No title

ECONO JOURNAL

OF THE ECONOMETRIC

SOCIETY

An InternationalSocietyfor the Advancementof Economic Theoryin its Relation to Statistics and Mathematics

CONTENTS Efficient Intra-Household M. BROWNING AND P. A. CHIAPPORI:

Allocations:

A General Characterization and Empirical Tests ..... ............. MARTIN L. WEITZMAN: The Noah's Ark Problem ...... ................ PETERC. B. PHILLIPS: New Tools for Understanding Spurious Regressions JOELL. HoRowITz:Bootstrap Methods for Median Regression Models ..... MATTHEW 0.

JACKSON AND THOMAS R. PALFREY: Efficiency

1241 ..

1279 1299 1327

and Voluntary

1353 Implementation in Markets with Repeated Pairwise Bargaining ..... Inference on Structural Parameters in InstruJIAHUI WANGANDERICZIVOT: mental Variables Regression with Weak Instruments ..... ......... 1389 NOTES AND COMMENTS:

EDIKARiI:Impartiality: Definition and Representation .....

........

1405

AARON S. EDLIN AND CHRIS SHANNON:Strict Single Crossing and the

Strict Spence-Mirrlees Condition: A Comment on Monotone ......................... Comparative Statics ...........

1417

ANNOUNCEMENTS................................................

1427 1427

NEWSNOTES...................................................

1435

NOMINATIONOF FELLOWS.

..........................................

PROGRAM OF THE FIFTEENTH LATIN AMERICAN MEETING OF THE ECONOMETRIC

SOCIETY .................................................. PROGRAM OF THE 1998 NORTH AMERICAN SUMMER MEETING OF THE ECONOMETRICSOCIETY.............................................

(1998 Volume Table of Contents is located on p. iii of this issue.)

VOL. 66, NO. 6-November, 1998

1439 1457

DOROTHY

J. HODGES, W7280 Oak Ridge Ct., Poynette,WI 53955, U.S.A. CO-EDITORS

RICHARD BLUNDELL, Dept. of Economics, University College, London, Gower St., LondonWC1E 6BT, U.K. ALAIN MONFORT, INSEE-CREST,15, BoulevardGabrielPeri-TimbreJ301, 92245 MalakoffCedex, France. ANDREW POSTLEWAITE, Dept. of Economics, University of Pennsylvania,3718 LocustWalk,Philadelphia,PA

19104, U.S.A. NANCY STOKEY,

Dept. of Economics, Universityof Chicago, 1126 E. 59th St., Chicago, IL 60637-1588, U.S.A. ASSOCIATEEDITORS

DONALDW. K. ANDREWS, Yale University HELMUTBESTER,Free University Berlin MARTINBROWNING, University of Copenhagen and

McMasterUniversity EDDIEDEKEL,Tel Aviv University RAYMOND J. DENECKERE, University of

Wisconsin-Madison MATHIASDEWATRIPONT, Universite Libre de Bruxelles J. DARRELL DUFFIE,Stanford University JEAN-MARIE DUFOUR,Universite de Montreal LARRYG. EPSTEIN,University of Rochester JoHN GEWEKE,University of Minnesota BRUCEHANSEN,University of Wisconsin-Madison GRANTHILLIER, University of Southampton BENGTR. HQLMSTROM, Massachusetts Institute of

Technology Bo E. HONORt,Princeton University JOELHOROWITZ, University of Iowa

Yuzo HoSoYA,TohokuUniversity MATHEW JACKSON, California Institute of

Technology SORENJOHANSEN, European University

LARRY NorthwesternUniversity JONES,

MICHIHIRO KANDORI,University of Tokyo DAVIDK. LEVINE,University of California,

Los Angeles ALBERTMARCET,Universitat Pompeu Fabra WHITNEYK. NEWEY,Massachusetts Institute of

Technology PIERREPERRON,Boston University MARTINE QuINzII, Universityof California, Davis PETER REISS, StanfordUniversity

ERIc RENAULT, INSEE-CREST JEANCHARLESROCHET,Universite des Sciences

Sociales, Toulouse JoHN RUST,Yale University LARRYSAMUELSON,University of Wisconsin-Madison MANUELS. SANTOS,University of Minnesota UZI SEGAL,University of Western Ontario RIcHARDJ. SMITH,University of Bristol JEANTIROLE,Universite des Sciences Sociales,

Toulouse JORGEN W. WEIBULL,Stockholm School of Economics ASHERWOLINSKY, Northwestern University WILLIAMR. ZAME,University of California,

Los Angeles

INGRIDSAYIED. EDITORIALSECRETARY:

SUBSCRIPTIONSCLAIMS DISTRIBUTION MEMBERSHIP,andMANUSCRIPTSUBMISSION(including style guidelines) informationis providedon the last page and inside back cover. Web Site Informationon the Society,how to join, programsof past meetingsand informationon futuremeetings,and muchadditionalmaterialcan be foundat the Society'sweb site:www.econometricsociety.org/es Typeset at Technical Typesetting Inc., Baltimore, Maryland 21227, U.S.A. Printed at Capital City Press, Montpelier,VT 05602, U.S.A. Copyright? 1998 by the Econometric Society. Permission to make digital or hard copies of part or all of this work for personal or classroomuse is grantedwithout fee providedthat copies are not made or distributed for profit or direct commercialadvantageand that copies show this notice on the first page or initial screen of a display along with the full citation, including the name of the author.Copyrightsfor components of this work owned by othersthan The EconometricSociety must be honored.Abstractingwith credit is permitted.To copy otherwise,;torepublish,to post on servers,to redistributeto lists, or to use any componentof this work in other works,requirespriorspecific permissionand/ora fee. Permissionsmay be requestedfrom Julie P. Gordon,Executive Directorand Secretary,The EconometricSociety, Dept. of Economics,NorthwesternUniversity,Evanston,IL 60208-2600, U.S.A. Econometricais publishedbi-monthly (ISSN 0012-9682) by the Econometric Society, Dept. of Economics, NorthwesternUniversity, Evanston, IL 60208-2600, U.S.A. Mailing agent: Capital City Press, P.O. Box 546, Montpelier,VT 05602. Periodicals postage paid at Evanston, Illinois, and additional mailing offices. U.S. POSTMASTER: Send all address changes to Econometrica, Blackwell Publishers Inc., Journals Dept., 350 Main St., Malden, MA 02148, U.S.A.

JOURNAL OF THE ECONOMETRIC SOCIETY An InternationalSocietyfor the Advancement of Economic Theoryin its Relation to Statistics and Mathematics

VOLUME66

1998

INDEX ARTICLES ANGRIST JOSHUAD.: Estimating the Labor Market Impact of Voluntary Military Service Using Social

249

Security Data on Military Applicants ............................................. BAI, JUSHAN, AND PIERRE PERRON: Estimating

and Testing

Linear

Models

with Multiple

Structural

47

Changes ................................................................... BLUNDELL,

RICHARD,

ALAN DUNCAN,

AND COSTAS MEGHIR:

Estimating

Labor Supply Responses

Using

............................................

Tax Reforms ...............

827

BROWNING,M., AND P. A. CHIAPPORI:Efficient Intra-Household Allocations: A General Characteriza-

tion and Empirical Tests ...............

1241

BUCHINSKY,MOSHE, AND JINYONG HAHN: An Alternative Estimator for the Censored Quantile Regres-

sion Model ..... CHEN,

.........................................................

XIAOHONG,AND XIAOTONGSHEN: Sieve Extremum

Q-CHIAPPORI,

Estimates

for Weakly

Dependent

Data

.....

P. A.: (See BROWNING)

(See ROCHET) Communication in Repeated Games with Imperfect Private Monitoring .... ..... DHAENE, GEERT, CHRISTIAN GOURIEROUX, AND OLIVIER SCAILLET: Instrumental Models and Indirect Encompassing .............................................................. DUFOUR, JEAN-MARIE, AND JAN F. KIVIET: Exact Inference Methods for First-Order Autoregressive Distributed Lag Models . ...................................................... DUFOUR, AND ERIC RENAULT: Short Run and Long Run Causality in Time Series: JEAN-MARIE, Theory .................................................................... DUNCAN, ALAN: (See BLUNDELL) EDLIN, AARON S., MARIO EPELBAUM, AND WALTER P. HELLER: Is Perfect Price Discrimination Really Efficient?: Welfare and Existence in General Equilibrium ...... ....................... ENGLE, ROBERT F., AND JEFFREY R. RUSSELL: Autoregressive Conditional Duration: A New Model Nfor Irregularly Spaced Transaction Data ....... ................................... EPELBAUM, MARIO: (See EDLIN) GILLESKIE,DoNNA: A Dynamic Stochastic Model of Medical Care Use and Work Absence GOURIEROUX, CHRISTIAN: (See DHAENE) GRANDMONT, JEAN-MICHEL: Expectations Formation and Stability of Large Socioeconomic Systems .................................................................... On the Role of the Propensity Score in Efficient Semiparametric Estimation of HAHN,JINYONG: Average Treatment Effects . ................................................... HAHN, JINYONG: (See BUCHINSKY) Scores for the HAJIVASSILIOU,VASSILIs A., AND DANIEL L. MCFADDEN: The Method of Simulated Estimation of LDV Models ........... ......................................... CHONE,

