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Intertemporal collective consumptiondecisions:

a revealed preference analysis

Laurens Cherchye, Bram De Rocky,Jeroen Sabbezand Ewout Verriestx

[preliminary draft - not to be quoted without permission]

February 12, 2010

Abstract

We provide a revealed preference methodology for analyzing in-tertemporal household consumption behavior in terms of the collec-tive model. Following Mazzocco (2007), we focus on the full commit-ment model and the no commitment model. We develop revealedpreference conditions that allow for testing data consistency withthese intertemporal consumption models in a nonparametric man-ner, i.e. while avoiding imposing (typically nonveriable) paramet-ric/functional structure for the household decision process. In turn,

CentER, Tilburg University, and Center for Economic Studies, University of Leu-ven. E. Sabbelaan 53, B-8500 Kortrijk, Belgium. E-mail: [email protected]. Laurens Cherchye gratefully acknowledges nancial support from the ResearchFund K.U.Leuven through the grant STRT1/08/004.

yECARES and ECORE, Universit Libre de Bruxelles. Avenue F.D. Roosevelt 50, CP114, 1050 Brussels, Belgium. E-mail: [email protected].

zCenter for Economic Studies, University of Leuven. E. Sabbelaan 53, B-8500 Kortrijk,

Belgium. E-mail: [email protected]. Jeroen Sabbe gratefully acknowl-edges the Fund for Scientic Research - Flanders (FWO-Vlaanderen) for nancial support.

xCenter for Economic Studies, University of Leuven. E. Sabbelaan 53, B-8500 Kor-trijk, Belgium. E-mail: [email protected]. Ewout Verriest gratefullyacknowledges the Fund for Scientic Research - Flanders (FWO-Vlaanderen) for nancialsupport.

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this implies an operational test for commitment, which can be inter-

esting from a policy point of view. An empirical application illustratesour methodology. In this application we also discuss an intermediatelimited commitment model that is situated between the full commit-ment and no commitment models.JEL Classication: D11, D12, D13, C14.Keywords: household economics, collective consumption model, in-tertemporal consumption model, commitment, revealed preferences.

1 Introduction

There is a growing consensus that a realistic modeling of household con-

sumption behavior must take into account preference heterogeneity withinthe household. In many cases, the household decision-making process cannotbe explained by the restrictive unitary framework, which models the house-hold as if it were a single decision maker.1 In addition, many householddecisions are inherently intertemporal. For example, such intertemporal in-terdependence typically applies to decisions on family savings, investmentin human capital, housing purchase, fertility decisions, etc. This directlyextends to a consumption setting. In many cases, consumption today im-pacts on consumption tomorrow and vice versa (e.g. because of (dis)savingpossibilities).

Mazzocco (2007) introduced a household consumption model that takesboth these aspects into account. He uses a collective model, which meansthat he explicitly recognizes that a multi-person household consists of mul-tiple members (decision makers) with their own preferences. As such, themodel considers the observed household consumption behavior as the re-sult of a within-household bargaining process between these members. It(only) assumes that this process yields a Pareto ecient within-householdallocation.2 Next, Mazzocco adopts the life cycle modeling of intertemporalconsumption decisions. Essentially, this obtains a collective extension of the

1 Many studies reject the empirical validity of the unitary model for multi-person house-hold behavior. See, for example, Lundberg (1988), Thomas (1990), Fortin and Lacroix(1997), Browning and Chiappori (1998), Chiappori, Fortin and Lacroix (2002), Duo(2003) and Cherchye and Vermeulen (2008).

2 See, for example, Chiappori (1988, 1992) for seminal contributions on the collectivemodel for multi-person household consumption and Apps and Rees (1988) for a closelyrelated approach.

