Cobb-Douglas Preferences under Uncertainty

Transcription

1 Inspirar para Transformar Cobb-Douglas Preferences under Uncertainty José Heleno Faro Insper Working Paper WPE: 278/2012

2 Cobb-Douglas Preferences under Uncertainty José Heleno Faro Insper Institute of Education and Research, Rua Quatá 300, , São Paulo, Brazil December 3, 2012 This paper axiomatizes Cobb-Douglas preferences under uncertainty. First, we extend the original Trockel (1989)'s axiomatic foundation to a general state space framework based on the Strong Homotheticity Axiom, obtaining also the incomplete case a la Bewley (2002). We show that this key axiom for the Cobb- Douglas expected utility specication is refuted by Ellsberg's uncertainty aversion behavioral pattern. Our main result provides a set of meaningful axioms characterizing Cobb-Douglas Min-Expected Utility preferences, an important class of uncertainty averse preferences for studying the consequences of ambiguity in nance and other elds. Finally, we present briey how to obtain more general representations like the variational case. JEL Classication: D81. Keywords: Cobb-Douglas preferences, Expected utility, Ellsberg paradox, Knightian uncertainty, Incomplete preferences, MEU preferences. Introduction The class of Cobb-Douglas preferences is one of the most popular objects in economic theory with applications that go far beyond any short description. The eld of choice under uncertainty presents a natural context where a decision maker can be assumed to behave in accordance with the Cobb-Douglas specication. However, usually the class of Cobb-Douglas preferences is The author wishes to thank Felipe de Araujo, Itzhak Gilboa, Faruk Gul, Marco Lyrio, Paulo Natenzon, Gil Riella, and Walter Trockel for their comments and suggestions. In special, thanks are also given to an anonymous referee for his/her valuable suggestions and comments. address: jhfaro@gmail.com (J.H. Faro)

3 simply viewed as an analytically convenient specication of homothetic preferences and its axiomatic foundation, although characterized by Trockel (1989) 1, received little attention in the economics of uncertainty. This paper begins by providing an axiomatic foundation for Cobb-Douglas (expected utility) preferences in the context of choice under uncertainty. This is done in the context of a general state space by showing how the Strong Homotheticity Axiom 2 is its key behavioral assumption. We also characterize the case of Cobb-Douglas incomplete preferences a la Bewley (2002). Nevertheless, the Strong Homotheticity Axiom has an important descriptive limitation since it violates the famous Ellsberg (1961)'s paradox. This motivates us to study the consequences of weakening the Strong Homothetic Axiom. In this way, our main result provides a set of meaningful axioms characterizing Cobb-Douglas Min-Expected Utility preferences, which lls the gap between the well known axiomatic foundation of multiple priors preferences of Gilboa and Schmeidler (1989) and its applications to general equilibrium and nance with unitary relative risk aversion, a special case of CRRA utility index 3. We also discuss how to obtain more general representations like the variational representation of Macherroni, Marinacci and Rustichini (2006). For the nite state space case an equivalent multiplicative representation is provided for Cobb-Douglas variational preferences using a condence function of Chateauneuf and Faro (2009). Finally, all the results in this paper can be useful when one wishes to ignore utility functions, perhaps because they cannot be measured. Framework Consider a set S of states of nature (world), endowed with a -algebra of subsets called events, and a set of consequences given by the interval (0; 1) =: R ++. We denote by B ++ 0 () [B 0 ()] the set of all simple real-valued -measurable functions ' : S! R ++ [R]. The norm in B 0 () is given by k'k 1 = sup s2s j' (s)j (called sup norm). The set B ++ 0 () is called the set of acts and the interpretation is that each act ' 2 B ++ 0 () promises a positive payoff in 1 Taking into account the consumer theory with a nite number of goods, Trockel (1989) provides a positive answer to the Grandmont (1987)'s question on whether Cobb-Douglas preferences are the only ones which are budget-invariant. Indeed, a scholium of Trockel (1989)'s results is an axiomatic foundation for Cobb-Douglas preferences. See also Trockel (1992). 2 This axiom says that given a triple of acts '; ; from the state space S to the outcome set (0; 1), if ' is as good as then ' is as good as, where the act ' is dened pointwise by (') (s) := ' (s) (s). 3 Recall that this utility on monetary payoffs is also mentioned as a special case of utility with hyperbolic absolute risk aversion (HARA). 2

