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Reference-Dependent Risk Attitudes

By BOTOND KOSZEGI AND MATTHEW RABlN*

We use Koszegi and Rabin's (2006) model of reference-dependent utility, and anextension of it that applies to decisions with delayed consequences, to study pref-erences over monetary risk. Because our theory equates the reference point withrecent probabilistic beliefs about outcomes, it predicts specific ways in which theenvironment influences attitudes toward modest-scale risk. It replicates "classical"prospect theory-including the prediction of distaste for insuring losses-whenexposure to risk is a surprise, but implies first-order risk aversion when a risk,and the possibility of insuring it, are anticipated. A prior expectation to take onrisk decreases aversion to both the anticipated and additional risk. For large-scalerisk, the model allows for standard "consumption utility" to dominate reference-dependent "gain-loss utility," generating nearly identical risk aversion across situ-ations. (JEL D81)

Daniel Kahneman and Amos Tversky's (1979)prospect theory, and the literature building fromit, provide theories of risk attitudes based on afew regularities. Most importantly, evaluationof an outcome is influenced by how it comparesto a reference point, with people exhibiting botha significantly greater aversion to losses thanappreciation of gains, and a diminishing sen-sitivity to changes in an outcome as it movesfarther from the reference point. In addition,people weight the probability of a prospect non-linearly, overweighting small probabilities andunderweighting high probabilities.

The implications of prospect theory havebeen studied with several different specifica-tions of the reference point, including the statusquo, lagged status quo, and the mean of the cho-sen lottery. These various approaches explainmany risk attitudes that are inconsistent with theclassical diminishing-marginalutility-of-wealth

* Koszegi: Department of Economics, University ofCalifornia, Berkeley, 549 Evans Hall, 3880, Berkeley, CA94720 (e-mail: [email protected]); Rabin: Depart-ment of Economics, University of California, Berkeley,549 Evans Hall, 3880, Berkeley, CA 94720 (e-mail: [email protected]). We are grateful to Paige Skiba andJustin Sydnor for research assistance, and to Erik Eyster,Vince Crawford, and three anonymous referees for detailedsuggestions. We also thank seminar participants at HarvardUniversity, London School of Economics, StockholmSchool of Economics, and UC Berkeley for comments.Rabin thanks the National Science Foundation for financialsupport under grant SES-0518758, and Koszegi thanks theCentral European University for its hospitality while someof this work was completed.

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model, but they also generate mutually inconsis-tent predictions that to our knowledge have notbeen formally reconciled. Under a status quospecification, loss aversion predicts the substan-tial dislike of modest-scale risks involving bothgains and losses that has been widely observed,and diminishing sensitivity predicts the risk lov-ingness in high-probability losses found by manyresearchers in the laboratory. 1 Under specifica-tions based on the lagged status quo-such asin Richard H. Thaler and Eric 1. Johnson (1990)and Francisco Gomes (2005)-diminishing sen-sitivity predicts the willingness to take unfavor-able risks to regain the previous status quo. This"disposition effect," which has been observedby Terrance Odean (1998) for small investorsand by David Genesove and Christopher Mayer(2001) for homeowners, is inconsistent with thesubstantial risk aversion predicted by a statusquo model for gambles involving both gains

1 As has been pointed out by many researchers and formal-ized by Rabin (2000), nontrivial modest-scale risk aversionis calibrationally inconsistent with the classical diminish-ing-marginal-utility-of-wealth model. For instance, aperson with $1 million in lifetime wealth who has CRRAutility and who rejects a "fifty-fifty lose $500 or gain $550"gamble would also turn down an equal-probability bet oflosing $4,000 or gaining $100,000,000,000,000. Whilenobody would turn down the large-scale bet, most peoplewould turn down the smaller gamble. Nicholas Barberis,Ming Huang, and Thaler (2006), for example, find that themajority of MBA students, financial advisors, and evenvery rich investors (with median financial wealth over $10million) reject the $500-$550 bet.

1048 THE AMERICAN ECONOMIC REVIEW SEPTEMBER 2007

and losses relative to the current status quo. Andunder specifications based on the chosen lottery'scertainty equivalent-such as in the disappoint-ment-aversion models of David E. Bell (1985),Graham Loomes and Robert Sugden (1986),and Faruk Gul (1991)-loss aversion impliessubstantial aversion to any risk. This strong riskaversion, which is apparent in consumers' choiceof low insurance deductibles and purchase ofextremely expensive extended warranties andautomobile service contracts, is inconsistent withthe risk lovingness in losses found in the lab andin the case of the disposition effect.

This paper uses the model from K6szegi andRabin (2006) and an extension to study monetaryrisk, unifying the seemingly different risk atti-tudes noted above as manifestations of the samepreferences in different domains, and makingnovel predictions about behavior in situationsnot studied in the related literature. Our model(a) combines the reference-dependent "gain-lossutility" with standard "consumption utility";(b) bases the reference point to which outcomesare compared on endogenously determinedlagged beliefs; and, to incorporate probabilisticbeliefs, (c) allows for stochastic reference points.Because of feature (a), our theory is consistentwith aversion to all large-scale risk as predictedby classical expected-utility theory. Because offeature (b), it predicts both risk lovingness inresponse to surprise modest-scale losses, and-since anticipated premium payments do notgenerate sensations of loss while bad outcomesin uncertain situations do-first-order risk aver-sion when a risk and the possibility to insure itare expected. Hence, our theory matches bothstatus quo prospect theory and disappointmentaversion in domains where these models havebeen applied, and more generally provides com-parative-statics predictions on the extent of risktaking as a function of the environment. Becauseof features (b) and (c), our theory predicts thatthe prior expectation of risk, even if it can nowbe avoided, decreases risk aversion. Unlike allthe theories above, therefore, it predicts lessrisk aversion when deciding whether to removeexpected risk than when deciding whether totake on that risk.

For a wealth level wand reference wealthlevel r, Section I specifies a person's utilityas u(wlr) == m(w) + 1L(m(w) - m(r)). Thereference-independent "consumption utility,"

m (w ), corresponds to the classical notionof outcome-based utility. Gain-loss utility,1L(m(w) - m(r)), depends on the differencebetween the consumption utility of the outcomeand of the reference level, with the shape of 1Lcorresponding to the loss aversion and dimin-ishing sensitivity of prospect theory. Some ofour results are established by assuming onlywhat is commonly taken to be the stronger ofthese two forces, loss aversion.

We assume that the reference point relativeto which a person evaluates an outcome is herrecent beliefs about that outcome. An employeewho had expected a $50,000 salary will assess asalary of $40,000 as a loss, and a taxpayer whohad expected to pay $30,000 in taxes will treata $20,000 tax bill as a gain. Because a personmay be uncertain about outcomes, our theoryallows for the reference point to be a distribu-tion G( . ), with a wealth outcome w then evalu-ated with "mixed feelings" as the average of thedifferent assessments u( wlr) generated by the rpossible under G( . ). For simplicity, we abstractfrom nonlinear decision weights: given a (sto-chastic or deterministic) reference point, a sto-chastic wealth outcome is evaluated accordingto its expected reference-dependent utility.

Our model of how utility depends on beliefscould be combined with any theory of how thesebeliefs are formed. As an imperfect but at thesame time disciplined and largely realistic firstpass, we assume that a person correctly pre-dicts her probabilistic environment and her ownbehavior in that environment, so that her beliefsfully reflect the true probability distribution ofoutcomes. We begin in Section II by considering"surprise" (low-probability) decisions, modelledin extreme form as situations where expectationsare given exogenously to the actual choice set.To illustrate implications for modest-scale risk,where consumption utility is approximately lin-ear, consider a person's decision on whether topay $55 to insure a 50 percent chance of hav-ing to pay $100. If she had expected to retainthe status quo of $0, our model makes the sameprediction as prospect theory: because of dimin-ishing sensitivity, she does not wish to insure therisk. If she had expected to pay $55 for insur-ance, however, paying that amount generates nogain or loss, while taking the gamble exposesher to a fifty-fifty chance of losing $45 or gain-ing $55. With a conventional estimate of two-

VOL. 97 NO.4 KOSZEGI AND RABIN: REFERENCE-DEPENDENT RISK ATTITUDES 1049

to-one loss aversion, she strongly dislikes thisgamble and buys the insurance. Yet if a personhad been expecting risk to start with, paying $0instead of $100 can decrease expected losses,and paying $100 might just decrease expectedgains, so the gamble is less aversive. When theex ante expected risk is the gamble itself, thisdecreased risk aversion can be interpreted asan endowment effect for risk. When the ex anteexpected uncertainty is very large, $100 cannotmuch change the extent to which money is eval-uated as a loss rather than a gain, so the personis close to risk neutral.

In Sections III and IV, we study attitudes toanticipated risks. We identify two implications ofour model: a person is more risk averse when sheanticipates a risk and the possibility to insure itthan when she does not-always displaying first-order risk aversion-and among such decisionsregarding anticipated risk, she is more risk aversewhen she can commit to insure ahead of time.

When a decision is made shortly before theoutcomes resulting from it occur, at that momentthe reference point is fixed by past expectations,so that the decision maker maximizes expectedutility taking the reference point as given. Beingfully rational, therefore, she can expect behavioronly if she is willing to follow it through, givena reference point determined by the expectationto do so. Formalizing this idea, in Section IIIwe import from K6szegi and Rabin (2006) theconcept of an "unacclimating personal equi-librium" (UPE), defined as behavior where thestochastic outcome generated by utility-maxi-mizing choices conditional on expectationscoincides with expectations. Positing that aperson can make any plans she knows she willfollow through, our analysis assumes that shechooses her favorite UPE, the "preferred per-sonal equilibrium" (PPE).

Applying PPE, we predict a very strong tastefor planning and executing the purchase of small-scale insurance. The reason is a formalizationand elaboration of some previous researchers'(e.g., Kahneman and Tversky 1984) psychologi-cal intuition that money given up in regular bud-geted purchases is not a loss. In our model, a badoutcome of an uncertain lottery is evaluated asa loss, but a fully expected premium payment isnot evaluated as a loss. Loss aversion thereforegenerates first-order risk aversion toward allinsurable risks.