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widely applied unitary life cycle consumption model.3

More specically, Mazzocco distinguishes between two versions of the in-tertemporal collective consumption model: the full commitment model andthe no commitment model. A main argument is that the two versions havevery dierent policy implications; and this makes it relevant to test (and com-pare) their empirical validity. Mazzocco also developed a characterization ofthe two versions, which eectively establishes such an empirical (commit-ment) test. We will briey recapture the full commitment model and theno commitment model in Section 2; this will also explain Mazzoccos keynotion of commitment. Essentially, commitment refers to the period of timeover which the relative bargaining weights of individual household membersremain constant (enforced by some explicit or implicit intrahousehold con-

tract).This paper presents a revealed preference characterization (and corre-

sponding tests) of Mazzoccos full commitment and no commitment models.In the tradition of Afriat (1967) and Varian (1982),4 we derive necessary andsucient conditions for household consumption data to be consistent witha particular model. These conditions then enable checking consistency of agiven data set with the model; in the spirit of Varian (1982), we refer to thisas testing data consistency with the model under study.5 More specically,we are able to verify these conditions in a fully nonparametric way, i.e. theassociated tests do not require an a priori (typically non-veriable) para-

metric specication of the intrahousehold decision process (e.g. individualpreferences). This contrasts with the tests that were originally proposed byMazzocco (2007), which do require such a parametric specication.

At this point, it is also interesting to relate our ndings to the earlier workof Browning (1989) and Cherchye, De Rock and Vermeulen (2007, 2009a,b).Browning provided a revealed preference characterization of the unitary lifecycle model (see also Crawford, 2009, for recent discussion).6 We extend

3 See, for example, Browning and Crossley (2001) for an extensive overview of theliterature on the unitary life cycle model.

4 See also Samuelson (1938), Houthakker (1950) and Diewert (1973) for seminal contri-butions on the revealed preference approach to analyzing consumption behavior.

5

As is standard in the revealed preference literature, the type of tests that we considerhere are sharp tests; either a data set satises the data consistency conditions or it doesnot.

6 Crawford (2009) extended Brownings (1989) characterization of the life cycle con-sumption model by accounting for habit formation in the household consumption. Inte-grating these insights in our revealed preference characterization is work for future research.

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this characterization to a collective setting. Next, Cherchye, De Rock and

Vermeulen established the revealed preference characterization of the staticcollective consumption model. We extend this characterization by accountingfor intertemporal consumption relations.

The rest of the paper unfolds as follows. The next section briey recap-tures Mazzoccos (2007) modeling of intertemporal collective consumption;we consider the case of full commitment and the case of no commitment. Wewill also discuss the policy relevance of empirical tests of commitment. Sec-tion 3 introduces the revealed preference characterization of the two versionsof the intertemporal collective consumption model. Section 4 provides theempirical results of the no commitment model, the full commitment modeland an intermediate limited commitment model, based on panel data drawn

from the Russian Longitudinal Monitoring Survey. Section 5 concludes andhints at future research topics. The Appendices contain the proofs of ourresults.

2 Intertemporal collective consumption with

and without commitment

We consider a household with M household members. The household has todecide over the consumption of a bundle of N private goods and a bundle of

K public goods. Given private and public consumption in the household, theutility of each member m is given by the function um(qm;Q), with qm 2 RN+the private consumption bundle of m, and Q 2 RK+ the public consump-tion bundle.7 Throughout, we will assume that each utility function um iscontinuous, concave, non-satiated and non-decreasing in its arguments.

The empirical analysis of household consumption starts from a data setS = fpt; Pt; qt; Qtjt = 1;:::;Tg, which pertains to a given number of

periods T. For each period t 2 f1;:::Tg, the vectors Qt and qt (=XT

t=1qmt )

represent the household bundles of public and private goods demanded at t;and we use pt 2 R

N++ for the price vector of the private commodities and

Pt 2 RK

++for the price vector of the public commodities. In what follows,

7 Throughout, we will abstract from externalities associated with privately consumedquantities. Importantly, however, our setting can actually account for such externalities.Specically, if an individual is the exclusive consumer of a particular private good, thenwe can account for externalities for this good by formally treating it as a public good.

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we will use yt = pt0qt + Pt0Qt.

As indicated above, we will adopt a life cycle approach to modeling in-tertemporal collective consumption behavior. In doing so, we will assumeexponential utility discounting. More specically, we will assume m repre-sents the (xed) personal utility discount factor for each individual m. Thesame assumption was made by Crawford (2009), who focused on revealedpreference characterizations of intertemporal consumption models in a uni-tary set-up.8 In what follows, we consider m as given. If this is not the casethen one can, as is suggested in Crawford (2009), perform a grid search thatconsiders dierent possible values of m. See also our empirical applicationin Section 4.

