Table Of ContentSpringer Texts in Business and Economics
Wolfgang Eichhorn
Winfried Gleißner
Mathematics
and Methodology
for Economics
Applications, Problems and Solutions
Springer Texts in Business and Economics
Moreinformationaboutthisseriesathttp://www.springer.com/series/10099
Wolfgang Eichhorn • Winfried Gleißner
Mathematics
and Methodology
for Economics
Applications, Problems and Solutions
123
WolfgangEichhorn WinfriedGleißner
KarlsruheInstituteofTechnology(KIT) UniversityofAppliedSciencesLandshut
Karlsruhe,Germany Landshut,Germany
ISSN2192-4333 ISSN2192-4341 (electronic)
SpringerTextsinBusinessandEconomics
ISBN978-3-319-23352-9 ISBN978-3-319-23353-6 (eBook)
DOI10.1007/978-3-319-23353-6
LibraryofCongressControlNumber:2016932103
SpringerChamHeidelbergNewYorkDordrechtLondon
©SpringerInternationalPublishingSwitzerland2016
Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof
thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,
broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation
storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology
nowknownorhereafterdeveloped.
Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication
doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneraluse.
Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook
arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor
theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany
errorsoromissionsthatmayhavebeenmade.
Printedonacid-freepaper
Springer International Publishing AG Switzerland is part of Springer Science+Business Media
(www.springer.com)
Preface
This book about mathematics and methodologyfor economicsis the result of the
lifelong teaching experience of the authors. It is written for university students
as well as for students of a university of applied sciences. It is completely self-
containedanddoesnotassumeanypreviousknowledgeofhighschoolmathematics.
At the end of all chapters and sections, there are exercises such that the reader
can test how familiar she or he is with the material of the preceding stuff. After
each set of exercises, the answers are given to encourage the reader to tackle the
problems.
The idea to write such a book was born in 1990 during an international
meeting on functional equations which took place at the University of Graz,
Austria. At this meeting a lot of fascinating applications of functional equations
to solve mathematically formulated economic problems inspired János Aczél,
DistinguishedProfessorofMathematics,UniversityofWaterloo,Ontario,Canada:
He proposed to one of us (W.E.) to start such an adventure in a form of a
textbookforbeginners.Sincethenhesupportedthetentativestepsintothisdirection
by a great wealth of brilliant scientific advices. Later on he became for both
of us the lodestar for our endeavour. Dear János, we owe you a great debt of
gratitude.
For a basic course Chaps.1 (sets, vectors, trigonometric functions, complex
numbers), 3 (mappings and functions), 4 (vectors, matrices, systems of linear
equations),6(functions,limits,derivations),7(importantnonlinearfunctions),and
10 (integration) are sufficient. If a later course will discuss discrete models of
economics,Chap.12(differenceequations)shouldbecovered,too.Forcontinuous
models,Chap.11(differentialequations)isnecessary.(However,wedecidednotto
goveryfarintodetails.)
Chapter 2 gives an introduction to linear optimisation and game theory using
productionsystems. These ideasare continuedin Chaps.5 and9, which discusses
the notion of a Nash Equilibrium. Chapter 8 deals with nonlinear optimisa-
tion.
Chapter13,astheconclusion,reflectsmethodologicallymostofallthatwhatwe
optimisticallyofferedinChaps.1,2,3,4,5,6,7,8,9,10,11and12.
v
vi Preface
ManythanksgotoThomasSchlinkfortypingmostofthemanuscriptinLATEX
very conscientiously and to Dr. Roland Peyrer for his inspiring drawings, which
weretransformedtoPSTricks,anadditionalpackageforgraphicsinLatex.
