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MATHEMATICS FOR ECONOMICS AND FINANCE MICHAEL HARRISON AND PATRICK WALDRON Mathematics for Economics and Finance The aim of this book is to bring students of economics and finance who have only an introductory background in mathematics up to a quite advanced level in the subject, thus preparingthemforthecoremathematicaldemandsofeconometrics,economictheory,quan- titative finance and mathematical economics, which they are likely to encounter in their final-yearcoursesandbeyond.Thelevelofthebookwillalsobeusefulforthoseembarking onthefirstyearoftheirgraduatestudiesinBusiness,EconomicsorFinance. Thebookalsoservesasanintroductiontoquantitativeeconomicsandfinanceformath- ematics students at undergraduate level and above. In recent years, mathematics graduates have been increasingly expected to have skills in practical subjects such as economics and finance, just as economics graduates have been expected to have an increasingly strong groundinginmathematics. Theauthorsavoidthepitfallsofmanytextsthatbecometootheoretical.Theuseofmath- ematicalmethodsintherealworldisneverlostsightofandquantitativeanalysisisbrought tobearonavarietyoftopicsincludingforeignexchangeratesandothermacrolevelissues. This makes for a comprehensive volume which should be particularly useful for advanced undergraduates,forpostgraduatesinterestedinquantitativeeconomicsandfinance,andfor practitionersinthesefields. Michael Harrison is Emeritus Senior Lecturer and Fellow of Trinity College Dublin, where he lectured from 1969 to 2009. He currently lectures in the School of Economics atUniversityCollegeDublin. Patrick Waldron is a graduate of the Universities of Dublin and Pennsylvania and a ResearchAssociateintheDepartmentofEconomicsatTrinityCollegeDublin. February12,2011 11:1 PinchedCrownA Page-i HarrWald February12,2011 11:1 PinchedCrownA Page-ii HarrWald Mathematics for Economics and Finance Michael Harrison and Patrick Waldron February12,2011 11:1 PinchedCrownA Page-iii HarrWald Firstpublished2011 byRoutledge 2ParkSquare,MiltonPark,Abingdon,Oxon,OX144RN SimultaneouslypublishedintheUSAandCanada byRoutledge 711ThirdAvenue,NewYork,NY10017 RoutledgeisanimprintoftheTaylor&FrancisGroup,aninformabusiness (cid:2)c 2011MichaelHarrisonandPatrickWaldron TypesetinTimesNewRomanbySunriseSettingLtd,Devon,UnitedKingdom PrintedandboundinGreatBritainbyTJInternationalLtd,Padstow,Cornwall Allrightsreserved.Nopartofthisbookmaybereprintedorreproduced orutilizedinanyformorbyanyelectronic,mechanical,orothermeans, nowknownorhereafterinvented,includingphotocopyingandrecording, orinanyinformationstorageorretrievalsystem,withoutpermissionin writingfromthepublishers. BritishLibraryCataloguinginPublicationData AcataloguerecordforthisbookisavailablefromtheBritishLibrary LibraryofCongressCataloginginPublicationData Acatalogrecordforthisbookhasbeenrequested ISBN978-0-415-57303-0(hbk) ISBN978-0-415-57304-7(pbk) ISBN978-0-203-82999-8(ebk) February12,2011 11:1 PinchedCrownA Page-iv HarrWald Contents Listoffigures ix Listoftables xi Foreword xiii Preface xv Acknowledgements xvii Listofabbreviations xviii Notationandpreliminaries xix Part I MATHEMATICS Introduction 3 1 Systemsoflinearequationsandmatrices 5 1.1 Introduction 5 1.2 Linearequationsandexamples 5 1.3 Matrixoperations 11 1.4 Rulesofmatrixalgebra 14 1.5 Somespecialtypesofmatrixandassociatedrules 15 2 Determinants 30 2.1 Introduction 30 2.2 Preliminaries 30 2.3 Definitionandproperties 31 2.4 Co-factorexpansionsofdeterminants 34 2.5 Solutionofsystemsofequations 39 3 Eigenvaluesandeigenvectors 53 3.1 Introduction 53 3.2 Definitionsandillustration 53 3.3 Computation 54 3.4 Uniteigenvalues 58 3.5 Similarmatrices 59 3.6 Diagonalization 59 February12,2011 11:1 PinchedCrownA Page-v HarrWald vi Contents 4 Conicsections,quadraticformsanddefinitematrices 71 4.1 Introduction 71 4.2 Conicsections 71 4.3 Quadraticforms 76 4.4 Definitematrices 77 5 Vectorsandvectorspaces 88 5.1 Introduction 88 5.2 Vectorsin2-spaceand3-space 88 5.3 n-DimensionalEuclideanvectorspaces 100 5.4 Generalvectorspaces 101 6 Lineartransformations 128 6.1 Introduction 128 6.2 Definitionsandillustrations 128 6.3 Propertiesoflineartransformations 131 6.4 LineartransformationsfromRn toRm 137 6.5 Matricesoflineartransformations 138 7 Foundationsforvectorcalculus 143 7.1 Introduction 143 7.2 Affinecombinations,sets,hullsandfunctions 143 7.3 Convexcombinations,sets,hullsandfunctions 146 