COMPTE,

PHILIPPE:

OLIVIER:

HECKMAN, JAMES, HIDEHIKOICHIMURA,JEFFREYSMITH, AND PETRATODD: Characterizing

Selection

597 673

79 1099

897 1127 1

741 315 863

Bias

1017

Using Experimental Data ................ (See PHILIPSON) HELLER, WALTER P.: (See EDLIN) HOROWITZ,JOEL L.: Bootstrap Methods for Median Regression Models . .... ................... ICHIMURA, HIDEHIKO: (See HECKMAN) Theoretic Approaches to IMBENS, GUIDO W., RICHARD H. SPADY, AND PHILLIPJOHNSON: Information Inference in Moment Condition Models ........ .................................. in Markets JACKSON, MATTHEWO., AND THOMAS R. PALFREY:Efficiency and Voluntary Implementation with Repeated Pairwise Bargaining ......... ..................................... JOHNSON, PHILLIP: (See IMBENS) HEDGES,

653 289

LARRY V.:

KANDORI, MICHIHIRO, AND MATSUSHIMA, HITOSHI: Private

Observation,

Communication

1327

333 1353

and

Collusion .................................................................. KIM,JAE-YOUNG: Large Sample Properties of Posterior Densities, Bayesian Information Criterion and the Likelihood Principle in Nonstationary Time Series Models .... .................... KIvIET, JAN F.: (See DUFOUR)

627 359

INDEX

iv

LEWBEL, ARTHUR:

Semiparametric Latent Variable Model Estimation with Endogenous or

Mismeasured

Regressors

105

......................................................

(See KANDORI) DANIEL L.: (See HAJIVASSILIOU) MCFADDEN, MEGHIR, COSTAS: (See BLUNDELL) PALFREY, THOMAS R.: (See JACKSON) PERRON, PIERRE: (See BAI) MATSUSHIMA,

HIToSHI:

PRELEC, DRAZEN : The ProbabilityWeightingFunction.....................................

381 1299 497

(See DUFOUR) (See VAN DEN BERG) ROBINSON, P. M.: Inference-Without-Smoothing in the Presence of Nonparametric Autocorrelation.................................................................

1163

PHILIPsoN, TOMAS, AND LARRYV. HEDGES: Subject

in Social

Evaluation

PHILLIPS,PETER C. B.: New Tools for Understanding

Spurious

Experiments

Regressions

....

....

.......... .................

RENAULT, ERIC: RIDDER,

GEERT:

ROCHET, JEAN-CHARLES,

AND PHILIPPE CHONE: Ironing,

Sweeping,

and Multidimensional

Screening.................................................................. ALVIN E.: (See SLONIM) RUSSELL, JEFFREY R.: (See ENGLE) SAKATA, SHINICHI, AND HALBERTWHITE: High BreakdownPoint ConditionalDispersion Estimation with Applicationto S&P 500 Daily ReturnsVolatility. ..............................

783

ROTH,

SANTOS,

MANUEL

S., AND JESUS VIGO-AGUIAR:

529

Analysis of a Numerical Dynamic Programming

AlgorithmApplied to Economic Models ....... ................................... (See DHAENE) SHEN, XIAOTONG: (See CHEN) SLONIM, ROBERT, AND ALVINE. ROTH: Learning in High StakesUltimatumGames:An Experimentin the Slovak Republic .......................................................... SMITH, JEFFREY: (See HECKMAN) SPADY, RICHARD H.: (See IMBENS) TODD, PETRA: (See HECKMAN)

409

SCAILLET, OLIVIER:

569

VAN DEN BERG, GERARD J., AND GEERTRIDDER: An Empirical Equilibrium Search Model of the Labor

Market .

1183

...................................................................

(See SANTOS) VOGELSANG,TIMOTHYJ.: TrendFunctionHypothesisTestingin the Presenceof Serial Correlation... WANG, JIAHUI, AND ERIC ZIVOT: Inference on Structural Parameters in InstrumentalVariables Regressionwith WeakInstruments............................................ .......................................... WEITZMAN, MARTIN L.: The Noah's Ark Problem WHITE, HALBERT: (See SAKATA) YANO, MAKOTO:On the Dual Stabilityof a von NeumannFacet andthe InefficacyofTemporaryFiscal Policy ..................................................................... ZIVOT, ERIC: (See WANG) VIGO-AGUIAR,

JESOS:

123 1389 1279

427

NOTES AND COMMENTS Serial Dictatorship and the Core from RandomEndowmentsin House Allocation Problems...... .......................... AUMANN, ROBERT J.: CommonPriors:A Reply to Gul...................................... ABDULKADIROGLU,ATILA, AND TAYFUN SONMEZ: Random

BROWN, BRYAN W., AND WHITNEY K. NEWEY: Efficient

Semiparametric

tations.................................................................... CHAPMAN, DAVID A.: HabitFormationandAggregateConsumption.....

Estimation

689 929

of Expec-

......................

453 1223

DAGAN, NIR, ROBERTO SERRANO, AND OSCAR VOLIJ: Comment on McLennan and Sonnenschein

'SequentialBargainingas a Non-CooperativeFoundationfor WalrasianEquilibrium' .......

1231

INDEX

DEKEL, EDDIE, BARTON L. LIPMAN, AND ALDO RuSTICHINI: Standard State-Space Unawareness

V

Models Preclude

159

...............................................................

EDLIN, AARON S., AND CHRIS SHANNON: Strict Single Crossing and the Strict Spence-Mirrlees Condi-

tion: A Commenton Monotone ComparativeStatics.1427 CointegrationMethodswhen RegressorsAlmost Have Unit Roots.. FARUK: A Commenton Aumann'sBayesianView.923

ELLIOTT,GRAHAM: On the Robustnessof GUL,

KARNI, EDI: Impartiality: LIPMAN,

BARTON

NEWEY,

WHITNEY

Definition

and Representation

149

.1415

L.: (See DEKEL) K.: (See BROWN)

(See DEKEL) (See DAGAN) .SHANNON, CHRIS: (See EDLIN) SONMEZ, TAYFUN: (See ABDULKADIROGLU) VOLIJ, OSCAR: (See DAGAN) RuSTICHINI, SERRANO,

ALDO:

ROBERTO:

ANNOUNCEMENTS AND NEWSNOTES 175,465,703,939,1234,1445 205 1997 ELECTION OF FELLOWS TO THE ECONOMETRIC SOCIETY . 713 ................................... FELLOWS OF THE ECONOMETRIC SOCIETY AS OF JANUARY, 1998.................................. 718 1998 FRISCH MEDAL AWARD ...................................................... 703 ANNOUNCEMENTS................................................ ECONOMETRICA

REFEREES

1996-1997....................................................

MEMBERSHIPDUES .......................

.........................................

175,465

182,473,709,945,1239,1453 NOMINATION OF FELLOWS .............. 175,465,1445 PROGRAM OF THE 1997 AUSTRALASIAN MEETING OF THE ECONOMETRIC SOCIETY .... 217 ................. 949 PROGRAM OF THE 1997 EUROPEAN MEETING OF THE ECONOMETRIC SOCIETY ..... ................... PROGRAM OF THE 1997 FAR EASTERN MEETING OF THE ECONOMETRIC SOCIETY ..... 231 ................. PROGRAM OF THE 1997 NORTH AMERICAN SUMMER MEETING OF THE ECONOMETRIC SOCIETY ............ 477 996 PROGRAM OF THE 1998 NORTH AMERICAN WINTER MEETING OF THE ECONOMETRIC SOCIETY .... ........ REPORT OF ACTIVITIES OF THE INDIA AND SOUTH EAST AsIA REGION ...... 214 ........................ REPORTOF THE EDITORS . 202 ............................................................. 211 REPORT OF THE EDITORS OF THE ECONOMETRIC SOCIETY RESEARCH MONOGRAPH SERIES .... ........... REPORTOF THE PRESIDENT . 711 ............................................................. REPORTOF THE SECRETARY. 185 ........................................................... 194 REPORTOF THETREASURER. ...................................................... SUBMISSIONOF MANUSCRIPTSTO THEECONOMETRICSOCIETYMONOGRAPHSERIES ..... ............... 213,947

NEWs NOTES

..............

Econometrica,

Vol. 66, No. 6 (November, 1998), 1241-1278

EFFICIENT INTRA-HOUSEHOLD ALLOCATIONS: A GENERAL CHARACTERIZATION AND EMPIRICAL TESTS BY M. BROWNINGAND P. A. CHIAPPORI1

The neoclassicaltheory of demandapplies to individuals,yet in empiricalwork it is usuallytaken as valid for householdswith manymembers.This paperexploreswhat the theoryof individualsimpliesfor householdsthat have more than one member.We make minimalassumptionsabout how the individualmembersof the householdresolve conflicts.All we assumeis that howeverdecisionsare made,outcomesare efficient.We refer to this as the collectivesetting.We showthat in the collectivesettinghouseholddemands must satisfya symmetryand rankconditionon the Slutskymatrix.We also presentsome furtherresultson the effects on demandsof variablesthat do not modifypreferencesbut that do affect how decisionsare made. We applyour theory to a series of surveysof householdexpendituresfrom Canada. The tests of the usualsymmetryconditionsare rejectedfor two-personhouseholdsbut not for one-personhouseholds.We also show that incomepoolingis rejectedfor two-person households.We then test for our collectivesetting conditionson the couplesdata.None of the collectivesettingrestrictionsare rejected.We concludethat the collectivesettingis a plausibleand tractablenext step to take in the analysisof householdbehavior. KEYWORDS:

Intra-householdallocation,householdbargaining,collectivemodel,Slutsky

mairix.

