A Representation Theorem for General Revealed Preference
General Revealed Preferences Cesar Martinelli and Mikhail Freer February 2016
Discussion Paper
Interdisciplinary Center for Economic Science 4400 University Drive, MSN 1B2, Fairfax, VA 22030 Tel: +1-703-993-4719 Fax: +1-703-993-4851 ICES Website: http://ices.gmu.edu ICES RePEc Archive Online at: http://edirc.repec.org/data/icgmuus.html
General Revealed Preferences C´esar Martinelli⇤1 and Mikhail Freer†1 1
Interdisciplinary Center for Economic Science, George Mason University February 2016
Abstract Following Richter (1966), we provide criteria under which a preference relation implied by a finite set of choice observations has a complete extension that can in turn be represented by a utility function. The usual properties of the original preference relation are inherited by the preference extension represented by the utility function. Noteworthy, our result relaxes the usual assumptions about the structure of budgets generating the observed choices. Against common wisdom, we show that (upper semi-)continuity does not come for free in general and it requires additional assumptions.
1
Introduction
The notion of rational behavior is central in economics. Historically there have been three ways to specify rational behavior. The first, introduced by Walras (1896), assumes that agents maximize some objective (utility) function. The second strand started back with Frisch (1926) and relies on the notion of preferences as a formal model of rational behavior. This approach was later developed and popularized by Debreu (1954) and Von Neumann and Morgenstern (1947). The third strand, starting with Samuelson (1938), and uses conditions on (finite sets of) observed choices to describe rational behavior. The general connection between utility functions and preference relation was originally studied1 by Debreu (1954) in the context of continuous utility functions. Rader (1963) and Jaffray (1975) relaxed the assumption of continuity and obtained semi-continuous utility rationalization results that were generalized by Bosi and Mehta (2002). Peleg (1970) shown the sufficient condition for existence of a continuous utility representation for incomplete Email adress: [email protected] Email adress: [email protected] 1 Earlier, utility representation theorems under certain additional assumptions were provided by Cantor (1895) and Von Neumann and Morgenstern (1947) ⇤ †
1
Preference Relation Frisch (1926) Szpilrajn (1930), . . .
Set of choices Samuelson (1938)
Debreu (1954), . . .
Afriat (1967), . . .
Utility Function Walras (1896)
Figure 1: Models of Rational Behavior preference relation. More recently Ok (2002), Evren and Ok (2011) have investigated a problem of existence of a vector-valued utility representation of preference relations. The basic result connecting the set of choices and preference relations was proven by Szpilrajn (1930). Szpilrajn shown that any acyclic preference relation has a complete and transitive extension. Demuynck (2009) generalized the result by providing a condition to test for existence of complete extension that has properties usually assumed by economists. As it is well-known, lexicographic preferences illustrate that not every complete and transitive relation can be represented by a utility function. The connection between finite data sets and the utility functions was originally investigated by Afriat (1967). Subsequent literature has constructed tests for consistency of the finite consumption data with various utility maximization hypothesis. Kannai (1977), Matzkin (1991), Matzkin and Richter (1991) and Forges and Minelli (2009) address the question of testing concavity of the utility representation. Varian (1983), Diewert and Parkan (1985), Echenique and Saito (2013) and Polisson et al. (2015) develop tests for separable utility representations including the context of choices under uncertainty. Crawford (2010) propose a test for habit formation models. In this paper we connect these three models of rational behavior in a parsimonious way. That is, we propose conditions under which a finite set of choices can be extended by a preference relation that in turn can be represented by a utility function. We generalize a classical result from Richter (1966), allowing a complete extension to satisfy properties deemed as desirable by economists. We show that, in our generalized framework, existence 2
of a utility function does not guarantee the existence of (upper semi-)continuous utility function. The remainder of this paper is organized as follows. In Section 2 we introduce basic definitions. In Section 3 we provide the criteria for the existence of a complete extension that can be represented by an (upper semi-)continuous utility function. In the Section 4, we illustrate our results by providing an example of application of our result to construct a revealed preference test.
2
Preliminaries
Consider a universal set X ✓ RN of alternatives. A set R ✓ X ⇥ X is said to be a preference relation. We denote a set of all preference relations on X by R. We denote the reverse relation R 1 = {(x, y)|(y, x) 2 R}. We denote the symmetric (indifferent) part of R by I(R) = R \ R 1 and the asymmetric (strict) part by P (R) = R \ I(R). We denote the incomparable part by N (R) = X ⇥ X \ (R [ R 1 ).
2.1
Preference Relations
A preference relation R is said to be • complete if (x, y) 2 R [ R
1
for all x, y 2 X (or equivalently N (R) = ;).
• transitive if (x, y) 2 R and (y, z) 2 R implies (x, z) 2 R for all x, y, z 2 X. • reflexive if for every x 2 X, (x, x) 2 R. • antisymmetric if (x, y) 2 R and (y, x) 2 R implies x = y for all x, y 2 X. • monotone2 if for any x, y 2 X x
y implies (x, y) 2 R.
The set LP (R) (x) = {y|(x, y) 2 P (R)} is called lower contour set of x. The set UP (R) (y) = {x|(x, y) 2 P (R)} is called upper contour set of y. A preference relation R is said to be upper semi-continuous if LP (R) (x) for all x 2 X are open. A preference relation R is said to be continuous if LP (R) (x) and UP (R) (x) for all x 2 X are open (or equivalently3 if LR (x) and UR (x) are closed). A tuple (X, R) where X is a set of alternatives and R is preference relation is said to be partially ordered set (or shortly poset) if R is transitive, antisymmetric and reflexive (usually called partial order). A (X, R) is said to be completely ordered set if R is transitive and complete. Note that any completely ordered set is also poset. 2 3
We use a definition that x y, if xi > yi for all i 2 {1, ..., N }. Definitions are equivalent for a complete preference relation on X ⇥ X, see e.g. Kreps (2012).
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Definition 1. A tuple (X, R) is said to be Z-separable for given Z ✓ X if for any (x, y) 2 P (R) there is z 2 Z, such that (x, z) 2 R and (z, y) 2 R. A tuple (X, R) is said to be order separable if there is at most countable Z such that (X, R) is Z-separable. Example. Denote by L ✓ RN ⇥ RN the lexicographic preference relation, i.e. (x, y) 2 L if there is k 2 {1, . . . , L} such that xi = yi for i < k and xk > yk . It is well-known that this relation is not order-separable (see e.g. Mas-Colell et al. (1995)). However, L is RN separable. Definition 2. A preference relation R0 is an extension of R, denoted R and P (R) ✓ P (R0 ).
R0 if R ✓ R0
The straight-forward example of extending a preference relation is taking a transitive closure.
