Economic Theory

, Volume 54, Issue 3, pp 485–500

Private versus public consumption within groups: testing the nature of goods from aggregate data

Authors

  • Laurens Cherchye
    • Center for Economic StudiesUniversity of Leuven
    • ECARES and ECOREUniversité Libre de Bruxelles
  • Vincenzo Platino
    • Centre d’Economie de la SorbonneParis School of Economics–Université Paris 1 Panthéon–Sorbonne
Research Article

DOI: 10.1007/s00199-012-0729-8

Cite this article as:
Cherchye, L., De Rock, B. & Platino, V. Econ Theory (2013) 54: 485. doi:10.1007/s00199-012-0729-8

Abstract

We study the testability implications of public versus private consumption in collective models of group consumption. The distinguishing feature of our approach is that we start from a revealed preference characterization of collectively rational behavior. Remarkably, we find that assumptions regarding the public or private nature of specific goods do have testability implications, even if one only observes the aggregate group consumption. In fact, these testability implications apply as soon as the analysis includes three goods and four observations. This stands in sharp contrast with existing results that start from a differential characterization of collectively rational behavior.

Keywords

Multi-person group consumptionCollective modelRevealed preferencesPublic goodsPrivate goodsConsumption externalities

JEL Classification

D11D12D13C14

1 Introduction

There is a growing consensus that multi-person group (e.g., household) consumption behavior should no longer be treated as if the group were a single decision maker that optimizes a group utility function subject to the group budget constraint. Indeed, this so-called unitary model of group consumption imposes empirically testable restrictions on the group demand function (e.g., Slutsky symmetry) that are frequently rejected when confronted with data on multi-person group consumption. See, for example, Browning and Chiappori (1998) and references therein.

Because of these empirical problems of the unitary model, the collective model has become increasingly popular to analyze group consumption behavior. Chiappori (1988, 1992) originally introduced this model for describing household labor supply behavior when all consumption is private, and consumption externalities are absent (i.e., individual preferences are egoistic). More recently, Browning and Chiappori (1998) suggested a most general collective consumption model, which does account for public consumption in addition to private consumption. This model also allows for externalities related to privately consumed quantities. In addition, Browning and Chiappori make the minimalistic assumption that the empirical analyst only observes the aggregate consumption and does not know which part of the quantities are publicly or privately consumed. Focusing on a ‘differential’ characterization of this general model, they establish that for two-person groups collectively rational group behavior requires a pseudo-Slutsky matrix that can be written as the sum of a symmetric negative semi-definite matrix and a rank one matrix.1

Building further on the original work of Browning and Chiappori (1998) and Chiappori and Ekeland (2006) particularly focused on the testability conclusions regarding the private and public nature of group consumption. More precisely, given that a priori all goods can be consumed privately, publicly, or both, there are several possibilities for specifying the collective model. For example, car use may be partly public (e.g., car use for a family trip) and partly private (e.g., car use for work), but the empirical analyst only observes the aggregate amount of car use of the group level; see also Browning et al. (2006) for additional discussion. The main conclusion of Chiappori and Ekeland (2006) is that, following a differential approach, “the private or public nature of consumption within the group is not testable from aggregate data on group behavior” . More specifically, they show that, when only observing the aggregate group consumption, the general collective consumption model has exactly the same testability implications as two more specific collective models, that is, a first benchmark model that assumes all consumption is public and a second benchmark model that assumes all consumption is private and preferences are egoistic (i.e., no consumption externalities).

In this paper, we complement the results of Chiappori and Ekeland. In the tradition of Afriat (1967) and Varian (1982),2 we investigate the same testability questions by focusing on the “revealed preference” characterization of the collective consumption model. Such a revealed preference characterization does not rely on any functional specification regarding the group consumption process; it typically focuses on revealed preference axioms that summarize the empirical implications of theoretical consumption models. Our study extends earlier work of Cherchye et al. (2007, 2010, 2011), who developed the revealed preference characterization of the (general and specific) collective consumption models mentioned above.

Remarkably, in contrast to the findings for the differential approach, we will provide artificial examples that allow us to conclude that our revealed preference approach does imply testability of privateness versus publicness of consumption, even if one only observes the aggregate group consumption. In addition, we will obtain that the model with all consumption public is independent from (or non-nested with) the model that assumes all consumption is private and preferences are egoistic: A data set that satisfies the revealed preference conditions for the first model does not necessarily satisfy the conditions for the second model and vice versa. Finally, our independence results are also confirmed by an empirical application on real-life data. Inter alia, this shows that our findings are not only a theoretical curiosity.

