ( t, {ˆπ}); t, {ˆπ})] < for all t and for all {ˆπ} atfyng Aumpton () E M x t, {ˆπ},c t, {ˆπ} < for all t and for all {ˆπ} atfyng Aumpton. 7
Optmal expectaton are the ubjectve probablte that maxmze expected total or lfetme well-beng, W, gven that agent are optmzng. 4 Followng Capln and Leahy (2000), we defne h T lfetme well-beng a the dcounted um of felcty of the agent over t lfe, E t= βt (M t + V t ). Gven agent optmzaton, felcty n perod t M t through antcpaton of future flow utlty n Vt behavor, x t. +Vt, o that belef mpact felcty drectly and ndrectly through ther effect on agent Defnton Optmal expectaton (OE) are a et of ubjectve probablte ˆπ OE ª t t that maxmze " X T W := E t= β t M x t, {ˆπ}, c t, {ˆπ} + V (x ( t, {ˆπ}); t, {ˆπ}) # (5) ubject to the four retrcton on ubjectve probablte (Aumpton ). Optmal expectaton ext f c OE ( t) and x OE ( t) are contnuou n probablte ˆπ t t that atfy Aumpton for all t and t, where for notatonal mplcty c OE ( t) :=c t, ˆπ OEª and x OE ( t) :=x t, ˆπ OEª. Th follow from the contnuty of V t and M t n probablte and control, Aumpton 2, and the compactne of probablty pace. For le regular problem, a for ratonal expectaton equlbra, optmal expectaton may or may not ext. A to unquene, optmal belef need not be unque, a wll be clear from the ubequent ue of th concept. In an economy wth multple agent, each agent belef maxmze equaton (5), where the tate varable x and control varable c are ndexed by, takng the belef and acton of the other agent a gven. Defnton 2 A compettve optmal expectaton equlbrum (OEE) a et of belef for each agent and an allocaton uch that () each agent ha optmal expectaton, takng a gven the tochatc proce for aggregate varable; () each agent olve equaton (4) at each t, takng a gven h belef and the tochatc proce 4 A ueful alternatve for nfnte horzon problem that belef maxmze average ntead of total felcty. 8
for aggregate varable; () market clear. Intutvely, an optmal expectaton equlbrum (OEE) cont of a et of belef for each agent and the correpondng equlbrum allocaton nduced by optmzaton gven thee belef. The optmal belef of each agent take a gven the aggregate dynamc, and the optmal acton take a gven the perceved aggregate dynamc. 2.3 Dcuon Before proceedng to the applcaton of optmal expectaton, t worth emphazng everal pont. Frt, becaue probablte, ˆπ OE t t, are choen once and forever, the law of terated expectaton hold wth repect to the ubjectve probablty meaure and tandard dynamc programmng can be ued to olve the agent optmzaton problem. An alternatve nterpretaton of optmal condtonal probablte ntead that the agent endowed wth optmal pror over the tate pace, ˆπ OE ( T ), and learn and update over tme accordng to Baye rule. 5 Thu agent are completely Bayean ratonal gven what they know about the economc envronment. Second, optmal expectaton are thoe that maxmze lfetme well-beng. The argument that tradtonally made for the aumpton of ratonal belef that uch belef lead agent to the bet outcome correct only f one aume that expected future utlty flow do not affect preent felcty. Th a omewhat chzophrenc vew: one part of the agent make plan that trade off preent and expected future utlty flow, whle another part of the agent actually enjoy utl but only from preent conumpton. 6 Under the Jevonan vew that an agent who care about the future ha felcty that depend on expectaton about the future, optmal expectaton gve agent the hghet lfetme utlty level. 5 The nterpretaton of the problem n term of optmal pror requre that one pecfy agent belef followng zero ubjectve probablty event, tuaton n whch Baye rule provde no retrcton. 6 See Loewenten (987) and the dcuon of the Samuelonan and Jevonan vew of utlty n Capln and Leahy (2000). 9
