Optimal Expectations - Princeton...Optimal Expectations By M ARKUS K. B RUNNERMEIER AND JONATHAN A. P ARKER * Forward-looking agents care about expected future utility ßows, and hence

Optimal Expectations

By MARKUS K BRUNNERMEIER AND JONATHAN A PARKER

Forward-looking agents care about expected future utility flows and hence havehigher current felicity if they are optimistic This paper studies utility-based biasesin beliefs by supposing that beliefs maximize average felicity optimally balancingthis benefit of optimism against the costs of worse decision making A smalloptimistic bias in beliefs typically leads to first-order gains in anticipatory utilityand only second-order costs in realized outcomes In a portfolio choice exampleinvestors overestimate their return and exhibit a preference for skewness in generalequilibrium investorsrsquo prior beliefs are endogenously heterogeneous In a con-sumption-saving example consumers are both overconfident and overoptimistic(JEL D1 D8 E21 G11 G12)

Modern psychology views human behavioras a complex interaction of cognitive and emo-tional responses to external stimuli that some-times results in dysfunctional outcomesModern economics takes a relatively simpleview of human behavior as governed by unlim-ited cognitive ability applied to a small numberof concrete goals and unencumbered by emo-tion The central models of economics allowcoherent analysis of behavior and economicpolicy but eliminate ldquodysfunctionalrdquo outcomesand in particular the possibility that individualsmight persistently err in attaining their goalsOne area in which there is substantial evidence

that individuals do consistently err is in theassessment of probabilities In particular agentsoften overestimate the probability of good out-comes such as their success (Marc Alpert andHoward Raiffa 1982 Neil D Weinstein 1980and Roger Buehler et al 1994)

We provide a structural model of subjectivebeliefs in which agents hold incorrect but opti-mal beliefs These optimal beliefs differ fromobjective beliefs in ways that match many of theclaims in the psychology literature about ldquoirra-tionalrdquo behavior Further in the canonical eco-nomic models that we study these beliefs leadto economic behaviors that match observed out-comes that have puzzled the economics litera-ture based on rational behavior and commonpriors Our approach has three main elements

First at any instant people care about currentutility flow and expected future utility flowsWhile it is standard that agents who care aboutexpected future utility plan for the future forward-looking agents have higher current felicity ifthey are optimistic about the future For exam-ple agents who care about expected future util-

Brunnermeier Department of Economics BendheimCenter for Finance Princeton University Princeton NJ08544 (e-mail markusprincetonedu) Parker Depart-ment of Economics Bendheim Center for Finance andWoodrow Wilson School Princeton University PrincetonNJ 08544 (e-mail jparkerprincetonedu) For helpfulcomments we thank Andrew Abel Roland Benabou An-drew Caplin Larry Epstein Ana Fernandes Christian Gol-lier Lars Hansen David Laibson Augustin Landier ErzoLuttmer Sendhil Mullainathan Filippos PapakonstantinouWolfgang Pesendorfer Larry Samuelson Robert ShimerLaura Veldkamp and three anonymous referees as well asseminar participants at the University of Amsterdam Uni-versity of California at Berkeley Birkbeck College Uni-versity of Bonn Boston University Carnegie-MellonUniversity University of Chicago Columbia UniversityDuke University the Federal Reserve Board Harvard Uni-versity the Institute for Advanced Study London BusinessSchool London School of Economics Massachusetts Insti-tute of Technology University of Munich New YorkUniversity Northwestern University University of Penn-sylvania Princeton University University of RochesterStanford University Tilburg University the University ofWisconsin Yale University and conference participants at

the 2003 Summer Meetings of the Econometric Society theNBER Behavioral Finance Conference April 2003 the 2003Western Finance Association Annual Meetings the NBEREconomic Fluctuations and Growth meeting July 2003 theMinnesota Workshop in Macroeconomic Theory 2003Bank of Portugal Conference on Monetary Economics2004 and CEPR Summer Symposium at Gerzensee 2004Both authors acknowledge financial support from the Na-tional Science Foundation Grants SES-021-4445 and SES-009-6076 and the Alfred P Sloan Foundation Parker alsothanks the NBER Aging and Health Economics Fellowshipthrough the National Institute on Aging (T32 AG00186)

1092

ity flows are happier if they overestimate theprobability that their investments pay off well ortheir future labor income is high

The second crucial element of our model isthat such optimism affects decisions and wors-ens outcomes Distorted beliefs distort actionsFor example an agent cannot derive utility fromoptimistically believing that she will be richtomorrow while basing her consumption-savingdecision on rational beliefs about future income

How are these forces balanced We assumethat subjective beliefs maximize the agentrsquos ex-pected well-being defined as the time-averageof expected felicity over all periods This thirdkey element leads to a balance between the firsttwomdashthe benefits of optimism and the costs ofbasing actions on distorted expectations

We illustrate our theory of optimal expecta-tions using three examples In general a smallbias in beliefs typically leads to first-order gainsdue to increased anticipatory utility and only tosecond-order costs due to distorted behaviorThus beliefs tend toward optimismmdashstateswith greater utility flows are perceived as morelikely Further optimal expectations are lessrational when biases have little cost in realizedoutcomes and when biases have large benefits interms of expected future happiness

More specifically in a portfolio choice prob-lem agents overestimate the return on theirinvestment and prefer skewed returns Henceagents can be risk-loving when investing inlottery-type assets and at the same time risk-averse when investing in nonskewed assetsSecond in general equilibrium agentsrsquo priorbeliefs are endogenously heterogeneous andagents gamble against each other We show inan example that the expected return on the riskyasset is higher than in an economy populated byagents with rational beliefs if the return is neg-atively skewed Third in a consumption-savingproblem with quadratic utility and stochasticincome agents are overconfident and overopti-mistic early in life they consume more thanimplied by rational beliefs

Optimal expectations matches other observedbehavior In a portfolio choice problem Chris-tian Gollier (2005) shows that optimal expecta-tions imply subjective probabilities that focuson the best and worst state This pattern ofprobability weighting is similar to that assumedby cumulative prospect theory to match ob-served behavior (see Daniel Kahneman and

Amos Tversky 1979 Tversky and Kahneman1992) Finally Brunnermeier and Parker (2002)show in a different economic setting that agentswith optimal expectations can exhibit a greaterreadiness to accept commitment regret and acontext effect in which nonchosen actions canaffect utility

Psychological theories provide many chan-nels through which the human mind is able tohold beliefs inconsistent with the rational pro-cessing of objective data First to the extent thatpeople are more likely to remember better out-comes they will perceive them as more likely inthe future leading to optimistic biases in beliefsas in our optimal expectations framework1 Sec-ond most human behavior is not based on con-scious cognition but is automatic processedonly in the limbic system and not the prefrontalcortex (John A Bargh and Tanya L Chartrand1999) If automatic processing is optimisticthen the agent may naturally approach problemswith optimistic biases However the agent mayalso choose to apply cognition to disciplinebelief biases when the stakes are large as in ouroptimal expectations framework

Our model of beliefs differs markedly fromtreatments of risk in economics While earlymodels in macroeconomics specify beliefs ex-ogenously as naive myopic or partially up-dated (eg Marc Nerlove 1958) since John FMuth (1960 1961) and Robert E Lucas Jr(1976) nearly all research has proceeded underthe rational expectations assumption that sub-jective and objective beliefs coincide There aretwo main arguments for this First the alterna-tives to rationality lack discipline But ourmodel provides precisely this discipline for sub-jective beliefs by specifying an objective forbeliefs The second argument is that agents havethe incentive to hold rational beliefs (or act as ifthey do) because these expectations makeagents as well off as they can be However thisrationale for rational expectations relies uponagents caring about the future but at the sametime having their expectations about the future

1 In Sendhil Mullainathan (2002) individuals have im-perfect recall and form expectations as if they did not InMichele Piccione and Ariel Rubinstein (1997) individualsunderstand that they have imperfect recall and in B Doug-las Bernheim and Raphael Thomadsen (2005) individualsadditionally can influence the memory process to increaseanticipatory utility

1093VOL 95 NO 4 BRUNNERMEIER AND PARKER OPTIMAL EXPECTATIONS

not affect their current felicity which we see asinconsistent Our approach takes into accountthe fact that agents care in the present aboututility flows that are expected in the future indefining what beliefs are optimal

Most microeconomic models assume thatagents share common prior beliefs This ldquoHar-sanyi doctrinerdquo is weaker than the assumptionof rational expectations that all agentsrsquo priorbeliefs are equal to the objective probabilitiesgoverning equilibrium dynamics But like ratio-nal expectations the common priors assumptionis quite restrictive and does not allow agents toldquoagree to disagreerdquo (Robert Aumann 1976)Leonard J Savage (1954) provides axiomaticfoundations for a more general theory in whichagents hold arbitrary prior beliefs so agents canagree to disagree But if beliefs can be arbitrarytheory provides little structure or predictivepower The theory of optimal expectations pro-vides discipline to the study of subjective be-liefs and heterogeneous priors Framed in thisway optimal expectations is a theory of priorbeliefs for Bayesian rational agents

The key assumption that agents derivecurrent felicity from expectations of futurepleasures has its roots in the origins of utilitar-ianism Detailed expositions on anticipatoryutility can be found in the work of BenthamHume Bohm-Barwerk and other early econo-mists More recently the temporal elements ofthe utility concept have reemerged in research atthe juncture of psychology and economics(George Loewenstein 1987 Kahneman et al1997 Kahneman 2000) and have been incor-porated formally into economic models in theform of belief-dependent utility by John Geana-koplos et al (1989) Andrew J Caplin and JohnLeahy (2001) and Leeat Yariv (2001)2

Several papers in economics study relatedmodels in which forward-looking agents distortbeliefs In particular George Akerlof and Wil-liam T Dickens (1982) model agents as choos-ing beliefs to minimize their discomfort fromfear of bad outcomes In a two-period modelagents with rational beliefs choose an industryto work in understanding that in the second

period they will distort their beliefs about thehazards of their work and perhaps not invest insafety technology Second Augustin Landier(2000) studies a two-period game in whichagents choose a prior before receiving a signaland subsequently taking an action based on theirupdated beliefs Unlike our approach belief dy-namics are not Bayesian common to our ap-proach agents tend to save less and beoptimistic about portfolio returns3 Third time-inconsistent preferences can make it optimal tostrategically ignore information (Juan D Car-rillo and Thomas Mariotti 2000) or distort be-liefs (Roland Benabou and Jean Tirole 20022004) In the latter unlike in our model multi-ple selves play intra-personal games with im-perfect recall and actions serve as signals tofuture selves Similarly concerns about self-reputations also play a central role in RichmondHarbaugh (2002) Finally there is a large liter-ature on bounded rationality and incompletememory Some of these models suggest mech-anisms for how individuals achieve optimal ex-pectations in the face of possibly contradictorydata

The structure of the paper is as follows InSection I we introduce and discuss the generaloptimal expectations framework In Sections IIthrough IV we use the optimal expectationsframework to study behavior in three differentcanonical economic settings Section II studiesa two-period two-asset portfolio choice prob-lem and shows that agents are biased toward thebelief that their investments will pay off welland prefer positively skewed payouts SectionIII shows that in a two-agent economy of thistype with no aggregate risk optimal expecta-tions are heterogeneous and agents gambleagainst one another Section IV analyzes theconsumption-saving problem of an agent withquadratic utility receiving stochastic labor in-come over time and shows that the agent isbiased toward optimism and is overconfidentand so saves less than a rational agent SectionV concludes The Appendix contains proofs ofall propositions

