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Econometrica, Vol. 83, No. 1 (January, 2015), 175–192 A CHARACTERIZATION OF RATIONALIZABLE CONSUMER BEHAVIOR PHILIP J. RENY University of Chicago, Chicago, IL 60637, U.S.A.

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Econometrica, Vol. 83, No. 1 (January, 2015), 175–192

A CHARACTERIZATION OF RATIONALIZABLE CONSUMER BEHAVIOR BY PHILIP J. RENY1 m For an arbitrary data set D = {(p x)} ⊆ (Rm + \ {0}) × R+ , finite or infinite, it is shown that the following three conditions are equivalent: (a) D satisfies GARP; (b) D can be rationalized by a utility function; (c) D can be rationalized by a utility function that is quasiconcave, nondecreasing, and that strictly increases when all its coordinates strictly increase. Examples of infinite data sets satisfying GARP are provided for which every utility rationalization fails to be lower semicontinuous, upper semicontinuous, or concave. Thus condition (c) cannot be substantively improved upon.

KEYWORDS: Utility representation, integrability, Afriat’s theorem.

1. INTRODUCTION WE REVISIT THE CLASSICAL PROBLEM OF CONSTRUCTING, from a single consumer’s demand data, a utility function rationalizing those data. When such a utility function exists, we say that the data set is rationalizable. We define terms as follows. m DEFINITIONS: A data set is an arbitrary subset D of (Rm + \ {0}) × R+ with generic element (p x). The interpretation is that when the consumer faces nonnegative prices p = 0 with income p · x, he chooses x. Implicit in this is the assumption that the consumer spends all his income. A utility function u : Rm + → R rationalizes the data set D if, for every (p x) ∈ , D and every y ∈ Rm +

p(y − x) ≤ 0

⇒

p(y − x) < 0

⇒

u(y) ≤ u(x) u(y) < u(x)

and 2

A data set, D, satisfies the Generalized Axiom of Revealed Preference (GARP) if, for every finite sequence (p1  x1 )     (pn  xn ) of points from D, p1 (x2 − x1 ) ≤ 0 ⇒

p2 (x3 − x2 ) ≤ 0



pn−1 (xn − xn−1 ) ≤ 0

pn (x1 − xn ) ≥ 0

1 Financial support from National Science Foundation Grants SES-1227506 and SES-0617884 is gratefully acknowledged. I thank Francoise Forges, Sergiu Hart, Andreu Mas-Colell, and Hal Varian for helpful comments, as well as a co-editor and four anonymous referees for very helpful comments that had a substantive and positive impact on the manner in which the material is presented. 2 Requiring strictly affordable bundles to yield strictly less utility rules out trivial rationalizations such as utility functions that are everywhere constant, or that are equal to 1 for all chosen bundles and equal to zero for all other bundles, etc. The literature often imposes instead the slightly more restrictive requirement of locally non-satiated utility. Given our eventual conclusion that rationalizing utility functions can always be chosen so that they strictly increase when all their coordinates strictly increase, our less restrictive requirement serves to strengthen our result.

© 2015 The Econometric Society

DOI: 10.3982/ECTA12345

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The data set satisfies the Strong Axiom of Revealed Preference (SARP) if the first n − 1 weak inequalities in this display always imply that the nth inequality is strict. Afriat (1967) showed that for finite data sets D, satisfaction of GARP is necessary and sufficient for D to be rationalized by a utility function and, moreover, a rationalizing utility function can always be constructed that is strictly increasing, continuous, and concave.3 And it is easy to see that satisfaction of GARP is necessary for an arbitrary data set D to be rationalized by a utility function. But, for arbitrary D, while sufficient conditions have been given,4 no single condition on D has been shown to be necessary and sufficient. This paper fills in this gap. We prove the following. THEOREM 1: For an arbitrary data set D, the following three conditions are equivalent: (a) D satisfies GARP. (b) D can be rationalized by a utility function. (c) D can be rationalized by a utility function that is quasiconcave, nondecreasing, and that strictly increases when all its coordinates strictly increase. The theorem implies, for example, that if a consumer’s yet-to-be-observed demand behavior is inconsistent with the utility maximization hypothesis, there exist some finite number of that consumer’s choices that would violate GARP.5 In terms of the analysis, one might hope that Afriat’s (1967) proof-technique extends to infinite data sets, but it does not.6 In fact, Afriat’s conclusion that a finite data set satisfying GARP is rationalized by a concave utility function is 3

See also Diewert (1973), Varian (1982), Chiappori and Rochet (1987), and Matzkin and Richter (1991). The term “GARP” was coined by Varian (1982). 4 See, for example, Houthakker (1950), Uzawa (1971), Richter (1966), Hurwicz and Richter (1971), Hurwicz and Uzawa (1971), Mas-Colell (1978), Sondermann (1982), Fuchs-Seliger (1983, 1996), and Jackson (1986). In several of these papers it is assumed, as in Samuelson (1938), that the data set is the entire graph of a demand function. 5 We do not wish to suggest that our main result has empirical significance beyond what is already contained in Afriat’s theorem. Empirical data sets are, after all, always finite. The point of our main result is to settle the rationalizability question in its most basic form and to unify the finite and infinite data set approaches to the problem. 6 But see Theorem 1 in Mas-Colell (1978) for an application of Afriat’s theorem to infinite data sets generated by continuous demand functions. Continuity of demand, an important special case, rules out many continuous preference relations (e.g., perfect substitutes and any continuous preference relation that is not strictly convex) that we do not wish to rule out, a priori, here. See Section 5.2.