When a person makes a committed decisionlong before outcomes occur, she affects the ref-erence point by her choice. For these situations,we introduce in Section IV the idea of a "choice-acclimating personal equilibrium" (CPE), definedas a decision that maximizes expected utilitygiven that it determines both the reference lotteryand the outcome lottery. Except that we specifythe reference point as a lottery's full distributionrather than its certainty equivalent, this conceptis similar to the disappointment-aversion modelsof Bell (1985), Loomes and Sugden (1986), andGul (1991). Like PPE, CPE predicts that the deci-sion maker strongly prefers to insure expectedrisks. But there is also an important difference.In some situations, a person would be better offwith the reduced uncertainty of expecting to buyexpensive insurance than not expecting to do so,but always choosing to avoid the expense of insur-ance at the last minute if not committed to buy-ing it. Hence, the distaste for the risk manifestsitself in behavior when the insurance decision ismade up front, as in CPE, but not when the deci-sion is made later, as in PPE. In such situations,environments where CPE is appropriate generategreater risk aversion and higher expected utilitythan environments where PPE applies.

The sensitivity of behavior to the economicenvironment described above applies only tomodest-scale choices, where risk attitudes arenecessarily dominated by the gain-loss compo-nent of preferences. In Section V, we investigateattitudes toward large-scale risk, where con-sumption utility cannot be assumed to be linear.We show that under reasonable conditions, thereference point has only a minor impact on howa person evaluates very large gambles. A personis therefore prone to exhibit risk aversion reflect-ing diminishing marginal utility of wealth inde-pendently of the environment.

Beyond helping to explain in a unified frame-work seemingly contradictory behavior, wehope the endogenous specification of the refer-ence point helps make our model readily por-table to many settings. To facilitate applications,in Appendix A we present an array of risk-characterization concepts and results. In SectionVI, we conclude the paper by discussing someof the shortcomings of our model, emphasizingespecially its failure to capture important waysthat reference-dependent risk attitudes reflectfailures of full rationality.


I. Reference-Dependent Utility

In this section we present the one-dimen-sional version of the utility function in K6szegiand Rabin (2006). As is standard for models ofrisky choice outside of full-fledged life-cycleconsumption models in macroeconomics, ourtheory takes as a primitive the choice set ofgambles a person focuses on, in isolation fromother risks and choices she faces.

For a riskless wealth outcome wEIR andriskless reference level of wealth r E IR, utility isgiven by u(wlr) == m(w) + 1L(m(w) - m(r)).2The term m( w) is intrinsic "consumption util-ity" usually assumed relevant in economics,and the term 1L(m(w) - m(r)) is the reference-dependent gain-loss utility. Our model assumesthat how a person feels about gaining or los-ing relative to a reference point depends onthe changes in consumption utility associatedwith such gains or losses. This separation andinterdependence of economic and psychologi-cal payoffs is analogous to assumptions madepreviously by Bell (1985), Loomes and Sugden(1986), and Veronika Kobberling and Peter P.Wakker (2005).

To accommodate our assumption below thatthe reference point is beliefs about outcomes, weallow for the reference point to be a probabilitymeasure Gover IR:

(1) U(wIG) = 1u(wlr) dG(r).

This formulation captures the notion that theevaluation of a wealth outcome is based on com-paring it to all possibilities in the support of the

2 Like many models, this paper assumes preferences areover monetary wealth rather than consumption. Strictlyspeaking, this means our model corresponds to a single-period setting. While we have not verified how our resultsextend to a multiperiod consumption model, the shortcutof using wealth may be appropriate even in that case, bothbecause people often experience sensations of gain andloss directly from wealth changes, and because wealth isa summary statistic for consumption and hence may gen-erate similar gain-loss sensations. As a person's wealthincreases, for example, her anticipation of increased con-sumption throughout the future is likely to generate a senseof gain similar to that presumed in our model. A developingbody of research on dynamic models of reference-depen-dent utility, such as Rebecca Stone (2005) and Koszegi andRabin (2007), may help extend and explore the robustnessof the results we develop in this paper.

reference lottery. For example, if the referencelottery is a gamble between $0 and $100, an out-come of $50 evokes a mixture of two feelings, again relative to $0 and a loss relative to $100.3

When w is drawn according to the probabilitymeasure F, utility is given by

(2) U(FIG) = 11u(wlr) dG(r)dF(w).

For simplicity and contrary to Kahneman andTversky (1979) and its extensions, we assumethat preferences are linear in probabilities. Thismeans that our model will get some predic-tions-e.g. regarding insurance of low-prob-ability losses-wrong.

Our utility function is closely related to thatof Sugden (2003). In his model, outcome lotter-ies are compared to reference lotteries state bystate, capturing a form of state-contingent dis-appointment missing from our theory.4 Anotheralternative to our formulation, pursued by Bell(1985), Loomes and Sugden (1986), Gul (1991),and Jonathan Shalev (2000), is to collapse thereference lottery into some type of certaintyequivalent.5 With such a specification, two

3 More than saying a person separately compares anoutcome to all components of the reference lottery, ourformulation of p,O below implies that losses relative to astochastic reference point count more than gains, so that the$50 above yields negative gain-loss utility. An alternativespecification is one where the relief of avoiding the $0 out-come outweighs the disappointment of not getting the $100outcome. This alternative seems difficult to reconcile withloss aversion relative to riskless reference points. It wouldalso seem to imply that people will seek to endow them-selves with risks because the chance for a pleasant relieffrom getting better outcomes outweighs the potential disap-pointment from getting bad outcomes.

4While this state-contingency seems in some cases to bemore realistic than our approach, and we do not know theextent to which the two models can be reconciled withina broader framework, our prediction of a state-indepen-dent disappointment when receiving worse-than-expectedoutcomes seems pervasively realistic, and is missing fromSugden's approach. In addition, Sugden's model (unlikeours) makes the implausible prediction that a personbecomes very risk loving when given the option to replacethe reference lottery with a riskless amount.

5 There is some suggestive evidence of mixed feelingswhen there are multiple counterfactuals relative to whichoutcomes can be evaluated. For instance, JeffT. Larsen et al.(2004) find that when a subject receives $5 from a lotterythat could have paid $5 or $9, she has both positive andnegative emotions-presumably from winning $5 and not

reference lotteries that have the same certaintyequivalent generate the same risk preferences.This is inconsistent with our theory's predictionthat a person is more inclined to accept a risk ifshe had been expecting risk, a prediction thatseems broadly correct based on the little avail-able evidence. But as we discuss below, the maindifference between all these models and ours isin the specification of the reference point.

We assume fL satisfies the following properties:

AO. fL(X) is continuous for all x, twice dif-ferentiable for x "* 0, and fL( 0) = o.

Ai. fL(X) is strictly increasing.

A2. Ify > x ~ 0, then fL(Y) + fL(-Y) < fL(X)+ fL(-X).

A3. fL"(X):::; 0 for x> 0 and fL"(X) ~ 0 forx<o.

A4. fL~(O)/fL~(O) == A > 1, where fL~(O) ==limHo fL' (lxl)and fL~(O) == limx....ofL'(-lxl).

Properties AO-A4, first stated by DavidBowman, Deborah Minehart, and Rabin (1999),correspond to Kahneman and Tversky's (1979)explicit or implicit assumptions about their"value function" defined on w - r. Loss aver-sion is captured by A2 for large stakes and A4for small stakes, and diminishing sensitivity iscaptured by A3. While the inequalities in A3 aremost realistically considered strict, to charac-terize the implications of loss aversion withoutdiminishing sensitivity as a force on behavior,we define a subcase of A3:

A3 I. For all x "* 0, fL" (x) = O.

When we apply A3' below, we will param-eterize fL as fL~(O) = TJ and fL~(O) = ATJ > TJ, sothat TJ can be interpreted as the weight attachedto gain-loss utility.

To determine behavior, the utility functionintroduced above needs to be combined witha theory of reference-point determination. Asa disciplined and largely realistic first pass, we

winning $9, respectively. Losing $5 when it is also possibleto lose $9 evokes similar mixed feelings.

assume that a person's reference point is therational expectations about the relevant out-come she held between the time she first focusedon the outcome and shortly before it occurs. Forexample, if an employee had been expecting asalary of $100,000, she would assess a salaryof $90,000, not as a large gain relative to herstatus quo wealth, but as a loss relative to herexpectations of wealth.6 As we explain in detailin K6szegi and Rabin (2006), our primary moti-vation for equating the reference point withexpectations is that this assumption helps unifyand reconcile existing intuitions and discus-sions. We assume rational expectations bothto maintain modelling discipline (much likein other rational-expectations theories), andbecause we feel in most situations people havesome ability to predict their own environmentand behavior. Unfortunately, relatively little evi-dence on the determinants of reference pointscurrently exists. Some existing evidence does,however, provide empirical support for expec-tations as a component of the reference point.In analyzing play in the huge-stakes game show"Deal or No Deal," for example, Thierry Post etal. (forthcoming) find evidence that past expec-tations affect behavior. In the game, a contestant"owns" a suitcase with a randomly determinedprize. Gradually, the contestant learns informa-tion about the prize in her bag (by opening otherbags and learning what is not in her bag). Ateach stage, a "bank" offers a riskless amount ofmoney to replace the amount in the bag. A con-testant's acceptance or rejection of the offer isan indication of her risk aversion. A key findingis that contestants reject better offers when theyhave received bad news in the last few rounds,suggesting that they are less risk averse in thesecontingencies.7

6 Our theory posits that preferences depend on laggedexpectations, rather than on expectations contemporaneouswith the time of consumption. This does not assume thatbeliefs are slow to adjust to new information or that peopleare unaware of the choices they have just made-but thatpreferences do not instantaneously change when beliefs do.When somebody finds out five minutes ahead of time thatshe will for sure not receive a long-expected $100, she pre-sumably immediately adjusts her expectations to the newsituation, but five minutes later she will still assess not get-ting the money as a loss.

7 For further examples of evidence of expectations-based counterfactuals affecting reactions to outcomes,


Many researchers have noted over the yearsthat the reference point may to some extent beinfluenced by expectations. But most previousformal models either equate the reference pointwith the status quo, or leave it unspecified, andnone explicitly equates it to recent beliefs aboutoutcomes. To our knowledge, the disappoint-ment-aversion models by Bell (1985), Loomesand Sugden (1986), and Gul (1991), come closestto saying that the reference point is recent expec-tations.8 But because these models assume thereference point is formed after choice, they treatsurprise situations differently from our theory,predicting in particular first-order risk aversionfor surprise losses. Our model can be thoughtof in part as unifying prospect theory and dis-appointment-aversion theory in one framework,while also proposing a solution concept (PPE)for situations where choices are anticipated butnot committed to in advance. We are not awareof any model that attempts such a unification.

