ECONOMIC RESEARCH REPORTS

C.V. STARR CENTERFOR APPLIED ECONOMICS

NEW YORK UNIVERSITYFACULTY OF ARTS AND SCIENCE

DEPARTMENT OF ECONOMICSWASHINGTON SQUARE

NEW YORK, NY 10003-6687

Cultural Transmission, Marriage andthe Evolution of Ethnic and

Religious Traits

by Alberto Bisinand

Thierry Verdier

RR# 98-40 November 1998

CulturalTransmission, M arriageandtheEvolutionofEthnicandReligiousTraits

A lbertoB isinN Y U

T hierryVerdierCER A S, D EL T A , CEPR

N ovember19 9 8¤

A bstract

T hispaperpresentsaneconomicanalysisoftheevolutionofreligiousandethniccharacteristicsinamodelofintergenerationaltransmissionofculturaltraitswhichoccursthroughfamilysocializationandmaritalsegregationdecisions.

T hemodelimpliesthatthefrequencyofintragroupmarriage(homogamy), aswellasthesocializationratesofreligiousandethnicgroups, dependsonthegroup’sshareofthepopulation;minoritygroups searchmoreintenselyforhomogamousmates, andspendmoreresourcestosocializetheiro¤spring.

W estudytheimplicationsofthemodelregardingthee¤ectofthesocialmatch-ingtechnology, divorceratesandthedegreeofculturaltolerancebetweengroupsontheevolutionofculturaltraits.

Existingempiricalevidencebearingdirectlyandindirectlyontheimplicationsofthemodelisdiscussed.

Keywords: CulturalTransmission, M arriageJEL : I2, Z 1, D 9

¤T hanks totworeferees and to A . Shleifer, whorecommended this cutfora paperon culturaltransmission, and to G . B ecker, J. B enhabib, L . Iannaccone, A . L izzeri, S. R osen, A . Schotter, G .Topaforcommentsandencouragement. T hanksalsotoA .CasagrandeandJ. G oldbergforassistance.T he…nancialand institutionalsupportoftheC. V . StarrCenterforA ppliedEconomics is gratefullyacknowledged.

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1 IntroductionH istorycontainsseveralexamplesofthestrikingpersistenceofethnicandreligioustraits.Basques, Catalans, Corsicans, IrishCatholics, inEurope, andQ uebecoisinCanada, haveallremainedstronglyattachedtotheirlanguages andculturaltraits eventhroughtheformationofpoliticalstateswhichdidnotrecognizetheirethnicandreligiousdiversity.Jewsofthediasporahavemaintainedatenaciousreligiousandethnicidentityresistingmanyattemptsatacculturationandevenextermination. M anysmallethnicandreligiousenclaves arehighlyresilient. Forexample, smallcommunities ofO rthodoxChristianA lbanians livingin thesouthofItalysincetheyemigratedthere inthe15th century,maintainedtheirlanguageandreligiousfaith. Similarly, smallwhitecommunitiesintheFrenchCaribbeanislands, ‘B lancs M atignons’, preservethemselves from racialmixing,andhavedonesosincethe18thcentury. L ong-liveddivisionsalongculturalandtriballinesarestillcommonplaceinmodernA frica(M cEvedy, 19 9 6).

T heseandthemanyotherexamples ofthepersistenceofculturaltraits inhistoryshouldobviouslynotbeinterpretedasevidencethatculturaltraitsarenecessarilymain-tainedagainstallodds. T herapidassimilationofJewsinItalyisaclearcounter-example(D ellaPergola, 19 7 2). B utthepersistenceofculturaltraitsandthedi¢cultyofaccul-turatingminoritiesdoesoftentakesociologistsandanthropologistswhostudyculturalevolution by surprise. Forinstance, mostsociologicalworkon A merican Jews in the50s and60s predictedthe‘extinction’ofJewishO rthodoxy, therebyfailingtoidentifytheroots ofthedemographic‘R enaissance’thatO rthodoxJewishculture in theU .S.has been experiencingsince the 7 0s (M ayer, 19 7 9 , Ch. 1). M ore generally, ClaudeL evi Strauss (19 9 7 ) has recentlyobservedthattherisks ofculturalassimilationhavebeenmuchover-statedintheanthropologicalandsociologicaldebateofthe50s, becausecultureshavedemonstrateda‘veryresilientstrongcore’. 1

T hepersistenceoftheevolutionofethnicandreligioustraitsismirroredinhighandpersistentintragroup marriage (homogamy) rates, even forsmallethnicand religiousgroups. Forinstance, ethnicintermarriagewasvirtuallyunknowntoimmigrants intheU .S.attheturnofthecentury. Pagnini-M organ(19 9 0), forinstanceestimatethatItalianandPolishimmigrantsaroundthebeginningofthecenturyintheU .S.weremorethan2 ;0 0 0 timesmorelikelytomarryintheirethnicgroup thantointer-marry. T heyalsonotelowlevelsofinter-marriagepersistinginsecondandthirdgenerationimmigrants.R eligioushomogamyisalsoverywelldocumentedformostreligiousgroups(e.g., Sander,

1W ecan even readthepoliticaldebateon assimilation in theU nited States as areaction tothepersistenceofculturaltraitsdespitethee¤ortsoftheassimilationandA mericanizationmovementborninthebeginningofthecenturytoincludeimmigrants, ando¤eroffullparticipation. Facinggrowingresistancetoassimilation(religiouslyandevenlinguisticallysegregatedparochialschools, forinstance,spread inthis period;seeTyack, 19 7 4), bythe 20 s themovementturned intoan increasinglyhostiledemandforimmigrants’acculturationtotheEnglishlanguageandA mericanculture(seeD avis, 19 20,foraselectionofearlyreadingsonassimilation). T heheateddebateovermulti-culturaleducationhascontinuedandhasrecentlytakenaradicalturn;seeG lazer(19 9 7 ).

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19 9 3, forCatholics, andJohnson, 19 80, Schoen-W einick, 19 9 3, forotherdenominations).InSection2wereportinsomedetailontwoethnographicexamplesofpopulationswhichadoptextremestrategiesofmarriagesegregationtopreservespeci…cculturaltraitsandtosocializechildren: aristocrats inFranceandO rthodoxJewsinN ewYork.

T hehistoricalexamples and evidenceofculturalpersistenceandhomogamyraiseimportantquestions regardingculturaltraits, particularly ethnicand religious traits,theirdeterminants intheshortrunandtheirlongrundistributioninthepopulation.

T hispaperstudiestheevolutionandpersistenceofculturaltraitsasdynamicproper-tiesofculturaltransmissionandsocializationmechanisms. Inparticular, weconcentrateon segregation bymarriagealongethnicand religious lines as amechanism tofavorsocializationatthefamilylevel.2

T hemodelofsocializationwedevelop and study is motivated byvarious stylizedfacts which emergefrom thesociologicalliterature (seethenextsection). W emodelculturaltransmission as amechanism which interacts socialization inside the familyandsocializationoutsidethefamilyinsocietyatlargeviaimitationandlearningfrompeers and rolemodels. In themodel, altruism motivates parents’e¤orts tosocializetheirchildren, andtotransmittheirownculturaltraits. (Socializatione¤ortconsists,forinstance, inthechoiceofneighborhood, school, orchurchattendance.) Families inwhich parents haveahomogeneous culturaltraitareadvantaged in the socializationprocess forthis trait, withrespecttoheterogamous families. Sinceeachparentwishestotransferhis owntraittohis children, thechoiceofamateinthemarriagemarketis functionaltothedesiretosocializetheeventualchildrenarisingfrom suchaunion.W hilethenperfectassortativematching(completehomogamy)wouldariseoptimallyintheabsenceofsearchcosts, wemodelthemarriageprocess as characterizedbysearchfrictions. M orespeci…cally, weassumethatbothmalesandfemalescansearchforamateinsomerestrictedpoolwhereadmissioniscostly, butwhereeverybodywhoisadmittedhasthesameculturaltrait(henceallmarriages inthepoolarehomogamous).

W ederiveimplicationsofsuchculturaltransmissionmechanisms intermsofdi¤er-entialbehaviorofculturalminoritiesandmajoritieswithrespecttotheire¤orttomarryhomogamouslyandtosocializechildrentotheirowntrait. W ealsostudythedynam-icsofthedistributionofculturaltraits inthepopulationimpliedbysuchtransmissionmechanisms, andthedeterminantsofthelongrunstabledistributionoftraits, withtheobjectiveofunderstandingtheobservedpersistenceofminorityculturaltraits. Someoftheresultsandtheintuitionbehindthemaresummarizedbelow.

2Culturaltransmission, butwith random matingand exogenous propensities forsocialization, isstudied in biology;seee.g., Cavalli Sforza-Feldman (19 81). T heroleofmarriageas an institutionoftransmission ofculturalvalues has been stressed in anthropology, forinstance by Boas (19 28) andL evi Strauss (19 49 ). Economists havemostlyconcentrated insteadontheagents’choiceoftheirownpreferences andvalues, as e.g., B ecker(19 9 6), B ecker-M ulligan (19 9 7 ), and, speci…cally forreligiouspreferences, Iannaccone(19 9 0, 19 9 8) . Forgeneticrathenthanculturaltransmissionmodels, seee.g.,Kockesen-O k-Sethi (19 9 7 ). W erefertoBowles(19 9 8) forasurveyandcompletereferences.

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M inorities, otherthings equal, havemorehighlysegregatedmarriagemarkets, andexercisemoree¤ortindirectlysocializingtheirchildren. T heyhave, infact, astrongerincentivetosegregate, tobe homogamous and tosocialize theirchildren. Sincethepopulation atlarge is mostly populated bymajority types, amemberofaminorityculturalgroup is likelytomarryheterogamouslyifhedoes notstructurehis lifesoastomeetmostlymates withthesametraits (e.g., bygoingtotheappropriatechurch,livingintherightneighborhood, etc.). M oreover, aminoritytypeinanheterogamousmarriagewillhavedi¢cultytransmittinghis owntraits, sincethespousewillfavoradi¤erentsetoftraits, andpeersandrolemodelswillbetakenfromapopulationmostlyofthemajoritytypes.

Patterns ofmaritalsegregations and socializationacross culturalgroups alsohavee¤ectsontheevolutionofculturaltraitsinsociety. W eshowthattheculturaltransmis-sionmechanismswestudygeneratedynamicsinthedistributionofculturaltraitswhichtendtomulticulturalpopulations andawayfrom completeassimilationofminorities.Culturalminoritiestendtoreactinequilibriumtotheprospectofculturalassimilationwithmarriage segregation, homogamous marriages, andwith more intense strategiesforthedirectsocializationofchildren. T houghmajorities havehigherhomogamyandsocializationrates overall, itis thesocialization e¤ort(which is higherforminorities)todeterminethetransitionaldynamics ofthedistribution oftraits whenonetraitisclosetoextinction. A s aconsequence, the fraction ofthepopulation ofagents withminoritytraits tends toincrease, minoritytraitsappearquitepersistent, andlong-runmulticulturalpopulationsarestable.