4 each state of nature 4, but ex ante the decision maker can't be sure about what state of nature will occur. Clearly, B 0 () is a topological vector space by considering, for instance, the norm topology. Let r be a positive real number, dene ' r 2 B ++ 0 () to be the constant act such that ' r (s) = r for all s 2 S: Hence, we can identify the set of positive numbers R ++ with the set of the constant acts in B ++ 0 (). Given '; 2 B ++ 0 () and a real number > 0, we dene the mapping ' 2 B ++ 0 () by, for all s 2 S ' (s) := ' (s) (s) ; In fact, we note that # Im ' # Im ' # Im and ' 1 k'k 1 kk 1. For an act ' 2 B 0 ++ (), we dene the function ln ' 2 B 0 (), by (ln ') (s) := ln (' (s)), for all s 2 S: Also, for a function a 2 B 0 (), we dene the act exp a 2 B ++ 0 (), by (exp a) (s) := exp a (s), for all s 2 S. If S is nite with #S = n then B ++ 0 () can be identied with R n ++ = fx = (x 1 ; :::; x n ) : x i > 0; i = 1; :::ng ; and each ' 2 B ++ 0 () is represented by some x 2 R n ++. We denote by := () the set of all (nitely additive) probability measures p :! [0; 1] endowed with the natural restriction of the well known weak topology (ba; B 0 ). A decision maker's preference is given by a binary relation % on the set of acts B Indeed, % is a subset of B 0 ++ B 0 ++ and ('; ) 2% means that the act ' is at least as good the act, and it will be denoted by ' %. For a binary relation % on B 0 ++, the symmetric part of % is given by a binary relation s on B 0 ++ where ' s if, and only if, ' % and % '. Also, the asymmetric part of % is given by a binary relation on B 0 ++ where ' if, and only if, ' % and not % '. The interpretation is that ' s captures indifference between ' and while ' means that the act ' is strictly better than the act. If for some two acts ' and we have that neither ' % nor % ' we then interpret it as a case of non-comparable acts. 4 Note that each act promises only a nite number of different payoffs across the set of state space S. Formally, given an act 2 B 0 (), the image of is dened by Im := f (s) : s 2 Sg. Hence, the cardinality of Im, denoted by # Im, satises # Im < 1. 3

5 Next, we present a list of properties that a preference relation might satisfy and that we will need as axioms in the next sections. Given a binary relation % on B ++ 0, we say that: % is Non-trivial if there exist '; 2 B ++ 0 () with '. % is Reexive when for all ' 2 B ++ 0 () we have that ' % '. % is Complete if for all '; 2 B ++ 0 (), ' % or % '. % is Transitive when for all '; ; 2 B ++ 0 (), if ' % and % then ' %. % is a Weak Order if % is Complete and Transitive. % is Monotone when for all '; 2 B ++ 0 (), if ' then ' %. % is Strictly Monotone if for all '; 2 B ++ 0 () s.t. ' and ' 6= then '. % is Locally Lower Continuous if there exists ' 2 B ++ 0 () such that f : ' % g is closed and f : ' g is open (in the norm topology). % is Continuous if given any sequence f(' n ; n )g n2n such that ' n % n for all n 1, if kk ' 1 n! ' 2 B ++ kk 0 () and 1 n! 2 B ++ 0 () then ' %. % is Indifference Invariant when for all '; 2 B ++ 0 () s.t. ', if ' 0 ' and 0 then ' 0 and ' 0. % is Strongly Homothetic if for all '; ; 2 B ++ 0 (), ' % ) ' %. % is Log-Convex if for all '; 2 B ++ 0 (), and 2 (0; 1) ; ' % ) ' 1 %. % is Power Invariant if for all '; 2 B ++ 0 (), and k > 0; ' % ) ' k % k. % is Homothetic if for all '; 2 B ++ 0 (), and k > 0; ' % ) k' % k. 4