2.2
Functions over Preference Relations
The transitive closure is natural example of a function over the set of preference relations. Denote it by T : R ! R, where (x, y) 2 T (R) if there is a sequence S = s1 , s2 , . . . sn , such that for any j = 1..n 1, (sj , sj+1 ) 2 R and s1 = x and sn = y. The transitive closure is monotone, i.e. the transitive closure of a subset of the relation is the subset of closure of the original relation. It is closed, i.e. any relation is a subset of its transitive closure. It is idempotent in the sense that the transitive closure of transitive relation is equal to the original relation. It is also compact-valued (or algebraic) in the sense that to obtain any element from transitive closure it is enough apply the transitive closure to a finite subset of original relation. The following definition specifies the family of functions that have the properties we just listed. Definition 3. A function F : R ! R is said to be an algebraic closure if (1) For all R, R0 2 R, if R ✓ R0 , then F (R) ✓ F (R0 ), (2) For all R 2 R, R ✓ F (R), (3) For all R 2 R, F (F (R)) ✓ F (R), (4) For all R 2 R and all (x, y) 2 F (R), there is a finite relation R0 ✓ R, s.t. (x, y) 2 F (R0 ). Conditions (1)-(3) define the closure, i.e. an operator that is monotone (1), closed (2) and idempotent (3) and condition (4) makes the closure algebraic. Remark. There is no algebraic closure that implies continuity. Denote continuity closure by C : R ! R, when (x, y) 2 C(R) if there is a sequence (xn , yn ) 2 R, such that xn ! x 4
and yn ! y. Let us construct a violation of (4). Let X = R and R = {( n1 , 1) : n 2 N}, then (0, 1) 2 C(R). However, there is no finite sub-relation R0 , such that (0, 1) 2 C(R0 ). For any function F : R ! R, let R⇤F = {R 2 R|R F (R)}. For any function F : R ! R and Z ✓ X, let RZF be a set of all Z-separable R, such that R F (R). Note that RZF ✓ R⇤F . Example. Let X = RN and Z = ;, then a set R⇤T (recall that T stays for transitive closure) will include all acyclic preference relations. The set R;T will consists only of relations R such that P (R) = ;, i.e. only equivalence relations on X. ˆ F ✓ R⇤ the set of all upper semi-continuous R : R F (R). Denote by R F Definition 4. A function F : R ! R is said to be separability-preserving if for Zseparable R, F (R) is also Z-separable. Note that separability preserving requires F (R) to be separable with respect to the same set as R. Moreover, transitive closure is an example of separability-preserving function. Recall that by definition for any (x, y) 2 T (R) there is a sequence S = s1 , s2 , . . . sn , such that for any j = 1..n 1, (sj , sj+1 ) 2 R and s1 = x and sn = y. It implies that for any (x, y) 2 P (T (R)) there is k, such that (sk , sk+1 ) 2 P (R). Hence, there is z 2 Z such that (x, z), (z, y) 2 T (R). The next condition is a slight variation of C7 from Demuynck (2009). Definition 5. A function F : R ! R is said to be expansive if for any R = F (R) and N (R) 6= ;, there is T ✓ N (R) such that R [ T 2 R⇤F and P (R) = P (R [ T ). Expansiveness of F means that we can add some indifferent points to any R = F (R), such that the new relation will be in RF⇤ . We claim that transitive closure is expansive, since it is possible to add some indifferent points such that the new relation is acyclic. We present a formal proof of the claim in Section 4. A function F : R ! R is said to be upper semi-continuous if the image of upper semi-continuous R is upper semi-continuous A function F : R ! R is said to be continuous if the image of continuous R is continuous In what follows, we will be using the fact that closure induces some properties. A function F induces property f (e.g. monotonicity, transitivity, convexity, etc) if for any fixed point4 R = F (R), R satisfies property f . Definition 6. A function F : R ! R is said to be a rational closure if it is an expansive algebraic closure that induces transitivity. Definition 7. A preference relation is said to be F-consistent if F (R) \ P
1
(R) = ;.
Note that F -consistency is the condition that guarantees that F (R) is an extension of R. From the definition of we know that R F (R), then P (R) ✓ P (F (R)). For R to 4
The fixed point approach for this purpose was originally used by Szpilrajn (1930) and implicitely by Richter (1966).
5
be extendable by F we need to show that F (R) does not contain elements from P 1 (R). Otherwise there will be some (x, y) 2 P (R) and (x, y) 2 / P (F (R)), which would imply, that R F (R). Example. Let the set of alternatives be X = {x1 , x2 , x3 } and preference relation 0 R = {(x1 , x2 ); (x2 , x3 ); (x3 , x1 )}. One can easily see that this relation is not transitive and is not T -consistent because (x1 , x3 ) 2 T (R) and (x3 , x1 ) 2 P (R). On other hand R = {(x1 , x2 ); (x2 , x3 )} from the very first example can is T -consistent. Moreover, T (R) is complete on X.
3
Results
The following theorem provides a convenient criteria for the existence of a utility function that rationalizes the complete extension of a given preference relation and preserves the properties induced by F . We do it showing the existence of a complete relation that is a fixed point of F . The separability-preserving property allows the fixed point to be order separable. Hence, the complete fixed point can be represented by a utility function. Theorem 1. Let F be a separability-preserving rational closure. An order-separable R 2 R has a complete extension R⇤ = F (R⇤ ) that can be represented by a utility function if and only if R is F-consistent. To prove Theorem 1 we need several supplementary results.
3.1
Lemmata
Let us start from Lemma 1 from Demuynck (2009), which shows that F -consistency is necessary and sufficient condition for F (R) to be an extension of R. Lemma 1 (Lemma 1 from Demuynck (2009)). For any function F : R ! R, R ✓ F (R). Then in order for R F (R) it is necessary and sufficient that F (R) \ P 1 (R) = ;. Expansiveness of F implies Demuynck (2009) condition C7, hence, following result will also hold: Lemma 2 (Lemma 4 from Demuynck (2009)). If F : R ! R is a rational closure, then: S (5) For every chain R0 ⇢ R1 ⇢ ... ⇢ R↵ ⇢ ..., where for all ↵ R↵ 2 R⇤F we have ↵ 0 R↵ is also in R⇤F .
Condition (5) is a technical restriction that allows us to apply Zorn’s lemma and show that the complete extension actually has the properties induced by F . In the case of finite sequences (5) is straight-forward, thus for finite sets of alternatives it is not binding, while in the case of infinite sets of alternatives it is restrictive. In order to show that there is a utility function that represents the complete extension we need to restate condition (5) in a more convenient form for the further proof. 6
Lemma 3. If F is a separability-preserving rational closure, then: S (6) For any chain R0 ✓ R1 ✓ ... ✓ R↵ ✓ ... in RZF we have ↵ 0 R↵ 2 RZF .
(7) For every R 2 RZF such that N (R) 6= ; there is a non-empty subset T of N (R) such that R [ T 2 RZF .
Proof. Let us start from showing that (6) is satisfied, i.e. For any chain R0 R1 ... S Z Z R↵ ... in RF then B = ↵ 0 R↵ 2 RF . Since F is rational closure we know that B 2 R⇤F . So, if there is (x, y) 2 P (B), then there is ↵ 0 such that (x, y) 2 R↵ , then there is z 2 Z, such that (x, z) 2 R↵ ✓ B and (z, y) 2 R↵ ✓ B. Not let us show that (7) is satisfied. Firstly, consider the case when R 6= F (R). Since F is separability-preserving rational closure, then if R 2 RZF , then T = F (R) \ R and R [ T is F consistent (because F is rational closure) and is Z-separable (F is separability-preserving) hence, R [ T 2 RZF . If R = F (R) and F is expansive we can add T such that R [ T is F -consistent and P (R [ T ) = P (R) hence it is also Z-separable. Thus R [ T 2 RZF . In order to get upper semi-continuous function we need to modify conditions (6) and (7) in continuity terms. Lemma 4. If F is an upper semi-continuous rational closure, then: (6’) For any chain R0
R1
...