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false for infinite data sets, which can sometimes be rationalized only by utility functions that are not concave.7 The approach taken here centers around a new and elementary method for constructing a nondecreasing and quasiconcave utility function that rationalizes a given finite data set. The construction resembles Afriat’s (1967) in that it defines utility in terms of sums of income differences. Importantly, however, we restrict these income differences to be nonpositive and we do not require any multipliers to act as weights. Intuitively, renormalizing the data set’s price vectors with multipliers (as in Afriat) to obtain subgradients of a concave utility function is unnecessary to obtain a quasiconcave utility function—it is enough to have a supporting hyperplane (supplied by the price vectors themselves) for the upper contour set at each demanded bundle. Without the need to find appropriate multipliers, our direct and explicit utility construction is considerably simpler than Afriat’s, and it has the additional advantage that it extends from finite to infinite data sets through an appropriate limiting argument.8 The next section contains the new utility construction and presents a simple result on rationalizing finite data sets satisfying GARP as well as a result on maintaining GARP when adding points to a data set. Section 3 contains the proof of Theorem 1 and Section 4 contains examples suggesting that the result cannot be substantively improved upon. Section 5 contains several additional observations, including results on the existence of rationalizing utility functions with additional properties—for example, unique maximizers over the data’s budget sets (Proposition 4) and upper semicontinuity (Proposition 5)—as well as a result for nonlinear budget sets (Proposition 6). 2. A UTILITY CONSTRUCTION Fix, throughout the paper, φ : Rm + → R to be any continuous, strictly increasing, quasiconcave function taking values in [0 1].9,10 For any x ∈ Rm + , say that an arbitrarily long but finite sequence (p1  x1 )     m (pn  xn ) of points in (Rm + \ {0}) × R+ is x-feasible if each of the quantities p1 (x − x1 ) p2 (x1 − x2 )     pn (xn−1 − xn ) is nonpositive. 7 See Example 3. See also Kannai (2004) and Apartsin and Kannai (2006) who provided conditions on a data set that are stronger than GARP and that are necessary for the existence of a concave rationalization. Levin (2005) provided necessary and sufficient conditions on a data set for the existence of a rationalizing utility function that is homogeneous of degree 1, and showed that under these conditions, which are stronger than GARP, utility can also be taken to be concave. 8 Fostel, Scarf, and Todd (2004) provided two simple proofs of Afriat’s theorem. Both proofs simplify the argument for the existence of a solution to the finite system of inequalities that is central to Afriat’s proof. Consequently, neither proof extends to infinite data sets. 9 For example, let φ(x) = 1 − e−1·x , where 1 · x is the sum of the coordinates of x. 10 Recall that a function f : Rm + → R is strictly increasing if f (x) > f (y) whenever x ≥ y and x = y.

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For any finite data set, F , and any x ∈ Rm + , define uF (x) as follows: • If p (x − x ) ≤ 0 for some (p  x ) ∈ F , then   (1) uF (x) := inf p1 (x − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn )  where the infimum is taken over all x-feasible sequences of points in F , that is, over all finite sequences (p1  x1 )     (pn  xn ) of points in F such that each term p1 (x − x1 ) p2 (x1 − x2 )     pn (xn−1 − xn ) appearing in the sum (1) is nonpositive. • If p (x − x ) > 0 for every (p  x ) ∈ F , then uF (x) := φ(x). The quantity uF (x) is well-defined because the set of x-feasible sequences of points in F is nonempty if and only if some (p  x ) ∈ F satisfies p (x − x ) ≤ 0. Indeed, any such (p  x ) is an x-feasible sequence of length 1, and the first term (p1  x1 ) in any x-feasible sequence satisfies p1 (x − x1 ) ≤ 0. REMARK 1: (a) Since there is no bound in (1) on the sequence length n, it can happen that uF (x) = −∞. On the other hand, because uF (x) either is the sum of nonpositive terms or is equal to φ(x) ∈ [0 1], uF (x) ≤ φ(x) ≤ 1 for every x. (b) Note that uF is not the pointwise infimum of a fixed collection of functions, because the collection of functions over which the infimum in (1) is taken depends upon x. Our first result shows that if F is a finite data set satisfying GARP, then uF is a utility function that rationalizes F , a conclusion that will be particularly useful in the sequel.11 Call a nondecreasing utility function u semi-strictly increasing if u(x) > u(y) whenever x  y, that is, whenever each coordinate of x is strictly greater than the corresponding coordinate of y. For example, the utility function for perfect complements, u(x) = min(x(1)     x(m)) where x(k) denotes the kth coordinate of x, is semi-strictly increasing. But it is not strictly increasing because increasing only one coordinate need not increase utility. PROPOSITION 2: If a finite data set, F , satisfies GARP, then uF (x) > −∞ for m every x ∈ Rm + . Moreover, uF : R+ → (−∞ 1] rationalizes F and is semi-strictly increasing, quasiconcave, and lower semicontinuous. PROOF: We first note the following: (2)

11

For any x ∈ Rm + , whenever p (x − x ) ≤ 0 for some (p  x ) ∈ F, the infimum in (1) can be achieved with an x-feasible sequence of distinct points in F.

