The Theory of Weak Revealed PreferenceWe are grateful to Charles Gauthier for useful comments. Hjertstrand thanks Jan Wallander och Tom Hedelius stiftelse, Marcus och Marianne Wallenberg stiftelse and Johan och Jakob Söderberg stiftelse for funding.

# The Theory of Weak Revealed Preference††thanks: We are grateful to Charles Gauthier for useful comments. Hjertstrand thanks Jan Wallander och Tom Hedelius stiftelse, Marcus och Marianne Wallenberg stiftelse and Johan och Jakob Söderberg stiftelse for funding.

Victor H. Aguiar Department of Economics, The University of Western Ontario, vaguiar@uwo.ca.    Per Hjertstrand Research Institute of Industrial Economics (IFN), Per.Hjertstrand@ifn.se.    Roberto Serrano Department of Economics, Brown University, roberto_serrano@brown.edu.
This version: May 2019
###### Abstract

We offer a rationalization of the weak generalized axiom of revealed preference (WGARP) for both finite and infinite data sets of consumer choice. We call it maximin rationalization, in which each pairwise choice is associated with a “local” utility function. We develop its associated weak revealed-preference theory. We show that preference recoverability and welfare analysis à la Varian (1982) may not be informative enough, when the weak axiom holds, but when consumers are not utility maximizers. We clarify the reasons for this failure and provide new informative bounds for the consumer’s true preferences.

JEL Classification: C60, D10.
Keywords: consumer choice; revealed preference; rationalization; maximin rationalization; reference-dependent utility; law of demand.

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## 1 Introduction

Rooted in the seminal work of Samuelson (1938), the weak (generalized) axiom of revealed preference (WGARP) has been seen as a minimal, normatively appealing, and potentially empirically robust consistency condition of choice.111Samuelson (1938) originally was interested in demand functions, and studied the Weak Axiom of Revealed Preference (WARP). Following Varian (1982) we argue that it is empirically more convenient to work with demand correspondences. To accomodate this we study both WGARP and WARP but focus on the former. WGARP states that, for any pair of observations and , when a consumer chooses with being affordable, it must be that is at least as expensive when she chooses . While a natural condition of choice consistency, no behavioral justification for WGARP has been given. The main contribution of this paper is to provide a new notion of rationality that is equivalent to WGARP, and to develop a comprehensive revealed-preference theory based on this notion. The proposed notion brings to the forefront the concept of reference points, as a framing relevant to each pairwise comparison. Indeed, the consumer acts as if any pairwise comparison colors her preferences over all possible choices she makes. The importance of our theoretical development is that it allows us to recover preferences, and to do welfare analysis on the basis of WGARP-consistent data sets that cannot be generated by standard utility maximization.

Standard utility maximization requires, in addition to WGARP-consistency, transitivity of preferences, against which there is abundant experimental and field evidence (Quah (2006)). The potential lack of robustness of the transitivity requirement on preferences motivated the seminal work of Kihlstrom et al. (1976), which essentially proposes to rewrite the entire theory of demand on the basis of WGARP alone. More recently, practitioners have recognized some difficulties surrounding the computational complexity of using standard utility maximization in setups of empirical interest (e.g., stochastic utility maximization is NP-hard to check, Kitamura and Stoye (2018)). In response, there has been a renewed interest in using WGARP as a minimalist version of the standard model of rationality, in works such as Blundell et al. (2008), Hoderlein and Stoye (2014), Cosaert and Demuynck (2018), and Cherchye et al. (2019). All these contributions are concerned with the use of WGARP to test rationality, the recovery of preferences, and the performance of welfare analysis.222In all of them, WGARP is usually stated without indifference, because the object of interest is a demand function, not a demand correspondence. We cover both here.

We find that a data set is consistent with WGARP if and only if it can be rationalized by a maximin preference function. We say that a data set (a collection of prices and commodity bundles) is weakly rationalized by a preference function if, for all , for any that is affordable at prices (and wealth ).

We now define a maximin preference function. Let be a finite index set, and the probability simplex on . A reference point consists of two arbitrary price-commodity bundle pairs where . Then, denotes a reference-dependent utility function. Note that each utility function has a double subscript, indicating the two situations in the reference point. We require that the order of such situations not matter (i.e., the reference-point pair refers to the bilateral comparison between choices and : ). We say that a data set is weakly rationalized by a maximin preference function if for any , we can write as:

 r(x,y) =maxμ∈Δ(U)minλ∈Δ(U)∑i∈U∑j∈Uλiμj(uij(x)−uij(y)).

Also, fixing a situation indexed by , consisting of the price-bundle , we require that for all reference points involving , it must be that for all such that . In this sense, the choice involved in the price-bundle can be “rationalized” according to the reference-dependent utility functions for all .

Maximin rationalization can be interpreted as an aggregation of preferences of an individual with multiple utility functions that are heterogeneous, due to reference dependence. If this consumer is asked to make up her mind about how to compare any pair of observations, how would she aggregate her different preferences if her behavior is consistent with WGARP? We show that this consumer has a preference function that is the maximum over the minimal difference among the local utilities of the two bundles. This consumer is cautious, in that she first looks at the smallest differences between utilities, and only then maximizes among them.

Alternatively, when focusing on the bilateral comparison between bundles and , which can be colored by their interactions with each situation or reference point , where the consumer commits to be pairwise consistent, the maximin preference function admits a game-theoretic interpretation. It can be interpreted as the outcome of two adversarial selves within the decision maker: (i) The former self, with a payoff function in their zero-sum game equal to , and choosing mixed-strategy over the choice situations; (ii) the latter self, with the negative of those payoffs, and choosing mixed-strategy . This is similar to the self-equilibrium notion in Kőszegi and Rabin (2006). The adversarial self captures the idea that this consumer is aware that her preferences may change strongly but wants to remain consistent in certain pairwise situations.