Our results confirm the “strong suspicion” of Chiappori and Ekeland (2006, p. 4) that their results do not hold globally. More precisely, Chiappori and Ekeland’s differential approach focuses on “local” conditions for collective rationality (which apply in a sufficiently small neighborhood of a given point). By contrast, the revealed preference conditions on which we focus are “global” by construction.3 In this interpretation, the global nature of the revealed preference conditions implies stronger testability conclusions.

We believe our results may have interesting implications from the viewpoint of practical applications. For example, they suggest that a practitioner may usefully apply revealed preference characterizations to verify if some given data set satisfies a particular specification of the collective model (in terms of publicly and/or privately consumed goods), prior to the actual empirical analysis. To illustrate this last point, we conduct an empirical application that uses experimental data originally presented in Bruyneel et al. (2012). This application complements our theoretical results in two ways. It focuses on less stylized data sets, which demonstrates that our results effectively do have practical relevance. Next, in our application, we make use of so-called assignable information, which means that we can assign parts of the observed group consumption to individual group members. As such, our application shows that our independence conclusions equally apply in settings characterized by assignable information.4

One final remark is in order before entering our analysis. Following a similar revealed preference approach, Cherchye et al. (2010) also considered testability of the private versus public nature of consumption within groups. A specific feature of their analysis is that it allowed for non-convex preferences of the individual group members. These authors obtain the same non-testability conclusion as Chiappori and Ekeland (2006). Our following analysis differs from the one of Cherchye et al. (2010) in that we make the extra assumption that individual preferences are convex (and represented by concave utility functions); this assumption of convex preferences follows the original analysis of Chiappori and Ekeland. As indicated above, we now do obtain different testability implications under alternative assumptions on the (public or private) nature of goods. When comparing this to the findings of Cherchye et al. (2010), we conclude that the assumption of convex preferences is crucial for obtaining our testability conclusions.

The remainder of the paper unfolds as follows. To set the stage, Sect. 2 defines collectively rational group consumption behavior as introduced in Browning and Chiappori (1998). Following Chiappori and Ekeland (2006, 2009), we focus on three collective consumption models. Section 3 discusses the revealed preference characterization of collectively rational group behavior that was obtained in Cherchye et al. (2007, 2011). Section 4 forms the core of our paper. It provides examples that demonstrate our testability results on public versus private consumption in the group. Section 5 presents our empirical application to experimental data. Section 6 summarizes our main conclusions.

2 Collective rationality

Following Chiappori and Ekeland (2006, 2009), we will concentrate on three collective consumption models in what follows. We will consider the general collective model (general-CR) of Browning and Chiappori (1998) as well as two specific benchmark models, that is, the collective model with all goods public (public-CR) and the collective model with all consumption private and egoistic preferences (egoistic-CR). In this section, we introduce the necessary concepts to study these three collective models.

Throughout, we consider groups (or households) that consist of two members.5 We assume a group that purchases the (non-zero) \(N\)-vector of quantities \(\mathbf q \in \mathbb R _{+}^{N}\) with corresponding prices \(\mathbf p \in \mathbb R _{++}^{N}\). As explained in detail in the Introduction, all quantities can be consumed privately, publicly, or both. For the general collective model, we will assume that the empirical analyst has no information on the decomposition of the observed \(\mathbf q \) into the bundles of private quantities \(\mathbf q ^{1},\mathbf q ^{2}\) and the bundle of public quantities \(\mathbf q ^{h}\). Therefore, we need to introduce (unobserved) feasible personalized quantities\(\widehat{\mathbf{q }}\) that comply with the (observed) aggregate quantities \(\mathbf q \). More formally, we define
$$\begin{aligned} \widehat{\mathbf{q }}=\left( \mathfrak q ^{1},\mathfrak q ^{2},\mathfrak q ^{h}\right) \text{ with} \mathfrak q ^{1},\mathfrak q ^{2},\mathfrak q ^{h}\in \mathbb R _{+}^{N}\quad \text{ and}\quad \mathfrak q ^{1}+\mathfrak q ^{2}+ \mathfrak q ^{h}=\mathbf q . \end{aligned}$$
Each \(\widehat{\mathbf{q }}\) captures a feasible decomposition of the aggregate quantities \(\mathbf q \) into member-specific private quantities (i.e., \(\mathfrak q ^{1}\) and \(\mathfrak q ^{2}\)) and public quantities (i.e., \(\mathfrak q ^{h}\)). This will be useful for modeling general preferences that depend on private consumption as well as public consumption. In the following, we consider feasible personalized quantities because we assume the minimalistic prior that only the aggregate quantity bundle \(\mathbf q \) and not the “true” personalized quantities are observed. Throughout, we will use that each \(\widehat{\mathbf{q }}\) defines a unique \(\mathbf q \).

