Foundations of mathematical & computational economics

547 Pages·2007·5.709 MB·English

by  Dadkhah, Kamran

Foundations of Mathematical and Computational
Economics
Second Edition
Kamran Dadkhah
Foundations of Mathematical
and Computational
Economics
Second Edition
123
KamranDadkhah
NortheasternUniversity
DepartmentofEconomics
Boston
USA
[email protected]
FirsteditionpublishedbyThomsonSouth-Western2007,ISBN978-0324235838
ISBN978-3-642-13747-1 e-ISBN978-3-642-13748-8
DOI10.1007/978-3-642-13748-8
SpringerHeidelbergDordrechtLondonNewYork
©Springer-VerlagBerlinHeidelberg2007,2011
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ToKarenandmydaughter,Lara
Preface
Mathematics is both a language of its own and a way of thinking; applying
mathematics to economics reveals that mathematics is indeed inherent to eco-
nomic life. The objective of this book is to teach mathematical knowledge and
computational skills required for macro and microeconomic analysis, as well as
econometrics.Inaddition,Ihopeitconveysadeeperunderstandingandappreciation
ofmathematics.
Examples in the following chapters are chosen from all areas of economics
and econometrics. Some have very practical applications, such as determining
monthly mortgage payments; others involve more abstract models, such as sys-
temsofdynamicequations.Someexamplesarefamiliarinthestudyofmicroand
macroeconomics;othersinvolvelesswell-knownandmorerecentmodels,suchas
realbusinesscycletheory.
Increasingly, economists need to make complicated calculations. Systems of
dynamic equations are used to forecast different economic variables several years
into the future. Such systems are used to assess the effects of alternative policies,
such as different methods of financing Social Security over a few decades. Also,
many theories in microeconomics, industrial organization, and macroeconomics
require modeling the behavior and interactions of many decision makers. These
types of calculation require computational dexterity. Thus, this book provides an
introductiontonumericalmethods,computation,andprogrammingwithExceland
Matlab. In addition, because of the increasing use of computer software such as
MapleandMathematica,sectionsareincludedtointroducethestudenttodifferenti-
ation,integration,andsolvingdifferenceanddifferentialequationsusingMapleand
totheconceptofcomputer-aidedmathematicalproof.
The second edition differs from the first in several respects. Parts of the book
are rearranged, some materials are deleted and some new topics and examples are
added. In the first edition most computational examples used Matlab and some
Excel.Inthepresentedition,ExcelandMatlabaregivenequalweights.Theseare
done in the hope of making the book more reader friendly. Similarly, more use
is made of the Maple program for solving non-numerical problems. Finally, many
errorshadcreptintothefirstedition,whicharecorrectedinthepresentedition.Iam
indebtedtostudentsinmymathandstatclassesforpointingoutsomeofthem.
vii
viii Preface
IwouldliketothankBarbaraFessofSpringer-Verlagforhersupportinpreparing
this new edition. I also would like to thank Saranya Baskar and her colleagues at
Integrafortheirexcellentworkinproducingthebook.
Asalways,mygreatestappreciationistoKarenChallberg,whoduringtheentire
projectgavemesupport,encouragement,andlove.
Contents
PartI BasicConceptsandMethods
1 Mathematics,Computation,andEconomics . . . . . . . . . . . . 3
1.1 Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.2 PhilosophiesofMathematics . . . . . . . . . . . . . . . . . . 9
1.3 WomeninMathematics . . . . . . . . . . . . . . . . . . . . . 11
1.4 Computation. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.5 MathematicsandEconomics . . . . . . . . . . . . . . . . . . 14
1.6 ComputationandEconomics . . . . . . . . . . . . . . . . . . 14
2 BasicMathematicalConceptsandMethods. . . . . . . . . . . . . 17
2.1 FunctionsofRealVariables . . . . . . . . . . . . . . . . . . . 17
2.1.1 VarietyofEconomicRelationships . . . . . . . . . . 22
2.1.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.2.1 SummationNotation(cid:2) . . . . . . . . . . . . . . . . 24
2.2.2 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.2.3 ConvergentandDivergentSeries . . . . . . . . . . . 27