A version of this result was first reported as Exercise 2.12 in Jehle and Reny (2011).

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Indeed, if in any x-feasible sequence some point appears more than once, GARP implies that the consecutive terms, pk (xk−1 − xk ), involved in the resulting cycle are all zero. Hence, these terms can be eliminated without affecting the overall sum on the right-hand side of (1) or x-feasibility. An obvious implication of (2) is that uF (x) is finite and nonpositive if p (x − x ) ≤ 0 for some (p  x ) ∈ F , and otherwise uF (x) = φ(x) ∈ [0 1]. Hence, uF : Rm + → (−∞ 1]. To see that uF is lower semicontinuous, suppose that xk →k x and that uF (xk ) converges (possibly to −∞). We must show that limk uF (xk ) ≥ uF (x). Without loss of generality (i.e., by considering a subsequence), either (i) there exists (p  x ) ∈ F such that p (xk − x ) ≤ 0 for all k, or (ii) p (xk − x ) > 0 for every (p  x ) ∈ F and every k. If (i) holds, then, because there are only finitely many sequences of distinct points in F , (2) implies that, without loss of generality (i.e., by considering a further subsequence), there is a fixed finite sequence of distinct points (p1  x1 )     (pn  xn ) in F that is xk -feasible for every k and such that uF (xk ) = p1 (xk − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) for every k. In particular, p1 (x − x1 ) = limk p1 (xk − x1 ) ≤ 0. Consequently, this fixed sequence of distinct points is x-feasible and uF (x) ≤ p1 (x − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn )   = lim p1 xk − x1 + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) k   = lim uF xk  k

On the other hand, if (ii) holds, then uF (xk ) = φ(xk ) for every k and so limk uF (xk ) = φ(x) ≥ uF (x) (see Remark 1(a)). Hence, uF is lower semicontinuous. To see that uF is semi-strictly increasing, suppose that x ≥ y. There are two cases. Either uF (x) = φ(x), in which case uF (x) = φ(x) ≥ φ(y) ≥ uF (y), with φ(x) > φ(y) if x  y; or uF (x) = p1 (x − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) for some x-feasible sequence (p1  x1 )     (pn  xn ), in which case uF (x) = p1 (x − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) ≥ p1 (y − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn )

a strict inequality if x  y

≥ uF (y) where the last inequality follows because x ≥ y implies that (p1  x1 )     (pn  xn ) is also y-feasible. In either case, uF (x) ≥ uF (y), with the inequality strict if x  y, and so uF is semi-strictly increasing. To see that uF is quasiconcave, let z = λx + (1 − λ)y. If uF (z) = φ(z), then uF (z) = φ(z) ≥ min(φ(x) φ(y)) ≥ min(uF (x) uF (y)). Otherwise, uF (z) = p1 (z − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) for some z-feasible sequence (p1  x1 )     (pn  xn ), where we may suppose without loss of generality that

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p1 x ≤ p1 y and so p1 x ≤ p1 z. But then, uF (z) = p1 (z − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) ≥ p1 (x − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) ≥ uF (x) where the last inequality follows because p1 x ≤ p1 z     (pn  xn ), being z-feasible, is also x-feasible. In min(uF (x) uF (y)), and so uF is quasiconcave. Finally, to see that uF rationalizes F , suppose p(y − x) ≤ 0. Since the singleton sequence (p x) ∈ F (2) implies that

implies that (p1  x1 ) either case, uF (z) ≥ that (p x) ∈ F and satisfies p(x − x) ≤ 0,

uF (x) = p1 (x − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) for some x-feasible sequence (p1  x1 )     (pn  xn ). But then, because p(y − x) ≤ 0, the sequence (p x) (p1  x1 )     (pn  xn ) is y-feasible and therefore, uF (y) ≤ p(y − x) + p1 (x − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) = p(y − x) + uF (x) Therefore, uF (y) ≤ uF (x), with the inequality strict if p(y − x) < 0.

Q.E.D.

REMARK 2: The proof above establishes that if F is finite and satisfies GARP, then either uF (x) = φ(x) ≥ 0 or uF (x) = p1 (x − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) for some x-feasible sequence (p1  x1 )     (pn  xn ) of distinct points in F , in which  case uF (x) ≥ −(p1 x1 + · · · + pn xn ). In either case, uF (x) ≥ −IF , where IF := (px)∈F px is defined to be the total income of all the data points in F . That is, uF is bounded below by −IF . REMARK 3: If, as in Diewert (1973), we restricted D so that the prices of all goods were always strictly positive, then uF would be strictly increasing (not merely semi-strictly increasing). The next lemma states that if a finite data set satisfies GARP, then to any consumption bundle x with strictly positive coordinates we may associate a price vector p so that when (p x) is added to the data set, the new data set also satisfies GARP.12 LEMMA 3: Let F be a finite data set satisfying GARP and let x0 be any point in m Rm ++ . Then there is a price vector p0 ∈ R+ such that p0 x0 = 1 and F ∪ {(p0  x0 )} satisfies GARP. 12

There need be no such p when the given data set is infinite.