The collective model explicitly recognizes the individual convex preferences of the group members. For the general model, these preferences may depend not only on the own private quantities and the public quantities, but also on the other individual’s private quantities. This allows for externalities between the group members. Formally, this means that the preferences of each group member \(m \left( m=1,2\right) \) can be represented by a well-behaved utility function of the form \(U^{m}(\mathfrak q ^{1},\mathfrak q ^{2}\), \(\mathfrak q ^{h}\)), with \(\mathbf q =\mathfrak q ^{1}+\mathfrak q ^{2}+\mathfrak q ^{h}\) and \(m=1,2\).6

Suppose then that we observe \(T\) choices of \(N\)-valued bundles. For each observation, \(t\) the vector \(\mathbf q _{t}\in \mathbb R _{+}^{N}\) records the quantities chosen by the group under the prices \(\mathbf p _{t}\in \mathbb R _{++}^{N}\) (with strictly positive components). We let \(S=\{( \mathbf p _{t},\mathbf q _{t});t=1,\ldots ,T\}\) be the corresponding set of \( T\) observations. A collective rationalization of a set of observations \(S\) requires the existence of utility functions \(U^{1}\) and \(U^{2}\) such that each observed quantity bundle can be characterized as Pareto efficient. The following definition provides a formal statement.

Definition 1

(general-CR) Let \(S=\{(\mathbf p _{t},\mathbf q _{t}); t=1,\ldots ,T\}\) be a set of observations. A pair of utility functions \(U^{1}\) and \(U^{2}\) provides a general-CR of \(S\) (i.e., a collective rationalization in terms of the general collective model), if for each observation \(t\) there exist feasible personalized quantities \(\widehat{\mathbf{q }}_{t}\) such that \( U^{m}\left( \widehat{\mathbf{z }}\right) >U^{m}\left( \widehat{\mathbf{q }} _{t}\right) \) implies \(U^{l}\left( \widehat{\mathbf{z }}\right) <U^{l}\left( \widehat{\mathbf{q }}_{t}\right) \) (\(m\ne l\)) for all feasible personalized quantities \(\widehat{\mathbf{z }}\) with \(\mathbf p _{t}\mathbf q _{t}\ge \mathbf p _{t}\mathbf z \).7

The two benchmark cases considered by Chiappori and Ekeland (2006, 2009) involve restrictions on the individual preferences and the nature of the goods. In the first case, we assume that all consumption is public. We formalize this by assuming individual preferences that are represented by a utility function \(U_{pub}^{m}(\mathfrak q ^{h})\). Clearly, in this case, we have \(\mathfrak q ^{h}=\mathbf q \) (or \(\mathfrak q ^{1}+ \mathfrak q ^{2}=\mathbf 0 \)), that is, the true personalized quantities are effectively observed. Given all this, Definition 1 directly leads to the following definition.

Definition 2

[public-CR] Let \(S=\{(\mathbf p _{t},\mathbf q _{t}); t=1,\ldots ,T\}\) be a set of observations. A pair of utility functions \(U_{pub}^{1}\) and \( U_{pub}^{2}\) provides a public-CR of \(S\) (i.e., a collective rationalization in terms of the collective model with only public consumption), if for each observation \(t\) there exist feasible personalized quantities \(\widehat{\mathbf{q }}_{t}\), with \(\mathfrak q ^{1}+\mathfrak q ^{2}=\mathbf 0 \), such that \(U_{pub}^{m}\left( \mathfrak z ^h\right) >U_{pub}^{m}\left(\mathfrak q ^h_{t}\right) \) implies \(U_{pub}^{l}\left( \mathfrak z ^h\right) <U_{pub}^{l}\left(\mathfrak q ^h_{t}\right) \) (\(m\ne l\)) for all feasible personalized quantities \( \widehat{\mathbf{z }}\) with \(\mathbf p _{t}\mathbf q _{t}\ge \mathbf p _{t}\mathbf z \) and \(\mathfrak z ^1+\mathfrak z ^2=\mathbf 0 .\)

The second benchmark case assumes that all consumption is private, that is, \( \mathfrak q ^{1}+\mathfrak q ^{2}=\mathbf q \) (or \(\mathfrak q ^{h}=\mathbf 0 \)). In addition, the individuals have egoistic preferences, which implies that they only care for their own consumption (i.e., no consumption externalities). We formalize this by assuming individual preferences that are represented by a well-behaved utility function \(U_{ego}^{m}(\mathfrak q ^{m}\)), with \(m=1,2\). The corresponding concept of collective rationality is as follows.

