Essential data, budget sets and rationalization

According to a minimalist version of Afriat’s theorem, a consumer behaves as a utility maximizer if and only if a feasibility matrix associated with his choices is cyclically consistent. An “essential experiment” consists of observed consumption bundles \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{1}, \ldots , x_{n})$$\end{document} and a feasibility matrix \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\alpha }$$\end{document}. Starting with a standard experiment, in which the economist has access to precise budget sets, we show that the necessary and sufficient condition for the existence of a utility function rationalizing the experiment, namely, the cyclical consistency of the associated feasibility matrix, is equivalent to the existence, for any budget sets compatible with the deduced essential experiment, of a utility function rationalizing them (and typically depending on them). In other words, the conclusion of the standard rationalizability test, in which the economist takes budget sets for granted, does not depend on the full specification of the underlying budget sets but only on the essential data that these budget sets generate. Starting with an essential experiment \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{1}, \ldots , x_{n}; \varvec{\alpha }$$\end{document}) only, we show that the cyclical consistency of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\alpha }$$\end{document}, together with a further consistency condition involving both \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{1}, \ldots , x_{n})$$\end{document} and \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\alpha }$$\end{document}, guarantees the existence of a budget representation and that the essential experiment is rationalizable almost robustly, in the sense that there exists a single utility function which rationalizes at once almost all budget sets which are compatible with \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(x_{1}, \ldots , x_{n}; \varvec{\alpha }$$\end{document}). The conditions are also trivially necessary.

a precise description of the budget sets) is just what is needed to test the consumer's rationality. This result will be formally stated as Proposition 3. However, testing the consumer's rationality is not the only content of Afriat's contribution: it also provides a way to construct an explicit utility function when it exists.
We are thus led to the following question: Given an essential experiment (x 1 , . . . , x n ; α) in which the feasibility matrix α is cyclically consistent, can we construct a utility function v which robustly rationalizes (x 1 , . . . , x n ; α), in the sense that v(x j ) maximizes v over B j , for any family (B j ) of budget sets compatible with (x 1 , . . . , x n ; α)?
Proposition 4 gives an answer to this question. The motivation for such a utility function v is clear: v would not be sensitive to those specific aspects of the budget sets that the consumer might not perceive.
The previous informal discussion hides some difficulties. First of all, we will show that the previous question is not meaningful unless the essential experiment satisfies some basic consistency requirement (independent of cyclical consistency) guaranteeing that there indeed exists (compact, monotonic) budget sets that are compatible with it. We introduce the property that the essential experiment "contains no contradictory statement" and show that it captures such a requirement. This result is formally stated as Proposition 2. Equipped with this tool, we can give a formal statement of the "good news" announced above: an essential experiment (x 1 , . . . , x n ; α) can be rationalized if and only if α is cyclically consistent and (x 1 , . . . , x n ; α) contains no contradictory statement.
Next, we construct an essential experiment (x 1 , x 2 ; α) ∈ R 2 + which contains no contradictory statement, where α is cyclically consistent, and which cannot be rationalized robustly. This simple example is by no means pathological and shows that, formulated exactly as above, the question cannot be answered positively.
Nonetheless, we prove that every essential experiment (x 1 , . . . , x n ; α) which contains no contradictory statement and where α is cyclically consistent can be rationalized in an almost robust way, in the sense that for every sufficiently small , there exists an almost largest family (B j ) of budget sets compatible with (x 1 , . . . , x n ; α) and a utility function v rationalizing (x 1 , . . . , x n ; α) over (B j ). It is not difficult to prove that, conversely, if (x 1 , . . . , x n ; α) can be rationalized in an almost robust way, then (x 1 , . . . , x n ; α) contains no contradictory statement and α is cyclically consistent. This is the main content of Proposition 4 given in Sect. 4.
As suggested above, our results can be interpreted in the standard framework where the economist has access to precise budget sets. From these and the observed consumption bundles, he can deduce the corresponding essential experiment. A by-product of Proposition 3 (partially contained in Lemma 1) is that the conclusion of the standard rationalizability test, in which the economist takes budget sets for granted, does not depend on the full specification of the underlying budget sets but only on the essential data that these budget sets generate; the economist's conclusion automatically applies to a whole family of budget sets. If the essential experiment passes the rationalizability test, Proposition 4 gives a way to construct an almost robust utility function, which rationalizes the consumer's choices for basically all budget sets that are consistent with these choices.
In a much less classical interpretation of our results, the economist does not have access to a precise description of the budget sets, which remain private information of the consumer. The only available data on the consumer could then be the essential experiment, as the outcome of a survey. The economist would be interested in checking whether the data could have been generated by a rational consumer, making choices in monotonic budget sets.
The paper is organized as follows: notations are made precise in the next subsection; Sect. 2 deals with a consumer choosing over finitely many items and defines cyclical consistency; Sect. 3 deals with a consumer choosing over consumption bundles and defines budget sets; Sect. 4 turns to rationalization and states the results listed above; most of the proofs are given in the appendix.