1. INTRODUCTION WHEN CONSIDERING HOUSEHOLDBEHAVIORand welfare it

is almost universally assumed that the many-personhousehold can be treated as though it has a single set of goals. The adoptionof this "unitary"model is very convenient,if only because standardtools of consumer analysiscan then be applied at the household level. Methodologically,however,it stands on weak grounds.Neoclassical utility theory applies to individualsand not to households.2There is also mounting empirical evidence that the unitary model does not hold. In particularthe fundamentalobservableimplicationof utility theory-symmetry of the Slutskymatrix-is regularlyrejectedon householddata (see, for example, Blundell, Pashardes, and Weber (1993) and Browning and Meghir (1991)). Further disquiet is given by the universal rejection of the "income pooling"

1We thank G. Becker, R. Blundell,F. Bourguignon,I. Ekeland,J. Heckman,G. Laroque,two referees,and the participantsat manyseminarsfor comments.The researchwas supportedin part by the CanadianSSHRC. 2Two majorcontributionshave triedto reconcilethe unitarymodelwith the fact that households may consist of more than one decision maker.However,Samuelson's(1956) idea of a household welfarefunctionrelies upon the ad-hocidea that the latter(and in particularthe respectiveweights given to each member'sutility)is independentof prices and incomes.While Becker's'rotten kid' theorem (see Bergstrom(1989) for a statement) appears sounder, it still requires transferable preferencesand a specificdecisionprocessto hold true.

1241

1242

M. BROWNING AND P. A. CHIAPPORI

property of the unitary model, that is, the implication that the source of householdincome should not have any effect on allocationsonce we condition on total expenditure(see, for instance, Thomas (1990), Schultz (1990), Bourguignon,Browning,Chiappori,and Lechene (1993), Phipps and Burton (1992), and Lundberg,Pollak, and Wales (1997)). These rejections have either been seen as a rejection of utility theory or have been attributedto specification problems(inadequatefunctionalforms, inappropriateseparabilityassumptions, misspecificationof the stochastic structure, and so on). Thus it has been concludedeither that utilitytheory is false or that it is untestable. Our answerto these "problems"with neoclassicalutilitytheoryis completely different.We claim that the theory has not been taken seriouslyenough. We start from the premisethat utilitytheorydoes apply,but only to individualsand not to households. In this paper we present a general characterizationof an alternative model of household behavior to the unitary model, namely the "collective"model suggestedin Chiappori(1988a and 1992). The two assumptions of the collective model are that each person in the household has his or her own preferencesand that collective decisions are Pareto efficient. Under these assumptions,we exhibit a set of theoretical propertiesthat have to be fulfilledby householddemands,and can thus be seen as a generalizationto the multi-personsettingof Slutskysymmetryin the unitaryframework.We then test the conditionson a sample of Canadianhouseholds. The idea of explicitlymodelling household behavior as a collective process can be traced back at least to Becker'sseminal work (see Becker (1991) for a recent exposition).Also, it has been clear for some time that a multi-,person approachmight well (and actuallyshould) lead to violations of the predictions from the unitarymodel. For instance,this point is emphasizedby Bourguignon (1984) within a noncooperativesetting and Pollak (1985) using a "transaction approach." Severalmodels have explicitlymodelledintra-householddecisionmakingas a cooperativeprocess.The Nash bargainingrepresentationof familydecisions,as initiated by Manser and Brown (1980) and McElroyand Horney (1981), is of particularinterestfor our presentpurpose.At the core of this approachare two interestingideas. One is that,withina collectiveframework,householddemands should be sensitive to the intra-householddistributionof resources,and more generallyto any environmentalvariablethat may influencethe decisionprocess. -say, througha shift in threat points ("EEP's"in McElroy(1990) terminology,' or "distributionfactors"in Browninget al. (1994)). This has given rise to the literatureon testing for "incomepooling"that was referencedabove. In previous papers with other co-authors(Bourguignonet al. (1993), Browninget al. (1994), and Bourguignon,Browning,and Chiappori(1995)) we contributedto this line of researchby investigatingwhat could be learned from conventional family expendituredata about what goes on inside the household.In the third paper, in particular,we showed that the collective setting imposes testable restrictions upon the way in which distributionfactors can enter demand

INTRA-HOUSEHOLD ALLOCATIONS

1243

equations;moreover,we investigatethe conditionsunderwhich the observation of household demandsenables us to identify individualEngel curves and the form of the decision process. The key point, however, is that this analysis requiresonly cross-sectionvariationin the data; that is, we did not exploit any price variation.

More relevantfor the present paper is the second intuitionput forth in the Nash-bargainingliterature-namely, that the repeated rejections of Slutsky symmetryin empiricalwork may occur because household decisionscannot be crammedinto an overly restrictiveunitaryframework.This suggests that the case where price variationscan be observed deserves careful investigation.In this framework,a very natural question arises: can one derive restrictive, testable implicationsof the Nash-bargaining frameworkupon demandfunctions, that could be seen as the counterpart(or, more precisely,the generalization)of Slutskysymmetryand negativenessin the unitary case? This is precisely the topic of the present paper. Importantas it is, it is fair to say that this question has not received a convincinganswerso far (see Chiappori(1988b,1991)and McElroyand Horney (1990)).One contributionof the presentpaperis to fill this gap. In what follows, we actually solve a more general problem-namely, what does the efficiency assumptionalone implyfor householddemands,and specificallyfor the form of the Slntskymatrix? Though we do not formallyjustify the efficiencyassumption,we do believe that it has a good deal of intuitiveappeal.For one thing,the householdis one of the preeminent examples of a repeated "game" so that we feel justified in assumingthat each person knows the preferences of the other people in the household. Given this symmetryof informationand the fact that the game is repeated, it is plausible that agents find mechanismsto supportefficient outcomes; as it is well known, cooperationoften emerges as a long-termequilibA second point is that efficiency rium of repeated noncooperativeframeworks.3 is probablythe most naturalgeneralizationto the multi-personsetting of utility maximizationin standardmodels.In particular,the collectivemodelwe consider includes the unitaryrepresentationas a (very) special case; hence, the conditions we derive generalize in a straightforwardway Slutskysymmetry-a fact that leads directlyto nested tests. Finally,axiomaticmodels of bargainingwith symmetricinformationgenerallyassumeefficientoutcomes.This is the case, for instance, of all models developed so far in the Nash-bargainingapproach.In other words,the "collective"frameworkwe considerin this paper encompasses all cooperativemodels existingin this literature.As a consequence,the condi3Thisis not to say, however,that we cannotenvisioncircumstancesthat wouldlead to inefficient outcomes.Clearly,if there is asymmetricinformation(for example,one partnercan consumesome goodswithoutthe otherpartnerknowing),then the case for efficiencyis weakened.In the end this is an empiricalmatter:what does the collectivesetting implyfor householdbehaviorand are these predictionsrejectedby the data?This paperis directedto these issues.

1244

M. BROWNING AND P. A. CHIAPPORI

tions we shall derivefrom the efficiencyassumptionalone apply,a fortiori,to all these models as well.4 Our main purposeis the derivationof testable implicationsof the collective framework.An immediateimplicationis that we must adopt the least restrictive set of assumptionspossible.Ideally,the conditionswe are seeking shouldresult from the efficiency axiom only, with no additional ("auxiliary")assumption required.In this spirit,we do not suppose that the econometriciancan determine which goods are private and which public within the household; any commoditymay be either public, or private, or both. Moreover,we do not assumethat the individualconsumptionof privategoods is observable.Similarly, we do not introduceany particularassumptionon individualpreferences,except that they can be representedby conventionalutilityfunctions.That is, we allow for intra-householdconsumptionexternalities,altruism,etc. Despite this explicitlyminimalistset of assumptions,we show that one can makevery specificpredictionsabout householdbehavior.The principaltheoretical result of the paper is that althoughSlutskysymmetryneed not hold in the collective setting, it can be generalizedin a straightforwardway; namely, the Slutskymatrixhas to be equal to the sum of a symmetricmatrixand a rankone matrix. This strong theoretical property is a consequence of the efficiency hypothesisalone. This basic result is presentedin Section 2. In Section 3 we extendthe analysis in three differentdirections.The most importantof these extensionsis to allow for distributionfactors(as alludedto above),formallydefinedas variableswhich do not enter individualutilities directlybut that do affect distributionwithinthe household. It turns out that the collective model implies that there is h close relationshipbetween the influence of such variables on demand and price responses. The second part of the paper is empirical, and is aimed at testing our predictionson household data. From a general viewpoint, the case of price variationsthat we considerhere has implicationsfor two areas:demandanalysis on time series of familyexpendituresurveys(for example,the U.K. FES or the U.S. CEX) and the analysis of labor supply on cross-sections(or panel data) where the prices that vary across individualsare wages. Although the latter is the more importantapplication,we have chosen initiallyto concentrateon the former since the analysisof labor supplyfor individualsraises many problems that are less pressingin the demandcase (for example,wages maybe nonlinear,endogenous,and unobservedfor some individuals). In Section 4 we present a flexible parametricdemandsystem and derive the implicationsof the predictionsof the previous sections for the parametersof this system.This includes a novel analysisof testing for the rank of a matrixin our context.In Section5 we presentempiricalresultsusingthe CanadianFamily 4The specificconcept of Nash-bargainingcan actuallybe viewed as a way of determiningthe location of the final outcome in the Pareto set. Whether this particularassumptionimplies additionalrestrictionsupon observedbehavioris still an open question.