R↵
ˆ F we have S ... of relations in R ↵
0
ˆF. R↵ 2 R
ˆ F such that N (R) 6= ; there is a non-empty subset T of N (R) such (7’) For every R 2 R ˆF. that R [ T 2 R Condition (6’) shows that if there is a chain of upper semi-continuous F-consistent relations, such that each element is extension of a previous one, then the upper bound of this sequence is also upper semi-continuous and F-consistent. Condition (7) tells us that each F-consistent upper semi-continuous preference relation can be extended by another F-consistent upper semi-continuous relation. Proof. Let us start from showing, that a rational closure satisfies (6’). Consider the relation S B = ↵ 0 R↵ . From (5) we know that it is in R⇤F , so in order to prove that (6’) holds it is enough to show that B is upper semi-continuous, i.e. for all x 2 X, LP (B) (x) are open. S Since Ra Rb , for any a b, then P (Ra ) ✓ P (Rb ), thus for any x LP (B) (x) = ↵ 0 LP (R↵ ) is a union of open sets, thus it is open as well. Now we will show, that a rational closure satisfies (7’). To show that F satisfies (7’) we need to consider two cases. First assume that R 6= F (R). Then let T = F (R) \ R, then R [ T = F (R) is upper semi-continuous, because R is an upper semi-continuous and F is upper semi-continuous. Now suppose instead that R = F (R) and we are adding only points that are indifferent, thus do not affect lower contour sets of R. Hence, the extension of it is upper semi-continuous. 7
Note that similar lemma can be obtained for the set of continuous relations. In order to prove Theorem 1 we need also the classical result from Debreu (1954). Lemma 5 (Lemma 2, from Debreu (1954)). Let (X, R) be a separable completely ordered set, then there is a utility function that represents R.
3.2
Main Result
Proof of Theorem 1. Let ⌦ be the set {R0 2 RZF |R R0 }. We know, that for any chain Z ˆF, B = S R0 ✓ R1 ✓ ... ✓ R↵ ✓ ... of relations in R ↵ 0 R↵ 2 RF . So, we are left to show that R B. Clearly, R ✓ B. If on the contrary there are elements x, y 2 X, for which (x, y) 2 P (R) and (y, x) 2 B, we have that there must be a relation R↵ in the chain for which (y, x) 2 R↵ . This contradicts the fact that R R↵ and we conclude that B 2 ⌦. Clearly (⌦, ✓) is a partially ordered set. So applying Zorn’s lemma we get that ⌦ has a maximal element. Let R⇤ be that maximal element. We omit the part of the proof that shows that R⇤ is complete and that R⇤ = F (R⇤ ) since it is similar to the proof of Lemma 2 in Demuynck (2009). We are left to show that there is a utility function that represents R⇤ = F (R⇤ ), since F is rational closure we know that R⇤ is transitive and complete. Hence we are left to show that R⇤ is order separable. R⇤ 2 RZF , hence, it is Z-separable with the same Z as original R which is order separable, R⇤ is order separable, because it is Z-separable and Z is no more than countable. Remark: One can consider this as an infinite construction algorithm. The algorithm is quite simple: (1) if R 6= F (R) use F (R) to add points and impose necessary properties; (2) if F (R) = R add indifference points (since we see that F (R) adds mainly strictly preferred bundles). In this sense Theorem 1 can be perceived as a proof that the algorithm converges. Construction Procedure is illustrated in Figure 2. Assume that purple area (open ball) is LP (R) (x). Then applying F (R) will extend the lower contour set and it will be the triangle without upper boundary. Then by completing relation R[T we will add "indifference curve" that is upper boundary of the triangle. It was shown by Rader (1963), that any upper semi-continuous relation, that can be represented by utility function, has upper semi-continuous utility representation.5 Hence we can provide a conditions under which preference relation have a complete extension that can be represented by an upper semi-continuous function. Note that is transitive, reflexive and antisymmetric relation, hence is a partial order. 5
We are referring to the theorem from Rader (1963) which states that a complete, transitive, upper semicontinuous relation can be represented by an upper semi-continuous utility function. It was mentioned by Mehta (1997) that the original Rader’s proof was incomplete, but the result was finally proven by Bosi and Mehta (2002).
8
x
x
R
)
x
F (R)
)
F (R) [ T
Figure 2: Illustration for construction algorithm Corollary 1. Let F be an upper semi-continuous rational closure. Then an upper semicontinuous R 2 R has a complete extension R⇤ = F (R⇤ ) that can be represented by an upper semi-continuous utility function if and only if R is F-consistent. ˆ F |R R0 }. We know, that for any chain R0 R1 ... Proof. Let ⌦ be the set {R0 2 R ˆF, B = S ˆ R↵ ... of relations in R B. ↵ 0 R↵ 2 RF . So, we are left to show that R Clearly, R ✓ B. If on the contrary there are elements x, y 2 X, for which (x, y) 2 P (R) and (y, x) 2 B, we have that there must be a relation R↵ in the chain for which (y, x) 2 R↵ . This contradicts the fact that R R↵ and we conclude that B 2 ⌦. Clearly (⌦, ) is a partially ordered set. So applying Zorn’s lemma we get that ⌦ has a maximal element. Let R⇤ be that maximal element. We omit the part of the proof that shows that R⇤ is complete and that R⇤ = F (R⇤ ) since it is similar to the proof of Lemma 2 in Demuynck (2009). In order to show that there is an upper semi-continuous utility function, that represents ⇤ ˆ F hence is also upper-semicontinuous, and because R⇤ = F (R⇤ ) R . We know that R⇤ 2 R it is transitive and complete, so by Rader (1963) there is an upper semi-continuous utility function that represents R⇤ . An upper semi-continuous function is enough for observed choices to be the points at which the utility function reaches supremum if the budgets are compact. However, it is common to assume that utility functions are continuous. The theorem from Debreu (1954) shows that any complete, transitive and continuous relation can be represented by a continuous utility function. Corollary 2. Let F be a continuous rational closure. Then a continuous R 2 R has a complete extension R⇤ = F (R⇤ ) that can be represented by a continuous utility function if and only if R is F-consistent
9
The proof of this Corollary is straight-forward, since we assume that R is continuous, then ¯ F as the set of all F-consistent continuous preference relations. Then Lemma one can define R ˆ F by R ¯ F . Hence, the complete extension of R will also be 4 will hold just by replacing R ¯ F . Since R⇤ = F (R⇤ ) is continuous, complete continuous, because it will be an element of R and transitive, then by Debreu (1954) theorem, there is a continuous utility function, that represents R⇤ .