Although our model is in the tradition of mostof economics in that it posits a utility functionwith certain properties, it differs from muchof the foundational literature on choice underuncertainty in that it does not derive the utilityfunction from axioms capturing those properties.It also differs from most economic theories inmaking explicit how utility depends on a mentalstate-beliefs-that is not directly observablein choice behavior. Obviously, neither of thesefeatures means that our model does not haveobservable or falsifiable implications. Indeed, inAppendix B we show how to extract the full util-ity function (m( . ) and f.L( . )) from behavior ina limited set of decision problems, and becauseour model provides a complete mapping fromdecision problems to possible choices, this com-pletely ties down predictions for all other deci-sion problems. And the many propositions in thepaper derive general restrictions that our modelimplies for observed behavior.

see Victoria Husted Medvec, Scott F. Madey, and ThomasGilovich (1995), Barbara Mellers, Alan Schwartz, and IlanaRitov (1999), and Hans C. Breiter et al. (2001).

8 In addition, the notion of "loss-aversion equilibrium"that Shalev (2000) proposed for multiplayer games canbe interpreted as saying that each player's reference pointis expectations. As we discuss below, rewriting Shalev'smodel using our specification of stochastic reference pointsand applying it to individual decision making correspondsto the UPE solution concept.

II. Risk Attitudes in Surprise Situations

In the next three sections, we investigate thedecision maker's attitudes toward modest-scalerisk, such as $100 or $1,000, where consump-tion utility can be taken to be approximatelylinear-and where we therefore derive formalresults under the assumption thatm(w) = w.9 InSection V, we return to an exploration of large-scale risk, where risk preferences can be sub-stantially influenced by diminishing marginalutility of wealth. We organize our results onmodest-scale risk into three sections accord-ing to the expectational environment the deci-sion maker faces, considering in turn "surprise,"PPE, and CPE situations. But a number ofthemes link the three sections. Propositions 1,2, 5, and 6 identify common ways in which thedecision maker becomes less risk averse if shehad been expecting, or is now facing, more risk.Proposition 3 shows that a person is first-orderrisk averse when she anticipates a risk as wellas the possibility to insure it. And Propositions4, 7, and 8 show that the more a risk can be pre-pared for, the greater is the risk aversion dis-played in behavior: the person's risk aversion isgreater when she expects an insurance decisionthan when she does not, and even greater whenshe can commit early to purchasing insurance.

This section begins the analysis with risk-tak-ing behavior when the reference point is fixed,considering both deterministic and stochasticreference points. The analysis is the limitingcase of UPE/PPE behavior when the decisionmaker finds herself in an ex ante low-probabil-ity situation, so that she has fixed expectationsformed essentially independently of the relevantchoice set.

9 If mO were not linear (but remained differentiable),some of our results would survive unchanged, and theothers could be modified by restating them in terms ofexpected consumption utilities instead of expected values.But because this would complicate our statements, andbecause consumption utility is so close to linear for mod-est stakes, we assume m(') is linear. To give a sense of thecalibrational appropriateness of this approximation, notethat even for a person who has a low $1 million in lifetimewealth and a very high consumption-utility coefficient ofrelative risk aversion of 10, winning or losing $1,000 (a dif-ference of $2,000 in wealth) changes marginal consump-tion utility by only 1.8 percent.


Proposition 2 of K6szegi and Rabin (2006)shows that when the decision maker expectsto keep the status quo, her behavior is identicalto that predicted by prospect theory modifiedto equate decision weights with probabilities.Hence, our model is consistent with much ofthe evidence motivating status quo prospecttheory. It can also be used to interpret the"disposition effect" found by Odean (1998)for stocks and Genesove and Mayer (2001) forhouses-whereby people appear disproportion-ately reluctant to sell an asset for less than theypaid.1O The intuition commonly invoked for thedisposition effect, formalized by Gomes (2005)in a version of prospect theory based on thelagged status quo, is that because the purchaseprice operates as a reference point for the sell-ing price, people will be risk-seeking in waitingfor a price to recover before selling. 11 While ourmodel cannot explain all aspects of the disposi-tion effect, by basing the reference point on theexpected resale price, it does help to determinewhen and how the effect is likely to be observed.Because home and stock owners usually expectto make money, they will be risk loving whenthese investments unexpectedly lose money. Butwhen a person foresees a good chance of losingsome money-for example, when investing in ahighly risky stock-she will be less willing totake chances to break even. And if she expectsan investment-such as a house in a boomingmarket or inventory a merchant expects to resellat a large margin-to make a large positivereturn, she may even be reluctant to sell at pricesinsufficiently above the purchase price.

While our model only replicates or qualifiesclassical prospect theory in the settings above,

10 The disposition effect is closely related to the "break-even effect" coined by Thaler and Johnson (1990) formonetary gambles. They predicted that following losses,gambles that offer a chance to break even become espe-cially attractive.

11 Barberis and Wei Xiong (2006) show that prospecttheory based on the lagged status quo does not necessarilyimply an increased risk lovingness after losses. Intuitively,for risks a person takes on voluntarily, losses are typicallysmaller than gains. Hence, a person is typically closer tothe reference point after a loss than after a gain, and dueto loss aversion this can mean greater aversion to substan-tial amounts of risk. In the Odean (1998) and Genesove andMayer (2001) studies, however, individuals can take moreincremental risk-waiting more or less exactly until theprice returns to the reference point.

it makes a set of novel predictions regarding theeffect of prior uncertainty on behavior, identi-fying senses in which the expectation of riskdecreases aversion to the expected as well asadditional risks. To state these results, we useH + H' to denote the distribution of the sum ofindependent draws from the distributions HandH'. (Thus, (H + H')(z) = fH(z - s) dH'(s).)When it creates no confusion, a real number willdenote both a deterministic wealth level and thelottery that assigns probability 1 to that amountof wealth. Proposition 1 says that under A3 I, aperson is no more willing to accept a given lot-tery if it is added to a riskless reference pointthan if it is added to a lottery and/or evaluatedrelative to a risky reference point.

PROPOSITION 1: Suppose m( . ) is linear andf.L( . ) satisfies A3'. For any lotteries F, G, andH and constant w, if U(w + Flw) ~ U(wlw),then U(H + FIG) ~ u(HIG).

Since m( . ) is linear and A3' is satisfied, asmall change in an outcome is evaluated solelyaccording to the previously expected probabili-ties of getting higher and lower outcomes, andnot according to the distance from those higherand lower outcomes. Now when F is added to ariskless reference point w, positive outcomes ofF are assessed as pure gains, and negative out-comes of F are assessed as pure losses. But whenF is instead added to a lottery H and is evaluatedrelative to a lottery G, positive outcomes of Fpartially eliminate losses suffered from H rela-tive to G, and are hence evaluated more favor-ably than pure gains; and negative outcomes ofF in part merely eliminate gains from H relativeto G, and are hence evaluated less unfavorablythan pure losses. For both these reasons, thedecision maker is more willing to accept F.

An important implication of Proposition 1-obtained by setting H = wand G = F -is a typeof endowment effect for risk: a person is less riskaverse in eliminating a risk she expected to facethan in taking on the same risk if she did notexpect it. This prediction of our model contrastswith previous theories of reference-dependentutility with which we are familiar. While littleevidence on the issue seems to be available,choice experiments by Jack L. Knetsch and 1.A. Sinden (1984) and evidence on hypotheticalchoices in Michael H. Birnbaum et al. (1992) do

indicate that people tend to be less risk aversewhen selling a lottery they are endowed withthan when buying the lottery.

A second way in which expecting riskdecreases risk aversion is that a person isapproximately risk neutral in accepting a lotterythat is "small" relative to the reference lottery.

PROPOSITION 2: Suppose m(') is linear. Forany lottery F with positive expected value:

(i) There exist A, e > 0 such that if the lot-tery G satisfies Prdr E (k - A, k + A)] <e for all constants k, then U(H + FI G) >U(HI G) > U(H - FI G)for any lottery H.

(ii) For any continuously distributed lotteryG, there is at> 0 such that for any t E(0, t] and any lottery H, U(H + t . FIG) >U(HIG) > U(H - t . FIG).

The proposition identifies attitudes towarda given lottery F with positive expected value,showing ways in which the decision maker is riskneutral when evaluating multiples of F. Part Isays that if the reference lottery is sufficientlywidely distributed, the decision maker will takeF and reject -F. Part 2 says that fixing a con-tinuously distributed reference lottery, she willtake a sufficiently small multiple of F and rejectthe same multiple of - F. Intuitively, if the extralottery is small relative to the reference lotteryG, it is unlikely to turn a gain or loss relativeto G into the opposite, and there is little dimin-ishing sensitivity over its range. Hence, neitherloss aversion nor diminishing sensitivity plays amajor role in its evaluation.

Part 2 of Proposition 2 is related to Proposi-tion I in Barberis, Huang, and Thaler (2006)showing that a loss-averse (in their terminology"first-order risk-averse") decision maker is onlysecond-order risk averse if she faces backgroundrisk. In addition, Proposition 2 demonstratesthat even if a person now faces no backgroundrisk (H is riskless), the mere prior expectation ofrisk makes her second-order risk averse.

Crucially for a number of later results, Propo-sitions I and 2 do not imply that a person isunbothered by risk she expects or faces, evenif it does make her less risk averse. In fact, thesame force that decreases risk aversion alsodecreases expected utility: when the decision

maker had been expecting risk or is alreadyfacing risk, she is already exposed to stochasticutility-decreasing sensations of loss, so takingadditional risk does not add so much exposureto losses.

To illustrate immediate and later points,Table I shows some of our model's implicationsin a parameterized example with consumptionutility m( w) = 10,000 In (w), and gain-loss util-ity f.L(x) = \IX for x ~ 0 and f.L(x) = -3V=Xfor x :::;0,12 We consider three fifty-fifty gamblesof different scales. Rows correspond to environ-ments in which the gambles might be evaluated.For each gamble, the first column identifies thepremium the decision maker is willing to payfor insurance against the risk in each environ-ment, calculated as the difference between thegamble's mean and its certainty equivalent. Thesecond column gives the coefficient of relativerisk aversion fJ that would be inferred from thegamble's certainty equivalent if one assumedreference-independent CRRA utility.

Gamble I is a small-stakes gamble that payseither $1,000,100 or $999,900. The first panelof the table considers surprise situations withvarious (exogenous) expectations. The decisionmaker displays extreme risk lovingness if thegamble is a small loss relative to expectations,and extreme risk aversion if it is a small gain orinvolves both a loss and a gain. Although lossaversion is a stronger force than diminishingsensitivity, the decision maker displays morerisk aversion for gains than for mixed gamblesbecause in the latter case she evaluates the cer-tainty equivalent as a lossY And as Propositions Iand 2 predict, the prior expectation of riskdecreases risk aversion, with expected risk onthe order of $1,000 or $10,000 already lowering

12 Note that, unless A3' holds, our model is not invariantto affine transformations of mO, so the appropriate speci-fication of mO and MO involves a substantive assumptionabout their relative scaling. This scaling amounts to anassumption about the speed of diminishing sensitivity ingain-loss utility. Indeed, the choice to specify our examplewith m(w) = 1O,OOOln(w)is designed to get gain-loss utilityto dominate for an appropriate range of small stakes, andconsumption utility to dominate for an appropriate rangeof large stakes.