Notations and terminology
R denote the Euclidean space of dimension ; For every 0} denote the nonnegative and positive orthant of R , respectively; Given a set A ∈ R , Fr A = {x ∈ A | {x k } + R ++ } ∩ A = ∅} and [A] + = A ∩ R + denote the frontier of A and the subset of the nonnegative elements of A, respectively; The vector 1 is the characteristic vector of R whose components are equal to 1; Let N = {1, . . . , n} be fixed once and for all.

Essential data
As announced in the introduction, we start with a feasibility matrix α = (α jk ) j,k∈N , that is, an n × n matrix which summarizes the affordability of n observed consumer choices: for every j, k ∈ N ; α jk ∈ {−1, 0, +1} and α j j = 0, α jk = −1 if item k is affordable at date j without exhausting the consumer's revenue; α jk = 0 if item k is affordable at date j and exhausts the consumer's revenue; α jk = +1 if item k is not affordable at date j.
The essential data determine a choice experiment in which the choice set reduces to the n items. A traditional question is to which extent the data are consistent with rational choice, namely whether there exists a rationalization of the data. This amounts to finding a number v j for every item j, such that v j ≥ v k for every item k that is affordable at date j, with strict inequality if item k does not exhaust entirely the revenue of the agent.
Definition 1 Utils (v j ) j∈N rationalize the feasibility matrix α, if, for every j ∈ N : v j ≥ v k for every k ∈ N such that α jk ≤ 0, v j > v k for every k ∈ N such that α jk < 0.
The following tractable condition of cyclical consistency is the usual test to check whether or not an experiment can be rationalized.
Definition 2 An n × n real matrix A = (a jk ) j,k∈N is cyclically consistent if for every chain j, k, , . . . , r , a jk ≤ 0, a k ≤ 0, . . . , a r j ≤ 0 implies all terms are 0.
The next proposition can be deduced from Afriat (1967)'s theorem, in which the feasibility matrix is actually implicit.

Proposition 1
The following conditions are equivalent: 1. The feasibility matrix α is cyclically consistent. 2. There exist utils (v i ) i∈N rationalizing the feasibility matrix α.
Proof [1. ⇒ 2.] is proved in Fostel et al. (2004, replacing A by α page 215). [2. ⇒ 1.] is proved in Ekeland and Galichon (2012, replacing R i j by α i j in the proof of 3. ⇒ 1., Theorem 0). (2012) propose another, "dual," interpretation of the matrix α in terms of a market with n traders and an indivisible good (house) to be traded [see also Shapley and Scarf (1974)]. In the autarky allocation, each trader j owns house j. The matrix α summarizes then the preferences of traders in the initial autarky allocation: α jk = 1 represents strict preference of his own house over house k; α jk = −1 represents strict preference of house k over his own house; α jk = 0 represents indifference of trader j between house k and his own house. In this dual interpretation, Proposition 1 actually amounts to: the autarky allocation is a no trade equilibrium allocation supported by prices π j = −v j (condition 2. of Proposition 1) if and only if it is Pareto optimal (condition 1. of Proposition 1).

