Savage in the Market 1

Transcription

1 Savage in the Market 1 Federico Echenique Caltech Kota Saito Caltech January 22, We thank Kim Border and Chri Chamber for inpiration, comment and advice. Matt Jackon uggetion led to ome of the application of our main reult. We alo thank eminar audience in Bocconi Univerity, Caltech, Collegio Carlo Alberto, Princeton Univerity, RUD 2014 (Warwick), Univerity of Queenland, Univerity of Melbourne, Larry Eptein, Eddie Dekel, Mark Machina, Maimo Marinacci, Fabio Maccheroni, John Quah, Ludovic Renou, Kyoungwon Seo, and Peter Wakker for comment. We are particularly grateful to the editor and three anonymou referee for their uggetion. The dicuion of tate-dependent utility and probabilitic ophitication in Section 4 and 5 follow cloely the uggetion of one particular referee.

2 Abtract We develop a behavioral axiomatic characterization of Subjective Expected Utility (SEU) under rik averion. Given i an individual agent behavior in the market: aume a finite collection of aet purchae with correponding price. We how that uch behavior atifie a revealed preference axiom if and only if there exit a SEU model (a ubjective probability over tate and a concave utility function over money) that account for the given aet purchae.

3 1. Introduction The main reult of thi paper give a revealed preference characterization of rik-avere ubjective expected utility. Our contribution i to provide a neceary and ufficient condition for an agent market behavior to be conitent with rik-avere ubjective expected utility (SEU). The meaning of SEU for a preference relation ha been well undertood ince Savage (1954), but the meaning of SEU for agent behavior in the market ha been unknown until now. Rik-avere SEU i widely ued by economit to decribe agent market behavior, and the new undertanding of rik-avere SEU provided by our paper i hopefully ueful for both theoretical and empirical purpoe. Our paper follow the revealed preference tradition in economic. Samuelon (1938) and Houthakker (1950) decribe the market behavior that are conitent with utility maximization. They how that a behavior i conitent with utility maximization if and only if it atifie the trong axiom of revealed preference. We how that there i an analogou revealed preference axiom for rik-avere SEU. A behavior i conitent with rik-avere SEU if and only if it atifie the trong axiom of revealed ubjective expected utility (SARSEU). (In the following, we write SEU to mean rik-avere SEU when there i no potential for confuion.) The motivation for our exercie i twofold. In the firt place, there i a theoretical payoff from undertanding the behavioral counterpart to a theory. In the cae of SEU, we believe that SARSEU give meaning to the aumption of SEU in a market context. The econd motivation for the exercie i that SARSEU can be ued to tet for SEU in actual data. We dicu each of thee motivation in turn. SARSEU give meaning to the aumption of SEU in a market context. We can, for example, ue SARSEU to undertand how SEU differ from maxmin expected utility (Section 6). The difference between the SEU and maxmin utility repreentation i obviou, but the difference in the behavior captured by each model i much harder to grap. In fact, we how that SEU and maxmin expected utility are inditinguihable in ome ituation. In a imilar vein, we can ue SARSEU to undertand the behavioral difference between SEU and probabilitic ophitication (Section 5). Finally, SARSEU help u undertand how SEU retrict behavior over and above what i captured by the more general model of tate-dependent utility (Section 4). The online appendix dicue additional theoretical implication. 1

4 Our reult allow one to tet SEU non-parametrically in an important economic deciion-making environment, namely that of choice in financial market. The tet doe not only dictate what to look for in the data (i.e SARSEU), but it alo ugget experimental deign. The yntax of SARSEU may not immediate lend itelf to a practical tet, but there are two efficient algorithm for checking the axiom. One of them i baed on linearized Afriat inequalitie, ee Lemma 7 of Section A. The other i implicit in Propoition 2). SARSEU i on the ame computational tanding a the trong axiom of revealed preference. Next, we decribe data one can ue to tet SARSEU. There are experiment of deciionmaking under uncertainty where ubject make financial deciion, uch a Hey and Pace (2014), Ahn et al. (2014) or Boaert et al. (2010). Hey and Pace, and Ahn et. al. tet SEU parametrically: they aume a pecific functional form. A nonparametric tet, uch a SARSEU, eem ueful becaue it free the analyi from uch aumption. Boaert et al. (2010) do not tet SEU itelf; they tet an implication of SEU on equilibrium price and portfolio choice. The paper by Hey and Pace fit our framework very well. They focu on the explanatory power of SEU relative to variou other model, but they do not tet how well SEU fit the data. Our tet, in contrat, would evaluate goodne of fit, and in addition be free of parametric aumption. The experiment by Ahn et al. and Boaert et al. do not fit the etup in our paper becaue they aume that the probability of one tate i known. In an extenion of our reult to a generalization of SEU (ee Appendix B), we how how a verion of SARSEU characterize expected utility when the probabilitie of ome tate are objective and known. Hence the reult in our paper are readily applicable to the data from Ahn et al. and Boaert et al. We dicu thi application further in Appendix B. SARSEU i not only ueful to teting SEU with exiting experimental data, but it alo guide the deign of new experiment. In particular, SARSEU ugget how one hould chooe the parameter of the deign (price and budget) o a to evaluate SEU. For example, in a etting with two tate, one could chooe each of the configuration decribed in Section 3.1 to evaluate where violation of SEU come from: tate-dependent utility or probabilitic ophitication. Related literature. The cloet precedent to our paper i the important work of Eptein (2000). Eptein etup i the ame a our; in particular, he aume data on 2

5 tate-contingent aet purchae, and that probabilitie are ubjective and unoberved but table. We differ in that he focue attention on pure probabilitic ophitication (with no aumption on rik averion), while our paper i on rik-avere SEU. Eptein preent a neceary condition for market behavior to be conitent with probabilitic ophitication. Given that the model of probabilitic ophitication i more general than SEU, one expect that the two axiom may be related: Indeed we how in Section 5 that Eptein neceary condition can be obtained a a pecial cae of SARSEU. We alo preent an example of data that are conitent with a rik avere probabilitically ophiticated agent, but that violate SARSEU. Polion and Quah (2013) develop tet for model of deciion under rik and uncertainty, including SEU (without the requirement of rik averion). They develop a general approach by which teting a model amount to olving a ytem of (nonlinear) Afriat inequalitie. See alo Bayer et al. (2012), who tudy different model of ambiguity by way of Afriat inequalitie. Non-linear Afriat inequalitie can be problematic becaue there i no known efficient algorithm for deciding if they have a olution. Another train of related work deal with objective expected utility, auming obervable probabilitie. The paper by Green and Srivatava (1986), Varian (1983), Varian (1988), and Kubler et al. (2014) characterize the dataet that are conitent with objective expected utility theory. Dataet in thee paper are jut like our, but with the added information of probabilitie over tate. Green and Srivatava allow for the conumption of many good in each tate, while we focu on monetary payoff. Varian and Green and Srivatava characterization i in the form of Afriat inequalitie; Kubler et. al. improve on thee by preenting a revealed preference axiom. We dicu the relation between their axiom and SARSEU in the online appendix. The yntax of SARSEU i imilar to the main axiom in Fudenberg et al. (2014), and in other work on additively eparable utility. 2. Subjective Expected Utility Let S be a finite et of tate. We occaionally ue S to denote the number S of tate. Let ++ = {µ R S ++ S =1 µ = 1} denote the et of trictly poitive probability meaure on S. In our model, the object of choice are tate-contingent monetary payoff, or monetary act. A monetary act i a vector in R S +. We ue the following notational convention: For vector x, y R n, x y mean that 3