To recat th pont, we can ak what objectve functon for belef would make ratonal expectaton optmal. Th the cae f the objectve functon for belef omt antcpatory or memory utlty, o that W = E T t= βt u (x t,c t ). Th mple that ether agent do not care about the future (V doe not contan future u) or belef do not maxmze lfetme well-beng (W defned over u ntead of V ). Becaue we vew th a mplauble, we retrct attenton to objectve functon for belef defned over V. Thrd, th dcuon alo make clear why lfetme well-beng, W, ue the objectve expectaton operator. Optmal belef are not thoe that maxmze the agent happne only n the tate that the agent vew a mot lkely. Intead, optmal belef maxmze the happne of the agent on average, acro repeated realzaton of uncertanty. The objectve expectaton capture th nce the actual unfoldng of uncertanty over the agent lfe determned by objectve probablte. Fourth, n all the example of th paper, the key reaon for belef dtorton that current felcty depend on antcpated future utlty flow. We enure th by tudyng tuaton n whch the objectve functon for belef evaluated for ratonal belef dentcal to the objectve functon of the agent. Th the cae when δ = β,whchwerefertoapreference contency. Wth preference contency, lfetme well-beng maxmzed by the acton of an agent wth ratonal expectaton. Wthout preference contency, there may be an addtonal ncentve for the dtorton of belef nce an agent rankng of utlty flow acro perod not tmenvarant (Capln and Leahy (2000)). Th dagreement over optmal acton n dfferent perod would gve an addtonal ncentve for belef to be dtorted to ncreae lfetme well-beng. That ad, for mot of our reult we do not need to pecfy the value of δ. In partcular, th allow for the cae of preference contency and the cae n whch δ =0. For the latter cae, the analy mplfe nce all experence utlty term M t vanh. Ffth, optmal expectaton could be derved from generalzed objectve functon. In partcular, an earler veron of th paper condered the poblty that the rate at whch future felcty dcounted n the objectve functon for belef dffered from the rate at whch the agent dcounted future utlty flow. Sxth, optmal ubjectve probablte are choen wthout any drect relaton to realty. Th 0
frctonle world provde nght nto the behavor generated by the ncentve to look forward wth optmm when belef dtorton lmted by the cot of poor outcome. In fact, t may be that belef cannot be dtorted far from realty for addtonal reaon. At ome cot n term of tractablty, the frctonle model could be extended to nclude contrant that penalze larger dtorton from realty. Belef would then bear ome relaton to realty even n crcumtance n whch there are no cot aocated wth behavor caued by dtorted belef. Seventh, th model alo extreme n that ˆπ OE t t are agned at every node, o that belef dtorton can vary gnfcantly acro tate wth mlar outcome. Agan at ome cot n term of tractablty, one could requre that belef dtorton be retrcted to be mooth or le on a coarer partton of the probablty pace. For example, one mght requre that belef bae be mlar for tate of the world that lead to mlar payout for a gven aet. Alternatvely, one could retrct the et that optmal belef are choen over to be a et of parmonou model of the envronment. For example, we mght requre that the agent beleve that ther ncome proceomefrt-order Markov proce rather than allow belef dtorton to be completely htory dependent. Whle uch retrcton are omewhat appealng, t dffcult to know how to dcplne the choce of uch retrcton. 7 Fnally, one mght be concerned that agent wth optmal expectaton mght be drven to extncton by agent wth ratonal belef. But evolutonary argument need not favor ratonal expectaton. Snce optmal expectaton repond to the cot of mtake, agent wth optmal expectaton are hard to explot. And many economc envronment favor agent who take on more rk (DeLong, Shlefer, Summer, and Waldmann (990)). Fnally, content wth our choce of W, there a bologcal lnk between happne and better health (Taylor and Brown (988)). 7 If the agent were aware that h pror/model choen from a et of parmonou model, then he mght queton thee belef. In th cae, t would make ene to mpoe the addtonal retrcton that only pror can be choen for whch the agent cannot detect the mpecfcaton, an approach beng purued n the lterature on robut control. By not retrctng the choce et over pror we avod thee complcaton.
3 ortfolo choce: optmm and gamblng In th ecton we conder a two-perod nvetment problem n whch an agent chooe between aet n the frt perod of lfe and conume the payoff of the portfolo n the econd perod of lfe. We how that agent are optmtc about the payout of ther own nvetment and do not hold perfectly dverfed portfolo. Th matche tylzed fact about nvetor behavor. The ubequent ecton place two of thee agent nto a general equlbrum model wth no aggregate rk, and how that thee agent dagree about the return of aet. 3. ortfolo choce gven belef There are two perod and two aet. In perod one, the agent allocate h unt endowment between a rk-free aet wth gro return R and a