4
Example
Let us illustrate how our results can be applied to construct the tests for existence of utility function that preserves certain properties. As an illustration we will use a classical result that shows existence of a utility function that rationalizes acyclic preference relation.6 We will use the transitivity closure T : R ! R already defined. Define revealed preference relation7 Rv obtained from a finite consumption experiment (xt , B t )Tt=1 , where xt are chosen points and B t are budget sets (xt , y) 2 Rv if y 2 B t \ {xt }. Denote by C = {xt }Tt=1 the set of all chosen points. Proposition 1. A revealed preference relation Rv obtained from a finite consumption experiment has a complete and transitive extension, that can be represented by a utility function if and only if Rv is T -consistent. To prove Proposition 1 we need to show that T is separability-preserving rational closure and that (X, Rv ) is order-separable in order to apply Theorem 1. So, let us start from proving properties of T . We omit the part of proof that T is an algebraic closure that induces transitivity, since it is in Demuynck (2009). We are left to prove that T is expansive and separability-preserving. For this purpose let us define the (x, y)-irreducible length sequence for any (x, y) 2 T (R) as a sequence S = s1 , ..., sn such that s1 = x; sn = y and (sj , sj + 1) 2 R, and there is no shorter sequence S 0 = s01 , . . . , sn0 such that s01 = x; s0n0 = y and (s0j , s0j + 1) 2 R. And from definition of T we know that there always is (x, y)-irreducible length sequence and that all sj are distinct elements, i.e. there is no i 6= j such that si = sj otherwise the length can be reduced. Proof. Let us start from proving that T is expansive. Consider a relation R = T (R) and assume that N (R) 6= ;. Take an element (x, y) 2 N (R) and consider the relation R0 = 6
Under more restrictive assumptions using other techniques it is possible to get stronger result. Existence of continuous utility representation on linear budgets was show by Afriat (1967). Existence of continuous utility representation on compact, monotone and balanced budgets was shown by Forges and Minelli (2009). Existence of continuous utility representation on compact budgets was shown by Heufer (2012) 7 Note that we are using stricter definition of revealed preference relation. Our result can be generalized to the standard definition (as one in Afriat (1967)) simply by adding one step to the proof which will eliminate inconsistency between definitions of revealed preference relation and general preference relation
10
S R {(x, y); (y, x)}. Let us show that R0 T (R0 ). It is clear that R0 ✓ T (R0 ). Therefore, by Lemma 1 from Demuynck (2009), we only need to show that T (R0 ) \ P 1 (R0 ) = ;. Assume, on the contrary, that there are elements z and w for which (z, w) 2 P (R0 ) and (w, z) 2 T (R0 ). From the definition of T , we know that there is a sequence s = {si }i=1,...,k , s.t. s1 = x, sk = y and 8i = 1, .., k 1 then (si , si+1 ) 2 R. Note that (z, w) 6= (x, y) and (z, w) 6= (x, y). Consider (z, w)-irreducible length sequence, then without loss of generality there is an i such that (si , si+1 ) = (x, y). Otherwise, we would have that (w, z) 2 T (R) = R, contradicting (z, w) 2 P (R0 ). Moreover, there is no j such that (sj , sj+1 ) = (y, x), otherwise sequence will not be the shortest. Let l be the highest integer such that (sl 1 , sl ) = (x, y) and let f be the smallest integer such that (sf , sf +1 ) = (x, y). Construct the sequence s0 = sl , sl+1 , ..., sk , s1 , ..., sf 1 , sf . If we apply the definition of T to this sequence, we have that (y, x) 2 T (R) = R, a contradiction. Now let us show that T is separability-preserving, i.e. if R is Z-separable, then T should also be Z-separable. If (x, y) 2 P (T (R)), then there is a chain S = s1 , . . . , sn , such that (sj , sj+1 ) 2 R for any j = 1, . . . , n 1 and at least for one k (sk , sk+1 ) 2 P (R), otherwise (x, y) 2 I(R). So, for (sk , sk+1 ) there is z 2 Z such that (sk , z) 2 R and (z, sk+1 ) 2 R and from transitivity of T (R) we know that (x, z) 2 T (R) and (z, y) 2 T (R). Hence T (R) is Z-separable and T is separability-preserving And since the consumption experiment is finite Z is finite as well, hence, (X, Rv ) is order-separable. The following Corollary is immediate. Corollary 3. A revealed preference relation Rv generated by a finite consumption experiment has a complete and transitive extension, that can be represented by a utility function if and only if Rv satisfied GARP. A modified version of Proposition 1 can be applied for transitive and monotone closure, just by defining Z = C [ Q. It will grant us a monotone utility function that represents the revealed preference relation. Corollary 3 is a version of a classical result dating back to Afriat (1967) and it is an example of the application of the results we shown above. Hence, the generalized technique we propose can be used to construct other revealed preference tests.
References Afriat, S.N., 1967. The construction of utility functions from expenditure data. International economic review 8, 67–77. Bosi, G., Mehta, G.B., 2002. Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof. Journal of Mathematical Economics 38, 311–328. 11
Cantor, G., 1895. Beitr¨age zur begr¨ undung der transfiniten mengenlehre. Mathematische Annalen 46, 481–512. Crawford, I., 2010. Habits revealed. The Review of Economic Studies 77, 1382–1402. Debreu, G., 1954. Representation of a preference ordering by a numerical function. Decision processes 3, 159–165. Demuynck, T., 2009. A general extension result with applications to convexity, homotheticity and monotonicity. Mathematical Social Sciences 57, 96–109. Diewert, W.E., Parkan, C., 1985. Tests for the consistency of consumer data. Journal of Econometrics 30, 127–147. Echenique, F., Saito, K., 2013. Savage in the market. Econometrica . ¨ Ok, E.A., 2011. On the multi-utility representation of preference relations. Journal Evren, O., of Mathematical Economics 47, 554–563. Forges, F., Minelli, E., 2009. Afriat’s theorem for general budget sets. Journal of Economic Theory 144, 135–145. Frisch, R., 1926. Sur un probl`eme d’´economie pure. Grøndahl & søns boktrykkeri. Heufer, J., 2012. Revealed preference and nonparametric analysis–continuous extensions and recoverability. Available at SSRN 2030705 . Jaffray, J.Y., 1975. Semicontinuous extension of a partial order. Journal of Mathematical Economics 2, 395–406. Kannai, Y., 1977. Concavifiability and constructions of concave utility functions. Journal of mathematical Economics 4, 1–56. Kreps, D.M., 2012. Microeconomic foundations I: choice and competitive markets. volume 1. Princeton University Press. Mas-Colell, A., Whinston, M.D., Green, J.R., et al., 1995. Microeconomic theory. volume 1. Oxford university press New York. Matzkin, R.L., 1991. Axioms of revealed preference for nonlinear choice sets. Econometrica: Journal of the Econometric Society , 1779–1786. Matzkin, R.L., Richter, M.K., 1991. Testing strictly concave rationality. Journal of Economic Theory 53, 287–303. Mehta, G.B., 1997. A remark on a utility representation theorem of rader. Economic Theory 9, 367–370. 12
Ok, E.A., 2002. Utility representation of an incomplete preference relation. Journal of Economic Theory 104, 429–449. Peleg, B., 1970. Utility functions for partially ordered topological spaces. Econometrica: Journal of the Econometric Society , 93–96. Polisson, M., Quah, J.K.H., Renou, L., 2015. Revealed preferences over risk and uncertainty. Institute for Fiscal Studies . Rader, T., 1963. The existence of a utility function to represent preferences. The Review of Economic Studies , 229–232. Richter, M.K., 1966. Revealed preference theory. Econometrica: Journal of the Econometric Society , 635–645. Samuelson, P.A., 1938. A note on the pure theory of consumer’s behaviour. Economica , 61–71. Szpilrajn, E., 1930. Sur l’extension de l’ordre partiel. Fundamenta mathematicae 1, 386–389. Varian, H.R., 1983. Non-parametric tests of consumer behaviour. The review of economic studies 50, 99–110. Von Neumann, J., Morgenstern, O., 1947. Theory of games and economic behavior (2d rev . ´ ements d’´economie politique pure; ou, Th´eorie de la richesse sociale. F. Walras, L., 1896. El´ Rouge.