Budget sets
From now on, we turn to the standard consumer problem, in which there are divisible consumption goods, and utility is thus defined by a function v : R + → R. Hence, the data contain consumption bundles in addition to the feasibility matrix. This leads to the following notion of experiment, which becomes the basic data in our revealed preference analysis.
Definition 3 An essential (consumer) experiment (x, α) consists of observed consumption bundles (x j ) j∈N , x j ∈ R ++ , and a feasibility matrix α.
We will distinguish such an essential experiment from a standard experiment which involves consumption bundles and budget sets. We follow Forges and Minelli (2009)'s model of budget sets, which is appropriate if finite bundles are consumed and free disposal is allowed. In particular, the formulation encompasses the following cases: classical linear budget sets; budget sets defined by the intersection of linear inequalities, as in Yatchew (1985); convex but nonlinear budget sets, as in Matzkin (1991). More generally, the budget set of the consumer can result from quantity constraints, taxes and other sources of nonconvexities.
Definition 4 A budget set is a compact and monotonic subset of R + .
The next definition is the natural extension of the classical notion of experiment with linear budget sets. In particular, the budget sets B j are implicitly assumed to be observed by the economist, who will make inferences over the consumer's choices. Furthermore, consumption choices entirely exhaust the consumer's available revenue, at each given date. Note that the latter fact is also implicitly assumed in the classical theory with linear budget sets defined by prices.
Definition 5 An experiment (x, B) consists of observed consumption bundles x j ∈ R ++ and of budget sets B j , such that x j ∈ Fr B j for every j ∈ N. 3 In the standard approach of revealed preference analysis, an experiment (x, B) is given. This formulation implicitly assumes that a rational consumer perfectly knows his budget set B j for every j ∈ N . The economist is interested in testing whether the consumer chooses every consumption bundle "rationally" given the budget sets at each date.
Next, we describe how to relate budget sets and the feasibility matrix α to perform the consumer's rationalizability test in terms of essential data only.

Definition 7 Given an experiment
An essential experiment (x, α) admits a budget representation if there exists a family of budget sets (B j ) j∈N such that (x, B) is an experiment and A x,B = α. A family (B j ) j∈N with this property is said to be compatible with (x, α).
Given an experiment (x, B), the economist can deduce the corresponding essential experiment by setting α = A x,B . Alternatively, let us imagine that the essential experiment (x, α) is the only available one. As explained in the introduction, this can happen in at least two different situations. In the first one, the economist has access to the full experiment (x, B) but he suspects that the consumer only used the partial description in the associated essential experiment (x, A x,B ). In the second situation, the economist has only access to survey data which take the form of an essential experiment (x, α).
If only the essential experiment (x, α) is available, (x, α) does not necessarily admit a budget representation. In the next section, we introduce a tractable necessary and sufficient condition, "no contradictory statement," for this property to hold (Proposition 2). For the time being, we just assume that (x, α) admits a budget representation, as it is the case if the essential experiment is simply deduced from some standard experiment (x, B).
The next result can be deduced from Proposition 3 in Forges and Minelli (2009 (x, B) where A x,B = α. Hence, v(x j ) ≥ v(x k ) for every k such that α jk ≤ 0; with strict inequalities if α jk < 1, using local nonsatiation. Then, [2. ⇒ 1.] of Proposition 1 gives the result.
Lemma 1 sheds further light on the standard rationalizability test, which is performed on the basis of the full experiment (x, B), but only uses the matrix A x,B , equal here to α. The economist designs the test with specific budget sets (B j ) j∈N in mind but ends up checking the cyclical consistency (or rationalization) of the matrix α, which is equivalent to the rationalization of a whole class of budget sets. By proceeding in this way, we get a different utility function for every family of compatible budget sets. One can therefore question the predictiveness of such a utility function, defined up to a family of budget sets. This motivates the next section, together with the issue of the existence of a budget representation. 4 The experiment (x, B) satisfies GARP if, for every j, k ∈ N , x k H x j implies x k / ∈ int B j , where H is the transitive closure of the direct revealed preference relation R: x k Rx j if x j ∈ B k . For easy constructive proofs of the equivalence between GARP and the existence of a rationalization, see, for example, Varian (1982) in the linear case and Forges and Minelli (2009) in the general case. 5 The matrix with entries (g B j (x k )) j,k∈N is cyclically consistent iff the matrix A x,B is cyclically consistent.

Existence of a budget representation and rationalization
Let us start with an essential experiment (x, α). Proposition 1 or Lemma 1 tells us which conclusion we can draw from the cyclical consistency of the matrix α but takes for granted that there exists a family of budget sets (B j ) j∈N compatible with (x, α).
As already observed, an essential experiment (x, α) cannot necessarily be generated by budget sets, which are monotonic, even if α is cyclically consistent. For instance, α = 0 1 1 0 is cyclically consistent, but if the associated bundles are x 1 = (1, 1) and x 2 = (2, 2), there do not exist any (monotonic) budget sets that are compatible with (x, α). This is not surprising as cyclical consistency characterizes the affordability of items without taking into account that they are consumption bundles. This motivates the following tractable condition, which expresses that the consumer understands that, at every date, free disposal is allowed.
Definition 8 An essential experiment (x, α) admits a contradictory statement if there exist j, k, k ∈ N such that either [α jk < α jk and x k ≥ x k ] or [α jk = α jk = 0 and The previous property is completely independent of cyclical consistency (recall the example above). The next proposition states that, in the same way as cyclical consistency is necessary and sufficient for rationalization (Proposition 1), the absence of contradictory statement is necessary and sufficient for budget representation. The proof is given in the appendix.