INTRA-HOUSEHOLD ALLOCATIONS

1245

ExpenditureSurvey(FAMEX)data on single personhouseholdsand households containingjust a marriedcouple. We first show that Slutskysymmetryis not rejectedfor singlesbut it is for couples.To the best of our knowledgethis is the first time that anyone has shown that symmetryis not rejected for singles. We then go on to test the predictionsof the collectivesetting derivedin Sections 2 and 3 on the couples data. We do not reject any of these restrictions.This provides strong, though preliminary,support for our view that the collective model is a viable alternativeto the unitarymodel. In the concludingsection we discusssome possible areas of future research. 2.

THEORY-THE

GENERAL CASE

2.1. The Collective Setting 2.1.1. Preferences

We consider a two person (A and B) household.Household purchases5are denoted by .the n-vector q with associated market price vector p. Household demandsare dividedbetween three uses: privateconsumptionby each person, qA and qB, and public consumption Q. Each good may serve several uses simultaneously;6public and privateconsumptionvectors are only linked by (2.1) \

qA + qB + Q

q.

The householdbudget constraintis (2.2)

pt(qA

+ qB + Q) = p'q=x

where x denotes total expenditure. As said before, we adopt a Beckerianframeworkin which each memberhas her or his own preferencesover the goods consumedin the household.Whether consumptionof a particulargood by a particularperson is, by nature, private, public,or both is irrelevantfor our results.Also, each member'spreferencescan depend on both members'privateand publicconsumption(the "altruistic"case in Bourguignonand Chiappori(1992)); this allows for altruism,but also for externalitiesor any other preferenceinteraction.Our resultsare consistentwith all possibleinteractions.We only assumethat preferences,definedon (qA, qB, Q), are "well-behaved"in the usual sense: 1: Member I's preferences (I = A, B) can be representedby a utility function of the form uI(qA, qB, Q) that is stronglyconcave and twice differentiablein in (q', Q). (qA, qB, Q), and strictlyincreasing AXIOM

5Formallypurchasescouldincludeleisure(so that the pricevectorincludesthe wages-or virtual wages for nonparticipants-of A and B). As alreadyindicated,we shall not be emphasizingthe implicationsof our resultsfor laborsupply.Also, we only considera static model, and assumethat all goods are nondurables. 6Forinstance,expenditureson "telephoneservices"includesa publicelement (the rental)and a privateelement (the actualuse of telephone).

1246

M. BROWNING AND P. A. CHIAPPORI

Note that we do not impose that u'( ) is increasingin qJ for J not equal to I; that is, we allow for selfishness or even negative consumption externalities between members. 2.1.2. The Decision Process

We now considerthe mechanismthat the householduses to decide on what to buy. Note, first,that if the functionsuA and uB representthe same preferences, then we are back in the conventional"unitary"model;then the commonutility is maximizedunder the budget constraint.Alternatively,we could assume that one of the partnerscan impose her (or his) preferencesand use the corresponding utility function in the traditionalway; this also yields a unitarymodel. But these are highlyspecificassumptions.In general,the "process"that takes place within the householdis more complex. As stated in the introduction,our approachat this point is axiomatic;we postulate efficiency,as expressedin the followingaxiom: AXIOM 2: The outcome of the household decision process is Pareto efficient; that is, for any price-income bundle (p, x), the consumption vector (qA, qB, Q) chosen by the household is such that no other vector (4A, qB, Q) in the budget set could make both members betteroff.

Following Chiappori (1992), we refer to models that allow for different preferences with efficiency as the "collective"setting. Finally, we add some structureby assumingthe following: AXIOM 3: There exists a differentiable,zero-homogeneousfunction ,u(p, x) such that, for any (p, x), the vectors (qA, qB, Q) are solutions to the program:

(2.3)

max qA, qB,

p.(p, x)

Q

subject to

uA(qA,

qB, Q) +

[1 - ,t(p,

x)] .uB(qA ,qB,Q)

p * (qA + q B+ Q) =x.

As it is well-known,any point on the Pareto frontier can be obtained as a solutionto a programof this type (for some well-chosen ,). Axiom 3 essentially postulatesthat the decision process alwayshas a unique,well-definedoutcomi or, in other terms,that there exists a demandfunction(and that, in addition,the latter is continuousand zero-homogeneous).Homogeneityis uncontroversial;it essentiallymeans that expressingprices and incomes in cents instead of dollars does not change actual behavior.The smoothnessassumptionis standard,and made for analyticalconvenience.7Uniqueness, on the other hand, is a real 7The key point that drives the results is that the set of efficient outcomes is one-dimensional-a property that could be derived without a smoothness (or even uniqueness) assumption. However, its more natural (and more tractable) development is in terms of (n - 1) dimensional manifolds, which requires smoothness.

INTRA-HOUSEHOLD ALLOCATIONS

1247

assumption, albeit not an extremely restrictive one. Two points should in particularbe emphasized: (i) The assumptionis consistentwith our generalframework,whichpostulates efficiency. Indeed, a natural (although not exclusive)justificationis that the members play some cooperativegame under symmetricinformation.In most cases, this should lead to a unique outcome.8 Note, in particular,that all bargainingmodels developedin the literatureexhibit the same property,since they are based upon a specific bargainingequilibriumconcept (Nash, KalaiSmorodinsky,...). (ii) From an appliedviewpoint,assumingthe existence of a demandfunction does not seem undulyrestrictive.To the best of our knowledge,most (if not all) existingempiricalwork on demandrelies upon a similarassumption. The "distribution"function ,u summarizesthe decision process. Take some given utility functions uA and uB. Then the budget constraintdefines, for any price-incomebundle,a Paretofrontier.FromAxiom 2, the final outcomewill be located on this frontier.Then ,u determinesthe final location of the demand vector on this frontier. The parameter,u has an obviousinterpretationas a "distributionof power" function. If , = 1 then the household behaves as though A alwaysget their way, whereas if u= 0 it is as though B is the effective dictator.For intermediate yalues, the householdbehaves as though each person has some decision power.Note that ,u will generallydependon pricesand total expenditures,since these environmentalvariablesinfluence the distributionof "power"within the household. Two additionalpoints may be noted at this stage. One is that, in general, ,u may also depend on other factors, such as the individualincomes of the two partners, or any factor of the household environmentthat may affect the decision process ("distributionfactors"in Browninget al. (1994)). This idea is explored in the next section; for the moment, let us first investigate the propertiesof the basic model. Also, assume preferencesare identical.Then we are back in the unitarysetting and ,u is not defined.However,we can then use the conventionthat , = 0 (or, as a matterof fact, any other convention). Any given (demand)function q(p, x) is said to be compatible with collective rationalityif and only if there exist functionsqA(p, x), qB(p, x), Q(p, x), solution of a programof the type (2.3), such that q(p,x) = qA(p, x) + qB(p, x) + Q(p, x). A first propertyof such functionsis given by the followingresult: PROPOSITION1: Assume that q(p, x) is compatible with collective rationality. Then it is zero-homogeneous, continuouslydifferentiable,and satisfies p'q(p, x) = x. 8This is in sharp contrast with noncooperativegames, or with models of bargainingunder asymmetricinformation-where multiplicityof equilibriais more difficultto rule out. Of course, such modelsare in generalincompatiblenot onlywith the uniquenessassumption,but withAxiom2 as well, since the outcomewill typicallyviolate efficiency.

1248

M. BROWNING AND P. A. CHIAPPORI

In the following,our goal is to deriveadditionalpropertiesof these functions. 2.1.3. Household Utility

The next step is to define what we shall call the household utility function. The latterwill be reminiscentof the unitarysetting,but with the differencethat it will depend on ,u. Formally,we have the followingdefinition. DEFINITION 1:

In the collectivesetting,the household utilityfunction is defined

as (2.4)

max guA(q,qB,

UH(q,g)=

qA

qB

Q) + (1

Q)

).uB(qA,qB,

Q

subjectto qA + qB +

Q=

q.