13
Discussion Paper
Interdisciplinary Center for Economic Science 4400 University Drive, MSN 1B2, Fairfax, VA 22030 Tel: +1-703-993-4719 Fax: +1-703-993-4851 ICES Website: http://ices.gmu.edu ICES RePEc Archive Online at: http://edirc.repec.org/data/icgmuus.html
General Revealed Preferences C´esar Martinelli⇤1 and Mikhail Freer†1 1
Interdisciplinary Center for Economic Science, George Mason University February 2016
Abstract Following Richter (1966), we provide criteria under which a preference relation implied by a finite set of choice observations has a complete extension that can in turn be represented by a utility function. The usual properties of the original preference relation are inherited by the preference extension represented by the utility function. Noteworthy, our result relaxes the usual assumptions about the structure of budgets generating the observed choices. Against common wisdom, we show that (upper semi-)continuity does not come for free in general and it requires additional assumptions.
1
Introduction
The notion of rational behavior is central in economics. Historically there have been three ways to specify rational behavior. The first, introduced by Walras (1896), assumes that agents maximize some objective (utility) function. The second strand started back with Frisch (1926) and relies on the notion of preferences as a formal model of rational behavior. This approach was later developed and popularized by Debreu (1954) and Von Neumann and Morgenstern (1947). The third strand, starting with Samuelson (1938), and uses conditions on (finite sets of) observed choices to describe rational behavior. The general connection between utility functions and preference relation was originally studied1 by Debreu (1954) in the context of continuous utility functions. Rader (1963) and Jaffray (1975) relaxed the assumption of continuity and obtained semi-continuous utility rationalization results that were generalized by Bosi and Mehta (2002). Peleg (1970) shown the sufficient condition for existence of a continuous utility representation for incomplete Email adress: [email protected] Email adress: [email protected] 1 Earlier, utility representation theorems under certain additional assumptions were provided by Cantor (1895) and Von Neumann and Morgenstern (1947) ⇤ †
1
Preference Relation Frisch (1926) Szpilrajn (1930), . . .
Set of choices Samuelson (1938)
Debreu (1954), . . .
Afriat (1967), . . .
Utility Function Walras (1896)
Figure 1: Models of Rational Behavior preference relation. More recently Ok (2002), Evren and Ok (2011) have investigated a problem of existence of a vector-valued utility representation of preference relations. The basic result connecting the set of choices and preference relations was proven by Szpilrajn (1930). Szpilrajn shown that any acyclic preference relation has a complete and transitive extension. Demuynck (2009) generalized the result by providing a condition to test for existence of complete extension that has properties usually assumed by economists. As it is well-known, lexicographic preferences illustrate that not every complete and transitive relation can be represented by a utility function. The connection between finite data sets and the utility functions was originally investigated by Afriat (1967). Subsequent literature has constructed tests for consistency of the finite consumption data with various utility maximization hypothesis. Kannai (1977), Matzkin (1991), Matzkin and Richter (1991) and Forges and Minelli (2009) address the question of testing concavity of the utility representation. Varian (1983), Diewert and Parkan (1985), Echenique and Saito (2013) and Polisson et al. (2015) develop tests for separable utility representations including the context of choices under uncertainty. Crawford (2010) propose a test for habit formation models. In this paper we connect these three models of rational behavior in a parsimonious way. That is, we propose conditions under which a finite set of choices can be extended by a preference relation that in turn can be represented by a utility function. We generalize a classical result from Richter (1966), allowing a complete extension to satisfy properties deemed as desirable by economists. We show that, in our generalized framework, existence 2
of a utility function does not guarantee the existence of (upper semi-)continuous utility function. The remainder of this paper is organized as follows. In Section 2 we introduce basic definitions. In Section 3 we provide the criteria for the existence of a complete extension that can be represented by an (upper semi-)continuous utility function. In the Section 4, we illustrate our results by providing an example of application of our result to construct a revealed preference test.
2
Preliminaries
Consider a universal set X ✓ RN of alternatives. A set R ✓ X ⇥ X is said to be a preference relation. We denote a set of all preference relations on X by R. We denote the reverse relation R 1 = {(x, y)|(y, x) 2 R}. We denote the symmetric (indifferent) part of R by I(R) = R \ R 1 and the asymmetric (strict) part by P (R) = R \ I(R). We denote the incomparable part by N (R) = X ⇥ X \ (R [ R 1 ).
2.1
Preference Relations
A preference relation R is said to be • complete if (x, y) 2 R [ R
1
for all x, y 2 X (or equivalently N (R) = ;).
• transitive if (x, y) 2 R and (y, z) 2 R implies (x, z) 2 R for all x, y, z 2 X. • reflexive if for every x 2 X, (x, x) 2 R. • antisymmetric if (x, y) 2 R and (y, x) 2 R implies x = y for all x, y 2 X. • monotone2 if for any x, y 2 X x
y implies (x, y) 2 R.
The set LP (R) (x) = {y|(x, y) 2 P (R)} is called lower contour set of x. The set UP (R) (y) = {x|(x, y) 2 P (R)} is called upper contour set of y. A preference relation R is said to be upper semi-continuous if LP (R) (x) for all x 2 X are open. A preference relation R is said to be continuous if LP (R) (x) and UP (R) (x) for all x 2 X are open (or equivalently3 if LR (x) and UR (x) are closed). A tuple (X, R) where X is a set of alternatives and R is preference relation is said to be partially ordered set (or shortly poset) if R is transitive, antisymmetric and reflexive (usually called partial order). A (X, R) is said to be completely ordered set if R is transitive and complete. Note that any completely ordered set is also poset. 2 3
We use a definition that x y, if xi > yi for all i 2 {1, ..., N }. Definitions are equivalent for a complete preference relation on X ⇥ X, see e.g. Kreps (2012).
3
Definition 1. A tuple (X, R) is said to be Z-separable for given Z ✓ X if for any (x, y) 2 P (R) there is z 2 Z, such that (x, z) 2 R and (z, y) 2 R. A tuple (X, R) is said to be order separable if there is at most countable Z such that (X, R) is Z-separable. Example. Denote by L ✓ RN ⇥ RN the lexicographic preference relation, i.e. (x, y) 2 L if there is k 2 {1, . . . , L} such that xi = yi for i < k and xk > yk . It is well-known that this relation is not order-separable (see e.g. Mas-Colell et al. (1995)). However, L is RN separable. Definition 2. A preference relation R0 is an extension of R, denoted R and P (R) ✓ P (R0 ).
R0 if R ✓ R0
The straight-forward example of extending a preference relation is taking a transitive closure.