Definition 3

[egoistic-CR] Let \(S=\{(\mathbf p _{t},\mathbf q _{t}); t=1,\ldots ,T\}\) be a set of observations. A pair of utility functions \(U_{ego}^{1}\) and \( U_{ego}^{2}\) provides an egoistic-CR of \(S\) (i.e., a collective rationalization in terms of the collective model with all consumption private and egoistic preferences), if for each observation \(t\) there exist feasible personalized quantities \(\widehat{\mathbf{q }}_{t}\), with \(\mathfrak q ^{h}=\mathbf 0 \), such that \(U_{ego}^{m}\left( \mathfrak z ^m\right) >U_{ego}^{m}\left( \mathfrak q ^m_{t}\right) \) implies \( U_{ego}^{l}\left(\mathfrak z ^l\right) <U_{ego}^{l}\left( \mathfrak q ^l_{t}\right) \) (\(m\ne l\)) for all feasible personalized quantities \(\widehat{\mathbf{z }}\) with \(\mathbf p _{t}\mathbf q _{t}\ge \mathbf p _{t}\mathbf z \) and \(\mathfrak z ^{h}=\mathbf 0 \).

3 Revealed preference characterization

In this section, we briefly recapture the revealed preference characterizations for the three models discussed in the previous section; see Cherchye et al. (2007, 2011) for a more detailed discussion of these characterizations. To formally define these revealed preference conditions, we will use the concept of feasible personalized prices\(\widehat{\mathbf{p }}^{1}\) and \(\widehat{\mathbf{p }} ^{2} \).
$$\begin{aligned}&\widehat{\mathbf{p }}^{1}=\left( \mathfrak p ^{1},\mathfrak p ^{2}, \mathfrak p ^{h}\right) \quad \text{ and} \quad \widehat{\mathbf{p }}^{2}=\left(\mathbf p -\mathfrak p ^{1}, \mathbf p -\mathfrak p ^{2},\mathbf p -\mathfrak p ^{h}\right) \quad \text{ with} \\&\mathfrak p ^{1},\mathfrak p ^{2},\mathfrak p ^{h}\in \mathbb R _{+}^{N} \quad \text{ and}\quad \mathfrak p ^{c}\le \mathbf p \quad \text{ for}\; c=1,2,h. \end{aligned}$$
This concept complements the concept of feasible personalized quantities defined above. The prices \(\widehat{\mathbf{p }}^{1}\) and \(\widehat{\mathbf{p }}^{2}\) capture the fraction of the price for the personalized quantities \( \widehat{\mathbf{q }}\) that is borne by the respective members: For each separate component of \(\widehat{\mathbf{q }}\), the corresponding personalized prices can be interpreted as Lindahl prices; by construction, they add up to the observed prices. More specifically, \(\mathfrak p ^{1}\) and \(\mathfrak p ^{2}\) refer to the private quantities and are used to express the willingness to pay for the externalities related to these private quantities. For example, if the members have egoistic preferences, then \( \mathfrak p ^{1}=\mathbf p \) (and thus \(\mathbf p -\mathfrak p ^{1}=\mathbf 0 \)) and \(\mathbf p -\mathfrak p ^{2}=\mathbf p \) (and thus \(\mathfrak p ^{2}= \mathbf 0 \)). Similarly, \(\mathfrak p ^{h}\) (respectively, \(\mathbf p - \mathfrak p ^{h}\)) refers to the public quantities and expresses member \(1\) ’s (respectively, member \(2\)’s) willingness to pay for these quantities.

The revealed preference conditions make use of the Generalized Axiom of Revealed Preference (GARP). Varian (1982) introduced the GARP condition for individually rational behavior; that is, he showed that it is a necessary and sufficient condition for the observed quantity choices to maximize a single utility function under the given budget constraint. We focus on the same condition in terms of feasible personalized prices and quantities; the next Proposition 1 will establish that collective rationality as defined in the above definitions requires GARP consistency for each individual member.

Definition 4

Consider feasible personalized prices and quantities for a set of observations \(S=\{(\mathbf p _{t},\mathbf q _{t}); t=1,\ldots ,T\}\). For \(m=1,2\), the set \(\{\left(\widehat{\mathbf{p }}_{t}^{m}, \widehat{\mathbf{ q }}_{t}\right); t=1,\ldots ,T\}\) satisfies GARP if there exist relations \(R_{0}^{m}, R^{m}\) that meet:
  1. (i)

    if \(\widehat{\mathbf{p }}^{m}_{s}\widehat{\mathbf{q }}_{s} \ge \widehat{\mathbf{p }}^{m}_{s}\widehat{\mathbf{q }}_{t}\) then \(\widehat{\mathbf{ q }}_{s} R_{0}^{m} \widehat{\mathbf{q }}_{t};\)

     
  2. (ii)

    if \(\widehat{\mathbf{q }}_{s} R_{0}^{m} \widehat{\mathbf{q }}_{u}\), \(\widehat{\mathbf{q }}_{u} R_{0}^{m} \widehat{\mathbf{q }}_{v}, \ldots , \widehat{\mathbf{q }}_{z} R_{0}^{m} \widehat{\mathbf{q }}_{t}\) for some (possibly empty) sequence \((u, v,\ldots , z)\) then \(\widehat{ \mathbf q }_{s} R^{m} \widehat{\mathbf{q }}_{t};\)

     
  3. (iii)

    if \(\widehat{\mathbf{q }}_{s} \)\(R^{m} \widehat{\mathbf{q }} _{t}\), then \(\widehat{\mathbf{p }}_{t}\widehat{\mathbf{q }}_{t} \le \widehat{ \mathbf p }_{t}\widehat{\mathbf{q }}_{s}\).

