Abstract We develop a framework that allows us to state precisely the relationshipbetween leading concepts of the theoretical and empirical research on reference-dependent preferences, namely the status quo bias, the endowment effect and thewillingness to accept (WTA)/willingness to pay (WTP) gap. We show that a monetaryversion of the status quo bias is a necessary condition for the WTA/WTP gap, andshow how to factor out the part of the gap due to income effects from the part ofthe gap due to the endowment effect. As a byproduct, we show that reference-depen-dent phenomena are generated by reference-independent factors, i.e., an underlyingreference-independent preference relation the properties of which are discussed atlength.
Keywords Reference-dependent preferences · WTA/WTP gap · Endowment effect ·Status quo bias
JEL Classification D03 · D11 · D81
I thank an anonymous referee audiences of the Centre d’Economie de la Sorbonne (University Paris I),Theory Seminar (University of California, Davis), Microeconomics Seminar (University of California,Berkeley), Riskattitude Conference (Montpellier), AFSE Conference on Behavioral Economics andExperimental Economics (Lyon), the first Meeting on Economic Perspectives (Bounded Rationality andIndividual Decision-Making, Universitat Autonoma de Barcelona), and in particular David Ahn, JoseApesteguia, Miguel Angel Ballester, Michèle Cohen, Bertrand Crettez, Jean-Christophe Vergnaud, EricDanan, Julie Le Gallo, Yusufcan Masatlioglu, Klaus Nehring, Neslihan Uhler.
Financial support from ANR Riskattitude is gratefully acknowleged.
R. Giraud (B)CRESE, University of Franche-Comté, 45 D, avenue del’observatoire, 25000 Besançon, Francee-mail: [email protected]
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1 Introduction
1.1 Motivation and contributions
One of the key ideas of prospect theory (PT) (Kahneman and Tversky 1979), and,perhaps, the least controversial, is that, contrary to what is usually assumed in tradi-tional microeconomics, preferences, and not only choice, may vary with the agent’scurrent endowment. They are reference dependent. This idea was introduced in main-stream economics relatively recently.1 It is related to two specific phenomena in behav-ioral economics: the status quo bias and the endowment effect.
The status quo bias is the general tendency to prefer to stick to the current position,only because it is the current position. Samuelson and Zeckhauser (1988) providedconvincing evidence for this tendency. The status quo bias is generally associated withthe endowment effect (Kahneman et al. 1991) which is the fact that people value anobject more when they own it than when they do not, all things being equal. The usualconsequence of the endowment effect is the willingness to accept (WTA)/willingnessto pay (WTP) gap, i.e., the fact that WTA is larger than WTP. Although consideredby most scholars a very robust stylized fact (see Schmidt and Traub 2009 for recentevidence), the reality of the WTA/WTP gap has recently been questioned (Plott andZeiler 2005, 2007). Even when it is observed, its interpretation as an endowment effect(i.e., as a consequence of the status quo bias) is not warranted, since it could arise fromclassical income or substitution effects rather than from the application of an intrin-sic psychological law. It is indeed well known that, if one accepts the identificationbetween WTP and WTA on the one hand and compensating and equivalent variationon the other, the gap between them disappears in the standard (reference-independent)framework if income effects are canceled by assuming quasilinear utility.
The first contribution of this paper is to address this question axiomatically in asgeneral a formal framework as possible and yet one that is sufficiently structured forthese different concepts to be unambiguously defined. We propose a formal frameworkthat we claim to be adequate to deal with the problem mentioned above. Specificallyit consists in endowing an arbitrary set with a monetary structure, i.e., allowing for acombination of objects of choice and money to be unambiguously defined.
Using this structure, we define a strong version of the status quo bias, as well asWTP and WTA functions, that can be applied in different contexts and we prove thefollowing results:
– The (appropriately strengthened) status quo bias is a necessary condition for theexistence of a WTA/WTP gap.
– When there is reference-dependence, canceling income effects is not enough toclose up the WTA/WTP gap. It can thus be attributed to the endowment effect and
1 The theoretical literature on reference-dependent preferences, i.e., the literature dealing with decision sit-uations where the reference point is explicit and variable, as opposed for instance to the standard literatureon prospect theory or sign-dependent models where the reference point is fixed, is not very extensive at thepresent time, compared to the considerable empirical literature on the same phenomenon, but is growingrapidly. Masatlioglu and Uler (2008) provide an insightful discussion of some of the leading models andan experimental test of their predictions.
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we show how to tell apart the fraction of the gap due to income effects and thefraction due to the endowment effect.
– The gap is then completely characterized by conditions that can be interpretedeither as strategic considerations pertaining to the announcement of a buying priceor a selling price or as a non-separability between monetary and non-monetaryassets not only in the classical sense of non-quasilinearity but also in a new sensethat concerns the role of the reference point: the way the reference point affectsthe decision through a status quo bias may be altered by addition of a particularamount of money.
One interesting question we may want to investigate, however, when consideringreference-dependent preferences, is to what extent they can be explained by reference-independent factors. To be more specific, in their strict behaviorist interpretation, pref-erences are but representations of choice behavior. If we allow for reference-dependentpreferences, we also allow for a weakening of the explanatory power of the notion ofpreference. If we could retrieve from these reference-dependent observed preferencesa reference-independent preference relation that in some sense generates or rational-izes them, while still having some structure, we would recover explanatory power. Thisis also achieved in the paper, although it is not its primary focus, and the properties ofthis rationalizing preference relation are extensively discussed.
1.2 Structure of the paper
In Sect.2, we describe the structural and decision-theoretic frameworks. In Sect. 3, wedefine the general concept of WTP and WTA functions and state the main theorem ofthe paper. In Sect. 4, we discuss the theorem, focusing on the properties of the ratio-nalizing preference relation and on the meaning of the theorem for the WTP/WTAgap. Additional material on the structural framework and proofs are gathered in theAppendix.
2 The framework
2.1 Structural framework: introducing real monetary spaces
The objective in this section is to precisely define the mathematical framework that isrelevant when one wants to deal with the endowment effect, or, put differently, withthe discrepancy between the willingness to pay and the willingness to accept. The ideais to define a mathematical structure in which the concept of an object x plus a certainamount of money λ, which we denote x ⊕ λ, has a precise meaning.
To that effect, we introduce the concept of real monetary spaces (r.m.s.).
Definition 1 (Real monetary space) A pair (X,⊕) is a real monetary space (r.m.s.) ifX is a set and ⊕ : X × R −→ X is a mapping such that (denoting ⊕(x, λ) by x ⊕ λ):
(i) For all x ∈ X, x ⊕ 0 = x ;(ii) For all λ,μ ∈ R, for all x ∈ X ,
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(x ⊕ λ) ⊕ μ = x ⊕ (λ + μ).
We denote x ⊕ (−λ) by x � λ.The primary example of such a structure2 (apart from R with its natural addition)
is the following. Let A be a nonempty set; then defining (a, w) ⊕ λ := (a, w + λ)
makes of (A × R,⊕) an r.m.s. We call this r.m.s. the canonical r.m.s. on A. Thisexample is important for two reasons: first, it justifies the term “monetary” in thename of the structure. Indeed, if we interpret a as the non-monetary assets of an agentand w as his/her wealth, then the operation of adding λ amounts to increasing wealthby the same amount, without affecting non-monetary assets. In particular, in the mugexperiment of Kahneman et al. (1990), we can take A = {0, 1}, with 0 meaning “notholding the mug” and 1 meaning “holding the mug”. Second, in the rest of the paperwe impose some axioms on preference relations defined on an r.m.s. and we show thatthese axioms force this r.m.s. to be isomorphic to the canonical r.m.s. on a certain setA. However, since this is a consequence of some but not of all the axioms, we think itpreferable not to impose this precise structure right from the outset.
Some examples, in turn, of this particular structure are related to pretty familiarsettings. In classical consumer theory, for instance, we may consider for X a set ofbundles of L goods, in non-necessarily nonnegative quantities. Often, one of thesegoods, say good 1, is considered to be the numeraire. We may, therefore, chooseA ⊆ R
L−1+ and X = A × R. We may also consider the set A = RL++ of price vectors
and define (X,⊕) as the canonical r.m.s. on A.The favored interpretation here, that underlies the intuition for the axioms, will be
that an element of X is a certain portfolio of liquid and illiquid assets, x = (a, w).In Appendix A, we discuss some other examples and properties of r.m.s. relevant
to this article.
2.2 The decision-theoretic framework
Let X be the non-empty set of objects on which preferences are defined. Throughoutthe paper, we assume that X is endowed with an operation3⊕ that makes (X,⊕) anr.m.s.
2 In mathematics, a mapping ⊕ satisfying (i) and (ii) is called an action of the group (R, +) on X . The propermathematical name for this structure is R-set. However, we use another name to emphasize the intendedinterpretation and for the sake of possible generalizations. If we replace (R,+) by another group G, or evena semi-group S, we get G-sets or S-sets. As far as we know, the only papers using this mathematical structurein decision theory are Lemaire and Le Menestrel (2004), and Le Menestrel and Lemaire (2006a,b). Thereis also a discussion of the related concept of vertically invariant functionals (which are homomorphismsfor a particular G-set) in the working paper version of Maccheroni et al. (2006).3 Essentially all results in this paper can be easily restated and proved under the more general assumptionthat for all r ∈ X , there exists a mapping ⊕r : X ×R → X such that (X, ⊕r ) is an r.m.s. An example of sucha structure is a subset X of a vector space V such that, for all r ∈ X , for all x ∈ X , for all λ ∈ R, x +λr ∈ Xso that one can define an r.m.s. by letting x ⊕r λ = x + λr .In order not to rig our formulas with subscripts and not to obscure the arguments, we work in the simplerframework where the group action is the same for all r ∈ X .
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We take here the approach pioneered by Tversky and Kahneman (1991)4of assum-ing that there exists a family (�r )r∈X of binary relations defined over X modeling theobserved choice behavior of the individual in a context where his or her reference pointis an element r of X . We call this family a profile of reference-dependent preferences,RDP profile for short.
The idea of an RDP profile on an r.m.s. can be better understood by coming back tothe interpretation of objects of X as pairs (a, w) of illiquid assets and monetary wealth.The reference point, r , can thus be identified with a pair (ar , wr ) which representsinitial monetary wealth and illiquid assets. Hence, if one writes
x ⊕ λ �r x ′ ⊕ μ,
one needs to bear in mind that this means:
(a, w + λ) �(ar ,wr ) (a′, w′ + μ)
i.e., that adding λ means modifying the liquid part of the portfolio without touchingthe rest. Most interpretations of axioms and results will rely on this.
The main objective of the paper is to study representations of RDP in terms ofwillingness to pay and willingness to accept. We work in the context of three basicaxioms that are adaptations to our setting of very standard rationality assumptions.The first axiom is the standard ordering assumption for observable preferences.