2 Caplin and Leahy (2004) and Kfir Eliaz and RanSpiegler (2003) show that the forward-looking nature ofutility raises problems for the revealed preference approachto behavior and the expected utility framework in the con-text of the acquisition of information

3 Also in Erik Eyster (2002) Matthew Rabin and Joel LSchrag (1999) and Yariv (2002) agents distort beliefs to beconsistent with past choices or beliefs

1094 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2005

I The Optimal Expectations Framework

We choose to maintain many of the assump-tions of canonical economic theory agents op-timize knowing the correct mapping fromactions to payoffs in different states of theworld But we allow agentsrsquo assessments ofprobabilities of different states to depart fromthe objective probabilities

This section defines our framework in twosteps First we describe the problem of theagent given an arbitrary set of beliefs At anypoint in time agents maximize felicity thepresent discounted value of expected flow util-ities Second we define optimal expectations asthe set of beliefs that maximize well-being inthe initial period Well-being is the expectedtime-average of the agentrsquos felicity and so is afunction of the agentrsquos beliefs and the actionsthese beliefs induce

A Optimization Given Beliefs

Consider a canonical class of optimizationproblems In each period from 1 to T agentstake their beliefs as given and choose controlvariables ct and the implied evolution of statevariables xt to maximize their felicity We con-sider a world where the uncertainty can be de-scribed by a finite number S of states4 Let(stst1) denote the true probability that statest S is realized after state history st1 (s1s2 st1) S t1 We depart from the ca-nonical model in that agents are endowed withsubjective probabilities that may not coincidewith objective probabilities Conditional andunconditional subjective probabilities are de-noted by (stst1) and (st) respectively andsatisfy the basic properties of probabilities (pre-cisely specified subsequently)

At time t the agent chooses control variablesct to maximize his felicity given by

(1) EUc1 c2 cT s t

where U() is increasing and strictly quasi-concave and Et is the subjective expectationsoperator associated with and given infor-

mation available at t The agent maximizes sub-ject to a resource constraint

(2) xt 1 gxt ct st 1

(3) hxT 1 0 and given x0

where g() gives the evolution of the state vari-able and is continuous and differentiable in xand c and h() gives the endpoint conditionDenote the optimal choice of the control asc(st ) and induced state variables as x(st)

While the agentrsquos problem is standard andgeneral we employ the specific interpretationthat E[U()st] is the felicity of the agent at timet The felicity of the agent depends on expectedfuture utility flows or ldquoanticipatoryrdquo utility sothat subjective conditional beliefs have a directimpact on felicity To clarify this point considerthe canonical model with time-separable utilityflows and exponential discounting In this casefelicity at time t

EUc t 1 ct cT s t

t 1 1

t 1

uct uct

E 1

T t

uct st

is the sum of memory utility from past con-sumption flow utility from current consump-tion and anticipatory utility from futureconsumption

B Optimal Beliefs

Subjective beliefs are a complete set of con-ditional probabilities after any history of theevent tree (stst1) We require that subjec-tive probabilities satisfy four properties

ASSUMPTION 1 (Restrictions on probabilities)

(i) yenstS (stst1) 1(ii) (stst1) 0

(iii) (st) (stst1)(st1st2) (s1)(iv) (stst1) 0 if (stst1) 0

4 Appendix A defines optimal expectations for the situ-ation with a continuous state space


Assumption 1(i) is simply that probabilities sumto one Assumptions 1(i)ndash(iii) imply that the lawof iterated expectations holds for subjectiveprobabilities Assumption 1(iv) implies that inorder to believe that something is possible itmust be possible That is agents understand theunderlying model and misperceive only theprobabilities For example consider an agentchoosing to buy a lottery ticket The states ofthe world are the possible numbers of the win-ning ticket An agent can believe that a givennumber will win the lottery But the agent can-not believe in the nonexistent state that she willwin the lottery if she does not hold a lotteryticket or even if there is no lottery Note that itis possible for the agent to believe that a possi-ble event is impossible But since we specifysubjective beliefs conditional on all objectivelypossible histories as in the axiomatic frame-work of Roger B Myerson (1986) the agentrsquosproblem is always well defined

We further consider the class of problems forwhich a solution exists and provides finite fe-licity for all possible subjective beliefs

ASSUMPTION 2 (Conditions on agentrsquosproblem)

EUc1 c2 cT s t for all st

and for all satisfying Assumption 1

Optimal expectations are the subjective prob-abilities that maximize the agentrsquos lifetimehappiness Formally optimal expectations max-imize well-being W defined as the expectedtime-average of the felicity of the agent

DEFINITION 1 Optimal expectations (OE) area set of subjective probabilities OE(stst1)that maximize well-being

(4) W E1

T t 1

T

EUc1 c2 cTstsubject to the four restrictions on subjectiveprobabilities (Assumption 1)

In addition to being both simple and naturalthis objective function is similar to that in Cap-lin and Leahy (2000) Further this choice of W

has the feature that under rational expectationswell-being coincides with the agentrsquos felicityso the agentrsquos actions maximize both well-beingand felicity We further discuss these issues inSection I C

Optimal expectations exist if cOE(st) andxOE(st) are continuous in probabilities (stst1)that satisfy Assumption 1 for all t and st1where cOE(st) c(st OE) and xOE(st) x(st OE) This follows from the continuityof expected felicity in probabilities and con-trols Assumption 2 and the compactness ofprobability spaces For less regular problemsoptimal expectations may or may not exist Asto uniqueness optimal beliefs need not beunique as will be clear from the subsequent useof this concept

Beliefs have an impact on well-being directlythrough anticipation of future flow utility andindirectly through their effects on agent behav-ior Optimal beliefs trade off the incentive to beoptimistic in order to increase expected futureutility against the costs of poor outcomes thatresult from decisions made based on optimisticbeliefs

How does this trade-off occur in practiceOne possible interpretation is that at first indi-viduals approach problems with overly optimis-tic beliefs (ldquoThis paper will be easy to writerdquo)and then choose how much to restrain theiroptimism by allocating scarce cognitive re-sources to the problemmdashasking themselveswhether the probabilities of a good outcome arereally as high as they would like to believe(ldquoAm I sure writing this paper will not stretchover yearsrdquo) As cognition is applied proba-bility assessments become more rational Weposit that the amount of cognition is directlyrelated to the true risks and rewards of biasedversus rational beliefs (ldquoI am hesitant to committo present the paper next week when I may nothave resultsmdashlet me think about itrdquo) This de-scription is consistent with the view that humanbehavior is determined primarily by the rapidand unconscious processing of the limbic sys-tem but that for important decisions people relymore on the slower conscious processing of theprefrontal cortex This description also matchesmany psychological experiments that find thatagents report optimistic probabilities particu-larly when these probabilities or their reports donot affect payoffs Probabilities tend to be moreaccurate and beliefs more rational when agents


have more to lose from biased beliefs5 Ouroptimal expectations framework is a simplemodel that captures some elements of this(and other) complex (and speculative) brainprocesses

We view these processesmdashthe mapping fromobjective to subjective probabilitiesmdashas hard-wired and subconscious not conscious Thuswhile the interaction between optimistic andrational forces can be viewed as a model of adivided self agents are unaware of this divisionand of the fact that their beliefs may be biasedThis lack of self-awareness implies that agentsare unable to figure out the true probabilitiesfrom the model and their subjective beliefs

So far we have focused on the optimizationproblem of a single agent In a competitiveeconomy each agent faces this maximizationproblem taking as given his beliefs and thestochastic process of payoff-relevant aggregatevariables In our notation xt

i includes thepayoff-relevant variables that agent i takes asgiven and so reflects the actions of all otheragents in the economy Each agent has optimalexpectations that maximize equation (4) wherethe states and controls are indexed by i takingthe actions of the other agents as given Inequilibrium markets clear

DEFINITION 2 A competitive optimal expec-tations equilibrium is a set of beliefs for eachagent and an allocation such that

(i) Each agentrsquos beliefs maximize equation(4) taking as given the stochastic processfor aggregate variables

(ii) Each agent maximizes equation (1) subjectto constraints taking as given his beliefsand the stochastic process for aggregatevariables

(iii) Markets clear

Intuitively optimal beliefs of each agent take asgiven the aggregate dynamics and the optimalactions take as given the perceived aggregatedynamics

C Discussion

Before proceeding to the application of op-timal expectations it is worth emphasizingseveral points First because probabilitiesOE(stst1) are chosen once and forever thelaw of iterated expectations holds with respectto the subjective probability measure and stan-dard dynamic programming can be used tosolve the agentrsquos optimization problem An al-ternative interpretation of optimal conditionalprobabilities is that the agent is endowed withoptimal priors over the state space OE(sT) andlearns and updates over time according toBayesrsquos rule6 Thus agents are completelyldquoBayesianrdquo rational given what they knowabout the economic environment

Second optimal expectations are those thatmaximize well-being The argument that is tra-ditionally made for the assumption of rationalbeliefsmdashthat such beliefs lead agents to the bestoutcomesmdashis correct only if one assumes thatexpected future utility flows do not affectpresent felicity This is a somewhat inconsistentview one part of the agent makes plans thattrade off present and expected future utilityflows while another part of the agent actuallyenjoys utils but only from present consump-tion7 Optimal expectations give agents thehighest average lifetime utility level under theJevonian view that the felicity of a forward-looking agent depends on expectations aboutthe future

To recast this point we can ask what objec-tive function for beliefs would make rationalexpectations optimal In the general frameworkthis is the case if well-being counts only thefelicity of the agent in the last period so thatW E[U(c1 c2 cT)] Alternatively in thecanonical time-separable model this is the caseif the objective function for beliefs omits antic-ipatory and memory utility so that W E[(1T) yen1

T 1u(c)]Third this discussion also makes clear why

well-being W uses the objective expectations

5 Sarah Lichtenstein et al (1982) surveys evidence onpeoplersquos overconfidence Professionals such as weatherforecasters or those who produce published gambling oddsmake very accurate predictions Note also that the predic-tions of professionals do not seem to be due to learning fromrepetition (Alpert and Raiffa 1982)

6 The interpretation of the problem in terms of optimalpriors requires that one specify agent beliefs following zerosubjective probability events situations in which Bayesrsquorule provides no restrictions

7 See Loewenstein (1987) and the discussion of the Sam-uelsonian and Jevonian views of utility in Caplin and Leahy(2000)


operator Optimal beliefs are not those thatmaximize the agentrsquos happiness only in thestates that the agent views as most likely In-stead optimal beliefs maximize the happinessof the agent on average across repeated real-izations of uncertainty The objective expecta-tion captures this since the actual unfolding ofuncertainty over the agentrsquos life is determinedby objective probabilities

Fourth the only reason for belief distortion isthat current felicity depends on expected futureutility flows There is no incentive to distortbeliefs to change actions In fact any change inactions caused by belief distortion reduces well-being To see this note that under rational ex-pectations the objective function for beliefsWis identical to the objective function of theagent E[U] Thus fixing beliefs to be rationalthe actions of the agent maximize well-being

To clarify this point consider a generalizedversion of current felicity at time t with time-separable utility and exponential discounting

