Mathematical Economics and Finance Michael Harrison Patrick Waldron December 2, 1998 CONTENTS i Contents ListofTables iii ListofFigures v PREFACE vii WhatIsEconomics? . . . . . . . . . . . . . . . . . . . . . . . . . . . vii WhatIsMathematics?. . . . . . . . . . . . . . . . . . . . . . . . . . . viii NOTATION ix I MATHEMATICS 1 1 LINEARALGEBRA 3 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 SystemsofLinearEquationsandMatrices . . . . . . . . . . . . . 3 1.3 MatrixOperations . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.4 MatrixArithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.5 VectorsandVectorSpaces . . . . . . . . . . . . . . . . . . . . . 11 1.6 LinearIndependence . . . . . . . . . . . . . . . . . . . . . . . . 12 1.7 BasesandDimension . . . . . . . . . . . . . . . . . . . . . . . . 12 1.8 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.9 EigenvaluesandEigenvectors . . . . . . . . . . . . . . . . . . . . 14 1.10 QuadraticForms . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.11 SymmetricMatrices . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.12 DefiniteMatrices . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2 VECTORCALCULUS 17 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 BasicTopology . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.3 Vector-valuedFunctionsandFunctionsofSeveralVariables . . . 18 Revised: December2,1998 ii CONTENTS 2.4 PartialandTotalDerivatives . . . . . . . . . . . . . . . . . . . . 20 2.5 TheChainRuleandProductRule . . . . . . . . . . . . . . . . . 21 2.6 TheImplicitFunctionTheorem . . . . . . . . . . . . . . . . . . . 23 2.7 DirectionalDerivatives . . . . . . . . . . . . . . . . . . . . . . . 24 2.8 Taylor’sTheorem: DeterministicVersion . . . . . . . . . . . . . 25 2.9 TheFundamentalTheoremofCalculus . . . . . . . . . . . . . . 26 3 CONVEXITYANDOPTIMISATION 27 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2 ConvexityandConcavity . . . . . . . . . . . . . . . . . . . . . . 27 3.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.2.2 Propertiesofconcavefunctions . . . . . . . . . . . . . . 29 3.2.3 Convexityanddifferentiability . . . . . . . . . . . . . . . 30 3.2.4 Variationsontheconvexitytheme . . . . . . . . . . . . . 34 3.3 UnconstrainedOptimisation . . . . . . . . . . . . . . . . . . . . 39 3.4 EqualityConstrainedOptimisation: TheLagrangeMultiplierTheorems . . . . . . . . . . . . . . . . . 43 3.5 InequalityConstrainedOptimisation: TheKuhn-TuckerTheorems . . . . . . . . . . . . . . . . . . . . 50 3.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 II APPLICATIONS 61 4 CHOICEUNDERCERTAINTY 63 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 4.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 4.4 OptimalResponseFunctions: MarshallianandHicksianDemand . . . . . . . . . . . . . . . . . 69 4.4.1 Theconsumer’sproblem . . . . . . . . . . . . . . . . . . 69 4.4.2 TheNoArbitragePrinciple . . . . . . . . . . . . . . . . . 70 4.4.3 OtherPropertiesofMarshalliandemand . . . . . . . . . . 71 4.4.4 Thedualproblem . . . . . . . . . . . . . . . . . . . . . . 72 4.4.5 PropertiesofHicksiandemands . . . . . . . . . . . . . . 73 4.5 EnvelopeFunctions: IndirectUtilityandExpenditure . . . . . . . . . . . . . . . . . . 73 4.6 FurtherResultsinDemandTheory . . . . . . . . . . . . . . . . . 75 4.7 GeneralEquilibriumTheory . . . . . . . . . . . . . . . . . . . . 78 4.7.1 Walras’law . . . . . . . . . . . . . . . . . . . . . . . . . 78 4.7.2 Brouwer’sfixedpointtheorem . . . . . . . . . . . . . . . 78 Revised: December2,1998 CONTENTS iii 4.7.3 Existenceofequilibrium . . . . . . . . . . . . . . . . . . 78 4.8 TheWelfareTheorems . . . . . . . . . . . . . . . . . . . . . . . 78 4.8.1 TheEdgeworthbox . . . . . . . . . . . . . . . . . . . . . 78 4.8.2 Paretoefficiency . . . . . . . . . . . . . . . . . . . . . . 78 4.8.3 TheFirstWelfareTheorem . . . . . . . . . . . . . . . . . 79 4.8.4 TheSeparatingHyperplaneTheorem . . . . . . . . . . . 80 4.8.5 TheSecondWelfareTheorem . . . . . . . . . . . . . . . 80 4.8.6 Completemarkets . . . . . . . . . . . . . . . . . . . . . 82 4.8.7 OthercharacterizationsofParetoefficientallocations . . . 82 4.9 Multi-periodGeneralEquilibrium . . . . . . . . . . . . . . . . . 84 5 CHOICEUNDERUNCERTAINTY 85 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 ReviewofBasicProbability . . . . . . . . . . . . . . . . . . . . 85 5.3 Taylor’sTheorem: StochasticVersion . . . . . . . . . . . . . . . 88 5.4 PricingState-ContingentClaims . . . . . . . . . . . . . . . . . . 88 5.4.1 Completionofmarketsusingoptions . . . . . . . . . . . 90 5.4.2 Restrictionsonsecurityvaluesimpliedbyallocationalef- ficiencyandcovariancewithaggregateconsumption . . . 91 5.4.3 Completingmarketswithoptionsonaggregateconsumption 92 5.4.4 Replicatingelementaryclaimswithabutterflyspread . . . 