rky aet wth gro return R + Z (Z the exce return of the rky aet over the rk-free rate). In perod two, the agent conume the payoff from h frt-perod nvetment. The agent chooe h portfolo hare to nvet n the rky aet, w, to maxmze expected utlty: V =max w.t. βê [u (c)] c = R + wz c 0 n all tate where u ( ) the utlty functon over conumpton, u 0 > 0, u 00 < 0 and atfyng the Inada condton. The econd contrant et by the market. Snce conumpton cannot be negatve, the contrant follow from the market requrng the agent to be able to meet h payment oblgaton n all future tate. Uncertanty characterzed by S tate, wth ex pot exce return Z and probablte π > 0 for =,...,S. Let the tate be ordered o that the larger the tate, the larger the payoff, Z + >Z,and R <Z < 0 <Z S <R, Z 6= Z 0 for 6= 0. Belef are gven by {ˆπ } S = atfyng Aumpton. Notng that the econd contrant can only bnd for the hghet or lowet payoff tate, the 2
agent problem can be wrtten a a Lagrangan wth multpler λ and λ S, max β X S ˆπ u (R + wz ) λ (R + wz ) λ S (R + wz S ). w = The neceary condton for an optmal w are SX 0 = ˆπ u 0 (R + w Z ) Z λ Z λ S Z S, = 0 = λ (R + w Z ), 0 = λ S (R + w Z S ). It turn out that optmal belef never lead the agent to chooe R + w Z =0. To ee th, uppoe that R + w Z =0for ome and conder an nfntemal change n probablty that reult n an ncreae of conumpton n th tate. By the Inada condton u 0 (0) =, th caue an nfnte margnal ncreae n lfetme well-beng. Thu, optmal expectaton mply R +w Z 6=0for any. By complementary lackne, λ =0for all, and the optmal portfolo unquely determned by SX 0= ˆπ u 0 (R + w Z ) Z w ({ˆπ}). (6) = 3.2 Optmal belef Optmal belef are a et of ˆπ for =,...,S wth ˆπ S = S = ˆπ that maxmze total well-beng, the expected dcounted um of felcte n perod and 2. Inperod, the agent felcty the ubjectvely expected (antcpated) utlty flow n the future perod, dcounted by β; n perod 2, the agent felcty the utlty flow from actual conumpton. max ˆπ max ˆπ max ˆπ E [V + βv2 ] h E βê [u (c)] + βu(c) β SX ˆπ u (R + w ({ˆπ}) Z )+β = SX π u (R + w ({ˆπ}) Z ) where w ({ˆπ}) gvenmplctlybyequaton(6). Thefrt-order condton for the choce of ˆπ are 0=βu 0 βu S + β SX = ˆπ OE u 0 (R + w dw Z ) Z dˆπ 0 3 = + β SX π u 0 (R + w dw Z ) Z. dˆπ 0 =
By the envelope condton, mall change n portfolo choce from the optmum caued by mall change n ubjectve probablte lead to no change n expected utlty, o that th condton mplfe to β (u S u 0)=β SX π u 0 (R + w dw Z ) Z. (7) dˆπ 0 = The left-hand de the margnal gan n dream utlty at t from ncreang ˆπ S at the expene of ˆπ 0 and alway potve; the rght-hand de the margnal lo n expected utlty n t + from the reultant change n the portfolo hare of the rky aet. In equlbrum, the gan n dream utlty balance the cot of dtortng actual behavor. Let w RE denote the optmal portfolo choce for ratonal belef. The followng propoton, proved n the appendx, tate that the agent wth optmal expectaton optmtc about the payout of h portfolo. Further, the agent wth optmal expectaton ether take an oppote poton relatve to the agent wth ratonal belef or more aggreve nvetng even more f the ratonal agent nvet, or hortng more f the ratonal agent hort. ropoton (Exce rk takng due to optmm) SX () f w OE > 0, (ˆπ π ) u 0 R + w OE SX Z Z > 0; fw OE < 0, (ˆπ π ) u 0 R + w OE Z Z < 0. = = () f E [Z] > 0, thenw RE > 0, w OE >w RE or w OE < 0; f E [Z] < 0, thenw RE < 0, w OE <w RE or w OE > 0; fe [Z] =0,thenw RE =0and w OE 6=0. The frt part of the propoton tate that agent wth optmal expectaton on average hold belef that are baed upward for tate n whch ther choen portfolo payout hgh and baed downward for tate n whch ther portfolo payout low. To ee th, note that u 0 > 0 for all, andz potve for large and negatve for mall. For w OE > 0, optmal expectaton on average ba up the ubjectve probablty for larger or potve exce return tate at the expene of maller or negatve exce return tate. The econd part of the propoton characterze behavor. Conder frt the tuaton when E [Z] =0. The ratonal agent chooe w RE =0nce the expected return on the rky aet theameatherk-freeaetandtherkyaetrky. Butadevatonofbeleffrom 4