     

We can now state the revealed preference characterization of the general collective model (i.e., general-CR) that has been derived in Cherchye et al. (2007).

Proposition 1

Let \(S=\{(\mathbf p _{t},\mathbf q _{t}); t=1,\ldots ,T\}\) be a set of observations. The following conditions are equivalent:
  1. (i)

    there exists a combination of well-behaved utility functions \( U^{1}\) and \(U^{2}\) that provide a general-CR of \(S;\)

     
  2. (ii)

    there exist feasible personalized prices and quantities such that for each member \(m=1, 2\), the set \(\{\left( \widehat{\mathbf{p }} ^{m}_{t}, \widehat{\mathbf{q }}_{t}\right) ; t=1,\ldots ,T\}\) satisfies GARP.

     

Essentially, condition (ii) states that collective rationality requires individual rationality (i.e., GARP consistency) of each member in terms of personalized prices and quantities. In general, however, the true personalized prices and quantities are unobserved. Therefore, it is only required that there must exist at least one set of feasible personalized prices and quantities that satisfies the condition.

The characterization in Proposition 1 is easily adapted to the two benchmark cases considered in the previous section; see also Cherchye et al. (2011) for more discussion. For a public-CR of the data, we need to include that all consumption is public. The implication is that only the willingness to pay for the public consumption will be relevant for the GARP test. This is contained in the following result.

Proposition 2

Let \(S=\{(\mathbf p _{t},\mathbf q _{t}); t=1,\ldots ,T\}\) be a set of observations. The following conditions are equivalent:
  1. (i)

    there exists a combination of well-behaved utility functions \( U_{pub}^{1}\) and \(U_{pub}^{2}\) that provide a public-CR of \(S;\)

     
  2. (ii)

    there exist feasible personalized prices and quantities, with \(\mathfrak q ^{1}_{t}+\mathfrak q ^{2}_{t}=\mathbf 0 \), such that for each member \(m=1, 2\), the set \(\{\left( \widehat{\mathbf{p }}^{m}_{t}, \widehat{ \mathbf q }_{t}\right) ; t=1,\ldots ,T\}\) satisfies GARP.8

     

Similarly, for an egoistic-CR of the data, we need to add to the second condition that all consumption is private (i.e., \(\mathfrak q _{t}^{h}= \mathbf 0 \)) and that the preferences are egoistic, which implies that the willingness to pay for externalities is zero.

Proposition 3

Let \(S=\{(\mathbf p _{t},\mathbf q _{t}); t=1,\ldots ,T\}\) be a set of observations. The following conditions are equivalent:
  1. (i)

    there exists a combination of well-behaved utility functions \( U_{ego}^{1}\) and \(U_{ego}^{2}\) that provide an egoistic-CR of \(S;\)

     
  2. (ii)

    there exist feasible personalized prices, with \(\mathfrak p ^{1}_{t}=\mathbf p _{t}\) and \(\mathfrak p _{t}^{2}=\mathbf 0 \), and feasible personalized quantities, with \(\mathfrak q ^{h}_{t}=\mathbf 0 \), such that for each member \(m=1, 2\), the set \(\{\left( \widehat{\mathbf{p }}^{m}_{t}, \widehat{\mathbf{q }}_{t}\right) ; t=1,\ldots ,T\}\) satisfies GARP.

     

4 Testing the nature of goods

This section contains our core findings. It shows that the nature of goods is testable, even if we only observe the aggregate group behavior. More specifically, we will prove two main results by means of example data sets. Firstly, we provide data sets for which there exists a general-CR but not a public-CR or, respectively, an egoistic-CR. This implies that consistency with the general model does not necessarily imply consistency with any of the specific benchmark models. Putting it differently, rejection of the specific benchmark models in these examples is caused by the corresponding assumptions on the nature of the goods and not by the Pareto efficiency assumption as such. Secondly, our example data sets will show that the two benchmark models are independent from (or non-nested with) each other, that is, data consistency with one benchmark model does not necessarily imply data consistency with the other benchmark model. As explained in the Introduction, these “global” testability results contrast with the “local” testability conclusions that have been obtained by Chiappori and Ekeland (2006).