6 x i y i for all i = 1,..., n; x < y mean that x y and x y; and x y mean that x i < y i for all i = 1,..., n. The et of all x R n with 0 x i denoted by R n + and the et of all x R n with 0 x i denoted by R n ++. Definition 1. A dataet i a finite collection of pair (x, p) R S + R S ++. The interpretation of a dataet (x k, p k ) K k=1 i that it decribe K purchae of a tatecontingent payoff x k at ome given vector of price p k, and income p k x k. A ubjective expected utility (SEU) model i pecified by a ubjective probability µ ++ and a utility function over money u : R + R. An SEU maximizing agent olve the problem max x B(p,I) µ u(x ) (1) S when faced with price p R S ++ and income I > 0. The et B(p, I) = {y R S + : p y I} i the budget et defined by p and I. A dataet i our notion of obervable behavior. The meaning of SEU a an aumption, i the behavior that are a if they were generated by an SEU maximizing agent. We call uch behavior SEU rational. Definition 2. A dataet (x k, p k ) K k=1 i ubjective expected utility rational (SEU rational) if there i µ ++ and a concave and trictly increaing function u : R + R uch that, for all k, y B(p k, p k x k ) S µ u(y ) S µ u(x k ). Three remark are in order. Firtly, we retrict attention to concave (i.e., rik-avere) utility, and our reult will have nothing to ay about the non-concave cae. In econd place, we aume that the relevant budget for the kth obervation i B(p k, p k x k ). Implicit i the aumption that p k x k i the relevant income for thi problem. Thi aumption i omewhat unavoidable, and tandard procedure in revealed preference theory. Thirdly, we hould emphaize that there i in our model only one good (which we think of a money) in each tate. The problem with many good i intereting, but beyond the method developed in the preent paper (ee Remark 4). 3. A Characterization of SEU Rational Data In thi ection we introduce the axiom for SEU rationality and tate our main reult. We tart by deriving, or calculating, the axiom in a pecific intance. In thi derivation, 4

7 we aume (for eae of expoition) that u i differentiable. In general, however, an SEU rational dataet may not be rationalizable uing a differentiable u; ee Remark 3 below. The firt-order condition for SEU maximization (1) are: µ u (x ) = λp. (2) The firt-order condition involve three unobervable: ubjective probability µ, marginal utilitie u (x ) and Lagrange multiplier λ The 2 2 cae: K = 2 and S = 2 We illutrate our analyi with a dicuion of the 2 2 cae, the cae when there are two tate and two obervation. In the 2 2 cae we can eaily ee that SEU ha two kind of implication, and, a we explain in Section 4 and 5, each kind i derived from a different qualitative feature of SEU. Let u impoe the firt-order condition (2) on a dataet. Let (x k 1, p k 1 ), (x k 2, p k 2 ) be a dataet with K = 2 and S = 2. For the dataet to be SEU rational there mut exit µ ++, (λ k ) k=k1,k 2 and a concave function u uch that each obervation in the dataet atifie the firt order condition (2). That i, for =,, and k = k 1, k 2. µ u (x k ) = λ k p k, (3) Equation (3) involve the oberved x and p, a well a the unobervable u, λ, and µ. One i free to chooe (ubject to ome contraint) the unobervable to atify Equation (3). We can undertand the implication of Equation (3) by conidering ituation in which the unobervable λ and µ cancel out: u (x k 1 ) u (x k 2 ) u (x k 2 ) u (x k 1 ) = µ 1u (x k1 µ 1 u (x k 2 ) ) µ 2 u (x k 2 ) µ 2 u (x k 1 ) = λk1 p k1 λ k2 p k2 λ k 2 p k 2 λ k 1 p k 1 = pk1 p k 2 p k2 p k 1 (4) Equation (4) i obtained by dividing firt order condition to eliminate term involving µ and λ: thi allow u to contrain the obervable variable, x and p. There are two ituation of interet. Suppoe firt that x k 1 > x k 2 and that x k 2 > x k 1. The concavity of u implie then that u (x k 1 ) u (x k 2 ) and u (x k 2 ) u (x k 1 ). Thi mean that the left hand ide of Equation (4) i maller than 1. Thu: x k 1 > x k 2 and x k 2 > x k 1 5 pk 1 p k2 p k 2 p k 1 1. (5)

8 x k 2 In econd place, uppoe that x k 1 > x k 1 while x k 2 > x k 2 (o the bundle x k 1 and are on oppoite ide of the 45 degree line in R 2 ). The concavity of u implie that u (x k 1 ) u (x k 1 ) and u (x k 2 ) u (x k 2 ). The far-left of Equation (4) i then maller than 1. Thu: x k 1 > x k 1 and x k 2 > x k 2 pk 1 p k2 p k 1 p k 2 1. (6) Requirement (5) and (6) are implication of rik-avere SEU for a dataet when S = 2 and K = 2. We hall ee that they are all the implication of rik-avere SEU in thi cae, and that they capture ditinct qualitative component of SEU (Section 4 and 5) General K and S We now turn to the general etup, and to our main reult. Firt, we hall derive the axiom by proceeding along the line uggeted above in Section 3.1: Uing the firt-order condition (2), the SEU-rationality of a dataet require that u (x k ) u (x k ) = µ λ k p k µ λ k The concavity of u implie omething about the left-hand ide of thi equation when x k > xk, but the right-hand ide i complicated by the preence of unobervable Lagrange multiplier and ubjective probabilitie. So we chooe pair (x k, x k ) with xk > x k uch that ubjective probabilitie and Lagrange multiplier cancel out. For example, conider p k x k 1 > x k 2, x k 3 > x k 1 3, and x k 2 3 > x k 3. By manipulating the firt-order condition we obtain that: u (x k ( 1 ) u (x k 2 ) u (x k3 ) u (x k 1 3 ) u (x k2 3 ) u (x k 3 ) = µ2 λ k1 p k ) ( 1 µ3 1 λ k3 p k ) 3 2 µ 1 λ k 2 p k 2 µ 2 λ k 1 p k 1 3. ( µ1 λ k2 µ 3 λ k 3 p k ) 2 3 p k 3 = pk 1 p k 2 p k3 p k2 3 p k 1 3 p k 3 Notice that the pair (x k 1, x k 2 ), (x k 3, x k 1 3 ), and (x k 2 3, x k 3 ) have been choen o that the ubjective probabilitie µ appear in the nominator a many time a in the denominator, and the ame for λ k ; hence thee term cancel out. condition (2) and (3) in the axiom below. Such canceling out motivate Now the concavity of u and the aumption that x k 1 > x k 2, x k 3 > x k 1 3, and x k 2 3 > x k 3 imply that the product of the price pk 1 p p p k 2 p k 1 3 p k 3 implication of SEU on price, an obervable entity. k 3 6 k 2 3 cannot exceed 1. Thu, we obtain an