Whatever the interpretation, it follows that maximin rationalization implies that the preference function is bounded above (resp., below) by the maximum (resp., minimum) of differences of utilities of any two bundles over reference points. This maximin aggregation of local utilities extends a partial, reflexive, and asymmetric order (the direct revealed-preference relation333We declare a bundle revealed preferred to whenever is chosen when is affordable. under WGARP) to a complete, reflexive, and asymmetric order on the grand-commodity set. In fact, we show that the maximin preference function is skew-symmetric, a key property of nontransitive consumers, first proposed by Shafer (1974) (i.e., for all ). In addition, for a given , the maximin is attained at a particular reference point, meaning that the model is locally equivalent to one where the consumer maximizes a utility function. Indeed,

 r(x,y)=ui∗j∗(x)−ui∗j∗(y),

for some .

The classical notion of rationalization by a utility function states that a data set can be rationalized by a utility function if for all such that is affordable at prices (and wealth ). Afriat (1967) and Varian (1982) show that the classical notion of rationality is equivalent to the Generalized Axiom of Revealed Preference (GARP). In the current work, we show that a data set that satisfies WGARP, but perhaps not GARP, is consistent with a local notion of rationalization. Of course, classical utility maximization is a special case of maximin rationalization when there is a (global) utility function that is capable of rationalizing the data set . In that case, for all ; in other words, for all .

Equipped with the equivalence between maximin rationalization and WGARP, we answer the four fundamental questions of nonparametric demand analysis posed by Varian (1982):

• (i) (Consistency) When is behavior consistent with maximin rationalization?

• (ii) (Recoverability/Welfare) How can we recover preferences or make welfare inferences on the basis of observed behavior?

• (iii) (Forecasting/Counterfactuals) How can we forecast what demand will be at a new price? and

• (iv) (Shape constraints) When is observed behavior consistent with maximizing a preference function that has specific shape constraints?

The foregoing discussion has already sketched the answer to item (i) , and stated the equivalence between WGARP and maximin rationalization (Theorem 1). Along similar lines, we also show that the strict version of maximin rationalization is equivalent to WARP444WARP states that, for any pair of observations and , when a consumer chooses with being affordable, then, when she chooses , it must be that is more expensive. (Theorem 2).

Next, we turn to item (ii), which we believe is one of our core contributions. We show that applying the tools developed by Varian (1982) to recover the preferences of a consumer that satisfies WGARP, but who fails GARP, may be ineffective. Formally, the bounds provided for preferences using the toolkit in Varian (1982) for these cases may provide uninformative bounds on preferences. Thus, the revealed-preference theory for WGARP developed herein is not a corollary of the one for GARP. Specifically, we provide in Theorem 3 such new informative bounds based on the notion of maximin rationalization for data sets consistent with WGARP. Our key innovation is to consider subsets of , consisting of pairs of observations, where we can apply Varian’s tools to recover local preferences that are combined to get bounds on the true global preferences.

In addition, we clarify the reasons behind the failure of Varian’s approach to recover preferences. In fact, WGARP alone cannot ensure that every possible commodity bundle can be supported, by some price, as the maximizer of a (maximin) preference function. Varian’s approach implicitly assumes this condition, which holds only when GARP holds.

In Figure 1, we observe an instance of this problem. There are three commodity bundles, depicted as budget shares, ( for , with goods), which satisfy WGARP, but not GARP.555Each arrow represents a direct revealed-preference relation, where the commodity at the origin is directly revealed preferred to the commodity at the end of the arrow. Then, we consider an out-of-sample bundle (). We note that the new bundle cannot be chosen, under any price, if the consumer remains consistent with WGARP (i.e., WGARP fails in the extended data set for all possible prices). The reason is that all bundles in are strictly and directly revealed preferred to . This is a consequence of being affordable at the prices at which the original commodity bundles were chosen. Nonetheless, for every possible price, is revealed preferred to at least one of the original commodity bundles.666The formal details of this figure are expanded in Example 1. If we apply Varian’s tools to this simple example, we cannot reject the hypothesis that is the worst possible bundle in . But that would be an absurd conclusion, since it dominates bundles such as .

As for item (iii), the question about counterfactual bounds on demand turns out to have a simple answer. In fact, the answers for WGARP and GARP are analogous. We can build counterfactual bounds for demand consistent with WGARP with a direct modification of the approach by Varian (1982). Evidently, WGARP is a weaker condition than GARP, and provides less informative bounds to demand for a new price. At the same time, such bounds are easier to compute, and possibly more robust, in that they require only local consistency. We provide Afriat-like inequalities for WGARP that facilitate testing this condition using modern linear/integer programming techniques. These inequalities seem to be new to the literature.

In contrast to item (iii), item (iv) –i.e., the question concerning shape constraints on the preference function– has an answer that is remarkably different from the classical work by Afriat (1967) and Varian (1982). In fact, we observe that convexity of preferences is a testable shape constraint in finite data sets: there are data sets satisfying WGARP that cannot be rationalized by preference functions that are quasiconcave in the first argument. This means that these data sets cannot be rationalized by convex preferences.777Informally, we say that preferences are convex if the set of commodities preferred to , , is convex. This finding provides a counterexample to Samuelson’s eternal darkness, which refers to the impossibility of testing convexity of preferences in the case of utility maximization. (Incidentally, the lack of convexity of some preference functions is precisely the reason that recovering preferences can no longer use the tools in Varian (1982).) Congruent with our analysis, John (2001) provides necessary and sufficient conditions for quasiconcavity of the preference function. And following Brown and Calsamiglia 2007; Allen and Rehbeck 2018, we next focus on the study of a quasilinear restriction on the local utilities in the maximin preference function representation, and show in Theorem 4 that this restriction is associated with the law of demand. This characterization may be of interest in its own right, due to the importance of the law of demand in both theoretical and applied literatures.