Axiom 1 (Weak order) �r is a weak order for all r ∈ X .
The second axiom is a continuity axiom. In essence it states that any object has amonetary equivalent in the sense that given his or her endowment the agent is indif-ferent between owning this object or owning a certain amount of money along withthe same initial endowment:
Axiom 2 (Solvability) For all (x, r) ∈ X2, there exists λ ∈ R such that x ∼r r ⊕ λ.
In the case X = A × R, this axiom states that for each (a, w) and each (ar , wr ),there exists λ such that
(a, w) ∼(ar ,wr ) (ar , wr + λ).
Assume that A = R+, that preferences �r can be represented by a differentiableutility function ur . Then using a first order approximation, we have
ur (r ⊕ λ) − ur (r) ≈ (a − ar )∂ur
∂a(r) + (w − wr )
∂ur
∂w(r).
4 …(A)nd pursued in Munro and Sugden (2002), Sugden (2003), Schmidt (2003), Schmidt et al. (2008),Bleichrodt (2007), Bleichrodt (2009), Sagi (2006) and Giraud (2006, 2010). This approach is differentfrom the one pioneered and developed by Masatlioglu and Ok (2005) in a series of papers (Masatlioglu andOk 2005, 2009) based on choice functions or correspondences instead of preference relations and seek-ing the smallest departure from the standard utility maximization paradigm compatible with empiricallyrelevant reference effects. Apesteguia and Ballester (2009) systematically explore bridges between the twoapproaches.
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Hence λ is a measure of the total effect, viewed from r , of moving from r to x . Wemay notice, moreover, that −λ is what Bateman et al. (1997) call the equivalent gain(i.e., the monetary gain equivalent to having x).
The third axiom simply reflects the fact that, other things being equal, more moneyis better than less.
Axiom 3 (Monotonicity) For all r ∈ X , for all λ,μ ∈ R,
λ > μ ⇒ r ⊕ λ �r r ⊕ μ.
Indeed, in the case X = A×R, this means that (ar , wr +λ) �r (ar , wr +μ) wheneverλ > μ.
A family of RDP that satisfies the three axioms above is henceforth referred to asa regular RDP profile.
As the reader may have probably already guessed, regular RDP profiles are repre-sentable by utility functions. Let us say that a family (ur )r∈X of real valued functionson X is normalized if
ur (r ⊕ λ) = λ, ∀r ∈ X, ∀λ ∈ R.
The example below illustrates the concept of a normalized family. It may appearsomewhat involved given what we want to illustrate. This, however, comes from thefact that it is designed to nontrivially satisfy axioms that will be introduced later on.
Example 1 Take X = R+ × R endowed with its natural r.m.s. structure. For r =(ar , wr ) and x = (a, w), one simple but trivial and degenerate example of a normal-ized family is
ur (x) = w − wr .
Another, non-degenerate example, is:
ur (x)=
⎧⎪⎪⎨
⎪⎪⎩
(1 − (a − ar )2)(w − wr + ar (a − ar )) if
⎧⎨
⎩
w−wr + ar (a − ar ) ∈ [−1, 0]and(a − ar )
2 < 1.
w − wr + ar (a − ar ) otherwise.
Then, since (r ⊕ λ) = (ar , wr + λ), ur (r ⊕ λ) = λ.
For a given r ∈ X , the orbit of r is the set
Xr = {r ⊕ λ | λ ∈ R}.
The concept of a normalized family of utility functions is used to formalize the ideathat at each reference point r , utility is renormalized so that this reference point haszero utility and its orbit is treated as the set of purely monetary assets, with the utility
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of a given amount of money being exactly equal to that amount. In particular, a nor-malized utility function cannot exhibit cardinal loss aversion for monetary outcomes,since at any reference-point r , and for all λ > 0, we have
ur (r ⊕ λ) = −ur (r � λ).
The analysis of the endowment effect that we are about to carry out, therefore,would be all the more convincing if this potential cause of endowment effect wereruled out, and this is why we seek a representation of preferences by a normalizedfamily of utility functions.
As it turns out, the three axioms that define a regular RDP profile are jointly neces-sary and sufficient for the existence of a unique normalized family of utility functionsfor this profile:
Proposition 1 (�r )r∈X is a regular RDP profile if and only if there exists a normalizedfamily (ur )r∈X such that for all r ∈ X,
x �r y ⇐⇒ ur (x) ≥ ur (y).
Moreover, this normalized family is unique.
Normalizing utility functions allows us, as will be seen later, to work with func-tions that are commensurable with monetary concepts like WTP and WTA. Normalizedutility functions will in fact assume the role played by surplus in classical consumertheory, and it is thus essential that this notion of surplus be well-defined. Uniquenessis, therefore, a desirable property. Since in general the purpose of a normalization is topin down a mathematical concept that might otherwise be too indeterminate to ensuresuch uniqueness, our concept of normalization would be inappropriately defined if itdid not deliver uniqueness. Fortunately, Proposition 1 shows that this is not the case.
Another important consequence of the axioms above is the following: let us saythat two r.m.s. (X,⊕) and (X ′,⊕′) are isomorphic if there exists a bijective mappingϕ : X → X ′ such that for all x ∈ X and λ ∈ R, ϕ(x ⊕ λ) = ϕ(x) ⊕′ λ. Then
Proposition 2 If (�r )r∈X is an RDP profile that satisfies monotonicity and such thatfor all r,�r is reflexive, then there exists a nonempty set A such that X is isomorphicto A × R.
As we said below, this proposition will be useful for the interpretation of the axi-oms, as they allow for a clear interpretation of x ⊕λ as the addition of some monetaryamount to the liquid part w of x = (a, w).
3 The main result
The main result of the present section is an axiomatization of behavior consistent withthe endowment effect. We first introduce a formal definition of the WTA and WTPfunctionals, and then proceed to introduce the axioms that characterize the endowmenteffect.
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3.1 The benefit and loss functions
Definition 2 Let (X,⊕) be an r.m.s. and let � be any binary relation defined on X .Then:
– the benefit function5 associated with � is the function:
b� : X2 −→ R
(x, r) �−→ sup B�(x, r)
where
B�(x, r) := {λ ∈ R | x � λ � r}.
– The loss function associated with � is the function s� defined by
s� : X2 −→ R
(x, r) �−→ inf S�(x, r),
where
S�(x, r) := {λ ∈ R | r ⊕ λ � x}.
As noted in Masatlioglu and Ok (2009), there is room for debate about the precisedefinition of the willingness to pay. They define it based on Tversky and Kahneman(1991)’s postulate that “the act of giving up money to buy goods is viewed by the agentnot as a loss, but as a foregone gain of money”. This yields the following definitionof WTP:
b�(x, r) = sup{λ ∈ R | x � r ⊕ λ}.
The corresponding definition for WTA would thus be
s�(x, r) = inf{λ ∈ R | r � x � λ}.
Our definition is based on the postulate that the result of the transaction in whichthe agent buys x for λ euros is x minus its cost. In particular, in the mug example whereX = {0, 1}×R, if r = (0, w) and x = (1, w), then obviously after buying the mug theindividual winds up having (1, w − λ) = x � λ. We favor this interpretation becauseb is based on a completely counterfactual comparison, that between x that would begiven away, on the one hand, and the endowment or initial situation augmented by the
5 The notion of benefit function was first introduced by Luenberger (1992). A number of authors havestudied the properties of this function (Courtault et al. 2004a,b), generalized it (Briec and Gardcres 2004)and used it in various contexts: consumer theory (Courtault et al. 2005), as was intended originally by Luen-berger, welfare theory (Luenberger, 1992, 1994, 1996; Courtault et al., 2007), production theory Chamberset al. (1995, 1998), decision under uncertainty (Quiggin and Chambers 1998).
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sum one should be paying for x (that is the idea of a foregone gain) r ⊕λ, on the otherhand. On the contrary, our definition is based on the two possible actual situations inwhich the decision maker might be given the choice of a price: either he/she buys, andends up with x �λ or he/she does not buy, and stays with r . Notice that our definitionis the same as that of Bateman et al. (1997).
These definitions cover a certain number of related concepts in economics.The first of these concepts, that are of obvious relevance for the present research,
are the standard WTP and WTA concepts. In classical consumer theory, when X isa set of bundles of L goods, in nonnecessarily nonnegative quantities, as discussedabove, the benefit function b�(x, r) is the WTP for bundle x when the endowment isbundle r . Similarly, the corresponding expression for WTA is the loss function.
Two closely related concepts of standard consumer theory are the Compensat-ing and Equivalent Variations. When X is the space of price vector-income levelpairs, and the consumer’s indirect utility function is v, consider x = (p1, w1), r =(p0, w0) and � defined by
(p, w) � (p′, w′) ⇐⇒ v(p, w) ≥ v(p′, w′).
Then b�(x, r) is the compensating variation for moving from (p0, w0) to (p1, w1)
while keeping the wealth unchanged, while s�(x, r) is the equivalent variation formoving from (p0, w0) to (p1, w1).
Other examples are discussed in “Examples of benefit functions” in Appendix.The benefit function satisfies the following property (the straightforward proof is
omitted):
Property 1 (Translation property or vertical equivariance) For all λ ∈ R,
b�(x ⊕ λ, r) = b�(x, r) + λ.
The interpretation of this property is that the willingness to pay for an object isonly linearly affected by the amount of money this object “contains”. It can actuallybe characterized by it, as shown in “Examples of benefit functions” in Appendix.
Since s�(x, r) = −b�(r, x), analogous results can be derived for the loss function.
3.2 Axioms for the endowment effect
We now consider the following axioms:
Axiom 4 (Buying price consistency (BPC)) For all (x, r) ∈ X2, for all λ ∈ R,
x � λ �r r ⇒ x �r r ⊕ λ.
We shall say that BPC holds strictly for the pair (x, r) if the weak preferencex �r r ⊕ λ is replaced by x �r r ⊕ λ for all λ.
The idea of this axiom is that, if the decision maker is willing to pay λ euros inorder to buy x when his current endowment is r , then this means x is worth at least
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λ to him from the point of view of r , i.e., he/she weakly prefers x to r even if he/shewere to receive a windfall amount of λ euros while keeping r . To give this axiom anoperational meaning, one should think of the situation discussed in this paper as areduced form: implicitly the decision maker is responding to a potential seller (in thecase of this axiom) or buyer. The amount λ could, therefore, in principle be observedif a transaction were to take place between them. However, we only model one sideof the transaction.
BPC could be rewritten as
x �r r ⇒ x ⊕ λ �r r ⊕ λ.
This shows the meaning of this property: if we observe that the decision maker,when comparing an alternative option with the reference point, prefers the alternativeoption, then we know that this will still be the case if he/she knows that, no matterwhat his final choice will be, he/she will be given an extra monetary amount λ. Theclause “with the reference point” is important here, as BPC does not require this typeof independence to hold when comparing two alternatives that are not the referencepoint.