Proposition 2
The two following conditions are equivalent: 1. The essential experiment (x, α) admits no contradictory statement.

The essential experiment (x, α) admits a budget representation.
We can now state our first main result: together, cyclical consistency and no contradictory statement are necessary and sufficient for an essential experiment to be consistent with the rational choices of a consumer facing budget sets. 6 (x, α) be an essential experiment. The following conditions are equivalent:

There exist a family of budget sets (B j ) j∈N compatible with (x, α) and a locally nonsatiated, continuous utility function v B rationalizing the experiment
The proof is given in the appendix. Recalling Proposition 1, Proposition 3 enables us to disentangle the conditions which ensure that an essential experiment (x, α) can be rationalized. The cyclical consistency of α is crucial to assess the rationality of choices over items, independently of the fact that these items might be consumption bundles. The structure of consumption bundles matters to define the absence of contradictory statement, and this property, together with cyclical consistency, characterizes a stronger form of rationalization (namely, 2. in Proposition 3 instead of 2. in Proposition 1).
What does Proposition 3 teach us in the "dual": framework of Ekeland and Galichon (2012) (recall Remark 1)? The analogue of consumption bundle x j could be a vector of "attributes" of house j, like its size, or other criteria that can be measured in real units. Proposition 3 would then say that if the n traders have monotonic preferences over the attribute vectors of the initial n houses, a continuous equilibrium price function can be constructed over the space of all attribute vectors. Such a price function seems to be of little practical use. 7 However, the construction of such a utility function over the whole consumption space is the heart of Afriat's result (as recalled in Forges and Minelli 2009, p. 139). But Proposition 3 is still not fully satisfactory in that respect, because, in condition 2., a different utility function v B is associated with every family (B j ) j∈N . This motivates our next section.

Robust rationalization
Taking again the essential experiment (x, α) as basic data, the following definition of robust rationalization naturally emerges from the discussion at the end of Sect. 3: the utility function v robustly rationalizes the experiment (x, α) if v rationalizes the experiment (x, B) for every family (B j ) j∈N compatible with (x, α). The existence of a robust rationalization amounts therefore to the existence of a largest family of budget sets compatible with the essential experiment. Unfortunately, even if (x, α) is well behaved (in particular, α is cyclically consistent), such a family may not exist as the next simple example illustrates.
The construction of the budget sets is given in Fig. 1.
To obtain a contradiction in the above construction we assumed that x 2 / ∈ {x 1 } + R + , which is by no means pathological. The previous essential experiment can be rationalized for any compatible family of budget sets, but we cannot hope for a robust rationalization. The previous example shows that, by enlarging gradually a family of budget sets which are compatible with a given essential experiment (x, α), we get at the limit budget sets which are well behaved but are not compatible with (x, α) anymore. We will nevertheless achieve an almost robust rationalization, a concept that we define precisely below.

Definition 9
Let (x, α) be an essential experiment. Let > 0, the pair ((B j ) j∈N , v ) where (B j ) j∈N is a family of budget sets and v is a utility function, is said to -robustly rationalize (x, α) if: The justification for the terminology is that (ii) implies that v rationalizes experiment (x, B), for every compatible family (B j ) j∈N included in (B j ) j∈N and, by (iii), every compatible family is almost included in (B j ) j∈N . To show the former statement, note that x j is such that for all x ∈ B j ; and since x j ∈ B j , it follows that v rationalizes the experiment (x, B).
We are now ready for our second main result which states that the conditions in Proposition 3 are also necessary and sufficient for the existence of an almost robust rationalization. (x, α) be an essential experiment. The following conditions are equivalent:
The proof is given in the "Appendix". It shows in particular that the utility function v is easy to construct. As already suggested above, the construction of an almost robust rationalization (as in condition 2. of Proposition 4) is justified if we want to recover Afriat's conclusion (namely, a well-behaved explicit rationalization) in a situation where we suspect that the consumer only has partial knowledge of the experiment (x, B).