ratonal belef can ncreae lfetme well-beng. A mall devaton lead the agent to chooe w 6= 0whch mple that conumpton no longer perfectly moothed acro future tate. The agent now beleve that he holdng (hortng) an aet wth potve (negatve) expected payoff. The cot of mperfect conumpton moothng are domnated by the gan n antcpated future utlty. Thu ˆπ OE 6= π and w OE 6= w RE =0. An mplcaton that, from the perpectve of objectve probablte, agent wth optmal expectaton are underdverfed. That, relatve to w OE, a portfolo wth the ame objectve expected return and le objectve rk avalable, nce E R + w OE Z = E R + w RE Z but Var R + w OE Z >Var R + w RE Z. For E [Z] > 0, the econd part of the propoton tate that the houehold ether nvet more than the ratonal agent n the rky aet or hort the rky aet, and vce vera for E [Z] < 0. Why would the agent take a poton n the oppote drecton to the ratonal agent, when th mple that he takng a negatve expected payoff gamble? Th occur when antcpatory utlty n the contraran poton uffcently large. For many utlty functon, th the cae when the aet ha the properte mlar to a lottery tcket, that when the aet kewed n the oppote drecton of the mean payoff. To llutrate th pont, conder a world wth two tate and an aet wth negatve expected exce payoff, E [Z] =: µ Z < 0. We pecfy the payoff Z and Z 2, uch that, a we vary probablte the mean and varance, σ 2 Z, tay contant, but kewne ncreae n π. State robablty Exce ayoff π Z = µ Z σ Z q π π 2 π Z 2 = µ Z + σ Z q π π When π large, the aet mlar to a lottery: the aet yeld a mall negatve return wth hgh probablty and a large potve return wth low probablty. ropoton 2 For unbounded utlty functon, there ext a π uch that for all >π >π () ˆπ <π and () w OE > 0 even though E [Z] < 0. For the agent hortng the aet, when π cloe to unty, ˆπ π near zero ubjectve belef are necearly near ratonal belef and w ({ˆπ}) near w RE. However, n th cae, f 5
the agent ntead optmtc about the payoff of the rky aet, ˆπ <π, then he can nvet n the aet and dream about the aet payng off well. In fact, for π near unty, ˆπ OE <π and w OE potve. Th type of behavor buyng tochatc aet wth negatve expected return and potve kewne wdely oberved n gamblng and bettng. 4 General equlbrum: endogenou heterogenou belef In th ecton, we conder an exchange economy wth no aggregate rk and derve endogenou heterogeneou belef. Further, n our model agent chooe to hold doyncratc rk and gamble agant one another n equlbrum when perfect conumpton nurance poble. Thee feature match tylzed fact about aet market. eople dagree about aet return, gamble, and do not perfectly nure conumpton. Fnally, the prce of the rky aet may dffer from that n an economy populated by agent wth ratonal belef. The economy cont of two agent wth the ame charactertc and facng the ame nvetment problem a n the prevou ecton. There are two tate n the econd perod and two aet (or tree ), denoted b (bond) and e (equty). Aet b pay unt of conumpton n both tate; we normalze the return on th tree to (R equal unty). Aet e return +Z and the ex pot return on aet e +Z = +ε where ndexe tate. We aume <ε <ε 2. Agent ntally endowed wth Xb bond and X e equty hare. There no aggregate uncertanty o that aet e n zero net upply. Aggregate conumpton n each tate thu the ame, 2 = X b = X b = C = C 2. Agent problem to take h belef, ˆπ ª, and the prce of equty,, a gven and chooe her portfolo to maxmze expected utlty, gven ntal wealth max β X w ˆπ u A +w Z A = X b + X e. The frt-order condton for portfolo choce are 0= X ˆπ u 0 c (( + ε ) ). 6
In aggregate, belef maxmze the lfetme well-beng for each agent max ˆπ X βˆπ u c + βπ u c ubject to the retrcton on probablte (Aumpton ), the budget contrant (the defnton of conumpton), and the agent frt-order condton for portfolo choce. The Lagrangan for each agent L = X h ³ X βˆπ u b + Xe ³ ³ +w +ε e λ " X # " X ˆπ µ + βπ u X b + X e +w +ε µ X ˆπ u 0 b + Xe µ µ # +w +ε (( + ε ) ) The equlbrum for the economy a a whole characterzed by the frt-order condton for each agent w : 0 = X µ " X βˆπ OE, ˆπ OE, ˆπ : 0 = βu c OE, λ : : 0 = λ " X + βπ u 0 c OE, µ +ε OE u 00 c OE, λ µ u 0 c OE, ˆπ OE, # (8a) µ # +ε 2 OE OE for =, 2 µ +ε OE OE for =, 2, =, 2 (8b) for =, 2 µ : 0 = µ " X ˆπ OE, u 0 c OE, ( + ε ) OE# for =, 2 (8c) and the aggregate reource contrant OE X X ew OE, = X X b woe,, (9) where c OE, = Xb + OE Xe +w OE, +ε. OE Before characterzng the equlbrum, we defne gamblng. Let x e be agent equlbrum holdng of equty, x e = w (Xb +X e). Agent are gamblng agant each other n an equlbrum ( ) f x e 6= x RE, e for =, 2. 7
