Essential Mathematics for Economic Analysis

766 Pages·2012·6.74 MB·English

by  Knut Sydsæter

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Knut Sydsæter is an Emeritus Professor of Mathematics in the Economics Department itayd
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Peter Hammond is currently a Professor of Economics at the University of Warwick, mnr
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He has taught mathematics for economists at both universities.
Knut Sydsæter & Peter Hammond
Arne Strøm has extensive experience in teaching mathematics for
economists in the Department of Economics at the University of Oslo. with Arne StrØm
www.pearson-books.com
ESSENTIAL MATHEMATICS FOR
EC ONOMIC
A N A LYSIS

ESSENTIAL MATHEMATICS FOR
E C O N O M I C
A N A LY S I S
FOURTH EDITION
Knut Sydsæter and Peter Hammond
with Arne Strøm
Pearson Education Limited
Edinburgh Gate
Harlow
Essex CM20 2JE
England
and Associated Companies throughout the world
Visit us on the World Wide Web at:
www.pearson.com/uk
First published by Prentice-Hall, Inc. 1995
Second edition published 2006
Third edition published 2008
Fourth edition published by Pearson Education Limited 2012
TothememoryofmyparentsElsie(1916–2007)and
© Prentice-Hall, Inc. 1995
© Knut Sydsæter and Peter Hammond 2002, 2006, 2008, 2012 Fred(1916–2008),myfirstteachersofMathematics,
The rights of Knut Sydsæter and Peter Hammond to be identified as authors of this work
basic Economics, and many more important things.
have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system,
or transmitted in any form or by any means, electronic, mechanical, photocopying, recording — Peter
or otherwise, without either the prior written permission of the publisher or a licence permitting
restricted copying in the United Kingdom issued by the Copyright Licensing Agency Ltd,
Saffron House, 6−10 Kirby Street, London EC1N 8TS.
Pearson Education is not responsible for the content of third-party internet sites.
ISBN 978-0-273-76068-9
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library
10 9 8 7 6 5 4 3 2 1
16 15 14 13 12
Typeset in 10/13 pt Times Roman by Matematisk Sats and Arne Strøm, Norway
Printed and bound by Ashford Colour Press Ltd, Gosport, UK
EssentialMath.forEcon.Analysis,4thedn EME4_A0100.TEX,17April2012,19:33 Pagev
TothememoryofmyparentsElsie(1916–2007)and
Fred(1916–2008),myfirstteachersofMathematics,
basic Economics, and many more important things.
— Peter
EssentialMath.forEcon.Analysis,4thedn EME4_A0100.TEX,17April2012,19:33 Pagev
C O N T E N T S
Preface xi 3.3 DoubleSums 59
3.4 AFewAspectsofLogic 61
1 Introductory Topics I: Algebra 1 3.5 MathematicalProofs 67
3.6 EssentialsofSetTheory 69
1.1 TheRealNumbers 1
3.7 MathematicalInduction 75
1.2 IntegerPowers 4
ReviewProblemsforChapter3 77
1.3 RulesofAlgebra 10
1.4 Fractions 14
4 Functions of One Variable 79
1.5 FractionalPowers 19
1.6 Inequalities 24 4.1 Introduction 79
1.7 IntervalsandAbsoluteValues 29 4.2 BasicDefinitions 80
ReviewProblemsforChapter1 32 4.3 GraphsofFunctions 86
4.4 LinearFunctions 89
2 Introductory Topics II: 4.5 LinearModels 95
Equations 35 4.6 QuadraticFunctions 99
2.1 HowtoSolveSimpleEquations 35 4.7 Polynomials 105
2.2 EquationswithParameters 38 4.8 PowerFunctions 112
2.3 QuadraticEquations 41 4.9 ExponentialFunctions 114
2.4 LinearEquationsinTwoUnknowns 46 4.10 LogarithmicFunctions 119
2.5 NonlinearEquations 48 ReviewProblemsforChapter4 124
ReviewProblemsforChapter2 49
5 Properties of Functions 127
3 Introductory Topics III:
5.1 ShiftingGraphs 127