2.1 Full commitment

As indicated above, the collective model assumes Pareto eciency. For oursetting with concave utility functions, a Pareto optimal allocation is usuallycharacterized as maximizing a weighted sum of individual utilities um subjectto the given budget constraint. The weights m associated with each um arethen typically interpreted as bargaining weights for the dierent membersm. In the collective set-up, these bargaining weights depend on the so-calleddistribution factors Z. These factors include, for example, exogenous incomeof individual household members, sex ratio within the relevant region, thecountrys divorce legislation, etc. (see, for example, Browning, Bourguignon,Chiappori and Lechene (1994) for more discussion). Importantly, in generalthe value of Z may vary over time periods t. As is currently standard in re-vealed preference tests, we will assume that the distribution factors Z are notobserved. However, the notion will be crucial for sketching the key dierencebetween the full commitment and the no commitment model. In Section 4,we will briey return to the possibility of using information on Z in empiricalapplications.

The concept of commitment crucially relates to the bargaining weightsmt associated with the dierent periods t. Essentially, it pertains to theextent to which the household members anticipate possible variation in the

weights mt when choosing their consumption path for the time horizon under

consideration. In the full commitment model, the bargaining weights mt are

8 Browning (1989) abstracted from utility discounting, which corresponds to m

= 1.Evidently, our model includes the assumption

m= 1 as a special case.

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held constant, i.e. mt = m for all t. The interpretation is as follows. At

t = 1, all members decide how to divide the power in the household for theentire future horizon, i.e. for each m they x the weight m that will prevailfor the entire time horizon. Once the members have decided on this relativebargaining power, they are fully committed to it and cannot renegotiateafterwards. Putting it dierently, the full commitment model assumes an(explicit or implicit) contract set up at time t = 1, which stipulates theindividual bargaining weights for all future periods. Inter alia, this meansthat the unanticipated variation in Z over time periods t is irrelevant for thehousehold consumption path that is chosen. While changes of Z that areanticipated at t = 1 can have an impact on the intrahousehold allocation;see our discussion in Section 2.3.

More formally, under full commitment the household chooses for eacht 2 f1;:::;Tg private consumption quantities qmt and public consumptionquantities Qt that satisfy the following condition:

q1t ;:::;q

Mt ;Qt

2 arg max

x1t;:::;xM

t;Xt

MXm=1

mTXt=1

t1m um(xmt ;Xt) (FC)

s:t:

TXt=1

pt0(

MXm=1

xmt ) + Pt0Xt 6

TXt=1

yt

Intuitively, this condition requires that the chosen consumption path eec-tively turns out (ex post) to be Pareto ecient for the given (xed) bargainingweights m.

The revealed preference approach checks whether it is possible to conceiveutility functions um that make observed behavior (captured by the set S)consistent with the condition (FC). If this is the case, then we say the set Sis Full Commitment (FC)-rationalizable.

Denition 1 Consider a data set S = fpt; Pt; qt; Qtjt = 1;:::;Tg and aspecication of m (m = 1;:::;M). The set S is FC-rationalizable if thereexist utility functions um, bargaining weights m, and private consumption

bundles qmt 2 RN+ , withXM

m=1 qmt = qt, such that each

q1t ;:::;qMt ;Qt

satises (FC).

This condition will form the basis for dening our revealed preferencecharacterization of the full commitment model in Section 3.

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the NC-rationalizability condition but not the FC-rationalizability condition,

then this means that the observed behavior can be modeled as Pareto e-cient for the life cycle consumption model under consideration, but not underthe assumption of constant bargaining weights (i.e. mt =

m). In this case,we empirically reject commitment within the household under consideration.

As discussed in Mazzocco (2007) such a test of commitment can haveparticular policy relevance. To see this, we recall that if consumers behaveaccording to the full commitment model, then unanticipated variation in thedistribution factors Z will not impact on the household consumption path.As such, a public policy change that aims at inuencing household behav-ior via the distribution factors is ineective. To be more precise, we mustmake the distinction between anticipated and unanticipated policy changes.

If household members anticipate a particular policy change (e.g. a changein divorce legislation at period t > 1), then they will take this informationinto account when dening the m at t = 1. On the other hand, if somepolicy measure is unanticipated, then the household cannot respond by rene-gotiating the bargaining weights. We conclude that only anticipated policychanges eectively impact on the household consumption (via the bargainingweights) in the full commitment model.