6 Remark 1 Given a binary relation % on B ++ 0 (), following Mas-Colell and Neuefeind (1977) and Grandmont (1987) 5, given an act we might dene the binary relation % on B ++ 0 () by, for all '; ' %, ' %. Clearly, the preference % on B ++ 0 () is Strongly Homothetic if and only if %=% for all 2 B ++ 0 (). Actually, following the denition of a budget-invariant preference used by Trockel (1989) in the context of a nite dimensional commodity space 6, we might say that a preference relation % on B ++ 0 () is act-invariant if %=% for all 2 B ++ 0 (). Also, Grandmont (1987) noted that the budget-invariant concept is identical to the notion of a "household equivalence scale" introduced by Barten (1964) 7. Remark 2 In the social welfare literature the Strongly Homothetic Axiom is known as "Scale Independence". This axiom, together with mild conditions including anonymity 8, characterize the Nash Collective Utility Function F : R n ++! R with F (u) = Y u i, that is, the social 1in welfare ordering induced by F admits a symmetric Cobb-Douglas social welfare representation 9. For two-person bargaining situations, Trockel (1999) provided an axiomatic foundation for the Nash product as a social planner's welfare function. Cobb-Douglas Subjective Expected Utility Preferences This Section presents the result that extends the Trockel (1989)'s axiomatic foundation of Cobb- Douglas preferences in consumer theory to our framework. Indeed, in the rst condition characterizing the Cobb-Douglas specication we impose only a minimal set of conditions in order to obtain the desired representation, motivated by Neuefeind and Trockel (1995). On the other 5 It is interesting to note that Mas-Colell and Neuefeind (1977) proposed such construction and used a trick with a Cobb-Douglas agent in order to obtain the result on the possibility of approximating an atomless pure exchange economy by economies having an excess demand function. For further discussions and results on the smoothing of aggregate demand see also Araujo and Mas-Colell (1978), Dierker, Dierker and Trockel (1984), Trockel (1984), and Grandmont (1987): 6 See, for instance, Trockel (1984) for a detailed treatment of budget-invariance based on group actions and G-spaces. 7 For the importance of Barten's model to econometric analysis in consumer theory, see Deaton and Paxson (1998, 2003), Gan and Vernon (2003), and Perali (2008). 8 For all u; v 2 R n +, if u is a permutation of v then u v. 9 See Kaneko (1984) and Moulin (1988) [Theorem 2.3] for such results in social welfare and also Østerdal (2005) for a health economics perspective. 5

7 hand, in the second list of necessary and sufcient conditions for obtaining the Cobb-Douglas representation, we list a set of axioms with apparently stronger conditions than the rst one, and it is done in order to facilitate the comparison of this theorem with the following ones. Denition 3 We say that % is a Cobb-Douglas Subjective Expected Utility preference when there exists a probability measure q 2 such that ' %, ln 'dq ln dq. Theorem 4 The following are equivalent: (i) A binary relation % is Non-Trivial, Reexive, Monotone, Locally Lower Continuous, Indifference Invariant, and Strongly Homothetic; (ii) A binary relation % is Non-Trivial, Monotone, Continuous, and a Strongly Homothetic Weak Order; (iii) A binary relation % is a Cobb-Douglas Subjective Expected Utility preference. Remark 5 Suppose that S is nite with #S = n and that % is strictly monotone (i:e:, for all x; y 2 R n ++, if x y and x 6= y then x y). In this case we obtain a representation where there exists a strictly positive probability q = (q 1 ; :::; q n ), q i > 0 8i, such that x % y () nx q l ln x l i=l nx q l ln y l. l=1 Clearly, x % y () Y 1l1 x q l l Y 1l1 y q l l. and, by continuity, we can extend the representation u (x) = Y 1l1 x q l l to the whole R n + obtaining that u (x) = 0 for all x R n + = x 2 R n + : x l = 0 for some l, that is x 0, for all x R n +. Cobb-Douglas Incomplete Preferences A well known criticism against well specied probabilities is the argument that the completeness condition is unrealistic because it forces the decision maker to be able to compare every 6

8 pair of acts 10. The next result obtains a Cobb-Douglas representation for incomplete preferences a la Bewley (2002) by dropping only completeness in the condition (ii) of Theorem Theorem 6 A binary relation % is a Non-trivial, Reexive, Transitive, Continuous, Monotone, Power Invariant, and Strongly Homothetic if and only if there exists a unique non-empty closed and convex set C of nitely additive probability measures such that for all '; 2 B 0 ++ (), ' %, ln 'dq ln dq, for all q 2 C. Moreover, when S is nite with #S = n the preference % can be extended to R n + and it satises, for any x; y 2 R n + x % y, ny l=1 x q l l ny l=1 y q l l, for all q 2 C. Ellsberg Paradox as a Violation of Strong Homotheticity Consider the Ellsberg (1961)'s three color urn, with 30 red balls and 60 balls either green or blue. Following the usual Ellsberg's bets, we propose the following simple variation in order to accommodate the restriction to strictly positive payoffs: betsncolor red green blue ' r e ' g 1 e 9 1 ' rb e 9 1 e 9 ' gb 1 e 9 e 9 where ' r pays e 9 dollars if a red ball is drawn and only one dollar otherwise, ' g pays one e 9 dollars if a green ball is drawn and one dollar otherwise, and so on. The well known Ellsberg argument says that most subjects rank these acts as ' r ' g and ' gb ' rb. Call ' b the act ' b := (1; 1; e 9 ), hence when % satises strongly homotheticity ' r ' g if and only if ' rb = ' r ' b ' g ' b = ' gb, which contradicts the Ellsberg's uncertainty averse pattern of behavior. Next section presents our axiomatic contribution in obtaining an uncertainty averse Cobb-Douglas representation. 10 See, for instance, Aumann (1962), Bewley (2002) and Rigotti and Shannon (2005). 11 In terms of condition (i) of Theorem 4, note that the incomplete preference case doesn't satisfy Locally Lower Continuity. See also Schmeidler (1971). On the other hand, it is clear that Cobb-Douglas incomplete preferences satisfy Indifference Invariance. 7 ;