13 Indeed, while not shown in the table, the risklessreference point that makes the person indifferent betweenthe reference point and the gamble-a situation where theinsurance premium is not evaluated as a loss-correspondsto a much higher, $48 premium for insurance.

TABLEI-ATTITUDES TOWARDTHREE50-50 GAMBLES

Gamble I Gamble II Gamble III$999,900/$1,000,100 $990,000/$1,010,000 $500,000/$1,500,000

WTP P WTP P WTP PSurprise with different (fixed) expectations$500,000 $0.01 1.01 $50 1.01 $135,840 1.02$990,000 0.02 3.4 329 6.5 132,477 0.99$999,900 19.2 3,936 469 9.4 132,248 0.99$1,000,000 9.17 1,844 432 8.6 132,246 0.99$1,000,100 -33.1 -7,171 397 7.9 132,244 0.99$1,010,000 -<0.01 -5.6 -712 -14 132,028 0.99$1,500,000 <0.01 0.97 49 0.97 128,429 0.96U[$999,900, $1,000, 100] 3.66 733 433 8.7 132,246 0.99U[$999,000, $1,001,000] 0.27 52 507 10.2 132,246 0.99U[$990,000, $1, 010, 000] 0.02 3.5 275 5.5 132,248 0.99

PPE with different amounts of background risknone $48.0 11,389 $757 15 $138,502 1.04U[-$100, $100] 8.54 1,717 779 14 138,467 1.04U[-$I,OOO, $1,000] 0.56 III 656 13 138,394 1.03U[-$10,000, $10,000] 0.03 6.0 259 4.7 138,176 1.03

CPE with different amounts of background risknone $71.0 23,348 $757 15 $138,502 1.04U[-$100, $100] 28.5 6,041 779 14 138,467 1.04U[-$I,OOO, $1,000] 1.84 368 656 13 138,394 1.03U[-$10,000, $10,000] 0.07 14 362 7.2 138,176 1.03

Notes: The table assumes a consumption utility of m (w) = 10, 000 In (w) and a gain-loss utility of p, (x) = \IX for x ~ 0 andp,(x) = -3v'"=X for x < O. For each gamble and environment, we calculated the gamble's certainty equivalent in that envi-ronment. WTP is the difference between the gamble's mean ($1,000,000 in each case) and the certainty equivalent. This isthe premium the decision maker is willing to pay for insurance. p is the coefficient of relative risk aversion that would beinferred from the gamble's certainty equivalent if one assumed reference-independent CRRA utility. U[x, y] refers to a dis-crete uniform distribution between x and y, with atoms on multiples of $50.

the decision maker's wiHingness to pay forinsurance to essentiaHy zero.

III. UPE and PPE Risk Attitudes

While the previous section considered riskattitudes in surprise situations, our primaryinterest is in behavior when the decision makercorrectly predicts the choice set she faces. Fromthe psychological hypothesis that the referencepoint for evaluating outcomes is equal to laggedrational beliefs about those outcomes, we developtwo reduced-form models that differ in whenthe decision maker makes a committed choice.In this section, we analyze her behavior in oneextreme possibility, when she anticipates thedecision she faces but cannot commit to a choiceuntil shortly before the outcome. Although ourmodel is ambiguous as to the interpretation of"shortly before," insurance choices on short-term rentals such as cars or skis probably bestcorrespond to such a case. The resulting model

is equivalent to the one-dimensional version ofthe model in K6szegi and Rabin (2006).

Suppose the decision maker has probabilis-tic beliefs over possible compact choice setsdescribed by {Dj, 1 - q; D2, q}, where choicesets Dj, D2 C Ll (IR) occur with probabilities1 - q and q, respectively. AH of our results inthe next two sections are for the case q = 0, sothat the decision maker knows the single choiceset she will face. But we keep our definition alittle more general because-as we explainbelow-this aHows us to capture the kinds ofsurprise situations discussed in the previous sec-tion as limiting cases of low-probability choicesets.

Since the person makes her decision shortlybefore the outcome resulting from it, at that timethe beliefs determining the reference point arepast and hence unchangeable. This means thatshe maximizes utility taking the reference pointas given, so that she can rationaHy expect to fol-low a plan of behavior only if she is wiHing to

follow it given a reference point generated by theexpectation to do so.

DEFINITION 1: A selection FI E DJo F2 E D2is an unacclimating personal equilibrium(UPE) if for each 1 E 1,2 and any F/ E Dz,U (Fzl(1 - q)FI + qF2) ~ U (FI'I (1 - q)FI +qF2).14

If the person expects to choose FI and F2from choice sets DI and D2, respectively, thengiven her expectations over possible choicesets, she expects the distribution of outcomes(1 - q)FI + qF2• Definition 1 states that withthose expectations as her reference point, sheshould indeed be willing to choose FI and F2from choice sets DI and D2•

UPE is closely related to the notion of "loss-aversion equilibrium" that Shalev (2000) definedfor multiplayer games as a Nash equilibrium fix-ing each player's reference point, where the ref-erence point is equal to the player's (implicitlydefined) reference-dependent expected utility.Although Shalev does not himself pursue thisdirection, reformulating his notion of loss-aver-sion equilibrium using our utility function andapplying it to individual decision making cor-responds to UPE.

In the decision between a fifty-fifty chanceof having to pay $100 out of current wealth wand buying insurance for $55, when is choosingthe lottery a UPE? If the lottery is the referencepoint, the following inequality indicates when itis preferred to paying $55:

(3) [~ (w - 100) + ~ w]

+ [~1L(100)+ ~1L( - 100)]

~ [w - 55]

14 Because each of our solution concepts is an exampleof personal equilibrium as first defined in Koszegi (2005),Theorem 1 of that paper implies that when (1 - q)D, +qD2 is convex and compact, UPE, as well as PPE and ePE,exist.

There can be multiple UPE in a given situa-tion-there can be multiple self-fulfilling expec-tations-and generically different UPE yielddifferent expected utilities. But the person'sexpectations are based on her own plans on whatto choose once the time comes. It seems likely,therefore, that she will choose the best plan sheknows she will follow through on.

DEFINITION 2: A selection FI E DJo F2 E D2is a preferred personal equilibrium (PPE) if it isa UPE, and U ((1 - q)F] + qF21(I-q) FI + qF2) ~ U((1 - q)F{ + qF21(1-q)F{ + qF2}forall UPE selections FI' E Db F{ E D2•

A major feature of UPE and PPE is the con-straint that choice must be optimal given expec-tations at the time. This means that the decisionmaker does not internalize the effect of her choiceon expectations, so-as we will illustrate-sheoften does not maximize ex ante expected utilityamong the choices available to her.

To begin our analysis of UPE and PPEbehavior, we note that the results in Section IIon expectations fixed independently of the per-son's choice set DI can be thought of as apply-ing UPE or PPE to situations where that choiceset is a surprise: if she had been expecting toface choice set D2 with near certainty and DIwith very small probability, and to chooseFE D2 (perhaps because D2 = {F}, so that shethought she would have no choice), her refer-ence point would be approximately F indepen-dently of DI or what she had been expecting tochoose from DI'

To study decisions for choice situations thatare anticipated, we first establish that a personhas a strong preference to insure modest-scalerisks-she is first-order risk averse-and thenshow that expecting risk at the start decreasesher aversion to additional risk.

PROPOSITION 3: Suppose m(') is linear.For any wEIR and mean-zero lquery F ¥- 0with bounded support, there exist k, t > 0 suchthat for any positive t < t,k <k, the uniquePPE with the choice set (w, w + t(F + k)} isto choose w.

Proposition 3 states that a person will choosea riskless w over a sufficiently small multipleof a better-than-fair but insufficiently attractive


bet.ls Intuitively, w is a UPE because when thedecision maker expects it, loss aversion leadsher to turn down the gamble. And w is a PPEbecause the gamble exposes the decision makerto a sense of loss from bad outcomes that out-weighs the sense of gain from good outcomes,and hence yields lower expected utility than w.

In predicting attitudes toward insuring mod-erate and high-probability losses, status quoprospect theory says both that loss aversion doesnot playa role, and that diminishing sensitivitypushes people toward not insuring. When it comesto insuring expected losses, our model reversesthis prediction: it says that loss aversion plays thecrucial role and pushes people toward insuring.As a key force behind this prediction, our modelcaptures an intuition regarding the differencebetween "costs" and "losses" that has sometimesbeen articulated (e.g., in Kahneman and Tversky1984 and Nathan Novemsky and Kahneman2005), but has not been formalized. In our model,beliefs about wealth take into account a plannedpremium payment, so that such a payment is notevaluated as a loss. By contrast, whether or nota person had expected to buy insurance, a badrealization of a stochastic lottery is evaluated asa loss. Because loss aversion therefore plays acentral role in the decision of whether to insure,first-order risk aversion results. Indeed, someconsumers' purchase of insurance against high-probability losses-such as extremely expensiveautomobile service contracts-seems consistentwith this prediction and is in direct contrast tostatus quo prospect theory.16

For low-probability losses-such as those cov-ered by extended warranties and low deductibleson homeowners' insurance-the overweightingof low probabilities in conventional status quoprospect theory can counteract diminishing

IS Proposition 11 in Appendix A identifies a precise con-dition of the attractiveness of a vanishingly small lotterythat determines whether the decision maker accepts thelottery.

16 An experiment by Antoni Bosch-Domenech andJoaquim Silvestre (2006) can also be interpreted as provid-ing evidence for risk aversion in high-probability losses.Subjects received money for their performance in the firstsession of the experiment, and were asked to come backweeks later for a second session. Subjects were warned thatthey could lose money during the second session. In thissession, a majority of students were risk averse for bothhigh- and low-probability losses.

sensitivity and create aversion to risk. But evenfor these decisions, the extent of risk aversionoften seems quantitatively much stronger thanpredicted by prospect theory. A calibrationalproblem is immediately apparent in intuitiveterms if we pose the deductible choices of hom-eowners analyzed by Justin Sydnor (2006) asgambles relative to the status quo. For the aver-age consumer with a $500 deductible, the pre-mium for a $1,000 deductible is about $600, andthe premium for a $500 deductible is about $100higher. The probability of making a claim in anygiven year is 0.05. Hence, the choice betweenthe two deductibles is equivalent to the choicebetween the gambles (-600,0.95; -1600,0.05)and (-700,0.95; -1200,0.05).17 We conjecturethat when asked in these terms, most individu-als would choose the former gamble, and indeedSydnor (2006) shows that typical parameteriza-tions of prospect theory make the same predic-tion. Yet the majority of consumers in his datasetchose the latter gamble.