The picture is very dierent for the no commitment model. If consumersbehave in accordance with this model (while the data reject full commit-ment), then policy changes do become an eective tool for inuencing house-

hold consumption behavior. In this case, even an unanticipated policy changecan be taken into account during the renegotiations after the exogenousshock. Thus, policy makers can now exert more control over the decisionprocess inside households by (unanticipated) changes of Z.

3 Revealed preference characterization

This section develops revealed preference conditions that are equivalent tothe conditions of the full commitment model in Denition 1 and the nocommitment model in Denition 2. Essentially, these revealed preference

characterizations will allow us to empirically test data consistency with thetwo models while avoiding a prior specication of the utility functions um

and the bargaining weights m (full commitment) and mt (no commitment).

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3.1 Full commitment

The following result gives the nonparametric characterization of the full com-mitment model.

Proposition 1 Consider a data set S = fpt; Pt; qt; Qtjt = 1;:::;Tg and aspecication of m (m = 1;:::;M). The following conditions are equivalent:(i) The set S is FC-rationalizable:(ii) There exist pricesPmt 2 R

K+ with

PMm=1P

mt = Pt, quantitiesq

mt 2 R

N+

withPM

m=1 qmt = qt, and numbers u

mt > 0 such that the following condition

holds for all s; t 2 f1;:::;Tg andm 2 f1;:::;Mg:

um

s um

t 6

1

t1m [(pt0q

m

s + Pm

t 0Qs) (pt0qm

t + Pm

t 0Qt)]

Condition (ii) in this proposition characterizes FC-rationalizability interms of so-called Afriat inequalities (see Varian, 1982; based on Afriat,1967). Our collective model implies that we get a set of Afriat inequali-ties for each member m. Intuitively, for each observation t and memberm, these inequalities dene a utility level umt , which can be interpreted asumt = u

m (qmt ;Qt). The prices Pmt then express the marginal willingness to

pay of each member m for the publicly consumed quantities Qt. They can beinterpreted as Lindahl prices since they must add up to the observed prices

(i.e.PM

m=1Pmt =

Pt). Cherchye, De Rock and Vermeulen (2007, 2009a)use a similar concept of Lindahl prices for the public goods in their revealed

preference characterization of the static collective consumption model.The characterization in condition (ii) is particularly attractive from a

computational point of view. For a given specication of m, it implies aset of inequalities that are linear in terms of the unknowns Pmt , q

mt and u

mt .

Thus, an operational test of the full commitment model can use standardlinear programming techniques to verify the inequalities.

As a nal remark, we can relate the characterization in Proposition 1 tothe revealed preference characterizations of Browning (1989) and Crawford(2009), who focused on unitary intertemporal consumption. These authors

characterize rational intertemporal behavior in terms of so-called cyclicalmonotonicity conditions. It is easy to verify that condition (ii) in Proposition1 can equally be expressed in terms of such cyclical monotonicity conditions;in this case, we will get such a condition for each m. Interestingly, thesecyclical monotonicity conditions will also imply a set of linear inequalities,

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which can thus be checked through linear programming. However, the veri-

cation of these conditions is computationally more complex than checkingthe Afriat inequalities in Proposition 1. Essentially, for a given data set thenumber of cyclical monotonicity inequalities will generally be (often substan-tially) higher than the number of Afriat inequalities. Therefore, we chooseto focus on the Afriat inequalities in this paper.

3.2 No commitment

Let us then provide the nonparametric characterization of the no commit-ment model.

Proposition 2 Consider a data set S = fpt; Pt; qt; Qtjt = 1;:::;Tg and aspecication of m (m = 1;:::;M). The following conditions are equivalent:(i) The set S is NC-rationalizable:(ii) There exist prices Pmt 2 R

K+ with

PMm=1P

mt = Pt, quantities q

mt 2

RN+ with

PMm=1 q

mt = qt, and numbers

mt ; u

mt > 0 such that the following

condition holds for all s; t 2 f1;:::;Tg andm 2 f1;:::;Mg:

ums umt 6

mt [(pt0q

ms + P

mt 0Qs) (pt0q

mt + P

mt 0Qt)]