As discussed in the introduction, in order to focus on behavior that is compatiblewith empirical data, it is reasonable to assume that behavior exhibits status-quo bias.Arguably the simplest, weakest and most direct way of doing this would be to imposethe following axiom:
Axiom 5 (Status quo bias (SQB)) For all (x, r) ∈ X2,
x �r r ⇒ x �x r.
This axiom conveniently expresses the idea that being the status quo gives an alter-native an extra power against other alternatives besides its intrinsic merit. It says thatif an alternative x beats the status quo r , then a fortiori x must beat r when x isthe status quo, since it is thus even more attractive than before. This axiom is alsoa consistency axiom that rules out a very direct form of preference reversal: prefer-ring x to y when y is the endowment and y to x when x is the endowment. Sincethis kind of preference reversal would make the decision maker vulnerable to moneypumps, ruling them out has been deemed necessary in the literature for a modelingof rational reference-dependent preferences, and, therefore, axioms akin to SQB havebeen introduced by many authors (Munro and Sugden 2002; Sagi 2006; Masatliogluand Ok 2005, 2009; Apesteguia and Ballester 2009). Vega-Redondo (1993) providesfoundations for such a behavior based on imprecise perception of the consequencesof actions that are not the current action. Vega-Redondo (1995) discusses dynamicimplications of the previous model.
As it turns out, however, this axiom, even together with the other axioms aboveis necessary but not sufficient to yield the endowment effect, defined by the fact thatWTP is smaller than WTA. We shall discuss this fact at some length further on. Fornow, let us introduce a stronger version of SQB, that we call wealth independent statusquo bias, that will suffice:
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Axiom 6 (Wealth independent status quo bias (WISQB)) For all (x, r) ∈ X2, for allλ ∈ R,
x �r r ⇒ x �x⊕λ r.
We shall say that WISQB holds strictly for the pair (x, r) if the weak preferencex �x⊕λ r is replaced by x �x⊕λ r for all λ.
When λ = 0, this is exactly SQB. What this axiom adds is that the status quo biasessentially depends only on the non-monetary part of the status quo.
Here again, we could require x ⊕ λ �x r ⊕ λ or x ⊕ λ �x⊕λ r ⊕ λ instead ofx �x⊕λ r . The first requirement, as above, enters the picture when we want to studyalternative definitions of WTP and WTA. The second, in turn, will not be studied herefor itself, as this would require a complete change of perspective with respect to theone adopted here. It does not have any implications for WTP or WTA as defined here.
One last word on the two axioms BPC and WISQB. The main conceptual differencebetween them is that BPC makes a prediction about observed behavior holding thereference point constant, while WISQB makes a prediction for a different referencepoint. This probably makes a significant difference from the point of view of testingthese axioms: in principle, and assuming the reference point is the endowment, testingBPC in an experimental setting seems easier. Indeed, we can imagine a protocol wherewe offer the subjects some alternative, ask them if they would be ready to accept thisalternative, and then, if they are, we can observe whether they would still accept itwere we to ask them to actually do so, telling them that they will receive a fee forparticipating in the experiment, and varying the fee among groups. If the answer to thefirst question correctly predicts the observations, BPC would not be rejected. On theother hand, testing WISQB requires manipulating the endowment before the actualchoice is made, and it is probably much more difficult to actually control what is goingon.
Let us turn now to the statement of the theorem.
Theorem 1 Let (�r )r∈X be a regular RDP profile and let (ur )r∈X be the unique nor-malized family that represents it. Then (�r )r∈X satisfies buying price consistency andWISQB iff there exists a reflexive binary relation � such that
(i) b�(x, r) ≤ ur (x) ≤ s�(x, r).
(ii) x �r r ⇐⇒ ur (x) ≥ 0 ⇐⇒ b�(x, r) ≥ 0.
Moreover, in (i),
– the first inequality is strict for the pair (x, r) iff BPC holds strictly for this pair;– the second inequality is strict for the pair (x, r) iff WISQB holds strictly for the
pair (r, x).
Before discussing this theorem, let us show that the function proposed in Example1 satisfies all its conditions:
Example 2 (Example 1 continued) Let us first compute b(x, r) for this example. Letλ be such that ur (x � λ) ≥ 0. Then we have
(1 − (a − ar )2)(w − wr + ar (a − ar )) ≥ (1 − (a − ar )
2)λ
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if w − λ − wr + ar (a − ar ) ∈ [−1, 0] and (a − ar )2 < 1 and
w − wr + ar (a − ar ) ≥ λ
otherwise.Since (a − ar )
2 < 1, both cases boil down to w − wr + ar (a − ar ) ≥ λ, hence
b(x, r) = w − wr + ar (a − ar ).
Now,
s(x, r) = −b(r, x) = w − wr − a(ar − a),
hence
s(x, r) − b(x, r) = −a(ar − a) − ar (a − ar ) = (a − ar )2 ≥ 0,
and, therefore, the endowment effect holds.Moreover, when w − wr + ar (a − ar ) ∈ [−1, 0] and (a − ar )
2 < 1,
b(x, r) − ur (x) = (a − ar )2(w − wr + ar (a − ar )) ≤ 0
and
ur (x) − s(x, r) = −(a − ar )2(1 + w − wr + ar (a − ar )) ≤ 0
and otherwise b(x, r) = ur (x) ≤ s(x, r), hence in all cases b(x, r) ≤ ur (x) ≤ s(x, r).Therefore, the conditions of the theorem must be met.
The theorem gives necessary and sufficient conditions for the existence of a binaryrelation � that “generates” the WTA and the WTP so as to satisfy conditions (i), theendowment effect, and (ii), that we may interpret as a normalization that makes theinterpretation of b�(x, r) as the WTP admissible, as it imposes that the WTP for x ofan agent holding r is positive if and only if x is an improvement relative to r (it wouldobviously be irrational to behave otherwise). To further interpret this binary relation,which we will refer to as the generating relation, we must address two questions:
1. The uniqueness of �.2. The conditions under which � has the standard properties of a preference relation:
completeness and transitivity.
We will address now the uniqueness problem. We will need some preliminary def-initions.
Definition 3 A binary relation � on X is
– upper ⊕-monotonic if for all x, y ∈ X, λ, μ ∈ R,
x ⊕ μ � y and λ ≥ μ ⇒ x ⊕ λ � y
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– upper ⊕-semicontinuous if for all x, y ∈ X , the set {λ | x ⊕ λ � y} is closed– upper ⊕-regular if it satisfies both conditions.
We denote �� the upper ⊕-regular closure of �, i.e., the smallest (in the sense ofinclusion) upper ⊕-regular binary relation extending �.
We can now state the uniqueness result relative to the main theorem. First, let �+be the binary relation defined by
x �+ y ⇐⇒ x �y y.
Let us call it the gain relation associated with the RDP profile. The result is thefollowing:
Proposition 3 Let (�r )r∈X be an RDP profile and let � be a binary relation. Then� satisfies conditions (i) and (ii) of the theorem iff �� =�+.
In view of this result, we may regard �+ as the canonical generating relation.We shall discuss some of its properties in the next section. Before proceeding to dothis, however, we must observe that this relation may be considered canonical in thestudy of reference-dependent preferences in a more general sense. In many studiesof reference-dependent behavior, when a reference-independent concept of prefer-ence relation is invoked, it usually coincides (details of the specific contexts aside)with �+. This is the case, for instance, in Masatlioglu and Ok (2005) and Apesteguiaand Ballester (2009), where �+ is the (unique in a suitably defined sense) partialorder that rationalizes6 reference-dependent choice functions. Recall that the settingintroduced in Masatlioglu and Ok (2005) and retained in Apesteguia and Ballester(2009) is that of choice functions c defined on the set
{(S, y) | ∅ �= S ⊆ X, y ∈ S or y = �},
where � is an object not belonging to X . In this context, �+ is defined by
x �+ y ⇐⇒ x ∈ c({x, y}, y).
The correspondence with our setting requires us to set:
x �r y ⇐⇒ x ∈ c({x, y, r}, r).
This, however, defines only a partial order on X : it is indeed possible to have c({x, y, r},r) = {r} with x �= r, y �= r and x �= y. Lemma 4.5 in Apesteguia and Ballester (2009)establishes that RDP profiles defined in this way rationalize a choice function if andonly if such a choice function satisfies the Weak Axiom of Revealed Preferences ateach reference point. Moreover, it is shown that two RDP profiles rationalizing achoice function have the same gain relation. Lemma 1 in Masatlioglu and Ok (2005)
6 Rationalization here means choosing maximal elements or undominated elements; we leave details aside.
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strengthens this result by dispensing with the RDP profile and using only the gainrelation and one of its completions to rationalize the choice function.7
Theorem 1 in Masatlioglu and Ok (2005) shows that in the finite case �+ generatesa set of criteria with respect to which an alternative must beat the status-quo in orderto be chosen over it. A similar result is obtained in an unpublished part of the workingpaper version of Sagi (2006), where a multi-utility representation is derived for �+and used to characterize a multicriteria representation of RDP profiles under risk.
These few examples show the pivotal importance of the gain relation for charac-terizing RDP profiles in different contexts. Let us now move on to properties of thegain relation in the context of the present paper.
4 Discussion of the theorem
4.1 Properties of the generating binary relation
From the uniqueness result, it follows that the canonical generating relation is upper⊕-regular (see “Exact benefit functions” in Appendix for details). Another obviousproperty that it has under the conditions of Theorem 1 is reflexivity (indeed, for allx ∈ X, x �x x , hence x �+ x). We will now investigate the conditions under which itsatisfies completeness and transitivity, and their consequences on the WTP and WTA.From now on, we shall write b(x, r) for b�+(x, r) and s(x, r) for s�+(x, r).
Transitivity of �+ has the following characterization in the context of the maintheorem:
Corollary 1 Let (�r )r∈X be an RDP profile satisfying the assumptions of Theorem1. Then the following statements are equivalent:
(i) �+ is transitive, i.e., for all x, y, z ∈ X, x �y y and y �z z implies x �z z.(ii) For all x, y, z ∈ X, min(b(x, y), b(y, z)) ≥ 0 ⇒ b(x, z) ≥ 0.
Similarly, the characterization of completeness is the following:
Corollary 2 Let (�r )r∈X be an RDP profile satisfying the assumptions of the theorem.Then the following statements are equivalent:
(i) �+ is transitive, i.e., for all x, y ∈ X, x �y y or y �x x.(ii) For all x, y ∈ X, max(b(x, y), b(y, x)) ≥ 0.