The direct predecessors of our work are the contributions of Kim and Richter (1986) and Quah (2006). Both works provide rationalizations of demand correspondences (or functions) consistent with WGARP (or WARP) with additional conditions on the invertibility of demand. The current work generalizes their results in two respects: (i) it provides a rationalization of WGARP/WARP for finite data sets, and (ii) for the case of infinite data sets, it relaxes a key invertibility assumption in both papers, i.e., we do not require that the set of prices at which any commodity bundle be chosen is nonempty.

This paper’s findings should be helpful for practitioners of revealed preferences since, from an empirical perspective, WGARP is significantly easier to work with than Varian’s GARP. In applications, it is common for practitioners to use WGARP as a synonymous of GARP. However, this is only true if price variation is limited, as shown in Cherchye et al. (2018). For example, it may happen that a finite data set of prices and observed consumption choices is consistent with WGARP, but cannot be rationalized by a utility function. If this occurs, the interpretation of the direct revealed-preference relation is unclear, yet we show that meaningful welfare analysis is possible.

The plan of the paper is as follows. After the central notions of revealed-preference theory are reviewed in section 2, sections 3 to 6 deal in turn with the questions posed on items (i) to (iv) discussed above. Section 7 extends the analysis to infinite data sets. Section 8 is a brief review of related literature, and section 9 concludes. Proofs are collected in an appendix.

## 2 Preliminaries

Suppose that a consumer chooses bundles consisting of goods in a market. We assume that we have access to a finite number of observations, denoted by , on the prices and chosen quantities of these goods, where observations are indexed by . Let denote the bundle of goods at time , which was bought at prices . We impose Walras’ law throughout: wealth at time is equivalent to , for all .888We use the following notation: The inner product of two vectors is defined as . For all , if for all ; if and ; and if for all . We denote and . We write to denote all price-quantity observations, and refer to as the data. In practice, the data describe a single consumer that is observed over time.

### 2.1 Revealed-Preference Axioms

We begin by recalling some key definitions in the revealed-preference literature.

###### Definition 1.

(Direct revealed preferred relations) We say that is directly revealed preferred to , written , when . Also, is strictly and directly revealed preferred to , written , when .

If is directly revealed preferred to , this means that the consumer chose and not , when both bundles were affordable. If is strictly and directly revealed preferred to , then she could also have saved money by choosing . These definitions only compare pairs of bundles. We can extend them to compare any subset of bundles by using the transitive closure of the direct relation:

###### Definition 2.

(Revealed preferred relations) We say that is revealed preferred to , written , when there is a chain with and such that . Also, is strictly revealed preferred to , written , when at least one of the directly revealed relations in the revealed preferred chain is strict.

Hence, the revealed preferred relation is the transitive closure of the direct revealed-preference relation . Next, we use these binary relations to define axioms that characterize different types of rational consumer behavior. We begin with Samuelson’s (1938) weak axiom of revealed preference:

###### Axiom 1.

(WARP) The weak axiom of revealed preference (WARP) holds if there is no pair of observations such that , and , with .

Kihlstrom et al. (1976) introduces a generalized version of WARP:999In contrast to only observing a finite number of prices and quantities, suppose that we knew the entire demand function. In this case, Kihlstrom et al. (1976) shows that if the demand function is differentiable and satisfies WGARP at every point in its domain, then the Slutsky substitution matrix derived from the demand function is negative semidefinite at every point.

###### Axiom 2.

(WGARP) The weak generalized axiom of revealed preference (WGARP) holds if there is no pair of observations such that , and .

Samuelson (1948) shows how WARP can be used to construct a set of indifference curves in the two-dimensional () case, but also recognizes that WARP is not enough to characterize rationality in the multi-dimensional () case. Responding to this challenge, Houthakker (1950) introduces the strong axiom of revealed preference (SARP), which makes use of transitive comparisons between bundles as implied by the revealed preferred relation:

###### Axiom 3.

(SARP) The strong axiom of revealed preference (SARP) holds if there is no pair of observations such that , and , with .

Varian (1982) notes that SARP requires single-valued demand functions, and argues that it is empirically more convenient to work with demand correspondences and “flat” indifference curves. To accomodate these properties, Varian introduces the generalized axiom of revealed preference (GARP):

###### Axiom 4.

(GARP) The generalized axiom of revealed preference (GARP) holds if there is no pair of observations such that and .

In the two-dimensional () case, the following equivalences are known:

Theorem A. (Equivalence of axioms) Let . Consider a finite data set :

• The data satisfies SARP if and only if satisfies WARP (Rose 1958).

• The data satisfies GARP if and only if satisfies WGARP (Banerjee and Murphy 2006).

### 2.2 Revealed-Preference Characterizations

In this section, we recall the main results from the revealed-preference literature that are needed in order to introduce our contribution. Consider the following definitions of rationalization:101010We say that a utility function is: (i) continuous if for any sequence for such that and with implies ; (ii) locally nonsatiated if for any and for any , there exists where such that ; (iii) strictly incresing if for , implies ; and (iv) concave if for any , we have , for , where is the subdifferential of .

###### Definition 3.

(Utility rationalization) Consider a data set and a utility function . For all and all such that ,

• the data is weakly rationalized by if .