9 Cobb-Douglas Min-Expected Utility Preferences We saw in the previous Section that the Ellsberg Paradox provides an objection to the Strong Homotheticity Axiom. Hence, a natural question is whether we might accommodate the Ellsberg pattern of behavior by weakening the Strong Homotheticity Axiom. Next, we present an axiomatic foundation of what we call Cobb-Douglas Min-Expected Utility preferences, a class of preferences in the tradition inspired by Knight (1921) and Ellsberg (1961) 12. It is worth to notice that we model uncertainty aversion in our framework by not assuming Strongly Homotheticity, but we impose the structure of a (continuous) weak order. Actually, in that context we assume some consequences of Strongly Homotheticity given by Power invariance, Log-Convexity, and Homotheticity 13. Hence, the fact that a ranking ' can change completely after the multiplicative action of some nonconstant act is the salient missing property in the next theorem, as compared to Theorem 4, leading to a multiple prior representation. Theorem 7 A binary relation % is a Non-trivial Weak Order, Continuous, Monotone, Log- Convex, Power Invariant, and Homothetic if and only if there exists a unique non-empty closed and convex set C of nitely additive probability measures such that ' %, min ln 'dp min ln dp: p2c p2c Moreover, when S is nite with #S = n the preference % can be extended to R n + and it satises, for any x; y 2 R n + x % y, min p2c Y 1ln x p l l min p2c Y 1ln y p l l. Remark 8 If we drop the Power Invariant Axiom from the list of axioms characterizing Cobb- Douglas Min-Expected Utility, then we can apply the Maccheroni, Marinacci and Rustichini (2006)'s main result for obtaining a Cobb-Douglas Variational Representation V for %: There 12 Gilboa and Schmeidler (1989) provided an axiomatic foundation for the so called maxmin expected utility (MEU) in the Anscombe and Aumann (1963) framework. In the context of monetary payoffs Chateauneuf (1991) provided an axiomatic foundation for MEU preferences with linear utility index (risk neutral agent). See, also, Ghirardato, Maccheroni, Marinacci, and Siniscalchi (2003), Casadesus-Masanell, Klibanoff, and Ozdenoren (2000), and Alon and Schmeidler (2011). Our result can be viewed as an axiomatic foundation for the special case of MEU preferences with log-index over consequences. See Gilboa and Marinacci (2012), Section 6, for many applications. In special, see Epstein and Miao (2003), and also Ohtaki and Ozaki (2012). 13 See Remark 13 in the Appendix for more details. 8

10 exists a grounded (attains zero), convex and lower semicontinuous function c :! [0; 1] such that V : B ++ 0 ()! R is given by V (') = min p2 ln 'dp + c (p). Furthermore, when S is nite with #S = n, we also have the multiplicative representation given by J (x) = exp V (x), which can be extended to R n + and J (x) = min p2 1 (p) Y 1ln where :! [0; 1] is a condence function a la Chateauneuf and Faro (2009, 2012) 14. Finally, if we drop the Power Invariant and Homothetic Axioms we can obtain the general uncertainty averse representation of Cerreia-Vioglio, Maccheroni, Marinacci and Montruchio (2011) with a log-index over consequences. x p l l ; Appendix For the proof of our results we always use an auxiliary binary relation % on B 0 () induced from % on B ++ 0 () in the following way: given a; b 2 B 0 () a % b, there exist '; 2 B ++ 0 () s.t. ' %, with a = ln ' and b = ln. We note that since ln : (0; +1)! R is a bijection, we have that % is a well dened Reexive and Non-Trivial binary relation whenever % is a Reexive and Non-trivial binary relation. Also, the following properties when imposed on % are inherited by % : completeness, transitivity, local lower continuity, continuity, and monotonicity 15. Also, for a binary relation % on B 0 () we dene: % is Mixture Invariant if for all a; b; c 2 B 0 () and 2 (0; 1), a % b if and only if a + (1 ) c % b + (1 ) c. % is Translation Invariant when for any a; b; c 2 B 0 (), if a % b then a + c % b + c. % is Convex when for all a; b 2 B 0 () and 2 (0; 1), if a % b then a+(1 ) b % b. 14 Clearly, () = exp ( c ()) is a regular (attains 1), quasi-concave and upper semicontinuous mapping. 15 Clearly, we change the environment from the set B ++ 0 () to the topological vector space B 0 (). 9