But the difference between anticipated andsurprise situations extends beyond attitudestoward losses. As indicated in the second panelof Table 1, the certainty equivalent for Gamble Iin our example is lower in a PPE situation withno background risk than in any of the sur-prise situations. Indeed, if a person is decidingbetween taking on a risk and fully insuring it,under A3' she is at least as risk averse in PPEas in a surprise situation, whatever her expecta-tions may have been in the latter case.

PROPOSITION 4: Suppose m(') is linear andf.L(.) satisfies A3'. If w + F is a PPE in thechoice set {w,w + F}, then for any lottery H,U(w + FIH) > u(wIH).

Because of the many ways it predicts thatbehavior in surprise situations differs from that inexpected situations, our theory cautions againstextrapolating from experimental results toocasually. Insofar as most experimental subjectsdo not have a clear idea about the tasks they aregoing to face, their behavior does not correspondto what they would do in a similar but expected

17 Strictly speaking, the two situations are not equivalentif some claims are between $500 and $1,000. But if so, ahigh deductible is all the more attractive.

situation in real life. Specifically, Proposition 4says that laboratory measurements of small-scalerisk aversion will often underestimate risk aver-sion for expected small-scale risks.

We conclude this section with results ontwo ways in which expecting risk at the startdecreases aversion to additional risk. Althoughour formal results identify the effect of facinga lottery G the decision maker cannot avoid,this can be the reduced-form representation of asituation where she could, but (because it is tooattractive) in equilibrium does not avoid G. First,the decision maker is no less likely to accept agiven lottery if she is already facing risk than ifshe is not.

PROPOSITION 5: Suppose m( . ) is linear andf.L( . ) satisfies A3'. For any lotteries G and Fand constant w, if choosing G is a UPE with thechoice set {G, G + F}, then choosing w is a PPEwith the choice set {w, w + F}.

And as a corollary to Proposition 2, a personis neutral to risks that are small relative to therisk she is already expecting.

PROPOSITION 6: Suppose m(') is linear. Forany lottery F with positive expected value:

(i) There exist A, e > 0 such that for anylottery G for which PrG(r E [k - A, k + AD< e for all constants k, the unique UPE withthe decision set {G,G + F} is to choose G +F, and the unique UPE with the decision set{G, G - F} is to choose G.

(ii) For any continuously distributed lotteryG, there is at> 0 such that for any t E (0, t],the unique UPE with the decision set {G, G +t . F} is to choose G + t . F, and the uniqueUPE with the decision set {G, G - t . F} is tochoose G.

The second panel of Table I quantifies Proposi-tions 5 and 6 for our parameterized example,showing that the decision maker approaches riskneutrality for Gamble I even for relatively limitedamounts of background risk. An unavoidablerisk on the order of a mere $100 decreases thepremium she is willing to pay for insurance from$48 to $9, and unavoidable risk on the order of$1,000 makes her virtually risk neutral.

IV. CPE Risk Attitudes

Beyond the distinction between surprise andanticipated exposure to risky decisions, ourunderlying assumptions also imply differencesin attitudes toward anticipated risks as a func-tion of how far in advance decisions are com-mitted to. We now analyze risk preferencesregarding outcomes that are resolved long afterall decisions are committed to, a situation thatapplies to most insurance choices. In this case,the expectations relative to which a decision'soutcomes are evaluated are formed after-andtherefore incorporate the implications of-thedecision.

DEFINITION 3: For any choice set D, FEDis a choice-acclimating personal equilibrium(CPE) if U(FIF) ~ U(F'IF')for all F' ED.

If the decision maker makes the choice FEDtoday, this will determine her reference point bythe time the relevant wealth outcome occurs.Thus, when evaluating her resulting expectedutility, both the reference and outcome lotteriesare equal to F. Note that CPE can naturally beextended to decisions where some uncertaintyis resolved today. In this case, immediatelyresolved uncertainty gets absorbed into the ref-erence point, so that the decision maker maxi-mizes the expectation of u(FIF) taken overthe possible lotteries F that capture solely theremaining uncertainty. Hence, for instance, ifsomebody is confronted with a choice betweengetting $10,000 for certain a month later andgetting $25,000 a month later if an immediatelyobserved coin flip comes up heads, we predictshe would evaluate her options solely in termsof expected consumption utility.

Our notion of CPE is related to the modelsof "disappointment aversion" of Bell (1985),Loomes and Sugden (1986), and Gul (1991),where outcomes are also evaluated relative to areference lottery that is identical to the chosenlottery. As we have mentioned, however, thesetheories differ from ours in two important ways.They assume that a person evaluates outcomesrelative to the reference lottery's certaintyequivalent instead of the full distribution. Andbecause they do not distinguish expectationalenvironments at all, they do not share many ofour model's predictions in other situations.


Returning to our example of choosingbetween a 50 percent chance of losing $100 andinsuring this risk for $55, selecting the lotteryis a CPE if

(4) [~(w - 100) + ~w]+ [± 1L(100) + ± 1L( - 100)]

~ [w - 55] +[0].

The difference between UPE and CPE is inthe right-hand sides of inequalities (3) and (4),which capture the decision maker's expectedutilities when deviating from the purportedUPE and CPE, respectively. In UPE, the refer-ence point does not adjust to the deviation, sopaying $55 is assessed partly as a loss of $55 andpartly as a gain of $45. In CPE, the referencepoint does adjust to the deviation, so there is nosensation of gain or loss when paying the $55.18

As with PPE, except in knife-edge cases therewill be a unique CPE. But unlike in PPE, wherethe decision maker can choose her favorite planonly from those that she would follow throughon, in CPE she can commit to her overall favor-ite lottery. Hence, there cannot be a divergencebetween behavior and welfare.

It bears emphasizing that UPE/PPE and CPEare not different theories of what outcomes peo-ple prefer. Indeed, in our example the expectedutility from choosing the lottery (the left-handside of inequality (3) or (4)) is independent ofwhether the choice is determined by UPE orCPE. Rather than reflecting different notions ofreference-dependent utility, the two concepts aremotivated by the same theory of preference, asmanifested differently depending on whether theperson can commit to her choice ahead of time.Of course, one weakness of our theory is that itdoes not specify the lag with which new beliefs

18 ePE implicitly assumes that the decision makermaximizes the expected future sensations generated oncea reference point determined by the decision is formedand outcomes are resolved. As we discuss in the conclu-sion, a person may not appreciate how her decision willaffect future sensations of gain or loss, and she may alsobe influenced by current anticipatory utility. Similarly toother models of reference-dependent utility, it seems to bean appropriate first approximation to ignore these issues.

are incorporated into the reference point, so thatit may be unclear whether a person's commit-ted decision is made sufficiently early that CPErather than PPE is appropriate for modeling herbehavior. In many situations, this will not be amajor problem, as the appropriate concept willbe intuitively clear. In addition, because the twoconcepts generate distinctive behavior, observedchoices can also be used to determine whichconcept applies.19

As the proofs in Appendix C establish, Prop-ositions 3 through 6 derived above for PPE alsoapply to CPE. Because anticipated risk exposesthe decision maker to sensations of loss, she isfirst-order risk averse. Furthermore, expectingrisk reduces her aversion to additional risk. Thislatter result follows partly from the force behindour analogous results above, that, when expect-ing and facing stochastic losses to start with,taking further risk does not increase exposure tolosses as much. When a person chooses both herreference and outcome lotteries, there is an addi-tional, parallel force acting in the same direc-tion: when expecting and facing stochastic lossesto start with, the expectation of further risk doesnot increase exposure to losses as much.

Despite these similarities, CPE is consistentwith risk aversion that is qualitatively differ-ent, not only from standard expected-utility-over-wealth models and prospect theory, butalso from the predictions of UPE, PPE, andany model where the reference point is takenas given at the moment of choice. Whereas inthese models people never choose stochasti-cally dominated options, they might do so inCPE. To illustrate this possibility, suppose 1Lsatisfies A3', and consider a lottery F that yieldsw + g > w with probability p ~ 0 and w withprobability 1 - p. Then U( w + Flw + F) =[p(w + g) + (1 - p)w] + [p(1 - p)1L(g) +p(1 - p)1L(- g)] = w + pg[1 - (1 - p)TJ(A -1)]. If TJ(A - 1) > 1, which is a calibrationally

19 For instance, suppose F, is a small binary lottery thatgenerates barely enough gains for w + F, to be chosenfrom {w, w + Ft!, and let F2 be a different small binarylottery with the same property. Let F be a mixture of F,and F2 with equal weights. Since in PPE the reference pointis fixed at the moment of choice, it is easy to show that inPPE the person would choose w + Ffrom {w, w + Fl. Butsince in ePE a person influences the reference point by herchoice, and she dislikes risky reference points, in ePE shewould choose w from {w, w + Fl.


plausible situation, the decision maker prefers p= 0 over a small p > 0.20 Intuitively, raisingexpectations of getting g makes an outcome ofno gain feel more painful. To avoid such disap-pointments, the person would rather give up thefragile hope of making gains. In fact, if gain-loss utility is sufficiently important, reducingexposure to sensations of loss is the decisionmaker's central concern.

PROPOSITION 7: Suppose m( . ) is linear andthe decision maker faces the finite choice set Dcontaining the deterministic outcome wand nogreater deterministic outcomes. For any givenfLo( . ) satisfying A2, there is an 1/ such that iffL( . ) = 'Y/fLo( . ) with 'Y/ > 1/, the unique CPEis to choose w.

While the tendency to choose a stochasticallydominated lottery may seem counterintuitive, itis consistent both with the flavor of some dis-cussions in the psychology literature on welfarein risky situations, and with some experimen-tal results on risk taking. Shane Frederick andGeorge Loewenstein (1999) discuss, for exam-ple, how a prisoner may be made worse off bya small chance of being released, because thatmakes the outcome of remaining in prison muchmore difficult to bear. In the domain being exam-ined in this paper, Uri Gneezy, John A. List, andGeorge Wu (2006) find situations where riskychoices are valued less than their worst pos-sible outcome. We also feel that the preferencefor a stochastically dominated lottery capturesin extreme form the strong risk aversion con-sumers display when purchasing insurance forlong-term modest-scale losses, choosing lowdeductibles on existing insurance, and selectingexpensive fixed-fee contracts for services.