7.4 Subsetsofn-dimensionalspaces 148 7.5 Basictopology 154 7.6 Supportingandseparatinghyperplanetheorems 157 7.7 Visualizingfunctionsofseveralvariables 158 7.8 Limitsandcontinuity 159 7.9 Fundamentaltheoremofcalculus 162 8 Differenceequations 167 8.1 Introduction 167 8.2 Definitionsandclassifications 167 8.3 Linear,first-orderdifferenceequations 172 8.4 Linear,autonomous,higher-orderdifferenceequations 181 8.5 Systemsoflineardifferenceequations 189 9 Vectorcalculus 202 9.1 Introduction 202 9.2 Partialandtotalderivatives 202 9.3 Chainruleandproductrule 207 9.4 Elasticities 211 9.5 Directionalderivativesandtangenthyperplanes 213 9.6 Taylor’stheorem:deterministicversion 217 9.7 Multipleintegration 224 9.8 Implicitfunctiontheorem 236 February12,2011 11:1 PinchedCrownA Page-vi HarrWald Contents vii 10 Convexityandoptimization 244 10.1 Introduction 244 10.2 Convexityandconcavity 244 10.3 Unconstrainedoptimization 257 10.4 Equality-constrainedoptimization 261 10.5 Inequality-constrainedoptimization 270 10.6 Duality 278 Part II APPLICATIONS Introduction 287 11 Macroeconomicapplications 289 11.1 Introduction 289 11.2 Dynamiclinearmacroeconomicmodels 289 11.3 Input–outputanalysis 294 12 Single-periodchoiceundercertainty 299 12.1 Introduction 299 12.2 Definitions 299 12.3 Axioms 301 12.4 Theconsumer’sproblemanditsdual 307 12.5 Generalequilibriumtheory 316 12.6 Welfaretheorems 323 13 Probabilitytheory 334 13.1 Introduction 334 13.2 Samplespacesandrandomvariables 334 13.3 Applications 338 13.4 Vectorspacesofrandomvariables 343 13.5 Randomvectors 345 13.6 Expectationsandmoments 347 13.7 Multivariatenormaldistribution 351 13.8 Estimationandforecasting 354 13.9 Taylor’stheorem:stochasticversion 355 13.10Jensen’sinequality 356 14 Quadraticprogrammingandeconometricapplications 371 14.1 Introduction 371 14.2 Algebraandgeometryofordinaryleastsquares 371 14.3 Canonicalquadraticprogrammingproblem 377 14.4 Stochasticdifferenceequations 382 15 Multi-periodchoiceundercertainty 394 15.1 Introduction 394 15.2 Measuringratesofreturn 394 February12,2011 11:1 PinchedCrownA Page-vii HarrWald viii Contents 15.3 Multi-periodgeneralequilibrium 400 15.4 Termstructureofinterestrates 401 16 Single-periodchoiceunderuncertainty 415 16.1 Introduction 415 16.2 Motivation 415 16.3 Pricingstate-contingentclaims 416 16.4 Theexpected-utilityparadigm 423 16.5 Riskaversion 429 16.6 Arbitrage,riskneutralityandtheefficientmarketshypothesis 434 16.7 Uncoveredinterestrateparity:Siegel’sparadoxrevisited 436 16.8 Mean–varianceparadigm 440 16.9 Othernon-expected-utilityapproaches 442 17 Portfoliotheory 448 17.1 Introduction 448 17.2 Preliminaries 448 17.3 Single-periodportfoliochoiceproblem 450 17.4 Mathematicsoftheportfoliofrontier 457 17.5 Marketequilibriumandthecapitalassetpricingmodel 478 17.6 Multi-currencyconsiderations 487 Notes 493 References 501 Index 505 February12,2011 11:1 PinchedCrownA Page-viii HarrWald List of figures 4.1 Parabolawithfocus(a,0)anddirectrixx=−a 72 4.2 Ellipsewithfoci(±a(cid:2),0)anddirectricesx=±a/(cid:2) 73 4.3 Hyperbolawithfoci(±a(cid:2),0)anddirectricesx=±a/(cid:2) 75 5.1 Vectorsin2-space 89 5.2 Additionofvectors 89 5.3 Multiplicationofvectorsbyaconstantandsubtractionofvectors 90 5.4 Vectorcoordinates 91 5.5 Vectoradditionwithcoordinates 91 5.6 Atranslation 92 5.7 Anothertranslation 92 5.8 A3-vector 93 5.9 Distancebetweenvectors 94 5.10 Anglesbetweenvectors 94 5.11 Orthogonalprojection 97 5.12 SubspaceofR2 103 5.13 Linearlydependentandlinearlyindependentvectors 107 5.14 OrthogonalprojectioninR3 115 5.15 Gram–SchmidtprocessinR2 116 5.16 Gram–SchmidtprocessinR3 117 5.17 Changeofbasisusingrectangularcoordinates 118 5.18 Changeofbasisusingnon-rectangularcoordinates 119 5.19 Orthogonaltransformation 123 ∗ 7.1 Thehyperplanethroughx withnormalp 149 7.2 UnitorstandardsimplexinR2 150 7.3 UnitorstandardsimplexinR3 151 7.4 Upperandlowerhemi-continuityofcorrespondences 163 7.5 Motivationforthefundamentaltheoremofcalculus,part(a) 164 7.6 Motivationforthefundamentaltheoremofcalculus,part(b) 164 8.1 Divergenttimepathof yt=1+2yt−1 from y1=1 177 8.2 Divergenttimepathof yt=1−2yt−1 from y1=1 178 8.3 Convergenttimepathof yt=1+12yt−1from y1=1to y∗=2 178 8.4 Convergenttimepathof yt=1−12yt−1from y1=1to y∗=23 179 9.1 Tangenthyperplanetothegraphofafunctionofonevariable 215 9.2 Indifferencemapandtwocross-sectionsofthegraphofthefunction f:R2→R:(x,y)(cid:5)→x2+y2 216 February12,2011 11:1 PinchedCrownA Page-ix HarrWald