Springer Texts in Business and Economics Norman Schofi eld Mathematical Methods in Economics and Social Choice Second Edition Springer Texts in Business and Economics Forfurthervolumes: www.springer.com/series/10099 Norman Schofield Mathematical Methods in Economics and Social Choice Second Edition NormanSchofield CenterinPoliticalEconomy WashingtonUniversityinSaintLouis SaintLouis,MO,USA ISSN2192-4333 ISSN2192-4341(electronic) ISBN978-3-642-39817-9 ISBN978-3-642-39818-6(eBook) DOI10.1007/978-3-642-39818-6 SpringerHeidelbergNewYorkDordrechtLondon ©Springer-VerlagBerlinHeidelberg2004,2014 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped.Exemptedfromthislegalreservationarebriefexcerptsinconnection with reviews or scholarly analysis or material supplied specifically for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. 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Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) DedicatedtothememoryofJeffrey Banks andRichard McKelvey Foreword The use of mathematics in the social sciences is expanding both in breadth and depthatanincreasingrate.Ithasmadeitswayfromeconomicsintotheothersocial sciences,oftenaccompaniedbythesamecontroversythatragedineconomicsinthe 1950s.Anditsusehasdeepenedfromcalculustotopologyandmeasuretheoryto themethodsofdifferentialtopologyandfunctionalanalysis. The reasons for this expansion are several. First, and perhaps foremost, mathe- maticsmakescommunicationbetweenresearcherssuccinctandprecise.Second,it helpsmakeassumptionsandmodelsclear;thisbypassesargumentsinthefieldthat are a result of different implicit assumptions. Third, proofs are rigorous, so math- ematics helps avoid mistakes in the literature. Fourth, its use often provides more insightsintothemodels.Andfinally,themodelscanbeappliedtodifferentcontexts withoutrepeatingtheanalysis,simplybyrenamingthesymbols. Ofcourse,theformulationofsocialsciencequestionsmustprecedetheconstruc- tionofmodelsandthedistillationofthesemodelsdowntomathematicalproblems, forotherwisetheassumptionsmightbeinappropriate. Aconsequenceofthepervasiveuseofmathematicsinourresearchisachange inthelevelofmathematicstrainingrequiredofourgraduatestudents.Weneedref- erenceandgraduatetextbooksthataddressapplicationsofadvancedmathematics toawideningrangeofsocialsciences.Thisbookfillsthatneed. Manyyearsago,BillRikerintroducedmetoNormanSchofield’sworkandthen toNorman.Heisuniqueinhisabilitytospanthesocialsciencesandapplyintegra- tivemathematicalreasoningtothemall.Theemphasisonhisworkandhisbookis onsmoothmodelsandtechniques,whilethemotivatingexamplesforpresentation ofthemathematicsaredrawnprimarilyfromeconomicsandpoliticalscience.The readeristakenfrombasicsettheorytothemathematicsusedtosolveproblemsat thecuttingedgeofresearch.Studentsineverysocialsciencewillfindexposureto this mode of analysis useful; it elucidates the common threads in different fields. SpeculationsattheendofChap.5providestudentsandresearcherswithmanyopen researchquestionsrelatedtothecontentofthefirstfourchapters.Theanswersare inthesechapters.Whenthefirsteditionappearedin2004,IwroteinmyForeword that a goal of the reader should be to write Chap. 6. For the second edition of the book,Normanhimselfhasaccomplishedthisopentask. St.Louis,Missouri,USA MarcusBerliant 2013 vii Preface to the Second Edition Forthesecondedition,Ihaveaddedanewchaptersix.Thischaptercontinueswith themodelpresentedinChap.3bydevelopingtheideaofdynamicalsocialchoice. In particular the chapter considers the possibility of cycles enveloping the set of socialalternatives. AtheoremofSaari(1997)showsthatforanynon-collegialset,D,ofdecisiveor winningcoalitions,ifthedimensionofthepolicyspaceissufficientlylarge,thenthe choiceisemptyunderDforallsmoothprofilesinaresidualsubspaceofCr(W,(cid:2)n). Inotherwordsthechoiceisgenericallyempty. However, we can define a social solution concept, known as the heart. When regarded as a correspondence, the heart is lower hemi-continuous. In general the heart is centrally located with respect to the distribution of voter preferences, and is guaranteed to be non-empty. Two examples are given to show how the heart is determinedbythesymmetryofthevoterdistribution. Finally,tobeabletousesurveydataofvoterpreferences,thechapterintroduces the idea of stochastic social choice. In situations where voter choice is given by a probability vector, we can model the choice by assuming that candidates choose policiestomaximisetheirvoteshares.Ingeneraltheequilibriumvotemaximising positionscanbeshowntobeattheelectoralmean.Thenecessaryandsufficientcon- ditionforthisisgivenbythenegativedefinitenessofthecandidatevoteHessians.In anempiricalexample,amultinomiallogitmodelofthe2008Presidentialelectionis presented,basedontheAmericanNationalElectionSurvey,andtheparametersof thismodelusedtocalculatetheHessiansofthevotefunctionsforbothcandidates. According to this example both candidates should have converged to the electoral mean. SaintLouis,Missouri,USA NormanSchofield June13,2013 ix Preface to the First Edition In recent years, the optimisation techniques, which have proved so useful in mi- croeconomictheory,havebeenextendedtoincorporatemorepowerfultopological anddifferentialmethods.Thesemethodshaveledtonewresultsonthequalitative behaviourofgeneraleconomicandpoliticalsystems.However,thesedevelopments