2.2
Functions over Preference Relations
The transitive closure is natural example of a function over the set of preference relations. Denote it by T : R ! R, where (x, y) 2 T (R) if there is a sequence S = s1 , s2 , . . . sn , such that for any j = 1..n 1, (sj , sj+1 ) 2 R and s1 = x and sn = y. The transitive closure is monotone, i.e. the transitive closure of a subset of the relation is the subset of closure of the original relation. It is closed, i.e. any relation is a subset of its transitive closure. It is idempotent in the sense that the transitive closure of transitive relation is equal to the original relation. It is also compact-valued (or algebraic) in the sense that to obtain any element from transitive closure it is enough apply the transitive closure to a finite subset of original relation. The following definition specifies the family of functions that have the properties we just listed. Definition 3. A function F : R ! R is said to be an algebraic closure if (1) For all R, R0 2 R, if R ✓ R0 , then F (R) ✓ F (R0 ), (2) For all R 2 R, R ✓ F (R), (3) For all R 2 R, F (F (R)) ✓ F (R), (4) For all R 2 R and all (x, y) 2 F (R), there is a finite relation R0 ✓ R, s.t. (x, y) 2 F (R0 ). Conditions (1)-(3) define the closure, i.e. an operator that is monotone (1), closed (2) and idempotent (3) and condition (4) makes the closure algebraic. Remark. There is no algebraic closure that implies continuity. Denote continuity closure by C : R ! R, when (x, y) 2 C(R) if there is a sequence (xn , yn ) 2 R, such that xn ! x 4
and yn ! y. Let us construct a violation of (4). Let X = R and R = {( n1 , 1) : n 2 N}, then (0, 1) 2 C(R). However, there is no finite sub-relation R0 , such that (0, 1) 2 C(R0 ). For any function F : R ! R, let R⇤F = {R 2 R|R F (R)}. For any function F : R ! R and Z ✓ X, let RZF be a set of all Z-separable R, such that R F (R). Note that RZF ✓ R⇤F . Example. Let X = RN and Z = ;, then a set R⇤T (recall that T stays for transitive closure) will include all acyclic preference relations. The set R;T will consists only of relations R such that P (R) = ;, i.e. only equivalence relations on X. ˆ F ✓ R⇤ the set of all upper semi-continuous R : R F (R). Denote by R F Definition 4. A function F : R ! R is said to be separability-preserving if for Zseparable R, F (R) is also Z-separable. Note that separability preserving requires F (R) to be separable with respect to the same set as R. Moreover, transitive closure is an example of separability-preserving function. Recall that by definition for any (x, y) 2 T (R) there is a sequence S = s1 , s2 , . . . sn , such that for any j = 1..n 1, (sj , sj+1 ) 2 R and s1 = x and sn = y. It implies that for any (x, y) 2 P (T (R)) there is k, such that (sk , sk+1 ) 2 P (R). Hence, there is z 2 Z such that (x, z), (z, y) 2 T (R). The next condition is a slight variation of C7 from Demuynck (2009). Definition 5. A function F : R ! R is said to be expansive if for any R = F (R) and N (R) 6= ;, there is T ✓ N (R) such that R [ T 2 R⇤F and P (R) = P (R [ T ). Expansiveness of F means that we can add some indifferent points to any R = F (R), such that the new relation will be in RF⇤ . We claim that transitive closure is expansive, since it is possible to add some indifferent points such that the new relation is acyclic. We present a formal proof of the claim in Section 4. A function F : R ! R is said to be upper semi-continuous if the image of upper semi-continuous R is upper semi-continuous A function F : R ! R is said to be continuous if the image of continuous R is continuous In what follows, we will be using the fact that closure induces some properties. A function F induces property f (e.g. monotonicity, transitivity, convexity, etc) if for any fixed point4 R = F (R), R satisfies property f . Definition 6. A function F : R ! R is said to be a rational closure if it is an expansive algebraic closure that induces transitivity. Definition 7. A preference relation is said to be F-consistent if F (R) \ P
1
(R) = ;.
Note that F -consistency is the condition that guarantees that F (R) is an extension of R. From the definition of we know that R F (R), then P (R) ✓ P (F (R)). For R to 4
The fixed point approach for this purpose was originally used by Szpilrajn (1930) and implicitely by Richter (1966).
5
be extendable by F we need to show that F (R) does not contain elements from P 1 (R). Otherwise there will be some (x, y) 2 P (R) and (x, y) 2 / P (F (R)), which would imply, that R F (R). Example. Let the set of alternatives be X = {x1 , x2 , x3 } and preference relation 0 R = {(x1 , x2 ); (x2 , x3 ); (x3 , x1 )}. One can easily see that this relation is not transitive and is not T -consistent because (x1 , x3 ) 2 T (R) and (x3 , x1 ) 2 P (R). On other hand R = {(x1 , x2 ); (x2 , x3 )} from the very first example can is T -consistent. Moreover, T (R) is complete on X.
3
Results
The following theorem provides a convenient criteria for the existence of a utility function that rationalizes the complete extension of a given preference relation and preserves the properties induced by F . We do it showing the existence of a complete relation that is a fixed point of F . The separability-preserving property allows the fixed point to be order separable. Hence, the complete fixed point can be represented by a utility function. Theorem 1. Let F be a separability-preserving rational closure. An order-separable R 2 R has a complete extension R⇤ = F (R⇤ ) that can be represented by a utility function if and only if R is F-consistent. To prove Theorem 1 we need several supplementary results.
3.1
Lemmata
Let us start from Lemma 1 from Demuynck (2009), which shows that F -consistency is necessary and sufficient condition for F (R) to be an extension of R. Lemma 1 (Lemma 1 from Demuynck (2009)). For any function F : R ! R, R ✓ F (R). Then in order for R F (R) it is necessary and sufficient that F (R) \ P 1 (R) = ;. Expansiveness of F implies Demuynck (2009) condition C7, hence, following result will also hold: Lemma 2 (Lemma 4 from Demuynck (2009)). If F : R ! R is a rational closure, then: S (5) For every chain R0 ⇢ R1 ⇢ ... ⇢ R↵ ⇢ ..., where for all ↵ R↵ 2 R⇤F we have ↵ 0 R↵ is also in R⇤F .
Condition (5) is a technical restriction that allows us to apply Zorn’s lemma and show that the complete extension actually has the properties induced by F . In the case of finite sequences (5) is straight-forward, thus for finite sets of alternatives it is not binding, while in the case of infinite sets of alternatives it is restrictive. In order to show that there is a utility function that represents the complete extension we need to restate condition (5) in a more convenient form for the further proof. 6
Lemma 3. If F is a separability-preserving rational closure, then: S (6) For any chain R0 ✓ R1 ✓ ... ✓ R↵ ✓ ... in RZF we have ↵ 0 R↵ 2 RZF .
(7) For every R 2 RZF such that N (R) 6= ; there is a non-empty subset T of N (R) such that R [ T 2 RZF .
Proof. Let us start from showing that (6) is satisfied, i.e. For any chain R0 R1 ... S Z Z R↵ ... in RF then B = ↵ 0 R↵ 2 RF . Since F is rational closure we know that B 2 R⇤F . So, if there is (x, y) 2 P (B), then there is ↵ 0 such that (x, y) 2 R↵ , then there is z 2 Z, such that (x, z) 2 R↵ ✓ B and (z, y) 2 R↵ ✓ B. Not let us show that (7) is satisfied. Firstly, consider the case when R 6= F (R). Since F is separability-preserving rational closure, then if R 2 RZF , then T = F (R) \ R and R [ T is F consistent (because F is rational closure) and is Z-separable (F is separability-preserving) hence, R [ T 2 RZF . If R = F (R) and F is expansive we can add T such that R [ T is F -consistent and P (R [ T ) = P (R) hence it is also Z-separable. Thus R [ T 2 RZF . In order to get upper semi-continuous function we need to modify conditions (6) and (7) in continuity terms. Lemma 4. If F is an upper semi-continuous rational closure, then: (6’) For any chain R0
R1
...