Inotherwords, linearextrapolationsofinter-marriagerates, socializationpractices,anddemographicdynamics, tendtounderestimatethepersistenceofculturaltraits, be-causeminoritiesreacttotheirassimilation. T heresilienceofmanyethnicandreligiousneighborhoodsinA mericancities, theincreasingdemandformulticulturaleducationinA merican society, aswellas inmanyW esternEuropeancountries, andthehistoryofmanyculturalcommunitiessuchas A mericanO rthodoxJewscanhoweverbeexplainedbythecomplexinteractionofmarriagesegregation, directculturalsocializationoffam-ilies, andchildren’sexposuretotheculturaltraitsofthemajorityofthepopulationatlarge.

O urmodelofculturaltransmissionalsoallows us tostudythedynamice¤ects onthecompositionofthepopulationwithrespecttoethnicandreligioustraitsofvariousinstitutionalarrangementswithinmarriage. Inparticularwestudythelong-rune¤ectsinthedynamicsduetostructuralchangesi) intheavailabilityofinter-culturalrelationships(due forinstancetourbanization orinformation technologies), ii) in the freedom tochooseones’mate, andiii) intheorganizationofthefamily(divorcerates, singleparentfamilies, femalelabormarketparticipation).

W eshowthatifthedistributionofthepopulationwithrespecttotheculturaltraitsissuchthatamajorityandaminoritytraitareidenti…able, thenahigheravailabilityof

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interculturalrelationships causes agents toreactwithhighere¤orts atmarriagesegre-gationandsocializationofchildren. A negativedirecte¤ectviarandommatchingandapositiveindirecte¤ectduetotheincreaseinthee¤orttomarryhomogamouslyandsocializecountereachotherina¤ectinghomogamyrates. T hedirecte¤ectonhomogamyratestendstobestronger, andhence, homogamydecreaseswitheasierinterculturalre-lationships. A sthehomogamyratesdecrease, theprobabilityofhomogamousmarriagesdecreases forbothgroupsbutlessrapidlyforthemajoritygroupthanfortheminoritygroup. A saconsequence, easierinter-culturalrelationshipsincrease, inthelong-run, thefractionofthepopulationwiththemajoritytrait.

G reaterfreedom tochooseones’mate, arguablyarelativelyrecenthistoricaltrendinmanyculturalpopulations, has similare¤ects. B y increasingthecosts ofmarriagesegregation, segregatione¤ortandhomogamyrates aredecreased. G reaterfreedom ofmatingchoiceincreases, inthelong-run, thefractionofthepopulationwiththemajoritytrait.

W ealsoshowthatahigherprobabilityofdivorcereducesatthemarginthevalueofhomogamousmarriage, therebydecreasinghomogamyrates inequilibrium. Theproba-bilityofdivorcereducestheresources spenttosegregateinmarriageandhencebringsmoreheterogamyintheshortrun. A s inthecaseofeasierinterculturalrelationships,theprobabilityofhomogamousmarriagesdecreasesforbothgroupsbutlessrapidlyforthemajoritygroupthanfortheminoritygroup. H igherprobabilitiesofdivorcetendtoincrease, inthelong-run, thefractionofthepopulationwiththemajoritytrait.

Similarly, changes in levels oftolerancebetween culturalgroups a¤ecthomogamyratesandthepersistenceofculturalminorities. Inparticular, anincreaseinculturaltol-erancebetweentwogroupsinduceslessmaritalsegregationandlessfamilysocializationin theshort-runtendingtobias culturalevolution, inthelong-run, towards thetraitofthemajoritygroup in society. (G reatertoleranceonthepartofthemajoritywithrespecttointermarriagewithminoritypopulationsgeneratesaformofacculturationoftheminority.)3

T hepaperisorganizedasfollows. First, inSection2.1, weintroducevariousstylizedfactsonculturaltransmissionandsocializationwhichmotivateourmodelling. InSection2.2 weanalyzetheimplicationsofthebasicmodelofmarriageandculturaltransmission,providesomeextensions, andperformsomecomparativestatics. Section2.3. introducessomeempiricalevidenceonhomogamy, socializationpractices, andsegregationanddi-vorce, o¤eringsupportfortheimplicationsofourmodelwithrespecttothetransmissionofethnicandreligioustraits. Finally, Section3studiesthedynamicsofthedistributionoftraits inthepopulationandderivesseveralcomparativedynamicimplications.

3T hegreatertoleranceofintermarriagewithJewsafterW orldW arII intheU .S.mightbethecauseofthetrendtowardacculturationofJews, asnotedbyseveralcontributorsinthecurrentdebateonthe‘Jewishidentity’intheU .S.;seeD ershowitz (19 9 7 ).

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2 M arriageandsocializationVariousstylizedfactsonmarriagesasculturaltransmissionmechanismscanbecollectedfromananalysisoftheempiricalliteratureinsociologyandsocialpsychology.

1. Culturaltraits are usually adopted in the early formative years ofchildren’spsychology. Family, peersandrolemodelsplayacrucialroleintheadoptionofculturaltraits. T hishasbeenextensivelydocumentedforreligiousandethnictraits, forinstance,byClark-W orthington(19 8 7 ), Cornwall(19 88), D eVaus(19 83), Erickson(19 9 2), H ayes-P ittelkow(19 9 3).4

2. Familiescareabouttheirchildren’sculturaltraitsandconsciouslyexercisee¤ortinanattempttosocializechildren. A lso, homogamous families (i.e., families inwhichparentshavehomogeneousculturaltraits)predominantlyfavorthetransmissionoftheirowntraits. Psychologicalstudiesofheterogamouscouplesconsistentlyreporttheircon-cernaboutthepossibleculturalattitudes ofchildrenwhendecidingtoform afamily(seeR osenblattetal., 19 9 5, forracialtraits, andM ayer, 19 85, Smith, 19 9 6, forethnicandreligioustraits). G ussinPaley(19 9 5) providesavividethnographicdocumentationofschoolchoiceofmiddle-class A frican-A mericanparentsinChicago’sSouthSide. T hemainissueinthechoiceconsists intradingo¤ thelowacademicqualityofthepredom-inantlyblackpublicschools andtheexposureto‘whiteculture’in integratedschools.O ’B rien-Fugita(19 9 1) documenttheperceivedimportanceforJapanesefamiliesofthedevelopmentofJapaneseschoolsafterW orldW arII intheU .S. (oftencontrarytothepreferencesoftheirchildren). Similarattitudesaredocumentedformanyethnicgroups(e.g., M ayer, 19 85, forJews, andTyack, 19 7 4, forG ermans, and, morerecently, G lazer,19 9 7 , forA frican-A mericans).

3. T he e¤ectiveness offamily socialization depends stronglyon parentalagree-mentonthetraittobetransmitted. Childrenofmixedreligiousmarriageshaveweakerreligiouscommitments thanthoseofreligiouslyhomogamousmarriages (H oge-Petrillo,19 7 8), andO zorak, 19 89 ). A lso, childrenofmixedreligiousmarriagesareless likelytoconformtoanyparentalreligiousideologies, andtopracticessuchaschurchattendance,orprescribedfertilitybehavior(H eaton, 19 86, H oge-Petrillo-Smith, 19 82, andO zorak,19 89 ).

T hesefacts motivateourmodelofsocialization. In particular, wemodelculturaltransmissionasamechanismwhichinteractssocializationinsidethefamilywithsocial-izationoutsidethefamily, insocietyatlarge. (Socializationinsidethefamilyisalsocalled‘directvertical’socialization, whilesocializationbysociety, whichoccurs viaimitationandlearningfrompeersandrolemodels, isalsocalled‘oblique’socialization.)5

4T hetransmission oftraitswhich areformedlaterin thepsychologicaldevelopementofchildren,though, isoftendominatedbytheroleofpeers;seee.g., East-Felice-M organ(19 9 3) forevidenceonthetransmissionofattitudestowardsexualbehavior.

5T histerminologyistakenfromCavalli Sforza-Feldman(19 81).

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W emodeldirectverticalsocialization through parents havingaltruisticattitudestowards theirchildren. A ltruism motivates parents toexerte¤ortto socialize theirchildren, andtotransmittheirownculturaltraits. A nimportantassumptionwemakeinthisrespectisthatparentswishtotransmittheirowntrait, anddonotjustinternalizetheirchildren’spreferencesorsomemeasureoftheirsuccess. Indirectevidenceforsuch‘paternalisticaltruism’comes, as alreadynoted, from studies ofparentalschoolchoicedecisions. A lso, an analysis ofnorms regardinginter-religious marriages reveals thatparents ofmostmajordenominations (from Catholics toBaptists andJews;butalsoforinstanceSeventh-D ayA dventistsandL utherans) atleasttendtowarnchildrennottointer-marry, justifyingtheirpositionwith aconcern aboutthereligious educationofgrand-children(Smith, 19 9 6)6. Someevidenceinsupportof‘paternalisticaltruism’canalsobederivedfrom socio-economicsurveys. Forinstance, inresponsetoN O RC’sG eneralSocialSurvey’squestion, ‘W hichthreeofthequalitieslistedwouldyousayarethemostdesirableforachildtohave?’, ‘obedience’iscitedonaverageacrossthesamplemorethan, (inorder) ‘self-control’, ‘success’, ‘studiousness’, ‘cleanliness’, andlessonlythan‘honesty’.7 8

T hisassumptionofparents’paternalisticattitudes is consistentwithourmodellingofculturaltraitsas‘pure’traits, withnodirecteconomice¤ect. Forinstance, weimplic-itlyassumethatagents’economicopportunities, e.g., theirexpectedpresentdiscountedincomeortheirhumancapitalaccumulationcosts, areindependentoftheirtrait. T hisis, ofcourse, anabstractionmeanttodisentangletheculturaltransmissionmechanismfromothereconomicconsiderations. Ethnicandreligioustraits, morethanotherculturaltraitsandattitudes, seemtoapproximatesatisfactorily‘pure’traits.9

W e…nallyassumethatfamilies inwhichparentshaveahomogeneousculturaltraitareadvantagedinthesocializationprocessforthistrait, withrespecttoheterogamousfamilies. Sinceeachparentwishes totransferhis owntraittohis children, thechoiceofamate in themarriagemarketis functionaltothedesiretosocializetheeventualchildrenfromsuchaunion. W hileassortativematchingwouldarisethenatequilibriumintheabsenceofsearchcosts (seeBecker, 19 7 3, 19 7 4), wemodelthemarriageprocess

6Forexample, the19 83 CodeofCanonlawfortheCatholicChurch says: ” W ithouttheexpresspermission ofthecompetentauthority, marriage is forbidden between twobaptized persons, one ofwhomwasbaptizedintheCatholicChurch...andtheotherofwhomisamemberofaChurch... whichisnotinfullcommunionwiththeCatholicChurch” (801). M oreover, thepermissioncannotbegrantedunless thefollowingcondition is full…lled: ”theCatholicpartydeclares thatheorsheis preparedtoremovedangersoffallingawayfromthefaithandmakesasincerepromisetodoallinhisorherpowertohaveallchildrenbaptizedandbroughtup intheCatholicChurch”.

7 A similarpatternofanswers is reportedtoasimilarquestion inthe N ationalSurveyofFamiliesandH ouseholds.

8Foranaturalselectionexplanationofpaternalisticformsofaltruism, seeB isin-Verdier(19 9 8).9 H owever, someevidenceon thee¤ectofreligious and ethnictraits oneconomicopportunities is

found in W arren (19 7 0) and Sowell(19 9 4). A …rstanalysis ofthesee¤ects oftraits canbefound inB isin-Verdier(19 96).