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PROOF: By Proposition 2, uF is quasiconcave and semi-strictly increasing. Hence, the set C = {x : uF (x) ≥ uF (x0 )} is convex with x0 on its boundary. By the separating hyperplane theorem, and since x0  0, there exists p0 ∈ Rm such that p0 x0 = 1 and such that p0 x0 ≤ p0 x for every x ∈ C, implying also that p0 ≥ 0 since (because uF is nondecreasing) x0 + ei ∈ C for every i, where ei is the / C and so uF (y) < uF (x0 ). ith unit vector. In particular, if p0 y < p0 x0 , then y ∈ It suffices now to show that p0 y = p0 x0 implies uF (y) ≤ uF (x0 ), since then uF rationalizes F ∪ {(p0  x0 )}, which must then satisfy GARP. So suppose that p0 y = p0 x0 . Then, for every λ ∈ (0 1), p0 (λy) < p0 x0 (since p0 x0 > 0), and so uF (λy) < uF (x0 ). Because, by Proposition 2, uF is nondecreasing and lower semicontinuous, limλ↑1 uF (λy) = uF (y) and so uF (y) ≤ Q.E.D. uF (x0 ). We are now prepared to prove our main result. 3. PROOF OF THEOREM 1 Since (c) ⇒ (b) ⇒ (a) is clear, it suffices to show (a) ⇒ (c). So, fix a data set D that satisfies GARP. The proof of (c) has two parts. In the first part, we define, for each positive integer k, and for each finite subset F of D, a set UFk of quasiconcave and semi-strictly increasing utility functions that rationalize the finite data set F . GARP is used here (and only here) to show that the sets UFk are nonempty. An important property implied by part (ii) of the definition of UFk is that, for any two points x and y that are coordinatewise strictly ordered, the difference in utility values assigned to x and y by any function in any of the sets UFk for all F and for all k large enough is bounded away from zero. This is critical because in the second part of the proof, the utility function we construct (and that rationalizes D) is a limit of functions in the sets UFk as k tends to infinity and as F increases setwise. The maintained gap in utility values ensures that the limit function is semi-strictly increasing. Let U denote the set of quasiconcave and semi-strictly increasing functions m from Rm + into [−1 1], and let y1  y2     be a dense sequence of points in R++ . m For every positive integer k, and every finite subset F of (R+ \ {0}) × Rm +, define the subset UFk of U as follows, where for any vector x, x(j) denotes its jth coordinate:  UFk := u ∈ U : (i) u rationalizes F and 

 x(j) (ii) u(x) ≤ k max − 1 + u(yk ) j yk (j) 2  m ∀k ≤ k ∀x ∈ R+ s.t. x  yk  1

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Clearly, the set UFk becomes smaller as F and k become larger. That is, if F ⊇ F and k ≥ k , we have UFk ⊆ UF k and so we also have (3)

inf u(x) ≥ inf u(x)

u∈UFk

u∈UF k

for every x ∈ Rm +

Before taking advantage of this useful property, let us show that UFk is nonempty whenever F is a finite subset of D. Let F = {(p 1  x 1 )     (p n  x n )} be any finite subset of D and let k be any positive integer. Since D satisfies GARP, F satisfies GARP. Therefore, since each y1      yk is in Rm ++ , k successive applications of Lemma 3 yields k nonnegative price vectors, q1      qk , such that each qi yi = 1 and the finite set {(q1  y1 )     (qk  yk ) (p 1  x 1 )     (p n  x n )} satisfies GARP. Since GARP is not affected when prices are multiplied by positive constants, the finite set     1 1 q1  y1      k qk  yk  F := 2 2     α1 αn p1  x1      k+n pn  xn 2k+1 2 satisfies GARP, where the αi > 0 are chosen so that αi p i x i ≤ 1 for every i.13 (Note that F need not be a subset of D.) Moreover, IF , the total income of the data points in F (see Remark 2), is no greater than 1 because each qi yi = 1 and each αi p i x i ≤ 1. Consequently, by Proposition 2 and Remark 2, uF : Rm + → [−1 1] is quasiconcave, semi-strictly increasing, and rationalizes the data set F . In particular, uF ∈ U and uF rationalizes F since every data point in F can be obtained by multiplying the price of some data point in F by a positive scalar. Consequently, uF satisfies (i) of the definition of UFk . We next show that uF also satisfies (ii), which implies that UFk is nonempty since uF is then a member. Suppose that k ≤ k and that x ∈ Rm + is such that x  yk . To show that uF satisfies (ii) of the definition of UFk , we must show that   x(j) 1 uF (x) ≤ k max − 1 + uF (yk ) j yk (j) 2 Because ( 21k qk  yk ) is a member of F , uF (yk ) is defined by (1). Hence, as shown in the proof of Proposition 2, uF (yk ) = p1 (yk − x1 ) + p2 (x1 − x2 ) + · · · + pn (xn−1 − xn ) for some yk -feasible sequence (p1  x1 )     (pn  xn ) of points in F . But then x  yk implies that qk (x − yk ) < 0 and that 13 The αi cannot necessarily be chosen so that each αi p i x i = 1 since we allow datapoints with zero income.