havealsoledtoanincreaseinthedegreeofformalisminpublishedwork.Thisfor- malismcanoftendetergraduatestudents.Myhopeisthattheprogressionofideas presentedintheselecturenoteswillfamiliarisethestudentwiththegeometriccon- cepts underlying these topological methods, and, as a result, make mathematical economics,generalequilibriumtheory,andsocialchoicetheorymoreaccessible. Thefirstchapterofthebookintroducesthegeneralideaofmathematicalstruc- ture and representation, while the second chapter analyses linear systems and the representationoftransformationsoflinearsystemsbymatrices.Inthethirdchapter, topological ideas and continuity are introduced and used to solve convex optimi- sation problems. These techniques are also used to examine existence of a “social equilibrium.”Chapterfourthengoesontostudycalculustechniquesusingalinear approximation,thedifferential,ofafunctiontostudyits“local”behaviour. The book is not intended to cover the full extent of mathematical economics orgeneralequilibriumtheory.However,inthelastsectionsofthethirdandfourth chaptersIhaveintroducedsomeofthestandardtoolsofeconomictheory,namely the Kuhn Tucker Theorem, together with some elements of convex analysis and proceduresusingtheLagrangian.Chapterfourprovidesexamplesofconsumerand produceroptimisation.Thefinalsectionofthechapteralsodiscusses,inaheuristic fashion,thesmoothorcriticalParetosetandtheideaofaregulareconomy.Thefifth andfinalchapterissomewhatmoreadvanced,andextendsthedifferentialcalculus ofarealvaluedfunctiontotheanalysisofasmoothfunctionbetween“local”vector spaces,ormanifolds.Modemsingularitytheoryisthestudyandclassificationofall suchsmoothfunctions,andthepurposeofthefinalchaptertousethisperspectiveto obtainagenericortypicalpictureoftheParetosetandthesetofWalrasianequilibria ofanexchangeeconomy. Sincetheunderlyingmathematicsofthisfinalsectionareratherdifficult,Ihave notattemptedrigorousproofs,butratherhavesoughttolayoutthenaturalpathof developmentfromelementarydifferentialcalculustothepowerfultoolsofsingular- itytheory.InthetextIhavereferredtoworkofDebreu,Balasko,Smale,andSaari, amongotherswho,inthelastfewyears,haveusedthetoolsofsingularitytheoryto xi xii PrefacetotheFirstEdition developadeeperinsightintothegeometricstructureofboththeeconomyandthe polity.Theseideasareattheheartofrecentnotionsof“chaos.”Somespeculations on this profound way of thinkingabout the world are offered in Sect.5.6. Review exercisesareprovidedattheendofthebook. I thank Annette Milford for typing the manuscript and Diana Ivanov for the preparationofthefigures. Iamalsoindebtedtomygraduatestudentsforthepertinentquestionstheyasked duringthecoursesonmathematicalmethodsineconomicsandsocialchoice,which IhavegivenatEssexUniversity,theCaliforniaInstituteofTechnology,andWash- ingtonUniversityinSt.Louis. Inparticular,whileIwasattheCaliforniaInstituteofTechnologyIhadthepriv- ilege of working with Richard McKelvey and of discussing ideas in social choice theorywithJeffBanks.Itisagreatlossthattheyhavebothpassedaway.Thisbook isdedicatedtotheirmemory. SaintLouis,Missouri,USA NormanSchofield Contents 1 Sets,Relations,andPreferences . . . . . . . . . . . . . . . . . . . . 1 1.1 ElementsofSetTheory . . . . . . . . . . . . . . . . . . . . . . 1 1.1.1 ASetTheory . . . . . . . . . . . . . . . . . . . . . . . . 2 1.1.2 APropositionalCalculus . . . . . . . . . . . . . . . . . 4 1.1.3 PartitionsandCovers . . . . . . . . . . . . . . . . . . . 6 1.1.4 TheUniversalandExistentialQuantifiers . . . . . . . . . 7 1.2 Relations,FunctionsandOperations . . . . . . . . . . . . . . . 7 1.2.1 Relations . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2.2 Mappings . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.2.3 Functions. . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.3 GroupsandMorphisms . . . . . . . . . . . . . . . . . . . . . . 12 1.4 PreferencesandChoices . . . . . . . . . . . . . . . . . . . . . . 24 1.4.1 PreferenceRelations . . . . . . . . . . . . . . . . . . . . 24 1.4.2 Rationality . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4.3 Choices . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.5 SocialChoiceandArrow’sImpossibilityTheorem . . . . . . . . 32 1.5.1 OligarchiesandFilters . . . . . . . . . . . . . . . . . . . 33 1.5.2 AcyclicityandtheCollegium . . . . . . . . . . . . . . . 35 FurtherReading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 2 LinearSpacesandTransformations . . . . . . . . . . . . . . . . . 39 2.1 VectorSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.2 LinearTransformations . . . . . . . . . . . . . . . . . . . . . . 45 2.2.1 Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2.2 TheDimensionTheorem . . . . . . . . . . . . . . . . . 49 2.2.3 TheGeneralLinearGroup . . . . . . . . . . . . . . . . . 53 2.2.4 ChangeofBasis . . . . . . . . . . . . . . . . . . . . . . 55 2.2.5 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . 59 2.3 CanonicalRepresentation . . . . . . . . . . . . . . . . . . . . . 62 2.3.1 EigenvectorsandEigenvalues . . . . . . . . . . . . . . . 63 2.3.2 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . 66 2.3.3 SymmetricMatricesandQuadraticForms . . . . . . . . 67 2.3.4 Examples. . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.4 GeometricInterpretationofaLinearTransformation . . . . . . . 73 xiii