Th ay that the amount of equty held by each agent dfferent n the ( ) equlbrum than n the ratonal expectaton equlbrum. Note that gamblng not mpled by a dfference between the equlbrum prce of equty under ratonal and optmal expectaton, OE 6= RE. Gamblng reult from dagreement about probablte. ut dfferently, at the optmal expectaton equlbrum we have for each agent X If agent are not gamblng, then X ˆπ OE, u 0 c OE, µ +ε OE =0. π u 0 c OE, µ +ε RE =0. Thu f optmal belef dffer from objectve belef, the prce of equty wll dffer n the ratonal expectaton and optmal expectaton equlbra even wthout gamblng. Mprcng doe not mply gamblng. We now tate our propoton. ropoton 3 (Gamblng wthout aggregate rk) () w RE, =0, x RE, e =0, () agent gamble, x OE, e 6= x RE, e 6= x OE,2 e, () n any equlbrum ˆπ OE, >π, ˆπ OE, 2 <π 2,w OE, < 0, c OE, >c OE, 2,andˆπ OE, 2 >π 2, ˆπ OE, <π, w OE, > 0, c OE, 2 >c OE, for =or 2. The frt pont tate that agent n the ratonal expectaton equlbrum perfectly nure ther conumpton by tradng o that nether agent hold the rky aet. The econd pont follow from the ame logc a n the partal equlbrum ecton. Agent have a lo from a mall amount of bettng agant one another, whch domnated by the gan n dream utlty from gamblng. The fnal pont emphaze the ource of the gamblng. Each agent beleve a dfferent tate more lkely than t really, and they trade o that each of them realze hgher conumpton n the tate they vew a unrealtcally lkely. Thu, a feature of the optmal expectaton equlbrum endogenou heterogenety n belef. Whle th model mple multple equlbrum, t natural to thnk that agent chooe to be optmtc about the tate n whch ther ntal endowment pay off more. Th equlbrum 8
mnmze tradng cot. Th content wth people belevng n the economc performance of ther own compane or countre. Recent tude of penon behavor fnd nvetor choong to hold a larger hare of ther wealth n the equty of ther employer than ratonal model ugget optmal. Smlarly, nvetor have a home ba, they hold a larger hare of ther wealth n the equty of ther own country than ratonal model ugget optmal. 8 5 Conumpton and avng over tme: underavng and overconfdence Th ecton conder the behavor of an agent wth optmal expectaton n a mult-perod conumpton-avng problem wth tochatc ncome. We how that the agent wth quadratc utlty overetmate the mean of future ncome and underetmate the uncertanty aocated wth future ncome. That, the agent both unrealtcally optmtc and overconfdent. Th content wth urvey evdence that how that growth rate of expected conumpton greater than that of actual conumpton. 5. Conumpton gven belef In each perod t =,...,T, the agent chooe conumpton and avng to maxmze the expected preent dcounted value of utlty flow from conumpton ubject to a budget contrant. ³ " o T t # X V A t ; ȳ t, nˆπ = maxê β τ u (c t+τ ) ȳ {c t t}.t. τ=0 TX t R τ (c t+τ y t+τ )=A t τ=0 u (c t+τ ) = ac t+τ b 2 c2 t+τ where ntal wealth A =0, a, b > 0 and βr =. The only uncertanty over ncome, y t. y t contnuouly dtrbuted, ha upport y, ȳ = Y where 0 <y< ȳ< bt a, ndependent over tme o that Π ³y t ȳ t = Π (y t ),anddπ(y t ) > 0 for all y Y. Subjectve probablte are 8 For urvey on overnvetng n the equty of one company and one country, ee oterba (2003) and Lew (999), repectvely. 9
denoted by ˆΠ ³y t ȳ t and do not have to be ndependently dtrbuted over tme. Appendx A tate the retrcton on probablte and objectve functon for a contnuou tate pace. T, Aumng an nteror oluton, the neceary condton for an optmum are, for t =to Z 0 = u 0 (c t ) βr o dv ³A t ; ȳ t, nˆπ Z = βr da t y t+ Y y t+ Y o dv ³A t+ ; ȳ t+, nˆπ dˆπ ³y t+ ȳ t da t+ o dv ³A t+ ; ȳ t+, nˆπ da t+ dˆπ ³y t+ ȳ t. Combnng thee condton and the aumpton of quadratc utlty gve the Hall random walk reult for conumpton but for ubjectve belef c t = hc Ê t+ ȳ t. (0) Subttutng back nto the budget contrant gve the optmal conumpton rule à TX t c t = A t + y t + R R (T t) τ= h! R τ Ê y t+τ ȳ t. () Optmal conumpton depend on ubjectve expectaton of future ncome and the htory of ncome realzaton through A t. Becaue quadratc utlty exhbt certanty equvalence from the perpectve of the agent, the problem mplfe gnfcantly. Gven the ubjectve expectaton of future ncome, the ubjectve varance (and hgher moment) of the ncome proce are rrelevant for the optmal conumpton-avng choce of the agent. 5.2 Optmal belef We want to chooe ˆΠ ³y t ȳ t for all ȳ t to maxmze lfetme well-beng ubject to the probablty condton and the agent optmal behavor gven belef. Wth quadratc utlty, the element of the objectve functon for belef are M t = V t = Xt µ δ r ac t r b c 2 2 t r r= T t X r=0 β r Ê ac t+r b c 2 2 t+r ȳt. 20