Miscellaneous 51
5.2 NewFunctionsfromOld 132
3.1 SummationNotation 51 5.3 InverseFunctions 136
3.2 RulesforSums. Newton’sBinomial 5.4 GraphsofEquations 143
Formula 55 5.5 DistanceinthePlane. Circles 146
EssentialMath.forEcon.Analysis,4thedn EME4_A0207.TEX,24April2012,22:12 Pagevii
C O N T E N T S
Preface xi 3.3 DoubleSums 59
3.4 AFewAspectsofLogic 61
1 Introductory Topics I: Algebra 1 3.5 MathematicalProofs 67
3.6 EssentialsofSetTheory 69
1.1 TheRealNumbers 1
3.7 MathematicalInduction 75
1.2 IntegerPowers 4
ReviewProblemsforChapter3 77
1.3 RulesofAlgebra 10
1.4 Fractions 14
4 Functions of One Variable 79
1.5 FractionalPowers 19
1.6 Inequalities 24 4.1 Introduction 79
1.7 IntervalsandAbsoluteValues 29 4.2 BasicDefinitions 80
ReviewProblemsforChapter1 32 4.3 GraphsofFunctions 86
4.4 LinearFunctions 89
2 Introductory Topics II: 4.5 LinearModels 95
Equations 35 4.6 QuadraticFunctions 99
2.1 HowtoSolveSimpleEquations 35 4.7 Polynomials 105
2.2 EquationswithParameters 38 4.8 PowerFunctions 112
2.3 QuadraticEquations 41 4.9 ExponentialFunctions 114
2.4 LinearEquationsinTwoUnknowns 46 4.10 LogarithmicFunctions 119
2.5 NonlinearEquations 48 ReviewProblemsforChapter4 124
ReviewProblemsforChapter2 49
5 Properties of Functions 127
3 Introductory Topics III:
5.1 ShiftingGraphs 127
Miscellaneous 51
5.2 NewFunctionsfromOld 132
3.1 SummationNotation 51 5.3 InverseFunctions 136
3.2 RulesforSums. Newton’sBinomial 5.4 GraphsofEquations 143
Formula 55 5.5 DistanceinthePlane. Circles 146
EssentialMath.forEcon.Analysis,4thedn EME4_A0207.TEX,24April2012,22:12 Pagevii
vviiiiii CCOONNTETNENTSTS CONTENTS ix
5.6 GeneralFunctions 150 9 Integration 293 12.5 ElasticityofSubstitution 428 15.5 TheTranspose 562
ReviewProblemsforChapter5 153 9.1 IndefiniteIntegrals 293 12.6 HomogeneousFunctionsof 15.6 GaussianElimination 565
TwoVariables 431 15.7 Vectors 570
9.2 AreaandDefiniteIntegrals 299
6 Differentiation 155 12.7 HomogeneousandHomothetic 15.8 GeometricInterpretationofVectors 573
9.3 PropertiesofDefiniteIntegrals 305
Functions 435 15.9 LinesandPlanes 578
6.1 SlopesofCurves 155 9.4 EconomicApplications 309
12.8 LinearApproximations 440 ReviewProblemsforChapter15 582
6.2 TangentsandDerivatives 157 9.5 IntegrationbyParts 315
9.6 IntegrationbySubstitution 319 12.9 Differentials 444
6.3 IncreasingandDecreasingFunctions 163
9.7 InfiniteIntervalsofIntegration 324 12.10 SystemsofEquations 449 16 Determinants and
6.4 RatesofChange 165
9.8 AGlimpseatDifferentialEquations 330 12.11 DifferentiatingSystemsofEquations 452 Inverse Matrices 585
6.5 ADashofLimits 169
9.9 SeparableandLinearDifferential ReviewProblemsforChapter12 458 16.1 DeterminantsofOrder2 585
6.6 SimpleRulesforDifferentiation 174
Equations 336 16.2 DeterminantsofOrder3 589
6.7 Sums,Products,andQuotients 178
13 Multivariable
ReviewProblemsforChapter9 341 16.3 DeterminantsofOrdern 593
6.8 ChainRule 184
Optimization 461
16.4 BasicRulesforDeterminants 596
6.9 Higher-OrderDerivatives 188
10 Topics in Financial 13.1 TwoVariables: NecessaryConditions 461 16.5 ExpansionbyCofactors 601
6.10 ExponentialFunctions 194
Economics 345 13.2 TwoVariables: SufficientConditions 466 16.6 TheInverseofaMatrix 604
6.11 LogarithmicFunctions 197
ReviewProblemsforChapter6 203 10.1 InterestPeriodsandEffectiveRates 345 13.3 LocalExtremePoints 470 16.7 AGeneralFormulafortheInverse 610
10.2 ContinuousCompounding 349 13.4 LinearModelswithQuadratic 16.8 Cramer’sRule 613
7 Derivatives in Use 205 10.3 PresentValue 351 Objectives 475 16.9 TheLeontiefModel 616