R↵
ˆ F we have S ... of relations in R ↵
0
ˆF. R↵ 2 R
ˆ F such that N (R) 6= ; there is a non-empty subset T of N (R) such (7’) For every R 2 R ˆF. that R [ T 2 R Condition (6’) shows that if there is a chain of upper semi-continuous F-consistent relations, such that each element is extension of a previous one, then the upper bound of this sequence is also upper semi-continuous and F-consistent. Condition (7) tells us that each F-consistent upper semi-continuous preference relation can be extended by another F-consistent upper semi-continuous relation. Proof. Let us start from showing, that a rational closure satisfies (6’). Consider the relation S B = ↵ 0 R↵ . From (5) we know that it is in R⇤F , so in order to prove that (6’) holds it is enough to show that B is upper semi-continuous, i.e. for all x 2 X, LP (B) (x) are open. S Since Ra Rb , for any a b, then P (Ra ) ✓ P (Rb ), thus for any x LP (B) (x) = ↵ 0 LP (R↵ ) is a union of open sets, thus it is open as well. Now we will show, that a rational closure satisfies (7’). To show that F satisfies (7’) we need to consider two cases. First assume that R 6= F (R). Then let T = F (R) \ R, then R [ T = F (R) is upper semi-continuous, because R is an upper semi-continuous and F is upper semi-continuous. Now suppose instead that R = F (R) and we are adding only points that are indifferent, thus do not affect lower contour sets of R. Hence, the extension of it is upper semi-continuous. 7
Note that similar lemma can be obtained for the set of continuous relations. In order to prove Theorem 1 we need also the classical result from Debreu (1954). Lemma 5 (Lemma 2, from Debreu (1954)). Let (X, R) be a separable completely ordered set, then there is a utility function that represents R.
3.2
Main Result
Proof of Theorem 1. Let ⌦ be the set {R0 2 RZF |R R0 }. We know, that for any chain Z ˆF, B = S R0 ✓ R1 ✓ ... ✓ R↵ ✓ ... of relations in R ↵ 0 R↵ 2 RF . So, we are left to show that R B. Clearly, R ✓ B. If on the contrary there are elements x, y 2 X, for which (x, y) 2 P (R) and (y, x) 2 B, we have that there must be a relation R↵ in the chain for which (y, x) 2 R↵ . This contradicts the fact that R R↵ and we conclude that B 2 ⌦. Clearly (⌦, ✓) is a partially ordered set. So applying Zorn’s lemma we get that ⌦ has a maximal element. Let R⇤ be that maximal element. We omit the part of the proof that shows that R⇤ is complete and that R⇤ = F (R⇤ ) since it is similar to the proof of Lemma 2 in Demuynck (2009). We are left to show that there is a utility function that represents R⇤ = F (R⇤ ), since F is rational closure we know that R⇤ is transitive and complete. Hence we are left to show that R⇤ is order separable. R⇤ 2 RZF , hence, it is Z-separable with the same Z as original R which is order separable, R⇤ is order separable, because it is Z-separable and Z is no more than countable. Remark: One can consider this as an infinite construction algorithm. The algorithm is quite simple: (1) if R 6= F (R) use F (R) to add points and impose necessary properties; (2) if F (R) = R add indifference points (since we see that F (R) adds mainly strictly preferred bundles). In this sense Theorem 1 can be perceived as a proof that the algorithm converges. Construction Procedure is illustrated in Figure 2. Assume that purple area (open ball) is LP (R) (x). Then applying F (R) will extend the lower contour set and it will be the triangle without upper boundary. Then by completing relation R[T we will add "indifference curve" that is upper boundary of the triangle. It was shown by Rader (1963), that any upper semi-continuous relation, that can be represented by utility function, has upper semi-continuous utility representation.5 Hence we can provide a conditions under which preference relation have a complete extension that can be represented by an upper semi-continuous function. Note that is transitive, reflexive and antisymmetric relation, hence is a partial order. 5
We are referring to the theorem from Rader (1963) which states that a complete, transitive, upper semicontinuous relation can be represented by an upper semi-continuous utility function. It was mentioned by Mehta (1997) that the original Rader’s proof was incomplete, but the result was finally proven by Bosi and Mehta (2002).
8
x
x
R
)
x
F (R)
)
F (R) [ T
Figure 2: Illustration for construction algorithm Corollary 1. Let F be an upper semi-continuous rational closure. Then an upper semicontinuous R 2 R has a complete extension R⇤ = F (R⇤ ) that can be represented by an upper semi-continuous utility function if and only if R is F-consistent. ˆ F |R R0 }. We know, that for any chain R0 R1 ... Proof. Let ⌦ be the set {R0 2 R ˆF, B = S ˆ R↵ ... of relations in R B. ↵ 0 R↵ 2 RF . So, we are left to show that R Clearly, R ✓ B. If on the contrary there are elements x, y 2 X, for which (x, y) 2 P (R) and (y, x) 2 B, we have that there must be a relation R↵ in the chain for which (y, x) 2 R↵ . This contradicts the fact that R R↵ and we conclude that B 2 ⌦. Clearly (⌦, ) is a partially ordered set. So applying Zorn’s lemma we get that ⌦ has a maximal element. Let R⇤ be that maximal element. We omit the part of the proof that shows that R⇤ is complete and that R⇤ = F (R⇤ ) since it is similar to the proof of Lemma 2 in Demuynck (2009). In order to show that there is an upper semi-continuous utility function, that represents ⇤ ˆ F hence is also upper-semicontinuous, and because R⇤ = F (R⇤ ) R . We know that R⇤ 2 R it is transitive and complete, so by Rader (1963) there is an upper semi-continuous utility function that represents R⇤ . An upper semi-continuous function is enough for observed choices to be the points at which the utility function reaches supremum if the budgets are compact. However, it is common to assume that utility functions are continuous. The theorem from Debreu (1954) shows that any complete, transitive and continuous relation can be represented by a continuous utility function. Corollary 2. Let F be a continuous rational closure. Then a continuous R 2 R has a complete extension R⇤ = F (R⇤ ) that can be represented by a continuous utility function if and only if R is F-consistent
9
The proof of this Corollary is straight-forward, since we assume that R is continuous, then ¯ F as the set of all F-consistent continuous preference relations. Then Lemma one can define R ˆ F by R ¯ F . Hence, the complete extension of R will also be 4 will hold just by replacing R ¯ F . Since R⇤ = F (R⇤ ) is continuous, complete continuous, because it will be an element of R and transitive, then by Debreu (1954) theorem, there is a continuous utility function, that represents R⇤ .
