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(cid:2) ESSENTIAL MATHEMATICS FOR E C O N O M I C A N A L Y S I S (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) ESSENTIAL MATHEMATICS FOR E C O N O M I C A N A L Y S I S FIFTH EDITION Knut Sydsæter, Peter Hammond, Arne Strøm and Andrés Carvajal (cid:2) (cid:2) (cid:2) (cid:2) PearsonEducationLimited EdinburghGate HarlowCM202JE UnitedKingdom Tel:+44(0)1279623623 Web:www.pearson.com/uk FirstpublishedbyPrentice-Hall,Inc.1995(print) Secondeditionpublished2006(print) Thirdeditionpublished2008(print) FourtheditionpublishedbyPearsonEducationLimited2012(print) Fiftheditionpublished2016(printandelectronic) ©PrenticeHall,Inc.1995(print) ©KnutSydsæter,PeterHammond,ArneStrømandAndrésCarvajal2016(printandelectronic) TherightsofKnutSydsæter,PeterHammond,ArneStrømandAndrésCarvajaltobeidentified asauthorsofthisworkhasbeenassertedbytheminaccordancewiththeCopyright,Designs andPatentsAct1988. Theprintpublicationisprotectedbycopyright.Priortoanyprohibitedreproduction,storagein aretrievalsystem,distributionortransmissioninanyformorbyanymeans,electronic, mechanical,recordingorotherwise,permissionshouldbeobtainedfromthepublisheror,where applicable,alicencepermittingrestrictedcopyingintheUnitedKingdomshouldbeobtained fromtheCopyrightLicensingAgencyLtd,Barnard’sInn,86FetterLane,LondonEC4A1EN. TheePublicationisprotectedbycopyrightandmustnotbecopied,reproduced,transferred, distributed,leased,licensedorpubliclyperformedorusedinanywayexceptasspecifically (cid:2) permittedinwritingbythepublishers,asallowedunderthetermsandconditionsunderwhichit (cid:2) waspurchased,orasstrictlypermittedbyapplicablecopyrightlaw.Anyunauthorised distributionoruseofthistextmaybeadirectinfringementoftheauthors’andthepublisher’s rightsandthoseresponsiblemaybeliableinlawaccordingly. PearsonEducationisnotresponsibleforthecontentofthird-partyinternetsites. ISBN:978-1-292-07461-0(print) 978-1-292-07465-8(PDF) 978-1-29-207470-2(ePub) BritishLibraryCataloguing-in-PublicationData AcataloguerecordfortheprinteditionisavailablefromtheBritishLibrary LibraryofCongressCataloging-in-PublicationData Names:Sydsaeter,Knut,author.|Hammond,PeterJ.,1945–author. Title:Essentialmathematicsforeconomicanalysis/KnutSydsaeterandPeterHammond. Description:Fifthedition.|Harlow,UnitedKingdom:PearsonEducation,[2016]|Includes index. Identifiers:LCCN2016015992(print)|LCCN2016021674(ebook)|ISBN9781292074610 (hbk)|ISBN9781292074658() Subjects:LCSH:Economics,Mathematical.Classification:LCCHB135.S8862016(print)| LCCHB135(ebook)|DDC330.01/51–dc23 LCrecordavailableathttps://lccn.loc.gov/2016015992 10987654321 2019181716 Coverimage:GettyImages Printeditiontypesetin10/13ptTimesLTProbySPi-Global,Chennai,India PrintedinSlovakiabyNeografia NOTETHATANYPAGECROSSREFERENCESREFERTOTHEPRINTEDITION (cid:2) (cid:2) To Knut Sydsæter (1937–2012), an inspiring mathematics teacher, as well as wonderful friend and colleague, whose vision, hard work, high professional standards, and