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ascharacterizedbysearchfrictions. M orespeci…cally, weassumethatwhilebothmalesandfemalescansearchforamateinsomerestrictedpoolwhereeveryoneadmittedhasthesameculturaltrait(henceallmarriages inthepoolarehomogamous), admissiontothepooliscostly. (W ethinkofdirectadmissioncosts, butalsoofthecosts intermsofotherunmodelleddesirablecharacteristicsofthematch, whichderivefromconstrainingoneselftosearchinarestrictedpool.)

M anydi¤erentinstitutionsdofunctionatleastpartiallyasmarriagepoolsrestrictedalongculturaltraits. ForinstanceKwon (19 9 7 ) documents thecentralityoftheKo-reanEthnicChurchinH oustonasamechanism forculturalidentityandasanetworkofcontacts among…rstandsecondimmigrationKoreanimmigrants. A similarpictureregardinglocalcatholicchurches isdrawnbyM atovina(19 9 5) fortheSpanish-speakingpopulationinSanA ntonio, Texas, between1821 to1860. Tobetterillustrateouranaly-sis ofthemarriageprocess as amechanism fortransmissionofculturaltraits, and inparticulartoisolatetheinstitutionswhichmayfunctionasrestrictedmarriagepools, weconsidertwoexamplesofpopulationswithratherextremesocializationpractices: aris-tocratsinFranceandO rthodoxJews inN ewYork.The Bottin M ondain andtheRallye. Various ethnographicstudies ofaristocrats haverevealedthe importanceoftheirattachmenttospeci…cculturalvalues andtheircon-cernfortheinter-generationaltransmissionoftheirsymbolicandculturalcapitalsuchas familynames, negativeattitudes towardsworkandmoney, andthe importanceoflandproperty(G range(19 9 6), M ension-R igau(19 9 3), P incon-P inconCharlot(19 8 9 ), deSaintM artin(19 9 3)).

B uthowarethesevaluestransmitted?InFrancethemostrelevantinstitutionswiththispurposearetheBottinM ondain, themainaristocracy’slistingbook, andtheR allye,achainofdancingparties(G range, 19 9 6).

Families canbelistedintheBottinonlyifinvitedbyfamiliesalreadylisted. M ostinformationpublishedintheBottin M undain isfamilyanddynasticoriented, andpro-fessionalindications arekepttoastrictminimum. 10 T heR allye, whichorganizes agatheringofbetween100 and500 youngpeopleeachmonth, consists insteadofagroupofyoungsinglewomen, whosefamilies arelisted intheBottin M ondain. T hefamilyofeachwoman, when subscribingtotheR allye, commits tohostapartyforalltheparticipantsoftheR allye.

A longwiththeBottin M ondain, theRallye is thereforean institution intendedtostimulatehomogamousaristocraticmating. Itinvolvessubstantialresourcesspentbythedi¤erentfamilies(partiesaregenerallyorganizedinsumptuouspalaces), andwellre‡ectsourvisionofarestrictedpoolinwhichresourcesarespenttoincreasetheprobabilityofbeingmarriedhomogamouslywithrespecttotherelevantculturaltrait.

Fromasurveyof39 14nuclearfamiliesintheBottinM ondainduringtheperiod19 03-10D atesofbirtharenotmentionedexceptforminorchildren, whichisusefulifthebookistobeused

asamarriagepool.

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19 8 7 , A rrondel-G range(19 9 3) estimatetheprobabilityofhomogamousmarriageforachildofafamilyintheBottinM ondain. T hey…ndasigni…cantrateofhomogamywellabovethatimpliedbyrandommatching. T heaverageprobabilityofbeingmarriedwithsomeoneoftheBottinM ondainforadaughterofacouplelistedintheBottinis44% (intheperiod19 50-19 69 ) and39 % (intheperiod19 7 0-19 83). Foryoungmalestheaverageestimatedprobabilityineitherperiods is 39 % . W henthetwoparents shareimportantaristocraticattributes (e.g., oldaristocracy, afamilycastle, ormembership inanaris-tocraticclub), this probabilityis over65% foryoungfemales, andover80% foryoungmales.TheShadchan. O rthodoxJews liveinmostlysegregatedneighborhoodsandadheretoveryextremenormstopreservetheirreligiousandculturaltraits(seetheethnographicstudiesofH eilman, 19 9 5, andM ayer, 19 7 9 ). Inareligiouscommunitywhosevariouspro-scriptionslimitcasualencounterbetweenthesexes, manymarriagesarearranged. T heethnographicstudyofO rthodoxJews inBoroPark, anO rthodoxJewishneighborhoodinBrooklyn, N ewYork, conductedbyM ayer(19 7 9 ) inthe 7 0s, surveysmatch-makers(shadchans). T hisstudyrevealsthatnotonlydoshadchans serveasgo-betweens(‘tele-phonenumbers’distributors’), butmostimportantlytheyalsoinform bothparties ofeachother’s adherencetoreligious norms, prescriptions andproscriptions (e.g., aboutthedress codeofthewoman, thetenureattherabbinicalseminaryoftheman, etc.).Essentially, theroleoftheshadchan inguaranteeingthepreservationofreligious andculturaltraitsinmarriageispreserved, evenifitshistoricalroleinprotectingandmatch-ingfamilies’assetshaslostmuchofits importanceandisnotanymoreanintegralpartofthetraditionalmarriagesystem.

A s importantasmatch-makingis (asarestrictedmarriagepool) inO rthodoxJew-ishcommunities, ‘love-marriages’areslowlyreplacingarrangedones. N onetheless, forinstanceinBoroPark, manyinstitutions, fromkosherpizzaparlorsandcafeteriasofthehundredsoftheYeshivas (religiousschools) intheneighborhood, toO rthodoxsummercamps, andYoungM en’s& W omen’s H ebrewA ssociation’sco-edactivities, operatetosubstitutetheshadchan infacilitatingmatingbyreligiousandculturaltraits(seeagainM ayer, 19 7 9 ).

T he institutions ofarrangedmarriages, segregatedlivingarrangements, segregatededucationinreligiousschool, andthecreationofrestrictedmarriagepoolslikesummercamps, has been exceptionally e¤ective in promotinghomogamy forO rthodox Jews.A ccordingtothe N ationalJews Population Survey, theintermarriagerate in 19 9 0 forO rthodox Jews was only3% , as opposed to37% forConservative Jews and 53% forR eform Jews.

T hepatternofhomogamyandsegregationobservedforFrencharistocratsandO rtho-doxJews, whileextreme, iscertainlynotunique. B aker(19 7 9 ), forinstance, documentssimilarratesofhomogamyforacommunityofU pperSilesianfarmerslivinginsegregatedneighborhoodsaroundanethniccatholicparishinTexas, from1850 to19 20. H omogamy

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rates inthiscommunityarestillveryhigh, ataround50 % .

2.1 TheA nalysisSupposetherearetwopossibletypes, fa;bg, ofculturaltraits in thepopulation. Inparticular, di¤erenttraitsmightcapturesomeaspectofethnictraitsorreligiousbeliefs.

Ineachperiodtherearetwostationary, equallysizedpopulationsofadultmalesandfemales. A gents livetwoperiods. Youngagentsarebornwithoutwellde…nedculturaltraits, whichtheyacquire(inawaydescribedbelow)beforebecomingadult. Inhisadultlife, amalegetsmatchedwithanadultfemale(inawaytobedescribedbelow)toformahousehold. Inordertomaintainthesizeofeachpopulationstationary, weassumethateachfamilyunionhastwochildren, amaleandafemale.

Parentsarealtruistictowardstheirchildrenandwanttosocializethemtotheirownspeci…cculturalmodel. L etV ijdenotetheutilityatypeiparentderivesfromatypejchild(i2fa;bg). W eassumethenV ii> V ij(andV jj> V ji).11

T he socialization process is modelled as follows. T hefraction ofindividuals withtraitiinthepopulationisdenotedqi. A ll‘naive’children, withoutde…nedpreferencesorculturaltraits, livinginafamilyinwhichbothparentshavethesametrait, are…rstexposedtotheirfamilytrait, sayi. ‘D irectvertical’socializationtotraitioccurswithprobability ¿i. W e imposetheextremeassumption thatonly families inwhich bothparentshavethesametraitcanverticallysocializetheirchildren. Childreninfamilieswitha‘mixed’traitpickthetraitofarolemodelchosenatrandom inthepopulation(i.e., they picktraitiwith probability qi and traitjwith probability qj = 1 ¡qi).Similarly ifa child from afamily with, say, traiti is notdirectly socialized, whichoccurswith probability 1¡¿i, hepicks thetraitofarolemodelchosenrandomly inthepopulation. Finally, socialization is costly. Socialization costs increasewith theprobabilityofsuccessfuldirectsocializationbyparents, andaredenotedH (̄ ¿i), fori2fa;bg (̄ isjustaparameterwhichweshalluseinthecomparativestaticsexercises). T hemarriagechoice, then, hasacruciale¤ectonthesocializationtechnologyavailable, andagents, choosingthebestmatetosocializechildren, aspireforhomogamyinmarriage.

11SupposeagentschoosewhenadultsomeabstractactionxinsomesetX . Childrenwithpreferencesoftypej, uj(x), thenwill, ingeneral, makeadi¤erentchoicethanparentsoftypeiwould, andviceversa.A ltruisticparentswillnecessarilypreferchildrenwiththeirowntypeofpreferences whenevaluatingtheirchildren’schoiceswiththeirown(theparents’) utilityfunction. Formally:

V ij=ui(xj); wherexj= argmaxx2X uj(x)

andhencetypicallyV ii > V ij (symmetricallyfortraitj). Itis importanttonotice, though, thatifthechoicesetX dependsonthepreferencetype, andforinstanceislargerforagentsoftypei (e.g. becausetype i agents arefavored inthelabourmarkets), thenparentsoftypej mightwanttosocializetheirchildrentotheoppositetypei. B yassumingthatV ii > V ij, asalreadymentioned, wee¤ectivelyrestricttherelevanceofouranalysis to‘pure’culturaltraitswhichhavenoe¤ectontheobjectiveeconomicsuccessoftheagents.

10

W emodelmarriagechoice inwhatfollows (wesetthenotation forthegeneralagenti2fa;bg).