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( 21k qk  yk ) (p1  x1 )     (pn  xn ) is x-feasible. Consequently, uF (x) is defined by (1) and uF (x) ≤

= ≤

1

qk (x − yk ) + p1 (yk − x1 ) + p2 (x1 − x2 ) + · · · 2k + pn (xn−1 − xn )

1 2k 1



qk (x − yk ) + uF (yk )

  x(j) − 1 + uF (yk )14 max yk (j) 2k j

Hence, uF ∈ UFk , which establishes that UFk is nonempty. We can now exhibit a utility function that we will show is semi-strictly increasing, quasiconcave, and rationalizes D. Define u∗ : Rm + → [−1 1] by u∗ (x) := sup inf u(x) Fk u∈UFk

where the sup is taken over all positive integers k, and all finite subsets F of D. That u∗ is well-defined and takes values in [−1 1] follow because all the UFk are nonempty and contain only functions taking values in [−1 1]. It remains to show that u∗ is semi-strictly increasing, quasiconcave, and rationalizes D. We consider each in turn. For the remainder of the proof, F F  F , etc. will always denote finite subsets of D, and k k  k , etc. will always denote positive integers. I. u∗ is semi-strictly increasing. Suppose first that x ≤ y. We must show that u∗ (x) ≤ u∗ (y). For any F and k, every u ∈ UFk is semi-strictly increasing (recall that UFk is a subset of U). Therefore, u(x) ≤ u(y) for every u ∈ UFk  Hence, inf u(x) ≤ inf u(y)

u∈UFk

u∈UFk

Since this holds for every F and k, it follows that u∗ (x) = sup inf u(x) ≤ sup inf u(y) = u∗ (y) Fk u∈UFk



Fk u∈UFk

We have used the fact that, for any y  0 and any x q ≥ 0 such that qy = 1, qx = x(j) (q(j)y(j)) ≤ maxj x(j) . (We thank Sergiu Hart for this simple proof of this inequality.) j y(j) y(j) 14

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Thus, we have shown that x ≤ y implies that u∗ (x) ≤ u∗ (y). Suppose next that x  y. We must show that u∗ (x) < u∗ (y). Since {y1  y2    } is a dense subset of Rm ++ , there exists k such that x  yk  y. Fix this k . ∗ ∗ By what we have just shown, u (yk ) ≤ u (y). Hence it suffices to show that u∗ (x) < u∗ (yk ). Consider any F and any k ≥ k . Since x  yk , part (ii) of the definition of UFk implies that   x(j) 1 − 1 + u(yk ) for every u ∈ UFk  u(x) ≤ k max j yk (j) 2 Hence, inf u(x) ≤

u∈UFk

  x(j) − 1 + inf u(yk ) max u∈UFk yk (j) 2k j 1

Since this holds for every F and every k ≥ k , we have   x(j) 1 − 1 + sup inf u(yk ) sup inf u(x) ≤ k max (4) j yk (j) 2 Fk≥k u∈UFk Fk≥k u∈UFk For any z ∈ Rm + , (3) implies that sup inf u(z) ≥ sup inf u(z)

Fk≥k u∈UFk

Fk u∈UFk

Since the reverse inequality is obvious (the sup on the left is over a smaller set of k’s than that on the right), we have sup inf u(z) = sup inf u(z) = u∗ (z)

Fk≥k u∈UFk

Fk u∈UFk

Applying this to (4) yields   x(j) 1 ∗ − 1 + u∗ (yk ) u (x) ≤ k max j yk (j) 2 − 1) < 0, we conclude that Finally, because x  yk implies that maxj ( yx(j) k (j) ∗ ∗ u (x) < u (yk ), as desired. II. u∗ is quasiconcave. For any F and k, every u ∈ UFk is in U and is therefore quasiconcave. Consequently, the function uFk defined by uFk (x) = infu∈UFk u(x) is quasiconcave, being the pointwise infimum of a collection of quasiconcave functions. Also, by the definition of u∗ , we have u∗ (x) = supFk uFk (x) for every x.

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Fix any x y ∈ Rm + and any λ ∈ [0 1], and let z = λx + (1 − λ)y. For any F and k, the definition of u∗ and the quasiconcavity of uFk imply that   u∗ (z) ≥ uFk (z) ≥ min uFk (x) uFk (y)  Hence, for all F F  k k , if we let F = F ∪ F and k = max(k k ), then   u∗ (z) ≥ min uF k (x) uF k (y)   ≥ min uFk (x) uF k (y)  where the second inequality follows by (3). Since F F  k, and k are arbitrary, this implies that

u∗ (z) ≥ min sup uFk (x) sup uF k (y) 

F k

Fk

 = min u∗ (x) u∗ (y) 

and we conclude that u∗ is quasiconcave. III. u∗ rationalizes D. Fix any (p x) ∈ D. For any y ∈ Rm + , we must show that if p(y − x) ≤ 0, then ∗ u (y) ≤ u∗ (x), and that if p(y − x) < 0, then u∗ (y) < u∗ (x). Suppose first that p(y − x) ≤ 0. For any F and k, every u ∈ UFk rationalizes F . Therefore, every u ∈ UFk will satisfy u(y) ≤ u(x) so long as F contains (p x). That is, for every k and every F containing (p x), u(y) ≤ u(x)

for every u ∈ UFk 

Hence, for every k and every F containing (p x), inf u(y) ≤ inf u(x)

u∈UFk

u∈UFk

It follows that (5)

sup

inf u(y) ≤

Fk:(px)∈F u∈UFk

sup

inf u(x)

Fk:(px)∈F u∈UFk

For every z ∈ Rm + , (3) implies that sup

inf u(z) ≥ sup inf u(z)

Fk:(px)∈F u∈UFk

Fk u∈UFk

But because the reverse inequality is obvious (the sup on the left is over a smaller set of F ’s than that on the right), we have sup

inf u(z) = sup inf u(z) = u∗ (z)

Fk:(px)∈F u∈UFk

Fk u∈UFk

Applying this to (5), we conclude that u∗ (y) ≤ u∗ (x).