Thus, we get a characterization in terms of Afriat inequalities that isclosely similar to the characterization of the full commitment model in Propo-

sition 1. The only dierence in the Afriat inequalities is the replacement ofthe given 1=t1m by the unknown mt . The main implication is that the Afriat

inequalities in Proposition 2 are no longer linear in terms of unknowns (incasu Pmt , q

mt ,

mt and u

mt ). This makes it dicult to directly verify these

Afriat inequalities for a given set S.However, the Afriat inequalities in Proposition 2 are exactly the ones

that apply to the static collective model studied by Cherchye, De Rock andVermeulen (2009a). These authors have shown that their conditions canequivalently be formulated in terms of mixed integer programming (MIP)constraints, which can be checked through standard MIP techniques. Essen-tially, they build on the well-known equivalence between the Afriat inequali-

ties conditions that we consider here and the so-called Generalized Axiom ofRevealed Preference (GARP) introduced by Varian (1982). Given this, weconclude that the nonlinear Afriat inequalities in Proposition 2 can equiva-lently be expressed in MIP terms. In turns, this implies operational tests fordata consistency with the no commitment model.

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One further remark relates to this formal similarity between the intertem-

poral Afriat inequalities in Proposition 2 and the static Afriat inequalitiesconsidered by Cherchye, De Rock and Vermeulen (2009a). This may appearsomewhat paradoxical given the intrinsically dierent denition of the struc-tural decision problems that underlie the two models. The interpretationpertains to the very nature of the no commitment model under considera-tion. As explained above, the no commitment model accounts for the possi-bility that the member specic bargaining weights change discretely betweenevery two periods t and t + 1. Intuitively, this possibility of (unanticipated)bargaining power shifts make it irrelevant to take into account intertemporalconsiderations when deciding upon the consumption at any period t. There-fore, the intertemporal optimization model generates empirical restrictions

that coincide with the ones of the static model, which eectively abstractsfrom intertemporal consumption relations.

Admittedly, the assumption of discrete variation in the bargaining weightsbetween any two periods is a strong one. Indeed, it may often be reasonable toassume a constant bargaining weight for at least a subset of periods. In fact, asimilar qualication applies to the full commitment model: when the numberof periods T gets large, it is often unrealistic to assume exactly the samebargaining weight in all periods t. Therefore, in the next section we will alsodiscuss the possibility of considering intertemporal collective consumptionmodels that are situated between the two extreme (full commitment and no

commitment) models.

4 Empirical application

We have introduced revealed preference conditions that allow for empiricallytesting data consistency with the intertemporal collective consumption modelunder full commitment and under no commitment. This eectively obtains atest of commitment, and we have argued that such a test has particular pol-icy relevance. Practical applications that use these conditions complementexisting empirical studies that have focused on revealed preference analysis

of household consumption behavior in terms of the static collective consump-tion model, which abstracts from intertemporal consumption behavior. SeeCherchye, De Rock, Sabbe and Vermeulen (2008) and Cherchye, De Rockand Vermeulen (2009a,c) for empirical studies of the static model that makeuse of a demand panel drawn from the Russian Longitudinal Monitoring Sur-

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vey (RLMS). The application in this paper will draw on the same data set

to assess the empirical performance of the intertemporal collective modelsunder consideration.

4.1 Data

The Russian Longitudinal Monitoring Survey is an extensive panel data setcontaining detailed consumption expenditure data of various categories ofcommodities for a large sample of households. We focus on two-person house-holds (M = 2). Every household in our sample is observed for eight non-consecutive years ranging from 1994 until 2003, with missing data points for1997 and 1999. In order to abate the concern of the non-separability between

consumption and leisure (see, e.g., Browning and Meghir (1991)), couples areexcluded from the sample if not every member is employed. These selectioncriteria result in an overall sample of 148 couples.

The commodity bundle under consideration consists of 21 nondurables.It comprises (1) food outside the home, (2) clothing, (3) car fuel, (4) woodfuel, (5) gas fuel, (6) luxury goods, (7) services, (8) housing rent, (9) bread,(10) potatoes, (11) vegetables, (12) fruit, (13) meat, (14) dairy products, (15)fat, (16) sugar, (17) eggs, (18) sh, (19) other food items, (20) alcohol and(21) tobacco. Throughout our empirical exercises, we will assume that woodfuel, gas fuel and housing rent represent public consumption, while all othergoods are assumed to be private. Real prices (pt;Pt) have been discountedusing compound real interest rates in Russia obtained from Thomson ReutersDatastream, taking into account the two years with lacking data. See Cher-chye, De Rock and Vermeulen (2009a,c) for a more detailed discussion of thedata (including our selection of private and public goods).