4
Example
Let us illustrate how our results can be applied to construct the tests for existence of utility function that preserves certain properties. As an illustration we will use a classical result that shows existence of a utility function that rationalizes acyclic preference relation.6 We will use the transitivity closure T : R ! R already defined. Define revealed preference relation7 Rv obtained from a finite consumption experiment (xt , B t )Tt=1 , where xt are chosen points and B t are budget sets (xt , y) 2 Rv if y 2 B t \ {xt }. Denote by C = {xt }Tt=1 the set of all chosen points. Proposition 1. A revealed preference relation Rv obtained from a finite consumption experiment has a complete and transitive extension, that can be represented by a utility function if and only if Rv is T -consistent. To prove Proposition 1 we need to show that T is separability-preserving rational closure and that (X, Rv ) is order-separable in order to apply Theorem 1. So, let us start from proving properties of T . We omit the part of proof that T is an algebraic closure that induces transitivity, since it is in Demuynck (2009). We are left to prove that T is expansive and separability-preserving. For this purpose let us define the (x, y)-irreducible length sequence for any (x, y) 2 T (R) as a sequence S = s1 , ..., sn such that s1 = x; sn = y and (sj , sj + 1) 2 R, and there is no shorter sequence S 0 = s01 , . . . , sn0 such that s01 = x; s0n0 = y and (s0j , s0j + 1) 2 R. And from definition of T we know that there always is (x, y)-irreducible length sequence and that all sj are distinct elements, i.e. there is no i 6= j such that si = sj otherwise the length can be reduced. Proof. Let us start from proving that T is expansive. Consider a relation R = T (R) and assume that N (R) 6= ;. Take an element (x, y) 2 N (R) and consider the relation R0 = 6
Under more restrictive assumptions using other techniques it is possible to get stronger result. Existence of continuous utility representation on linear budgets was show by Afriat (1967). Existence of continuous utility representation on compact, monotone and balanced budgets was shown by Forges and Minelli (2009). Existence of continuous utility representation on compact budgets was shown by Heufer (2012) 7 Note that we are using stricter definition of revealed preference relation. Our result can be generalized to the standard definition (as one in Afriat (1967)) simply by adding one step to the proof which will eliminate inconsistency between definitions of revealed preference relation and general preference relation
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S R {(x, y); (y, x)}. Let us show that R0 T (R0 ). It is clear that R0 ✓ T (R0 ). Therefore, by Lemma 1 from Demuynck (2009), we only need to show that T (R0 ) \ P 1 (R0 ) = ;. Assume, on the contrary, that there are elements z and w for which (z, w) 2 P (R0 ) and (w, z) 2 T (R0 ). From the definition of T , we know that there is a sequence s = {si }i=1,...,k , s.t. s1 = x, sk = y and 8i = 1, .., k 1 then (si , si+1 ) 2 R. Note that (z, w) 6= (x, y) and (z, w) 6= (x, y). Consider (z, w)-irreducible length sequence, then without loss of generality there is an i such that (si , si+1 ) = (x, y). Otherwise, we would have that (w, z) 2 T (R) = R, contradicting (z, w) 2 P (R0 ). Moreover, there is no j such that (sj , sj+1 ) = (y, x), otherwise sequence will not be the shortest. Let l be the highest integer such that (sl 1 , sl ) = (x, y) and let f be the smallest integer such that (sf , sf +1 ) = (x, y). Construct the sequence s0 = sl , sl+1 , ..., sk , s1 , ..., sf 1 , sf . If we apply the definition of T to this sequence, we have that (y, x) 2 T (R) = R, a contradiction. Now let us show that T is separability-preserving, i.e. if R is Z-separable, then T should also be Z-separable. If (x, y) 2 P (T (R)), then there is a chain S = s1 , . . . , sn , such that (sj , sj+1 ) 2 R for any j = 1, . . . , n 1 and at least for one k (sk , sk+1 ) 2 P (R), otherwise (x, y) 2 I(R). So, for (sk , sk+1 ) there is z 2 Z such that (sk , z) 2 R and (z, sk+1 ) 2 R and from transitivity of T (R) we know that (x, z) 2 T (R) and (z, y) 2 T (R). Hence T (R) is Z-separable and T is separability-preserving And since the consumption experiment is finite Z is finite as well, hence, (X, Rv ) is order-separable. The following Corollary is immediate. Corollary 3. A revealed preference relation Rv generated by a finite consumption experiment has a complete and transitive extension, that can be represented by a utility function if and only if Rv satisfied GARP. A modified version of Proposition 1 can be applied for transitive and monotone closure, just by defining Z = C [ Q. It will grant us a monotone utility function that represents the revealed preference relation. Corollary 3 is a version of a classical result dating back to Afriat (1967) and it is an example of the application of the results we shown above. Hence, the generalized technique we propose can be used to construct other revealed preference tests.
References Afriat, S.N., 1967. The construction of utility functions from expenditure data. International economic review 8, 67–77. Bosi, G., Mehta, G.B., 2002. Existence of a semicontinuous or continuous utility function: a unified approach and an elementary proof. Journal of Mathematical Economics 38, 311–328. 11
Cantor, G., 1895. Beitr¨age zur begr¨ undung der transfiniten mengenlehre. Mathematische Annalen 46, 481–512. Crawford, I., 2010. Habits revealed. The Review of Economic Studies 77, 1382–1402. Debreu, G., 1954. Representation of a preference ordering by a numerical function. Decision processes 3, 159–165. Demuynck, T., 2009. A general extension result with applications to convexity, homotheticity and monotonicity. Mathematical Social Sciences 57, 96–109. Diewert, W.E., Parkan, C., 1985. Tests for the consistency of consumer data. Journal of Econometrics 30, 127–147. Echenique, F., Saito, K., 2013. Savage in the market. Econometrica . ¨ Ok, E.A., 2011. On the multi-utility representation of preference relations. Journal Evren, O., of Mathematical Economics 47, 554–563. Forges, F., Minelli, E., 2009. Afriat’s theorem for general budget sets. Journal of Economic Theory 144, 135–145. Frisch, R., 1926. Sur un probl`eme d’´economie pure. Grøndahl & søns boktrykkeri. Heufer, J., 2012. Revealed preference and nonparametric analysis–continuous extensions and recoverability. Available at SSRN 2030705 . Jaffray, J.Y., 1975. Semicontinuous extension of a partial order. Journal of Mathematical Economics 2, 395–406. Kannai, Y., 1977. Concavifiability and constructions of concave utility functions. Journal of mathematical Economics 4, 1–56. Kreps, D.M., 2012. Microeconomic foundations I: choice and competitive markets. volume 1. Princeton University Press. Mas-Colell, A., Whinston, M.D., Green, J.R., et al., 1995. Microeconomic theory. volume 1. Oxford university press New York. Matzkin, R.L., 1991. Axioms of revealed preference for nonlinear choice sets. Econometrica: Journal of the Econometric Society , 1779–1786. Matzkin, R.L., Richter, M.K., 1991. Testing strictly concave rationality. Journal of Economic Theory 53, 287–303. Mehta, G.B., 1997. A remark on a utility representation theorem of rader. Economic Theory 9, 367–370. 12
Ok, E.A., 2002. Utility representation of an incomplete preference relation. Journal of Economic Theory 104, 429–449. Peleg, B., 1970. Utility functions for partially ordered topological spaces. Econometrica: Journal of the Econometric Society , 93–96. Polisson, M., Quah, J.K.H., Renou, L., 2015. Revealed preferences over risk and uncertainty. Institute for Fiscal Studies . Rader, T., 1963. The existence of a utility function to represent preferences. The Review of Economic Studies , 229–232. Richter, M.K., 1966. Revealed preference theory. Econometrica: Journal of the Econometric Society , 635–645. Samuelson, P.A., 1938. A note on the pure theory of consumer’s behaviour. Economica , 61–71. Szpilrajn, E., 1930. Sur l’extension de l’ordre partiel. Fundamenta mathematicae 1, 386–389. Varian, H.R., 1983. Non-parametric tests of consumer behaviour. The review of economic studies 50, 99–110. Von Neumann, J., Morgenstern, O., 1947. Theory of games and economic behavior (2d rev . ´ ements d’´economie politique pure; ou, Th´eorie de la richesse sociale. F. Walras, L., 1896. El´ Rouge.
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