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Thus, we have so far shown that u∗ (y) ≤ u∗ (x) for every y ∈ Rm + such that p(y − x) ≤ 0. Suppose next that y ∈ Rm + is such that p(y − x) < 0. Then there exists y such ∗ that y  y and p(y − x) < 0. By what we have already shown, u (y ) ≤ u∗ (x). But because u∗ is semi-strictly increasing, u∗ (y) < u∗ (y ). Hence, u∗ (y) < Q.E.D. u∗ (x), as desired. 4. EXAMPLES We present four examples of infinite data sets, each satisfying GARP. The first example admits no lower semicontinuous rationalization, the second no upper semicontinuous rationalization, the third no concave rationalization, and the fourth, whose data set satisfies SARP, admits no strict rationalization.15 In each example, there are two goods, that is, m = 2, and a typical bundle will be denoted by x = (a b) ∈ R2+ . EXAMPLE 1—No Lower Semicontinuous Rationalization: Suppose that, for every n ≥ 2, the bundle x0 = (1 0) is chosen at the price vector pn = (1 − n1  1), and the bundle x1 = (0 1) is chosen at the price vector p = (1 2). Let D denote the resulting data set. It is easily checked that D satisfies GARP because only two bundles, x0 and x1 , are ever chosen and x1 is never affordable when x0 is chosen. For each k ≥ 1, let xk = (0 1 − k1 ). If v is any utility function rationalizing D, then we must have v(x0 ) < v(x1 ) because px0 < px1 , and for each k we must have v(xk ) < v(x0 ) because pn xk < pn x0 for n > k. Consequently, limk v(xk ) ≤ v(x0 ) < v(x1 ). Since xk converges to x1 , v is not l.s.c. at x1 . EXAMPLE 2—No Upper Semicontinuous Rationalization: Suppose that bundles x0 = (1 0) and x1 = (0 1) are each chosen at the price vector p0 = (1 1), and for each n ≥ 2 the bundle xn = ( n3  1 − n2 ) is chosen at the price vector p = (1 2). Let D denote the resulting data set. It is easily checked that D is rationalized by the increasing utility function u defined by u(a b) = a + b if a + b ≤ 1, and u(a b) = a + 2b otherwise. Consequently, D satisfies GARP. If v is any utility function rationalizing D, then we must have v(x0 ) < v(x2 ) < v(xn ) for every n because p0 x0 < px2 < pxn , and we must have v(x1 ) = v(x0 ) because p0 x0 = p0 x1 . Consequently, v(x1 ) = v(x0 ) < v(x2 ) ≤ limn v(xn ). Since xn converges to x1 , v is not u.s.c. at x1 . EXAMPLE 3—No Concave Rationalization: √ Consider the strictly increasing and quasiconcave utility function u(a b) = b + a + b2 . Its indifference curves are straight lines connecting the axes, though they are not parallel. Their slopes 15 A utility function, u, strictly rationalizes the data if, for each data point (p x), u(x) > u(y) holds for every y = x such that py ≤ px.