sense of humour were all essential in creating this book. —Arne, Peter and Andrés To Else, my loving and patient wife. —Arne (cid:2) (cid:2) TothememoryofmyparentsElsie(1916–2007)and Fred(1916–2008),myfirstteachersofMathematics, basic Economics, and many more important things. —Peter To Yeye and Tata, my best ever students of “matemáquinas”, who wanted this book to start with “Once upon a time ...” —Andrés (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) (cid:2) C O N T E N T S Preface xi 3 Solving Equations 67 3.1 SolvingEquations 67 Publisher’s 3.2 EquationsandTheirParameters 70 Acknowledgement xvii (cid:2) 3.3 QuadraticEquations 73 (cid:2) 3.4 NonlinearEquations 78 1 Essentials of Logic and 3.5 UsingImplicationArrows 80 Set Theory 1 3.6 TwoLinearEquationsinTwo 1.1 EssentialsofSetTheory 1 Unknowns 82 1.2 SomeAspectsofLogic 7 ReviewExercises 86 1.3 MathematicalProofs 12 4 Functions of One Variable 89 1.4 MathematicalInduction 14 ReviewExercises 16 4.1 Introduction 89 4.2 BasicDefinitions 90 2 Algebra 19 4.3 GraphsofFunctions 96 2.1 TheRealNumbers 19 4.4 LinearFunctions 99 2.2 IntegerPowers 22 4.5 LinearModels 106 2.3 RulesofAlgebra 28 4.6 QuadraticFunctions 109 2.4 Fractions 33 4.7 Polynomials 116 2.5 FractionalPowers 38 4.8 PowerFunctions 123 2.6 Inequalities 43 4.9 ExponentialFunctions 126 2.7 IntervalsandAbsoluteValues 49 4.10 LogarithmicFunctions 131 2.8 Summation 52 ReviewExercises 136 2.9 RulesforSums 56 5 Properties of Functions 141 2.10 Newton’sBinomialFormula 59 2.11 DoubleSums 61 5.1 ShiftingGraphs 141 ReviewExercises 62 5.2 NewFunctionsfromOld 146 (cid:2) (cid:2) viii CONTENTS 5.3 InverseFunctions 150 8.6 LocalExtremePoints 305 5.4 GraphsofEquations 156 8.7 InflectionPoints,Concavity,and 5.5 DistanceinthePlane 160 Convexity 311 5.6 GeneralFunctions 163 ReviewExercises 316 ReviewExercises 166 9 Integration 319 6 Differentiation 169 9.1 IndefiniteIntegrals 319 6.1 SlopesofCurves 169 9.2 AreaandDefiniteIntegrals 325 6.2 TangentsandDerivatives 171 9.3 PropertiesofDefiniteIntegrals 332 6.3 IncreasingandDecreasingFunctions 176 9.4 EconomicApplications 336 6.4 RatesofChange 179 9.5 IntegrationbyParts 343 6.5 ADashofLimits 182 9.6 IntegrationbySubstitution 347 6.6 SimpleRulesforDifferentiation 188 9.7 InfiniteIntervalsofIntegration 352 6.7 Sums,Products,andQuotients 192 9.8 AGlimpseatDifferentialEquations 359 6.8 TheChainRule 198 9.9 SeparableandLinearDifferential 6.9 Higher-OrderDerivatives 203 Equations 365 6.10 ExponentialFunctions 208 ReviewExercises 371 6.11 LogarithmicFunctions 212 ReviewExercises 218 10 Topics in Financial Mathematics 375 7 Derivatives in Use 221 10.1 InterestPeriodsandEffectiveRates 375 (cid:2) 7.1 ImplicitDifferentiation 221 (cid:2) 10.2 ContinuousCompounding 379 7.2 EconomicExamples 228 10.3 PresentValue 381 7.3 DifferentiatingtheInverse 232 10.4 GeometricSeries 383 7.4 LinearApproximations 235 10.5 TotalPresentValue 390 7.5 PolynomialApproximations 239 10.6 MortgageRepayments 395 7.6 