Obviously, in view of these characterizations, �+ is a weak order if and only ifboth conditions on b hold. Though this provides the exact characterization we werelooking for, the property we obtain for b is rather weak. We can strengthen it by usingthe monetary structure of X . Consider the following axiom
Axiom 7 (Unpacking bias) For all x, y, z ∈ X ,
– for all λ, λ′ ∈ R, if x � λ �y y and y � λ′ �z z, then x � λ �z z or x � λ′ �z z,– there exists μ ∈ R such that x � μ �y y or y � μ �z z and z �z x � μ.
7 See (Apesteguia and Ballester, 2009, Theorem 5.1 and Corollary 5.2) for details.
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This axiom compares how much the decision maker values a given object x , given areference object z, in terms of his or her WTP, if he/she acquires it indirectly, using anintermediate step, or if he/she acquires it directly. To better understand this let us recalla phenomenon often observed in contingent valuation studies, and known as extensionneglect, whereby subjects report a higher value for complex goods if these goods arebroken down into less complex components. For instance, if one asks subjects howmuch they are willing to pay to save whales, and then how much they are willing topay to save sea mammals as a whole, the value may be larger but not that much larger,and if these two questions are asked of two different groups, the values will tend tobe very similar. This happens essentially because they value the part and the wholeidentically, the values being in both cases based on a prototypical example (see, e.g.,Kahneman et al. 1999). Similarly college students may give a different answer to thequestion “are you happy right now” if asked in isolation or if asked after the question“how many dates did you have in the last 6 months?”
The axiom imposes some minimal rationality condition in this context while allow-ing some bias to take place: the value of the whole in the direct question (x) cannotbe smaller than the value of all the separate “elements” but cannot be larger than thevalue of all of them either (notice that this interpretation is more compelling whenone considers more intermediate steps than just one as stated in the axiom). Relatedphenomena are a key feature in the behavioral theory of belief known as Support The-ory (see, e.g., Tversky and Koehler 1994) where they are referred to as the unpackingprinciple, hence the name of the axiom.
We have the following result:
Proposition 4 Let (�r )r∈X be an RDP profile satisfying the assumptions of the the-orem. Then the following statements are equivalent:
(i) Unpacking bias holds;(ii) For all x, y, z ∈ X, max(b(x, y), b(y, z)) ≥ b(x, z) ≥ min(b(x, y), b(y, z)).
Moreover, in that case �+ is a weak order.
One natural question that we may ask is whether the fact that �+ is completeimplies that there is no reference-dependence. We shall provide an answer to thisquestion in the context of Theorem1, but we must defer it as of now, since the answeris simultaneously an answer to the question of how our result relates to the standardones on the WTP/WTA gap in classical consumer theory.
4.2 The WTA/WTP gap
The previous theorem provides some insights into the conditions under which aWTA/WTP gap due to an endowment effect appears. It is thus interesting to investigatethe conditions under which it disappears, as then failure of these conditions to obtainmay be considered the reason for its appearance. It is well-known that in standard con-sumer theory the WTA/WTP gap is only due to income effects that disappear whenthe utility function is assumed to be quasilinear. Quasilinearity can be captured in oursetting by the following axiom:
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Axiom 8 (Income independence (II)) For all r ∈ X , for all x, y ∈ X , for all λ ∈ R,
x �r y ⇐⇒ x ⊕ λ �r y ⊕ λ.
Traditionally, the permanent income hypothesis as defined by Friedman (1957)states that the individual’s consumption behavior is not affected by unexpected gains(windfalls) or losses. Imposing the previous axiom can be seen as stating essentiallythe same idea if one interprets r as the individual’s permanent income.8
Income independence obviously implies BPC. On the other hand, BPC impliesincome independence only for the special case where y = r . It is, therefore, a strictweakening.
The following corollary holds:
Corollary 3 Let (�r )r∈X be a regular RDP profile and let (ur )r∈X be the uniquenormalized family that represents it. Then the following are equivalent:
(i) the preference profile (�r )r∈X satisfies wealth independent status quo bias andincome independence.
(ii) there exists a reflexive upper ⊕-monotonic and upper ⊕-semicontinuous binaryrelation � on X such that(a) b�(x, r) = ur (x) ≤ s�(x, r).
(b) x � r ⇐⇒ ur (x) ≥ 0 ⇐⇒ b�(x, r) ≥ 0.
Corollary 3 shows when the willingness to pay can be used as a utility function, asis the usual practice in Industrial Organization and Partial Equilibrium Analysis. It iswell-known that in classical consumer theory this is possible if and only if preferencesare quasilinear. Corollary 3 extends to an abstract setting and generalizes this resultto the case of reference-dependent preferences. This in turn leads to potential gen-eralizations of extant models of Industrial Organization to settings where objects ofconsumption cannot easily be described in terms of commodity bundles. Corollary 3also extends the classical result that when preferences are quasilinear the compensat-ing variation is a sound measure of the welfare variation experienced by the consumer(see Mas-Colell et al. 1995, p.83).
Corollary 3, however, suggests that when there may be reference dependence, thestandard way of canceling income effects (income independence or quasi-linearity asit is more standardly called) is not sufficient in itself to eliminate the WTP/WTA gap,and the remaining one can thus be called the endowment effect proper, and be attrib-uted to reference-dependence effects. This is in line with the experimental findingsin Schmidt and Traub (2009), which are that the WTP/WTA gap persists even whenincome effects are reversed, so as to imply, in the standard theory, that WTP is largerthan WTA.
What is the exact nature of the reference-dependence effects involved? To figurethis out, we must note that, when we introduce a reference point, there is a potentialincome effect that can come into the picture when windfalls are integrated in the current
8 This interpretation would be consistent with Koszegi and Rabin (2006)’s conception of the referencepoint as expected future consumption.
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endowment. This would lead us to the following counterpart to income independence,wealth independence:
Axiom 9 (Wealth independence (WI)) For all x, y ∈ X , for all r ∈ X , for all λ ∈ R,
x �r y ⇐⇒ x �r⊕λ y.
A very natural interpretation of this axiom is that the wealth level of individualsdoes not affect their preferences. It, therefore, captures the cancellation of a differentform of income effects from the one captured by income independence. It is quitecounterintuitive: for instance, richer people are less risk averse than poorer people,since for identical lotteries what is at stake is lower. We do not claim that it has eithernormative or descriptive relevance in general, our intention is here merely to ana-lyze its consequences. These are summed up in the following corollary. Consider thefollowing axiom:
Axiom 10 (Inverse status quo bias (ISQB)) For all (x, r) ∈ X2,
x �x r ⇒ x �r r.
Then:
Corollary 4 Let (�r )r∈X be a regular RDP profile and let (ur )r∈X be the uniquenormalized family that represents it. Then the following are equivalent:
(i) the preference profile (�r )r∈X satisfies buying price consistency, status quobias, inverse status quo bias and wealth independence;
(ii) there exists a reflexive, upper ⊕-monotonic and upper ⊕-semicontinuous binaryrelation � on X such that(a) b�(x, r) ≤ ur (x) = s�(x, r).
(b) x �r r ⇐⇒ ur (x) ≥ 0 ⇐⇒ s�(x, r) ≥ 0.
As can be seen, this is exactly symmetrical to the previous corollary, as now, theelimination of the second kind of income effects justifies using the WTA as a util-ity function but does not eliminate the WTP/WTA gap altogether. This eliminationrequires controlling for both types of income effects. This implies that, if in experi-mental settings steps are taken towards eliminating standard income effects, i.e., tocreate the conditions of II, and if the gap still appears, it must be the result of refer-ence-dependence effects, i.e., the presence of SQB together with a failure of WI, evenfor small amounts.9
We may now, as promised, come back to the question of the link between the factthat �+ is a weak order and reference independence. Consider the following axioms:
Axiom 11 (Income independence for gains (IIG)) For all x, y ∈ X , for all λ ∈ R,
x �y y ⇐⇒ x ⊕ λ �y⊕λ y ⊕ λ.
9 Notice that Corollary 4 makes use of the SQB axiom and not the Monetary SQB axiom. This is becauseWI is strong enough to supplement for the monetary aspect of MSQB, and then some, as is obvious fromthe proof.
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Axiom 12 (Weak order for gains (WOG)) �+ is complete and transitive.
Axiom 13 (Existence of a buying price (EBP)) For all x, r ∈ X , there exists λ ∈ R
such that
x � λ �r r.
Axiom 14 (Reference independence (RI)) For all x, y, r, r ′ ∈ X ,
x �r y ⇐⇒ x �r ′ y.
Say that a function v : X → R is quasilinear if it satisfies v(x ⊕λ) = v(x)+λ. Thenwe have the following proposition:
Theorem 2 Let (�r )r∈X be a regular RDP profile. Then the following are equivalent:
(i) (�r )r∈X satisfies BPC, WISQB, IIG, WOG and EBP.(ii) (�r )r∈X satisfies II, SQB, ISQB, WI, WOG and EBP.
(iii) (�r )r∈X satisfies BPC and RI.(iv) (�r )r∈X satisfies II and RI.(v) There exists a quasilinear function v : X → R such that, for all x, y, r ∈ X,
x �r y ⇐⇒ v(x) ≥ v(y).
Moreover, b(x, r) = s(x, r) = v(x) − v(r) for all (x, r) ∈ X2.
Theorem 2 achieves several results.First, it provides a characterization of reference independence in the context of the
main theorem. It shows in particular that reference independence is equivalent to prop-erties of the gain relation �+, namely the fact that it is a weak order satisfying a formof income independence (which corresponds to quasilinearity in traditional consumertheory). This also helps clarify the question of the implications of completeness ofthe gain relation in the context of the theorem: it could be hypothesized that it wouldimply reference independence, but something more is needed for that, namely thequasilinearity property, or as point (ii) shows, the absence of income effects. Note,incidentally, that in (ii), WOG could be weakened to transitivity only, since all theother axioms taken together would imply completeness of the gain relation.
Second, Theorem 2 explains why quasilinearity is equivalent to the disappearanceof the WTA/WTP gap in the standard setting, since this setting is implicitly reference-independent. As we have seen, the disappearance of the gap requires controlling forboth types of income effects captured by II and WI. Quasilinearity takes care of II,while reference independence takes care of WI.
To pursue our analysis of the causes of the endowment effect, let us introduce thefollowing definition:
Definition 4 k ∈ X is crisp at r if it satisfies the following two conditions:
(i) For all λ ∈ R, k � λ �r r ⇐⇒ k �r r ⊕ λ or, equivalently, k �r r ⇐⇒k ⊕ λ �r r ⊕ λ.
(ii) For all λ ∈ R, r ⊕ λ �r k ⇐⇒ r ⊕ λ �k k or, equivalently, r �k k ⇐⇒r �r⊕λ k.
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Crisp elements can be characterized in a way that justifies their name.
Proposition 5 Assume (�r )r∈X satisfies weak order, solvability and monotonicity.Then k is crisp at r only if b�(k, r) = s�(k, r), where � is as in Theorem 1.