10.4 GeometricSeries 353 13.5 TheExtremeValueTheorem 482 ReviewProblemsforChapter16 621
7.1 ImplicitDifferentiation 205 10.5 TotalPresentValue 359 13.6 ThreeorMoreVariables 487
7.2 EconomicExamples 210 10.6 MortgageRepayments 364 13.7 ComparativeStaticsandthe 17 Linear Programming 623
7.3 DifferentiatingtheInverse 214 10.7 InternalRateofReturn 369 EnvelopeTheorem 491 17.1 AGraphicalApproach 623
7.4 LinearApproximations 217 10.8 AGlimpseatDifferenceEquations 371 ReviewProblemsforChapter13 495 17.2 IntroductiontoDualityTheory 629
7.5 PolynomialApproximations 221
ReviewProblemsforChapter10 374 17.3 TheDualityTheorem 633
7.6 Taylor’sFormula 225 14 Constrained Optimization 497
17.4 AGeneralEconomicInterpretation 636
7.7 WhyEconomistsUseElasticities 228 11 Functions of Many 14.1 TheLagrangeMultiplierMethod 497 17.5 ComplementarySlackness 638
7.8 Continuity 233 Variables 377 14.2 InterpretingtheLagrangeMultiplier 504 ReviewProblemsforChapter17 643
7.9 MoreonLimits 237
11.1 FunctionsofTwoVariables 377 14.3 SeveralSolutionCandidates 507
7.10 IntermediateValueTheorem. Appendix: Geometry 645
11.2 PartialDerivativeswithTwoVariables 381 14.4 WhytheLagrangeMethodWorks 509
Newton’sMethod 245
11.3 GeometricRepresentation 387 14.5 SufficientConditions 513
The Greek Alphabet 647
7.11 InfiniteSequences 249
11.4 SurfacesandDistance 393 14.6 AdditionalVariablesandConstraints 516
7.12 L’Hôpital’sRule 251
11.5 FunctionsofMoreVariables 396 14.7 ComparativeStatics 522 Answers to the Problems 649
ReviewProblemsforChapter7 256
11.6 PartialDerivativeswithMoreVariables 400 14.8 NonlinearProgramming:
11.7 EconomicApplications 404 ASimpleCase 526 Index 739
8 Single-Variable
11.8 PartialElasticities 406 14.9 MultipleInequalityConstraints 532
Optimization 259
ReviewProblemsforChapter11 408 14.10 NonnegativityConstraints 537
8.1 Introduction 259 ReviewProblemsforChapter14 541
12 Tools for Comparative
8.2 SimpleTestsforExtremePoints 262
Statics 411 15 Matrix and Vector
8.3 EconomicExamples 266
Algebra 545
8.4 TheExtremeValueTheorem 270 12.1 ASimpleChainRule 411
8.5 FurtherEconomicExamples 276 12.2 ChainRulesforManyVariables 416 15.1 SystemsofLinearEquations 545
8.6 LocalExtremePoints 281 12.3 ImplicitDifferentiationalonga 15.2 MatricesandMatrixOperations 548
8.7 InflectionPoints 287 LevelCurve 420 15.3 MatrixMultiplication 551
ReviewProblemsforChapter8 291 12.4 MoreGeneralCases 424 15.4 RulesforMatrixMultiplication 556
EssentialMath.forEcon.Analysis,4thedn EME4_A0207.TEX,24April2012,22:12 Pageviii EssentialMath.forEcon.Analysis,4thedn EME4_A0207.TEX,24April2012,22:12 Pageix
viii CONTENTS CCOONNTTEENNTTSS iixx
5.6 GeneralFunctions 150 9 Integration 293 12.5 ElasticityofSubstitution 428 15.5 TheTranspose 562
ReviewProblemsforChapter5 153 9.1 IndefiniteIntegrals 293 12.6 HomogeneousFunctionsof 15.6 GaussianElimination 565
TwoVariables 431 15.7 Vectors 570
9.2 AreaandDefiniteIntegrals 299
6 Differentiation 155 12.7 HomogeneousandHomothetic 15.8 GeometricInterpretationofVectors 573
9.3 PropertiesofDefiniteIntegrals 305
Functions 435 15.9 LinesandPlanes 578
6.1 SlopesofCurves 155 9.4 EconomicApplications 309
12.8 LinearApproximations 440 ReviewProblemsforChapter15 582
6.2 TangentsandDerivatives 157 9.5 IntegrationbyParts 315
9.6 IntegrationbySubstitution 319 12.9 Differentials 444
6.3 IncreasingandDecreasingFunctions 163
9.7 InfiniteIntervalsofIntegration 324 12.10 SystemsofEquations 449 16 Determinants and
6.4 RatesofChange 165
9.8 AGlimpseatDifferentialEquations 330 12.11 DifferentiatingSystemsofEquations 452 Inverse Matrices 585