• the data is strictly rationalized by if whenever .

Afriat’s (1967) fundamental theorem is well known:

Theorem B. (Afriat’s theorem, Varian 1982) Consider a finite data set . The following statements are equivalent:

1. The data can be weakly rationalized by a locally nonsatiated utility function.

2. The data satisfies GARP.

3. There exist numbers and for all such that the Afriat inequalities:

 Ut−Us≥λtpt(xt−xs),

hold for all .

4. There exist numbers for all such that the Varian inequalities:

 if pt(xt−xs) ≥ 0 then, Vt−Vs≥0, if pt(xt−xs) > 0 then, Vt−Vs>0,

hold for all .

5. The data can be weakly rationalized by a continuous, strictly increasing, and concave utility function.

There are several interesting features of Afriat’s theorem.111111 Statements (i), (ii), (iii), and (v) comprise Varian’s (1982) original formulation of Afriat’s theorem. Statement (iv) is rather new to the revealed preference literature. See Demuynck and Hjertstrand (2019) for a motivation and intuition of the Varian inequalities in statement (iv). Statements (ii), (iii), and (iv) give testable conditions that are easy to implement in practice (See, e.g., Demuynck and Hjertstrand 2019). Perhaps the most interesting theoretical implication of Afriat’s theorem is that statements (i) and (v) are equivalent, which means that continuity, monotonicity, and concavity are nontestable properties. In other words, separate violations of any of these properties cannot be detected in finite data sets.

Varian (1982) shows that the numbers and in statement (iii) can be interpreted as measures of the utility level and marginal utility level of income at observation . Analogously, Demuynck and Hjertstrand (2019) shows that the numbers in statement (iv) can be interpreted as measures of the utility levels at the observed demands.

Matzkin and Richter (1991) provides an analogous result for strict rationalization, by showing that SARP is a necessary and sufficient condition for a data set to be strictly rationalized by a continuous, strictly increasing, and strictly concave utility function:

Theorem C. (Matzkin and Richter 1991) Consider a finite data set . The following statements are equivalent:

1. The data can be strictly rationalized by a locally nonsatiated utility function.

2. The data satisfies SARP.

3. There exist numbers and for all such that the inequalities

 if xt ≠ xs then, Ut−Us>λtpt(xt−xs), if xt = xs then, Ut−Us=0,

hold for all .

4. There exist numbers for all such that the inequalities:

 if xt ≠ xs and pt(xt−xs)≥0 then, Vt−Vs>0, if xt = xs then, Vt−Vs=0,

hold for all .

5. The data can be strictly rationalized by a continuous, strictly increasing and strictly concave utility function.

Matzkin and Richter’s (1991) theorem mirrors Afriat’s, in the sense that it shows that continuity, monotonicity, and strict concavity are nontestable properties.121212Matzkin and Richter’s (1991) original formulation consists of statements (i), (ii), (iii), and (v). Talla Nobibon et al. (2016) proves the equivalence of statements (ii) and (iv). Although much is known about the types of consumer behavior that characterize finite data sets satisfying SARP and GARP, there are no analogous characterizations for WARP and WGARP. The current paper, starting with the next section, fills this gap.

## 3 Characterizations of WGARP and WARP

In this section, we provide revealed-preference characterizations, analogous to the ones in Afriat’s and Matzkin and Richter’s theorems, for WGARP and WARP. We begin by introducing the maximin preference model, which, to the best of our knowledge, is a new model of consumer behavior.

### 3.1 The Maximin Preference Model

###### Definition 4.

(Preference function) A preference function is a mapping , that maps ordered pairs of commodity bundles to real numbers.

A preference function is a numerical representation of a consumer’s preferences. If then the consumer prefers bundle to . Similarly, if , we say that is strictly preferred to .

In what follows, we focus on a particular representation of , namely the maximin model. Let be a finite index set that enumerates a collection of reference points. We define a reference point as a pair of observations, consisting of two price vectors and the corresponding vectors of commodity bundles for . We consider a collection of reference-dependent utility functions , with the property that if for all , then it must be the case that , for all . We assume that the reference-dependent utility functions are independent of permutations of the indices in the subscript, so that for all . Let be the probability simplex defined on .

###### Definition 5.

(Maximin (strict) preference model) We say that the preference function is a maximin (strict) preference function if, for any , it can be written as:

 r(x,y) =maxμ∈Δ(U)minλ∈Δ(U)∑i∈U∑j∈Uλiμj(uij(x)−uij(y)),

where, for any reference point indexed by , the ’local’ utility function, , is continuous, strictly increasing, and (strictly) concave.

The maximin preference function assigns a numerical value to the comparison of any pair of commodity bundles , by aggregating over local preferences that are defined for any reference point in the set . More specifically, the aggregation is a maximin function, which in the first dimension takes the maximal difference between the utility gains of over , and in the second dimension, takes the minimal value of that difference, over the different ’local’ utility functions.

The maximin aggregation structure is appealing because the preference function effectively extends the incomplete direct revealed preference relation when it is asymmetric (i.e., when WGARP holds) to the commodity space . As such, this representation provides us with an interpretation of the direct revealed preference relation when rationality –GARP– fails, but WGARP is satisfied.

For any reference point indexed by , the local utility function, , is continuous, strictly increasing, and (strictly) concave. Moreover, the maximin is achieved at a particular reference point. Thus, for any pairwise comparison, a consumer behaves as if they are rational. However, note from the definition of that, for any distinct pairwise comparison that is not a permutation, the preferences may change, in which case, preferences are not stable across all observations. In the maximin (strict) preference model, a consumer still behaves locally “as if she were rational” in that behavior according to this model rules out binary inconsistencies. More precisely, we show that a data set satisfies (WARP) WGARP if and only if the maximin (strict) preference model (strictly) rationalizes the data.