Clearly,the maximizationof the householdutility under the budget constraint will lead to the same demandfunctionas program(2.3) above.Two points must be stressedhere: (i) The household utility function uH will depend on prices and income as soon as ,u is a function of these variables.So we are in a case of price-dependent preferences, which explains why the usual results of consumer theory (Slutskysymmetry,etc.) will no longer hold true in the collectivecontext. (ii) However, prices and income enter only through the scalar fuinction ,u. The same will also be true of any other variablethat affects the decisionprocessbut not preferences.This remarkwill be crucial in the derivationof the results below. 2.2. Dual Representationsof the CollectiveProgram

Given utilityfunctionsfor the two people we can define a dual representation of "household"preferences.This can be done in two equivalentways.First,for any ,u, define the householdindirectutilityfunction V(p,x, ,) as the maximand of the initial optimizationproblemabove: (2.5)

V(p, x,

)

=

max qA ,qB,

Q

subjectto p *(qA + which can also be writtenas

+

(1

_ g)uB(qA

qB, Q)

q B+ Q) =x,

V(p, x, ,) = max uH(q, ,)

(2.6)

Q)

,UuA(qA,qB,

subjectto

q

p'q =x.

In what follows, let q = f(p, x, ,) denote the solution of this program-that is, the collectivecounterpartof Marshalliandemands;note that f() is a functionof ,u as well.

INTRA-HOUSEHOLD ALLOCATIONS

1249

Now, we know,from the envelope theorem,that 8V(p,x,)/8pi

8V(p, x, ,)4/dx

which is the equivalent,in the collectivesetting, of Roy's identityin the unitary case. This means that, for any constant ,u, an infinitesimal change in one price,

say dpi, can be "compensated"(in the sense that the householdutilitywill not change) by a change in income exactly equal to dx = qi dpi. Of course, each member'sutilitywill, in general,change. The correspondingexpenditurefunctionwill be definedas (2.7)

)

E(p, u,

=

min p (q + qAq qB,

L.UA(qA,

qB

Q) +

Q

(1 -)*u

qB

+ Q)

B(qA,qB,

subject to Q) ? U.

The analogywith traditionalduality theory can in fact be pushed somewhat further.First,the expenditurefunction E(y) is linear homogeneousand concave in p. Also, let h(p, u, ,) denote the solution of program (2.7). Note that h(p,u, ,) can be interpretedas a compensateddemandfunction(since it is the demand that obtains holding household utility constant). It is important to stress, however,that h(-) is definedas a functionof the "distributionof power" index ,u()-that is, ,u must also be kept constant. Again from the envelope theorem,we have dE(p, u, u) dpi

Dualitybetween programsimpliesthat f(p, E(p, u, ,), ,) = h(p, u, ,u).

It follows that df1 dpj

dff ax f

dhi dpj

This is equivalentto Slutskyconditionsin the unitarycase. In particular,the matrix X with general term

07ij-

df1 dfx -1 dp + dx

can be interpretedas the partial derivativesof demandswith respect to prices, holding both household utility and the "distributionof power" index ,u constant.

1250

M. BROWNING AND P. A. CHIAPPORI

2.3. Restrictionson Demands

We now derive our main result, which characterizesthe properties of observed demand functions. What has to be emphasizedhere is that we never observe the function f(p, x, ,). Indeed, by definition, f(p, x, ,) describes how demands respond to independentvariations of p, x, and ,t. But we do not

observe such independentvariation.For any given price-incomebundle (p, x), the behaviorwe observe correspondsto one specificvalue of ,u-namely, the value ,(p, x) taken at this point by the specific distributionfunction that characterizesthe householdat stake. In otherwords,whatwe actuallyobserveis the demandfunction g definedby (p, x) = f(p, x, ,u(p, x)).

The question, now, is which predictionsdoes the collective setting imply for observeddemandfunctions {(p, x)? A first, elementarypropertywas given in Proposition1 above: demands {(p, x) are zero-homogeneousand continuously differentiablein (p, x) and satisfyadding-up: V' (P, x) = x.

Of course,we are interestedin deeper and more structuralproperties.To derive these, we first define the pseudo-Slutskymatrixassociatedwith {(p, x) as S=

+ {x

where gp is the (n x n) Jacobianmatrixof partialsof g with respect to p, and {x the vector of partialsof g with respect to x. In the unitarysetting, S would be symmetricand negative semi-definite.In the collective model, this property generalizesas follows: PROPOSITION 2: In the collective setting, the Pseudo-Slutskymatrix S is the sum of a symmetricand negative semi-definite matrix X and an outerproduct:

S = X + uv'

where u and v are n-vectors with ui =

dfj d,u

and

v =

d,u +-f

dpi

d,u . dxJ

The interpretationof this formulagoes as follows. Assume that the price of good j is changed by an infinitesimalamount dpj, the change being compensated by an increase in income dx = qj*dpj. What will be the effect of this on the demandfor good i? The formulasays that this effect can be decomposedinto two components.One correspondsto a substitutioneffect: holdingboth household utility V and power index ,u constant, the change in price will induce a reallocationof consumption,as definedby the correspondingterm in matrix 3. But, on the top of this, such a changewill also modify ,u;precisely, d,=d*dpj+a*dx

=

-+

q

INTRA-HOUSEHOLD ALLOCATIONS

1251

and hence the v vector. This, in turn,will change consumptionof good i by an amount dfj d,u

dqi =-* -d, as indicatedby the u vector. The following corollary states a consequence that will be useful in the following. COROLLARY 1 (SR1 Property): In the collective setting, the pseudo-Slutsky MatrixS is the sum of a symmetric, negative semi-definitematrix I and a matrixR that has at most rank one.

This SR1 ("symmetricplus rank one") condition obviouslygeneralizes the unitarymodel (since R = 0 in the latter).This propertyis somewhatreminiscent of the Diewert-Mantelaggregationrestrictionsfor economieswith more goods than agents;see Shaferand Sonnenschein(1982) for an overview. A geometricinterpretationof SR1 is the following.Remember,first,that for any given pair of utilities,the budget constraintdefinesthe Pareto frontieras a functionof the price-incomebundle;then ,u determinesthe locationof the final outcomeon the frontier.Assume,now, that pricesand income are changed.This has two consequences.For one thing, the Pareto frontierwill move. Keeping ,u constant,this would change demandin a way describedby the I matrix.Note, however,that this changewill not violate Slutskysymmetry;that is, its natureis not differentfromthe traditional,unitaryeffect. The second effect is that , will also change;this will introducean additionalmove of demand along the (new) frontier. This change (as summarizedby the R matrix) does violate Slutsky symmetry(in general). But moves along a one-dimensionalmanifoldare quite restricted.For instance,the set of price-incomebundlesthat lead to the same ,u is likely to be quite large in general;indeed, under our smoothnessassumption, it is an (n - 1)-dimensionalmanifold.Consideringthe lineartangentspaces,this means that there is a whole hyperplanesuch that, if the (infinitesimal)changein prices and income belongs to that hyperplane,then no deviationfrom Slutsky symmetry can be observed. In other words, the SRI condition is a direct consequence of the fact that, in a 2-person household, the Pareto frontier is of dimension 1, whateverthe number of commodities. 2.4. Testingfor SRI

How can a propertylike SR1 be tested? The resultwe exploitis that a matrix S is SR1 if and only if the antisymmetric matrix M = S - S' is of rank at most 2 (rememberthat a matrix M is antisymmetricif M'= -M). A more precise

statementis the following: LEMMA1: (i) Let S be some SR1 matrix: S = X + uv'

1252

M. BROWNING AND P. A. CHIAPPORI

and assume that S is not symmetric.Then vectors u and v are linearlyindependent; the matrix M = S - S' is of rank 2, and Im(M) (the subspace spanned by the columns of M) is spanned by u and v. (ii) Conversely, let M be an antisymmetricmatrix of rank 2, and let iu and v be arbitraryindependent vectors of Im(M). There exists a scalar A # 0 that M = A(uv' - vW')= uv' - vu' where u = Au, v = v. In particular, for any symmetric matrix I, the matrix S = X + uv' is such that M = S - S'.

What is importantfor our purposeis that, accordingto this result, testing for the collectivemodel amountsto testingfor the rankof matrixM = (S - S'). The collectivemodel (withtwo decisionmakers)predictsthis rankshouldbe at most two, while it would be zero in the unitary case. This will be crucial in the empiricalsections below. A final remarkis that antisymmetryhas specificimplicationsfor the rank of M. These are given by the followingLemmas: LEMMA 2: All the nonzero eigenvalues of a real antisymmetric matrix are imaginary. In particular, a real antisymmetricmatrix has even rank. LEMMA 3: Let M= (mik) be any nonzero, real antisymmetric matrix, and assume, without loss of generality, that Mi12 is not equal to 0. Then M has rank 2 if and only if, for all (i, k) such that k > i > 2, mlUm2k - mlkm2i Mik=

l

Thus the elements of rows3 to n in M are functionsof the elements of the first two rows(the same is true for columns).Since this characterizationonly involves parametricrestrictionsof the familiarsort it is easy to test. Note that for an (n x n) matrix this involves (n - 2)(n - 3)/2 restrictions. As a benchmark, testingfor Slutskysymmetryinvolvesn(n - 1)/2 restrictions.So, thoughSlutsky symmetryis of course more restrictive,the numberof restrictionsis of the same orderwhen n is large. Our findingscan be summarizedin the followingproposition,that underliethe empiricalanalysisof the next sections: 3: Let S denote the pseudo-Slutsky matrix, and let M = S - S'. Then, in the collective setting: (i) M has rank zero or two. (ii) If M has rank zero, the unitarycase cannot be ruled out. (iii) If M has rank 2, then M = uv' - vu' for two vectors u and v that span Im(M). Moreover, for any vector w orthogonal to Im(M) (that is, such that w'v = w'u = 0), then w'Sw i > 2. Conversely, if the relationship given in the lemma holds, then we can write row i for i > 2 as (M13M2 - M23M1)/M12and hence M has rank 2. Q.E.D. PROOFOF PROPOSITION 3: Only the final statement is new. But for any vector w orthogonal to Im(M), we have w'Sw = wXw < 0, since X is negative semi-definite. Q.E.D.