M atchingofadultindividuals isorganizedviaamarriagegame. Theprobabilityofenteringanhomogamousmarriageisendogenouslychosenbyeachagent. M oreprecisely,weassumetherearetworestrictedmarriagematchingpools(oneforeachculturaltrait)whereindividualswiththesametraitcanpossiblymatchinmarriage. W ithprobability®ianagentoftypeienters therestrictedpooland is marriedhomogamously. W ithprobability1¡®ianagentoftypeidoesnotgetmarriedintherestrictedpool. H ethenentersacommonpoolmadeofallindividualswhohavenotbeenmatchedinmarriageintheirownrestrictedpools. Inthiscommonpoolindividualsmatchrandomly. IfAiisthefractionofindividualsoftypeiwhoarematchedintheirrestrictedpool(inequilibrium,bysymmetry, allindividualswiththesametraitbehaveidenticallyandhence®i= Ai)the probability an individualoftype i in the common unrestrictedmarriagepoolismatched inmarriagewithan individualofthesametype is then (1¡A

i)qi

(1¡Ai)qi+ (1¡Aj)(1¡qi);andhencetheprobabilityofhomogamousmarriageofan individualoftypeiis givenby:

¼i(®i;Ai;Aj;qi) = ® i+ (1¡®i) (1¡Ai)qi

(1¡Ai)qi+ (1¡Aj)(1¡qi) (1)

W eassumethatindividualsoftypeicana¤ecttheprobabilityofbeingmatchedintheirrestrictedpoolbychoosing®iatacostC (±®i), where± is justaparameterwhichwewilluseinthecomparativestaticsexercises. T hetypicalproblem ofamaleoftypeiwillbetochoosetheprobabilityofmatchingintherestrictedmarriagepoolknowingthat, ifheismatched inanhomogamous household, hehas access toatechnologytosocializehis children. A nagentwithtraitichooses ®i2 [0 ;1], forgivenAi, Aj, qi, tomaximize

¼i(® i;Ai;Aj;qi)W i(qi)+ [1¡¼i(®i;Ai;Aj;qi)][qiV ii+ (1¡qi)V ij]¡C (±®i) (2)

whereqiV ii+ (1¡qi)V ijrepresents theexpectedutilityofatypeiparentinanhet-erogamousmarriage(inwhichthesocializationofthechildrenisdeterminedbyrandommatchingonly);whileW i(qi)representsthecorrespondingexpectedutilityinanhomog-amousmarriage. Sincehomogamousmarriagesareendowedwithadirectsocializationtechnology, W i(qit) dependsontheparents’choiceofsocializatione¤ort, ¿i, aswellasonmatchingprobabilityqi:

W i(qit) = max¿i [¿i+ (1¡¿i)qi]V ii+ (1¡¿i)(1¡qi)V ij]¡H (̄ ¿i) (3)

N otethatagentsiandjinteractnon-triviallyinthemarriagegame: agent’simax-imization problem depends (via ¼i(:)) on Aj, the fraction ofagents oftypejin therestrictedpool. Infactthemoreagentsoftypejintherestrictedpool, thelessofthemintheresidualpopulation, andthemorefavorableforagentsoftypeithestrategyofnotenteringtheirownrestrictedpool(andbeingmatchedinthecommonresidualpool).

11

A symmetricN ashequilibrium ofthemarriagegameis thenrepresentedbymappings®i(qi)whichare…xedpoints ofthebestreplies ofagents i2 fa;bg derivedfrom themaximizationofequation(2). T heprobabilityofhomogamousmarriageforagents oftypeiistheninequilibrium justfunctionofqi, andisdenoted¼i(qi).

Proposition1 U nderconvexityandregularityassumptionsoncostsC (±® i)andH (̄ ¿i)(explicitelystatedintheappendix),

ThereexistauniquesymmetricN ashequilibriumofthemarriagegame, denoted[® (qi)]=[® i(qi)]i2fa;bg;moreover, ®(qi) is acontinuousmapping.

Thesolutionofthesocializatione¤ ortchoiceofhomogamousfamilies, i.e. ofthemax-imizationinequation(3), denoted [¿i(qi)]i2fa;bg, is acontinuousmapping.

2.2 ImplicationsIn this section we study severalimplication ofthemarriage and socialization modelforagivendistribution oftraits in thepopulation, qi. T he implieddynamics ofthedistributionoftraits is studiedinthenextsection.

Proposition2 The equilibrium probabilityofmatchingin the restricted pool, ® i(qi),andtheequilibrium socializatione¤ ortofhomogamousfamilies, ¿i(qi), aredecreasinginqi, fori2fa;bg.

T heprobabilityofmatchingintherestrictedpoolandthechoiceofsocializatione¤ortofhomogamousfamiliesarehigherforminorities, otherthingsequal.12 M inoritieshaveastrongerincentivetosegregate, tobehomogamous, andtosocializechildren. Infact,anindividualinaculturalminorityhasalargeprobabilityofmakinganheterogamousmarriage ifhedoes notentertherestrictedpool, sincethecommonunrestrictedpoolwouldbemostlypopulatedbymajoritytypes. M oreover, aminoritytypeinanheterog-amousmarriagewillnothaveaccesstothetechnologyofsocializationandhischildrenwillbesocializedtotheexternalculturalenvironment, thatis, withlargeprobability,themajoritytrait. T hismotivatesagentswithminoritytraitstohomogamy. O nceho-mogamous, familieswithaminoritytraitstillhavelargeincentivestodirectlysocializetheirchildrenbecauseifdirectsocializationisunsuccessful, onceagain, childrenwillbesocializedtotheexternalculturalenvironment, i.e. mostprobablytothemajoritytrait.

12Itis importanttostress thatthis cross-sectionalinterpretationofProposition 2 requires culturaltraitsnottoomuchdi¤erentintermsoftolerancetoeachother, i.e. intermsofV ii¡V ij=¢V i;i2fa;bg. M oreover, theidenti…cationofculturalminoritiesandmajorities isonlypossibleifthedynamicsofthedistributionofculturaltraits is notatits stationarystate, sinceotherwisethepopulationwilltendtobeevenlydistributedacrossculturaltraits;seeSection3. T hecross-sectionalinterpretationiscentraltoouranalysis becausemostoftheempiricalevidenceavailableonmarriageandsocializationis infactcross-sectional;seeSection2.3.

12

Itisalsoeasilyshownthatforanygivendistributionoftraits, qi, both®i(qi)and¿i(qi)aredecreasingin socialization costs, ¯ , and increasingin thegain from socialization,V ii¡V ij= ¢ V i. A lso, ®iisdecreasingin(while¿iisuna¤ectedby)marriagesegregationcosts, parametrized by ±. A positive change in the costofdirectsocialization, notsurprisingly, negativelya¤ects directsocialization e¤ort, butitalsonegativelya¤ectsentrytotherestrictedmarriagepoolsincethebene…tsoftherestrictedpoolconsistintheoptiontousethedirectsocializationtechnology, which is nowmorecostly. Inthesameway, highergainsfromsocializationpositivelya¤ectbothdirectsocializatione¤ortandentry intotherestrictedmarriagepool, whilehighermarriagesegregationcosts ±negativelya¤ectmarriagesegregation, withouthavinganye¤ectonfamilysocialization,whichispossibleonlyforhomogamousfamilies.

T heequilibriumhomogamyrateofthetypeipopulationatequilibrium isgivenby:

¼i(® i;®i;®j;qi) = ¼i(qi) = ®i(qi)+ (1¡®i(qi)) [1¡®i(qi)]qi

[1¡®i(qi)]qi+ [1¡®j(1¡qi)](1¡qi)(4)

H owdohomogamyratesdependonthecompositionofthepopulation?H omogamyratesofminoritypopulationsre‡ectthetrade-o¤ ofstrongermarriagesegregationstrategies(®i(qi) is decreasinginqibyProposition2)withtheadversee¤ectduetotheirhigherinterculturalmatchinginthecommonpool, wherematchingisrandomandhencere‡ectsrelativepopulationsizes. A saconsequence, thedependenceof¼ionqiisnotmonotonic.Itcanbeshownthat, ifpopulationiis intolerantenough(i.e. forhighenough¢ V i), itshomogamyrate…rstdecreasesandthenincreases inqi. M inoritypopulationshencewilltendtohavehomogamyrates inverselyrelatedtotheirshareinthewholepopulation.

Socializationrates, asmeasuredbytheprobabilityofanhomogamous familywithtraitiofhavingachildofthesametrait, Pii= ¿i(qi)+ (1¡¿i(qi))qi, alsodonotdependmonotonicallyonqi. Itiseasytoshowthat, aswiththehomogamyrate, thesocializationrateofgroup i…rstdecreasesandthenincreases inqi, if¢ V i is largeenough.

2.2.1 Extensions

T hemarriagemodeljustintroducedcanbeextendedinvariousdirectionswiththeob-jectiveofderivingricherempiricalimplications. W e summarily reporthereon someextensions wepursued. (T hecompleteanalysis is reported in an A ppendix availablefromtheauthorsuponrequest.)

Supposeafractionofthepopulation, thesameacrossgenderandculturaltype, cannot(ordoesnotwantto)havechildren. W eassume, forsimplicity, thatsuchagentsformamarriagepoolbythemselves. Sincetheonlyadvantageofhomogamyinourset-up liesinthetechnologyofchildren’s socialization, theyhaveinfactnointerestinhomogamyalongtheculturaltraitdimension. Insuchamodel, thedi¤erentialhomogamyoffamilies

13

withchildrenwithrespecttofamilieswithoutchildrenismeasuredbyas

¢ H M i(qi) = H M i(qi)¡[qi]2

where [qi]2 is theprobabilityofhomogamyforanagentwithtraitiifhecannothavechildren, calculatedfrompurerandommatching.

Itcanbeshownthatsuchhomogamydi¤erential, ¢ H M i(qi), is positiveinequilib-rium.

Supposemarriageinthecommonpoolis biasedinfavorofhomogamousmatching.Forinstance, thebias couldarisefrom segregatedneighborhoods inthepopulation, orfromtheexistenceofinstitutionswhichfunctionasrestrictedmarriagepoolsandwhoseentry is free. W ewritethe probability ofan individualoftype ibeingmatched inmarriagewithanindividualofthesametype(thehomogamyrateoftypei) as

¼i(®i;Ai;Aj;qi;°) = ® i+ (1¡®i)(1¡Ai)qi+ (1¡Aj)(1¡qi)°

(1¡Ai)qi+ (1¡Aj)(1¡qi) (5)

wherethesecondterm ontheright-hand-sideof(5) represents thefractionoftypeiindividuals homogamouslymatchedinthecommonresidualmarriagepool, giventhatthere is abiasedmatchingprocess parametrizedby°2 [0 ;1]. W hen°= 0 ; there israndom matchingin thecommon pool. W hen°= 1; individuals matchwith prob-ability 1 tosomeoneofthe sametype in thecommonpool: there is perfectassorta-tivematchingforeachcommunityindependentoftheexistenceofrestrictedpools (i.e.¼i(® i;Ai;Aj;qi;1) = 1 forany® i).

Forthisextensionofthemarriagemodel, comparativestaticsexercisesshowthat®iisdecreasingin°. A nincreaseinsegregationofthepopulationoutsideoftherestrictedpool, (i.e., apositivechange in°), reduces the incentives foragents toenterthere-strictedpool. T hee¤ectonhomogamyrates is, ontheotherhand, ambiguous, becausethechgangein°hasalsoadirecte¤ectonhomogamyrates(homogamousmarriagesbyrandommatchingarenoweasier). U ndersomeweakconditions(detailedintheA ppen-dix), itcanbeshownthatthedirecte¤ectonhomogamyrates is stronger, andhencethatapositivechangein°hasapositivee¤ectonequilibriumhomogamyrates.13

T helastextensionweconsiderinvolvesaddinganexogenousprobabilityofdivorce.Supposeeach familyhas aprobabilitycofseparating. W eassumeseparation occursafterchildrenareborn, butbeforetheyaresocializedtotheculturaltraits. Ifseparationoccurs, weassumethatoneoftheparents is chosenrandomlytoform asingleparentfamily. W ealsoassumethatsocializationismorecostlyforsingleparentfamilies (see

13O bviously ° hasnoe¤ecton¿i and¼i. Itcanalsobeshownthat¢HM i decreaseswith ° .A similaranalysis, with qualitatively similarcomparative statics results, can be carried overfor

distortionswhichfavortheparents’traitintheobliquephaseofsocialization.