The converse holds if (�r )r∈X satisfies all the conditions of Theorem 1.
Given this proposition, we may say that the WTA/WTP gap rooted in the endowmenteffect is based on strategic considerations: if an object is not crisp, and so exhibits anendowment effect, this is either because the decision maker announces a WTP smallerthan the true monetary equivalent of the object (violation of crispness condition (i))or a higher WTA (violation of (ii)). Talking about strategic considerations implicitlyinvolves a game-theoretic setting. Indeed, it seems a little bit artificial to talk aboutWTP or WTA without implicitly referring to a commercial context involving at leasttwo agents, one buyer and one seller. Their interests are not aligned, even in the casewhere they value the object at exactly the same price: the seller wants to make a profit,while the buyer wants to make a surplus. This divergence of interests may be thereason for the WTA/WTP gap, even when the strategic environment remains implicitand potential.
4.3 Alternative formulations of the axioms and the main theorem
We wish here to discuss three questions:
1. What is the effect of imposing SQB only instead of WISQB?2. What are the appropriate preference conditions under which b and s can be
replaced by their variants b and s as defined earlier.3. What is the effect of relaxing some of the regularity properties of regular RDP
profiles?
4.3.1 The pure effect of the status quo bias
The answer to the first question is given by the following proposition (in the statementof which we omit the clauses pertaining to the case where the inequalities are strict,as they can be easily guessed from their counterpart in Theorem 1):
Proposition 6 Let (�r )r∈X be a regular RDP profile and let (ur )r∈X be the uniquenormalized family that represents it. Then (�r )r∈X satisfies buying price consistencyand SQB iff there exists a reflexive binary relation � such that
(i) b�(x, r) ≤ ur (x),(ii) ur (x) ≥ 0 ⇒ s�(x, r) ≥ 0,
(iii) x �r r ⇐⇒ ur (x) ≥ 0 ⇐⇒ b�(x, r) ≥ 0.
As can be seen, by weakening the axiom we get the weaker result that, while theWTP still has to be smaller than the utility function, the WTA does not have to be largerthan the WTP, or even than the normalized utility function. However, it cannot be neg-ative if either of them is nonnegative. The three conditions maintain some consistencyof evaluations: if ur (x) represents the true subjective value of object x for the decision
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maker, it is not economically rational to pay more for it than this value (condition (i))and it is rational to be willing to pay a nonnegative amount for it if and only if its truevalue is nonnegative (condition (iii)). Moreover, it would seem inconsistent to allowthe decision maker to pay to get rid of an object he/she otherwise values positively(condition (ii)). However this proposition allows selling at a loss, since the WTA canbe smaller than the WTP. The following example illustrates this case.
Example 3 We consider again the setting of example 1, i.e., X = R+ × R endowedwith its natural r.m.s. structure. For r = (ar , wr ) and x = (a, w), let
ur (x) = w − wr ear −a .
It is easily checked that this defines a normalized family of utility functions. Let usfirst compute b(x, r) for this family. For all λ ∈ R,
ur (x � λ) ≥ 0 ⇐⇒ w − wr ear −a ≥ λ,
hence
b(x, r) = w − wr ear −a = ur (x),
therefore, conditions (i) and (iii) are met. Now,
s(x, r) = −b(r, x) = wea−ar − wr ,
and some computations show that
s(x, r) − b(x, r) = (we−ar − wr e−a)(ea − ear ),
which can be negative, so that the endowment effect does not always hold.However,
b(x, r) ≥ 0 ⇐⇒ wea ≥ wr ear ⇐⇒ s(x, r) ≥ 0,
so that condition (ii) in the proposition is met.
4.3.2 The endowment effect with alternative definitions of WTA and WTP
Our second variation is to consider the conditions under which b and s can be replacedby their variants b and s as defined earlier. Although we have given arguments for thedefinition favored here, we will now, for the sake of completeness, provide and discussthe axioms that should replace BPC and WISQB in order to obtain a theorem similarto Theorem 1.
Consider the following axioms:
Axiom 15 (Wealth independence for gains (WIG)) For all x, r ∈ X, x �r r ⇒x �r⊕λ r for all λ ∈ R.
Axiom 16 (Income independent status-quo bias (IISQB)) For all x, r ∈ X, x �r r ⇒x ⊕ λ �x r ⊕ λ for all λ ∈ R.
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Recall that the condition in BPC can be rewritten as
x �r r ⇒ x ⊕ λ �r r ⊕ λ.
Axiom 15 is, therefore, a variation on BPC where the conclusion x ⊕ λ �r r ⊕ λ
is replaced by x �r⊕λ r . Insensitivity to the ex ante addition of a monetary amount tothe reference point is thus replaced by insensitivity with respect to ex post addition ofa monetary amount to both the reference point and the alternative. Axiom 16 performsthe opposite substitution for axiom WISQB.
The following holds:
Proposition 7 In the main theorem, statements (i) and (ii) hold with b� in lieu of b�iff BPC is replaced by WIG, and with s� in lieu of s� iff WISQB is replaced by IISQB.
This result shows that the existence of a gap between WTP and WTA can be char-acterized even in the absence of a consensus on the precise definition of these notions.Note, furthermore, it suggests that the different definitions of WTP and WTA are dualin the sense that they reverse the role of the relevant monetary invariance (ex ante orex post) that they require for the existence of a gap between them.
The property required in BPC could also be replaced by
x �r r ⇒ x ⊕ λ �r⊕λ r ⊕ λ.
This version requires insensitivity with respect to both ex ante and ex post addi-tion of the same monetary amount. Thus modified the axiom would require that theperception of an improvement with respect to the reference point depends only on thenon-monetary part of both the reference point and the alternative. Let us call uniformglobal independence for gains (UGIG) this modification of BPC. Theorem 2 aboveshows the implications of this property as it is involved in axiom IIG (however, thisaxiom is stronger than GIG since it requires an equivalence and not only an implica-tion).
UGIG, WIG and BPC are all weakenings of the following axiom:
Axiom 17 (Global independence for gains (GIG)) For all x, r ∈ X , for all λ,μ ∈ R,
x �r r ⇒ x ⊕ λ �r⊕μ r ⊕ λ.
It suffices to choose μ = λ for UGIG, λ = 0 for WIG and μ = 0 for BPC. Moreprecisely we have the following result between these properties:
Proposition 8 The following statements are equivalent for a given RDP profile:
(i) UGIG and WIG hold.(ii) UGIG and BPC hold
(iii) GIG holds.
Moreover, neither one of BPC,WIG and UGIG implies the other two.
4.4 Nonregular RDP profiles
In some situations, the decision maker might be unsure of how valuable a given objectis, in monetary terms. In such cases, the Solvability axiom cannot hold. This may be
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because he/she is unsure of how valuable it is in utility terms. In a companion paper(Giraud 2010), we explore the consequences of relaxing this axiom, as well as theWeak Order axiom, while essentially retaining both BPC and WISQB. This leads toa characterization of the WTP/WTA gap as a monetary translation of the imprecisionin the utility evaluation of a given object. We refer the reader to this paper for furtherdetails.
5 Concluding remarks
We have developed here a framework that allows us to precisely state the relation-ship between leading concepts of the theoretical and empirical research on reference-dependent preferences, namely the status quo bias, the endowment effect and theWTA/WTP gap. We have shown that a strengthened version of the status quo biasis a necessary condition for the WTA/WTP gap, and we have provided a method tofactor out the share of the gap due to income effects from the share of the gap dueto the endowment effect proper. We also showed that WTA and WTP are based on areference-independent preference relation that is complete and transitive whenever arestricted form of unpacking bias exists.
Many research paths can be followed at this point.First, in a companion working paper, we develop empirical predictions of the theory
spelled out here. In particular, we define, and characterize preference for liquidity andshow a link between preference for liquidity and preference reversals. These predic-tions need to be tested.
The second is to apply our results to fields of the economic literature where notionslike the willingness to pay play an important role, such as auction theory, electoralcompetition theory and models of product differentiation, where it might be of interestto study the impact of introducing reference-dependence and status quo effects.
The third research path is related to the formal framework we introduced. Thisframework can be generalized to allow for different notions of money, i.e., for a dif-ferent numeraire circulating between the objects of choice: one can think of time,information or decision frames. Replacing R by another group or semi-group andshowing how this can lead to fruitful analysis of these other contexts is thus the sub-ject of our future research.
Appendix
A. Additional material on r.m.s.
Examples
Example 4 Take V a vector space and g ∈ V, g �= 0. Assume X is a subset of V suchthat, for all x ∈ X , for all λ ∈ R, x + λg ∈ X . Then we can define an r.m.s. by lettingx ⊕ λ = x + λg.
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Example 5 Let (X,⊕) be an r.m.s. and Y be a non-empty set. Then:
– The set XY of all mappings from Y to X can be endowed with an r.m.s. structuredefining f ⊕ λ by
( f ⊕ λ)(y) = f (y) ⊕ λ.
– The set Y X of all mappings from X to Y can be endowed with an r.m.s. structuredefining f ⊕ λ by
( f ⊕ λ)(x) = f (x � λ).
Example 6 In particular, the set �0(X) of all simple probability measures on X can beendowed with an r.m.s. structure letting (p ⊕λ)(x) = p(x �λ). If {x1, . . . , xn} is thesupport of p, then {x1 ⊕ λ, . . . , xn ⊕ λ} is the support of p ⊕ λ, as (p ⊕ λ)(xi ⊕ λ) =p((xi ⊕ λ) � λ) = p(xi ⊕ (λ − λ)) = p(xi ) > 0.
Homomorphisms
Definition 5 Let (X,⊕) and (X ′,⊕′) be two r.m.s. A homomorphism from X to X ′is a mapping ϕ : X → X ′ such that for all x ∈ X and λ ∈ R,
ϕ(x ⊕ λ) = ϕ(x) ⊕′ λ.
X and X ′ are said to be isomorphic as r.m.s if there exists a bijective homomorphismfrom X to X ′.
Definition 6 Let (X,⊕) be an r.m.s. Then ⊕ is free if for all x ∈ X, λ ∈ R,
x ⊕ λ = x ⇒ λ = 0.
A r.m.s. (X,⊕) is free if ⊕ is a free action.
The following result is classical; we provide the proof for completeness.
Proposition 9 Let (X,⊕) be an r.m.s.. The following statements are equivalent:
(i) ⊕ is free.(ii) There exists a set A such that (X,⊕) is isomorphic to the canonical r.m.s. on A.
Proof (ii) ⇒ (i): Trivial.(i) ⇒ (ii): Consider the relation ≈ defined on X by
x ≈ y ⇐⇒ ∃λ ∈ R, y = x ⊕ λ.
Clearly this is an equivalence relation. The equivalence classes are called orbits.Let A be the set of orbits, i.e., A := X/ ≈. For each a ∈ A, by the axiom of choice,it is possible to fix xa ∈ a. We consider it fixed for the rest of the proof.