1275

INTRA-HOUSEHOLD ALLOCATIONS

PROOF OF PROPOSITION4: From Lemma 1, S satisfies SR1 iff M = S - S' is of rank zero or two.

But M is antisymmetric; from Lemma 2, its rank must be even. It follows that, if n < 3, the (n x n) matrix M cannot be of rank more than two, so that SR1 is fulfilled. Assume, now, that n = 4. Then M can be of rank zero, two, or four. But homogeneity plus adding-up implies that M-p = 0, so that M cannot be of full rank. Hence, it can only be of rank zero or two and SR1 is fulfilled. Q.E.D.

PROOF OF PROPOSITION7: (i) From the proof of Proposition 2 we have that S = X + fu( ,up + From {(p,x,y)=f(p,x,,u(p,x,y)) we have {Y(p,x,y)=f f,uy. Thus S= X+ y(up+ /uxq)'(1/uy) = X + Yv'. ,u/q)'.

(ii) If M = S - S' has rank 2, then {y and v in part (i) are linearly independent. Take any vector w that is orthogonal to {y but not to v. Then Mw = {Yv'w so that {y is in the column space of M. Q.E.D. PROOF OF PROPOSITION8: From {(p, x, Yl, Y2 *--Yn) = f(P, x, U(p, X, Yl' Y2 f

yYi

=

Ym)) we have

=Y

Q.E.D.

( yi/Yl){Yl-

PROOF OF PROPOSITION9: Consider the vector v in Proposition since (dpu/dPi) = 0, v is colinear to {.

2 as a function of (P, X, Y). Then Q.E.D.

PROOFOF PROPOSITION 10: From equation (4.4) we have that S takes the form S = F + R( F') + X where the matrix R = 2( 1 + 2 A(x/b(p)))p' and X is symmetric. If S is SR1 for all (p, x) then set prices equal to unity so that p = 0 and R = 0. Then S = F+ X,

whichimpliesthat F is SR1. Conversely, if F is SR1 then we can write it as

F=

+*? uv' where X * is symmetric. Then:

M= S-S' = (I + R)(uv'-vu') + (uv'-vu')R'. Since R has at most rank1, (uv'- vu')R'has at most rank1. Thus M is the sum of matriceswith at most rank2 and 1 respectivelyso that it has at most rank3. Since it is antisymmetric, by Lemma2 it has rank0 or 2, consequentlyS is SR1,fromLemma1. Q.E.D.

APPENDIX

B: DATA APPENDIX

The CanadianFAMEX is a multi-stagedstratifiedclusteredsurveythat collects informationon annualexpenditures,incomes,laborsupply,and demographicsfor individualhouseholds.The survey is run in the Springafter the surveyyear (that is, the informationfor 1978was collectedin Spring 1979).All of the informationis collectedby interviewso that the expenditureand incomedata are subjectto recall bias. Althoughthis may give rise to problems,the FAMEXsurveyingmethodhas the great advantagethat informationon annual expendituresis collected.Thus the FAMEX has much less problemwith infrequencybias than do surveysbased on short diaries.For example,the proportionof householdsreportingzero expenditureon clothing is about 3% in the FAMEX whereasit is over 50%in the U.K. FES. It is also the case that since the surveyyear coincideswith the taxyear (Januaryto December)the incomeinformationis thoughtto be unusuallyreliablesince it is collected at about the time that Canadiansare filing their (individual)tax returns.These are often explicitlyreferencedby the enumerators. Prices are taken from StatisticsCanada.When compositecommoditiesare created, the new compositecommodityprice is the weightedgeometricmean of the componentpriceswith budget

1276

M. BROWNING AND P. A. CHIAPPORI TABLE DI SAMPLE SELECTION Single Females

Single Males

Couples

7,343 2,229 2,179 2,179 2,173 2,173 2,173

4,653 2,084 2,052 2,051 2,048 2,044 2,044

12,237 2,512 2,458 2,449 2,442 2,440 2,428

Full sample In full-time employment Age < 65 Incomes positive Education level given Reasonable expenditures Reasonable earnings

Sampleyears: 1974, 1978, 1982, 1984, 1986, 1990, 1992.

shares averaged across the strata (couples, single males, and single females) for weights. Thus, the weights are not the individual household budget shares. Table DI gives the sample selection path followed; the principal selection is on all agents being in full-time employment and under the age of 65. As well, we select on the education level being observed, net household income being positive, and, for couples, gross earnings being above $2981 (in 1992 terms) (see "reasonable earnings" in the Table). Finally, in 1978 expenditures on recreational vehicles are not given separately from other spending on recreation. This lead to a small number of very high values for the latter in 1978; these have been deleted (see "reasonable expenditures" in the Table). Experiments were also made with "cleaner" samples than those reported (for example, households with very low net incomes or high budget shares for some goods were excluded). In no case were the qualitative results different.

TABLE DII DESCRIPTIONOF BUDGET SHARESAND INCOMES Couples Budget Shares

Food at Home (F) Food Outside (E) Men's Clothing (M) Women's Clothing (W) Hhold Operations (H) Recreation (R) Transport (T) Vices (V)

Total Expenditurea Hhold Net Incomea Gross Earnings (Husb.)a Gross Earnings (Wife)a

Price of Vices

Single Females

Single Males

Mean

# Zeros

Mean

# Zeros

Mean

# Zeros

.202 .104 .054 .084 .125 .107 .245 .078 Mean

1 27 14 6 1 10 9 79 Std. Dev.

.205 .104 0 .149 .169 .098 .209 .065 Mean

15 74 2,173 6 0 46 11 266 Std. Dev.

.174 .148 .085 0 .101 .123 .247 .122

37 58 23 2,044 1 33 21 121

Mean

Std. Dev.

23,815 55,324 41,262 29,318 Mean

8,501 19,248 20,015 13,201 Range

12,162 26,039

5,004 10,101

15,027 31,478

6,357 14,192

Mean

Range

Mean

Range

0.53

0.175-1.04

0.52

0.175-1.04

0.54

0.175-1.04

aAllvalues in 1992 Canadiandollars($1 Canadian= $0.75 U.S. ?0.50 U.K.).

1277

INTRA-HOUSEHOLD ALLOCATIONS TABLE DIII MEANSOF DEMOGRAPHIC VARIABLES

Atlantic Quebec Prairies B.C. Car Owner Homeowner City Dweller Agea More than High Schoola Francophonea Allophonea White Collara Age of Wife Wife White Collar

Couples

Single Females

Single Males

.145 .192 .294 .103 .949 .645 .810 37.3 .190 .196 .112 .376 35.0

.150 .188 .320 .100 .634 .228 .845 38.6 .174 .187 .095 .406

.137 .169 .312 .131 .773 .281 .811 36.6 .226 .172 .113 .376

.345

aRefers to husband for couples.

Tables DII and DIII present sample means and other statistics for all of the variables used in the analysis (except for the homogeneous prices).