(5) EUt c t 1 ct cT s t

t 1 1

t 1

uct uct

E 1

T t

uct stwhere the agent discounts past utility flows atrate 0 1 If 1 then thisexample fits into the framework we have as-sumed so far we refer to this case as preferenceconsistency If 1 the agentrsquos memory utilitydecays through time which has more intuitiveappeal In this case however an agentrsquos rank-ing of utility flows across periods is not time-invariant under rational expectations (Caplinand Leahy 2000) Thus there is an incentive todistort beliefs in order to distort actions so as toincrease well-being In Section IV we assumetime-separable utility and exponential discount-ing While the behavior of agents depends on the qualitative behavior characterized by ourpropositions holds for any 1

Fifth one might be concerned that agentswith optimal expectations might be driven toextinction by agents with rational beliefs But

evolutionary arguments need not favor rationalexpectations Since optimal expectations re-spond to the costs of mistakes agents withoptimal expectations are harder to exploit thanagents with fixed biases Further many eco-nomic environments favor agents who take onmore risk (J Bradford DeLong et al 1990)Finally from a longer-term perspective andconsistent with our choice of W there is abiological link between happiness and betterhealth (Janice K Kiecolt-Glaser et al 2002Sheldon Cohen et al 2003)

Before turning to the applications we discussthree generalizations of our approach First op-timal expectations could be derived from a moregeneral objective function than a simple time-average of felicities In particular an earlierversion of this paper defined well-being as aweighted average of the agentrsquos felicities

Second optimal subjective probabilities arechosen without any direct relation to realityThis frictionless world provides insight into thebehaviors generated by the incentive to lookforward with optimism when belief distortion islimited by the costs of poor outcomes In fact itmay be that beliefs cannot be distorted far fromreality for additional reasons At some cost interms of simplicity the frictionless model canbe extended to include constraints that penalizelarger distortions from reality Beliefs wouldthen bear some relation to reality even in cir-cumstances in which there are no costs associ-ated with behavior caused by distorted beliefs

What sort of restrictions might be reasonableto impose One could require that belief distor-tions be restricted to be ldquosmoothrdquo or lie on acoarser partition of the probability space so thatbelief distortions are similar for states with sim-ilar outcomes Alternatively one could restrictthe set of feasible beliefs to be consistent with aset of parsimonious models For example theagent might be able to bias beliefs only throughhis belief about his own ability level Or onemight require that the agent believe that hisincome process is some first-order Markov pro-cess rather than allow belief distortions to becompletely history dependent8

8 If the agent were aware that his priormodel is chosenfrom a set of parsimonious models then he might questionthese beliefs In this case it would make sense to impose theadditional restriction that only priors for which the agentcannot detect the misspecification can be chosen an ap-


Finally returning to the first point of ourdiscussion we maintain the assumption thatconditional probabilities are fixed through timeAs an alternative one might consider beliefs asbeing reset in each period to maximize well-being given the new information that has ar-rived We describe the relationship betweenthese different approaches at the end of SectionIV B

II Portfolio Choice Optimism and a Preferencefor Skewness

In this section we consider a two-period in-vestment problem in which an agent choosesbetween assets in the first period and consumesthe payoff of the portfolio in the second periodWe show that the agent is optimistic about thepayout of his own investment and prefers assetswith positively skewed returns The subsequentsection places a continuum of these agents intoa general equilibrium model with no aggregaterisk and shows that agents disagree and howskewness affects asset prices

A Portfolio Choice Given Beliefs

There are two periods and two assets Inperiod one the agent allocates his unit endow-ment between a risk-free asset with gross returnR and a risky asset with gross return R Z (Zis the excess return of the risky asset overthe risk-free rate) In period two the agentconsumes the payoff from his first-periodinvestment

In period one the agent chooses his portfolioshare to invest in the risky asset in order tomaximize felicity in the first period E[U(c)]

max

s 1

S

sucs

st cs R Zs

cs 0

where u() is the utility function over consump-

tion u 0 u 0 u(0) and u(0) limcn0u(c) The second constraint cs 0 alsoholds for states with zero subjective probabilitybecause the market requires that the agent isable to meet his payment obligations in all fu-ture states

Uncertainty is characterized by S states withex post excess return Zs and probabilities s 0 for s 1 S Let the states be ordered sothat the larger the state the larger the payoffZs1 Zs Z1 0 ZS and Zs 13 Zs for s 13s Beliefs are given by ss1

S satisfying As-sumption 1

Noting that the second constraint can bindonly for the highest or lowest payoff state theagentrsquos problem can be written as a Lagrangianwith multipliers 1 and S

max

s 1

S

suR Zs 1R Z1

SR ZS

The necessary conditions for an optimal are

0 s 1

S

s uR Zs Zs 1 Z1 S ZS

0 1 R Z1

0 S R ZS

It turns out that optimal beliefs are never suchthat cs 0 (or R Zs 0) for any s To seethis suppose that R Zs 0 for some s andconsider an infinitesimal change in probabilitiesthat results in an increase of consumption in thisstate Since u(0) this causes an infinitemarginal increase in well-being Thus optimalexpectations imply R Zs 13 0 for all s Bycomplementary slackness s 0 for all s andthe optimal portfolio is uniquely determined by

(6) 0 s 1

S

s uR Zs Zs f

B Optimal Beliefs

Optimal beliefs are a set of probabili-ties that maximize well-being the expected

proach being pursued in the literature on robust control Bynot restricting the choice set over priors we avoid thesecomplications


time-average of felicities in the first and sec-ond period

W 12

EE1 Uc E2 Uc

In period one the agentrsquos felicity is the subjec-tively expected (anticipated) utility flow in thefuture period in period two the agentrsquos felicityis the utility flow from actual consumptionSubstituting for our utility function and con-sumption and writing out the expectationsOE solve

max

12

s 1

S

suR Zs

12

s 1

S

suR Zs

subject to the restrictions on probabilities (As-sumption 1) and where () is given im-plicitly by equation (6)

To characterize optimal beliefs first note thats

OE 0 for at least one state s with Zs 0and one state s with Zs 0 If this were not thecase the agent would view the risky asset as amoney pump and would invest or short asmuch of the asset as possible so that cs 0 fors 1 or for s S which contradicts ourprevious argument Now consider the first-ordercondition associated with moving d from thelow-payoff state s to the high-payoff state swhere both states have positive subjective prob-ability By the envelope condition smallchanges in portfolio choice from the optimumcaused by small changes in subjective probabil-ities lead to no change in expected utility sothat this condition is

(7) 12

us us

12

s 1

S

suR ZsZs

d

d

where us u(R Zs) The left-hand side isthe marginal gain in anticipatory utility in thefirst period from increasing s at the expense ofs the right-hand side is the marginal loss inexpected utility in the second period from the

resultant change in the portfolio share of therisky asset At the optimum the gain in antici-patory utility balances the costs of distortingactual behavior

Let RE denote the optimal portfolio choicefor rational beliefs The following propositionproved in the Appendix states that the agentwith optimal expectations is optimistic aboutthe payout of his portfolio Further the agentwith optimal expectations either takes a positionopposite that of the agent with rational beliefs oris more aggressivemdashinvesting even more if therational agent invests or shorting more if therational agent shorts

PROPOSITION 1 (Excess risk taking due tooptimism)

(i) Optimal beliefs on average are biased up-ward (downward) for states in which anagentrsquos chosen portfolio payout is positive(negative)

if OE 13 0

yens1S s suR OEZsZs 0

if OE 0

yens1S (s s)uR OEZsZs 0

(ii) An agent with optimal expectations investsmore aggressively than an agent with ra-tional expectations or in the oppositedirection

if EZ 13 0 then RE 13 0

and OE 13 RE or OE 0

if EZ 0 then RE 0

and OE RE or OE 13 0

if EZ 0 and S 13 2

then RE 0 and OE 13 0

To understand the first part of the propositionnote that marginal utility is always positive and


(excess) payoff Zs is positive for large s andnegative for small s Thus when the agent isinvesting in the asset (OE 0) optimal expec-tations on average bias up the subjective prob-ability for positive excess return states at theexpense of negative excess return states

The second part of the proposition character-izes behavior When the expected payoff is pos-itive E[Z] 0 the rational agent chooses toinvest in the asset RE 0 since the expectedexcess return on the risky asset is positiveStarting from rational beliefs again consider asmall increase in the probability of state s at theexpense of state s where s s The left-handside of equation (2) shows that this leads to afirst-order gain in anticipatory utility The mar-ginal cost of this distortion shown on the right-hand side of equation (2) is zero because thecost of a small change in portfolio allocationaway from the rational optimum is only ofsecond order as shown in equation (6) ThusOE 13 and OE 13 RE

Further the individual either invests morethan the rational agent in the risky asset orshorts the risky asset for E[Z] 0 and viceversa for E[Z] 0 Why would the agent take aposition in the opposite direction to the rationalagent when this implies that he is taking anegative expected payoff gamble This occurswhen anticipatory utility in the contrarian posi-tion is sufficiently large For many utility func-tions this is the case when the asset has theproperties similar to a lottery ticket that iswhen the asset is skewed in the opposite direc-tion of the mean payoff

To illustrate this point consider a world withtwo states and an asset with negative expectedexcess payoff E[Z] Z 0 We specify thepayoffs Z1 and Z2 such that as we vary prob-abilities the mean and variance Z

2 stay con-stant but skewness decreases in 2

State Probability Excess Payoff

1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

When the probability of the good state 2 issmall the asset is similar to a real-world lotterythe asset yields a small negative return withhigh probability and a large positive return with

low probability The following propositionshows that agents with optimal expectations canhave a preference for skewness that is strongerthan their aversion to risk

PROPOSITION 2 (Preference for skewness)For unbounded utility functions there existsa 2 such that for all greater levels ofskewness 2 2 (i ) the agent is opti-mistic about the asset 2

OE 2 and (ii)invests in the asset OE 0 even thoughE[Z] 0

If the agent were to short the asset when it isvery skewed that is when 2 is close to zerothen 2 2 and so 2 2 is near zeromdashsubjective beliefs are necessarily near rationalbeliefsmdashand () is near RE In this casehowever if the agent is optimistic about thepayoff of the risky asset 2 2 then he caninvest in the asset and dream about the assetpaying off well This type of behaviormdashbuyingstochastic assets with negative expected returnand positive skewnessmdashis widely observed ingambling and betting The preference forskewness may also explain the design of se-curities with highly skewed returns such asRussian Belgian and Swedish lottery bondsand the German ldquoPS-Lotteriesparenrdquo9

Returning to Proposition 1(ii) when E[Z] 0 the risky asset has the same expected returnas the risk-free asset and the cost and benefit ofa marginal change in beliefs from rational be-liefs are both of second order However weshow for S 2 that the gains in anticipatoryutility always dominate the costs10 HenceOE 13 and the agent with optimal beliefsholds or shorts an asset that a rational agentwould not An implication is that from the

9 Russia financed the majority of its national debt in the1870s using bonds with random maturities and lottery-typecoupon payments (Andrey D Ukhov 2004) Also Swedishlottery bonds make large coupon payments only to theholders of a few randomly selected bonds The GermanldquoPS-Lotteriesparenrdquo is a commitment savings plan with abank that includes a lottery For example at the nationalnetwork of savings banks (Sparkassen) one-fifth of themonthly savings is taken by the bank in exchange forparticipation in their lottery for 10000 euros