9 In general, the aumption of SEU rationality require that, for any collection of equence a above, appropriately choen o that ubjective probabilitie and Lagrange multiplier will cancel out, the product of the ratio of price cannot exceed 1. Formally: Strong Axiom of Revealed Subjective Utility (SARSEU): For any equence of pair (x k i i, x k i i) n i=1 in which 1. x k i i > x k i i for all i; 2. each appear a i (on the left of the pair) the ame number of time it appear a i (on the right); 3. each k appear a k i (on the left of the pair) the ame number of time it appear a k i (on the right): The product of price atifie that n i=1 p k i i p k i i 1. Theorem 1. A dataet i SEU rational if and only if it atifie SARSEU. It i worth noting that the yntax of SARSEU i imilar to that of the main axiom in Kubler et al. (2014), with rik-neutral price playing the role of price in the model with objective probabilitie. The relation between the two i dicued further in Appendix B. We conclude the ection with ome remark on Theorem 1. The proof i in Section A. Remark 1. In the 2 2 cae of Section 3.1, Requirement (5) and (6) are equivalent to SARSEU. Remark 2. The proof of Theorem 1 i in Section A. It relie on etting up a ytem of inequalitie from the firt-order condition of an SEU agent maximization problem. Thi i imilar to the approach in Afriat (1967), and in many other ubequent tudie of revealed preference. The difference i that our ytem i nonlinear, and mut be linearized. A crucial tep in the proof i an approximation reult, which i complicated by the fact that the unknown ubjective probabilitie, Lagrange multiplier, and marginal utilitie, all take value in non-compact et. Remark 3. Under the following aumption on the dataet: x k x k if (k, ) (k, ), 7

10 SARSEU implie SEU rationality uing a mooth rationalizing u. Thi condition on the dataet play a imilar role to the aumption ued by Chiappori and Rochet (1987) to obtain a mooth utility uing the Afriat contruction. Remark 4. In our framework we aume choice of monetary act, which mean that conumption in each tate i one-dimenional. Our reult are not eaily applicable to the multidimenional etting, eentially becaue concavity i in general equivalent to cyclic monotonicity of upergradient, which we cannot deal with in our approach. In the one-dimenional cae, concavity require only that upergradient are monotone. The condition that ome unknown function i monotone i preerved by a monotonic tranformation of the function, but thi i not true of cyclic monotonicity. If one et up the multidimenional problem a we have done, then one loe the property of cyclic monotonicity when linearizing the ytem. Finally, it i not obviou from the yntax of SARSEU that one can verify whether a particular dataet atifie SARSEU in finitely many tep. We how that, not only i SARSEU decidable in finitely many tep, but there i in fact an efficient algorithm that decide whether a dataet atifie SARSEU. Propoition 2. There i an efficient algorithm that decide whether a dataet atifie SARSEU. 1 We provide a direct proof of Propoition 2 in Section C. Propoition 2 can alo be een a a reult of Lemma 7 together with the linearization in the proof of Theorem 1. The reulting linear ytem can be decided by uing linear programming. 4. State-dependent Utility SEU aert, among other thing, the exitence of a (concave) tate-dependent utility (i.e., an additively eparable utility acro tate). SEU require more, of coure, but it i intereting to compare SEU with the weaker theory of tate-dependent utility. We hall trace the aumption of tate-dependent utility to a particular weakening on SARSEU. The tate-dependent utility model ay that an agent maximize U d (x) = S u (x ), an additively eparable utility, for ome collection (u ) S of concave and trictly increaing tate-dependent function, u : R + R. We hould emphaize that here, a in the ret of the paper, we retrict attention to concave utility. 1 Efficient mean that the algorithm run in polynomial time. 8

11 x 2 x 2 x k 2 x k 2 x k 1 x k 1 x 1 x 1 Figure 1: A violation of Requirement (5) 4.1. The 2 2 cae: K = 2 and S = 2 We argued in Section 3.1 that requirement (5) and (6) are neceary for SEU rationality. It turn out that (5) alone capture tate-dependent utility. A dataet that violate Requirement (5) can be viualized on the left of Figure 1. The figure depict choice with x k 1 > x k 2 and x k 2 > x k 2, but where Requirement (5) i violated. Figure 1 preent a geometrical argument for why uch a dataet i not SEU rational. Suppoe, toward a contradiction, that the dataet i SEU rational. Since the rationalizing function u i concave, it i eay to ee that optimal choice mut be increaing in the level of income (the demand function of a rik-avere SEU agent i normal). At the right of Figure 1, we include a budget with the ame relative price a when x k 2 wa choen, but where the income i larger. The larger income i uch that the budget line pae through x k 1. Since her demand i normal, the agent choice on the larger (green) budget line mut be larger than at x k 2. The choice mut lie in the line egment on the green budget line that conit of bundle larger than x k 2. But uch a choice would violate the weak axiom of revealed preference (WARP). Hence the (counterfactual) choice implied by SEU at the green budget line would be inconitent with utility maximization, contradicting the aumption of SEU rationality. It i ueful to emphaize that Requirement (5) i a trengthening of WARP, omething we hall return to below. So SEU rationality implie (5), which i a trengthening of WARP. Now we argue that tate-dependent utility implie (5) a well. To ee thi, uppoe that the agent maximize u 1 ( ) + u 2 ( ), where u i i concave. A in Section 3.1, aume that u i i differentiable. 9