There are ways in which our result must bequalified, however. The preference for domi-nated lotteries clearly arises only when suchlotteries reduce exposure to gain-loss sensa-tions. In addition, diminishing sensitivity cansubstantially reduce a person's dislike of risk formodest stakes: as the gain g in the lottery aboveincreases, the sensation of loss from comparingnothing to g increases more and more slowly,

20 It is easy to check that 7) (A - 1) > 1 wheneverobserved loss aversion is at least two-to-one-whenever

whereas consumption utility increases linearly.And as we discuss in the conclusion, people mayunderappreciate how coming to expect the gainincreases future sensations of loss, and hencemay not be averse to choosing the lottery.

The possibility of rejecting a probabilisticgain under CPE but not under PPE reflects amore general sense in which people are morerisk averse under CPE than PPE. Proposition 8establishes that under A3', if the decision makerchooses the less risky one of two lotteries in PPE,she does not choose the riskier one in CPE.

PROPOSITION 8: Suppose m( . ) is linear, A3'holds, andfor different lotteries F and F' in thedecision maker's choice set, F' is a mean-pre-serving spread of F + k for some constant k. IfF is a PPE, F' is not a CPE.

The intuition derives directly from the deci-sion maker's desire to avoid risky expectations.That F is a PPE implies that holding the refer-ence point fixed at F, the person prefers F to F'.When her choice affects the reference lottery inaddition to the outcome lottery, as in CPE, herdislike of risky reference points leads her to pre-fer F to F' even more.

The most important implication of Proposi-tion 8 is that people will be more risk aversewhen decisions are committed to well inadvance than when people are uncommitted.But the same results also say that the availabil-ity of some risky options-exactly those that thedecision maker takes in PPE but would not takein CPE-decreases welfare in PPE. Intuitively,in PPE the decision maker realizes that she willtake a lottery that is attractive fixing expecta-tions, and because she incorporates the possibil-ity of good outcomes into her reference point, hersense of loss from low outcomes is increased.

Both the similarities and differences betweenPPE and CPE are illustrated in Table 1.Comparing the second and third panels forGamble I, for any given amount of backgroundrisk, CPE choices are more risk averse thanPPE choices. Nevertheless, CPE behavior alsoapproaches risk neutrality with even moderateamounts of background risk.

overall sensitivity to losses, 1 + 7)A, is at least twice as highas overall sensitivity to gains, 1 + 7).

V. Immodest Risk

We now illustrate some of our model's impli-cations for attitudes toward large-scale risk,showing that for such stakes fL( . ) can becomelargely irrelevant in determining risk prefer-ences. Combined with the previous three sec-tions, this means that our model can reconcilehighly context-dependent behavior in modest-stakes gambles with context-independent "clas-sical" predictions for larger stakes.

Proposition 9 shows that, when diminishingsensitivity is a significant-enough feature ofgain-loss utility, the expected utility from veryrisky outcomes is little influenced by the refer-ence point.

PROPOSITION 9: Suppose m(·) has full rangeand limHoo fL'(X) = limH-oo fL'(X) = O. Forany r, r' E IR, r > r' and e > 0, there is a8 > 0 such that if F is continuously distrib-uted with density less than 8 everywhere, then0:::; U(Flr') - U(Flr) < e.

If F is a very risky gamble, most of its out-comes are far from both rand r'. Since sensi-tivity of gain-loss utility to changes approacheszero for comparisons far apart, for a typical out-come of F it makes little difference whether it isbeing compared to r or r'.

Beyond this analytical result, our model'spredictions for large-scale risks can be illus-trated by applying the parameterized version wehave considered above to larger stakes. Table 1performs this exercise for two gambles. GambleII is a fifty-fifty gamble that yields either$990,000 or $1,010,000, and Gamble III is afifty-fifty gamble that yields either $500,000or $1,500,000. While the qualitative patternsof behavior are similar for the $100-stakesGamble I and the $IO,OOO-stakes Gamble II,the sensitivity of behavior to the environmentis significantly smaller in the latter case. Andfor the $500,000 gamble, risk attitudes are closeto what they would be without gain-loss utility,being largely determined by m( .) indepen-dently of the environment.

The wide range of If's in Table 1 replicatesan observation that an increasing number ofresearchers have become aware of: that the wildvariation in measured risk aversion inferredby use of the prevalent reference-independent

model is due to a systematic misspecificationof that model, and a great deal of the variationreflects reference-dependent utility. In additionto the patterns identified in previous sections, ourmodel predicts a systematic relationship betweenthe inferred coefficient of risk aversion and thescale at which it is measured: when measured onlarge-stakes data, single- and double-digit coef-ficients will be found; when measured on mod-est-stakes data, triple-digit coefficients will befound; and when measured on small-stakes data,coefficients too embarrassingly large to reportwill be found. Indeed, Raj Chetty (2005) showsthat existing evidence on the income elasticityof labor supply comfortably bounds the coeffi-cient of relative risk aversion from above by two.Based on hypothetical choices between largegambles on lifetime wealth, Robert B. Barskyet al. (1997) measure an average coefficient ofrelative risk aversion of around five. Fitting anunemployment model with consumption andsearch-effort choices to data on unemploymentdurations, Chetty (2003) finds a coefficient ofrelative risk aversion of around seven. RajnishMehra and Edward C. Prescott (1985) estimatethat to explain the historical equity premium,investors must have a coefficient of relative riskaversion well in the double digits. The deduct-ible choices of American homeowners analyzedby Sydnor (2006) and the risk-taking behav-ior of Paraguayan farmers analyzed by LauraSchechter (2005) imply coefficients of relativerisk aversion in the triple digits. And althoughrisk attitudes are often measured in small-stakeslaboratory experiments, coefficients calculatedusing the same mathematics as those above aretypically not reported; if they were, they wouldbe extremely high.21

VI. Caveats, Discussion, and Conclusion

Several complications limit the degree to whichour model moves us to a full understanding of

21 Many papers that report small coefficients of risk aver-sion for small stakes do so by dint of using the same termi-nology to describe mathematically different measures. Byvariously defining wealth as monthly income or potentialincome over the course of a one-hour experiment ratherthan lifetime wealth, these papers report figures that areorders of magnitude different from measures using wealthlevels similar to those in the estimates above.


reference-dependent risk preferences. A weak-ness that our theory shares with other modelsis that it takes as one of its primitives the set ofdecisions and risks a person is considering, asdistinct from all the decisions and risks she isfacing. In fact, as Barberis, Huang, and Thaler(2006) emphasize in their setting, Propositions 2and 6 can be interpreted as saying that if peopleincorporated all the risks they are facing intotheir expectations, reference dependence wouldessentially not affect risk attitudes-becausepeople would essentially be neutral to smallrisks. Although psychological evidence doesindicate that people often "narrowly bracket"-they isolate individual decisions and risks fromrelevant other decisions-relatively little isknown about the extent, patterns, and effects ofsuch bracketing phenomena.22

Another major limitation of our model con-cerns welfare. Although the analysis empha-sized behavioral implications of our model, awelfare interpretation was implicit throughoutand explicit at times. Insofar as the hedoniceffects of choice include both gain-loss sensa-tions and consumption utility, our utility func-tion may provide a useful welfare measure. Fortwo specific reasons, however, we are more hesi-tant about our model's welfare implications thanabout its behavioral implications. First, the nar-row bracketing discussed above may lead peopleto care too much prospectively about a gain orloss whose effects are likely to be eliminatedafterward by an offsetting loss or gain. Second,evidence indicates that people underestimatehow quickly the reference point will adjust to achoice, and hence put too much weight on gain-loss sensations when making decisions.23 For

22 For papers highlighting the role of bracketing inthis and other domains, see Kahneman and Dan Lovallo(1993), Shlomo Benartzi and Thaler (1995), Daniel Read,Loewenstein, and Rabin (1999), Thaler (2000), andBarberis, Huang, and Thaler (2006).

23 For interpretation of evidence along these lines, seeKahneman's (2003) discussion of the "transition heuristic,"and Loewenstein, Ted O'Donoghue, and Rabin's (2003)formulation of "projection bias."

both these reasons, the appropriate welfare mea-sure is likely to be closer to consumption utilitythan we assume in this paper. This could sig-nificantly alter some of our welfare conclusions.For example, although we showed above thatthe availability of a lottery can (in PPE) lowerwelfare defined to include gain-loss disutility,the same analysis implies that this would not bepossible for welfare based solely on consump-tion utility.

The underestimation of changes in the refer-ence point also has behavioral implications. If aperson underestimates the effect of changes inher expectations on her preferences, she may notappreciate fully how she can rid herself of sensa-tions of loss by ridding herself of risky expecta-tions. Because our predictions of first-order riskaversion are driven partly by a person's desire toavoid risky expectations, this underappreciationcan reduce risk aversion.

Our model also glosses over a set of issuesrelated to what outcomes a person pays attentionto. In contrast to our formal model, differentoutcomes resulting from the same choice oftendiffer in salience. As noted in Sydnor (2006), forinstance, having to pay for repairing an unin-sured house is a very salient loss, but not havingto pay is unlikely to result in a salient sensationof gain. A person who focuses mostly on suchlosses presumably has an even stronger taste forinsurance than our model predicts.

Finally, our model ignores a source of util-ity-anticipatory emotions-that seems impor-tant in many risky situations. For instance, aninvestor's anxiety about funding her child'seducation is likely to affect both her welfareand many of her financial decisions. Insofaras anticipatory feelings are about future con-sumption and gain-loss utilities, our qualitativeresults would not be affected by adding themto the model. Quantitatively, however, anticipa-tory emotions can affect the degree of risk aver-sion; Andrew Caplin and John Leahy (2001), forinstance, show that the attempt to avoid anxietyabout uncertain outcomes can increase a per-son's preference for riskless options.

ApPENDIX A: FURTHER DEFINITIONS AND RESULTS

In this appendix we present an array of concepts and results that may be of practical use in apply-ing our model, but that are not key to any of the main points of the paper. The first result identifies a

condition such that with the reference point being the status quo, the decision maker rejects all fairgambles.

PROPOSITION 10: Suppose m (. ) is linear and the reference point is $0. The decision maker rejectsall fair gambles if and only iflimHoo fL' (-x) ~ fL'+(0).