10 When there are only two states and E[Z] 0 weknow of one special case for which OE RE in this caseskewness is zero (both states are equally likely) We thankErzo Luttmer and Christian Gollier for this example


perspective of objective probabilities the agentwith optimal expectations holds an underdiver-sified portfolio That is there exists a portfoliowith the same objective expected return and lessobjective risk since E[R OEZs] E[R REZs] but Var[R OEZs] Var[R REZs]

III General Equilibrium EndogenousHeterogenous Beliefs

In this section we place the portfolio choiceproblem into an exchange economy with iden-tical agents and no aggregate risk In an optimalexpectations equilibrium agents choose to holdidiosyncratic risk and gamble against one an-other even though perfect consumption insur-ance is possible These features match stylizedfacts about asset markets In addition the price ofthe risky asset may differ from that in an econ-omy populated by agents with rational beliefs

The economy consists of a continuum of agentsof mass one with the same utility function andfacing the same investment problem as in theprevious section As before there are two periodswith S states in the second period There is onetechnology bonds that is risk free and gives nor-malized gross return 1 (R 1) There is also anasset in zero net supply equity which gives ran-dom gross return 1 Z with realized returns 1 Zs (1 s)P where P is the equilibrium priceof equity and 1 2 S Each agent i isinitially endowed with one unit of bonds Sinceequity is in zero net supply Cs i cs

idi 1 inall states s

Agent i takes her beliefs si and the price

of equity P as given and chooses her portfolioto maximize expected utility

maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0

As before the first-order conditions for portfo-lio choice deliver a unique optimal portfolioshare

(8) 0 s 1

S

siu1 Zs Zs f i

Optimal beliefs maximize the well-being ofeach agent

maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi

subject to the restrictions on probabilities (As-sumption 1) where cs(i) 1 (i)Zsand (i) is given by equation (8) Notethat since Zs [(1 s)P] 1 optimalbeliefs and asset demand depend on P

An optimal expectations equilibrium is a setof beliefs and an allocation of assets character-ized by each agent holding beliefs that maxi-mize her well-being subject to constraints andmarket clearing Letting OE denote values in anoptimal expectations equilibrium (eg ZOE [(1 )POE] 1) and RE denote values in arational expectations equilibrium we have thefollowing proposition

PROPOSITION 3 (Heterogeneous beliefs andgambling)

(i) An optimal expectations equilibriumexists

(ii) For S 2 agents have heterogenouspriors such that some agents hold therisky asset and some agents short therisky asset

bull there exists a subset of the agents Isuch that for all i I j IOEi 13 OEj and Ei[ZOE] 0OEi 0 and Ej[ZOE] 0 OEj 0

bull OEi 13 RE 0 OEi 13 forall i

Since there is no aggregate risk in the ratio-nal expectations equilibrium no agent holdsany of the risky asset and all agents have thesame consumption in all states In contrast in anoptimal expectations equilibrium agents haveheterogeneous beliefs and some agents hold theasset and some short it Consequently agentsgamble against each other and choose to bearconsumption risk


To gain intuition for this result consider thefollowing example with only two states u(c) [1(1 )]c1 with 3 1 025 2 075 1 06 2 02 We choose the riskyasset to have negative skewness like daily re-turns on the US stock market The rationalexpectations equilibrium has PRE 1 so thatE[Z] 0 and no agent holds the risky asset Atthis price because the payoff of the asset isnegatively skewed agents with optimal expec-tations would be pessimistic about the payout ofthe asset and short the asset This is shown inFigure 1 the dashed line plots well-being Was a function of 2 for the rational expectationsprice P 1 At this price the market for therisky asset does not clear because demand is toolow

At lower prices E[Z] 0 and Proposition 1implies that agents with optimal expectationseither hold more of the asset than the agent withrational expectations or short the asset If theprice were far below PRE then the asset wouldhave such a high expected return that it wouldbe optimal for all agents to be optimistic aboutthe return on the asset and to hold the asset (thedotted line in Figure 1) so again the marketwould not clear The unique optimal expecta-tions equilibrium occurs at a price of 0986 Atthis equilibrium price each agent holds one oftwo beliefs each of which gives the same levelof well-being These correspond to the two localmaxima of the solid line in Figure 1 One set ofagents has optimistic beliefs about the return on

the asset and holds the asset (2OEi 082 and

OEi 019) the remaining agents have pes-simistic beliefs and short the asset (2

OE j 038and OE j 067) The market for the riskyasset clears when 78 percent of the agents areoptimistic and the remaining 22 percent arepessimistic No agents hold rational beliefs

From an economic perspective an interestingresult in this example is that the optimal expecta-tions equilibrium has a 14-percent-higher equitypremium than the rational expectations equilib-rium In the example the more negativelyskewed the asset the greater is the equity pre-mium For the case in which the asset is posi-tively skewed by symmetry of the problemPOE PRE For the knife-edge case in whichthe asset is not skewed (1 05 and 1 2) agents hold rational expectations and theoptimal expectations price is equal to the ratio-nal expectations price But this result is quitespecific to this example since by Proposition 1this is not the case if S 2 or if there isaggregate risk

This negative relationship between skewnessand expected returns is also observed more gen-erally and in almost the exact setting we studyin the payoffs and probabilities in pari-mutuelbetting at horse tracks As in our example inpari-mutuel betting there is no aggregate riskand there are risky assets The longer the oddson a horse the more positively skewed is thepayoff Joseph Golec and Maurry Tamarkin(1998) document that the longer the odds the

FIGURE 1 WELL-BEING AS A FUNCTION OF SUBJECTIVE BELIEFS


lower the expected return on the bet or equiv-alently the higher the price of the asset Moregenerally initial public offerings (IPOs) ofstock have both low (risk-adjusted) return andpositive skewness during the first year follow-ing public trading11

To summarize when returns are positivelyskewed as in pari-mutuel betting or the IPOexample our model predicts lower expectedreturns than the rational model when returnsare negatively skewed as in the US stockmarket our model predicts higher expected re-turns than the rational model

It is also worth noting that in an optimalexpectations equilibrium there is significanttrading volume while there is no trade in therational expectations equilibrium Further thereis more trading when the asset is more skewedThis is consistent with the empirical findings inJoseph Chen et al (2001) We note the caveatthat the choice of endowments drives this resultFinally let us speculate about a setting withheterogeneous endowments In such a setup itis natural to pick an equilibrium in which agentsare more optimistic about the payoffs of theirinitial endowments This choice minimizes trad-ing Interpreting endowment as labor incomethe model suggests that most agents are opti-mistic about the performance of the companiesthey work in or the countries they live inHence investors overinvest in the equity oftheir employer and of their country relative tothe predictions of standard rational models con-sistent with the data (see for example JamesM Poterba 2003 on pension underdiversifica-tion and Karen K Lewis 1999 on the homebias puzzle)

IV Consumption and Saving over TimeUndersaving and Overconfidence

This section considers the behavior of anagent with optimal expectations in a multi-period consumption-saving problem with sto-chastic income and time-separable quadratic

utility We show that the agent overestimatesthe mean of future income and underestimatesthe uncertainty associated with future incomeThat is the agent is both unrealistically optimis-tic and overconfident This is consistent withsurvey evidence that shows that the growth rateof expected consumption is greater than that ofactual consumption We also use this exampleto make four general points about the dynamicchoices of agents with optimal expectations

A Consumption Given Beliefs

In each period t 1 T the agent choosesconsumption to maximize the expected dis-counted value of utility flows from consumptionsubject to a budget constraint

maxct

E 0

T t

uct yt

st 0

T t

Rct yt At

where u(ct) act (b2)ct2 initial

wealth A1 0 a b 0 R 1 and yt denotesthe history of income realizations up to t Theagentrsquos felicity at time t is given by equation(5) so that the agent has time-separable utilityand discounts the future and the past exponen-tially Equation (5) allows the rate at which pastutility flows are discounted to differ from theinverse of the rate at which future flows arediscounted We note again that the choice of does not affect the agentrsquos actions givenbeliefs

The only uncertainty is over income In-come yt has cumulative distribution function(ytyt1) with support [y y] and d(yt) 0for all y Y where 0 y y a(bT) Weassume income is independently distributedover time and so (ytyt1) (yt) Agentscan believe however that income is seriallydependent so subjective distributions are de-noted by (ytyt1)

Assuming an interior solution the necessaryconditions for an optimum imply the Hall Mar-tingale result for consumption but for subjec-tive beliefs

(9) ct Ect 1 yt

11 See Jay Ritter (1991) Alon Brav and James B Heaton(1996) and Nicholas C Barberis and Ming Huang (2005)derive a preference for IPOs that are skewed from theexogenous probability weighting (ldquodecision weightsrdquo) ofprospect theory Gollier (2005) shows how the overweight-ing of extreme eventsmdashassumed by prospect theorymdashis anendogenous outcome of optimal expectations


Substituting back into the budget constraintgives the optimal consumption rule

(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt

Optimal consumption depends on subjective ex-pectations of future income and on the historyof income realizations through At Because qua-dratic utility exhibits certainty equivalence inthe optimal choice of consumption the subjec-tive variance (and higher moments) of the in-come process are irrelevant for the optimalconsumption-saving choices of the agent Onlythe subjective means of future incomes matter

B Optimal Beliefs

Optimal expectations maximize well-beingsubject to the agentrsquos optimal behavior givenbeliefs and the restrictions on expectations As-sumption 1 in Appendix A states the restric-tions on expectations for a continuous statespace We incorporate optimal behavior directlyinto the objective function and characterize con-sumption choices ct

OE implied by optimalbeliefs Optimal beliefs EOE and OE imple-ment these consumption choices given optimalbehavior on the part of the agent12

Since the objective is a sum of utility func-tions it is concave in future consumption Andsince the agentrsquos behavior depends only on thesubjective certainty-equivalent of future in-come optimal beliefs minimize subjective un-certainty Thus future income is optimallyperceived as certain which is an extreme formof overconfidence

Using the fact that optimal beliefs are cer-tain and the consumption Euler equation

E[u(ct)yt] u(E[ctyt]) u(ct ) the agentrsquos

felicity at time t can be written as

EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t

Subjective expectations are chosen to yield thepath of ct that maximizes well-being

1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT

subject to the budget constraint Collectingterms the objective simplifies to

(11)1

TE

t 1

T

t uct where t t1(1 yen1

Tt ( ()))Notice that regardless of the average con-sumption path of agents is not constant Only ifthe objective for beliefs were to ignore antici-patory utility and memory utility ( 0) so thatt t1 would beliefs be rational and theexpected consumption path standard

Under optimal expectations the first-ordercondition implies that expected consumptiongrowth between t and t is given by

uctOE

t

tREuct

OE yt

which substituting for the quadratic utilityfunction implies that

12 In taking this approach we are assuming that the optimalchoice of consumption and thus EOE[ytyt] does not requireviolation of Assumptions 1 (iv) which can be checked Thatis if the support of yt is small belief distortion may beconstrained by the range of possible income realizations Toincorporate these constraints directly one would solve foroptimal E[ytyt] by replacing c(yt OE) using equation(10) and impose Assumption 1


(12) cOEyt a

b

t

tRa

b EcOEyt yt

Level consumption is recovered by substitutinginto the budget constraint and taking objectiveconditional expectations