12 Then, x k 1 > x k 2 and x k 2 > x k 1 imply that u (x k 1 ) u (x k 2 ) and u (x k 2 ) u (x k 1 ). The firt order condition are u (x k ) = λ k p k ; hence p k 2 p k 1 Indeed, thi i Requirement (5). p k1 = u p k 2 ) (x k 2 u (x k 1 ) u (x k 1 ) u (x k 2 ) General K and S A dataet (x k, p k ) K k=1 i tate-dependent utility (SDU) rational if there i an additively eparable function U d uch that, for all k, y B(p k, p k x k ) U d (y) U d (x k ). In the 2 2 cae, we have een that Requirement (5) i neceary for rationalization by a tate-dependent utility. More generally, there i a natural weakening of SARSEU that capture rationalization by a tate-dependent utility. Thi weakening i trong enough to imply WARP. Concretely, if one ubtitute condition (2) in SARSEU with the tatement i = i the reulting axiom characterize SDU rationality: Strong Axiom of Revealed State-dependent Utility (SARSDU): For any equence of pair (x k i i, x k i i) n i=1 in which 1. x k i i > x k i i 2. i = i. for all i; 3. each k appear a k i (on the left of the pair) the ame number of time it appear a k i (on the right): The product of price atifie that n i=1 p k i i p k i i 1. It hould be obviou that SARSEU implie SARSDU. SARSDU i equivalent to SDU rationality. Theorem 3. A dataet i SDU rational if and only if it atifie SARSDU. 10

13 The proof that SARSDU i neceary for tate-dependent utility i imple, and follow along the line developed in Section 3. The proof of ufficiency i imilar to the proof ued for the characterization of SEU, and i omitted. Note that, in the 2 2 cae, SARSDU and Requirement (5) are equivalent. Hence, (5) characterize tate-dependent utility in the 2 2 cae. By the theorem above, we know that SARSDU implie the weak axiom of revealed preference (WARP), but it may be ueful to preent a direct proof of the fact that SARSDU implie WARP. Definition 3. A dataet (x k, p k ) K k=1 atifie WARP if there i no k and k uch that p k x k p k x k and p k x k > p k x k. Propoition 4. If a dataet atifie SARSDU, then it atifie WARP. Proof. Suppoe, toward a contradiction, that a dataet (x k, p k ) K k=1 atifie SARSDU but that it violate WARP. Then there are k and k uch that p k x k p k x k and p k x k > p k x k. It cannot be the cae that x k x k for all, o the et S 1 = { : x k < x k } i nonempty. Chooe S 1 uch that p k p k pk p k for all S 1. Now, p k x k p k x k implie that (x k 1 xk ) p k p (x k k x k ). We alo have that p k x k > p k x k, o 0 > p k (x k x k ) + p k (xk xk ) p k (x k x k ) + pk pk p k pk = p k (1 pk / S 1 p k )(x k x k p k (x k x k ) ) } {{ } A + S 1 \{ } p k pk p k pk (1 pk )(x k x k ) } {{ } B We hall prove that A 0 and that B 0, which will yield the deired contradiction.. 11

14 a x k For all / S 1 we have that (x k x k ) 0. Then SARSDU implie that < xk SARSDU. Hence A 0. p k pk p k pk 1, o that the equence {(xk, xk ), (xk, x k )} atifie (1), (2), and (3) in Now conider B. By definition of, we have that pk (x k x k ) < 0 implie that for all S 1. Hence B 0. ) (1 pk pk (x k p k x k ) 0, pk p k p k p k 1 for all S 1. Then, We make ue of thee reult in the online appendix, where we how how SARSDU and SARSEU rule out violation of Savage axiom. The online appendix alo include a condition on the data under which SEU and SDU are obervationally equivalent. 5. Probabilitic Sophitication We have looked at the apect of SARSEU that capture the exitence of an additively eparable repreentation. SEU alo affirm the exitence of a unique ubjective probability meaure guiding the agent choice. We now turn to the behavioral counterpart of the exitence of uch a probability. We do not have a characterization of probability ophitication. In thi ection, we imply oberve how SARSEU and Requirement (6) are related to the exitence of a ubjective probability. 2 We alo how how SARSEU i related to Eptein neceary condition for probability ophitication The 2 2 cae: K = 2 and S = 2 Conider, a before, the 2 2 cae: S = 2 and K = 2. We argued in Section 3.1 that Requirement (5) and (6) are neceary for SEU rationality, and in Section 4 that Requirement (5) capture the exitence of a tate-dependent repreentation. We now how that Requirement (6) reult from impoing a unique ubjective probability guiding the agent choice. 2 In the online appendix, we alo how that Requirement (6) rule out violation of Savage P4, which capture the exitence of a ubjective probability (Machina and Schmeidler (1992)). 12

15 x 2 x 2 x k 2 x k 2 x k 1 x k 1 x 1 x 1 Figure 2: Violation of Requirement (6) Figure 2 exhibit a dataet that violate Requirement (6). We have drawn the indifference curve of the agent when chooing x k 2. Recall that the marginal rate of ubtitution (MRS) i µ 1 u (x 1 )/µ 2 u (x 2 ). At the point where the indifference curve croe the 45 degree line (dotted), one can read the agent ubjective probability off the indifference curve becaue u (x 1 )/u (x 2 ) = 1, and therefore the MRS equal µ 1 /µ 2. So the tangent line to the indifference curve at the 45 degree line decribe the ubjective probability. It i then clear that thi tangent line (depicted in green in the figure) mut be flatter than the budget line at which x k 2 wa choen. On the other hand, the ame reaoning reveal that the ubjective probability mut define a teeper line than the budget line at which x k 1 wa choen. Thi i a contradiction, a the latter budget line i teeper than the former General K and S In the following, we focu intead on the relation with probabilitic ophitication, namely the relation between SARSEU and the axiom in Eptein (2000). Eptein tudie the implication of probabilitic ophitication for conumption dataet. He conider the ame kind of economic environment a we do, and the ame notion of a dataet. He focue on probabilitic ophitication intead of SEU, and importantly doe not aume rik averion. Eptein how that a dataet i inconitent with probabilitic ophitication if there exit, t S and k, ˆk K uch that (i) p k p k t and pˆk pˆk t, with at leat one trict inequality; and (ii) x k > x k t and xˆk < xˆk t. Of coure, an SEU rational agent i probabilitically ophiticated. Indeed, our next reult etablihe that a violation of Eptein condition implie a violation of SARSEU. 13