11 Weak Translate Invariant when for all a; b 2 B 0 () and c 0 2 R, if a % b then a + c 0 % b + c 0. Remark 9 It is worth to notice that a Reexive and Transitive binary relation % on B 0 () is Mixture Invariant if, and only if, a % b implies that a + c % b + c for all > 0 and c 2 B 0 (). See Gilboa, Maccheroni, Marinacci and Schmeidler (2010) [Lemma 1]. Remark 10 Note that if % on B 0 () is a Translation Invariant and Continuous Weak Order then % is also Homothetic (For all a; b 2 B 0 () and 0, if a % b then a % b). See, for instance, Candeal-Haro and Indurain-Eraso (1995) [Remark 1, p. 520]. It is also evident from the proof of the main result in Neuefeind and Trockel (1995) [p. 355, item (c)]. The following lemma give us some basic properties relating % and %, which will be useful for next results. Lemma 11 Suppose that % is a Reexive and Non-trivial binary relation on B ++ 0 () and consider its induced binary relation % on B 0 () as dened above. (i) % is Power Invariant iff % is Homothetic; (ii) % is Log-Convex iff % is Convex; (iii) % is Homothetic iff % is Weak Translate Invariant; (iv) % is Strongly Homothetic iff % is Translation Invariant. Proof. (i) Let a; b 2 B 0 () such that a % b, that is, exp (a) % exp (b). Since % is Power Invariant we obtain that (exp (a)) k % (exp (b)) k, for all k > 0. Hence exp (ka) % (exp (kb)) which entails that ka % kb for all k > 0. For the converse, let '; 2 B ++ 0 () such that ' %, that is, ln ' % ln. Since % is Homothetic it follows that for all k > 0; ln ' k = k ln ' % k ln = ln k, i.e., ' k % k. (ii) Let a; b 2 B 0 () and 2 (0; 1) such that a % b. Since % is Log-Convex then (exp (a)) (exp (b)) 1 % exp (b) and then ln (exp (a)) (exp (b)) 1 % ln (exp (b) ), that is, a + (1 ) b % b. For the converse, let '; 2 B 0 ++ (), and 2 (0; 1) ; such that ' %. Hence, ln ' % ln and since % is Convex we obtain that ln ' + (1 ) ln % ln, i.e., ln ' 1 % ln which implies that ' 1 %. (iii) Let a; b 2 B 0 () and c 0 2 R. If a % b then exp (a) % exp (b) and since % is Homothetic, by considering the constant k := e c 0 which implies that a + c 0 = ln (exp (a) k) % ln (exp (b) k) = b + c 0 : 10 > 0, we obtain that k exp (a) % k exp (b),

12 For the converse, let '; 2 B ++ 0 () and k > 0. Assume that ' %, hence ln ' % ln and since % is Weak Translate Invariant we obtain that which give us that k' % k. ln ('k) = ln ' + ln k % ln + ln k = ln (k) ; (iv) Let a; b; c 2 B 0 () such that a % b. So, we have that exp (a) % exp (b) and since % is Strongly Homothetic we obtain that exp (a + c) = exp (a) exp (c) % exp (b) exp (c) exp (a + c) = exp (a) exp (c) % exp (b) exp (c) = exp (b + c) ; that is, a + c % b + c. For the converse, let '; ; 2 B ++ 0 () such that ' %. This implies that ln ' % ln and since % is Translation Invariant we obtain that ln ' + ln % ln + ln that is, ' %. ln (') = ln ' + ln % ln + ln = ln () The proof of our rst important result that characterizes "Cobb-Douglas Subjective Expected Utility" relies heavily on Neuefeind and Trockel (1995). The Neuefeind and Trockel's theorem provides a very general result on continuous linear representability of binary relations on topological vector spaces. Since we deal with the special case of B 0 (), it seems to us that it might be useful to have an explicit result about a linear representation for a preference over B 0 () that follows as a corollary of Neuefeind and Trockel's theorem, as we have below: Lemma 12 [Based on Neuefeind and Trockel, 1995] A binary relation % on B 0 () is a Non- Trivial, Reexive, Locally Lower Continuous, Indifference Invariant, and Translate Invariant if and only if there exists a nitely additive measure 2 ba () such that for all a; b 2 B 0 (), a % b, ad bd. Theorem 4: Proof. The part (ii) ) (i) is trivial and (iii) ) (ii) is straighforward. Let us show the part (i) ) (iii). Suppose that % is a Non-Trivial, Reexive, Monotone, Locally Lower Continuous, Indifference Invariant, and Strong Homotheticity binary relation on B ++ 0 (), hence the induced binary relation % on B 0 () is a Non-Trivial, Locally Lower Continuous, Indifference Invariant and Translation Invariant binary relation. 11