Taylor’sFormula 243 10.7 InternalRateofReturn 399 7.7 Elasticities 246 10.8 AGlimpseatDifferenceEquations 401 7.8 Continuity 251 ReviewExercises 404 7.9 MoreonLimits 257 7.10 TheIntermediateValueTheoremand Newton’sMethod 266 11 Functions of Many 7.11 InfiniteSequences 270 Variables 407 7.12 L’Hoˆpital’sRule 273 11.1 FunctionsofTwoVariables 407 ReviewExercises 278 11.2 PartialDerivativeswithTwoVariables 411 11.3 GeometricRepresentation 417 8 Single-Variable 11.4 SurfacesandDistance 424 Optimization 283 11.5 FunctionsofMoreVariables 427 8.1 ExtremePoints 283 11.6 PartialDerivativeswithMore 8.2 SimpleTestsforExtremePoints 287 Variables 431 8.3 EconomicExamples 290 11.7 EconomicApplications 435 8.4 TheExtremeValueTheorem 294 11.8 PartialElasticities 437 8.5 FurtherEconomicExamples 300 ReviewExercises 439 (cid:2) (cid:2) CONTENTS ix 12 Tools for Comparative 14.9 MultipleInequalityConstraints 569 Statics 443 14.10 NonnegativityConstraints 574 ReviewExercises 578 12.1 ASimpleChainRule 443 12.2 ChainRulesforManyVariables 448 15 Matrix and 12.3 ImplicitDifferentiationalongaLevel Vector Algebra 581 Curve 452 12.4 MoreGeneralCases 457 15.1 SystemsofLinearEquations 581 12.5 ElasticityofSubstitution 460 15.2 MatricesandMatrixOperations 584 12.6 HomogeneousFunctionsofTwo 15.3 MatrixMultiplication 588 Variables 463 15.4 RulesforMatrixMultiplication 592 12.7 HomogeneousandHomothetic 15.5 TheTranspose 599 Functions 468 15.6 GaussianElimination 602 12.8 LinearApproximations 474 15.7 Vectors 608 12.9 Differentials 477 15.8 GeometricInterpretationofVectors 611 12.10 SystemsofEquations 482 15.9 LinesandPlanes 617 12.11 DifferentiatingSystemsofEquations 486 ReviewExercises 620 ReviewExercises 492 16 Determinants and 13 Multivariable Inverse Matrices 623 Optimization 495 16.1 DeterminantsofOrder2 623 13.1 TwoChoiceVariables:Necessary 16.2 DeterminantsofOrder3 627 (cid:2) Conditions 495 16.3 DeterminantsinGeneral 632 (cid:2) 13.2 TwoChoiceVariables:Sufficient 16.4 BasicRulesforDeterminants 636 Conditions 500 16.5 ExpansionbyCofactors 640 13.3 LocalExtremePoints 504 16.6 TheInverseofaMatrix 644 13.4 LinearModelswithQuadratic 16.7 AGeneralFormulafortheInverse 650 Objectives 509 16.8 Cramer’sRule 653 13.5 TheExtremeValueTheorem 516 16.9 TheLeontiefModel 657 13.6 TheGeneralCase 521 ReviewExercises 661 13.7 ComparativeStaticsandtheEnvelope Theorem 525 17 Linear Programming 665 ReviewExercises 529 17.1 AGraphicalApproach 666 17.2 IntroductiontoDualityTheory 672 14 Constrained 17.3 TheDualityTheorem 675 Optimization 533 17.4 AGeneralEconomicInterpretation 679 14.1 TheLagrangeMultiplierMethod 533 17.5 ComplementarySlackness 681 14.2 InterpretingtheLagrangeMultiplier 540 ReviewExercises 686 14.3 MultipleSolutionCandidates 543 14.4 WhytheLagrangeMethodWorks 545 Appendix 689 14.5 SufficientConditions 549 14.6 AdditionalVariablesandConstraints 552 Solutions to the Exercises 693 14.7 ComparativeStatics 558 14.8 NonlinearProgramming:ASimple Index 801 Case 563 (cid:2)