4.1 General-CR does not imply public-CR

The following example contains a data set for which there exists a general-CR but not a public-CR. The “Appendix” proves our claims in the examples.

Example 1

Suppose that the data set \(S\) contains the following 3 observations of bundles consisting of 3 quantities:
$$\begin{aligned}&\mathbf q _{1}=(5,2,2)^{\prime },\quad \mathbf q _{2}=(2,5,2)^{\prime }, \quad \mathbf q _{3}=(2,2,5)^{\prime }; \\&\mathbf p _{1}=(4,1,1)^{\prime },\quad \mathbf p _{2}=(1,4,1)^{\prime }, \quad \mathbf p _{3}=(1,1,4)^{\prime }. \end{aligned}$$
This data set \(S\) satisfies the conditions in Proposition 1 (i.e., there exists a general-CR), but it rejects the conditions in Proposition 2 (i.e., there does not exist a public-CR).

This example has two important implications. Firstly, as discussed in the introduction, it contrasts with the results of Chiappori and Ekeland (2006): Following a (local) differential approach, these authors show that the general collective model and the collective model with only public consumption are indistinguishable if one only observes aggregate group behavior. Example 1 illustrates that this is no longer the case if one adopts the (global) revealed preference approach.

Secondly, the example demonstrates that we need only three goods and three observations to obtain our conclusion. In fact, these numbers provide absolute lower bounds on the number of goods and observations for the collective models to have testable implications. Indeed, it can be verified that the conditions in Propositions 1 and 2 cannot be rejected if the number of observations or the number of goods is smaller than three.9 Thus, as soon as collective rationality can be rejected, we can distinguish the specific model with all consumption public from the general collective consumption model. In this respect, it is also worth noting that the differential approach needs at least five goods for verifying the testable implications of the collective consumption model characterized in Propositions 2 ; see Browning and Chiappori (1998) and Chiappori and Ekeland (2006). The fact that our revealed preference approach requires a smaller number of goods provides a further illustration of the fact that the (global) revealed preference approach that we follow here can effectively yield stronger testability conclusions than the (local) differential approach.

4.2 General-CR does not imply egoistic-CR

We next provide an example with a data set for which there exists a general-CR but not an egoistic-CR.

Example 2

Suppose that the data set \(S\) contains the following 4 observations of bundles consisting of 4 quantities:
$$\begin{aligned}&\mathbf q _{1}=(1,0,0,0)^{\prime },\quad \mathbf q _{2}=(0,1,0,0)^{ \prime },\quad \mathbf q _{3}=(0,0,1,0)^{\prime },\quad \mathbf q _{4}=(0,0,0,1)^{\prime }; \\&\mathbf p _{1}=(7,4,4,4)^{\prime },\quad \mathbf p _{2}=(4,7,4,4)^{ \prime },\quad \mathbf p _{3}=(4,4,7,4)^{\prime },\quad \mathbf p _{4}=(4,4,4,7)^{\prime }. \end{aligned}$$
This data set \(S\) satisfies the conditions in Proposition 1 (i.e., there exists a general-CR), but it rejects the conditions in Proposition 3 (i.e., there does not exist an egoistic-CR).

Two remarks are in order. Similar to before, we conclude that the general collective model and the model with only private consumption and egoistic preferences are distinguishable from each other. Inter alia, this implies that the private nature of the goods is testable. Again, this conclusion contrasts with the one for the differential approach. Next, for mathematical elegance, we have used four goods in Example 2.10 Similar (but less elegant) examples exist for data sets that only consider three goods.

A final observation applies to the number of observations in Example . We have now used four observations, which contrasts with Example . In fact, in general, we need minimally four observations for the collective model with private goods and egoistic preferences to be distinguishable from the general collective model. This result is formalized in the following proposition, which we prove in the “Appendix”.

Proposition 4

Let \(S=\{(\mathbf p _{t},\mathbf q _{t}); t=1, 2, 3\}\) be a set of three observations. Suppose that there exists a general-CR of \(S\), then there also exists an egoistic-CR of \(S\).

4.3 Independence of egoistic-CR and public-CR

So far, we have shown that the general collective model is distinguishable from the two specific benchmark models. In the “Appendix,” we argue that a similar conclusion also holds for the two benchmark cases. More precisely, we show that there exists an egoistic-CR for the data set considered in Example 1 and a public-CR for the data set considered in Example 2. Generally, this obtains that data consistency with one benchmark model does not necessarily imply data consistency with the other benchmark model.

Another interesting implication of this result is that we need no more than four observations and three goods to distinguish between the three collective consumption models under study. This conclusion directly carries over to “intermediate” collective models that are situated between the two benchmark cases, that is, models which assume that part of the goods is privately consumed (without externalities) while all other goods are publicly consumed. See Cherchye et al. (2011) for a detailed discussion (including revealed preference characterizations) of these intermediate models.