13 Hence, by the Lemma 12 based on Neuefeind and Trockel (1995), there exists a nitely additive measure 2 ba () such that for all a; b 2 B 0 (), a % b, ad bd. Moreover, since % is also monotone, we obtain that (E) 0 for all E 2. Note that (S) > 0: Since % is Non-trivial there exist a; b 2 B 0 () with a b, and by translation invariance a b 0, so 0 < (a b) d ka bk 1 (S). Now, taking q := (S) 1 2, we obtain that for all a; b 2 B 0 (), a % b, adq bdq. Also, ' %, ln ' % ln, ln 'dq ln dq. Theorem 6: Proof. We note that % is a Non-trivial, Reexive, Transitive, Continuous, Monotone, Power Invariant, and Strong Homotheticity binary relation on B ++ 0 () if and only if % is a Nontrivial, Reexive, Transitive, Continuous, Monotone, and Mixture Invariant binary relation on B 0 (). By the Proposition A.2 (p. 157) in Ghirardato, Maccheroni and Marinacci (2004) % is characterized by the existence of a non-empty, weak closed and convex subset C of such that a % b, adp bdp for all p 2 C. Hence, ' %, ln ' % ln, ln 'dp ln dp for all p 2 C. Remark 13 In the proof below concerning Cobb-Douglas Min-Expected Utility representation we assume mild basic conditions and use some axioms that are weaker than the Strong Homotheticity Axiom. Actually, assume that a Continuous Weak Order is Strongly Homothetic (SH). First, it is trivial that is Homothetic. That is Power Invariant follows from the fact 12

14 that the induced binary relation % is Weak Translate Invariant, and by Remark 10 we know that % is Homothetic, that is, is Power Invariant. Finally, is also Log-Convex. Indeed, let '; 2 B ++ 0 (), and 2 (0; 1) such that ' %. Hence, SH implies '= % 1 and Power Invariance give us ('=) % 1. SH again entails that ('=) % ) ' 1 %. Proof. Theorem 7: Suppose that % is a Non-trivial Weak Order, Continuous, Monotone, Log-Convex, Power Invariant, and Homothetic binary relation on B ++ 0 (). Hence the induced binary relation % on B 0 () is a Non-trivial, Continuous, Monotone, and Homothetic Weak Order. Also, we note that % is a Convex and Weak Translate Invariant binary relation. Since % is a Nontrivial Weak Order, Continuous and Monotone then each act a admits a unique % -certainty equivalent a 2 R 16. Hence, we can dene the functional I : B 0 ()! R where for all a 2 B 0 (), I (a) := a. It is immediate to see that I represents %. Now, we can invoke the fundamental lemmas of Gilboa and Schmeidler (1989) [Lemmas 3.5 and 3.5, p ] in order to conclude that there exists a closed and convex set C of nitely additive probability measures on such that I (a) = min p2c adp. Finally, given '; 2 B ++ 0 ' %, ln ' % ln, min p2c For the converse, the proof is straighforward. ln 'dp min p2c ln dp. References and Notes 1. Alon, S. and D. Schmeidler (2011): Purely Subjective Maxmin Expected Utility, mimeo,tel-aviv University. 2. Anscombe. F.J. and R. Aumann (1963): A denition of subjective probability, Annals of Mathematical Statistics 34, Araujo, A., and A. Mas-Colell (1978): Notes on the smoothing of aggregate demand. Journal of Mathematical Economics 5, See, for instance, the proof of Lemma 28 in Maccheroni, Marinacci and Rustichini (2006). 13

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