Assumption A3 allows the decision maker's risk lovingness in losses to be much stronger than herrisk aversion in gains, which may even lead her to accept unfair gambles given a reference point of$0. Proposition 10 says that when m ( .) is linear, a necessary and sufficient additional condition torule out such possibilities is that sensitivity to losses is everywhere greater than sensitivity to gains.When m ( .) is concave, this condition is of course sufficient, but not necessary.

For our results on PPE and CPE behavior, we introduce three definitions characterizing the riski-ness of lotteries. Our first definition is the conventional one of second-order stochastic dominance,except that it allows comparisons of lotteries with different means.

DEFINITION 4: A lottery F is less risky than the lottery F' if F' is a mean-preserving spread ofF + k for some constant k E IR.

Because it turns out to be an especially pertinent measure of riskiness in our model, we also intro-duce a more specific concept that is (to our knowledge) undefined and unexplored in the literature onrisk preferences.

DEFINITION 5: The average self-distance of a lottery F is

S(F) ==Jf Ix - YI dF(x )dF(y).

The average self-distance of a lottery is the average distance between two independent draws fromthe lottery. A lower self-distance is a necessary but not sufficient condition for one lottery to be unam-biguously less risky than another.

LEMMA 1: If F is less risky than F', then F has lower average self-distance than r

Finally, we introduce a measure for how a lottery's possible gains compare to its possible losses.

DEFINITION 6: For a lottery F, let F+ = EF[max{x,O}] and F_= EF[max{-x,O}]. The favorabilityof F is defined as (F)== 1 if F+ = F_ = 0, (F)= 00 if F+ > 0, and F_ = 0 and (F) == F+/F_otherwise.

The favorability of a lottery is the ratio of the average gain of the lottery (relative to zero) and theaverage loss. Using the concepts above, Proposition II precisely identifies the extent of the decisionmaker's first-order risk aversion, generalizing Arrow's theorem to reference-dependent risky choice.This extends the limit result of Proposition 3 in the text, which said that small bets that are insuf-ficiently better than fair will be rejected.

PROPOSITION II:Suppose wEIR, and F is a lottery with bounded support.

(i) If (F) < (1 + fL ~ (0) )/( 1 + fL ~ (0)), then there exists a "i > 0 such that for any positivet < "i, the unique PPE in the choice set {w, w + t· F} is to choose w.

If (F)> (1 + fL~(O))/(1 + fL~(O)), then there exists a "i> 0 such that for any positivet < "i, the unique PPE in the choice set {w, w + t· F} is to choose w + t· F.

(ii) lf2E[F] < (iL'-(O) - iL'+(O))S[F], then there exists a i > 0 such thatfor any positive t < i, theunique CPE in the choice set {w, w + t· F} is to choose w. If2E[F] > (iL'-(O) - iL'+(O)S[F], thenthere exists a i > 0 such thatfor any positive t < i, the unique CPE in the choice set {w, w + t . F}is to choose w + t . F.

Part (i) states that when applying PPE, small bets will be accepted if and only if their favorabilityis greater than the "coefficient of loss aversion" associated with u( wlr) -which is the slope ratio atthe kink in u( wlr) at w = r. Part (ii) says that when applying CPE, a small bet will be accepted if andonly if twice its expected value is greater than the product of its average self-distance and the differ-ence in the decision maker's sensitivity to small losses and small gains.

Proposition 12 shows that when consumption utility is linear and A3' holds, we can characterize aperson's CPE attitude toward a lottery purely in terms the lottery's mean and average self-distance.

PROPOSITION 12: Suppose m(w) = wand iL( . ) meets A3'. Then,

(i) For any lottery F,1

U(FIF) = E[F] - 21/ (A - 1)S[F].

(ii) For afinite choice set D, define y(D) = (F E DIVF' ED, either (a) S[F] < S[F'], or (b) S[F]= S[F'] and E[F] ~ E[F']}. For a sufficiently high 1/, y(D) is the set ofCPE.

Part (ii) represents a lexicographic ranking of lotteries by their lowest average self-distance, andthen the highest mean. There will generally be a unique lottery with minimal average self-distance,in which case that lottery is chosen when gain-loss utility is very important-even if it is stochasti-cally dominated by other options.

The role of average self-distance in determining a person's CPE choices when A3' holds is bestseen with a simple but striking observation. In a personal equilibrium of any sort, every possiblesensation of gain-say from comparing an outcome x to a counterfactual y < x -is matched by anequally likely and equally large loss-from comparing y to x. Because the losses are more heavilyfelt, net gain-loss utility will be proportional to the negative of the average of these distances.

Finally, we show two properties of risky choice in our model that it shares with the standard model.Although expected risk decreases aversion to risk under either of our solution concepts, withoutdiminishing sensitivity it never eliminates the risk aversion completely.

PROPOSITION 13: Suppose m( . ) is linear, A3' holds, and F second-order stochastically dominatesF' *" F. Then both the unique PPE and the unique CPEfrom the choice set {F, F'} is to choose F.

We also note that if F first-order stochastically dominates F: then for any given reference lotterythe decision maker prefers F to r Hence, choosing F' cannot be a UPE. While mathematicallytrivial, this result is of interest because it contrasts with some of our results for CPE.

PROPOSITION 14:Suppose the decision maker faces the choice set D. IfF E D first-order stochas-tically dominates F', then F' is not a UPE.24

ApPENDIX B: EXTRACTING m(') AND iL(') FROM BEHAVIOR

In this appendix, we provide an algorithm that identifies a person's full utility function from herbehavior (as long as the consumption utility m(') inferred in the first step is continuous). Once this

24 In fact, this result relies solely on Assumption AI, guaranteeing that u(wlr) is increasing in w, and on no other feature of fl,.


utility function is identified, our theory provides a prediction for behavior in any decision over mon-etary risk. To determine f.L( . ), our algorithm works whenever m( . ) has unbounded support; minormodifications cover the opposite case as well.

As we note in Section IV, when a person makes decisions over risk with immediate resolution anddelayed consequences, the natural extension of CPE predicts that she maximizes expected consump-tion utility. This means that we can identify m( . ) up to an affine transformation using standardrevealed-preference techniques. Once m( . ) is identified and a normalization is chosen (e.g., settingm (1,000,001) - m (1,000,000) = I), the next two parts of our procedure allow us to infer f.L(a) andf.L( - a) for any given a > O. Carrying out these steps for all a > 0 identifies the entire functionf.L( . ).

For a given a, we find wealth levels WO,WI,W2 such that m(w2) - m(wI) = m(wI) - m(wo) = a.Then, we find the wealth level WCE such that in CPE the person is indifferent between a deterministicWCE and a fifty-fifty gamble that pays either Wo or WI. This certainty equivalent will satisfy

(5)

so that we have identified f.L(a) + f.L( - a). Intuitively, because the ex ante evaluation of risksinvolved in CPE pairs any possible loss with a corresponding equally weighted gain, CPE behaviorallows us to infer the difference between the pain from a loss and the pleasure from the same-sizedgain. But for the same reason, CPE behavior cannot identify how large the pain and pleasure are.

To identify how big they are, we can identify the ratio of f.L(a) and 1f.L( - a)1 by eliciting the prob-ability p > 1/2 such that a surprise chance p of a gain of a compensates the person for a surprisechance 1 - P of a loss of a. Specifically, we find the probability p such that when expecting WJ, theperson is indifferent between WI and a gamble that pays W2 with probability p and Wo with probability1 - p. To do this, we tell her that she will get WI with probability I - e and the choice with prob-ability e, and we let e """"* O. Then, p satisfies

(6)

and if A2 is satisfied, p > 1/2. Equation (6) can be rearranged to show that

which identifies f.L( - a) and f.L(a) based solely on the person's observed choices.

ApPENDIX C: PROOFS

PROOF OF PROPOSITION I:Let F+ = EF[max{x,O}] and F_= EF[max{-x, a}]. F+ and F_ are respectively the expected gains

and losses of lottery F relative to O.Clearly, U(w + Flw) ~ u(wlw) if and only if (1 + 'Y/)F+ ~ (1 + 'Y/A)F _0 Now U(H + FIG) ~

u(HIG) is equivalent to

Jff [w + Wi + f.L(w + Wi - r)] dG(r)dH(w)dF(w') ~ Jf [w + f.L(w - r)] dG(r)dH(w)

1066

or

THE AMERICAN ECONOMIC REVIEW

Iff [w' + 1L(W + w' - r) - 1L(W - r)] dG(r)dH(w)dF(w') ~ O.

SEPTEMBER 2007

Noticethatforanyw' ~ O,1L(W + w' - r) - 1L(W - r) ~ 'l1w',andforanyw' :::;O,1L(W + w' - r)- 1L(W - r) ~ 'l1Aw'. Hence

Iff [w' + 1L(W + w' - r) - 1L(W - r)] dG(r)dH(w)dF(w')

~ Iff [w' + 'l1max{w', O} + 'l1Amin{w',O}] dG(r) dH(w)dF(w')

This completes the proof.

PROOF OF PROPOSITION 2:For each part of the proposition, we prove the first inequality (U(H + FIG) > u(HIG) in part (i),

and U(H + t· FIG) > u(HIG) in part (ii)). The other inequality is analogous.Let F' be the mean-zero lottery that satisfies F' + v = F for a constant v > O.(i) We prove that for any 81 > 0, there are A,8 > 0 such that if PrG(r E [k - A, k + AD < 8 for

all k E IRthen for any a E IR,

(7)

This is sufficient because it implies that U(H + F'IG) - u(HIG) > - 81' and henceU(H + FIG) - u(HIG) ~ U(H + F'IG) - u(HIG) + v > v - 81 > 0 for 81 sufficiently small.

Since 1L is differentiable other than at zero and is concave in gains and convex in losses, bothlimHoo 1L'(x) and limH-oo 1L'(x) exist. This implies that for any 82 > 0, there is an A such that

for any Ihl > A. h is also bounded; let its bound be M. Notice that

If(1L(a + w - r) -1L(a - r)) dF'(w)dG(r) = 1h(a - r) dG(r).

Since Prda - r E [-A, A]] < 8 and h( . ) > - 82 outside this range, the integral above is greaterthan - 8M - 82' Therefore, we can choose A, 8, and 82 such that if G satisfies the conditions of theproposition, the integral above is greater than - 81'

(ii) Whenever G( . ) is a continuous distribution, the function

h(') =1u('lr) dG(r)

is differentiable everywhere. Hence, for any wEIR,

f[h(w + t·w') - h(w)] dF'(w')lim ------------= 0t->oo t

VOL. 97 NO.4

everywhere. Hence

KOSZEGI AND RABIN: REFERENCE-DEPENDENT RISK ATTITUDES 1067

lim_U_(H_+_t_·F_'_IG_)_-_U_(_H1_G_)= J [lim_I_[h_(W_+_t._W_')_-_h(_W_)]_d_F_'(W_'_)]dH(W)= O·t->co t t->co t

This implies that

. U(H + t· FIG) - u(HIG)hm--------- > 0,t->co t

completing the proof.