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Now take x ∈ X . There exists a unique a ∈ A such that x ∈ a. There exists,therefore, by definition a w ∈ R such that x = xa ⊕w. Let therefore ϕ(x) = (a, w).Does this properly define a mapping? If w′ is such that x = xa ⊕ w′, then, because(X,⊕) is a free r.m.s., w = w′. Therefore, ϕ is well-defined. It is clearly one-to-one.To see that it is onto, take (a, w) ∈ A × R. Then ϕ(xa ⊕ w) = (a, w). Therefore,ϕ is bijective. Let us show that it is a morphism. Take x ∈ X and λ ∈ R. Then,by definition, if x ∈ a, then x ⊕ λ ∈ a as well. Moreover, if x = xa ⊕ w, thenx ⊕ λ = xa ⊕ (w + λ). Therefore,
ϕ(x ⊕ λ) = (a, w + λ) = (a, w) ⊕ λ = ϕ(x) ⊕ λ.
This completes the proof. ��It is easy to deduce from this proposition that the first two examples of r.m.s. we
gave are isomorphic to a canonical r.m.s. However, this is not necessarily true for ther.m.s. discussed in Example 3; this depends on the underlying r.m.s. The followingexample shows that not all r.m.s. are free, and, therefore, not all r.m.s. are representableby a canonical r.m.s.
Example 7 Let X = R∗+, and define x ⊕ λ = x2λ
. Then x ⊕ 0 := x1 = x and
(x ⊕ λ) ⊕ μ = (x2λ)2μ = x2λ+μ = x ⊕ (λ + μ). This r.m.s. is not free because
1 ⊕ λ = 1 for all λ ∈ R.
The next proposition provides a simple test for the fact that an r.m.s. is free. If(X,⊕) is an r.m.s., say that f : X → R is a ⊕-quasilinear mapping if it is a homo-morphism from X to R endowed with its natural r.m.s. structure. Call X⊕ the set of⊕-quasilinear functionals. Then:
Proposition 10 The r.m.s. (X,⊕) is free if and only if X⊕ �= ∅.
Proof Assume X is free. Then, X is isomorphic to A × R for some set A. Therefore,w.l.o.g. we can write x = (a, w) for a unique pair (a, w). Define f : X → R lettingf (x) = w. Then for all λ ∈ R, f (x ⊕ λ) = f (a, w + λ) = w + λ = f (x) + λ.Therefore, f ∈ X⊕.
Conversely, if X⊕ �= ∅, take f ∈ X⊕, x ∈ X and λ ∈ R such that x ⊕ λ = x .Then, f (x ⊕ λ) = f (x) + λ = f (x), so that λ = 0. ��
B. Additional material on the benefit function
Examples of benefit functions
Luenberger’s benefit function
Luenberger’s original definition of the benefit function was set in the context of stan-dard consumer theory, and is as follows: let X be a subset of R
L+ bounded from below
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and g ∈ X ; let u : X → R be a utility function for preference ordering � and α ∈ R.Then, Luenberger’s definition of the benefit function (Luenberger 1992) is:
b(x, α; g) := sup{λ ∈ R | u(x − λg) ≥ α}.
Given r ∈ X , it is also possible to define what Luenberger calls the compensatingbenefit function, namely cb(x, r; g) = b(x, u(r); g). If � is the preference order-ing generated by u, Luenberger’s compensating benefit function is none other thanb�(x, r) when the r.m.s. operation is defined by x ⊕ λ := x + λg. Luenberger doesnot define the equivalent of the loss function, but it is straightforward to see how itcould be defined.
Shephard and Deaton’s distance function
In a very similar spirit, Shephard and Deaton’s distance function (Shephard 1953;Deaton 1979) defined, for some binary relation � on X , by:
D�(x, r) = sup{μ > 0 | 1μ
x � r}
can be associated with a benefit function. For λ > 0, define
x ⊕ λ := exp(λ)x .
Then, we have
D�(x, r) = exp(b�(x, r)).
Again, Shephard and Deaton do not define a notion corresponding to the loss func-tion, but it is straightforward to see how this could be done.
Value-at-risk
A standard tool for risk measurement and financial decision-making is value-at-risk.It can be related to the benefit function as we have defined it. Let (�,F , P) be aprobability space and let X be the set L∞(�,F , P) of bounded real random vari-ables modeling financial assets. X can be endowed with an r.m.s. structure defining,for Z ∈ X and λ ∈ R, Z ⊕λ = Z +λ. Let α ∈ [0, 1] and Z ∈ X . Then, the α-quantileof Z , qα(Z) is defined by
The idea of this notion is that it measures the maximal amount of money one iswilling to pay to hedge the risk involved in R by buying Z instead. It is the buyingprice for a swap contract between Z and R. In other words, defining the binary relation�P on X by
Z �P Z ′ ⇐⇒ P(Z ≤ 0) ≤ P(Z ′ ≤ 0),
we have
qR(Z) = b�P (Z , R).
Z �P Z ′ can be interpreted as Z being less risky than Z ′.Now, we can analogously define the value-at-risk relative to the benchmark asset
R as
V a R(Z , R) = −qR(Z) = −b�P (Z , R).
It is easy to see that, if P is non-atomic, then for all α there exists Rα ∈ X such thatP(Rα ≤ 0) = α, and then the standard value-at-risk is a special case of the notion wedefined here.
Value-at-risk is a special case of the concept of capital requirement defined as fol-lows: let A be a proper subset of X such that Z ≥ Z ′ ∈ A implies Z ′ ∈ A – Ais called an acceptability set. Then the capital requirement associated to A , ρA , isdefined as
ρA (Z) = inf{α ∈ R | Z + α ∈ A }
Now assume that each benchmark asset R defines an acceptability set AR and definethe capital requirement associated to R as
ρ(Z , R) := ρAR (Z).
Then, defining a relation � on X by
Z � Z ′ ⇐⇒ Z ∈ AZ ′ ,
it is easy to see that ρ(Z , R) = −b�(Z , R). Coherent risk measures introduced byArtzner et al. (1999) and convex risk measures introduced by Föllmer and Schied(2002) are special cases with respectively convex cones and convex sets as accept-ability sets.
Moreover, the translation property links benefit functions with the concept of capi-tal requirement (Artzner et al. 1999). Let an acceptability set be a proper subset A of
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some r.m.s. (X,⊕) such that for all λ ≥ 0 and x ∈ A , x ⊕ λ ∈ A . Then the capitalrequirement ρA : X → R is defined as:
ρA (x) = inf{α ∈ R | x ⊕ α ∈ A }.
Proposition 11 Let b : X2 → R. The following are equivalent:
(i) b satisfies the translation property.(ii) there exists a binary relation � on X such that b(x, r) = b�(x, r)
(iii) there exists an acceptability set A (r) for all r ∈ X such that b(x, r) =−ρA (r)(x).
Proof
(i)⇒ (ii) Define � by x � y if and only if b(x, y) ≥ 0. Then by the Transla-tion property B�(x, r) = {λ | b(x, r) ≥ λ}, and b(x, r) ∈ B�(x, r), therefore,b�(x, r) = b(x, r).(i)⇒ (iii) Defined A (r) = {x ∈ X | x � r}, with � as above. Notice that the Trans-lation Property implies that A (r) is an acceptability set since λ ≥ 0 and x ∈ A (r)
imply that b(x ⊕ λ, r) = b(x, r) + λ ≥ 0 and moreover b(x � (b(x, r) + 1), r) =−1 < 0 so that A (r) must be a proper subset of X . Now by the previous argumentb(x, r) = b�(x, r), and b�(x, r) = −ρ(x, r), so that b(x, r) = −ρ(x, r).(ii)⇒ (i) If b = b�, then
b(x ⊕ λ, r) = b�(x ⊕ λ, r)
= sup{λ′ ∈ R | (x ⊕ λ) � λ′ � r}= sup{λ′ ∈ R | x ⊕ (λ − λ′) � r}= sup{λ′ ∈ R | x � (λ′ − λ) � r}= sup{μ + λ | μ ∈ R, x � μ � r}= sup{μ ∈ R | x � μ � r} + λ
= b�(x, r) + λ
= b(x, r) + λ.
(iii)⇒ (i) Proved similarly. ��
Exact benefit functions
The purpose here is to study the relationship between a benefit function and the binaryrelation that generates it. It is easy to see that for all x, y ∈ X ,
x � y ⇒ b�(x, y) ≥ 0,
but does the converse hold however? It turns out that this is not always the case, as wenow show. Specifically, call exact a benefit function that has this property; recall that
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an upper ⊕-monotonic binary relation � on X is such that for all x, y ∈ X, λ, μ ∈ R,
x ⊕ μ � y and λ ≥ μ ⇒ x ⊕ λ � y
and it is upper ⊕-semicontinuous if for all x, y ∈ X , the set {λ | x ⊕ λ � y} isclosed. Let us say that � is upper ⊕-regular if it satisfies both conditions. We havethe following first result:
Lemma 1 A benefit function b� is exact if and only if � is upper ⊕-regular.
Proof If: Assume � is upper ⊕-monotonic and upper ⊕-semicontinuous andtake x, y ∈ X such that b�(x, y) ≥ 0. Since b�(x, y) ≥ 0, B�(x, y) �= ∅.We must show that there exists λ ∈ B�(x, y) such that λ ≥ 0. Assumefirst that b�(x, y) < +∞. Then, by upper semi-continuity, the supremumof B�(x, y), is reached, so that x � b�(x, y) � y. If b�(x, y) = +∞, thenit is impossible that for all λ ∈ R, λ ∈ B�(x, y) implies λ < 0, becauseotherwise we would have b�(x, y) ≤ 0, a contradiction. Take, therefore, aλ ≥ 0 in B�(x, y). Then, since −λ ≤ 0, upper monotonicity implies x � y.
Only if: Let (λn) be a sequence converging to λ ∈ R and such that x ⊕ λn � y.Then, by definition b�(x, y) ≥ −λn , and, therefore, b�(x, y) ≥ −λ, i.e.,b�(x ⊕ λ, y) ≥ 0 and, since b� is exact, x ⊕ λ � y. Therefore, � is upper⊕-semicontinuous. To show upper monotonicity, take x, y ∈ X, λ, μ ∈ R,
such that x ⊕ μ � y and λ ≥ μ. Then b�(x, y) ≥ −μ ≥ −λ, therefore,since b� is exact, x ⊕ λ � y.
��Can we say more? Let � be a binary relation and let �� be the upper ⊕-regular clo-
sure of �, i.e., the smallest (in the sense of inclusion) upper ⊕-regular binary relationextending �. Moreover, let �b be the binary relation defined by
x �b y ⇐⇒ b�(x, y) ≥ 0.