16 x 2 x 2 p k 2 p k 1 x k 1 x k 2 x 1 x 1 Figure 3: Probabilitically ophiticated violation of SARSEU Propoition 5. If a dataet (x k, p k ) K k=1 hold for ome, t S, k, ˆk K atifie SARSEU, then (i) and (ii) cannot both Proof. Suppoe that, t S, k, ˆk K are uch that (ii) hold. Then {(x k, x k t ), (xˆk t, xˆk ))} atifie the condition in SARSEU. Hence, SARSEU require that pk pˆk t p k t pˆk p k p k t or pˆk pˆk t. Hence, (i) i violated. 1, o that Propoition 5 raie the iue of whether SEU and probabilitic ophitication are ditinguihable. In the following, we how that we can indeed ditinguih the two model: We preent an example of a dataet that violate SARSEU, but that i conitent with a rik-avere probabilitically ophiticated agent. Hence the weakening in going from SEU to probabilitic ophitication ha empirical content. Let S = {, }. We define a dataet a follow. Let x k 1 = (2, 2), p k 1 = (1, 2), x k 2 = (8, 0), and p k 2 = (1, 1). It i clear that the dataet violate SARSEU: x k 2 > x k 1 and x k 1 > x k 2 while p k 2 p k1 = 2 > 1. p k 1 p k 2 Oberve, moreover, that the dataet pecifically violate Requirement (5), not (6). Figure 3 illutrate the dataet. We hall now argue that thi dataet i rationalizable by a rik-avere probabilitically ophiticated agent. Fix µ ++ with µ 1 = µ 2 = 1/2, a uniform probability over S. Any 14

17 vector x R 2 + induce the probability ditribution on R + given by x 1 with probability 1/2 and x 2 with probability 1/2. Let Π be the et of all uniform probability meaure on R + with upport having cardinality maller than or equal to 2. We hall define a function V : Π R that repreent a probabilitically ophiticated preference, and for which the choice in the dataet are optimal. 3 We contruct a monotone increaing and quaiconcave h : R 2 + R +, and then define V (π) = h( x π, x π ), where x π i the mallet point in the upport of π, and x π i the larget. A a conequence of the monotonicity of h, V repreent a probabilitically ophiticated preference. The preference i alo rik-avere. The function h i contructed o that (x 1, x 2 ) h(max{x 1, x 2 }, min{x 1, x 2 }) ha the map of indifference curve illutrated on the left in Figure 3. There are two important feature of the indifference curve drawn in the figure. The firt i that indifference curve exhibit a convex preference, which enure that the agent will be rik-avere. The econd i that indifference curve become le convex a one move up and to the right in the figure. A a reult, the line that i normal to p k 1 upport the indifference curve through x k 1, while the line that i normal to p k 2 upport the indifference curve through x k 2. It i clear from Figure 3 that the contruction rationalize the choice in the dataet. 6. Maxmin Expected Utility In thi ection, we demontrate one ue of our main reult to tudy the difference between SEU and maxmin expected utility. We how that SEU and maxmin are behaviorally inditinguihable in the 2 2 cae, but ditinguihable more generally. The maxmin SEU model, firt axiomatized by Gilboa and Schmeidler (1989), poit that an agent maximize min µ M µ u(x ), S where M i a cloed and convex et of probabilitie. A dataet (x k, p k ) K k=1 i maxmin expected utility rational if there i a cloed and convex et M ++ and a concave and trictly increaing function u : R + R uch that, for all k, y B(p k, p k x k ) min µ u(y ) min µ u(x k ). µ M µ M S 3 It i eay to how that there i a probabilitically ophiticated weak order defined on the et of all probability meaure on R 2 + with finite upport, uch that V repreent on Π. The detail of the example are technical and left to the online appendix. 15 S

18 Note that we retrict attention to rik-avere maxmin expected utility. Propoition 6. Let S = K = 2. Then a dataet i maxmin expected utility rational if and only if it i SEU rational. The proof of Propoition 6 i in the online appendix. The reult in Propoition 6 doe not, however, extend beyond the cae of two obervation. In the online appendix, we provide an example of data from a (rik-avere) maxmin expected utility agent that violate SARSEU. A. Proof of Theorem 1 We firt give three preliminary and auxiliary reult. Lemma 7 provide nonlinear Afriat inequalitie for the problem at hand. A verion of thi lemma appear, for example, in Green and Srivatava (1986), Varian (1983), or Bayer et al. (2012). Lemma 8 and 9 are verion of the theorem of the alternative. Lemma 7. Let (x k, p k ) K k=1 be a dataet. The following tatement are equivalent: 1. (x k, p k ) K k=1 i SEU rational. 2. There are trictly poitive number v k, λ k, µ, for = 1,..., S and k = 1,..., K, uch that µ v k = λ k p k, x k > x k vk v k. (7) Proof. We hall prove that (1) implie (2). Let (x k, p k ) K k=1 be SEU rational. Let µ RS ++ and u : R + R be a in the definition of SEU rational dataet. Then (ee, for example, Theorem 28.3 of Rockafellar (1997)), there are number λ k 0, k = 1,..., K uch that if we let v k = λk p k µ then v k u(x k ) if x k > 0, and there i w u(x k ) with v k w if x k = 0 (here we have u(0) ). In fact, ince u i trictly increaing, it i eay to ee that λ k > 0, and therefore v k > 0. By the concavity of u, and the conequent monotonicity of u(x k ) (ee Theorem 24.8 of Rockafellar (1997)), if x k > x k > 0, vk u(x k ), and v k x k > x k = 0, then w u(xk ) with vk w. So vk w v k 16 u(xk ), then vk v k. If.

19 In econd place, we how that (2) implie (1). Suppoe that the number v k, λ k, µ, for = 1,..., S and k = 1,..., K are a in (2). Enumerate the element of X in increaing order: y 1 < y 2 <... < y n. Let y i = min{v k : x k = y i } and ȳ i = max{v k : x k = y i }. Let z i = (y i + y i+1 )/2, i = 1,..., n 1; z 0 = 0, and z n = y n + 1. Let f be a correpondence defined a follow: [y i, ȳ i ] if z = y i, f(z) = max{ȳ i : z < y i } if y n > z and i(z y i ), (8) y n /2 if y n < z. Then, by the aumption placed on v k, and by contruction of f, y < y, v f(y) and v f(y ) imply that v v. Then the correpondence f i monotone, and there exit a concave function u for which u = f (ee e.g. Theorem 24.8 of Rockafellar (1997)). Given that v k > 0 all the element in the range of f are poitive, and therefore u i a trictly increaing function. Finally, for all (k, ), λ k p k /µ = v k u(v k ) and therefore the firt-order condition to a maximum choice of x hold at x k. Since u i concave the firt-order condition are ufficient. The dataet i therefore SEU rational. We hall ue the following lemma, which i a verion of the Theorem of the Alternative. Thi i Theorem in Stoer and Witzgall (1970). We hall ue it here in the cae where F i either the real or the rational number field. Lemma 8. Let A be an m n matrix, B be an l n matrix, and E be an r n matrix. Suppoe that the entrie of the matrice A, B, and E belong to a commutative ordered field F. Exactly one of the following alternative i true. 1. There i u F n uch that A u = 0, B u 0, E u There i θ F r, η F l, and π F m uch that θ A + η B + π E = 0; π > 0 and η 0. The next lemma i a direct conequence of Lemma 8: ee Lemma 12 in Chamber and Echenique (2014) for a proof. Lemma 9. Let A be an m n matrix, B be an l n matrix, and E be an r n matrix. Suppoe that the entrie of the matrice A, B, and E are rational number. Exactly one of the following alternative i true. 17