4.2 Results

Intertemporal collective rationality is tested for each household separately,thereby avoiding preference homogeneity assumptions across households. Thissubsection describes the setup of the testing phase and the results for the

commitment models discussed in Section 3. Given the disparity in the em-pirical results of these two extreme (full commitment and no commitment)models, we will conclude the section with an introductory discussion on anintermediate limited commitment model.

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4.2.1 No commitment

It was already stated in Section 3 that the revealed preference implicationsof the no commitment model are identical to those of the static collectivemodel. In this respect, Cherchye, De Rock and Vermeulen (2009a) foundthat all households are consistent with the static collective model for the dataunder study. Thus, we conclude that the data also satisfy the intertemporalcollective model that assumes no commitment; the pass rate is 100%.

However, to adequately test the empirical validity of the model, oneshould complement the testing exercise with a power analysis. For a givendata set, such a power analysis quanties the probability of detecting be-havior that is not consistent with the model subject to testing. Evidently,

a high pass rate has little value if the model is hard to reject empirically,i.e. the model has low discriminatory power. We refer to Bronars (1987)and Andreoni and Harbaugh (2006) for detailed discussions on power issuesrelated to revealed preference tests such as ours.

In this respect, we can again refer to Cherchye, De Rock and Vermeulen(2009a). These authors show that the discriminatory power tends to berather low for the static collective model that is empirically equivalent tothe no commitment model under study.9 As argued by Beatty and Crawford(2009), this puts the pass rate of 100% somewhat into perspective.

4.2.2 Full commitment

Let us then consider the full commitment model. Conditional on the discountfactors m, the characterization in Proposition 1 implies a set of inequalitiesthat is linear in the unknowns (um, P

mt , q

mt ). Together with the linear

aggregation constraints on public good prices and private good quantities,FC-rationalizability can thus be tested by a standard linear programmingalgorithm. However, in our application the value of m is not given priorto the analysis. To solve this problem, we perform a double grid searchprocedure, which allows the discount factors for the two individuals (1 and

9 To be more precise, Cherchye, De Rock and Vermeulen concluded that the static col-lective consumption model has reasonable discriminatory power if assignable informationis available for the privately consumed quantities. If no such assignable quantity informa-tion is available, however, then power turns out to be rather low. The no commitmentmodel under consideration complies with the static model without assignable quantityinformation.

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2) to vary discretely between 0.50 and 1 (with steps of 0.01). This requires

solving about 2600 linear programs per household.10

The tests show that the full commitment model is rejected for all 148couples, i.e. no household behavior is FC-rationalizable. These results arerobust for any values of1 and 2 that we consider. Given the data set underconsideration, this should not be very surprising. The underlying assumptionthat the relative bargaining weights can be fully predetermined over a timespan of 10 years (with 8 observations) is indeed a very demanding premise.

From these results , we can conclude that full commitment is not a very re-alistic working assumption for real-life situations with a constantly changingenvironment. Similarly, Browning (1989) rejected perfect foresight over theentire time horizon when testing the nonparametric unitary life-cycle model.

The results also suggest that the collective model with full commitment hassubstantial power, which contrasts with our ndings for the no commitmentmodel. We see a more detailed power analysis of the full commitment modelunder consideration as an interesting avenue for further research.

4.2.3 Limited commitment

The above results indicate that neither of the two extreme (full commitmentand no commitment) models provides a completely satisfactory descriptionof observed couples behavior: the no commitment model has little power,and the full commitment model is rmly rejected. This section provides a

rst exploration of an intermediate collective model that puts more structureon the data than the no commitment model but less structure than thefull commitment model. Specically, we consider a limited commitmentmodel that focuses on continuous subsets of periods that satisfy the fullcommitment restrictions.11 The interpretation is that the bargaining weightis held constant within these subsets of periods while it can vary between thesubsets. Intuitively, the model assumes that households can only commit fora limited period of time.