We have the following result:
Lemma 2 Let � be a binary relation. Then �� =�b and b�� = b�.
Proof For the first property, since �⊆�b, we must show two things: that �b is upper⊕-regular and that for any upper ⊕-regular binary relation �′ such that �⊆�′, wehave �b⊆�′.
Let (λn) be a sequence converging to λ ∈ R and such that x ⊕ λn �b y. Then, bydefinition b�(x, y) ≥ −λn , and, therefore, b�(x, y) ≥ −λ, i.e., b�(x ⊕λ, y) ≥ 0 andx ⊕ λ �b y. Therefore, �b is upper ⊕-semicontinuous. To show upper monotonicity,take x, y ∈ X, λ, μ ∈ R, such that x ⊕ μ �b y and λ ≥ μ. Then b�(x, y) ≥ −μ ≥−λ, therefore, x ⊕ λ �b y.
Now let �′ be an upper ⊕-regular binary relation such that �⊆�′ and assumex �b y. Then b�(x, y) ≥ 0, or, specifically sup B�(x, y) ≥ 0. But, since �⊆�′,B�(x, y) ⊆ B ′
Proof (Proposition 1)We begin by the existence part.By Solvability, there exists λ ∈ R such that: x ∼r r ⊕ λ. Monotonicity implies
that it is unique.Moreover, monotonicity and weak order imply that, for all λ,μ ∈ R,
λ ≥ μ ⇐⇒ r ⊕ λ �r r ⊕ μ.
Indeed, the first implication follows immediately from Monotonicity if λ > μ andfrom reflexivity if λ = μ. Conversely, if r ⊕ λ �r r ⊕ μ, it is impossible to haveμ > λ because, by Monotonicity, this would imply r ⊕ μ �r r ⊕ λ, which, by weakorder, is contrary to the assumption.
It suffices, therefore, to let ur (x) := λ. This clearly defines a normalized family ofutility functions. This completes the proof of sufficiency. Necessity is straightforward.
Now for the uniqueness part. Assume there exists another normalized family thanthe one just constructed, (vr )r∈X . Then, since x ∼r r ⊕ ur (x) by definition and sincevr is a utility function,
vr (x) = vr (r ⊕ ur (x)) = ur (x).
��Proof (Proposition 2) Assume that x = x ⊕ λ for some x and λ. Then, by reflexivity,x ∼x x ⊕ λ, hence, by Monotonicity, λ = 0. Therefore, (X,⊕) is a free r.m.s. and weconclude by Proposition 9. ��Proof (Theorem 1)
Sufficiency of the axioms: Define � by x � y ⇔ x �y y and let s := s� andb := b�. We want to show now that s(x, r) ≥ ur (x) ≥ b(x, r) for all x, r ∈ X .Take μ such that r ⊕μ �x x . Then, by WISQB, r ⊕μ �r x . But x ∼r r ⊕ur (x).Therefore, by Weak Order, r ⊕μ �r r ⊕ur (x) and, by Monotonicity, μ ≥ ur (x)
and, finally, s(x, r) ≥ ur (x). Take λ such that x � λ �r r . Then, by Buying
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Price Consistency, x �r r ⊕ λ. But x ∼r r ⊕ ur (x). Therefore, by Weak Order,r ⊕ur (x) �r r ⊕λ and, by Monotonicity, ur (x) ≥ λ and, finally, ur (x) ≥ b(x, r).
As for the second claim, assume first ur (x) ≥ 0. Then x �r r , implying, bydefinition of b(x, r), b(x, r) ≥ 0. But, since b(x, r) ≤ ur (x), we have ur (x) ≥ 0.This shows in particular by Lemma 1 that � is reflexive, upper ⊕-monotonic andupper ⊕-semicontinuous.
Necessity of the axioms: Assume the conclusion of the theorem holds. Given Prop-osition 1, the only thing to prove is that WISQB and Buying Price Consistencyhold.
For WISQB, assume x ⊕ λ �r r . Then, s(r, x) ≤ λ. But s(r, x) ≥ ux (r), sothis implies ux (r) ≤ λ = ux (x ⊕ λ), that is x ⊕ λ �x r .
For Buying Price Consistency, assume x � λ �r r . Then b(x, r) ≥ λ. But, asur (x) ≥ b(x, r), it follows that ur (x) ≥ λ = ur (r ⊕ λ): x �r r ⊕ λ.
Now, assume that BPC holds strictly for (x, r) and that b(x, r) = ur (x). Thenx � ur (x) �r r , and, since BPC holds strictly, x �r r ⊕ ur (x), a contradiction withthe definition of ur (x). Conversely, assume ur (x) > b(x, r) and x � λ �r r . Then,b(x, r) ≥ −λ, hence ur (x) > −λ, therefore, x �r r � λ, i.e., BPC holds strictly for(x, r).
Assume that WISQB holds strictly for (r, x) and that ur (x) = s(x, r). Then r ⊕ur (x) �x x , and, since WISQB holds strictly, r ⊕ ur (x) �r x , a contradiction withthe definition of ur (x). Conversely, assume ur (x) < s(x, r) and r ⊕ μ �x x . Then,s(x, r) ≤ μ, hence ur (x) < μ, therefore, r ⊕ μ �r x , i.e., WISQB holds strictly for(r, x). ��Proof (Proposition 3)
Assume first that � satisfies conditions (i) and (ii) of the theorem. Then condition(ii) and Lemma 2 imply that �� =�+.
Conversely, assume that �� =�+. Then, lemma 2 and the assumption then implythat b� = b�+ , hence also by duality s� = s�+ , so that (i) automatically holds. Asfor (ii), we have, by Lemma 2,
x �y y ⇔ x �+ y ⇔ x ��y ⇔ b�(x, y) ≥ 0.
��Proof (Proposition 6)
In view of the proof of Theorem 1, the only thing to prove is that SQB is a necessaryand sufficient condition for (ii), given that all the other axioms hold.
Assume that SQB holds. Then, if ur (x) ≥ 0, then x �r r , hence x �x r , henceb(r, x) ≤ 0, i.e., s(x, r) ≥ 0. Conversely, if (ii) holds, then if x �r r , then ur (x) ≥ 0,hence s(x, r) ≥ 0, hence x �x r . ��Proof (Proposition 7)
We shall focus on proving (i), since (ii) does not rely on the definition of WTP orWTA used.
Assume that WIG holds. Let (x, r) ∈ X2 and λ ∈ R such that x �r⊕λ r ⊕λ. Then,by WIG, we have x �r r ⊕ λ, hence r ⊕ ur (x) �r r ⊕ λ, which implies ur (x) ≥ λ,
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and, therefore, ur (x) ≥ b�(x, r). Conversely, assume (i) holds with b� in lieu of b�and suppose x �r r . Then, b�(x, r) ≥ 0, hence for all λ, b�(x, r)−λ ≥ −λ. Now, itis easy to show that b�(x, r ⊕λ) = b�(x, r)−λ. Hence, we have b�(x, r ⊕λ) ≥ −λ,therefore, by (i), ur⊕λ(x) ≥ −λ = ur⊕λ(r), hence WIG holds.
Assume that IISQB holds. Let (x, r) ∈ X2 and λ ∈ R such that r �x�λ x�λ. Then,by IISQB, we have r ⊕ λ �r x , hence r ⊕ λ �r r ⊕ ur (x), which implies λ ≥ ur (x),and, therefore, s�(x, r) ≥ ur (x). Conversely, assume (i) holds with s� in lieu of s�and suppose x �r r . Then, s�(r, x) ≤ 0, hence for all λ, s�(r, x) + λ ≤ λ. Now, it iseasy to show that s�(x ⊕ λ, r) = s�(x, r) + λ. Hence, we have s�(r ⊕ λ, x) ≤ λ,therefore, by (i), ux (r ⊕ λ) ≤ λ = ux (x ⊕ λ), hence IISQB holds. ��
Proof (Corollary 1) Let x, y, z ∈ X .Assume �+ is transitive and min(b(x, y), b(y, z)) ≥ 0. Then b(x, y) ≥ 0 and
b(y, z) ≥ 0, hence, by Theorem 1, x �y y and y �z z, hence, by transitivity of�+, x �z z, which implies b(x, z) ≥ 0.
Conversely, assume that min(b(x, y), b(y, z)) ≥ 0 ⇒ b(x, z) ≥ 0 and thatx �y y and y �z z. Then b(x, y) ≥ 0 and b(y, z) ≥ 0 by the theorem, hencemin(b(x, y), b(y, z)) ≥ 0, hence b(x, z) ≥ 0 and, by the theorem, x �z z. ��
Proof (Corollary 2) Let x, y ∈ X .Assume �+ is complete. Then x �y y or y �x x , that is b(x, y) ≥ 0 or b(y, x) ≥ 0.
Hence max(b(x, y), b(y, x)) ≥ 0.Conversely, assume that max(b(x, y), b(y, x)) ≥ 0. Then b(x, y) ≥ 0 or b(y, x) ≥
0, that is x �y y or y �x x , hence �+ is complete. ��
Proof (Proposition 4) Let x, y, z ∈ X .Assume the axiom holds. Assume first that b(x, y) and b(y, z) are finite. Then,
since x � b(x, y) �y y and y � b(y, z) �z z:
x � b(x, y) �z z or x � b(y, z) �z z.
By the definition and properties of b, however, this implies, in the context of themain theorem, since �+ must upper ⊕-monotonic, x � min(b(x, y), b(y, z)) �z z,hence b(x, z) ≥ min(b(x, y), b(y, z)). If, either b(x, y) or b(y, z) is infinite, by The-orem 1 it must be −∞, hence min(b(x, y), b(y, z)) = −∞ ≤ b(x, z). Moreover, takeμ as in the axiom. Since x � μ �y y or y � μ �z z, b(x, y) ≥ μ or b(y, z) ≥ μ,i.e., max(b(x, y), b(y, z)) ≥ μ. On the other hand, z �z x � μ, hence b(x, z) ≤ μ,so that max(b(x, y), b(y, z)) ≥ b(x, z).
Conversely, assume max(b(x, y), b(y, z)) ≥ b(x, z) ≥ min(b(x, y), b(y, z)) andx � λ �y y and y � λ′ �z z. Then b(x, y) ≥ λ ≥ min(λ, λ′) and b(y, z) ≥λ′ ≥ min(λ, λ′), hence min(b(x, y), b(y, z)) ≥ min(λ, λ′), therefore, by assumptionb(x, z) ≥ min(λ, λ′), hence x �min(λ, λ′) �z z. But min(λ, λ′) can be either λ or λ′,so the first part of the axiom holds. Moreover, since max(b(x, y), b(y, z)) ≥ b(x, z),there exists μ such that max(b(x, y), b(y, z)) ≥ μ ≥ b(x, z), hence b(x, y) ≥ μ orb(y, z) ≥ μ and x � μ �y y or y � μ �z z and μ ≥ b(x, z), hence z �z x � μ.