Given this characterization of optimal be-havior agents are optimistic at every timeand state Define human wealth at t as thepresent value of current and future labor in-come Ht yen0

Tt Ryt

PROPOSITION 4 (Overconsumption due tooptimism) For all t 1 T 1

(i) On average agents revise down theirexpectation of human wealth over timeEOE[Ht1yt] E[EOE[Ht1yt1]yt]

(ii) On average consumption falls over timect

OE E[ct1OE yt]

(iii) Agents are optimistic about their futureconsumption EOE[ct1

OE yt] E[ct1OE yt]

The first point of the proposition states thatagents overestimate their present discountedvalue of labor income and on average revisetheir beliefs downward between t and t 1 Thesecond point states that consumption on averagefalls between t and t 1 Because on averagethe agent revises down expected future income

on average consumption falls over time Theproof follows directly from the expected changein consumption given by equation (12) and not-ing that (ab) ct

OE(yt) 0 and (t1t)R 1 Finally the optimal subjective expectation offuture consumption exceeds the rational expec-tation of future consumption This is optimismPart (iii) follows from part (ii) and equation (9)In sum households are unrealistically optimis-tic and in each period are on average surprisedthat their incomes are lower than they expectedso on average household consumption declinesover time

Figure 2 summarizes these results qualita-tively The agent starts life optimistic aboutfuture income At each point in time the agentexpects that on average consumption will re-main at the same level Over time the agentobserves on average that income is less than heexpected and consumption typically declinesover his life Note that the agent updates hisbeliefs according to but does not learn overtime that is incorrect because he does notknow that his income is identically and indepen-dently distributed over time The agent merelyobserves one realization of income at each age and(on average) believes that he was unlucky

While quadratic utility makes this examplequite tractable the agentrsquos overconfidence isextreme Before each period the agent is certainabout what his future income will be and this

FIGURE 2 AVERAGE LIFE-CYCLE CONSUMPTION PROFILES


belief is contradicted by the realization But asseen in equation (10) less extreme overconfi-dence would not alter consumption choicesonly reduce the agentrsquos felicity early in life

This optimism matches survey evidence ondesired and actual life-cycle consumption pro-files Robert Barsky et al (1997) find thathouseholds would choose upward-sloping con-sumption profiles But survey data on actualconsumption reveal that households have down-ward-sloping or flat consumption profiles(Pierre-Olivier Gourinchas and Parker 2002Orazio Attanasio 1999) In our model house-holds expect and plan to have constant marginalutility since R 1 0 On average howevermarginal utility rises at the age-specific rate(tt1) 1 0 Thus in the model thedesired rate of increase of consumption exceedsthe average rate of increase as in the real worldIn addition the model matches observed house-hold consumption behavior in that average life-cycle consumption profiles are concavemdashconsumption falls faster (or rises more slowly)later in life

In general in consumption-saving problemsthe relative curvatures of utility and marginalutility determine what beliefs are optimal Un-certainty about the future enters the objectivefor beliefs both through the expected futurelevel of utility and through the agentrsquos behaviorwhich depends on expected future marginalutility For utility functions with decreasing ab-solute risk aversion greater subjective uncer-tainty leads to greater precautionary savingthrough the curvature in marginal utility Thishas some benefit in terms of less distortion ofconsumption In such cases optimal beliefsmay consist of a positive bias for both expectedincome and its variance

We conclude this section by using our con-sumption-saving problem to make four pointsabout the dynamic choices of agents with opti-mal expectations First given that in expecta-tion the consumption of the agent is alwaysdeclining the costs of optimism early in lifecould be extreme for long-lived agents Butillustrating a general point optimal expectationsdepend on the horizon in a way that mitigatesthese possible costs The behavior of an agentwith a long horizon is close to that of an agentwith rational expectations For large but finite Tan agent with optimal expectations consumes asmall amount more for most his life leading to

a significant decline in consumption at the endof life As the horizon becomes infinite at anyfixed age the consumption choice of the agentwith optimal beliefs converges to that of theagent with rational beliefs as the subjectiveexpectation of human wealth converges to therational expectation As shown in AppendixB6 for any t as T 3 ct

OE(yt) 3 ctRE(yt)

EOE[Ht1yt] 3 E[Ht1yt] and ctOE(yt) 3

E[ct1OE (yt1)yt] Beliefs become more rational

as the stakes become largerSecond an agent with optimal expectations

may choose not to insure future income whenoffered an objectively fair insurance contractFormally let the agent face an additional binarydecision in period one whether or not to ex-change all current and future income for B E[H1y1] A rational agent would always takethis contract while the agent with optimal ex-pectations may choose not to insure consump-tion Interestingly since beliefs affect whetherthe agent insures or not the addition of thepossibility of insurance may change what be-liefs are optimal

Optimal expectations are either the beliefsthat maximize well-being conditional on induc-ing the agent to reject the insurance or thebeliefs that maximize well-being conditional oninducing the agent to accept the insurance Theformer are the optimal expectations from Prop-osition 4 These beliefs are optimal for the prob-lem without the constraint and the agent rejectsthe insurance because both income streams areperceived as certain and E[H1y1] E[H1y1] B Well-being in this case is

1

T t 1

T

t Euct y1

with c characterized by equation (12) The latteroptimal beliefs conditional on accepting the insur-ance are irrelevant for well-being provided thatthe agent believes that E[H1y1] is small enoughandor the process for y uncertain enough that heaccepts the insurance13 Well-being in this case is

1

T t 1

T

t ucFIy1

13 While nothing formally requires this it seems naturalto assume that expectations are rational in this case


where cFI(y1) [(R 1)(R R1T)]E[yent1T

R1 tyty1]Risk determines which expectations are opti-

mal Well-being decreases in objective incomerisk when the agent rejects the insurance whileit is invariant to risk if he accepts the insuranceIf objective income risk is small then the cost ofdistorted beliefsmdashvariable future consumptionmdashis small and optimal expectations are optimis-tic If objective income risk is large optimalexpectations are more rational and induce theagent to insure his future income

Third at the start of life the agent facing theproblem with the option to insure income mayhave a lower level of felicity than the agentfacing the problem without this option Infor-mally we might think of an agent approachinglife blithely optimistic about the future Givenno choice of insurance this is indeed optimal Ifplaced in an environment with large amounts ofincome risk however an agent given the op-portunity to insure considers his life more real-istically puts more weight on possible badstates of the world and chooses insuranceSince cOE(y1) cFI(y1) the agent who has andaccepts the option to insure is less happyinitially14

Finally what if beliefs were chosen in eachperiod to maximize well-being Suppose thatthe agent in each period chooses his actionstaking as given his own beliefs in the futurewhich are possibly different The agentrsquos felic-ity is the present discounted value of utilityflows evaluated using his own subjective be-liefs This can be viewed as if the agent in eachperiod were a different self that knows the con-ditional beliefs of his future selves The well-being function for optimal beliefs at time twould then be W t E[(1T) yen1

T E[U(c1c2 cT)s]st] where E[ s] denotes the be-liefs of the agent at time under this alterna-tive assumption Et[ s] maximizes W t givenE[ s]t and the future decision rules thatthese beliefs induce

Because the objective function changesthrough time typically it is not the case that theagent updates probabilities according to Bayesrsquos

law That is Et[ s] varies across an agentrsquosselves in different periods t In the appli-cation of this section however an agentrsquosselves agree

PROPOSITION 5 (Time consistency of beliefs)In this consumption-saving problem optimalexpectations are time consistent EOEt[ s] isindependent of t for all possible histories and t

This result obtains here because of the ex-tremity of overconfidence Consider first thechoice of beliefs at time t following an event att s viewed subjectively as having zero prob-ability These beliefs do not influence either theactions or anticipatory utility of the agent attime t they influence only the actions and an-ticipatory utility of the agent at time t s Thusbeliefs following realizations of income otherthan the expected one are chosen simply tomaximize the expected utility of that agent inthat period The perspective from which onechooses these optimal beliefs is irrelevant If theincome realization matches the expected levelthen consumption remains constant and theagent continues to hold the certain beliefs thatthey held in the previous period (by Bayesrsquoslaw) Note that this argument pins down theprofile of optimal income expectations E[yt]which increases in a pattern opposite the aver-age consumption profile E[ct]

V Conclusion

This paper introduces a model of utility-serving biases in beliefs While our applicationshighlight many of the implications of our the-ory many remain to be explored

First the specification of possible eventsseems to be more important in a model withoptimal expectations than it is in a model withrational expectations For example an optimalexpectations equilibrium in a world with onlycertain outcomes is different from the equilib-rium in the same world with an available sun-spot or public randomization device With therandomization device agents can gambleagainst one another

Second agents with optimal expectations canbe optimistic about uncertain events and there-fore can be better off with the later resolution ofuncertainty For instance you tell someone that

14 On average the agent who has and accepts the optionto insure has greater levels of felicity later in life This isbecause lifetime well-being with the option to insure isgreater than or equal to lifetime well-being without theoption


he is going to receive gifts on his birthday butyou do not tell him what those gifts are until thebirthday15 More generally because more infor-mation can change the ability to distort beliefsagents can be better off not receiving informa-tion despite the benefits of better decision mak-ing However without relaxing the assumptionsof expected utility theory and Bayesian updat-ing agents would not choose that uncertainty beresolved later because agents take their beliefsas given

Third we conjecture that the agent who facesthe same problem again and again and so facesthe possibility of large losses from an incorrect

specification of probabilities will in our frame-work have a better assessment of probabilitiesThus optimal expectations agents are not easyto turn into ldquomoney pumpsrdquo although they mayexhibit behavior far from that generated by ra-tional expectations in one-shot games

Fourth and closely related to what extent dooptimal beliefs give an evolutionary advantageor disadvantage relative to rational beliefs Onthe one hand agents with optimal expectationsmake poorer decisions On the other handagents with optimal expectations may take onmore risk which can lead to an evolutionaryadvantage

Finally optimal expectations has promisingapplications in strategic environments In a stra-tegic setting each agentrsquos beliefs are set takingas given the reaction functions of other agents

APPENDIX

A Optimal Expectations When the State Space Is Continuous

In the main text we define optimal expectations when the state space is finite and discrete Toconsider random variables with continuous distributions we extend our definitions Let ST F denote the state space -algebra and objective probability measure Let F F0 FT be afiltration Let and E denote the subjective probability measure and expectation respectivelyFirst agent optimization given continuously distributed random variables is standard Second it ismathematically simpler to state the restrictions on subjective beliefs in terms of subjective condi-tional expectations Thus one solves for optimal expectations by choosing E[AFt] for any Ft in thefiltration F and any event A ST to maximize the functional objective and Assumption 1 is replacedby

ASSUMPTION 1 (Restrictions on probabilities for a continuous state space) For every Ft F

(i) E[STFt] 1(ii) E[ f Ft] 0 for any nonnegative function f ST R which is Ft-measurable(iii) E[AFt] E[E[AFt]Ft] for any 0 and any event A(iv) is a dominating measure of

B Proofs of Propositions

B1 PROOF OF PROPOSITION 1Part (i) We prove the case for OE 0 the case for OE 0 is analogous For OE RE when

the asset pays off poorly marginal utility is higher (lower) for the agent with the higher (lower) shareinvested in the risky asset