Karlsruhe,Germany WolfgangEichhorn
Landshut,Germany WinfriedGleißner
Summer,2015
Contents
1 Sets,NumbersandVectors ................................................ 1
1.1 Introduction......................................................... 1
1.2 Basics................................................................ 1
1.2.1 Exercises................................................... 7
1.2.2 Answers.................................................... 7
1.3 Subsets,OperationsBetweenSets ................................. 8
1.3.1 Exercises................................................... 11
1.3.2 Answers ................................................... 12
1.4 CartesianProductsofSets,Rn,Vectors ........................... 12
1.4.1 Exercises................................................... 18
1.4.2 Answers ................................................... 19
1.5 Operations for Vectors, Linear Dependence
andIndependence................................................... 19
1.5.1 Sums,Differences,LinearCombinationsofVectors.... 19
1.5.2 LinearDependence,Independence....................... 21
1.5.3 InnerProduct.............................................. 24
1.5.4 Exercises................................................... 25
1.5.5 Answers ................................................... 26
1.6 GeometricInterpretations.Distance.OrthogonalVectors ........ 26
1.6.1 Exercises................................................... 30
1.6.2 Answers ................................................... 30
1.7 Complex Numbers;the Cosine, Sine, Tangent
andCotangent ...................................................... 31
1.7.1 MultiplicationofComplexNumbers..................... 31
1.7.2 TrigonometricFormofComplexNumbers;
Sine,Cosine ............................................... 34
1.7.3 DivisionofComplexNumbers;Equations............... 40
1.7.4 Tangent,Cotangent........................................ 42
1.7.5 Exercises................................................... 43
1.7.6 Answers ................................................... 43
vii
viii Contents
2 ProductionSystemsProductionProcesses,Technologies,
Efficiency,Optimisation ................................................... 45
2.1 Introduction......................................................... 45
2.2 Basics................................................................ 46
2.2.1 Exercises................................................... 49
2.2.2 Answers ................................................... 49
2.3 LinearProductionModels,LinearOptimisationProblems....... 49
2.3.1 Exercises................................................... 52
2.3.2 Answers ................................................... 53
2.4 SimpleApproachestoLinearOptimisationProblems............ 53
2.4.1 Exercises................................................... 59
2.4.2 Answers ................................................... 60
3 Mappings,Functions....................................................... 61
3.1 Introduction......................................................... 61
3.2 Basics. Domains, Ranges, Images
(Codomains). Mappings (Binary Relations),
Functions,Injections,Surjections,Bijections.Graphs............ 63
3.2.1 Exercises................................................... 72
3.2.2 Answers ................................................... 72
3.3 FunctionsofnVariables,n-DimensionalIntervals,
CompositionofFunctions.......................................... 73
3.3.1 Exercises................................................... 77
3.3.2 Answers.................................................... 78
3.4 MonotonicandLinearlyHomogeneousFunctions.
MaximaandMinima................................................ 78
3.4.1 Exercises................................................... 84
3.4.2 Answers ................................................... 85
3.5 Convex(Concave)Functions.ConvexSets........................ 85
3.5.1 Exercises................................................... 92
3.5.2 Answers ................................................... 92
3.6 Quasi-convexFunctions............................................ 93
3.6.1 Exercises................................................... 99
3.6.2 Answers ................................................... 100
3.7 Functionsinthe“StatisticalTheory”ofPriceIndices ............ 100
3.7.1 Exercises................................................... 103
3.7.2 Answers ................................................... 104
4 Affine and Linear Functions and Transformations
(Matrices),LinearEconomicModels,SystemsofLinear
EquationsandInequalities ................................................ 105
4.1 Introduction......................................................... 105
4.2 Proportionality,Linear and Affine Functions.
Additivity,LinearHomogeneity,Linearity........................ 107
4.2.1 Exercises................................................... 112
4.2.2 Answers.................................................... 113
Contents ix
4.3 Additivity, Linear Homogeneity, Linearity
ofVector-VectorFunctions,Matrices.............................. 113
4.3.1 Exercises .................................................. 117
4.3.2 Answers.................................................... 117
4.4 MatrixAlgebra...................................................... 118
4.4.1 Exercises .................................................. 124
4.4.2 Answers.................................................... 125
4.5 LinearEconomicModels:Leontief,vonNeumann............... 126
4.5.1 Exercises................................................... 133
4.5.2 Answers.................................................... 134
4.6 Systems of Linear Equations. Solution
byElimination.Rank.NecessaryandSufficientConditions ..... 135
4.6.1 Exercises................................................... 154
4.6.2 Answers.................................................... 155
4.7 Determinant,Cramer’sRule,InverseMatrix...................... 156
4.7.1 Exercises................................................... 164
4.7.2 Answers.................................................... 165
4.8 Applicationsof Functionsof Vector Variables:
AggregationinEconomics ......................................... 165
4.8.1 Exercises................................................... 174
4.8.2 Answers.................................................... 176
5 LinearOptimisation,Duality:Zero-SumGames....................... 177
5.1 Introduction......................................................... 177
5.2 LinearOptimisationProblems ..................................... 179
5.2.1 Exercises................................................... 192
5.2.2 Answers.................................................... 192
5.3 Duality............................................................... 194
5.3.1 Exercises................................................... 200
5.3.2 Answers.................................................... 201
5.4 Two-PersonZero-SumGames ..................................... 201
5.4.1 Exercises................................................... 207
5.4.2 Answers.................................................... 207
6 Functions,TheirLimitsandTheirDerivatives ......................... 209
6.1 Introduction......................................................... 209
6.2 Limits,InfinityasLimit,LimitatInfinity,Sequences:
TrigonometricFunctions,Polynomials,RationalFunctions...... 211
6.2.1 Exercises................................................... 220
6.2.2 Answers.................................................... 221
6.3 Continuity,SectionalContinuity,LeftandRightLimits.......... 221
6.3.1 Exercises .................................................. 226
6.3.2 Answers ................................................... 227
6.4 Derivative,Derivation .............................................. 227
6.4.1 Exercises .................................................. 233
6.4.2 Answers ................................................... 234