5 Empirical application

To demonstrate that our results of the previous section are not only theoretical curiosities, we illustrate the independence of different specifications of the collective model by means of an empirical application. In this application, we use the experimental data originally presented in Bruyneel et al. (2012). It has been argued that revealed preference testing tools are especially useful within an experimental context; a particularly convincing case is provided by Sippel (1997), who focused on individual rationality. The laboratory nature of experiments allows one to avoid often controversial preference homogeneity assumptions, and data measurement problems that are associated with using real-life data. Moreover, the experimental setup allows for obtaining information on consumption quantities for the individual group members; such information is typically not available in real-life data sets.

Bruyneel et al. (2012) conducted an experiment with 102 undergraduate students. As the experiment was designed to analyze collective choice behavior, both men and women were asked to sign up for an experimental session together with either a male or a female friend or a romantic partner. This led to a sample of 51 dyads (i.e., groups of 2 members).11 These dyads were confronted with 9 choice problems in which they each time had to allocate their budget of 10 Euros over three different quantities: red wine, orange juice, and M&Ms (i.e., a type of chocolate candy). Each choice problem was characterized by a different price regime. After the dyads had made their choices, they were asked to indicate which percentage of their demands was intended for each individual. They were truthfully told that, as a reward, the dyad would receive one of the 9 chosen bundles and that the indicated percentages would then be used to determine the member-specific allocation. For more discussion about the experimental setup and data set, we refer to Bruyneel et al. (2012).

This rich data set allows us to include so-called assignable quantity information into our analysis. Such information pertains to goods that are assigned to individual members as privately consumed goods without externalities. The remaining (unassigned) goods should then be interpreted as private consumption without externalities (i.e., egoistic-CR) or as private consumption with externalities (i.e., public-CR).12 It is straightforward to adapt our notations introduced above to take this assignable information into account. Essentially, using such information boils down to restricting the feasible set of (unobserved) personalized prices \(\widehat{\mathbf{p }}\) and quantities \(\widehat{\mathbf{q }}\).13 In our following analysis, we will not consider the general collective model. The GARP conditions in statement (ii) of Proposition 1 are nonlinear in personalized prices and quantities, which makes it cumbersome to test them in practice.14

To demonstrate the impact of using assignable information on the independence conclusions, we consider different assignability scenarios. The basic differences between the alternative scenarios pertain to the (number of) goods that we assign to the individual members.15 Table 1 specifies the different scenarios and gives the corresponding test results. In particular, for each scenario the table reports the pass rate for our sample of dyads, which is defined as the fraction of dyads that pass the corresponding revealed preference conditions.
Table 1

Pass rates for different assignability scenarios

 

Egoistic-CR

Public-CR

Difference

No good assigned

100.00

100.00

0.00

One good assigned

   

   Wine assigned

88.24

88.24

0.00

   Orange juice assigned

94.12

94.12

0.00

   M&Ms assigned

98.04

98.04

0.00

Two goods assigned

   

   Wine not assigned

90.20

88.24

1.96

   Orange juice not assigned

86.27

86.27

0.00

   M&Ms not assigned

84.31

80.39

3.92

All goods assigned

72.55

/

/

A first observation from Table 1 is that the pass rates generally decrease when we include more assignable information, going from a 100 % pass rate if we do not assign any good down to a pass rate of 72.55 % if all goods are assigned. Actually, this should not be very surprising given that assignable information restricts the feasible set of personalized prices and quantities, which in turn obtains more stringent revealed preference conditions.

Table 1 also makes clear that it matters which good(s) we assign pass rates differ depending in the nature of the assignable good(s). The maximum difference between pass rates amounts to about 10 % for a single assigned good and to about 6 % for two assigned goods. One may be inclined to think that these differences follow from different budget shares of the different goods. For example, if the budget share of wine is much bigger than the budget shares of the other two goods, then we may reasonably expect that assigning wine effectively restricts the feasible prices and quantities more stringently than assigning orange juice or M&Ms. However, Bruyneel et al. (2012) report that, for the sample under study, the average budget shares of the three goods are actually very close to each other. As such, our results should be explained by some other characteristic of the assigned good. We believe this is an intriguing finding, which may stimulate follow-up research.

Finally, and perhaps most importantly, Table 1 reveals that we effectively do have some dyads that pass the egoistic-CR but not the public-CR conditions, which shows the independence of the two models. Specifically, we obtain differing pass rates for the scenarios with two assignable goods. Although the differences are small, we find it quite remarkable that we do obtain them in a simple consumption setting such as the one we consider here, with dyads having to make decisions over small budgets and only 9 choice observations. In general, we expect our non-nestedness conclusions to become more relevant for real-life settings involving more substantial group consumption decisions and characterized by larger data sets.