20 1. There i u R n uch that A u = 0, B u 0, and E u There i θ Q r, η Q l, and π Q m uch that θ A + η B + π E = 0; π > 0 and η 0. A.1. Neceity Lemma 10. If a dataet (x k, p k ) K k=1 i SEU rational, then it atifie SARSEU. Proof. Let (x k, p k ) K k=1 be SEU rational, and let µ ++ and u : R + R be a in the definition of SEU rational. By Lemma 7, there exit a trictly poitive olution v k, λ k, µ to the ytem in Statement (2) of Lemma 7 with v k u(x k ) when x k > 0, and v k w u(x k ) when x k = 0. Let (x k i i, x k i) n i=1 be a equence atifying the three condition in SARSEU. Then x k i i > i x k i. Suppoe that x k i i > 0. Then, v k i i i u(x k i i ) and v k i u(x k i i By the concavity of u, i). it follow that λ k i µ i p k i i λ k i µi p k i (ee Theorem 24.8 of Rockafellar (1997)). Similarly, i if x k i i = 0, then v k i i u(x k i i ) and v k i i w u(x k i ). So λ k i µ i i p k i i λ k i µi p k i i Therefore, 1 n i=1 λ k i µ i p k i i λ k iµ i p k i i = n i=1 p k i i p k i i, a the equence atifie Condition (2) and (3) of SARSEU; and hence the number λ k and µ appear the ame number of time in the denominator a in the numerator of thi product. A.2. Sufficiency We proceed to prove the ufficiency direction. An outline of the argument i a follow. We know from Lemma 7 that it uffice to find a olution to the Afriat inequalitie (actually firt order condition), written a tatement (2) in the lemma. So we et up the problem to find a olution to a ytem of linear inequalitie obtained from uing logarithm to linearize the Afriat inequalitie in Lemma 7. Lemma 11 etablihe that SARSEU i ufficient for SEU rationality when the logarithm of the price are rational number. The role of rational logarithm come from our ue of a verion the theorem of the alternative (ee Lemma 9): when there i no olution to the linearized Afriat inequalitie, then the exitence of a rational olution to the dual 18

21 ytem of inequalitie implie a violation of SARSEU. The bulk of the proof goe into contructing a violation of SARSEU from a given olution to the dual. The next tep in the proof (Lemma 12) etablihe that we can approximate any dataet atifying SARSEU with a dataet for which the logarithm of price are rational, and for which SARSEU i atified. Thi tep i crucial, and omewhat delicate. One might have tried to obtain a olution to the Afriat inequalitie for perturbed ytem (with price that are rational after taking log), and then conidered the limit. Thi doe not work becaue the olution to our ytem of inequalitie are in a non-compact pace. It i not clear how to etablih that the limit exit and are well-behaved. Lemma 12 avoid the problem. Finally, Lemma 13 etablihe the reult by uing another verion of the theorem of the alternative, tated a Lemma 8 above. The tatement of the lemma follow. The ret of the paper i devoted to the proof of thee lemma. Lemma 11. Let dataet (x k, p k ) k k=1 atify SARSEU. Suppoe that log(pk ) Q for all k and. Then there are number v k, λ k, µ, for = 1,..., S and k = 1,..., K atifying (7) in Lemma 7. Lemma 12. Let dataet (x k, p k ) k k=1 atify SARSEU. Then for all poitive number ε, there exit q k [p k ε, p k ] for all S and k K uch that log q k Q and the dataet (x k, q k ) k k=1 atify SARSEU. Lemma 13. Let dataet (x k, p k ) k k=1 atify SARSEU. Then there are number vk, λ k, µ, for = 1,..., S and k = 1,..., K atifying (7) in Lemma 7. A.2.1. Proof of Lemma 11 We linearize the equation in Sytem (7) of Lemma 7. The reult i: log v k + log µ log λ k log p k = 0, (9) x k > x k log vk log v k. (10) In the ytem compried by (9) and (10), the unknown are the real number log v k, log µ, log λ k, for k = 1,..., K and = 1,..., S. Firt, we are going to write the ytem of inequalitie (9) and (10) in matrix form. 19

22 We hall define a matrix A uch that there are poitive number v k, λ k, µ the log of which atify Equation (9) if and only if there i a olution u R K S+K+S+1 to the ytem of equation A u = 0, and for which the lat component of u i trictly poitive. Let A be a matrix with K S row and K S +S +K +1 column, defined a follow: We have one row for every pair (k, ); one column for every pair (k, ); one column for every ; one column for each k; and one lat column. In the row correponding to (k, ) the matrix ha zeroe everywhere with the following exception: it ha a 1 in the column for (k, ); it ha a 1 in the column for ; it ha a 1 in the column for k; and log p k in the very lat column. Matrix A look a follow: (1,1) (k,) (K,S) 1 S 1 k K p (1,1) log p (k,) log p k (K,S) log p K S Conider the ytem A u = 0. If there are number olving Equation (9), then thee define a olution u R K S+S+K+1 for which the lat component i. If, on the other hand, there i a olution u R K S+S+K+1 to the ytem A u = 0 in which the lat component (u K S+S+K+1 ) i trictly poitive, then by dividing through by the lat component of u we obtain number that olve Equation (9). In econd place, we write the ytem of inequalitie (10) in matrix form. There i one row in B for each pair (k, ) and (k, ) for which x k > x k. In the row correponding to x k > x k we have zeroe everywhere with the exception of a 1 in the column for (k, ) and a 1 in the column for (k, ). Let B be the number of row of B. In third place, we have a matrix E that capture the requirement that the lat component of a olution be trictly poitive. The matrix E ha a ingle row and K S+S+K +1 column. It ha zeroe everywhere except for 1 in the lat column. To um up, there i a olution to ytem (9) and (10) if and only if there i a vector 20