93 5.5 TheExpectedUtilityParadigm . . . . . . . . . . . . . . . . . . . 93 5.5.1 Furtheraxioms . . . . . . . . . . . . . . . . . . . . . . . 93 5.5.2 Existenceofexpectedutilityfunctions . . . . . . . . . . . 95 5.6 Jensen’sInequalityandSiegel’sParadox . . . . . . . . . . . . . . 97 5.7 RiskAversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.8 TheMean-VarianceParadigm . . . . . . . . . . . . . . . . . . . 102 5.9 TheKellyStrategy . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.10 AlternativeNon-ExpectedUtilityApproaches . . . . . . . . . . . 104 6 PORTFOLIOTHEORY 105 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2 Notationandpreliminaries . . . . . . . . . . . . . . . . . . . . . 105 6.2.1 Measuringratesofreturn . . . . . . . . . . . . . . . . . . 105 6.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3 TheSingle-periodPortfolioChoiceProblem . . . . . . . . . . . . 110 6.3.1 Thecanonicalportfolioproblem . . . . . . . . . . . . . . 110 6.3.2 Riskaversionandportfoliocomposition . . . . . . . . . . 112 6.3.3 Mutualfundseparation . . . . . . . . . . . . . . . . . . . 114 6.4 MathematicsofthePortfolioFrontier . . . . . . . . . . . . . . . 116 Revised: December2,1998 iv CONTENTS 6.4.1 Theportfoliofrontierin(cid:60)N: riskyassetsonly . . . . . . . . . . . . . . . . . . . . . . 116 6.4.2 Theportfoliofrontierinmean-variancespace: riskyassetsonly . . . . . . . . . . . . . . . . . . . . . . 124 6.4.3 Theportfoliofrontierin(cid:60)N: riskfreeandriskyassets . . . . . . . . . . . . . . . . . . 129 6.4.4 Theportfoliofrontierinmean-variancespace: riskfreeandriskyassets . . . . . . . . . . . . . . . . . . 129 6.5 MarketEquilibriumandtheCAPM . . . . . . . . . . . . . . . . 130 6.5.1 Pricingassetsandpredictingsecurityreturns . . . . . . . 130 6.5.2 Propertiesofthemarketportfolio . . . . . . . . . . . . . 131 6.5.3 Thezero-betaCAPM . . . . . . . . . . . . . . . . . . . . 131 6.5.4 ThetraditionalCAPM . . . . . . . . . . . . . . . . . . . 132 7 INVESTMENTANALYSIS 137 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 ArbitrageandPricingDerivativeSecurities . . . . . . . . . . . . 137 7.2.1 Thebinomialoptionpricingmodel . . . . . . . . . . . . 137 7.2.2 TheBlack-Scholesoptionpricingmodel . . . . . . . . . . 137 7.3 Multi-periodInvestmentProblems . . . . . . . . . . . . . . . . . 140 7.4 ContinuousTimeInvestmentProblems . . . . . . . . . . . . . . . 140 Revised: December2,1998 LISTOFTABLES v List of Tables 3.1 Signconditionsforinequalityconstrainedoptimisation . . . . . . 51 5.1 PayoffsforCallOptionsontheAggregateConsumption . . . . . 92 6.1 The effect of an interest rate of 10% per annum at different fre- quenciesofcompounding. . . . . . . . . . . . . . . . . . . . . . 106 6.2 Notationforportfoliochoiceproblem . . . . . . . . . . . . . . . 108 Revised: December2,1998 vi LISTOFTABLES Revised: December2,1998 LISTOFFIGURES vii List of Figures Revised: December2,1998 viii LISTOFFIGURES Revised: December2,1998 PREFACE ix PREFACE This book is based on courses MA381 and EC3080, taught at Trinity College Dublinsince1992. Comments on content and presentation in the present draft are welcome for the benefitoffuturegenerationsofstudents. An electronic version of this book (in LATEX) is available on the World Wide Web at http://pwaldron.bess.tcd.ie/teaching/ma381/notes/ althoughitmaynotalwaysbethecurrentversion. Thebookisnotintendedasasubstituteforstudents’ownlecturenotes. Inparticu- lar,manyexamplesanddiagramsareomittedandsomematerialmaybepresented inadifferentsequencefromyeartoyear. In recent years, mathematics graduates have been increasingly expected to have additional skills in practical subjects such as economics and finance, while eco- nomicsgraduateshavebeenexpectedtohaveanincreasinglystronggroundingin mathematics. The increasing need for those working in economics and finance to have a strong grounding in mathematics has been highlighted by such layman’s guides as ?, ?, ? (adapted from ?) and ?. In the light of these trends, the present book is aimed at advanced undergraduate students of either mathematics or eco- nomicswhowishtobranchoutintotheothersubject. The present versionlacks supportingmaterialsin Mathematicaor Maple, such as areprovidedwithcompetingworkslike?. Before starting to work through this book, mathematics students should think about the nature, subject matter and scientific methodology of economics while economics students should think about the nature, subject matter and scientific methodology of mathematics. The following sections briefly address these ques- tionsfromtheperspectiveoftheoutsider. What Is Economics? This section will consist of a brief verbal introduction to economics for mathe- maticiansandanoutlineofthecourse. Revised: December2,1998