6 Conclusions

We have shown that the revealed preference approach implies different testability conclusions for collective consumption models with alternative assumptions on the (public or private) nature of goods. In particular, we obtain different testable implications as soon as we have three goods and four observations. Interestingly, these conclusions stand in sharp contrast with the existing results for the differential approach. As indicated before, this confirms the “strong suspicion” of Chiappori and Ekeland (2006) that the local testability conditions obtained by their differential approach do no hold globally.

Our empirical results suggest that the practitioner may fruitfully apply revealed preference conditions to verify if the data satisfy a particular specification of the collective model that (s)he wants to use in the empirical analysis. We illustrated this point by means of an empirical application to group consumption data gathered in the context of a laboratory experiment. As for practical applications, it is also useful to refer to recent work in Cherchye et al. (2008, 2011). These authors consider the empirical implementation of the revealed preference characterizations of the collective consumption model under alternative specifications of the (public or private) nature of the goods (possibly including assignable quantities, as in our own application in Sect. 5). In doing so, they also address a number of empirical issues that are typically associated with such implementation, such as dealing with measurement error and optimization error and assessing the discriminatory power of the revealed preference tests.

Footnotes
1

The term differential refers to the fact that the characterization is obtained by integrating and/or differentiating the functional specifications of the fundamentals (e.g., the individual preferences of the group members) of the model.

 
2

See also Samuelson (1938), Houthakker (1950), Diewert (1973) and Fostel et al. (2004) for seminal contributions on the revealed preference approach to analyzing consumption behavior.

 
3

See for example Hurwicz (1971) and Pollak (1990) for discussions on the difference between the global revealed preference approach and the local differential approach.

 
4

Other possible applications could combine our results with those of Gersbach and Haller (2005), who focus on the impact of externalities on intra-household allocations, and with those of Ekeland and Galichon (2012), who focus on the duality between the housing market and revealed preference theory.

 
5

The results below can be generalized toward the setting of \(M\) members, with \(M\ge 2\). However, we believe that the core arguments underlying our results are better articulated for this simple case.

 
6

As in the differential approach, we say that a function is well behaved if it is concave, differentiable and monotonically increasing.

 
7

For ease of exposition, the scalar product \(\mathbf p ^{\prime }_{t}\mathbf q _{t}\) is written as \(\mathbf p _{t}\mathbf q _{t}\).

 
8

Given that \(\mathfrak q ^{1}_{t}+\mathfrak q ^{2}_{t}=\mathbf 0 \), the specification of \(\mathfrak p ^1_t\) and \(\mathfrak p ^2_t\) is irrelevant for verifying GARP. In other words, only the willingness to pay for the public goods (i.e., \(\mathfrak p ^h_t\) and \(\mathbf p _t-\mathfrak p ^h_t\)) is important. A similar remark holds for Proposition 3.

 
9

If \(T=2\), one can easily verify that \(\mathfrak p _{1}^{h}=\mathbf p _{1}\) and \(\mathfrak p _{2}^{h}=\mathbf 0 \) is a solution for the GARP conditions in Proposition 2 (and thus a fortiori also for the GARP conditions in Proposition 1). Next, if \(N=2\), one can verify that member 1 paying for the first good (i.e., \((\mathfrak p _{t}^{h})_{1}=(\mathbf p _{1})_{1})\) for all observations \(t\) and, similarly, member 2 paying for the second good (i.e., \((\mathfrak p _{t}^{h})_{2}=0\)) for all observations \(t\) obtains again a solution for the GARP conditions in Proposition 2.

 
10

A similar qualification applies to the use of zeroes in Example 2.

 
11

Our sample contains four types of dyads, namely, two male friends (12 in total), two female friends (14 in total), a male friend, and a female friend who do not have a romantic relationship together (13 in total), and a male and a female who are in a romantic relationship together (12 in total).

 
12

Although, by definition, a private good with externalities and a public good is not the same, they are very related. As in the case of the public good, the private good with externalities enters simultaneously in both utility functions. This implies that both members have a willingness to pay for the consumption of this good, which is in our approach captured by the personalized prices. As such, if the private good with externalities is exclusively consumed by one member, we can reinterpret a public-CR as a collective rationalization with private consumption with externalities.

 
13

See Cherchye et al. (2011) for a detailed discussion of this point.

 
14

See Cherchye et al. (2007, 2008) for discussions on the difficulties associated with testing the general collective consumption model.

 
15

We have used the integer programming methodology introduced by Cherchye et al. (2011) to test the conditions in Propositions 2 and 3 (while accounting for assignable quantities if applicable). See also our discussion in the concluding section, on the practical implementation of the revealed preference conditions under consideration.

 

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