(A1) uR OEZs uR REZs for s such that Zs 0

uR OEZs uR REZs for s such that Zs 13 0

Combining this with the first-order condition of the agent with rational expectations

15 A surprise party for an agent raises the possibility inthe agentrsquos mind that he might get more surprise parties inthe future and he enjoys looking forward to this possibility


sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0

Subtracting this from the first-order condition of the agent with optimal expectations gives thedesired inequality

(A2) s 1

S

sOE suR OEZsZs 13 0

Thus if we can show that OE 0 implies OE RE the proof of (i) is complete This follows fromthe second point of the proposition which we now prove

Part (ii) The proof of the sign of RE in each case is standard and omitted We first treat the caseof E[Z] 0 and RE 0 the case of E[Z] 0 and RE 0 is analogous and we treat E[Z] 0and RE 0 subsequently

We show that an agent with arbitrary beliefs invests more in the risky asset (or shorts itless) as the subjective probability of a state s with Zs 0 is increased relative to a state swith Zs 0 Examine the agentrsquos first-order condition for and consider moving d froms to s

0 uR Zs Zs uR Zs Zs d s 1

S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0

since the denominator is negative and Zs 0 ZsSuppose for purposes of contradiction that 0 OE RE As in the proof of part (i) we have

uR OEZs uR REZs for s Zs 0

uR OEZs 13 uR REZs for s Zs 13 0

which implies from the first-order condition of the agent with rational expectations

(A3) s 1

S

s uR OEZsZs 13 0

Now to establish the contradiction the first-order condition for beliefs equation (7) implies


signus us sign s 1

S

suR OEZsZs

d

d

From (dd) 0 and equation (A3) the sign of the right-hand side is strictly negative while thefact that we are assuming OE 0 implies that us us and the left-hand side is strictly positivea contradiction Therefore either OE RE or OE 0 The final step is to rule out OE 0 IfOE 0 then (us us) 0 so that by the first-order condition for beliefs

(A4) 0 s 1

S

s uR OEZsZs

But OE 0 cannot solve equation (A4) because equation (A4) is the same as the first-ordercondition for the optimal portfolio choice of the rational agent and the objective of the rational agentis globally concave with a unique RE satisfying equation (A4)

Finally we prove that when E[Z] 0 and RE 0 OE 13 0 Suppose that instead OE weresuch that OE 0 which occurs if and only if EOE[Z] 0 These beliefs actually satisfy thefirst-order condition for optimal expectations because (a) there is no gain to the marginal beliefdistortion since us us and (b) starting from RE the first-order cost of a small change in optimalportfolio choice is zero We show however that the second-order condition is violated for somebeliefs such that E[Z] 0 which means that there is a deviation from this set of beliefs that increaseswell-being and therefore OE 13 0

The second-order condition for the same d that moves an infinitesimal probability from s to swhere Zs 0 Zs 0 and s 0 is

d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2

Since W is linear in probabilities (d2Wdd) 0 Now omitting 1frasl2 from all terms

W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs

We evaluate this second-order condition for such that E[Z] 0 so that 0 and u and uare independent of s yielding


d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d

where the third equality makes use of yen sZs 0 and yen sZs 0 Thus any such that yens1S

sZs2 yens1

S sZs2 and E[Z] 0 has (d2Wdd) 0 and so there exists a deviation that increases

well-being completing the proof This final step requires S 2 for S 2 the second-ordercondition is necessarily zero and there are cases where OE RE We conjecture that OE RE

occurs only for S 2 Z1 Z2 and 1 2 1frasl2

B2 PROOF OF PROPOSITION 2To avoid arbitrage we consider only 1 large enough such that Z2 0 We show that as 1 3 1

well-being when investing in the asset is higher than when shorting the asset We do this by con-structing a lower bound for well-being when investing in the asset (W(1)) and an upper boundwhen shorting the asset (W (1)) and showing that lim13W(1) lim13W (1) Definewell-being as a function of subjective and objective beliefs given optimal agent behavior as

W1 1 12

1 1uR Z1 12

2 1 1uR Z2

where Zs Zs(1) and (1 Z1(1) Z2(1))Step 1 lim131W( ) for 0Consider an optimistic belief 1 0 1 1 such that the agent invests in the asset 0

Since may be suboptimal well-being with this belief is a lower bound for the well-being of theagent conditional on 0 Define W(1) W(1 1) W(1

OE(1) 1) Taking the limitas skewness goes to infinity

lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2


since Z1 3 Z Z2 3 and limc3u(c) Step 2 lim131W( ) for 0Define an upper bound for well-being by choosing the portfolio and subjective beliefs subject only

to the conditions that the agent shorts the asset that the agent is pessimistic about the payout andthat the portfolio is feasible

W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0

W(1) is an upper bound since we do not restrict to be the optimal agentrsquos choice given 1 Theoptimal 1 1 and the first and third constraints become (RZ2) 0 (which is not the nullset since Z2 0) so that this can be rewritten as

lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR

where the second line follows from substituting the best portfolio choice in each state separately andlim131(Z1Z2) 0

The proof follows from lim131W(1) lim131W(1) and Proposition 1

B3 PROOF OF PROPOSITION 3Part (i) Write the well-being of an agent as a function of subjective beliefs and the price given

and optimal agent behavior

W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs

This function is well-defined for the set of prices and beliefs such that the agent chooses positiveconsumption in every state Denote this set M In M the function W is continuous in prices andsubjective probabilities because is continuous in subjective probabilities and prices M is notclosed but we now show that POE and all OE do not lie outside M or in the setClosure(M)M First consider prices such that the lowest payoff Z1 0 or the highest payoffZS 0 In this case all agents would have an identical arbitrage opportunity for any possiblesubjective beliefs except possibly for 1 1 or S 1 Hence this cannot constitute an equilibriumbecause agents would all choose to buy or all choose to short the risky asset and so the market forthe risky asset would not clear Thus the equilibrium price must lie on the interior of the set P (1 1 1 S) Second consider beliefs such that an agent chooses cs 0 for some s Becauseu(0) in some state a marginal increase in 1 or S leads to cs 0 for all s and results in aninfinite increase in well-being

We will argue that at a low enough price W( P) is maximized by beliefs such that


E[Z] 13 0 and 0 and at a high enough price W( P) is maximized by beliefs such thatE[Z] 0 and 0 Then by continuity of W( P) either at some intermediate price thereare multiple global maxima some with E[Z] 13 0 and some with E[Z] 0 or at someintermediate price there is a unique global maximum with E[Z] 0 and 0 By Proposition1 this second alternative cannot occur for S 13 2 Thus for S 13 2 the unique equilibrium inwhich markets clear has a fraction of agents believing E[Z] 0 and shorting the asset and afraction of agents believing E[Z] 13 0 and buying the asset The fractions are such that theaggregate demand for the asset is zero

We now show that there exists a low enough price such that optimal beliefs always induce theagent to buy the asset We do this by showing that for a low enough price an upper bound on thewell-being of an agent who shorts the asset is lower than the well-being of an agent with rationalbeliefs who buys the asset Consider an agent who shorts the asset and consider lower and lowerprices for the asset Since the agent shorts he must believe E[Z] 0 As P n 1 1 E[Z] 0implies 1 m 1 and Z1 m 0 so that

limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1

where the inequality follows from the fact that in the limit the risky asset becomes dominated by therisk-free asset For P yens1

S s(1 s) we have E[Z] 0 and so for an agent with rational beliefswell-being is

W P s 1

S

s u1 Zs

where 0 Since the rational agent chooses to maximize his objective yields higher utilitythan 0 so

s 1

S

s u1 Zs 13 u1

Thus there is a low enough price such that the beliefs that maximize the well-beingfunction have E[Z] 0 and 0 The problem is symmetric so that there is a completelyanalogous argument that in the limit as P m 1 S optimal expectations have E[Z] 0 and 0

Part (ii) The proof for the rational expectations equilibrium is standard and omittedFor E[ZOE] 0 Proposition 1 directly implies OEi 13 RE 0 because in this case RE

RE(POE) RE(yens1S s(1 s)) 0 where RE(POE) denotes the portfolio choice of an agent who

has beliefs equal to the objective probabilities and faces the optimal expectations equilibrium priceof equity By market clearing and since agents either short or hold the asset some agents must beshorting and some agents must be holding the asset which implies the result Again by Proposition


1 for E[ZOE] 0 each price-taking agent has OEi RE(POE) 0 or OEi 0 so OEi 13 0and since OEi 13 RE(POE) OEi 13 Again by market clearing and since agents either shortor hold the asset some agents must be shorting and some agents must be holding the asset whichimplies the result The case of E[ZOE] 0 is analogous

B4 PROOF OF PROPOSITION 4Part (ii) Subtract E[ct1

OE yt] from each side of equation (12) with 1

ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt

Since the support of the income process does not admit a plan such that E[ctOE yt] ab for any

the second term is positive The following demonstrates that the first term is positive

t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0

and we have the resultPart (iii) From the agentrsquos consumption Euler equation

Ect 1OE yt ct

OE 13 Ect 1OE yt

where the inequality follows from part (ii)Part (i) the consumption rule at t 1 is

ct 1OE

1 R1

1 RT t At 1 EHt 1yt 1

where Ht1 yen0Tt1 Ryt1 From part (iii) we have


Ect 1OE yt 13 Ect 1

OE yt

E 1 R1

1 RT t At 1 EHt 1 yt 1 yt 13 E 1 R1

1 RT t At 1 EHt 1 yt 1 ytand by the law of iterated expectations

EHt 1 yt 13 EEHt 1 yt 1 yt

B5 PROOF OF PROPOSITION 5This result obtains because t1t depends only on the number of periods until T T t

t 1

t

1 1

T t1

1 1

T t

Thus optimal beliefs from the perspective of any time period imply the same rationally expectedpercent change in the profile of marginal utility between any two periods Since the budget constraintdetermines the level of the profile and all plans exhaust the resources the levels are necessarily thesame Since income is perceived as certain if that income actually occurs beliefs are bound byBayesrsquo rule In this case there is no change in beliefs about future incomes and the subjectiveexpectation of consumption coincides with actual consumption

B6 PROOF OF CONSUMPTION BEHAVIOR AS T 3 It can be seen that c1

OE 3 c1RE by taking conditional expectations of the budget constraint and

repeatedly substituting E[ctOE yt] for all from the Euler equation (12) to solve for c1

OE and notingthat (tt)R 3 1 as T 3 and (tt)R t for T 3 This together with equation(10) implies that EOE[H2 y1] 3 E[H2 y1] Again using the fact that (t t) R 3 1 as T 3 equation (12) implies that for finite t ct

OE 3 ctRE so that the first two results also hold for

any finite t (not just t 1)

REFERENCES

Akerlof George A and Dickens William T ldquoTheEconomic Consequences of Cognitive Disso-nancerdquo American Economic Review 198272(3) pp 307ndash19

Alpert Marc and Raiffa Howard ldquoA ProgressReport on the Training of Probability Asses-sorsrdquo in Daniel Kahneman Paul Slovic andAmos Tversky eds Judgment under uncer-tainty Heuristics and biases CambridgeCambridge University Press 1982 pp 294ndash305

Attanasio Orazio P ldquoConsumptionrdquo in John BTaylor and Michael Woodford eds Hand-book of macroeconomics Volume 1B Am-sterdam Elsevier Science North-Holland1999 pp 741ndash812