Snce the objectve concave n future conumpton and nce the agent behavor depend only on the ubjectve certanty-equvalent of future ncome, optmal belef mnmze ubjectve uncertanty. Thu, future ncome optmally perceved a certan, whch an extreme form of overconfdence. We ncorporate optmal behavor drectly nto the value functon and characterze conumpton choce mpled by optmal belef, c OE ª t.optmalbelef, nˆπoeo,mplementthee conumpton choce gven optmal behavor on the part of the agent. In takng th approach, we are aumng that the optmal choce of conumpton and thu hy Ê t+τ ȳ t doe not requre volaton of the aumpton on probablty, whch can be checked. 9 To proceed, optmal behavor ummarzed by the agent Euler equaton. Ung the Euler h equaton and the fact that ubjectve certanty mple Ê u c t+τ ȳt = u ³Ê hc t+τ ȳ t,the value functon over future conumpton become V t = TX t β r u ³Ê hc t+r ȳ t r=0 TX t = u (c t ) β r. Subjectve expectaton are choen to yeld the path of {c t } that maxmze u (c ) T τ= βτ + βu(c {z } ) δ + βu(c {z } 2) T τ= βτ + β 2 u (c {z } ) δ 2 + u (c 2) δ + {z } V E M2 V2 M3 β 2 u (c 3) T 2 τ= {z βτ +... + β T T τ= } δt τ u (c τ ) + β T u (c T ) {z } {z } V3 MT VT " X T # = E t= ψ t u (c t ) r=0 ³ ubject to the budget contrant and where ψ t = β t + T t τ= (βτ +(βδ) τ ). If there were no memory utlty (δ =0), and the objectve for belef gnored antcpatory utlty, then ψ t = β t. In th cae, the optmal conumpton path tandard, and belef would be ratonal. 9 If the upport of y t mall, belef dtorton may be contraned by the range of poble ncome realzaton. n o To ncorporate thee contrant drectly, one ntead replace c ³ȳ t, ˆΠ OE ung equaton () andearche for optmal Ê hy t+τ ȳ t whle mpong the contrant mpoed by Aumpton (v). (2) 2
Under optmal expectaton, expected conumpton growth gven by the frt-order condton u 0 Z c OE ψ t = t+τ R τ ψ t ȳ t+τ ȳ t τ Y u 0 c OE t+τ dπ ³ȳ t+τ ȳ t, whch mple that, for any equence of ncome realzaton, ³ c OE ȳ t = a b ψ ³ t+τ R τ a h ³ ψ t b E c OE ȳ t+τ ȳ t. (3) Level conumpton recovered by ubttutng nto the budget contrant after takng objectve expectaton. Gven th characterzaton of optmal behavor, agent are optmtc at every tme and tate. Defne human wealth a the preent value of current and future labor ncome at t, H t = T t τ=0 R τ y t+τ. ropoton 4 (Overconumpton due to optmm) For all t {,...,T }, () hh Ê t+ ȳ t >EhÊ hh t+ ȳ t+ ȳ t ; h () c OE t >E c OE t+ ȳ ; h th () Ê c OE t+ ȳ >E c OE t t+ ȳ. t The frt pont of the propoton tate that agent overetmate ther preent dcounted value of labor ncome and on average reve ther belef downward between t and t +. Th downward revon of expected lfetme wealth can come about drectly due to y t+ beng on average le than expected, or due to new that the realzed y t+ brng on average about expectaton of future ncome. The econd pont tate that conumpton on average fall between t and t +. Becaue on average the agent reve down expected future ncome, on average conumpton fall over tme. The proof follow drectly from the expected change n conumptongvenbyequaton(3) and notng that a b coe t ³ȳ t > 0 and ψ t+ ψ R<. Fnally, the t optmal ubjectve expectaton of future conumpton exceed the ratonal expectaton of future conumpton. Th optmm. art () follow from part () and equaton (0). In um, houehold are unrealtcally optmtc, and, n each perod, are on average urpred that ther ncome are lower than they expected, and o, on average, houehold conumpton declne over tme. 22
c(t) + + + + + + + Ratonal expectaton profle ( ) Optmal expectaton profle (+) + + expected future conumpton path 2 3 4 5 T- T t Fgure : Average Lfe-cycle Conumpton rofle Fgure ummarze thee reult. The agent tart lfe optmtc about future ncome. At each pont n tme the agent expect that on average conumpton wll reman at the ame level. Over tme, the agent learn on average that ncome le than he expected, and conumpton typcally declne over the lfe. Th optmm matche urvey evdence on dered and actual lfe-cycle conumpton profle. Barky, Juter, Kmball, and Shapro (997) fnd that houehold would chooe upward lopng conumpton profle. But urvey dataet on actual conumpton reveal that houehold have downward lopng or flat conumpton profle (Gourncha and