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decrease as one moves outward from the origin. It is well known (Fenchel (1953), Arrow and Enthoven (1961), Aumann (1975)) that this utility function cannot be concavified.16 That is, there is no strictly increasing function f defined on the range of u such that f ◦ u is concave. Consequently, if D consists of all (p x) such that p is the gradient of u at x, then u rationalizes D. Hence D satisfies GARP but has no concave rationalization. EXAMPLE 4—SARP and No Strict Rationalization: Suppose that, for every λ > 0 and every n ≥ 2, the bundle (λ 2λ) is chosen at the price vector pn = (1 + n1  1) and the bundle (2λ λ) is chosen at the price vector p = (1 1). Let D be the resulting data set. To see that D satisfies SARP, note that if (a b) is affordable when the distinct bundle (a  b ) is chosen, then a + b > a + b in all cases but those in which (a b) = (λ 2λ) and (a  b ) = (2λ λ).17 This is sufficient to ensure that, within the data set, there can be no cycles in the “directly revealed preferred to” relation, proving SARP.18 Let v rationalize D. We must show that v does not strictly rationalize D. If λ > λ, then (2λ λ) is strictly affordable when (2λ  λ ) is chosen and so we must have v(2λ  λ ) > v(2λ λ). Hence, v(2λ λ) is a strictly increasing function of λ > 0 and so it is continuous except at perhaps countably many points. Let λ∗ > 0 be a continuity point. Since (2λ∗  λ∗ ) is chosen at the price vector (1 1) when (λ∗  2λ∗ ) could have been chosen, we must have v(2λ∗  λ∗ ) > v(λ∗  2λ∗ ) if v is to strictly rationalize D. Thus, it suffices to show that v(2λ∗  λ∗ ) ≤ v(λ∗  2λ∗ ). For any λ < λ∗ , pn (2λ λ) < pn (λ∗  2λ∗ ) for some n large enough. Consequently, (2λ λ) is affordable when (λ∗  2λ∗ ) is chosen at the price vector pn and so v(2λ λ) ≤ v(λ∗  2λ∗ ). Since this inequality holds for any λ < λ∗ , we may take the limit as λ converges to λ∗ from below. The continuity of v(2λ λ) at λ∗ then yields v(2λ∗  λ∗ ) ≤ v(λ∗  2λ∗ ). 5. ADDITIONAL OBSERVATIONS 5.1. Strict Rationalizability Call a data set D connected (in the sense of Richter (1966)) if (p x), (q y) ∈ D implies (p  tx + (1 − t)y) ∈ D for some price vector p and some t ∈ (0 1). Assuming that D is connected and satisfies SARP, Hurwicz and 16 See Reny (2013) for a general nonconcavifiability result that includes this function as a special case. 17 The only nontrivial cases are those in which (a b) = (2λ λ) is affordable when (a  b ) = (λ  2λ ) is chosen at some price p = (1 + n1  1). Then, (1 + n1 )2λ + λ ≤ (1 + n1 )λ + 2λ , which, after factoring out λ and λ on each side, implies that λ < λ . Then, adding n1 λ to the left-hand side and the larger n1 2λ to the right-hand side gives (1 + n1 )(2λ + λ) < (1 + n1 )(λ + 2λ ) and division by (1 + n1 ) yields the desired conclusion. 18 One bundle is directly revealed preferred to another if the one is affordable at a price vector at which the other is chosen.

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Richter (1971) showed that D can be strictly rationalized by a preference relation that is monotone and convex on the set of chosen bundles.19 Under the same conditions, Sondermann (1982) showed that D can be strictly rationalized by a utility function, though not necessarily nondecreasing or quasiconcave even when restricted to the set of chosen bundles. Theorem 1 permits us to connect these two results under our maintained assumption (which neither Hurwicz and Richter nor Sondermann assume) that all income is spent. PROPOSITION 4: If D is connected and satisfies SARP, then D can be strictly rationalized by a utility function that is strictly increasing and strictly quasiconcave on the set of chosen bundles.20 PROOF: By Theorem 1, D is rationalized by a semi-strictly increasing and quasiconcave utility function u taking values in [−1 1]. Moreover, if for (p x) (q y) ∈ D we have y = x and py = px, we claim that u(y) < u(x). Indeed, by connectedness, (p  tx + (1 − t)y) ∈ D for some price p and some t ∈ (0 1). Letting x¯ = tx + (1 − t)y, px¯ = px implies that p (tx + (1 − t)y) < p x ¯ ≤ u(x). Hence, p y < p x¯ and so u(y) < u(x) ¯ ≤ u(x) by SARP and that u(x) as claimed. So, redefining u to be −2 for bundles that are never chosen, this redefined u strictly rationalizes D. Finally, observe that strict rationalization implies that the redefined u is strictly increasing and strictly quasiconcave on the set of chosen bundles. Q.E.D. As in Hurwicz and Richter (1971), one can, in addition, ensure that the rationalizing utility function is upper semicontinuous at every chosen bundle by redefining u so that its value at any chosen bundle is the limsup of the values of nearby bundles.21 We use a similar idea in the next section to obtain an everywhere upper semicontinuous utility function. 5.2. Upper Semicontinuous Rationalizability By Theorem 1 in Mas-Colell (1978), every continuous demand function satisfying SARP can be strictly rationalized by an upper semicontinuous, monotone, and convex preference relation. An immediate consequence is that every 19 A preference relation, , (i) strictly rationalizes a data set D if (p x) ∈ D implies x  y for every y = x such that py ≤ px; (ii) is monotone on Z if x y ∈ Z and x ≥ y implies x  y; and (iii) is convex on Z if x y ∈ Z and x  y implies tx + (1 − t)y  y for every t ∈ [0 1] such that tx + (1 − t)y ∈ Z. 20 A utility function u is (i) strictly increasing on Z if x y ∈ Z, x ≥ y, and x = y, imply u(x) > u(y); and (ii) is quasiconcave on Z if x y ∈ Z implies u(tx + (1 − t)y) ≥ min(u(x) u(y)) for every t ∈ [0 1] such that tx + (1 − t)y ∈ Z. 21 Sondermann (1982, Remark) mistakenly redefined u at every bundle as the limsup of the utility values of nearby bundles. The resulting upper semicontinuous utility function need not rationalize the given data set. (The proof of rationalizability in Sondermann’s Remark fails to check bundles that are never chosen.) This error does not affect Sondermann’s main result.