14

T homson-M cL anahan-Curtin, 19 9 2, forsomeevidenceonthis point). N otethatsingleparentfamilies, as opposed toheterogamous families, have atechnology tosocializechildren;noambiguityonwhichtraittotransmitarises inthiscase.

Inthiscasethetypicalproblemofanindividualoftypeiisbecomestomaximize

¼i(®i;Ai;Aj;qi)[(1¡c)W im (qi)+ cW is(qi)]+[1¡¼i(®i;Ai;Aj;qi)][(1¡c)W ih(qi)+ cW is(qi)]¡C (±® i)

whereW im (qi), W is(qi), andW ih (qi) denote, respectively, thegains from socializingchil-dreninsideanhomogamousmarriage, asingleparentfamily, andanheterogamousmar-riage. G ivenourassumptionsaboutthesocializationtechnologiesofthedi¤erentfamilytypes, thegainsfromsocializationaregivenby:

W im (qi) = max¿i [¿i+ (1¡¿i)qi]V ii+ (1¡¿i)(1¡qi)V ij]¡H m (̄ ¿i) (6)

W is(qi) = max¿i [¿i+ (1¡¿i)qi]V ii+ (1¡¿i)(1¡qi)V ij]¡H s(̄ ¿i) (7 )

W h (qi) = [qiV ii+ (1¡qi)V ij]withH m (̄ ¿i)andH s(̄ ¿i)beingthesocializationcostfunctionsofhomogamouscouplesandsingleparentfamily. W eassumeH m (̄ ¿i) < H s(̄ ¿i), forall¿i2 (0 ;1).

T hesolutiontothesocializationproblemsprovidessocializatione¤ortsforhomoga-mousparents, ¿im (qi), andsingleparentfamilies, ¿is(qi), withthepropertythat¿im (qi) >¿is(qi);homogamous families haveabetterdirectsocialization technologythan singleparentfamilies, andhenceinequilibrium theyactuallydosocializetheirchildrenmoreintensely. Comparativestatics exercises showthathigherdivorcerates inequilibriumimplylowersegregationrates inrestrictedmarriagepools, lowerhomogamyrates, andlowerdi¤erentialsinhomogamywithrespecttoagentswhocannothavechildren. W henlookingforamate, agents anticipatethatthemarriagemightfail. T hevalueofthehomogamyinmarriageisthenreduced, because, ifthemarriageends, childrenwillbesocializedwith arelatively ine¢cienttechnology. A gents’ incentives toenterthere-strictedmarriagepool, i.e., tolookforanhomogamousmate, arelowerthehighertheprobabilityofdivorce, c.

2.3 EvidenceT his section collects someoftheexistingempiricalevidence, mostlydrawn from thesociologicalliterature, ontheimplicationsofthemodelregardinghomogamyandsocial-izationwithrespecttoethnicandreligioustraits.

2.3.1 H omogamy

H igh rates ofhomogamy alongculturaldimensions and positive di¤erentials in ho-mogamywith respecttofamilies which cannothavechildren arecertainlyafact, at

15

leastalongthereligiousandtheethnicdimensions. T hehomogamyofnewimmigrantsintheU .S. attheturnofthecenturywas ‘almostcastelike’, andquitepersistentoversuccessivegenerations. H igh rates ofhomogamybyethnicgroup aremoregenerallydocumentedbyPeach(19 80). T heexamplesofFrencharistocrats, O rthodoxJews, andU pperSilesianfarmersreportedinSection2 alsosupporttheseobservations. R eligioushomogamyisalsopervasive(seeSander, 19 9 3, forCatholics, andJohnson, 19 80, Schoen-W einick, 19 9 3, forotherdenominations).14 H omogamyrateswellabovethoseimpliedbyrandom matching, ofcourse, mightwellhavemanyexplanationsotherthanthedesiretopreserveoneorseveralculturaltraits inchildren. M easuresofpsychologicalcostsofreligiousintermarriagearequitelow, bothintermsofcostsbornebyspouses(e.g., mar-italinstability;seeL ehrer-Chiswick, 19 9 3, H eaton, 19 9 4) andbychildren(e.g., anomie,lackofself-esteem;seeA ellen-L ambert, 19 69 , Johnson-N agoshi, 19 86, Stephan-Stephan,19 9 1), therebysupportingtheargumentthatthesocializationofchildrenisanimportantdeterminantoftheobservedreligioushomogamy.

Themarriagemodelwehave developedmore speci…cally implies thathomogamyrates should behigherforfamilies which expecttohavechildren. In particular, ho-mogamyratesshouldbehigherinmarriageunionsthanincohabitations, sincefertilityexpectationsofcohabitersarenotstatisticallydi¤erentfromthoseofsingleindividuals,asdocumentedbyR indfuss-VandenH euvel(19 9 0). Consistently, 51% ofmarriagesintheN ationalSurveyofFamiliesandH ouseholds(19 8 7 -88)arereligiouslyhomogamous, com-paredtoonly37% ofcohabitations (Schoen-W einick, 19 9 3). R elatedly, L ehrer(19 9 6)reportshigherintendedfertilityforreligiouslyhomogamouscouples.

O urmodelofculturaltransmissionhas itsmostimportantclass ofimplicationsforthebehaviorofminorities. M inorities, otherthings equal, shouldexercisemoree¤ortin marriage segregation. W hile e¤ortin marriage segregation is di¢culttomeasuredirectly, populations withminoritytraits, such as O rthodox Jews orA mish, seem tosegregatemoreintenselyandtodevelop institutionsforsegregatedmarriages. Evenforapopulationwithlessextremehomogamypatterns, Japanese-A mericans, O ’Brien-Fugita(19 9 1) reportthatculturalandethnicinstitutionsandclubs(whichwewouldinterpretasrestrictedmarriagepools) aremostprevalentinareaswhereJapanese-A mericanareminorities.

FormalevidenceonhomogamyratesforreligioustraitshasbeenreportedandstudiedbyJohnson(19 80). U singdatafrom thepooled19 7 3-7 6N O RC G eneralSocialSurvey,the1960 G rowthofFamilies Survey, andothersources, Johnson(19 80) constructsmar-riagetablesforsixreligiousgroups.15 H ethenestimatesalog-linearmodelofmarriage

14Indirectevidencefortheperceivedimportanceofreligioushomogamyinmarriagedecisionscomesfromthestudyofconversions: bothW arren(19 7 0) andG reeley(19 7 9 ) foundthatmostreligious iden-ti…cationchangeswereattributabletotheconversionsofspousestoestabilishhomogamyinreligiouslyheterogamousmarriages.

15T hemaindenominationsineachofthesixgroupsare: B aptists, M ethodists, Presbyterians, L uther-ans, Catholics, andO thers.

16

frequencies foreachreligiousgroup to…tthemarriagetables, identifyingtwomainex-planatoryfactors intheanalysis ofassortativemarriage: thereligious compositionofthepopulation, and the‘intrinsicendogamy’ofeachreligious group, where‘intrinsicendogamy’isameasureofthegroup’se¤ortinmarriagesegregation, i.e., ameasureof®i inournotation. Boththeestimatesofthemodelrelativetothenationalandthere-gionallevel(i.e., relativetothenationalortheregionalcompositionofthepopulationbyreligiousgroup), showthattheintrinsichomogamycoe¢cientsaregenerallyhigherforthegroupswhichcompriseasmallerproportionofthepopulation, asourresults imply.A tthenationallevel, forinstance, thesmallestgroup, ‘O thers’(theresidualgroup), hasthehighestintrinsichomogamy, whilethelargestgroups, BaptistsandCatholics, havethelowest. A ttheregionallevel, also, thesmallestintrinsichomogamyforCatholics isin theN orth-East, whereCatholics comprisemorethan 4 5% ofthepopulation, whilethelargest(morethanthreetimesaslarge) is intheSouth, whereCatholicsconstituteonly10 % ofthepopulation.

2.3.2 Socializationandsocializatione¤ort

Socializationtotraitiinthemodeldependspositivelyonsocializatione¤ort, ¿i, andontheshareofthepopulationwithtraiti, qi. M oreover, sinceweassumedheterogamousfamilies arenotendowedwithasocializationtechnology, homogamyshouldproxyforsocialization e¤ort, ¿i. Consistentlywith these implications ofouranalysis, there isevidencethatsuccessfulsocialization occurs morefrequently in homogamous families(H ayes-P ittelkow, 19 9 3, H eaton, 19 86, H oge-Petrillo, 19 7 8 , H oge-Petrillo-Smith, 19 82,O zorak, 19 89 ). A lso, M ayer(19 85) constructed a survey ofmixed Jewish-Christianmarriages in19 83, comparingseveralmeasuresofsocializationsuccessofconversionarymarriages (in which the Christian spouse converted toJudaism atmarriage) tothesamemeasuresforheterogamousmarriages. H eestimatesthatchildrenofconversionarymarriagesaremorethanthreetimesaslikelytoidentifythemselvesasJewsthanchildrenofheterogamousmarriages.M oreimportantly, intheirstudyofreligiousbeliefinA ustralia, H ayes-P ittelkow(19 9 3)…ndthatthee¤ectofhomogamyonsocializationvanisheswhenameasureofsocializatione¤ort(e.g., ‘parentaldiscussionofreligiousbeliefs’) is introducedintheregression. T hisisconsistentwithourmodel’simplicationthathomogamya¤ectssocializationonlyasaproxyforhighersocializatione¤ort.

Interm ofdirectsocializatione¤ort, ourmodelimplies thathomogamous familiesexercisemoree¤orton children socialization (because they haveabettertechnologytothis e¤ect), and families withminority culturaltraits exercisehighersocializatione¤orts, coeteris paribus. T he presence ofhighersocialization e¤ortforhomogamousfamilieswithchildren is suggestedbytheanalysis ofthesurveypanelconstructedbyThornton-A xinn-H ill(19 9 2)onD etroitfamiliesbetween1962 and19 80. M arriedfamiliesinthepanelengagemoreinreligiousactivities(proxyingforreligioussocialization), after

17

conditioningforreligiosityatthemomentinwhichthefamilyis formed, thanfamiliesin cohabitation (as alreadynoted, cohabitations aremuch less fertileandmuchmoreheterogamousthatmarriages).D irectevidence forthe socialization behaviorofminorities is ratherscarce. B arber(19 9 4), however, doesdocumentthatblackandH ispanicfamiliesmoreaggressivelyso-cializetheirchildren: theybothhavehigherstandardsforbehaviorandarebetterabletoenforcethosestandard.O therinterestingevidenceonsocializatione¤ortcanbeobtainedbyanalyzingneigh-borhoodsegregationbyethnicandreligiousgroup, insofarasneighborhoodsegregationisendogenouslydeterminedpartlybythedesiretosocializeo¤spring. Ethnicneighbor-hoodshavebeenadominantaspectofA mericansocietysinceitsearlyhistory, especiallysincethemassmigrationstotheU .S. inthelastcentury. A searlyas17 03, forinstanceN ewYorkstreetswere identi…edas eitherD utchorBritish (H omberger, 19 9 4). A lso,extremeresidentialsegregation byethnicityofturnofthecentury immigrants iswelldocumented, e.g., byD uncan-L ieberson (19 59 ), Peach (19 80). W hileadjustmentcostexplanations arealsoconsistentwithhigh segregationlevels alongethniclines of…rstgenerationmigrants, suchexplanations canhardlybeextended, inouropinion, tosig-ni…cantlevelsofethnicsegregationofneighborhoodswhichpersistafterthesecondandthirdgenerations. Inthisrespect, using19 7 0 CensusD ata, Borjas(19 9 5) estimatesthattheprobabilitythatasecondgenerationethnicfamilygroup lives nearfamilygroupsofthesameethnicoriginismuchhigherthanonewouldexpectiffamilieswerespreadacrossneighborhoods independentlyoftheirethnicorigin. Forinstance, amongsecondgenerationworkers, thetypicalfamilyofPolishancestrylives inaneighborhoodthatis7:8% Polish, eventhough…rstandsecondgenerationPolishmakeuponly1:7% ofthepopulation. Similarly, secondgeneration Italians live in 12 :1% Italianneighborhoods,eventhough Italians …rstandsecondgeneration immigrants accountforonly 2 :8% ofthepopulation.16 M oreover, accordingtoBorjas (19 9 5), highsegregationrates persistforthirdgenerationimmigrants, andthereislittleevidencethatonlyeconomicallydis-advantagedgroupsaregeographicallysegregated.