23 u R K S+S+K+1 that olve the following ytem of equation and linear inequalitie A u = 0, S1 : B u 0, E u 0. Note that E u i a calar, o the lat inequality i the ame a E u > 0. The entrie of A, B, and E are either 0, 1 or 1, with the exception of the lat column of A. Under the hypothee of the lemma we are proving, the lat column conit of rational number. By Lemma 9, then, there i uch a olution u to S1 if and only if there i no vector (θ, η, π) Q K S+B+1 that olve the ytem of equation and linear inequalitie S2 : θ A + η B + π E = 0, η 0, π > 0. In the following, we hall prove that the non-exitence of a olution u implie that the dataet mut violate SARSEU. Suppoe then that there i no olution u and let (θ, η, π) be a rational vector a above, olving ytem S2. By multiplying (θ, η, π) by any poitive integer we obtain new vector that olve S2, o we can take (θ, η, π) to be integer vector. Henceforth, we ue the following notational convention: For a matrix D with K S + S + K + 1 column, write D 1 for the ubmatrix of D correponding to the firt K S column; let D 2 be the ubmatrix correponding to the following S column; D 3 correpond to the next K column; and D 4 to the lat column. Thu, D = [D 1 D 2 D 3 D 4 ]. Claim 14. (i) θ A 1 +η B 1 = 0; (ii) θ A 2 = 0; (iii) θ A 3 = 0; and (iv) θ A 4 +π E 4 = 0. Proof. Since θ A + η B + π E = 0, then θ A i + η B i + π E i = 0 for all i = 1,..., 4. Moreover, ince B 2, B 3, B 4, E 1, E 2, and E 3 are zero matrice, we obtain the claim. We tranform the matrice A and B uing θ and η. Define a matrix A from A by letting A have the ame number of column a A and including: (i) θ r copie of the rth row when θ r > 0; (ii) omitting row r when θ r = 0; and (ii) θ r copie of the rth row multiplied by 1 when θ r < 0. We refer to row that are copie of ome r with θ r > 0 a original row, and to thoe that are copie of ome r with θ r < 0 a converted row. 21

24 Similarly, we define the matrix B from B by including the ame column a B and η r copie of each row (and thu omitting row r when η r = 0; recall that η r 0 for all r). Claim 15. For any (k, ), all the entrie in the column for (k, ) in A 1 are of the ame ign. Proof. By definition of A, the column for (k, ) will have zero in all it entrie with the exception of the row for (k, ). In A, for each (k, ), there are three mutually excluive poibilitie: the row for (k, ) in A can (i) not appear in A, (ii) it can appear a original, or (iii) it can appear a converted. Thi how the claim. Claim 16. There exit a equence (x k i i, x k i ) n i=1 of pair that atifie Condition (1) in i SARSEU. Proof. Define X = {x k k K, S}. We define uch a equence by induction. Let B 1 = B. Given B i, define B i+1 a follow: Denote by > i the binary relation on X defined by z > i z if z > z and there i at leat one copy of a row correponding to z > z in B i : there i at leat one pair (k, ) and (k, ) for which (1) x k > x k, (2) z = xk and z = x k, and (3) the row correponding to x k > x k in B had trictly poitive weight in η. The binary relation > i cannot exhibit cycle becaue > i >. There i therefore at leat one equence z1, i... zl i i in X uch that zj i > i zj+1 i for all j = 1,..., L i 1 and with the property that there i no z X with z > i z1 i or zl i i > i z. Oberve that B i ha at leat one row correponding to z i j > i z i j+1, for all j = 1,..., L i 1. Let the matrix B i+1 be defined a the matrix obtained from B i by omitting one copy of a row correponding to z i j > z i j+1, for each j = 1,... L i 1 The matrix B i+1 ha trictly fewer row than B i. There i therefore n for which B n +1 would have no row. The matrix B n B n will remove all row of B n. Define a equence (x k i i, x k i i ha row, and the procedure of omitting row from ) n i=1 of pair by letting x k i i = z1 i and x k i = z L i i i. Note that, a a reult, x k i i > x k i for all i. Therefore the equence (x k i i i, x k i ) n i=1 of pair atifie Condition i (1) in SARSEU. We hall ue the equence (x k i i, x k i ) n i=1 of pair a our candidate violation of SARSEU. i Conider a equence [ of matrice ] A i, i = 1,..., n defined a follow. Let A 1 = A, B 1 = B, and C 1 = Claim 14. A 1 B 1. Oberve that the row of C 1 add to the null vector by 22

25 We hall [ proceed ] by induction. Suppoe that A i ha been defined, and that the row of C i = A i B i add to the null vector. Recall the definition of the equence x k i i = z i 1 >... > z i L i = x k i i. There i no z X with z > i z i 1 or z i L i > i z, o in order for the row of C i to add to zero there mut be a 1 in A i 1 in the column correponding to (k i, i) and a 1 in A i 1 in the column correponding to (k i, i ). Let r i be a row in A i correponding to (k i, i ), and r i be a row correponding to (k i, i). The exitence of a 1 in A i 1 in the column correponding to (k i, i), and a 1 in A i 1 in the column correponding to (k i, i ), enure that r i and r i exit. Note that the row r i i a converted row while r i i original. Let A i+1 be defined from A i by deleting the two row, r i and r i. Claim 17. The um of r i, r i, and the row of B i which are deleted when forming B i+1 (correponding to the pair z i j > z i j+1, j = 1,..., L i 1) add to the null vector. Proof. Recall that z i j > i z i j+1 for all j = 1,..., L i 1. So when we add row correponding to z i j > i z i j+1 and z i j+1 > i z i j+2, then the entrie in the column for (k, ) with x k = z i j+1 cancel out and the um i zero in that entry. Thu, when we add the row of B i that are not in B i+1 we obtain a vector that i 0 everywhere except the column correponding to z i 1 and z i L i. Thi vector cancel out with r i + r i, by definition of r i and r i. Claim 18. The matrix A can be partitioned into pair of row (r i, r i), in which the row r i are converted and the row r i are original. Proof. For each i, A i+1 differ from A i in that the row r i and r i are removed from A i to form A i+1. We hall prove that A i compoed of the 2n row r i, r i. Firt note that ince the row of C i add up to the null vector, and A i+1 and B i+1 are obtained from A i and B i by removing a collection of row that add up to zero, then the row of C i+1 mut add up to zero a well. We now how that the proce top after n tep: all the row in C n are removed by the procedure decribed above. By way of contradiction, uppoe that there exit row left after removing r n and r n. Then, by the argument above, the row of the matrix C n +1 mut add to the null vector. If there are row left, then the matrix C n +1 i well defined. By definition of the equence B i, however, B n +1 i an empty matrix. Hence, row remaining in A n +1 1 mut add up to zero. By Claim 15, the entrie of a column (k, ) of A 23