arker (2002), Attanao (999)). In our model, houehold expect and plan to have contant margnal utlty nce βr =0. ψ However, on average margnal utlty re at the age-pecfc rate t ψ > 0. Thu, n the t+ model, the dered rate of ncreae of conumpton exceed the average rate of ncreae, a n the real world. Alo matchng oberved houehold conumpton behavor, the model produce average lfe-cycle conumpton profle that are concave conumpton fall fater (or re more lowly) later n lfe. In general, n conumpton-avng problem, the relatve curvature of utlty and margnal utlty determne what belef are optmal. Uncertanty about the future enter the objectve for belef both through the expected future level of utlty and through the agent behavor whch 23
depend on expected future margnal utlty. For utlty functon wth decreang abolute rk averon, greater ubjectve uncertanty lead to greater precautonary avng through the curvature n margnal utlty. Th ha ome beneft n term of le dtorton of conumpton. In uch cae optmal belef may cont of a large potve ba for both expected ncome and t varance. We conclude th ecton by ung our conumpton-avng problem to make three pont about the dynamc choce of agent wth optmal expectaton. Frt, gven that n expectaton the conumpton of the agent alway declnng, the cot of optmm early n lfe could be extreme for long-lved agent. But, llutratng a general pont, optmal expectaton depend on the horzon n a way that mtgate thee poble cot. The behavor of an agent wth a long horzon cloe to that of an agent wth ratonal expectaton. For T large but fnte, an agent wth optmal expectaton conume a mall amount more for mot h lfe, leadng to a gnfcant declne n conumpton at the end of lfe. A the horzon become nfnte, at any fxed age, the agent wth optmal belef chooe a level of conumpton arbtrarly cloe to that of the agent wth ratonal expectaton and the ubjectve expectaton of future labor ncome become nearly ratonal. Formally, for any t, at, c t ³ȳ t c RE t ³ȳ t, h Ê hh t+ ȳ t E hh t+ ȳ t, c t ³ȳ t E c t+ ³ȳ t+ ȳ t. 0 Belef become more ratonal a the take become larger. Second, the agent wth optmal expectaton may chooe not to nure future ncome gven an objectvely far nurance contract. Formally, let the agent face an addtonal bnary decon n perod : whether or not to exchange all current and future ncome for B = E [H y ]. A ratonal agent would alway take th contract, whle the agent wth optmal expectaton may chooe not to nure conumpton. Interetngly, nce belef affect whether the agent nure or not, the addton of the poblty of nurance may change what belef are optmal. Optmal expectaton are ether the belef that maxmze lfetme well-beng condtonal 0 It can be een that c ³ȳ t c RE ³ȳ t by repeatedly ubttutng the Euler equaton (3) nto the budget contrant to olve for c ³ȳ t and notng that ψ t+τ ψ R τ a T and ψ t t ψ R τ <ψ t+τ t for τ,t. Th together wth equaton () mplythatê hh 2 ȳ E hh 2 ȳ. Agan ung the fact that ψ t+τ R τ a ψ t T, equaton (3) mple that for fnte t, c t ³ȳ t c RE t ³ȳ t o that the frt two reult alo hold for any fnte t (not jut t = ). 24
on nducng the agent to accept the nurance, or the belef that maxmze lfetme well-beng condtonal on nducng the agent to reject the nurance. The former are the optmal expectaton from ropoton 4. Thee belef are optmal for the problem wthout the contrant, and the agent reject the nurance becaue both ncome tream are perceved a certan and Ê [H y ] >E[H y ]=B. Lfetme well-beng n th cae from equaton (2). TX ψ t E [u (c t ) y ] t= morerealtcaboutthencomeproce. The latter, optmal belef condtonal on acceptng the nurance, are Theactualbelefarerrelevantforlfetmewellbeng provded that the agent beleve that Ê [H y ] mall enough and/or the proce for {y} uncertan enough that he accept the nurance. Lfetme well-beng n th cae where c FI (y )= R R R T E TX ψ t E u c FI (y ) y t= h T t= R t y t y. Rk determne whch expectaton are optmal. Lfetme well-beng decreae n objectve ncome rk when the agent reject the nurance, whle t nvarant to rk f he accept the nurance. If objectve ncome rk mall, then the cot of dtorted belef varable future conumpton mall, and optmal expectaton are optmtc. The agent dream about future ncome and reject conumpton nurance. If objectve ncome rk large, optmal expectaton are more ratonal and nduce the agent to nure h future ncome. Thrd, at the tart of lfe, the agent facng the problem wth the opton to nure ncome may have a lower level of felcty than the agent facng the problem wthout th opton. Informally, we mght thnk of an agent approachng ther lfe blthely optmtc about ther future. Gven no choce of nurance, th ndeed optmal. However, when faced wth the opportunty to nure and n an envronment wth large amount of ncome rk, the agent conder ther lfe more realtcally, puttng more weght on poble bad tate of the world, and chooe nurance. Mot nteretngly, nce c (y ) >c FI (y ), the agent who chooe to nure made le happy Whle nothng formally requre th, t eem natural to aume that expectaton are ratonal n th cae. 25