6.5 ADashofLimits 169
9.9 SeparableandLinearDifferential ReviewProblemsforChapter12 458 16.1 DeterminantsofOrder2 585
6.6 SimpleRulesforDifferentiation 174
Equations 336 16.2 DeterminantsofOrder3 589
6.7 Sums,Products,andQuotients 178
13 Multivariable
ReviewProblemsforChapter9 341 16.3 DeterminantsofOrdern 593
6.8 ChainRule 184
Optimization 461
16.4 BasicRulesforDeterminants 596
6.9 Higher-OrderDerivatives 188
10 Topics in Financial 13.1 TwoVariables: NecessaryConditions 461 16.5 ExpansionbyCofactors 601
6.10 ExponentialFunctions 194
Economics 345 13.2 TwoVariables: SufficientConditions 466 16.6 TheInverseofaMatrix 604
6.11 LogarithmicFunctions 197
ReviewProblemsforChapter6 203 10.1 InterestPeriodsandEffectiveRates 345 13.3 LocalExtremePoints 470 16.7 AGeneralFormulafortheInverse 610
10.2 ContinuousCompounding 349 13.4 LinearModelswithQuadratic 16.8 Cramer’sRule 613
7 Derivatives in Use 205 10.3 PresentValue 351 Objectives 475 16.9 TheLeontiefModel 616
10.4 GeometricSeries 353 13.5 TheExtremeValueTheorem 482 ReviewProblemsforChapter16 621
7.1 ImplicitDifferentiation 205 10.5 TotalPresentValue 359 13.6 ThreeorMoreVariables 487
7.2 EconomicExamples 210 10.6 MortgageRepayments 364 13.7 ComparativeStaticsandthe 17 Linear Programming 623
7.3 DifferentiatingtheInverse 214 10.7 InternalRateofReturn 369 EnvelopeTheorem 491 17.1 AGraphicalApproach 623
7.4 LinearApproximations 217 10.8 AGlimpseatDifferenceEquations 371 ReviewProblemsforChapter13 495 17.2 IntroductiontoDualityTheory 629
7.5 PolynomialApproximations 221
ReviewProblemsforChapter10 374 17.3 TheDualityTheorem 633
7.6 Taylor’sFormula 225 14 Constrained Optimization 497
17.4 AGeneralEconomicInterpretation 636
7.7 WhyEconomistsUseElasticities 228 11 Functions of Many 14.1 TheLagrangeMultiplierMethod 497 17.5 ComplementarySlackness 638
7.8 Continuity 233 Variables 377 14.2 InterpretingtheLagrangeMultiplier 504 ReviewProblemsforChapter17 643
7.9 MoreonLimits 237
11.1 FunctionsofTwoVariables 377 14.3 SeveralSolutionCandidates 507
7.10 IntermediateValueTheorem. Appendix: Geometry 645
11.2 PartialDerivativeswithTwoVariables 381 14.4 WhytheLagrangeMethodWorks 509
Newton’sMethod 245
11.3 GeometricRepresentation 387 14.5 SufficientConditions 513
The Greek Alphabet 647
7.11 InfiniteSequences 249
11.4 SurfacesandDistance 393 14.6 AdditionalVariablesandConstraints 516
7.12 L’Hôpital’sRule 251
11.5 FunctionsofMoreVariables 396 14.7 ComparativeStatics 522 Answers to the Problems 649
ReviewProblemsforChapter7 256
11.6 PartialDerivativeswithMoreVariables 400 14.8 NonlinearProgramming:
11.7 EconomicApplications 404 ASimpleCase 526 Index 739
8 Single-Variable
11.8 PartialElasticities 406 14.9 MultipleInequalityConstraints 532
Optimization 259
ReviewProblemsforChapter11 408 14.10 NonnegativityConstraints 537
8.1 Introduction 259 ReviewProblemsforChapter14 541
12 Tools for Comparative
8.2 SimpleTestsforExtremePoints 262
Statics 411 15 Matrix and Vector
8.3 EconomicExamples 266
Algebra 545
8.4 TheExtremeValueTheorem 270 12.1 ASimpleChainRule 411
8.5 FurtherEconomicExamples 276 12.2 ChainRulesforManyVariables 416 15.1 SystemsofLinearEquations 545
8.6 LocalExtremePoints 281 12.3 ImplicitDifferentiationalonga 15.2 MatricesandMatrixOperations 548
8.7 InflectionPoints 287 LevelCurve 420 15.3 MatrixMultiplication 551
ReviewProblemsforChapter8 291 12.4 MoreGeneralCases 424 15.4 RulesforMatrixMultiplication 556
EssentialMath.forEcon.Analysis,4thedn EME4_A0207.TEX,24April2012,22:12 Pageviii EssentialMath.forEcon.Analysis,4thedn EME4_A0207.TEX,24April2012,22:12 Pageix