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BROWN,DONALD,ANDROSAMATZKIN (1995): "Estimation of Nonparametric Functions in Simultaneous Equations Models, with an Application to Consumer Demand," Mimeo, Northwestern University. ANDVALE-RIE LECHENE PIERRE-ANDRE CHIAPPORI, BROWNING, MARTIN,FRANCOIS BOURGUIGNON, (1994): "Income and Outcomes: A Structural Model of Intrahousehold Allocation," Journal of Political Economy, 102, 1067-1096. BROWNING, MARTIN,ANDCOSTASMEGHIR(1991): "The Effects of Male and Female Labor Supply on Commodity Demands," Econometrica, 59, 925-951. CHIAPPORI, PIERRE-ANDRE(1988a): "Rational Household Labor Supply," Econometrica, 56, 63-90. (1988b): "Nash Bargained Household Decisions: A Comment," International Economic Review, 29, 791-796. (1990): "La Fonction de Demande Agregee en Biens Collectifs: Theorie et Application," Annales d'Economie et de Statistiques, 19, 27-42. (1991): "Nash Bargained Household Decisions: A Rejoinder," International Economic Review, 32, 761-762. (1992): "Collective Labor Supply and Welfare," Journal of Political Economy, 100, 437-467. BERNARD (1998): "Household Labor Supply, CHIAPPORI, PIERRE-ANDkE, FORTIN,ANDGuY LACROIX Sharing Rule and the Marriage Market," Mimeo, University of Chicago. ANDGuY LACROIX FORTIN,BERNARD, (1997): "A Test of the Neo-Classical and Collective Models of Labour Supply," Economic Journal, forthcoming. FRY, VANESSA,ANDPANOSPASHARDES (1992): "An Almost Ideal Quadratic Logarithmic Demand System for the Analysis of Micro Data," Oxford Applied Economics Discussion Paper Series: 145. LEWBEL, ARTHUR(1991): "The Rank of Demand Systems: Theory and Non-parametric Estimation," Econometrica, 59, 711-730. ANDTERENCEWALES(1997): "Do Husbands and Wives LUNDBERG, SHELLY,ROBERTA. POLLAK, Pool Their Resources? Evidence From the U.K. Child Benefit," Journal of Human Resources, 32, 463-480. AND MURRAYBROWN(1980): "Marriage and Household Decision Making: A MANSER,MARILYN, \ Bargaining Analysis," InternationalEconomic Review, 21, 31-44. McELROY,MARJORIE B. (1990): "The Empirical Content of Nash Bargained Household Behavior," Journal of Human Resources, 25, 559-583. B., ANDMARYJEANHORNEY(1981): "Nash Bargained Household Decisions: McELRoY, MARJORIE Toward a Generalization of the Theory of Demand," InternationalEconomic Review, 22, 333-349. (1990): "Nash Bargained Household Decisions: Reply," International Economic Review, 31, 237-242. ANDPETERBURTON(1992): "What's Mine is Yours? The Influence of Male and PHIPPS,SHELLEY, Female Incomes on Patterns of Household Expenditure," Working Paper 92-12, Department of Economics, Dalhousie University. ROBERTA. (1985): "A Transaction Cost Approach to Families and Households," Journal of POLLAK, Economic Literature,23, 581-608. PAUL(1956): "Community Indifference Curves," QuarterlyJournal of Economics, 70, SAMUELSON, 1-22. T. PAUL(1990):"Testing the Neoclassical Model of Family Labor Supply and Fertility," SCHULTZ, Journal of Human Resources, 25, 599-634. SHAFER,WAYNE,ANDHUGOSONNENSCHEIN (1982): "Market Demand and Excess Demand Functions," Ch. 14 in Handbook of Mathematical Economics, Volume 2, ed. by Kenneth Arrow and Michael Intriligator. Amsterdam: North Holland. DUNCAN(1990): "Intra Household Resource Allocation: An Inferential Approach," JourTHOMAS, nal of Human Resources, 25, 635-664.

Econometrica,

Vol. 66, No. 6 (November, 1998), 1279-1298

THE NOAH'S ARK PROBLEM BY MARTINL. WEITZMAN This paperis aboutthe economictheoryof biodiversitypreservation.A cost-effectiveness methodologyis constructed,whichresultsin a particularformulathat can be used as a criterionto rankprojects.The rankingcriterionis sufficientlyoperationalto be usefulin suggestingwhat to look at when determiningactualconservationprioritiesamongendanrigorous geredspecies.At the same time, the formulais firmlyrootedin a mathematically optimizationframework,so that its theoreticalunderpinningsare clear. The underlying model,called the "Noah'sArk Problem,"is intendedto be a kind of canonicalformthat hones down to its analyticalessence the problemof best preservingdiversityunder a limitedbudgetconstraint. endangeredspecies,diversitytheory. KEYWORDS:Biodiversity,

INTRODUCTION

THE PRESERVATION OF BIODIVERSITY is plagued by the absence of a workable

cost-efectiveness framework,withinwhich, at least in principle,basic questions can be posed and answered.Currentapproachesto endangeredspecies protection seem almost completely lacking in theoretical underpinningsthat might reasonablyguide policy.As a result,we do not have rigorouslygroundedcriteria for choosing among biodiversity-preserving alternatives,and it is difficult to evaluateperformance. There are several motivations for this paper. Essentially, I am trying to introduce an analytical frameworkthat represents a useful way of thinking about the economicsof diversitythroughthe mediumof an abstractmathematical model. The "Noah's Ark Problem"is a parable intended to be a kind of canonicalform of the simplestpossibleway of representinghow best to preserve biodivers-tyunder a limitedbudget constraint.Throughoutthe paper,emphasis is laid upon simplicityof structure,in order to get at the analyticalessence of the problem. While several themes are developed in this paper, the main motivating question is how to determine basic priorities for maintainingor increasing diversity.The centralissue is to developa cost-effectivenessformulaor criterion that can be used to rank prioritiesamong biodiversity-preserving projects.The formulashould be operationalenough to be useful in suggestingwhat to look at when actually determiningconservationpriorities,e.g., among endangered species. At the same time, the methodology should be firmly rooted in a mathematicallyrigorousoptimizationmodel, so that the theoreticalfoundation for the cost-effectivenessrankingcriterionis made very clear. In talking about biodiversitypreservation,there is always a question about what is the appropriatelevel of discourse.In principle,the basic unit could be at 1279

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the level of the molecule, cell, organ,individual,species, habitat,ecosystem,or so forth.For the purposesof this paper,I take the underlyingunit of analysisto be the "species,"althoughI think that the same basic issues and themes of the paper will arise at any level. In this spirit, the Noah's Ark Problem could be thoughtof as a generic formulationof "the" core problemof optimalbiodiversity preservationunder a budget constraint. Suppose, then, that the underlyingconservationunit i essentiallystands for "species i." In the context of this paper, it is useful to conceptualize a "conservationproject"as follows. "Projecti" is some preservationaction that increasesthe probabilityof survivalof species i by APi at a cost of Ci. Let Ub representthe directutilityof how muchwe like or value the existenceof species i. (After all, most of us like Pandasmore than mosquitoes.)Withouttryingto be precise about it at this stage, let the "distinctiveness"of i, meaningroughlyits differenceor distancefromits closest resemblingunit, be Di. (Imaginesomehow that Di has been made commensuratewith Ui.) Then the following formula might appear on heuristicgroundsto be conveyingroughlythe right priorities for rankingalternatives: (1)

Ri = (Di + U)

CL).

As a rankingcriterion,Ri is a measureof the "expectedmarginaldistinctiveness plus utility per dollar." Interpreted loosely, when making preservation decisionsthe conservationauthoritiesare asked to look at the four factors:Di, Ui, APi, Ci, which seems reasonable,combinedin such a way as describedby formula(1), which also seems intuitivelyplausible.The proposed formula has some heuristicappeal as a rule of thumb,and it seems to make sense generally, but can it be groundedin any theoreticalconstruct?Does the rankingcriterion (1) have a formaljustification? The primarytask of the paperis to present a model and a scenariowhere the use of Ri as a rankingcriterioncan be given a rigorousunderpinning.The main question I am asking is this: If the "right assumptions"are made, can a reasonableoptimizingmodel be producedthat providesa rigorousjustification for using Ri to select conservationpriorities?The short answeris "yes,"and the underlyingmodel that providesthe "rightassumptions"I call the "Noah'sArk Problem." Tryingto answerthe above, rathernarrowlyposed, question about an appropriate ranking criterion forces a researcher to confront a number of basic theoretical and conceptual issues that go to the core of modeling diversity preservation.My hope is that the basic conceptual frameworkand the set of ideas introducedhere for thinkingabout the economics of diversitywill prove useful even beyond the specificapplicationsof this paper. The model itself will be at a high level of abstraction.My aim is to build the simplestpossible analyticalstructurethat capturesthe mathematicalessence of the underlyingproblem.In so doing,I entertainthe standardeconomictheorist's

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hope that basic insightsinto policywhich emerge from the bare-bonescanonical-formmodel will remain a useful point of departureeven in a more complicated world.

THE

SPECIES/LIBRARY"

MODEL OF DIVERSITY

A key point of departurefor this paper involvesconceptualizingthe underlying conservationunit-the "species"-as if it were a "library."Concentrating Qnthe question of how best to preservelibrariesallowsfor a crisp formulation of the generic problem of optimallyconservingdiversityunder a budget constraint.In this section, then, the basic unit is the library.Each individuallibrary stands,at an appropriatelevel of abstraction,for a particularspecies. Let the index i = 1,2,..., n denote a particularlibrary.The set of all n librariesis denoted S. The value of a libraryconsists of two components:the buildingitself and the collection of books that it houses. Libraryi is housed in a building that has some inherent value as a structure-let this utility be denoted Ui. The other, presumablyprimary,value of library i is its book collection-suppose this consists of Mi differentbooks. (In the species interpretation,Ui representsthe direct\utilityof how much we like or value the existence of species i, while the Mi"books"are very roughlyanalogousto the gene pool of the species itself.) The book collections in various librariesmay overlap to some degree. The definitionof the diversityof S, denoted V(S), is the numberof differentbooks in the overall library collection S. In other words, the diversityfunction V(S) measuresthe size of the set that consistsof the union of all the differentbooks in all the librariesof S. A simple two-libraryexample may help to illustrate the basic idea. Let S consist of the two libraries 1 and 2. Let the number of books held jointly in commonbetween libraries1 and 2 be denoted J. Let E1 stand for the number of books distinctiveor unique to library1, while E2 is the number of books distinctiveto library2. Then M1 = E1 + J, while M2 = E2 + J. In the case of two libraries,it is alwayspossible to give a tree representation of the book structure.The tree, depicted in Figure 1, has a main branch of length J that dividesinto two sub-branchesof lengths E1 and E2. (One could tell an evolutionarystory "as if' the two libraries evolved by "descent with modification"from a common ancestor.)The diversityfunction in this case is V(S) = J + E1 + E2. Note that diversityhere has the interpretationof being the total (vertical)branchlength of the correspondingtree. We defined diversityfor the particularassemblageS of libraries.But essentially the same idea holds for any assemblageof libraries.For any subset of librariesQ (c S), define the diversityof Q, denoted V(Q), to be the numberof differentbooks in all the librariesof Q. In other words, the diversityfunction V(Q) is the size of the set consistingof the union of all books in the librariesof Q, where Q may representessentiallyany assemblageof libraries.