Aumann Robert J ldquoAgreeing to Disagreerdquo An-nals of Statistics 1976 4(6) pp 1236ndash39

Barberis Nicholas C and Huang Ming ldquoStocksas Lotteries The Implications of ProbabilityWeighting for Security Pricesrdquo UnpublishedPaper 2005

Bargh John A and Chartrand Tanya L ldquoTheUnbearable Automaticity of Beingrdquo Ameri-can Psychologist 1999 54(7) pp 462ndash79

Barsky Robert B Juster Thomas F KimballMiles and Shapiro Matthew D ldquoPreferenceParameters and Behavioral HeterogeneityAn Experimental Approach in the Health andRetirement Studyrdquo Quarterly Journal ofEconomics 1997 112(2) pp 537ndash79

Benabou Roland and Tirole Jean ldquoSelf-Confidenceand Personal Motivationrdquo Quarterly Journal ofEconomics 2002 117(3) pp 871ndash915


Benabou Roland and Tirole Jean ldquoWillpowerand Personal Rulesrdquo Journal of PoliticalEconomy 2004 112(4) pp 848ndash86

Bernheim B Douglas and Thomadsen RaphaelldquoMemory and Anticipationrdquo Economic Jour-nal 2005 115(503) pp 271ndash304

Brav Alon and Heaton James B ldquoExplaining theUnderperformance of Initial Public Offer-ings A Cumulative Prospect Utility Ap-proachrdquo Unpublished Paper 1996

Brunnermeier Markus K and Parker JonathanA ldquoOptimal Expectationsrdquo Princeton Uni-versity Woodrow Wilson School DiscussionPaper in Economics No 221 2002

Buehler Roger Griffin Dale and Ross MichaelldquoExploring the ldquoPlanning Fallacyrdquo WhyPeople Underestimate Their Task Comple-tion Timesrdquo Journal of Personality and So-cial Psychology 1994 67(3) pp 366ndash81

Caplin Andrew and Leahy John ldquoPsychologicalExpected Utility Theory and AnticipatoryFeelingsrdquo Quarterly Journal of Economics2001 116(1) pp 55ndash79

Caplin Andrew and Leahy John ldquoThe SocialDiscount Raterdquo National Bureau of Eco-nomic Research Inc NBER Working Pa-pers No 7983 2000

Caplin Andrew and Leahy John ldquoThe Supplyof Information by a Concerned Expertrdquo Eco-nomic Journal 2004 114(497) pp 487ndash505

Carrillo Juan D and Mariotti Thomas ldquoStrate-gic Ignorance as a Self-Disciplining DevicerdquoReview of Economic Studies 2000 67(3) pp529ndash44

Chen Joseph Hong Harrison and Stein JeremyC ldquoForecasting Crashes Trading VolumePast Returns and Conditional Skewness inStock Pricesrdquo Journal of Financial Econom-ics 2001 61(3) pp 345ndash81

Cohen Sheldon Doyle William J TurnerRonald B Alper Cuneyt M and SkonerDavid P ldquoEmotional Style and Susceptibilityto the Common Coldrdquo Psychosomatic Med-icine 2003 65(4) pp 652ndash57

De Long J Bradford Shleifer Andrei SummersLawrence H and Waldmann Robert J ldquoNoiseTrader Risk in Financial Marketsrdquo Journal ofPolitical Economy 1990 98(4) pp 703ndash38

Eliaz K and Spiegler Ran ldquoAnticipatory Feel-ings and Choices of Information SourcesrdquoUnpublished Paper 2003

Eyster Erik ldquoRationalizing the Past A Tastefor Consistencyrdquo Unpublished Paper 2002

Geanakoplos John Pearce David and StacchettiEnnio ldquoPsychological Games and SequentialRationalityrdquo Games and Economic Behav-ior 1989 1(1) pp 60ndash79

Golec Joseph and Tamarkin Maurry ldquoBettorsLove Skewness Not Risk at the HorseTrackrdquo Journal of Political Economy 1998106(1) pp 205ndash25

Gollier Christian ldquoOptimal Illusions and Deci-sion under Riskrdquo University of ToulouseIDEI Working Papers No 2376 2005

Gourinchas Pierre-Olivier and Parker JonathanA ldquoConsumption over the Life CyclerdquoEconometrica 2002 70(1) pp 47ndash89

Harbaugh Rick ldquoSkill Reputation ProspectTheory and Regret Theoryrdquo ClaremontGraduate University Institute of EconomicPolicy Studies Working Paper No 2002-032002

Kahneman Daniel ldquoExperienced Utility andObjective Happiness A Moment-Based Ap-proachrdquo in Daniel Kahneman and AmosTversky eds Choices values and framesCambridge Cambridge University Press2000 pp 673ndash92

Kahneman Daniel and Tversky Amos ldquoProspectTheory An Analysis of Decision underRiskrdquo Econometrics 1979 47(2) pp 263ndash91

Kahneman Daniel Wakker Peter P and SarinRakesh ldquoBack to Bentham Explorations ofExperienced Utilityrdquo Quarterly Journal ofEconomics 1997 112(2) pp 375ndash405

Kiecolt-Glaser Janice K McGuire LyanneRobles Theodore F and Glaser Ronald ldquoPsy-choneuroimmunology and PsychosomaticMedicine Back to the Futurerdquo Psychoso-matic Medicine 2002 64(1) pp 15ndash28

Landier Augustin ldquoWishful Thinking A Modelof Optimal Reality Denialrdquo Unpublished Pa-per 2000

Lewis Karen K ldquoTrying to Explain Home Biasin Equities and Consumptionrdquo Journal ofEconomic Literature 1999 37(2) pp 571ndash608

Lichtenstein Sarah Fischhoff Baruch and Phil-lips Lawrence D ldquoCalibration of Probabili-ties The State of the Art to 1980rdquo in DanielKahneman Paul Slovic and Amos Tverskyeds Judgment under uncertainty Heuristicsand biases Cambridge Cambridge Univer-sity Press 1982 pp 306ndash34

Loewenstein George ldquoAnticipation and the Val-


uation of Delayed Consumptionrdquo EconomicJournal 1987 97(387) pp 666ndash84

Lucas Robert E Jr ldquoEconometric PolicyEvaluation A Critiquerdquo in Karl Brunnerand Alan H Meltzer eds The Phillipscurve and labor markets Carnegie Roch-ester Conference Series on Public PolicyVol I Amsterdam North-Holland 1976pp 19 ndash 46

Mullainathan Sendhil ldquoA Memory-BasedModel of Bounded Rationalityrdquo QuarterlyJournal of Economics 2002 117(3) pp735ndash74

Muth John F ldquoOptimal Properties of Exponen-tially Weighted Forecastsrdquo Journal of theAmerican Statistical Association 196055(290) pp 229ndash305

Muth John F ldquoRational Expectations and theTheory of Price Movementsrdquo Econometrica1961 29(3) pp 315ndash35

Myerson Roger B ldquoAxiomatic Foundations ofBayesian Decision Theoryrdquo NorthwesternUniversity Center for Mathematical Studiesin Economics and Management Science Dis-cussion Papers No 671 1986

Nerlove Marc ldquoAdaptive Expectations andCobweb Phenomenardquo Quarterly Journal ofEconomics 1958 72(2) pp 227ndash40

Piccione Michele and Rubinstein Ariel ldquoOn theInterpretation of Decision Problems with Im-perfect Recallrdquo Games and Economic Be-havior 1997 20(1) pp 3ndash24

Poterba James M ldquoEmployer Stock and 401(k)

Plansrdquo American Economic Review 2003(Papers and Proceedings) 93(2) pp 398ndash404

Rabin Matthew and Schrag Joel L ldquoFirst Im-pressions Matter A Model of ConfirmatoryBiasrdquo Quarterly Journal of Economics1999 114(1) pp 37ndash82

Ritter Jay ldquoThe Long-Run Performance of Ini-tial Public Offeringsrdquo Journal of Finance1991 46(1) pp 3ndash27

Savage Leonard J The foundation of statisticsNew York John Wiley 1954

Tversky Amos and Kahneman Daniel ldquoAd-vances in Prospect Theory CumulativeRepresentation of Uncertaintyrdquo Journal ofRisk and Uncertainty 1992 5(4) pp 297ndash323

Ukhov Andrey D ldquoPreferences toward Risk andAsset Prices Evidence from Russian LotteryBondsrdquo Unpublished Paper 2004

Weinstein Neil D ldquoUnrealistic Optimism AboutFuture Life Eventsrdquo Journal of Personalityand Social Psychology 1980 39(5) pp806ndash20

Yariv Leeat ldquoBelieve and Let Believe Axi-omatic Foundations for Belief DependentUtility Functionalsrdquo Yale University CowlesFoundation Discussion Papers No 13442001

Yariv Leeat ldquoIrsquoll See It When I Believe ItmdashASimple Model of Cognitive ConsistencyrdquoYale University Cowles Foundation Discus-sion Papers No 1352 2002


ity flows are happier if they overestimate theprobability that their investments pay off well ortheir future labor income is high

The second crucial element of our model isthat such optimism affects decisions and wors-ens outcomes Distorted beliefs distort actionsFor example an agent cannot derive utility fromoptimistically believing that she will be richtomorrow while basing her consumption-savingdecision on rational beliefs about future income

How are these forces balanced We assumethat subjective beliefs maximize the agentrsquos ex-pected well-being defined as the time-averageof expected felicity over all periods This thirdkey element leads to a balance between the firsttwomdashthe benefits of optimism and the costs ofbasing actions on distorted expectations

We illustrate our theory of optimal expecta-tions using three examples In general a smallbias in beliefs typically leads to first-order gainsdue to increased anticipatory utility and only tosecond-order costs due to distorted behaviorThus beliefs tend toward optimismmdashstateswith greater utility flows are perceived as morelikely Further optimal expectations are lessrational when biases have little cost in realizedoutcomes and when biases have large benefits interms of expected future happiness

More specifically in a portfolio choice prob-lem agents overestimate the return on theirinvestment and prefer skewed returns Henceagents can be risk-loving when investing inlottery-type assets and at the same time risk-averse when investing in nonskewed assetsSecond in general equilibrium agentsrsquo priorbeliefs are endogenously heterogeneous andagents gamble against each other We show inan example that the expected return on the riskyasset is higher than in an economy populated byagents with rational beliefs if the return is neg-atively skewed Third in a consumption-savingproblem with quadratic utility and stochasticincome agents are overconfident and overopti-mistic early in life they consume more thanimplied by rational beliefs

Optimal expectations matches other observedbehavior In a portfolio choice problem Chris-tian Gollier (2005) shows that optimal expecta-tions imply subjective probabilities that focuson the best and worst state This pattern ofprobability weighting is similar to that assumedby cumulative prospect theory to match ob-served behavior (see Daniel Kahneman and

Amos Tversky 1979 Tversky and Kahneman1992) Finally Brunnermeier and Parker (2002)show in a different economic setting that agentswith optimal expectations can exhibit a greaterreadiness to accept commitment regret and acontext effect in which nonchosen actions canaffect utility