As a special case, when all local utility functions are the same, i.e., for all , the maximin preference model reduces to standard utility maximization. Then, for all . As such, traditional utility maximization is a case of global rationality, as opposed to the new notion of local rationality introduced here.

Next, we present some properties of the preference function. We begin with a property that turns out to be key in our characterizations of WARP and WGARP:

###### Definition 6.

(Skew-symmetry) We say that a preference function is skew-symmetric if for all .

Skew-symmetry means that the preference function induces a preference order on that is complete and asymmetric. Note that, when a data satisfies WGARP, then the direct preference relation is an (incomplete) asymmetric relation on , the preference function extends it if it is a maximin function. We have the following result:

###### Lemma 1.

If is a maximin preference function, then for any , we have:

 r(x,y) =maxμ∈Δ(U)minλ∈Δ(U)∑i∈U∑j∈Uλiμj(uij(x)−uij(y)) =minλ∈Δ(U)maxμ∈Δ(U)∑i∈U∑j∈Uλiμj(uij(x)−uij(y)),

and moreover, is skew-symmetric.

The proof of this lemma is omitted, as it follows directly from the classical von Neumann’s minimax theorem. It is easy to see that the maximin preference function is skew-symmetric, making this model a variant of the general nontransitive consumer model, considered in Shafer (1974).

The next definition lists some additional important properties of preference functions, which will connect with the maximin preference model in the following section:

###### Definition 7.

Consider a preference function . We say that:

1. is continuous if for all and any sequence of elements in that converges to it must be that .131313We state the weaker versions of these properties, as all we need is to work with movements in one of the arguments.

2. is locally nonsatiated if for any such that and for any , there exists a such that .

3. is strictly increasing if for all , implies .

4. is quasiconcave if for all and any we have , where , and strictly quasiconcave if, for any , the inequality is strict whenever .

5. is concave if for all and any we have , and strictly concave if, for any , the inequality is strict whenever .

6. is piecewise concave if there is a sequence of concave functions in the first argument for in a compact set such that , and strictly piecewise concave if there is a similar sequence of strictly concave functions.

Some remarks are in order: Continuity is a technical condition that is convenient to ensure existence of a maximum in the constrained maximization of the preference function. Local nonsatiation rules out thick indifference curves: if we take an arbitrarily small neighborhood of a bundle that is indifferent to a given bundle , the neighborhood contains bundles that are dominated by . Strict monotonicity simply means that "more is better". Quasiconcavity says that for a fixed reference point , a mixture of two bundles is at least as good as the worst of the two bundles, according to the preference function. Concavity is a cardinal version of quasiconcavity. Quasiconcavity and concavity are important properties because they ensure well-behaved optimization problems. More precisely, quasiconcavity guarantees that a function defined on a compact set has a convex set of maxima points, while a strictly concave function defined on a compact set always has a unique global maximum.

Piecewise concavity and its strict version are new properties, which turn out to be especially important for our characterizations of WGARP/WARP.141414See Zangwill (1967) for a detailed discussion of piecewise concave functions. The property says that for a fixed , a mixture of two bundles is at least as good as the worst one of the two bundles, but only if are close enough. In other words, this is a local version of concavity, implying local quasiconcavity.

### 3.2 Preference Function Rationalization

We now introduce the notion of (strict) rationalization by a preference function, which is analogous to utility rationalization in Definition 3:

###### Definition 8.

(Preference function rationalization) Consider a data set and a preference function . For all and all  such that ,

• the data is weakly rationalized by if .

• the data is strictly rationalized by if whenever .

The next definition introduces the concept of rationalization in terms of the maximin preference model:

###### Definition 9.

(Maximin preference rationalization) The data is weakly (strictly) rationalized by a maximin (strict maximin) preference function if is a maximin (strict maximin) preference function as stated in Definition 5, and is weakly (strictly) rationalized by as stated in Definition 8.

In the subsequent two sections, we show that WGARP (WARP) is a necessary and sufficient condition for a data set to be rationalized by the maximin (strict maximin) preference model.

### 3.3 Wgarp

The next theorem provides a revealed-preference characterization of WGARP for finite data sets. This result mirrors Afriat’s theorem in terms of preference function rationalization (as opposed to utility rationalization):

###### Theorem 1.

Consider a finite data set . The following statements are equivalent:

1. The data can be weakly rationalized by a locally nonsatiated and skew-symmetric preference function.

2. The data satisfies WGARP.

3. There exist numbers and  for all with and such that inequalities:

 Rt,s≥λttspt(xt−xs),

hold for all .

4. There exist numbers for all with  such that inequalities:

 if pt(xt−xs)≥0 then, Wt,s≥0, if pt(xt−xs)>0 then, Wt,s>0,

hold for all .

5. The data can be weakly rationalized by a maximin preference function.

6. The data can be weakly rationalized by a continuous, strictly increasing, piecewise concave, and skew-symmetric preference function.

The equivalence of statements (i) and (vi) shows that, if the data can be weakly rationalized by any nontrivial preference function at all, it can, in fact, be weakly rationalized by a preference function that satisfies continuity, monotonicity, and piecewise concavity. Put differently, separate violations of these three properties cannot be detected in finite data sets.

The numbers and in statement (iii) have a similar interpretation as in Afriat’s theorem for each reference point; that is, if we consider , then is a measure of the utility difference for that particular pairwise data set, while is a measure of the marginal utility level of income at observation for the pairwise data set.