Appendix
The proofs of Propositions 2, 3 and 4 are deduced from the next theorem and its proof: -The statements of Propositions 3 and 4 coincide with the equivalence given below.
Proof [1. ⇒ 2.] 9 To show the cyclical consistency of α, we can proceed as in the proof of Lemma 1 (2. ⇒ 1.). To show the property of no contradictory statement, suppose, first, on the contrary that there exist j, k, k ∈ N such that [α jk < α jk and together with x k ≥ x k , but this contradicts the monotonicity property required in the definition of a budget set (see Definition 4 and also Sect. 1.1). Second, suppose on the contrary that there exist j, k, k ∈ N such that [α jk = α jk = 0 and x k x k ]. Since (B j ) j∈N is compatible with (x, α), we obtain that x k , x k ∈ Fr B j but this contradicts x k x k . [2. ⇒ 3.] Let m > 0 be such that x j ≤ m1 for every j ∈ N and define the following family (B j ) j∈N (see also Fig. 2): ))) c Let us check first the condition (iii) of Definition 9.
Claim For every compatible family (B j ) j∈N with (x, α), it holds that B j ⊆ (1+ )B j for any j ∈ N and any > 0.
Proof Let x ∈ B j and assume that 1 1+ x / ∈ B j . Then, it must be true that, for some k ∈ N , either α jk = 0 together with 1 1+ x x k or α jk = 1 together with 1 1+ x 1 1+ x k . In both cases, the fact that (B j ) j∈N is compatible with (x, α) and the monotonicity property implies that x / ∈ B j , which is a contradiction.
The next two claims establish that, for any > 0 sufficiently small, the family of budget sets (B j ) j∈N is compatible with (x, α) (i.e., condition (i) of Definition9). 10 Claim (x, B ) is an experiment for any > 0 sufficiently small.
Proof By construction, each B j is a budget set since it is the intersection of two monotonic subsets of R + , one of those being compact. Suppose next that there exists j ∈ N such that x j / ∈ Fr B j for all > 0. Thus, by construction, there exists necessarily k such that either α jk = 0 and x k x j or α jk = 1 and 1 1+ x k x j , for all > 0. This contradicts the fact that (x, α) admits no contradictory statement, by using that tends to 0 if necessary in the latter case.
Claim A x,B = α for any > 0 sufficiently small.
Proof Let j, k ∈ N be such that α jk = −1. Suppose that there exists k such that x k ≥ x k with α jk = 0. Then, it is a contradictory statement, which cannot be the case. Suppose then that there exists k such that x k ≥ 1 1+ x k with α jk = 1 for all > 0. As tends to 0, this contradicts again the fact that (x, α) admits no contradictory statement. Therefore, for any > 0 sufficiently small x k / ∈ (∪ i∈N ,α ji =0 ({x i } + R + )) (∪ i∈N ,α ji =1 ({ 1 1+ x i } + R + )). Thus, x k ∈ int B j , that is a x,B jk = −1. Let j, k ∈ N be such that α jk = 0. Suppose that there exists k such that x k x k with α jk = 0. Then, it is a contradictory statement, which cannot be the case. Suppose then that there exists k such that x k ≥ 1 1+ x k with α jk = 1 for all > 0. As tends to 0, this contradicts again the fact that (x, α) admits no contradictory statement. Therefore for any > 0 sufficiently small x k / ∈ int((∪ i∈N ,α ji =0 ({x i } + R + )) (∪ i∈N ,α ji =1 ({ 1 1+ x i }+R + ))) but since x k ∈ ∪ i∈N ,α ji =0 ({x i }+R + ), it follows that x k ∈ Fr B j , that is a x,B jk = 0. Let j, k ∈ N be such that α jk = 1. Then clearly, for all > 0, x k ∈ int((∪ i,α ji =0 ({x i } + R + )) (∪ i∈N ,α ji =1 ({ 1 1+ x i } + R + ))) since x k 1 1+ x k . That is to say x k / ∈ B j , that is, a x,B jk = 1.
It remains to be proved that one can construct a well-behaved utility function v with the desired properties (i.e., condition (ii) of Definition 9). Using condition 2. and the fact that (B j ) j∈N is compatible with (x, α) (for any > 0 sufficiently small), Lemma 1 establishes the existence of a locally nonsatiated, continuous utility function v rationalizing (x, B ).