2.3.3 Segregationanddivorce

O urmodelimpliesthatabiasinfavorofhomogamousmarriageintheunrestrictedpoolhas anegativee¤ectoneachculturalgroup’s e¤ortinmarriage segregation, becauseagentsreacttothebiasbydecreasingtheire¤orttoentertherestrictedpool. Johnson(19 80) …ndshigher‘intrinsichomogamy’rates inurbanenvironmentthaninruralenvi-ronments, whichisconsistentwiththeimplicationsofourmodelifurbanenvironmentsarecharacterizedbyeasierinterculturalrelationships.

W hen extendingtheanalysis tosingleparentfamilies anddivorce, weexpectless16EvenstrongersegregationpatternsbyethnicityarerevealedintheN ationalL ongitudinalSurveyof

Youth (B orjas, 19 9 5).

18

directsocializatione¤ortforsingleparentfamiliesthanforhomogamousfamilies. T hisisconsistentwithT homson-M cL anahan-Curtin(19 9 2), which…ndweakercontrolofandfewerdemandplacedonchildrenofsingleparentfamilies (independentoftheparent’sgender). W ewouldalsoexpectadeclineinhomogamyratesespeciallystartingfromthe60s, asaconsequenceofhigherdivorcerates(seeforexampleD avis, 19 85, forasurveyofthemain trends in marriagerelationships in the U .S.). T his is broadlyconsistentwiththetrendstowardcohabitations intheU .S.fromthe60s(Spanier, 19 85), since, aswealreadynoted, cohabitations arerelativelyheterogamous. A lso, andmoredirectly,theaveragefractionofreligiouslyhomogamousmarriages inthe G eneralSocialSurveysampleslowlydeclinesfrom :9 inthe20sto:83 inthe60s, andthendropsto:75 inthe7 0s (andremainsconstantinthe80s).

3 ThedynamicsofthedistributionofculturaltraitsIntheprevioussections, weanalyzedhowmaritalstrategiesacrossculturalcommunitiesarea¤ectedbytheagents’concernfortransmittingculturalvalues, assumingthedistri-butionofculturaltraits inthepopulation, qi, was exogenouslydetermined. H owever,patternsofmaritalsegregationsandsocializationacrossculturalgroupshavee¤ectsonthedynamicsofculturaltraits insociety, oronthedynamicsofqi.

Takingthedynamicsofculturaltraitsexplicitlyintoaccountallowsustoaskques-tionslike: W hatdistributionoftraitswillprevailinthelong-run?D oesthepopulationremainmulticulturalinthelimit, ordoweobserveatendencytowardsculturalhomo-geneity?W hatarethee¤ectsofvariousstructuralchangesininstitutionalarrangementswithinmarriage?Inthissectionweinvestigatetheseissuesbyanalyzingtheexplicitdynamicsofculturaltraits inthepopulation, andthedependenceofthedynamics onvarious historicalin-stitutionalchanges inmarriagerelationships, likeagreateravailabilityofinterculturalrelationships (dueforinstancetourbanizationorinformationtechnologies);agreaterfreedomofchoosingones’mate;agreateracceptanceofdivorce, singleparentfamilies,andfemalelabormarketparticipation.

L etus…rstconsiderthemodelwithabiasinthecommonpool(°6= 0)butnodivorceandsingleparentfamilies (c= 0). T heprobabilitythatachildwithafatherwithtraitiwilldeveloptraitiis

Pii= ¼i(qi;°)[¿i(qi;°)+ (1¡¿i(qi;°))qi]+ [1¡¼i(qi;°)]qi

where¼i(qi;°) is theequilibrium homogamyrateprobabilityofpopulationi. W enoteitsdependenceinequilibriumontheparameter°. Similarly, theprobabilitythatachildwithafatherwithtraitjwilldeveloptraitiis

Pji= ¼j(qj;°)[(1¡¿j(qj;°))qi]+ [1¡¼j(qj;°)]qi

19

L etqitdenotethefractionofthepopulationwithtraitiattimet(weomittheindextwhennotnecessary). T hedynamics ofthepopulationofagentswithtraitiis thendeterminedbythedi¤erenceequation:

qit+ 1 = Piitq

it+ P

jit (1¡qit) = qit+ qit(1¡qit)[¼i(qit;°)¿i(qit)¡¼j(qjt;°)¿j(qjt)]

T hisdynamicalprocesshascornerstationarystates, qi= 0 andqi= 1, andpossiblyinteriorstationarystates, qi¤, whichsatisfy

¼i(qi¤;°)¿i(qi¤) = ¼j(1¡qi¤;°)¿j(1¡qi¤); i;j2fa;bg; i6= j (8)

T hefollowingresultstates thatcornerstationarystates areunstable, andthatthereexistsatleastoneinteriorlocallystablestationarystate.

Proposition3 Thecornerstationarystates, qi = 0 andqi = 1, are locallyunstable.Therealways exists oneinteriorsteadystateqi¤, which, underconvexityconditions oncostfunctions (intheA ppendix), is locallystable. 17

T hemechanismofmarriageandculturaltransmissionwestudygeneratesdynamicsofthedistributionofculturaltraitswhichtendtomulticulturalpopulationsandawayfromcompleteassimilationofminorities. T hisisbecausethetransmissionmechanismhasthepropertythatculturalminoritiestendtoreactinequilibriumtotheprospectofculturalassimilationwithmarriagesegregation, homogamousmarriages, andwithmoreintensestrategies forthedirectsocializationofchildren. Eventhoughmajorities havehighersocializationrates, duesimplytothee¤ectofpeers androlemodels, thedynamics ofthedistributionoftraitsinthepopulation, whenonetraitsisclosetobecomingextinct,dependsessentiallyondirectsocializatione¤ort, whichishigherforminorities.18

Itis importanttostressthatsucharesultdependsonthetraits nothavinge¤ectsontheagents’economicopportunities. T his is, ofcourse, anabstraction. T heresultsofProposition 3 aremostproperly interpretedas identifyingaform ofpersistence inthedynamics ofculturaltraits, anon-linearity in thedegreeofculturalassimilation.Suchpersistenceoftraits, andthedi¢culty inacculturationofminorities, whilehardtomeasureanddocument, is evidentinmanyhistoricalandethnographicaccounts oftheevolutionofethnicandreligious traits, as discussed intheIntroduction.19 O neofthefeweconometricattemptsatmeasuringthepersistenceofculturaltraits is Borjas’s

1 7 M ultiple interiorstationarystates mightarise. T hereason is thatwhiletheprobabilityofbeingmarried in the restricted pool, ®i(qi), is a decreasingfunction ofthe frequency ofthe traitin thepopulation, theprobabilityofbeinghomogamouslymarried, ¼i(qi), maybeincreasingwithqi. A s aconsequenceequation(8)mayhavemorethanonesolutioninqi¤.

18Foranexampleofhowinsteadpeerpressureandsocialinteractionsmightleadtohomogeneity, seeG laeser-Sacerdote-Scheinkman, 19 9 6.

19 T he‘R enaissance’ofO rthodoxJews is onesuchaccounts (M ayer, 19 85). Itis notjustexplainedbyextremehomogamyrates, butalsobyrelativelyhighfertilityrates. T heaveragenumberofchildrenperfamilyofO rthodoxJews in19 9 0, accordingtothe N ationalJewishPopulation Survey, wasabove

20

19 9 5 studyoftheassimilationofimmigrants’‘ethniccapital’intheU .S. Consistentlywithourresults, he…ndsquiteslowratesofculturalconvergence, explainedmainlybyneighborhood…xede¤ects, whichweinterpretasaproxyforhomogamyratesanddirectsocializatione¤ort.

H owwillchanges inthemaritalandsocialenvironmenta¤ectthelong-rundistrib-utionofculturaltraits? W ewillconsiderthreesuchchanges. First, observethattheconditionforaninteriorstationarystate, equation(8), canberestatedas:

qa¤¡H T ab(qa¤;°)(1¡qa¤)¡H Tab(qa¤;°) =

qa¤

1¡qa¤¿b(1¡qa¤)¿a(qa¤)

where H Tab(qa¤;°) measures theheterogamyofthepopulation in equilibrium atthestationarystatefractionofpopulationwithtraita, qa¤. T his equation is representedinFigure1, wheretheLL and R R curvesrepresentrespectivelytheleftandtherighthand-sideofequation(8) asafunctionofqa¤.i) D ecreasein°;increasein ±. Consideranegativechangein°, thedistortiontowardshomogamy in theunrestricted pool. Typically, urbanization and thedevelopmentofcommunication and transportation technologies should beassociatedwith anegativechangein°;assuchstructuralchangestendtoincreaseandfacilitateinterculturalcon-tacts. A negativechange in°increases equilibrium marriagesegregation, ®i, ofbothculturalgroups. Itgenerallyincreasesheterogamy, H Tab, anditdoesnota¤ecttheso-cializatione¤ortofhomogamousmarriages(theR R curveinFigure(1)doesnotmove).N otethatanincreaseinH Tab shiftsup(down)theLL curve, toLL0, whenqa¤ islarger(smaller) than 12 (seeFigure1). T hereason is thatan increasein H T

ab decreases forbothgroupstheprobability¼i(qi;°) ofgettinganhomogamousmarriage. H oweverthedecreaseismorepronouncedfortheminoritygroup thanforthemajoritygroup, sincerandommatchingintheunrestrictedpoolfavorsbyde…nitionhomogamyofthemajoritygroup. A s theLL curverepresents theratioofhomogamous marriages ingroup a togroup b, itisthenincreasing(decreasing)withH T ab whena isthemajority(minority)group(i.e., qa¤larger(smaller)than1=2 ). A reductionin°, thoughleadingintheshortruntohighere¤orttomaritalsegregationbybothgroups, generallytends toincreaseheterogamyinsociety, and, as showninFigure1, favors inthelimitthemajoritytrait(thestationarystatefrequencyofthemajoritygroup, i.e., thegroup iwith qi¤ > 12 ,increases).