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MARTIN

L. WEITZMAN

IJ

El |lE2 1 2 FIGURE 1.

A critical aspect of the underlyingproblem is uncertainty.To continue the metaphor,a librarycan burn down, with possible loss of the buildingand the book collection that it houses. Variouspreventivemeasurescan be undertaken that lower the probabilityof a fire-at a cost. The essence of the conservation problem is how best to allocate scarce fire prevention resources among the variouslibraries. Uncertaintyis introducedas follows. Let Pi be the probabilityof survivalof libraryi, while 1 - Pi is the probabilityof extinctionof i-e.g., the probability that library i burns down. For analyticalsimplicity,I assume throughoutthe paper that all probabilitiesare independent,since this is the easiest case to analyze. As previouslynoted, the direct utility of library building i is Ui, \which representshow much the librarybuildingitself is liked irrespectiveof its book contents. Without loss of generality,suppose that the coefficient Ui is normalized relativeto the value of each differentbook,whichis set equal to one. Then the expectedvalue of all the librarybuildingsis (2)

U(P) = EPiLi.

Turningnow to the book collections,the definitionof expected diversityis a natural extension of the original deterministicconcept to the case of uncertainty. The appropriatestochastic concept here is the probability-weighted diversity of all subsets. The expected diversityfunction, denoted W(P), is the average number of different books. More formally,

(3)

W(P)-

r: Pi)( k E=S\Q

5?(

QCS jE=Q

-P)VQ

In the two-libraryexample, (4)

W(P)

= P1P2 (M1 + M2-J) +P1(l-P2) Jr

P2(1

-

PO4

-P pl(1 M2 + (1

M1 -

P2

-0

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which can be rewrittenmore simplyas (5)

W(P1, P2) = MlPl

+ M2P2 -JP1P2

Note that U(P) representsthe expected utility of the librarybuildings,while W(P) representsthe expectedvalue of the overall collection of books. Of the two functions, W(P) is by far the more novel or unorthodox,and the really innovativepart of this paper deals with it. The expected diversity function W(P) expresses the average number of dieferentbooks in the collection.Now we mightlike manydifferentbooks per se, just as we might like manydifferentcolors simplybecause of the more colorful world their sheer diversitycreates. This would be a kind of aestheticvalue of diversity.Or, we might want to have differentbooks for the utilitarianreason that they are a potential source of new future ideas about medicines,foods, or whatever.This might be called the informationcontent of a book collection. These two concepts are actually closely related to each other. As the next section shows, at a sufficientlyhigh level of abstractionthe aestheticvalue of diversityis essentiallythe same concept as the informationcontentof diversity. EXPECTED DIVERSITY AS INFORMATION CONTENT

This section develops a strict isomorphismbetween "diversityas aesthetic value" and "diversityas informationcontent."It will be provedthat when they are appropriatelymodeled, the two concepts are formallyidentical. Suppose we take the average number of books W(P) as an appropriate measure of the aesthetic value of diversity.What should be the appropriate measureof the informationcontent of diversity? A "book"in the present model is a metaphorfor a containerof information. Supposethat a researcheris lookingfor a particularpiece of useful information. There are three possibilities:the informationmight not exist in any of the books; or it might be located in just one book; or it might be redundantly contained in more than one book. Suppose that the probabilitythat any one book contains this particularpiece of useful informationis ?. A vivid image is that e standsfor the probabilitythat in any book drawnat randomis contained the alchemist'ssecret of how to turn lead into gold. In a more realisticscenario, the "useful information"being sought might be about a particularcure for a certaindisease or a specificnew food source. Considerany assemblageof librariesQ. Then the probabilitythat Q contains useful informationis the probabilitythat at least one book of Q contains the useful information.Since there are V(Q) different books in Q, this equals one minus the probabilitythat all V(Q) books fail to contain the useful information.Thus, the probabilitythat Q containsuseful informationis ,6

__ H(Q

. - (1 -

v

Q).

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MARTIN L. WEITZMAN

Note that takingthe derivativeof (6) with respectto e and evaluatingat e = 0 yields the relationship = V(Q).

7H(Q;e)

The expression(6) for H(Q; 8) representsthe probabilitythat the deterministic libraryassemblageQ containsuseful information.But underuncertainty,the libraryassemblagesthemselvesexist only in a probabilisticsense. Libraryi exists with probabilityPi. The libraryassemblageQ (c S) exists with probability (8) jcQ_

P

(1-Pk)j -kcS\Q

It is naturalto define formallythe informationcontentI (as a function of P and e) to be the probabilitythat the overallstochasticlibrarycollectioncontains useful informationon a particulartopic. Metaphorically,informationcontent is just the probabilitythat in at least one librarythat has not burned down is containedat least one book in which is writtenthe alchemist'ssecret of how to turn lead into gold. Combining(6) with (8), the informationcontent of the stochastic library collection is

(9)

I

)

(l)(kT1 jE=Q

QcS

(1-Pk)) [H(Q; )].

kcE-S\Q

What is the relationship between diversity and information content? Essen-

tially, the basic result here is that diversityis the first derivativeof information content with respect to uncertainty.Takingthe derivativeof (9) with respect to 8, using (7) to evaluate it at

8

= 0, and comparing with definition (3) yields the

basic result: (10)

MI(P;e) d8

=o

-W(P)

Since it is readilyverifiedthat I(P; 0) = 0, equation(10) means that W(P) e is the appropriatefirst order approximationof I(P; 8) in the formalsense tha,t (11)

J(P; 8) = W(P).

*

+

o(82)

where O(e2) standsfor squaredor higher order terms in 8. The conclusionis that, for small e, maximizingthe expecteddiversityfunction W(P) is "essentially"the same as maximizinginformationcontent I(P; 8). We state this result somewhatinformallyas the following: THEOREM 1: The expected diversityof a set of librarycollections is "essentially" the same concept as the information content of the same set of librarycollections.

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What do we want to do with expecteddiversity(equals informationcontent)? Presumablywe want to pick conservationpolicies that maximizethe sum of the expected diversityfunction of the books W(P) plus the expected utility of the librarybuildingsU(P), takingaccountof the relevantbudget-likeconstraintson P. The simplestcanonicalform of a preservationprobleminvolvingoptimizing under budget constraints,which at the same time conveyssome useful content, is what I call here the "Noah'sArk Problem." THE NOAH S ARK PROBLEM

The "Noah's Ark Problem"is intended to be an allegory or parable that renders a vivid image of the core problem of maximizingdiversityunder a budgetconstraint.What is treated here is actuallynot the most generalform of the underlyingmathematicalproblem.Some slight generalizationsare possible, but they would come at the expense of diluting a crisp version of the basic paradigm. Noah knows that a flood is coming. There are n existing species/libraries, indexed i = 1,2,... , n. Using the same notation as before, the set of all n species/libraries is denoted S. An Ark is available to help save some species/libraries. In a world of unlimitedresources,the entire set S might be saved.,Unfortunately,Noah'sArk has a limitedcapacityof B. In the Bible, B is given as 300 x 50 x 30 = 450,000cubits3.More generally,B standsfor the total size of the budgetavailablefor biodiversitypreservation. In either case, Noah, or society, must face the central problemof choice.A basic choice question must be answered. Which species/libraries are to be afforded more protection-and which less-when there are not enough resources aroundto fully protect everything?I present here the simplestway that I know to model the analyticalessence of this choice problem. If species/library i is not put on board the Ark, but left unprotected,its probabilityof survivingthe flood unaidedis Pi. If species/libraryi is boardedon the Ark, and thereby afforded some protection, its survival probabilityis enhancedto Pi. Essentially,boardingon the Ark is a metaphorfor investingin a conservationproject, like habitat protection, that improves survivabilityof a particularspecies/library. A particularlygrimversionof the Noah'sArk Problemwouldmake the choice a matter of life or death, meaning that Pi= 0 and Pi= 1. This specificationis perhapsclosest to the old testamentversion,so I am takingliterarylicense here by extending the metaphor to less stark alternatives.The only mathematical restrictionhere is then (12)

0