### 3.4 Warp

The next theorem provides a revealed-preference characterization of WARP for finite data sets, and mirrors Matzkin and Richter’s (1991) theorem:

###### Theorem 2.

Consider a finite data set . The following statements are equivalent:

1. The data can be strictly rationalized by a locally nonsatiated and skew-symmetric preference function.

2. The data satisfies WARP.

3. There exist numbers and  for all with and such that inequalities:

 if xt ≠ xs then, Rt,s>λttspt(xt−xs), if xt = xs then, Rt,s=0,

hold for all .

4. There exist numbers for all with  such that inequalities:

 if xt ≠ xs and pt(xt−xs)≥0 then, Wt,s>0, if xt = xs then, Wt,s=0,

hold for all .

5. The data can be strictly rationalized by a maximin strict preference function.

6. The data can be strictly rationalized by a continuous, strictly increasing, piecewise strictly concave, and skew-symmetric preference function.

Analogous to Theorem 1, this result shows that separate violations of continuity, monotonicity, and strict piecewise concavity cannot be detected in finite data sets.

## 4 Recoverability of Preferences

In this section, we tackle the question of when one can use the direct revealed-preference relation elicited from the observed consumer behavior in order to make inferences about her true preferences. We begin by showing that recovering preferences using WGARP does not follow as a trivial corollary of the original approach proposed by Varian (1982). We next propose an alternative method to recover bounds of preferences using WGARP.

### 4.1 Varian’s Approach to Recover Bounds on Preferences Using WGARP

At this point, it is useful to briefly recall the classical approach from Varian (1982), which finds upper and lower bounds to the true preferences of a consumer. The object of interest is the true preference of the consumer, captured by the strict upper contour set of a commodity bundle according to the true preference function :

###### Definition 10.

(Set of strictly better alternatives) We define the set of strictly better alternatives than a (possibly unobserved) commodity bundle as:

 Ur(x)={y∈X:r(y,x)>0},

for the true preference function .

Varian (1982) defines the supporting set of prices for any new commodity bundle so that the extended data set, , satisfies GARP as:

 S(x)={p∈P:OT∪{p,x}\>satisfies GARP}.

Varian then uses the set to create upper and lower bounds for the set of interest . We need to define two new sets. The revealed worse set is

 RW(x)={y∈X:∀p∈S(x),x≻R,DOT∪{(p,x)}y}

for , defined on the extended data set . The nonrevealed worse set is the complement of . The revealed preferred set is

 RP(x)={y∈X:∀p∈S(y),y≻R,DOT∪{(p,y)}x}.

Varian (1982) shows that, in the case of utility maximization (i.e., ), for some and all ) we have:

 RP(x)⊆Ur(x)⊆NRW(x).

One would be tempted to use the same construction for WGARP by replacing the definition of the supporting set with one where the extended data set satisfies WGARP. Of course, when the data consists of two goods (i.e., ), this does not cause any problems since, in such a case, WGARP and GARP are equivalent (See Theorem A). However, if , as we show, performing such an exercise is generally not advisable. In particular, we illustrate this by means of an example that, in some cases, yields an uninformative upper bound set .

We begin the elaboration of these points by using an example in Keiding and Tvede (2013):

###### Example 1.

(Keiding and Tvede 2013, Example 1, p.467) Consider a data set with prices , , , and bundles , , . It is easy to verify that this data set satisfies WGARP. The objective is to recover the preferences of this consumer for a new commodity bundle, , given this observed behavior. Suppose that the unobserved commodity bundle is:

 xT+1=13(x1+x2+x3)=(222)′.

If the analyst were to use the methods in Varian (1982), she would need to recover all prices for which the extended data set satisfies WGARP. In this extended data set, we have , for all . However, there is no for which , for all . This implies that this extended data set fails WGARP. This presents a problem, if we want to recover preferences using Varian’s (1982) approach, because it implicitly assumes that the analyst can always find at least one such vector of prices.

In this example, Varian’s supporting set is empty, i.e., . Moreover, it directly follows that any monotonically dominated bundle such as cannot be ruled out from the set . Consequently, the upper bound of is uninformative, i.e., . Thus, any analysis based on this approach is problematic, since the data set can be rationalized by a preference function that is strictly increasing (in the first argument). In other words, Varian’s method to bound preferences does not yield any valuable information in this example.

Yet the method proposed by Varian (1982) to recover preferences for GARP is often useful. Our analysis simply says that recoverability of preferences in terms of WGARP does not follow as a trivial corollary of Varian’s results. We can also clarify the source of this failure. Consider Example 1 again, and note that the original data satisfies WGARP, which implies that there is a preference function that rationalizes the data. Moreover, we have that , , and . In addition, we know that the new bundle, , is a convex combination of the observed bundles. Given the equivalence between GARP and convexity of preferences, if we followed Varian’s approach then we would be implicitly assuming that preferences are convex. But, in our case, convexity of preferences is implied by the assumption that the preference function is quasiconcave in its first argument, in which case, we must have . This implies that must be revealed to be weakly better than at least one of the three observed bundles , and . However, note that, for all , we have . This implies that, if the consumer is maximizing a quasiconcave preference function, then all observed commodity bundles must be strictly preferred to the new bundle, i.e., for all . Hence, the extended data set cannot be weakly rationalized by a quasiconcave and skew-symmetric preference function.

Interestingly, this example also shows that quasiconcavity of the preference function is, in fact, a testable property in finite data sets. As such, this is also a counterexample to Samuelson’s eternal darkness conjecture, that any finite data set always can be rationalized by a convex preference relation. Summarizing these results, the lack of convexity of preferences, which can be inferred from behavior consistent with WGARP, limits the applicability of the tools developed in the classical treatment by Varian (1982). In section 6.1, we will return to the study of the testable implications of convexity.