A nincreasein±, ameasureofthecostofmarriagesegregation, captures, forinstance,greaterfreedom inchoosingones’mate, arelativelyrecentdevelopmentinmarriagein-

4, as opposedtoless than 2 forConservativeandR eform Jews. O urmodelofsocialization, extendedtoendogenous fertility, would explain such positivecorrelation between fertility, homogamyand so-cialization, sincehighlevelofhomogamyandsocializationareequivalenttohighexpected‘quality’ofchildren(B isin-Verdier, 19 9 6). A s alreadynoted, higherintendedfertilityforhomogamous couples isalsodocumentedbyL ehrer(19 9 6).

21

stitutionsacrossmanyethnicgroupsatleastinthewesternworld(seee.g., D avis, 19 85).A nincreasein± decreasesequilibriummarriagesegregation, ®i, therebyincreasingH T ab,whileitdoesnota¤ectthesocializatione¤ortofhomogamousmarriages(theR R curveinFigure) doesnotmove). A n increasein ±, implyingan increasein H Tab, decreasesforboth groups the probability ¼i(qi;°) ofgettingan homogamous marriagematch.H owever, asbefore, thedecreaseismorepronouncedfortheminoritygroupthanforthemajoritygroup. A n increasein ±, thenfavors themajoritytraitinthelimit(thesta-tionarystatefrequencyofthemajoritygroup, i.e., thegroupiwithqi¤> 12 , increases).ii) Increasein ¯ . Itis alsointerestingtoconsidertheimpactofchanges inthecostofdirectfamilysocialization, ¯ . Suchchangesmaybeassociatedwithstructuralchangesingenderroles insidethefamily, liketheincreasedfemaleparticipationrateinthela-bormarket. T his phenomenon increases theopportunitycosts towomenofspendingtimesocializingchildren insidethefamily, andtherefore, shouldbeassociatedwithapositivechangeinthecostofdirectculturalsocialization. A notherhistoricalstructuralchangemovingin thesamedirection, is theshiftfrom afamily-laborbasedeconomytowards amarket-wagebasedeconomy, makingitagainmorecostlyforthefamilytodirectlytransmititsownculturaltrait. Formally, theimpactofalesse¢cientsocializa-tiontechnology, bychangingdirectlyfamilysocialization, ¿i(qi), andindirectlymaritalsegregationstrategies, ®i(qit), a¤ectsboththeLL andtheR R curves inFigure1. A s itinducesareductioninthemaritalsegregationstrategy, ®i, andinthefamilysocializatione¤ort, ¿i;theimpactonthemarriagegameistoinducealargerequilibriumheterogamy,H T ab. A sbefore, this e¤ectdecreasestheprobabilityofhomogamousmatchingfortheminoritygroup morethan forthemajoritygroup, hence implyingalargerfractionofagentsofthemajoritygroup inthelongrundistributionofthepopulation. H owever,thereis nowinprincipleanothere¤ectemanatingfrom thedirectdecreasein ¿istim-ulatedbytheparameter’s change. Ifthe increase in socializationcosts, ¯ , a¤ects thetechnologyoffamilysocializationinthesamewayforbothgroups, theratioofsocializa-tione¤orts ¿

b(qb)¿a(qa) isnota¤ected: theR R curvedoesnotshift. T hee¤ectofanincrease

inthecostofdirectfamilysocialization, ¯ , is equivalent, then, toadecreasein°: itincreasesinthelimitthefractionofagentswiththemajoritytrait(i.e., qi¤, forthetraitisuchthatqi¤> 12 ).iii) Changes in ¢ V i. A s inthecaseofchanges inthecosts offamilysocialization, achangeintheperceivedculturaldistanceofgroupi, ¢ V i, withrespecttotheothergroupwilla¤ectboththeLL andR R curves. Forinstance, iftheminoritygroup(say, groupb)tendstobemoretoleranttowardsthemajoritygroup (¢ V b decreases), thenthatgroupbecomeslesshomogamousandtheequilibrium heterogamyrateH Tab consequentlyin-creases, meaninganupwardshiftofLL. A tthesametime, familysocialization¿b(qb) isalsoreduced, implyingadownwardshiftofR R . Bothe¤ectstendtoincreasethefrac-tionofthemajoritygroup a, andthe‘culturalassimilation’oftheminoritygroup. A nincreaseinculturaltoleranceofthemajoritygroup, group a, similarlyincreases inter-marriagebetweenthetwocommunities. O ntheotherhand, italsoimpliesareduction

22

oftheintensityoffamilysocializationofthatgroup. T he…rste¤ectpositivelya¤ectsthelongrun proportion ofthemajoritygroup, whilethesecond e¤ecttends, on thecontrary, tofavortheminority. T hetotale¤ectis ambiguous. H owever, itis easytoseethatwhenthemajoritygroup is largeenough (qa¤ closeenoughto1), the impactofachangein ¢ V a onlymarginallya¤ects thesocializatione¤ort, ¿a(qa), leavingtheR R curvealmostuna¤ected. In thatcase, onlythepositive impactofadecrease in¢ V a onLL remains, implyinganincreaseinthesteadystatefrequencyofthemajoritygroup, and, conversely, asmallersizedminoritygroup. Forexample, as noted intheIntroduction, A mericanJews’fasteracculturationsincetheendofW orldW arII mightbeexplainedbyothermajorreligiousgroups’increasingtoleranceofinter-marriagewithJews(seeD ershowitz, 19 9 7 ).

Finally, webrie‡ydiscuss the implications ofincreasingtheprobabilityofdivorce, c,onthelongrundistributionofculturaltraitsinthepopulation. T hedynamicsequationisnowgivenby:

qit+ 1¡qit= qit(1¡qit)£(1¡c)[¼i(qit)¿im (qit)¡¼j(qjt)¿jm (qjt)]+ c[¿is(qit)¡¿js(qjt)]

¤(9 )

Inequation(9 )weseethattheculturalselectionforcesoperatethroughtwosocializationchannels. H omogamous couples whohave notdivorced (in proportion (1 ¡c)¼ a(qt)and (1 ¡c)¼ b(1 ¡qt)) socialize theirchildren with directfamily socialization e¤ort,¿im (qit). D ivorcedcouplesandsingleparentfamilies(inproportionc¼a(qt)andc¼b(1¡qt))socializetheirchildrenwithdirectfamilysocializatione¤ort¿is(qit). Clearlywhenthetotale¤ective socialization e¤ortofgroup i(homogamous couples plus single parentfamilies) is largerthantheothergroup’s, thenthefrequencyoftraitiincreases inthepopulation. A swehavepointedoutinSection2.3, anincreaseincreducestheresourcesspenttomatchintherestrictedpoolsandbringsmoreheterogamyintheshort-run. A shomogamyforbothculturalgroups decreases, homogamyrates alsodecreaseforbothgroups, butlessrapidlyforthemajoritygroupthanfortheminoritygroup. T hisimpliesabias in theevolutionoftraits which favors themajoritygroup. A tthesametime,an increase in divorcerates increases the importanceofsingleparentsocialization inthedynamicsofthedistributionoftraits. Clearly, itbringsanadvantagetothegroupwhich is more successfulatsocializingin single parentcontexts. W hen both groupsareequallysuccessfulatsocializingtheirchildreninsingleparentfamilycontexts (i.e.,theyhaveaccesstothesametechnologiesofsocialization), thise¤ecttendstofavortheminorityculturalgroup, asagentsofthatgrouphavelargerincentivestospendresourcesforculturaltransmission. H ence, itappearsthatwhilereducinghomogamytheoveralle¤ectofhigherdivorceratesonpreferencesisambiguous. W hen, however, singleparentfamiliesarenotabletosigni…cantlybiastheculturaltransmissionprocess, wegetsomeclearerimplications. Inthiscasethetermc(¿as(q)¡¿bs(1¡q)) isclosetozeroinequation(9 ). W earethereforeleftwithonlythee¤ectofconthegroups’maritalstrategies. T hatis, an increaseintheprobabilityofdivorce, c, reducesmaritalsegregationstrategiesof

23

bothgroupsandincreasesheterogamy. Thisfavors, inthelong-run, theculturaltraitofthemajoritygroup.

4 ConclusionT hispaperanalyzesmaritalsegregationdecisionsandtheirimpactonthetransmissionofethnicandreligious traits. W econcentrateontheinteractionbetweendirectfamilysocializationandobliquesocializationbyteachers, peersandrolemodels. W hilemostre-searchonculturaltransmissionhasstressedthisinteraction(e.g., CavalliSforza-Feldman,19 81, B oyd-R icherson, 19 85), wecomplementthisemphasisbymodellingmarriageanddirectfamilysocializationaseconomicdecisionsofagents. T hiseconomicapproachgen-eratesmanyinterestingrestrictions, aswellastestableimplications, whichweattempttoidentifyandstudyinthepaper.

O uranalysis ofsocialization is relativelyabstract, and, hence, in principle, canbeextendedtoanalyzetheevolutionofotherculturaltraitsordi¤erentsocializationmech-anisms. H owever, theassumptionthatculturaltraitsare‘pure’, ordonothaverelevante¤ectsonagents’economicopportunities, isquiterestrictive. T hisassumptionneedstoberelaxedinparticulartoapplyouranalysis tostudytheevolutionofmanyinterest-ingculturaltraitsandpreferenceparameters, likepoliticalattitudes, riskaversion, andintertemporaldiscounting. Suchtraits, infact, a¤ecthowagents interacteconomicallyandsocially, especiallyinstrategicenvironments.

24

A ppendix

T heproblemofanindividualoftypeiistochoose®i2 [0 ;1], foragivenAi, Aj, qi,tomaximize

¼i(®i;Ai;Aj;qi)W i(qi)+ [1¡¼i(®i;Ai;Aj;qi)]V i(qi)¡C (±®i) (10)

whereW i(qi) isgivenby

W i(qi) = max¿i [¿i+ (1¡¿i)qi]V ii+ (1¡¿i)(1¡qi)V ij]¡H (̄ ¿i)

andV i(qi) = qiV ii+ (1¡qi)V ij

W eassume:

A ssumptionA .Fori2fa;bg, C (±®i) andH (̄ ¿i) aremonotonicincreasing, ofclassC3, andconvex. M oreover,

A -i) @3C@®i3·0

A -ii) ± @C@®i(±) > [W i(0 )¡V i(0 )];

A -iii) (1¡®i) @2 C@®i)2

¡ @C@®i > 0 at®i= ®imax suchthat± @C@®i(±® imax) = W i(0 )¡V i(0 ):

A ssumptions A -1) - A -iii) providesu¢cientconditionsfortheexistenceandunique-ness ofthe N ash equilibrium in themarriagegame. A -i) requires thatthemarginalcostofmarriagesegregationisincreasingandconcave. A -ii) ensuresthatmatchingwithprobability1 intherestrictedpoolisprohibitivelycostly. Finally, A -iii)requiresthat, atsomelargestpossiblerestrictedpoolmatchingprobability, ® imax; thecostfunctionC (:)isconvexenough.