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continuous demand function satisfying SARP can be strictly rationalized by an upper semicontinuous, nondecreasing, and quasiconcave utility function. As we have already noted, continuity of demand, while standard, rules out all nonstrictly-convex preferences, even if continuous. In particular, perfect substitutes are ruled out. Theorem 1 will be used below to obtain a similar result under weaker conditions that do not rule out linear indifference curves, for example. Of course, whenever linear indifference curves are permitted, strict rationalizability cannot be expected. Say that a data set D exhibits locally non-maximal income if, for every (p x) ∈ D, there is a sequence (p xn ) ∈ D with xn → x such that pxn > px for every n. Note that any data set generated by a demand function (which might be a selection from a demand correspondence) that, holding prices fixed, is continuous in income, exhibits locally non-maximal income. PROPOSITION 5: Suppose that D satisfies GARP and exhibits locally nonmaximal income. Then D is rationalized by an upper semicontinuous, quasiconcave, semi-strictly increasing utility function. PROOF: By Theorem 1, D is rationalized by a semi-strictly increasing and ˆ = limn u(x + 1/n), quasiconcave utility function u. For every x ∈ Rm + , let u(x) where 1 = (1     1) ∈ Rm . It is straightforward to show that uˆ is upper semicontinuous, semi-strictly increasing, and quasiconcave. To see that uˆ rationalizes D, suppose that (p x) ∈ D and py ≤ px. Then, by locally non-maximal income, there is a sequence (p xn ) ∈ D with xn → x such that py ≤ px < pxn for every n. For any k, we may first choose n sufficiently large so that xn  x + 1/k, and then choose m sufficiently large so that p(y + 1/m) < pxn . Hence, u(y + 1/m) ≤ u(xn ) ≤ u(x + 1/k). That is, for every k, u(y + 1/m) ≤ ˆ ˆ u(x + 1/k) holds for all m large enough. Consequently, u(y) ≤ u(x). Moreˆ ˆ over, as a consequence, (p x) ∈ D and py < px implies u(y) < u(x) because uˆ is semi-strictly increasing. Q.E.D. 5.3. Preference-Relation versus Utility-Function Rationalizability If a data set can be rationalized by a preference relation, then the data set must satisfy GARP. Consequently, Theorem 1 implies that a data set can be rationalized by a preference relation if and only if it can be rationalized by a semi-strictly increasing and quasiconcave utility function. So, for example, it is not a coincidence that, when all prices are positive, the demand behavior induced by lexicographic preferences can be rationalized by a utility function (e.g., u(a b     c) = a) even though the preferences themselves have no utility representation.

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5.4. Nonlinear Budget Sets Forges and Minelli (2009) proved a version of Afriat’s theorem for finite data sets with nonlinear budget sets.22 Our proof-technique allows us to appropriately extend their Propositions 3 and 4 to infinite data sets, although again, in the infinite case one cannot ensure continuity or concavity of the rationalizing utility function.23 Define a nonlinear data set D to be an arbitrary collection of ordered pairs m (g x), where g : Rm + → R is continuous and semi-strictly increasing, x ∈ R+ , and 24 g(x) = 0. For (g x) ∈ D, the interpretation is that when the consumer faces the budget set {y : g(y) ≤ 0} he chooses x. Implicit in this is the assumption that the consumer’s choice always satisfies the constraint exactly. A utility function u rationalizes D if, for every (g x) ∈ D and every y ∈ Rm +, g(y) ≤ 0

⇒

u(y) ≤ u(x)

and

g(y) < 0

⇒

u(y) < u(x)

A nonlinear data set, D, satisfies the Generalized Axiom of Revealed Preference (GARP) if, for every finite sequence (g1  x1 )     (gn  xn ) of points from D, g1 (x2 ) ≤ 0

g2 (x3 ) ≤ 0



gn−1 (xn ) ≤ 0

⇒

gn (x1 ) ≥ 0

The proof of Theorem 1 can be adapted to yield the following result.25 PROPOSITION 6: For an arbitrary nonlinear data set D, the following three conditions, (a), (b), (c), are equivalent: (a) D satisfies GARP. (b) D can be rationalized by a utility function. (c) D can be rationalized by a semi-strictly increasing utility function. Furthermore, if, for each (g x) ∈ D, the function g is quasiconcave, that is, if the complement of each budget set is convex, then the following fourth condition is equivalent to the previous three: 22

See also Matzkin (1991). We are grateful to Francoise Forges for raising the question of whether our techniques apply also to nonlinear budget sets. 24 Forges and Minelli (2009) used the term increasing to describe functions that we have been calling semi-strictly increasing. 25 There are two adaptations for the equivalence of (a)–(d). The first is to treat (g x) ∈ D in the nonlinear case like (p x) ∈ D in the linear case, in the sense that instances of p(y − x) in the original proof are replaced by g(y). The second is to replace ( 21k qk  yk ) in the set F defined after (3) in the original proof with ( 21k gk  yk ), for an appropriate choice of the constraint function gk . For the equivalence of (a), (b), (c), use gk (z) = maxj ( yz(j) − 1); and for the equivalence of (a)– k (j) (c) and (d), use gk (z) = qk (z − yk ), where the yk ’s and qk ’s are as in the original proof. For the equivalence of (d) and (e), it is straightforward to show that (d) implies (e) under quasiconvexity and differentiability. The reverse implication follows because Theorem 1 implies that the data set defined in (e) can be rationalized by a semi-strictly increasing quasiconcave utility function, which also rationalizes D by the quasiconvexity, differentiability, and nonzero gradient assumptions. 23

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Dept. of Economics, University of Chicago, 1126 East 59th Street, Chicago, IL 60637, U.S.A.; [email protected]. Manuscript received March, 2014; final revision received August, 2014.