### 4.2 A New Approach to Recover Bounds on Preferences Using WGARP

In this subsection, we use the new notion of maximin preference rationalization as a way to provide new informative bounds of the true preferences. We show that these new bounds escape the problems associated with Varian’s approach.

We have shown, in the proof of Theorem 1, that, without loss of generality, we can identify the index set of reference point situations with the set of observations , so that we have a local true utility for all , in which case the true global preferences for any are given by:

 r(x′,x)=maxμ∈Δ(T)minλ∈Δ(T)∑t∈T∑s∈Tλsμt(ust(x′)−ust(x)).

We have also shown that any data set satisfying WGARP can be broken into pairwise data sets , and have argued that each one of these data sets satisfies GARP. Define the ’local’ (Varian) support set as for any . For a data set of observations, note that we have a collection of such ’local’ support sets, and that everyone of these sets is never empty:

###### Definition 11.

(WGARP-robust revealed preferred set) For each let

 RPst(x)={y∈X:∀p∈Sst(y),py>px}

be the pairwise revealed preferred set. We define the (WGARP-)robust revealed preferred set as:

 RPW(x)=∪s∈T∩t∈TRPst(x).

Next, we argue that the robust revealed preferred set is a lower bound of for all . If , this implies that for all and for some . Thus, it must be the case that, for and for all , . This means that . Hence, we can conclude that, if , then , which is the same as saying that .

###### Definition 12.

(WGARP-robust (non)revealed worse set) For each , let

 RWst(x)={y∈X:∀p∈Sst(x),px>py}

be the pairwise revealed worse set. Let be the complement of . We define the (WGARP-)robust nonrevealed worse set as

 NRWW(x)=∩s∈T∪t∈TNRWst(x).

From this definition, it directly follows that, if , then . To see this, note that, if , then there must be some such that for all . This implies, by Varian 1982, that for all . By Definition 12, we must have that . Hence, we can conclude that . The following theorem summarizes these results, confirming that the bounds recovered using Varian’s approach in this context are not sharp:

###### Theorem 3.

The upper contour set of the true preferences at any given is:

 RPW(x)⊆Ur(x)⊆NRWW(x).

Moreover, (i) the upper bound, , recovered using Varian’s approach is not sharp, i.e., for all (with strict containment for some ); and (ii) the lower bound, , recovered using Varian’s approach is not sharp, i.e., for all (with strict containment for some ).

We note that, in the context of Example 1, does not contain the dominated commodity . In fact, excludes all commodity bundles that are monotonically dominated by , which is a desirable property, lacking in Varian’s analogous set . Similar statements can be made about the set.

Theorem 3 has some important implications. The first part shows that the new method of using subsets of data sets for bound computation yields informative bounds. The second part highlights that a naive application of the methodology in Varian (1982), when the assumption of convex preferences does not hold, is problematic.

## 5 Demand Counterfactuals

In this section, we show how to perform counterfactual analysis. That is, for a new (possibly unobserved) price vector we propose a simple method to provide a bound for the demanded bundle under WGARP. This method is a simplification of the procedure for the same purpose under GARP, proposed by Varian (1982).

###### Definition 13.

(W-Demand Set) We define the W-demand set, or the set of all commodity bundles compatible with WGARP, by

 D(pT+1)={x∈X:OT∪{pT+1,x} satisfies WGARP and pT+1x=1}.

The W-demand set for a new price-vector can be formulated as a linear program by making use of the following result:

###### Corollary 1.

The bundle is in if and only if it satisfies:

1. ,

2. , for all , for which ,

3. , for all , for which .

The first condition is a normalization of wealth and imposes Walras’ law. The second condition imposes the restriction that, if the observed bundles are cheaper than the new bundle at the new prices, then the new bundle cannot be affordable at the observed prices. The third condition strengthens the second, for the case of a strict inequality. Note that, if the new bundle is cheaper at the new prices than the observed bundles, then the second and third conditions are redundant as WGARP is trivially satisfied for the extended data set.

Recently, Adams (2019) highlights some technical difficulties in the construction of the W-demand set, when predictions are made at more than a single price regime. For that case, Adams (2019) proposes a mixed integer linear programming (MILP) problem, which can be simplified for the case of WGARP using the results in Theorem 1. In particular, the inequalities in statement (iv) of Theorem 1 are useful for this purpose.

Cherchye et al. (2015) shows how the Varian inequalities in statement (iv) in Theorem B can be formulated as a MILP problem. This MILP problem can be modified in a straightforward way to check whether the inequalities in statement (iv) in Theorem 1 have a solution. Specifically, it follows from Theorem 4 in Cherchye et al. (2015) that statement (iv) in our Theorem 1 holds if and only if there exist numbers , with , and binary variables such that, for all observations , the following linear inequalities (in and ) hold:

 Rt,s

As a final remark, in contrast to recoverability of preferences for data sets consistent with WGARP, the construction of demand counterfactuals is just a simplification of the procedure outlined in Varian (1982). The main reason is that the construction of the W-demand set does not depend on duality arguments, which fail when quasiconcavity of the preference function does not hold.

## 6 Shape Constraints: Concave Rationalization and the Law of Demand

We have shown, in sections 3.3 and 3.4, that WGARP (WARP) is a necessary and sufficient condition for a data set to be rationalized by a continuous, strictly increasing, skew-symmetric, and piecewise (strictly) concave preference function. In this section, we consider conditions that are necessary and sufficient for a data set to be rationalized under stronger shape restrictions.