Foranindividualoftypei, the…rstorderconditionforthechoiceof®i is:

±@C@®i

(±®i) = pi(Ai;Aj;qi)[W i(qi)¡V i(qi)]; (11)

withpi(Ai;Aj;qi) =

(1¡Aj)(1¡qi)(1¡Ai)qi+ (1¡Aj)(1¡qi)

A symmetricN ashequilibrium ofthemarriagegamehastheproperythatallagentsoftypeichoosethesame® i, andisrepresentedbymappings ® i(qi)whichare…xedpointsofthebestrepliesofagentsi2fa;bg derivedfrom themaximizationofequation(10).Bestrepliesmustthensatisfyequation(11), whichcanberewrittenas

±@C@®i

(±® i) =(1¡®j)(1¡qi)

(1¡®i)qi+ (1¡®j)(1¡qi)[Wi(qi)¡V i(qi)]= 0 (12)

25

fori;j2fa;bg andi6= j:

ProofofProposition 1 (underA ssumption A ). A ta symmetric N ash equilibrium®i = Aiandthe…rstorderconditionofan individualoftypeiforthechoiceof® i isequation(12). D enoteby© i( ® i;®j;qi)thelefthandsideofequation(12). T hen

@© i

@® i=

@ 2 C@®i2

(±® i)¡ (1¡®j)(1¡qi)qi

[(1¡® i)qi+ (1¡®j)(1¡qi)]2[W i(qi)¡V i(qi)]

and

@ 2 © i

@®i2=

@3C@®i3

(±® i)¡ 2 (1¡®j)(1¡qi)(qi)2

[(1¡® i)qi+ (1¡®j)(1¡qi)]3[W i(qi)¡V i(qi)]< 0 ;

becauseofA -i). H ence © i is continuous and concave in ®i forany (®j;qi) 2 [0 ;1]2 .A lso© i(0 ;®j;qi) · 0 and © i(1;®j;qi) > 1, becauseofA -ii). H ence, forany (®j;qi)2 [0 ;1]2 ;thereexistsaunique® i2 [0 ;1]satisfying© i(® i;®j;qi) = 0:L etusdenotesuch®iby v®

i(®j;qi). v®

i(®j;qi) canbeviewedas abestresponsefunctionofthemarital

segmentatione¤ortofgroup i. B ecauseoftheconcavityof© i, asimpleargumentbycontradiction

shows that, at v®i(®j;qi), necessarily @© i@®i(

v®i;:::) > 0:A lso, equation (12) implies

0 < v®i(0 ;qi) < 1 and v®

i(1;qi) = 0 . Finally,

@v®i(®j;qi)@®j

= ¡@© i@®j(

v®i;:::)

@© i@®i(

v®i;:::)

;

whichhasthesignof¡@© i@®j(v®i;:::):B ut

¡@©i

@®j(v®i;:::) = ¡ (1¡®

i)qi(1¡qi)[(1¡®i)qi+ (1¡®j)(1¡qi)]2

[W i(qi)¡V i(qi)]< 0 ;

andtherefore v®i(®j;qi) is adecreasingfunctionof®j. D i¤erentiationofequation(12)

showsthat

@© i

@qi=

(1¡®j)(1¡®i)[(1¡®i)qi+ (1¡®j)(1¡qi)]2

[W i(qi)¡V i(qi)]

¡ (1¡®j)(1¡qi)

(1¡®i)qi+ (1¡®j)(1¡qi)d [W i(qi)¡V i(qi)]

d qi;

butW i(qi)¡V i(qi) = ¿i(qi)[V ii¡V ij](1¡qi)¡H (¿i(qi)). A pplyingtheEnvelopeThe-oremwehave d [W

i(qi)¡V i(qi)]d qi < 0 . H ence

@© i@qi > 0 , implyingthat

@v®i

@qi < 0 .

26

Considernowthemapping- (® a), de…nedon [0 ;1]andgivenby - (® a) = v®a[v®

b(® a)]:A

symmetricN ashequilibriumofthemarriagegameisa…xedpointofthismapping. A sbothbestresponses functions v®

a(® b) and v®

b(® a) arecontinuous functions from [0 ;1]

into [0 ;1], - (® a) is alsoacontinuousmappingfrom [0 ;1] into [0 ;1]:H encetheKaku-tani FixedPointTheorem impliestheexistenceofasymmetricN ashequilibrium inthemarriagegame.

Toprove uniqueness ofthe symmetric N ash equilibrium itsu¢ces to showthat- (® a)¡® a is strictlydecreasingin® a:Continuityof®i(qi) thenfollowsdirectly.Since - (® a) is di¤erentiable, - (® a)¡® a is strictlydecreasingin ® a i¤ - 0(® a) < 1;or,moreprecisely,

@ v®a

@® b£@

v®b

@® a< 1

L ettingD = (1¡® a)qa + (1¡® b)(1¡qa)andK i(qai) = [W i(qi)¡V i(qi)], wehave

@ v®a

@® b£@

v®b

@® a=

(1¡®a)qa(1¡qa)D2 K

a

±@2 C@®a 2 (±® a)¡

(1¡®b)(1¡qa)qaD2 K b

£±@ 2 C@®b2 (±®

b)¡(1¡® a)(1¡qa)qaD2 K a(1¡®a)qa(1¡qa)

D2 K b

W iththisnotation, the…rstordercondition, equation(12), canberewrittenas

±@C@®i

(±®i) =(1¡qi)(1¡®j)

DK i(qi)

Substituting, weobtainthat@v®a

@®b £ @v®b

@®a < 1 holdsi¤

@C@®b

(1¡qa)D

K aK b

d 2 Cd ® a2 ¡

qaD

@C@® a

£@2 C@®b2 ¡

(1¡qa)D

@C@®b

@C@®a

qaD

K bK a

< 1;

whichisequivalentto:

qa@ 2 C@® b2

@C@® a

+ (1¡q)@2 C

@® a 2@C@® b

< D£ @2 C

@® a 2£ @

2 C@® b2

A saconsequence @v®a

@®b £ @v®b

@®a < 1 holds i¤

@ 2 C@® b2

qa [(1¡® a)@2 C

@® a 2¡ @C@® a

]+@ 2 C@® a 2

(1¡qa)[(1¡® b)@2 C

@® b2¡ @C@® b

]> 0 ;

whichissatis…edunderA ssumptionA . M oreprecisely, as @3C@®i3·0 , (1¡®i)@2 C

@®i2 ¡ @C@®i isdecreasingin ® i, and isthereforepositiveforallrelevant®i, since, bycondition A -iii),itispositiveforthelargestpossible® i, ®imax, givenby± @C@®i(±®

imax) = K i(0 ).

27

T hechoiceof¿i isderivedfromthefollowingoptimizationproblem:

W i(qi) = max¿i [¿i+ (1¡¿i)qi]V ii+ (1¡¿i)(1¡qi)V ij]¡H (̄ ¿i)

whichisaconvexproblemunderA ssumptionA .T hisimmediatelyimpliesthecontinuityofthesolutionasafunctionoftheparameters, ¿i(qi). }

ProofofProposition 2 (underassumption A ). N ote that @® a@qa = ¡@-@qa@-@®a¡1

has the

signof @ -@qa:U singthefactthatv®a(® b;qa) isdecreasingin® b, and @

v®a

@qa < 0 , itiseasyto

seethat @ -@qa =@v®a

@qa +@v®a

@®b £@v®b

@qa < 0:H encetheresultthat®a(qa) isdecreasinginqa. By

asymmetricargument® b(qb) isdecreasinginqb = 1¡qa:T he…rstorderconditionforthechoiceof¿i, is

¯@H@¿i

(̄ ¿i) = [V ii¡V ij](1¡qi)

BecauseoftheconvexityofH (:), thesecondorderconditionis satis…edanddi¤erenti-atingthepreviousequation, weget

@¿i

@qi= ¡[V

ii¡V ij]¯ 2 @ 2 H@¿2

< 0: }

T hecomparativestatics results inSection2.2, andtheextensionsandthecompar-ative statics analysis ofSection 2.2.1, arestudied in an A ppendixavailable from theauthors.

W enowstudythedynamics ofthedistributionoftraits, followingtheanalysis ofSection3, where°6= 0 andc= 0:T hegeneralcaseinwhichc6= 0 is studied intheA ppendixavailablefromtheauthors. T heequationforthedynamicsofthedistributionoftraits inthepopulationis

qit+ 1 = qit¼

i(qit)Piim + q

it(1¡¼i(qit))Piih + + qjt¼j(qjt)Pjim + qjt(1¡¼j(qjt))Pjih ; (13)

wherePiim ; Piih ;(resp. Pjim ; Pjih ) arethetransition probabilities foraparentoftypei

(resp. j)ofanhomogamousandheterogamousfamily, ofhavingchildrenoftypei;thatis,

Piim = ¿im + (1¡¿im )qit; Piih = qit

Pjim = (1¡¿jm )qit; Pjih = qitSubstitutingthesetransitionprobabilities inequation(13), andsubtractingqitonbothsides, weget, afterrearrangement,

qit+ 1¡qit= qit(1¡qit)£¼i(qit)¿

im (q

it)¡¼j(qjt)¿jm (qjt)

¤

28

ProofofProposition3. L etqitdenotethefractionofthepopulationwithtraitiattimet.i)Fromthe…rstorderconditionsofthesocializationprobleminequation(6), ¿im (1) = 0 .A lso¿im (0 ) > 0 , ¼i(1) = 1;¼i(0 ) = ® i(0 )+ (1¡® i(0 ))°> 0:H ence"

@¡qat+ 1¡qat

¢

@qat

#

qa= 0

= [¼a(0 )¿am (0 )¡¼b(1)¿bm (1)]

= [¼a(0 )]¿am (0 ) > 0

and "@¡qat+ 1¡qat

¢

@qat

#

qa= 1

= ¡£[¼ a(1)¿am (1)¡¼ b(0 )¿bm (0 )]

¤

= [¼b(0 )]¿bm (0 ) > 0

T hetwopreviousexpressionsensurethatthecornerstationarystatesqa¤= 0 andqa¤= 1arelocallyunstable.ii) Considerthefunction

£ (qa) = ¼ a(qa)¿am (qa)¡¼b(1¡qa)¿bm (1¡qa)

T hisfunctioniscontinuouson [0 ;1]:M oreover

£ (0 ) = ¼ a(0 )¿am (0 ) > 0

and£ (1) = ¡¼b(0 )¿bm (0 ) < 0

Bycontinuityof£ (:)thereexistsaninteriorpointqa¤2 (0 ;1) suchthat£ (qa¤) = 0 and£ 0(qa¤) < 0 . Suchapointisaninteriorstationarystateandsatis…es

¼a(qa¤)¿am (qa¤) = ¼ b(1¡qa¤)¿bm (1¡qa¤)

iii) A ninteriorstationarystateqa¤willbelocallystableif"@¡qat+ 1¡qat

¢

@qat

#

q= qa¤

= qa¤(1¡qa¤)£ 0(qa¤) 2 (¡2 ;0 )

But·@(qat+ 1¡qat)

@qat

¸

q= qa¤< 0 is ensuredby£ 0(qa¤) < 0 . M oreover,

·@(qat+ 1¡qat)

@qat

¸

q= qa¤> ¡2

canberewrittenasqa¤(1¡qa¤)j£ 0(qa¤)j< 2 (14)

A su¢cientconditionforequation(14) tobesatis…edis: j£ 0(qa¤)j< 8, whichinturnissatis…edif @¿i@qai and

@®i@qai aresu¢cientlybounded, i.e., ifH (̄ ¿

i) andC (±® i) areconvexenoughin¿iand®i:}

29

R eferences

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