10 As already stated in Section 2, Browning (1989) assumes = 1 when testing theunitary life-cycle model. Next, Crawford (2009) considers a grid search procedure with

ranging from 0.95 to 1, but the tests are run on quarterly instead of yearly data. Whencompared to these studies, we may safely conclude that our analysis is quite exible byconsidering the range 0.5 - 1.

11 A similar model was considered by Browning (1989) in his revealed preference analysisof the life cycle consumption model for the unitary context.

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Table 1 presents pass rates for this limited commitment model. It reports

the number (# pass) and percentage (% pass) of households that satisfy thecommitment restrictions for the rst consecutive T periods, starting fromperiod 1 (1994). The column with heading T = 8 corresponds to the fullcommitment model, which has been discussed before. As indicated above,the pass rate for this model is zero. We see that relaxing the constancy of thebargaining weight leads to ever increasing pass rates, up to the point wherethe model is no longer rejectable if T = 1.

Generally, the results in Table 1 suggest that intermediate models situatedbetween the full and no commitment models can be useful for empiricalanalysis of collective consumption behavior. First, we nd that the degree ofcommitment (measured by the number of consecutive periods for which full

commitment holds) can vary substantially across households. Subsequentresearch may relate this variation to dierences in the distribution factors Zacross households. Next, our results also indicate that limited commitmentmodels may have reasonable power. For example, we nd that only about43% of all households satisfy the model restrictions for T = 2, and the passrate decreases quite rapidly for higher T.

Table 1 : Limited commitment over the rst T periods (couples)Periods T = 8 T = 7 T = 6 T = 5 T = 4 T = 3 T = 2 T = 1# pass 0 1 5 11 37 47 63 148% pass 0 0.68 3.38 7.43 25 31.8 42.6 100

5 Summary and conclusions

We have developed a revealed preference methodology for analyzing house-hold behavior in terms of collective models that account for intertemporalconsumption relations. By focusing on Mazzoccos (2007) concept of com-mitment, we have provided revealed preference tests for the full commit-ment model and the no commitment model. These tests are intrinsicallynonparametric, i.e. they avoid imposing (typically nonveriable) paramet-ric/functional structure for the household decision process. We have argued

that comparing the test results for the two model specications can be usefulfrom a policy perspective.

We have applied our methodology to panel data drawn from the RussianLongitudinal Monitoring Survey (RLMS). For these data, we conclude thatthe no commitment model is not rejected but has low discriminatory power.

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By contrast, the more powerful full commitment model is rejected for all

households in our sample. Given this, we also provided a rst exploration ofan intermediate limited commitment model that is situated between the twoextreme (full commitment and no commitment) models. Our results suggestthat this limited commitment model has satisfactory power. In addition,it can provide a useful basis for analyzing varying degrees of commitmentacross households.

Given our results, we see a more extensive revealed preference analysisof collective models with limited commitment as an interesting avenue forfurther research. In doing so, one may also think of enriching the modelspecication by adding intertemporal aspects such as habit formation or ra-tional addiction. Crawford (2009) has provided a revealed preference analysis

of these aspects in a unitary setting.

Appendix 1: Proof of Proposition 1

(i) ) (ii): Suppose condition (i) of Proposition 1 holds for the allocationq1t ;:::;q

Mt ;Qt

, with t 2 f1;:::;Tg. Let denote the Lagrange multi-

plier associated with the household budget constraint. We get the followingrst order constraints for the household optimization problem under FC-rationalizability (with

@um(qmt;Qt)

@qmt

and@um(qm

t;Qt)

@Qt(m = 1;:::;M) the subgra-

dients for the function um at bundle (qmt ;Qt)):

@um(qmt ;Qt)

@qmt6

1

t1m

mpt; and

MXm=1

mt1m@um(qmt ;Qt)

@Qt6 Pt.

Under concavity of the individual utility functions um, we have

um(qms ;Qs)um(qmt ;Qt) 6

@um(qmt ;Qt)

@qmt

0(qms q

mt )+

@um(qmt ;Qt)

@Qt

0(QsQt)

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Now dene for each t

Pmt =m

t1m

@um(qmt ;Qt)@Qt

, m = 1;:::;M 1,

PMt = Pt M1Xm=1

Pmt , and

m =

m, m = 1;:::;M:

Then we obtain

um

(qm

s ;Qs) um

(qm

t ;Qt) 6

1

t1m m

pt0(qm

s qm

t ) +

1

t1m m

P

m

t 0(Qs Qt)()

um(qms ;Qs) um(qmt ;Qt) 6

1

t1mm [(pt0q

ms +P

mt 0Qs) (pt0q

mt + P

mt 0Qt)]

Using um(qms ;Qs)=m = ums , this gives condition (ii) of Proposition 1.

