MathematicalEconomicsandFinance
MichaelHarrisonPatrickWaldron
December2,1998

Contents
ListofTablesiii
ListofFiguresv
PREFACEvii
WhatIsEconomics?...........................vii
WhatIsMathematics?...........................viii
NOTATIONix
IMATHEMATICS1
1LINEARALGEBRA3
1.1Introduction.............................3
1.2SystemsofLinearEquationsandMatrices.............3
1.3MatrixOperations..........................7
1.4MatrixArithmetic..........................7
1.5VectorsandVectorSpaces.....................11
1.6LinearIndependence........................12
1.7BasesandDimension........................12
1.8Rank.................................13
1.9EigenvaluesandEigenvectors....................14
1.10QuadraticForms..........................15
1.11SymmetricMatrices.........................15
1.12De?niteMatrices..........................15
2VECTORCALCULUS17
2.1Introduction.............................17
2.2BasicTopology...........................17
2.3Vector-valuedFunctionsandFunctionsofSeveralVariables...18
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ii CONTENTS
2.4PartialandTotalDerivatives....................20
2.5TheChainRuleandProductRule.................21
2.6TheImplicitFunctionTheorem...................23
2.7DirectionalDerivatives.......................24
2.8Taylor!ˉsTheorem:DeterministicVersion.............2
2.9TheFundamentalTheoremofCalculus..............26
3CONVEXITYANDOPTIMISATION27
3.1Introduction.............................27
3.2ConvexityandConcavity......................27
3.2.1De?nitions.........................27
3.2.2Propertiesofconcavefunctions..............29
3.2.3Convexityanddifferentiability...............30
3.2.4Variationsontheconvexitytheme.............34
3.3UnconstrainedOptimisation....................39
3.4EqualityConstrainedOptimisation:
TheLagrangeMultiplierTheorems.................43
3.5InequalityConstrainedOptimisation:
TheKuhn-TuckerTheorems....................50
3.6Duality...............................58
IIAPPLICATIONS61
4CHOICEUNDERCERTAINTY63
4.1Introduction.............................63
4.2De?nitions..............................63
4.3Axioms...............................66
4.4OptimalResponseFunctions:
MarshallianandHicksianDemand.................69
4.4.1Theconsumer!ˉsproblem..................6
4.4.2TheNoArbitragePrinciple.................70
4.4.3OtherPropertiesofMarshalliandemand..........71
4.4.4Thedualproblem......................72
4.4.5PropertiesofHicksiandemands..............73
4.5EnvelopeFunctions:
IndirectUtilityandExpenditure..................73
4.6FurtherResultsinDemandTheory.................75
4.7GeneralEquilibriumTheory....................78
4.7.1Walras!ˉlaw.........................7
4.7.2Brouwer!ˉs?xedpointtheorem...............7
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CONTENTS iii
4.7.3Existenceofequilibrium..................78
4.8TheWelfareTheorems.......................78
4.8.1TheEdgeworthbox.....................78
4.8.2Paretoef?ciency......................78
4.8.3TheFirstWelfareTheorem.................79
4.8.4TheSeparatingHyperplaneTheorem...........80
4.8.5TheSecondWelfareTheorem...............80
4.8.6Completemarkets.....................82
4.8.7OthercharacterizationsofParetoef?cientallocations...82
4.9Multi-periodGeneralEquilibrium.................84
5CHOICEUNDERUNCERTAINTY85
5.1Introduction.............................85
5.2ReviewofBasicProbability....................85
5.3Taylor!ˉsTheorem:StochasticVersion...............8
5.4PricingState-ContingentClaims..................88
5.4.1Completionofmarketsusingoptions...........90
5.4.2Restrictionsonsecurityvaluesimpliedbyallocationalef-
?ciencyandcovariancewithaggregateconsumption...91
5.4.3Completingmarketswithoptionsonaggregateconsumption92
5.4.4Replicatingelementaryclaimswithabutter?yspread...93
5.5TheExpectedUtilityParadigm...................93
5.5.1Furtheraxioms.......................93
5.5.2Existenceofexpectedutilityfunctions...........95
5.6Jensen!ˉsInequalityandSiegel!ˉsParadox..............
5.7RiskAversion............................99
5.8TheMean-VarianceParadigm...................102
5.9TheKellyStrategy.........................103
5.10AlternativeNon-ExpectedUtilityApproaches...........104
6PORTFOLIOTHEORY105
6.1Introduction.............................105
6.2Notationandpreliminaries.....................105
6.2.1Measuringratesofreturn..................105
6.2.2Notation..........................108
6.3TheSingle-periodPortfolioChoiceProblem............110
6.3.1Thecanonicalportfolioproblem..............110
6.3.2Riskaversionandportfoliocomposition..........112
6.3.3Mutualfundseparation...................114
6.4MathematicsofthePortfolioFrontier...............116
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iv CONTENTS
N
6.4.1Theportfoliofrontierin :
riskyassetsonly......................116
6.4.2Theportfoliofrontierinmean-variancespace:
riskyassetsonly......................124
N
6.4.3Theportfoliofrontierin :
riskfreeandriskyassets..................129
6.4.4Theportfoliofrontierinmean-variancespace:
riskfreeandriskyassets..................129
6.5MarketEquilibriumandtheCAPM................130
6.5.1Pricingassetsandpredictingsecurityreturns.......130
6.5.2Propertiesofthemarketportfolio.............131
6.5.3Thezero-betaCAPM....................131
6.5.4ThetraditionalCAPM...................132
7INVESTMENTANALYSIS137
7.1Introduction.............................137
7.2ArbitrageandPricingDerivativeSecurities............137
7.2.1Thebinomialoptionpricingmodel............137
7.2.2TheBlack-Scholesoptionpricingmodel..........137
7.3Multi-periodInvestmentProblems.................140
7.4ContinuousTimeInvestmentProblems...............140