### 6.1 Concave Rationalization

John (2001) provides conditions under which a data set can be weakly rationalized by a continuous, strictly increasing, skew-symmetric, and concave preference function:

Theorem D. (John 2001) Consider a data set . The following statements are equivalent:

1. The data can be weakly rationalized by a locally nonsatiated, concave, and skew-symmetric preference function.

2. There exist numbers and  for all with such that the inequalities:

 Rt,s≥λtpt(xt−xs),

hold for all .

3. The data can be weakly rationalized by a continuous, strictly increasing, concave, and skew-symmetric preference function.

John (2001) provides additional testable conditions, which he shows are equivalent to condition (ii). By comparing John’s (2001) results with our Theorem 1, it is easy to see that his results imply any of the conditions in our theorem, but not vice versa. In particular, note that, in contrast to statement (iii) in Theorem 1, the indices are constant across all pairs . This ensures that John’s rationalizing preference function is indeed concave.

Of course, by Afriat’s theorem and the equivalence between WGARP and GARP (See Theorem A), concavity is a nontestable property when . However, by Theorem 1 and Theorem D, when , concavity is a testable property. This raises the question of whether it is at all possible to further strenghthen the results in Theorem 1, by showing that WGARP is equivalent to weak rationalization by means of a quasiconcave preference function. However, from the discussion in section 4.1, Example 1 shows that this is not the case. In fact, this is a counterexample showing that the data cannot be rationalized by a quasiconcave preference function. Thus, with a finite number of observations, Example 1 shows that quasiconcavity is a testable property.

### 6.2 The Law of Demand and Quasilinear Preference Functions

This subsection derives necessary and sufficient conditions for a data set to be rationalized by a continuous, strictly increasing, skew-symmetric, concave, and quasilinear preference function. Interestingly, we show that one such condition is the law of demand, and consequently, this is equivalent to rationalization by a maximin quasilinear preference function. Before presenting these results, we briefly recall the revealed-preference characterization for quasilinear-utility maximization.

#### 6.2.a The Quasilinear Utility Maximization Model

First, we consider the definition of quasilinear utility maximization:

###### Definition 14.

(Quasilinear utility maximization) Consider a locally nonsatiated utility function . We say that a consumer facing prices and income is a quasilinear utility maximizer if she solves

 maxx∈Xu(x)+w−px⟺maxx∈X,y∈Ru(x)+ys.t.px+y≤w.

As in standard applications of quasilinear utility maximization, we allow the numeraire to be negative in order to avoid technicalities related to corner solutions.151515Allen and Rehbeck (2018) shows the equivalence of the unconstrained quasilinear maximization and the constrained version with a numeraire in Definition 14. Consider the following definition of quasilinear utility rationalization for a finite data set :

###### Definition 15.

(Quasilinear utility rationalization) Consider a data set and a utility function . For all , the data is rationalized by a locally nonsatiated and quasilinear utility function if, for all :

 u(xt)−ptxt≥u(x)−ptx.

Brown and Calsamiglia (2007) shows that the axiom of the strong law of demand is a necessary and sufficient condition for a data set to be rationalized by a continuous, strictly increasing, concave, and quasilinear utility function.161616Brown and Calsamiglia (2007) refers to the strong law of demand as cyclical monotonicity, while Geanakoplos (2013) calls it additive GARP (AGARP).

###### Axiom 5.

(Strong law of demand) The strong law of demand holds if, for all distinct choices of indices :

 p1(x1−x2)+p2(x2−x3)+⋯+pn(xn−x1)≤0.

The next theorem recalls the revealed-preference characterization of quasilinear utility maximization from Brown and Calsamiglia (2007) and Allen and Rehbeck (2018):

Theorem E. (Brown and Calsamiglia 2007; Allen and Rehbeck 2018) Consider a finite data set . The following statements are equivalent:

1. The data can be rationalized by a locally nonsatiated and quasilinear utility function.

2. The data satisfies the strong law of demand.

3. There exist numbers for all such that the inequalities:

 Ut−Us≥pt(xt−xs),

hold for all .

4. The data can be rationalized by a continuous, strictly increasing, concave, and quasilinear utility function.

#### 6.2.b The Law of Demand and the Quasilinear Preference Function Maximization Model

In this subsection, we provide analogous results for the quasilinear preference function model, and show that the law of demand is a necessary and sufficient condition for a data set to be rationalized by a continuous, strictly increasing, concave, skew-symmetric, and quasilinear preference function. The axiom of the law of demand is formally defined as:

###### Axiom 6.

(Law of demand) The law of demand holds if, for all observations :

 (pt−ps)(xt−xs)≤0.

For any sequence consisting of only two (distinct) observations , it is easy to see that the strong law of demand and the law of demand are equivalent. Thus, in a sense, for the quasilinear model, the relation between the law of demand and the strong law of demand mirrors the relation between WGARP (WARP) and GARP (SARP). Before stating our revealed-preference characterization of the law of demand, we introduce the maximin quasilinear preference function, which is a particular representation of a quasilinear preference function:

###### Definition 16.

(Maximin quasilinear preference model) We say that the preference function is a maximin quasilinear preference function if, for any , it can be written as:

 r(x,y) =maxμ∈Δ(U)minλ∈Δ(U)∑i∈U∑j∈Uλiμj(uij(x)−uij(y)),

where, for any reference point indexed by , the ’local’ utility function is continuous, strictly increasing, concave, and quasilinear.

The next two definitions detail the notion of rationalization, in terms of the quasilinear preference function, and in the specific case of the maximin quasilinear preference function model:

###### Definition 17.

(Quasilinear preference function rationalization) Consider a data set and a preference function . For all and all , the data is rationalized by a locally nonsatiated and quasilinear preference function if

 r