(ii) ) (i): Suppose condition (ii) of Proposition 1 holds. Dene

um(xm;X) = mins

ums +

s1m [ps0(x

m qms ) + Ps0(X Qs)]

Varian (1982) has shown that u

m

(qm

t ;Qt) = u

m

t .Then consider any consumption path xmt ;Xt such that

TXt=1

MXm=1

[pt0(xmt q

mt ) + P

mt 0(Xt Qt)] 6 0;

then we need that

MXm=1

mTXt=1

1

t1mum(xmt ;Xt) 6

MXm=1

mTXt=1

1

t1mum(qmt ;Qt)

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By denition, we have that

MXm=1

mTXt=1

1

t1mum(xmt ;Xt)

6

MXm=1

mTXt=1

1

t1m

umt +

t1m [pt0(x

mt q

mt ) + Pt0(Xt Qt)]

Without losing generality, we concentrate on m = 1 for every m = 1;:::;M,which obtains

M

Xm=1

mT

Xt=1

1

t

1m

um(xmt ;Xt)

6

MXm=1

mTXt=1

1

t1mumt

+

MXm=1

TXt=1

[pt0(xmt q

mt ) + Pt0(Xt Qt)]

6

MXm=1

mTXt=1

1

t1mumt

Since um(qmt ;Qt) = umt , we obtain that condition (i) of Proposition 1 is

satised.

Appendix 2: Proof of Proposition 2

(i) ) (ii): Suppose condition (i) of Proposition 2 holds for the allocationq1t ;:::;q

Mt ;Qt

, with t 2 f1;:::;Tg. Again, we let denote the Lagrange

multiplier associated with the household budget constraint, so that we getthe rst order constraints (with

@um(qmt;Qt)

@qmt

and@um(qm

t;Qt)

@Qt(m = 1;:::;M) the

subgradients for the function um at bundle (qmt ;Qt))

@um(qmt ;Qt)

@qm

t

61

t1m

m

t

pt, and

MXm=1

mt t1m

@um(qmt ;Qt)

@Qt6 Pt:

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Next dene for each t

Pmt =m

t1m

@um(qmt ;Qt)@Qt

, m = 1;:::;M 1,

PMt = Pt M1Xm=1

Pmt , and

mt =

mt, m = 1;:::;M:

Like before, we can use concavity of the utility functions um to obtain

um(qms ;Qs) um(qmt ;Qt) 6

mt pt0(q

ms q

mt ) +

mt P

mt 0(Qs Qt)

()um(qms ;Qs) u

m(qmt ;Qt) 6 mt [(pt0q

ms + P

mt 0Qs) (pt0q

mt + P

mt 0Qt)]

Using um(qms ;Qs) = ums obtains condition (ii) of Proposition 2.

(ii) ) (i): Similar to before, we dene

um(xm;X) = mins

ums +

s1m

ms [ps0(x

m qms ) + Ps0(XQs)]

;

with um(qmt ;Qt) = umt .

We again consider xmt ;Xt such that

TXt=1

MXm=1

[pt(xmt q

mt ) + P

mt (Xt Qt)] 6 0

In this case, we need

MXm=1

TXt=1

mt1

t1mum(xmt ;Xt) 6

MXm=1

TXt=1

mt1

t1mum(qmt ;Qt):

Without losing generality, we concentrate on mt =1mt

. Then we obtain

by denition that

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MXm=1

TXt=1

1

t1mmt u

m(xmt ;Xt)

6

MXm=1

TXt=1

1

t1mmt

umt +

t1m

mt [pt0(x

mt q

mt ) + Pt0(Xt Qt)]

6

MXm=1

TXt=1

1

t1mmt u

mt

+

MXm=1

TXt=1

[pt0(xmt q

mt ) + Pt0(Xt Qt)]

6

M

Xm=1

T

Xt=1

1

t1

m

mt umt

:

Since um(qmt ;Qt) = umt , we obtain that condition (i) of Proposition 2 is

satised.

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