today by the opton to completely nure ncome. The agent happer when the choce et maller. 2 6 Concluon Th paper ntroduce a model of utlty-ervng bae n belef. Optmal expectaton provde a tructural model of non-ratonal but optmal belef. Whle our applcaton hghlght many of the mplcaton of our theory, many reman to be explored. Frt, the pecfcaton of poble event eem to be more mportant n a model wth optmal expectaton than t n a model wth ratonal expectaton. For example, an optmal expectaton equlbrum n a world wth only certan outcome dfferent from the equlbrum n the ame world wth an avalable unpot or publc randomzaton devce. Wth the randomzaton devce agent can gamble agant one another. Second, agent wth optmal expectaton can be optmtc about uncertan envronment, and therefore can be better off wth the later reoluton of uncertanty. For ntance, you tell omeone that they are gong to receve gft on ther brthday but not what thoe gft are untl ther brthday. 3 More generally, becaue more nformaton can change the ablty to dtort belef, agent can be better off not recevng nformaton depte the beneft of better decon makng. It, however, not the cae that agent would ever chooe that uncertanty be reolved later becaue agent take ther belef a gven. Bayean agent never prefer the later reoluton of uncertanty. Thrd, we conjecture that the agent who face the ame problem agan and agan, and o face the poblty of large loe from an ncorrect pecfcaton of probablte, wll naturally have a better aement of probablte. Thu, optmal expectaton agent are not eay to turn nto money pump, although they may exhbt behavor far from that generated by ratonal expectaton n one-hot game. 2 The agent wth the opton who accept the opton ha greater level of felcty later n lfe on average. Th becaue, lfetme well-beng wth the opton to nure greater than or equal to lfetme well-beng wthout the opton n th model. 3 A urpre party for an agent rae the poblty n the agent mnd that he mght get more urpre parte n the future and he enjoy lookng forward to th poblty. 26
Fourth, and cloely related, to what extent do optmal belef gve an evolutonary advantage or dadvantage relatve to ratonal belef? On the one hand, agent wth optmal expectaton make poorer decon. On the other hand, agent wth optmal expectaton may take on more rk, whch can lead to an evolutonary advantage. Fnally, optmal expectaton ha promng applcaton n trategc envronment. In a trategc ettng, each agent belef are et takng a gven the reacton functon of other agent. 27
Appendxe A Optmal expectaton when the tate pace contnuou In the man text, we decrbe optmal expectaton when the tate pace fnte and dcrete. Th appendx defne equlbrum when uncertanty contnuou. Let Π t t denote the condtonal cumulatve probablty dtrbuton functon of the vector t S and ˆΠ t t the ubjectve veron. When the tate pace contnuou, equaton (4) become ³ o Z V x t ; t, nˆπ ³ o =max u (x t,c t )+β V g (x t,c t, t+ ); t+, t, nˆπ dˆπ ( t+ ). c t Aumpton replaced by Aumpton (Retrcton on probablte, contnuou tate pace) Z () dˆπ t t = t S () dˆπ t t 0 () ˆΠ ( 0 t) =ˆΠ 0 0 t ˆΠ 0 0 t t t 2 ˆΠ ( 0 ) (v) dˆπ t t =0f dπ t t =0. Fnally, optmal expectaton are determned by choong contnuou probablty functon to maxmze the functonal objectve " T max {ˆΠ( τ τ )} E X ³ β t M t= ³ ³ o x t, nˆπ ³ o, c t, nˆπ ³ ³ o o # + V x t, nˆπ ; t, nˆπ. B roof of ropoton B. roofofropoton roof: () We prove the cae for w OE > 0; thecaeforw OE < 0 analogou. If w OE >w RE,then u 0 R + w OE Z u 0 R + w RE Z for Ä Z 0 (B.) u 0 R + w OE Z < u 0 R + w RE Z for Ä Z > 0 When the aet pay off poorly, margnal utlty hgher for the agent wth the hgher hare nveted n the rky aet. The agent wth ratonal expectaton ha frt order condton X π u 0 R + w RE Z Z + X π u 0 R + w RE Z Z =0 ÄZ 0 ÄZ >0 28