PROOF OF PROPOSITION 3:(The proof below establishes the result for both PPE and CPE (Definition 3 in Section IV). The

proof uses the concept of average self-distance defined in Appendix A.)Let G = F + k. As in the proof of Proposition I, let G+ = Edmax{x,O}] and G_ = Edmax{ - x,

O}].Let k be defined by G +/G _ = (1 + iL'- (0) )/(1 + iL~ (0)). Clearly, k> O.Then for any k < kwehaveG+/G_ < (1 + iL'-(O))/(1 + iL~(O)).WeprovethatforanysuchG,thereisatsatisfyingthe statement of the proposition.

We have

. U(w + tGlw) - u(wlw)hm-------- =t->co t

f[max{O,tw'} + iL(max{O,tw'})] dG(w') + f[min{O,tw'} + iL(min{O,tw'})] dG(w')t

Hence, there is a t such that for t < i, choosing w is a UPE in the choice set {w, w + tG}.Also:

U(w + tGlw + tG - U(wlw) J JJiL(t(W' - r))dG(w') dG(r)lim---------- = lim w' dG(w') + ----------t->O t t->O t

=Jw' dG(w') + lim rriL(lt(w' - r)l) + iL( - It(w' - r)l) dG(w')dG(r)t->O JJ 2t

1=J w' dG(w') + JJ2(iL'+(O) - iL'-(O))I w' - rl dG(w')dG(r)

where the second-to-last inequality is true because G+ ~ G_. This establishes that w is both a PPEandCPE.

PROOF OF PROPOSITION 4:(The proof below establishes the result for both PPE and CPE (Definition 3 in Section IV).)By the proof of Proposition 8, ifw is a UPE, then u(wlw) > U(w + Flw + F), so that w is the

PPE. This means that if w + F is a PPE, then it must be the case that w is not a UPE. This is equiva-lent to U(w + Flw) > U(wlw). Then, by Proposition 1, the result is immediate.

To establish the result for CPE, suppose w + F is a CPE. By Proposition 8, w is not a PPE. Then,by the same logic as above, it must be the case that w is not a UPE. From here, the proof is the sameas above.

PROOF OF PROPOSITION 5:(The proof below establishes the result for both PPE and CPE (Definition 3 in Section IV).)By a trivial modification of the proof of Proposition 1, U( w + Flw) > U(wlw) implies

U(G + FIG) > u(GIG). Hence, U(G + FIG) :::; u(GIG) implies U(w + Flw) :::; u(wlw).Therefore, choosing w is a UPE in the choice set {w, w + F}. Then, by the proof of Proposition 8,u(wlw) > U(w + Flw + F), so w is the PPE.

We now prove the same statement for CPE. We wantto prove that if U( GIG) ~ U( G + FIG + F),then U( wlw) ~ U( w + Flw + F). Since w just shifts both sides of the latter inequality by a constant,it is sufficient to prove for w = O. We will prove that u(FIF) :::; U(G + FIG + F) - u(GIG).

Notice that the two sides are equal in consumption utility (which is equal to the expectation of F).Hence, we prove the inequality for the gain-loss-utility component. We take advantage of a geometricanalogy: for any distribution H, the negative of the gain-loss utility part of u(HIH) is proportionalto the average self-distance of H. Therefore, the statement above is equivalent to the following: whenthe distribution F is added to the distribution G, the increase in the average self-distance is lower thanthe average self-distance of F. To show this, consider any two realizations a and b of G, and any tworealizations x and y of F. By the triangle inequality,

I(a + x) - (b + y)1 :::; la - bl + Ix - yl

or

I(a + x) - (b + y)1 - la - bl :::; Ix - yl.

completing the proof.

PROOF OF PROPOSITION 6:(The proof below establishes the result for both PPE and CPE (Definition 3 in Section IV).)As in the proof of Proposition 2, for each part of the proposition we prove the first part of the

statement. Also as in the proof of that proposition, define F' as the mean-zero lottery such thatF' + v = F for a constant v > O.

(i) As an obvious implication of Proposition 2, there are A, e > 0 such that the unique UPE, andhence also the PPE, is to choose G + F.

The proof for CPE is only slightly more complicated. By the proof of Proposition 2, for any ej > 0thereareA,e >0 such thatifPrG(rE [k - A, k +AJ)< dorallkE IR, then U(G + F'IG) - u(GIG)> - ej• Applying a similar argument, there areA, e >0 such that if Pr G(r E [k - A, k +AJ) <dor allkE IR, U(G + F'IG + F') - U(G + F'IG) > - el.Hence, thereareA.€ > OsuchthatifPrG(rE[-A,AJ) <e,U(G + FIG + F) - u(GIG) = U(G + F'IG + F') - u(GIG) + v > v - 2el > 0for a sufficiently small ej.

(ii) By Proposition 2, there is at> 0 such that for t < t, the unique UPE, and hence also the PPE,in the choice set {G, G + tF} is to choose G + tF.

We now prove for CPE. By the same argument as in the proof of Proposition 2,

so that

. U(G + t·FIG + t·F') - u(GIG)hm ------------= 0,t~O t

. U(G + t·FIG + t·F) - u(GIG)lim ------------ = v > O.t~O t

PROOF OF PROPOSITION 7:For any lottery F,

(8) U(FIF) = E[F] + 1/ If(fLo(w' - r) dF(r)dF(w')

= E[F] + ~1/ If(fLo(lw' - rl) + fLo(-lw' - rl)) dF(r)dF(w')...•....... --"~

== - n(F)

By A2, n (F) >0 for any nondeterministic lottery F. Letx = minFED,Nw n (F) and y = max FED,NwE[F] .If 1/ > 2(y - w)/x, the unique CPE is to choose w.

PROOF OF PROPOSITION 8:We prove that if F is a UPE, then u(FIF) > u(F'IF'), so that F' is not a CPE. If F' = F + k for

some k, then the result is immediate, since in that case we would otherwise have to have k < O.That F is a UPE when F' is available implies that u(FIF) ~ u(F'IF). Hence, there is a constant

k' ~ OsuchthatforF" = F' + k',wehaveU(FIF) = u(F"IF).WeprovethatU(FIF) > u(F"IF"),which is sufficient because u(F"IF") ~ u(F'IF').

The condition that U(F" IF) = u(FIF) can be written as

(9) 1w dF"(w) + If fL(W - r) dF(r)dF"(w) = 1w dF(w) + If fL(W - r) dF(r)dF(w).

Clearly, we must have 1w dF( w) < 1w dF" (w). Otherwise, because fL is strictly increas-i~,g and concave and F" is riskier than F, we would have 11fL(W - r) dF(r)dF"(w) <f.J fL(W - r) dF(r)dF(w), a contradiction.

We want to prove that

1w dF"(w) + If fL(W - r) dF"(r)dF"(w) <1w dF(w) + If fL(W - r) dF(r)dF(w).

Given equation (9), this is equivalent to

If fL(W - r) dF"(r)dF"(w) < If fL(W - r) dF(r)dF"(w).

Now, since the mean of F" is greater than that of F, there is a kIf > 0 such that F" - kIf and F havethe same mean. Notice that

If fL(W - r) dF"(r)dF"(w) < If fL(W - r) d(F" - k")(r)dF"(w),

which in turn is less than or equal to

J! 1L(w - r) dF(r)dF"(w)

because F" - kIf is a mean-preserving spread of F and 1L is concave. This completes the proof.

Proof of PROPOSITION 9:LetM = -1L(-(m(r) - m(r'))). By propertiesA2 andA3 of 1L( . ), 1L(x + m(r) - m(r')) -1L(x)

:::;M for any x E IR.Since limHoo 1L'(x) = limH-001L'(x) = 0, for any 81 > 0, there is an A such that if Ixl > A, then

1L' (x) < 81' Furthermore, since m( . ) has full range, for all 82 > 0 there is a 8 > 0 such that if thedensity of Fis less than 8 everywhere, PrF[m(w) E [m(r') - A, m(r) + A]] < 82' Denote the interval[m(r') - A, [m(r) - A] by B. Under these conditions,

U(Flr') - U(Flr) = J[1L(m(w) - m(r')) - 1L(m(w) - m(r))] dF(w)

J [1L(m(w) - m(r')) - 1L(m(w) - m(r))] dF(w) +m(w)EB

J [1L(m(w) - m(r')) - 1L(m(w) - m(r))] dF(w)m(w)fl-B

which is less than 8 for appropriately chosen 81,82'

PROOF OF PROPOSITION 10:Obvious.

PROOF OF LEMMA I:Since constant shifts in a distribution clearly leave the average self-distance unchanged, we can

assume without loss of generality that F and F' have the same mean. Then, using that the absolute-value function is convex,

J!lx - YI dF(x)dF(y):::; J!lx - YI dF'(x)dF(y)

= J!lx - YI dF(y)dF'(x):::; J!lx - YI dF' (y)dF' (x).

Proof of PROPOSITION II:Obvious from the proof of Proposition 3.

Proof of PROPOSITION 12:Part (i) follows from equation (8). Part (ii) is trivial from part (i).

PROOF OF PROPOSITION 13:Notice that under A3', the utility function U('IF) = fU('lr) dF(r) is weakly concave. Hence,

u(FIF) ~ u(F'IF), so that choosing F is a UPE.

We now prove that u(FIF) > u(F'IF'), both completing the proof that F is a PPE and provingthat it is a CPE. Since the expected consumption utilities cancel, this inequality is equivalent to

!! 1L(w - r) dF(w)dF(r) > !! 1L(w - r) dF'(w)dF'(r),

or S(F) < S(F'). To show this, note that by the convexity of the absolute-value function,

!!Iw - rl dF(w)dF(r)::; !!Iw - rl dF'(w)dF(r) = !!Iw - rl dF(r)dF'(w).

Using the same reasoning and that F' -=1=F, the above is strictly less than

!!Iw - rl dF'(r)dF'(w) = !!Iw - rl dF'(w)dF'(r).

PROOF OF PROPOSITION 14:For any reference lottery G, the function

U('IG) =!u('lr) dG(r)

is strictly increasing. Hence, for any two distributions F, F' such that F first-order stochasticallydominates F', u(FIG) > u(F'IG). This implies that F' cannot be a UPE when F is available.

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