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Finally, z = x in the first inequality implies max(b(x, y), b(y, x)) ≥ 0, hence�+ is complete and, if min(b(x, y), b(y, z)) ≥ 0, then the second inequality impliesb(x, z) ≥ 0, hence �+ is transitive. ��Proof (Corollary 3) In all of this proof we consider a normalized family (ur ) of utilityfunctions.
(i)⇒ (ii) Given Theorem 1, the only thing to be proved is that b�(x, r) = ur (x),where � is defined as in Theorem 1, since WISQB implies that s�(x, r) ≥ ur (x).Now, for all x, r ∈ X, x ∼r r ⊕ ur (x), therefore, by II, x � ur (x) ∼r r , and,therefore, b�(x, r) ≥ ur (x). Conversely, if λ is such that x � λ �r r , then by IIagain x �r r ⊕ λ, and, therefore, ur (x) ≥ λ, and since this holds for any suchλ, ur (x) ≥ b�(x, r). Therefore b�(x, r) = ur (x).(ii)⇒ (i) This follows from Theorem 1 and the Translation Property.
��Proof (Corollary 4) In all of this proof we consider a normalized family (ur ) of utilityfunctions.
(i)⇒ (ii) Given Theorem 1, the only thing to be proved is that s�(x, r) = ur (x),where � is defined as in Theorem 1, since BPC implies that b�(x, r) ≤ ur (x).Now, for all x, r ∈ X, x ∼r r ⊕ ur (x), therefore, by WI, x ∼r⊕ur (x) r ⊕ ur (x),and by ISQB x �x r ⊕ur (x); therefore, s�(x, r) ≤ ur (x). Conversely, if μ is suchthat r ⊕ μ �x x , then, by SQB, r ⊕ μ �r⊕μ x , and, by WI, r ⊕ μ �r x , therefore,ur (x) ≤ μ, and since this holds for any such μ, ur (x) ≤ s�(x, r). Therefore,s�(x, r) = ur (x).(ii)⇒ (i) This follows from Theorem 1, the Translation Property and the fact thats�(x, r) = −b�(r, x).
��Proof (Theorem 2)
(i)⇒ (v) Let us first construct v. Fix x0 ∈ X . For any x ∈ X , consider the sets
U (x) := {λ ∈ R | x0 ⊕ λ �+ x}
and
L(x) := {λ ∈ R | x �+ x0 ⊕ λ}.
By EBP, U (x) is non-empty. Moreover, since �+ is upper ⊕-continuous, U (x) isclosed.By IIG, L(x) = {λ ∈ R | x � λ �+ x0}. Therefore, by the same arguments it isnonempty and closed.By WOG, U (x) ∪ L(x) = R.Therefore, since R is connected, U (x) ∩ L(x) �= ∅: there exists λ such that x ∼+x0 ⊕ λ. λ is unique: if there is another one λ′, then by WOG x0 ⊕ λ′ ∼+ x0 ⊕ λ,hence by IIG x0 ⊕(λ′−λ) ∼+ x0, i.e., x0 ⊕(λ′−λ) ∼x0 x0, hence by Monotonicity
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λ′ = λ. We, therefore, denote v(x) this number. Clearly, since x ∼+ x0 ⊕ v(x)
implies by IIG x ⊕ λ ∼+ x0 ⊕ (v(x) + λ) for all λ, v(x ⊕ λ) = v(x) + λ.Now, let us show that v represents �+. We have:
v(x) ≥ v(y) ⇐⇒ v(x) − v(y) ≥ 0
⇐⇒ x0 ⊕ (v(x) − v(y)) �+ x0
⇐⇒ x0 ⊕ v(x) �+ x0 ⊕ v(y) by IIG
⇐⇒ x �+ y by WOG.
Now, by EBP, b(x, r) �= −∞ and x � b(x, r) �r r implies x � b(x, r) �+ r ,hence v(x � b(x, r)) ≥ v(r) which implies v(x) − v(r) ≥ b(x, r). Conversely,v(x � (v(x)− v(r)))− v(r) = v(x)− v(x)+ v(r)− v(r) = 0, hence x � (v(x)−v(r)) ∼+ r , i.e., x � (v(x) − v(r)) ∼r r , hence b(x, r) ≥ v(x) − v(r). There-fore, b(x, r) = v(x)−v(r). Furthermore, s(x, r) = −b(r, x) = −(v(r)−v(x)) =v(x)−v(r) as well, therefore, b(x, r) = s(x, r). Since we are in the context of The-orem 1, this implies ur (x) = b(x, r) = v(x) − v(r). We can, therefore, concludethat
x �r y ⇐⇒ v(x) − v(r) ≥ v(y) − v(r) ⇐⇒ v(x) ≥ v(y).
(v)⇒ (iv): Trivial.(iv)⇒ (iii): Trivial.(iii)⇒ (ii): RI trivially implies SQB, ISQB and WI. For II, assume x �r y. Thenby RI, x �y y and, by BPC, for all λ ∈ R, x ⊕ λ �y y ⊕ λ, hence by RI again,x ⊕λ �r y ⊕λ. Conversely, if x ⊕λ �r y ⊕λ for some λ, then by RI, this impliesx ⊕ λ �y⊕λ y ⊕ λ, hence by BPC, x �y⊕λ y, and by RI again, x �r y. For WOG,notice that by RI x �+ y iff x �y y iff x �r y for some (actually, for all) r . Sinceeach �r is a weak order, so is �+. For EBP, let x, r ∈ X and notice that, since RIimplies automatically SQB, ISQB and WI, we have by (the proof of) Corollary 4s(r, x) = ux (r). Therefore, b(x, r) = −ux (r), hence b(x, r) �= −∞ and settingλ = −ux (r), which exists since the RDP profile is regular, makes do.(ii)⇒ (i): II implies BPC. SQB and WI imply WISQB. For IIG, suppose x ⊕λ �y⊕λ
y ⊕ λ. Then by BPC, we have x �y⊕λ y, and by WI x �y y. Conversely, supposex �y y. Then, by BPC, we have x ⊕ λ �y y ⊕ λ, and by WI x ⊕ λ �y⊕λ y ⊕ λ.
��Proof (Proposition 5) Since (�r )r∈X satisfies Weak Order, Solvability and Monoto-nicity, fix a normalized family of utility functions (ur )r∈X .
Let k be crisp at r . We must show that b�(k, r) = ur (k) = s�(k, r). For simplicitywe omit the reference to �. Take λ ∈ R such that k � λ �r r . Then k �r r ⊕ λ,therefore, r ⊕ur (k) �r r ⊕λ, so that ur (k) ≥ λ, yielding ur (k) ≥ b(k, r). Conversely,k ∼r r ⊕ ur (k) implies k � ur (k) �r r : ur (k) ≤ b(k, r). Therefore, b(k, r) = ur (k).To show that s(k, r) = ur (k), note that k ∼r r ⊕ ur (k) implies that r ⊕ ur (k) �r kand, therefore, since k is crisp at r, r ⊕ ur (k) �k k: ur (k) ≥ s(k, r). Conversely, forall μ such that r ⊕ μ �k k, we have r ⊕ μ �r k ∼r r ⊕ ur (k), so that μ ≥ ur (k):s(k, r) ≥ ur (k).
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336 R. Giraud
Conversely, assume that all the conditions of Theorem 1 hold and that b(k, r) =s(k, r). Then this implies that b(k, r) = s(k, r) = ur (k). Let us first prove crisp-ness condition (i). Since Buying Price Consistency holds, we only have to prove thatk �r r ⊕λ implies k �λ �r r . k �r r ⊕λ implies that uk(r) ≥ λ, i.e., uk(r)−λ ≥ 0.But since b(k, r) = ur (k), this implies b(k, r) − λ = b(k � λ, r) ≥ 0, and, therefore,k � λ �r r by condition (ii) of Theorem 1. Let us now prove crispness condition(ii). Since WISQB holds, we only have to prove that r ⊕ λ �r k ⇒ r ⊕ λ �k k.Assume, therefore, that r ⊕ λ �r k. Then λ ≥ ur (k) = s(k, r) = −b(r, k), therefore,b(r ⊕ λ, k) ≥ 0 and r ⊕ λ �k k. This completes the proof. ��Proof (Proposition 8)
(i) ⇒ (iii) Assume x �r r . Then, by UGIG, x ⊕λ �r⊕λ r ⊕λ for all λ ∈ R,and by WIG, x ⊕ λ �r⊕μ r ⊕ λ for all μ ∈ R.
(ii) ⇒ (iii) Assume x �r r . Then, by UGIG, x ⊕ μ �r⊕μ r ⊕ μ for allμ ∈ R, and by BPC, x ⊕ λ �r⊕μ r ⊕ λ for all λ ∈ R.
The other implications are trivial. The next examples show the independence of allthe properties with respect to each other.
Example 8 (BPC implies neither WIG nor UGIG) Consider again the case X = R+ ×R with, for r = (ar , wr ) and x = (a, w),
ur (x) = w − wr ear −a .
That BPC holds is a consequence of the fact that this utility function satisfies con-dition (i) of Proposition 6 (see Example 3). To see that neither WIG nor UGIG hold,take x = (1, 3), r = (2, 1) and λ = 2(3−e)
Example 9 (WIG implies neither BPC nor UGIG) Consider the case X = N×R withits canonical r.m.s. structure and let, for r = (ar , wr ) and x = (a, w),
ur (x) = wa−ar .
Since ur (x) does not depend on wr , WIG is satisfied by construction. Moreover,for the same reason, BPC is satisfied iff UGIG is satisfied. Now,
ur (x ⊕ λ) ≥ ur (r ⊕ λ) ⇔ (w + λ)ar −a ≥ 1
⇔⎧⎨
⎩
λ ≥ 1 − w if a > ar ,
λ ≤ 1 − w if a < ar ,
λ ∈ R if a = ar .
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Money matters 337
It is, therefore, obvious that neither BPC nor UGIG hold in general.
Example 10 (UGIG implies neither BPC nor WIG) Consider again the case X =R+ × R with, for r = (ar , wr ) and x = (a, w),
ur (x) ={ war −wr a
ar −a if ar �= a2w − wr otherwise
.
That UGIG holds is a consequence of the fact that
ur⊕λ(x ⊕ λ) = ur (x) + λ.
To see that WIG does not hold, take x = (1, 3), r = (4, 1) and λ = −5. Then
ur (x) = 11
3> 1 = ur (r),
hence x �r r , but
ur⊕λ(x) − ur⊕λ(r) = −2
3< 0,
hence r �r⊕λ x .To see that BPC does not hold, take x = (1, 3), r = (4, 1) and λ = 5. Then
ur (x ⊕ λ) − ur (r ⊕ λ) = −7
3< 0,
hence r �r⊕λ x .��
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