26 are alway of the ame ign. Moreover, each row of A ha a non-zero element in the firt K S column. Therefore, no ubet of the column of A 1 can um to the null vector. Claim 19. (i) For any k and, if (k i, i ) = (k, ) for ome i, then the row correponding to (k, ) appear a original in A. Similarly, if (k i, i) = (k, ) for ome i, then the row correponding to (k, ) appear converted in A. (ii) If the row correponding to (k, ) appear a original in A, then there i ome i with (k i, i ) = (k, ). Similarly, if the row correponding to (k, ) appear converted in A, then there i i with (k i, i) = (k, ). Proof. (i) i true by definition of (x k i i, x k i i). (ii) i immediate from Claim 18 becaue if the row correponding to (k, ) appear original in A then it equal r i for ome i, and then (k i, i ) = (k, ). Similarly when the row appear converted. Claim 20. The equence (x k i i, x k i) n i=1 atifie Condition (2) and (3) in SARSEU. i Proof. By Claim 14 (ii), the row of A 2 add up to zero. Therefore, the number of time that appear in an original row equal the number of time that it appear in a converted row. By Claim 19, then, the number of time appear a i equal the number of time it appear a i. Therefore Condition (2) in the axiom i atified. Similarly, by Claim 14 (iii), the row of A 3 add to the null vector. Therefore, the number of time that k appear in an original row equal the number of time that it appear in a converted row. By Claim 19, then, the number of time that k appear a k i equal the number of time it appear a k i. Therefore Condition (3) in the axiom i atified. Finally, in the following, we how that n a the equence (x k i i, x k i Claim 21. n p k i i i=1 p k i i i > 1. i=1 p k i i p k i i > 1, which finihe the proof of Lemma 11 ) n i=1 would then exhibit a violation of SARSEU. Proof. By Claim 14 (iv) and the fact that the ubmatrix E 4 equal the calar 1, we obtain n 0 = θ A 4 + πe 4 = ( (r i + r i)) 4 + π, 24 i=1

27 where ( n i=1 (r i + r i)) 4 i the (calar) um of the entrie of A 4. Recall that log p k i i i the lat entry of row r i and that log p k i i the lat entry of row r i, a r i i converted and r i original. Therefore the um of the row of A 4 are n i=1 log(pk i/p k i i i ). Then, n i=1 log(pk i /p k i i i ) = π < 0. Thu n p k i i i=1 p k i i > 1. i A.2.2. Proof of Lemma 12 For each equence σ = (x k i i, x k i i) n i=1 that atifie Condition (1), (2), and (3) in SARSEU, we define a vector t σ N K2 S 2. For each pair (x k i i, x k i i), we hall identify the pair with ((k i, i ), (k i, i)). Let t σ ((k, ), (k, )) be the number of time that the pair (x k, x k ) appear in the equence σ. One can then decribe the atifaction of SARSEU by mean of the vector t σ. Define T = { t σ N K2 S 2 σ atifie Condition (1), (2), (3) in SARSEU }. Oberve that the et T depend only on (x k ) K k=1 in the dataet (xk, p k ) K k=1. It doe not depend on price. For each ((k, ), (k, )) uch that x k > x k, define ˆδ((k, ), (k, )) = log ( p k p k ). And define ˆδ((k, ), (k, )) = 0 when x k x k. Then, ˆδ i a K 2 S 2 -dimenional real-valued vector. If σ = (x k i i, x k i i) n i=1, then ( n ˆδ t σ = ˆδ((k, ), (k, ))t σ ((k, ), (k, )) = log ((k,),(k, )) (KS) 2 i=1 So the dataet atifie SARSEU if and only if ˆδ t 0 for all t T. Enumerate the element in X in increaing order: y 1 < y 2 < < y N. And fix an arbitrary ξ (0, 1). We hall contruct by induction a equence {(ε k (n))} N n=1, where ε k (n) i defined for all (k, ) with x k = y n. By the denene of the rational number, and the continuity of the exponential function, for each (k, ) uch that x k = y 1, there exit a poitive number ε k (1) uch that log(p k ε k (1)) Q and ξ < ε k (1) < 1. Let ε(1) = min{ε k (1) x k = y 1 }. In econd place, for each (k, ) uch that x k = y 2, there exit a poitive ε k (2) uch that log(p k ε k (2)) Q and ξ < ε k (2) < ε(1). Let ε(2) = min{ε k (2) x k = y 2 }. p k i i p k i i ). 25

28 In third place, and reaoning by induction, uppoe that ε(n) ha been defined and that ξ < ε(n). For each (k, ) uch that x k = y n+1, let ε k (n + 1) > 0 be uch that log(p k ε k (n + 1)) Q, and ξ < ε k (n + 1) < ε(n). Let ε(n + 1) = min{ε k (n + 1) x k = y n }. Thi define the equence (ε k (n)) by induction. Note that ε k (n + 1)/ε(n) < 1 for all n. Let ξ < 1 be uch that ε k (n + 1)/ε(n) < ξ. For each k K and S, let q k = p k ε k (n), where n i uch that x k = y n. We claim that the dataet (x k, q k ) K k=1 atifie SARSEU. Let δ be defined from (q k ) K k=1 in the ame manner a ˆδ wa defined from (p k ) K k=1. For each pair ((k, ), (k, )) with x k > x k, if n and m are uch that xk x k = y m, then n > m. By definition of ε, Hence, ε k (n) (m) < εk (n) ε(m) < ξ < 1. ε k = y n and δ ((k, ), (k, )) = log pk ε k (n) p k ε k (m) < log pk p k + log ξ < log pk = p ˆδ(x k, x k k ). Thu, for all t T, δ t ˆδ t 0, a t 0 and the dataet (x k, p k ) K k=1 atifie SARSEU. Thu the dataet (x k, q k ) K k=1 atifie SARSEU. Finally, note that ξ < εk (n) < 1 for all n and each k K, S. So that by chooing ξ cloe enough to 1 we can take the price (q k ) to be a cloe to (p k ) a deired. A.2.3. Proof of Lemma 13 Conider the ytem compried by (9) and (10) in the proof of Lemma 11. Let A, B, and E be contructed from the dataet a in the proof of Lemma 11. The difference with repect to Lemma 11 i that now the entrie of A 4 may not be rational. Note that the entrie of E, B, and A i, i = 1, 2, 3 are rational. Suppoe, toward a contradiction, that there i no olution to the ytem compried by (9) and (10). Then, by the argument in the proof of Lemma 11 there i no olution to Sytem S1. Lemma 8 with F = R implie that there i a real vector (θ, η, π) uch that θ A + η B + π E = 0 and η 0, π > 0. Recall that B 4 = 0 and E 4 = 1, o we obtain that θ A 4 + π = 0. Let (q k ) K k=1 be vector of price uch that the dataet (xk, q k ) K k=1 atifie SARSEU and log q k Q for all k and. (Such (q k ) K k=1 exit by Lemma 12.) Contruct matrice A, B, and E from thi dataet in the ame way a A, B, and E i contructed in the proof 26