Psychological theories provide many chan-nels through which the human mind is able tohold beliefs inconsistent with the rational pro-cessing of objective data First to the extent thatpeople are more likely to remember better out-comes they will perceive them as more likely inthe future leading to optimistic biases in beliefsas in our optimal expectations framework1 Sec-ond most human behavior is not based on con-scious cognition but is automatic processedonly in the limbic system and not the prefrontalcortex (John A Bargh and Tanya L Chartrand1999) If automatic processing is optimisticthen the agent may naturally approach problemswith optimistic biases However the agent mayalso choose to apply cognition to disciplinebelief biases when the stakes are large as in ouroptimal expectations framework

Our model of beliefs differs markedly fromtreatments of risk in economics While earlymodels in macroeconomics specify beliefs ex-ogenously as naive myopic or partially up-dated (eg Marc Nerlove 1958) since John FMuth (1960 1961) and Robert E Lucas Jr(1976) nearly all research has proceeded underthe rational expectations assumption that sub-jective and objective beliefs coincide There aretwo main arguments for this First the alterna-tives to rationality lack discipline But ourmodel provides precisely this discipline for sub-jective beliefs by specifying an objective forbeliefs The second argument is that agents havethe incentive to hold rational beliefs (or act as ifthey do) because these expectations makeagents as well off as they can be However thisrationale for rational expectations relies uponagents caring about the future but at the sametime having their expectations about the future

1 In Sendhil Mullainathan (2002) individuals have im-perfect recall and form expectations as if they did not InMichele Piccione and Ariel Rubinstein (1997) individualsunderstand that they have imperfect recall and in B Doug-las Bernheim and Raphael Thomadsen (2005) individualsadditionally can influence the memory process to increaseanticipatory utility

















(1) EUc1 c2 cT s t







EUc t 1 ct cT s t

t 1 1

t 1

uct uct

E 1

T t

uct st


B Optimal Beliefs














(4) W E1

T t 1

T















(iii) Markets clear


C Discussion














t 1 1

t 1

uct uct

E 1

T t















max

s 1

S

sucs

st cs R Zs

cs 0






max

s 1

S

suR Zs 1R Z1

SR ZS


0 s 1

S


0 1 R Z1

0 S R ZS


(6) 0 s 1

S

s uR Zs Zs f

B Optimal Beliefs





W 12

EE1 Uc E2 Uc


max

12

s 1

S

suR Zs

12

s 1

S

suR Zs




(7) 12

us us

12

s 1

S

suR ZsZs

d

d






if OE 13 0


if OE 0





if EZ 0 then RE 0


if EZ 0 and S 13 2










1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES








































































(1) EUc1 c2 cT s t







EUc t 1 ct cT s t

t 1 1

t 1

uct uct

E 1

T t

uct st


B Optimal Beliefs














(4) W E1

T t 1

T















(iii) Markets clear


C Discussion














t 1 1

t 1

uct uct

E 1

T t















max

s 1

S

sucs

st cs R Zs

cs 0






max

s 1

S

suR Zs 1R Z1

SR ZS


0 s 1

S


0 1 R Z1

0 S R ZS


(6) 0 s 1

S

s uR Zs Zs f

B Optimal Beliefs





W 12

EE1 Uc E2 Uc


max

12

s 1

S

suR Zs

12

s 1

S

suR Zs




(7) 12

us us

12

s 1

S

suR ZsZs

d

d






if OE 13 0


if OE 0





if EZ 0 then RE 0


if EZ 0 and S 13 2










1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES































































(1) EUc1 c2 cT s t







EUc t 1 ct cT s t

t 1 1

t 1

uct uct

E 1

T t

uct st


B Optimal Beliefs














(4) W E1

T t 1

T















(iii) Markets clear


C Discussion














t 1 1

t 1

uct uct

E 1

T t















max

s 1

S

sucs

st cs R Zs

cs 0






max

s 1

S

suR Zs 1R Z1

SR ZS


0 s 1

S


0 1 R Z1

0 S R ZS


(6) 0 s 1

S

s uR Zs Zs f

B Optimal Beliefs





W 12

EE1 Uc E2 Uc


max

12

s 1

S

suR Zs

12

s 1

S

suR Zs




(7) 12

us us

12

s 1

S

suR ZsZs

d

d






if OE 13 0


if OE 0





if EZ 0 then RE 0


if EZ 0 and S 13 2










1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES
































































(4) W E1

T t 1

T















(iii) Markets clear


C Discussion














t 1 1

t 1

uct uct

E 1

T t















max

s 1

S

sucs

st cs R Zs

cs 0






max

s 1

S

suR Zs 1R Z1

SR ZS


0 s 1

S


0 1 R Z1

0 S R ZS


(6) 0 s 1

S

s uR Zs Zs f

B Optimal Beliefs





W 12

EE1 Uc E2 Uc


max

12

s 1

S

suR Zs

12

s 1

S

suR Zs




(7) 12

us us

12

s 1

S

suR ZsZs

d

d






if OE 13 0


if OE 0





if EZ 0 then RE 0


if EZ 0 and S 13 2










1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES
































































(iii) Markets clear


C Discussion














t 1 1

t 1

uct uct

E 1

T t















max

s 1

S

sucs

st cs R Zs

cs 0






max

s 1

S

suR Zs 1R Z1

SR ZS


0 s 1

S


0 1 R Z1

0 S R ZS


(6) 0 s 1

S

s uR Zs Zs f

B Optimal Beliefs





W 12

EE1 Uc E2 Uc


max

12

s 1

S

suR Zs

12

s 1

S

suR Zs




(7) 12

us us

12

s 1

S

suR ZsZs

d

d






if OE 13 0


if OE 0





if EZ 0 then RE 0


if EZ 0 and S 13 2










1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES





























































t 1 1

t 1

uct uct

E 1

T t















max

s 1

S

sucs

st cs R Zs

cs 0






max

s 1

S

suR Zs 1R Z1

SR ZS


0 s 1

S


0 1 R Z1

0 S R ZS


(6) 0 s 1

S

s uR Zs Zs f

B Optimal Beliefs





W 12

EE1 Uc E2 Uc


max

12

s 1

S

suR Zs

12

s 1

S

suR Zs




(7) 12

us us

12

s 1

S

suR ZsZs

d

d






if OE 13 0


if OE 0





if EZ 0 then RE 0


if EZ 0 and S 13 2










1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES































































max

s 1

S

sucs

st cs R Zs

cs 0






max

s 1

S

suR Zs 1R Z1

SR ZS


0 s 1

S


0 1 R Z1

0 S R ZS


(6) 0 s 1

S

s uR Zs Zs f

B Optimal Beliefs





W 12

EE1 Uc E2 Uc


max

12

s 1

S

suR Zs

12

s 1

S

suR Zs




(7) 12

us us

12

s 1

S

suR ZsZs

d

d






if OE 13 0


if OE 0





if EZ 0 then RE 0


if EZ 0 and S 13 2










1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES


























































W 12

EE1 Uc E2 Uc


max

12

s 1

S

suR Zs

12

s 1

S

suR Zs




(7) 12

us us

12

s 1

S

suR ZsZs

d

d






if OE 13 0


if OE 0





if EZ 0 then RE 0


if EZ 0 and S 13 2










1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES































































1 1 2 Z1 Z Z 2

1 2

2 2 Z2 Z Z1 2

2

















maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES
































































maxi

s 1

S

siucs

i

st csi 1 iZs

csi 0


(8) 0 s 1

S

siu1 Zs Zs f i


maxi

12

s 1

S

siucs

i

12

s 1

S

sucsi



























maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES










































































maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES

































































maxct

E 0

T t

uct yt

st 0

T t

Rct yt At





(9) ct Ect 1 yt




(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES


























































(10) ct 1 R1

1 RT t

At yt 1

T t

REyt yt


B Optimal Beliefs







EUt c1 c2 cT yt

t 1 1

t 1

uct uct t

T

t


1

TE uc1

1

T

1

E U1 y1

uc1 uc2 2

T

2E U2

y2

T 1 1

T 1

ucT ucTEUT

yT


(11)1

TE

t 1

T




uctOE

t

tREuct

OE yt




(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES

























































(12) cOEyt a

b

t

tRa

b EcOEyt yt



Tt Ryt




OE E[ct1OE yt]


OE yt] E[ct1OE yt]













OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES






























































OE(yt) 3 ctRE(yt)






1

T t 1

T

t Euct y1


1

T t 1

T

t ucFIy1













V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES



































































V Conclusion











APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES






























































APPENDIX













sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES

























































sZs0

s uR REZsZs sZs0

suR REZsZs 0

yields

sZs0

s uR OEZsZs sZs0

suR OEZsZs 0


(A2) s 1

S

sOE suR OEZsZs 13 0





S

s uR Zs Zs2d

d

d

uR Zs Zs uR Zs Zs

s 1

S

suR ZsZs2

13 0





(A3) s 1

S

s uR OEZsZs 13 0



signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES

























































signus us sign s 1

S

suR OEZsZs

d

d


(A4) 0 s 1

S

s uR OEZsZs




d2Wdd

22W

d

d

2W

2W2 d

d 2

W

d2

d2


W

s 1

S

s s us Zs

2W2

s 1

S

s s u s Zs2

2W

us Zs us Zs



d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES

























































d2Wdd

2uZs Zs u s 1

S

s s Zs2 u

u

Zs Zs

s 1

S

s Zs2 d

d u

s 1

S

s s Zs

d2

d2

2

s 1

S

s s Zs2

s 1

S

s Zs2 Zs Zs u

d

d u

s 1

S

s Zs s 1

S

s Zsd2

d2

1

s 1

S

s Zs2

s 1

S

s Zs2 Zs Zs u

d

d


sZs2 yens1






W1 1 12

1 1uR Z1 12

2 1 1uR Z2




lim131

W1 12

1 1lim131

uR Z1 12

1 1lim131

uR Z2

12

1 1uR Z 12

1 1lim131

uR Z2




W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES



























































W 1 12

sup1

1 1uR Z1 2 1 1uR Z2

st 0

1 13 1

R Z2 0


lim131

W1 1

2 lim

131

supRZ2 0

1 1uR Z1 1 1uR Z2

1

2 lim

1311 1uR

R

Z2Z1 1 1uR uR





W P 12

s 1

S

s u1 Zs 12

s 1

S

s u1 Zs






limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES



























































limPn11

W P limPn11

12

s 1

S

su1 Zs 12

s 1

S

su1 Zs

12

u1 12

limPn11

s 1

S

su1 Zs

12

u1 12

u1 u1



W P s 1

S

s u1 Zs


s 1

S

s u1 Zs 13 u1









ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES




























































ctOE Ect 1

OE yt a

b

t 1

tRa

b Ect 1

OE yt Ect 1OE yt

1 t 1

tRa

b Ect 1

OE yt



t 1

tR

t

t 1 R

1 1

T t1

1 1

T t

R

1 1

T t1

1 1

T t1

T t T t

1

therefore

ctOE Ect 1

OE yt 13 0


Ect 1OE yt ct

OE 13 Ect 1OE yt


ct 1OE

1 R1




Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES

























































Ect 1OE yt 13 Ect 1

OE yt

E 1 R1





t 1

t

1 1

T t1

1 1

T t








REFERENCES



























































































































Documents

Optimal Expectations - Princeton...Optimal Expectations By M ARKUS K. B RUNNERMEIER AND JONATHAN A. P ARKER * Forward-looking agents care about expected future utility ßows, and hence