2.2.4 ArithmeticProgression . . . . . . . . . . . . . . . . 29
2.2.5 GeometricProgression . . . . . . . . . . . . . . . . . 31
2.2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 34
2.3 Permutations, Factorial, Combinations, and the
BinomialExpansion . . . . . . . . . . . . . . . . . . . . . . . 35
2.3.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 37
2.4 LogarithmandExponentialFunctions . . . . . . . . . . . . . 38
2.4.1 Logarithm . . . . . . . . . . . . . . . . . . . . . . . 38
2.4.2 BaseofNaturalLogarithm,e . . . . . . . . . . . . . 40
2.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 41
2.5 MathematicalProof . . . . . . . . . . . . . . . . . . . . . . . 42
2.5.1 Deduction, Mathematical Induction, and
ProofbyContradiction. . . . . . . . . . . . . . . . . 42
2.5.2 Computer-AssistedMathematicalProof . . . . . . . . 44
2.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 45
ix
x Contents
2.6 Trigonometry . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.6.1 CyclesandFrequencies . . . . . . . . . . . . . . . . 50
2.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7 ComplexNumbers . . . . . . . . . . . . . . . . . . . . . . . 51
2.7.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 56
3 BasicConceptsofComputation . . . . . . . . . . . . . . . . . . . 57
3.1 IterativeMethods . . . . . . . . . . . . . . . . . . . . . . . . 57
3.1.1 NamingCellsinExcel . . . . . . . . . . . . . . . . . 60
3.2 AbsoluteandRelativeComputationErrors . . . . . . . . . . . 61
3.3 EfficiencyofComputation . . . . . . . . . . . . . . . . . . . 62
3.4 oandO . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.5 SolvinganEquation . . . . . . . . . . . . . . . . . . . . . . . 66
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4 BasicConceptsandMethodsofProbabilityTheoryandStatistics 69
4.1 Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
4.2 RandomVariablesandProbabilityDistributions . . . . . . . . 72
4.3 MarginalandConditionalDistributions . . . . . . . . . . . . 74
4.4 TheBayesTheorem . . . . . . . . . . . . . . . . . . . . . . . 79
4.5 TheLawofIteratedExpectations . . . . . . . . . . . . . . . . 81
4.6 ContinuousRandomVariables . . . . . . . . . . . . . . . . . 82
4.7 CorrelationandRegression . . . . . . . . . . . . . . . . . . . 85
4.8 MarkovChains . . . . . . . . . . . . . . . . . . . . . . . . . 88
4.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
PartII LinearAlgebra
5 Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
5.1 VectorsandVectorSpace . . . . . . . . . . . . . . . . . . . . 96
5.1.1 VectorSpace . . . . . . . . . . . . . . . . . . . . . . 100
5.1.2 NormofaVector. . . . . . . . . . . . . . . . . . . . 102
5.1.3 Metric . . . . . . . . . . . . . . . . . . . . . . . . . 104
5.1.4 Angle Between Two Vectors
andtheCauchy-SchwarzTheorem . . . . . . . . . . 105
5.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2 OrthogonalVectors . . . . . . . . . . . . . . . . . . . . . . . 109
5.2.1 Gramm-SchmidtAlgorithm . . . . . . . . . . . . . . 109
5.2.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 111
6 MatricesandMatrixAlgebra . . . . . . . . . . . . . . . . . . . . 113
6.1 BasicDefinitionsandOperations . . . . . . . . . . . . . . . . 113
6.1.1 SystemsofLinearEquations . . . . . . . . . . . . . 118
6.1.2 ComputationwithMatrices . . . . . . . . . . . . . . 121
6.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 122
6.2 TheInverseofaMatrix . . . . . . . . . . . . . . . . . . . . . 123
6.2.1 ANumberCalledtheDeterminant . . . . . . . . . . 127
Contents xi
6.2.2 RankandTraceofaMatrix . . . . . . . . . . . . . . 132
6.2.3 AnotherWaytoFindtheInverseofaMatrix . . . . . 133
6.2.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 135
6.3 SolvingSystemsofLinearEquationsUsingMatrixAlgebra. . 137
6.3.1 Cramer’sRule . . . . . . . . . . . . . . . . . . . . . 139
6.3.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 142
7 AdvancedTopicsinMatrixAlgebra . . . . . . . . . . . . . . . . . 143
7.1 Quadratic Forms and Positive and Negative
DefiniteMatrices . . . . . . . . . . . . . . . . . . . . . . . . 143
7.1.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 146
7.2 GeneralizedInverseofaMatrix. . . . . . . . . . . . . . . . . 147
7.2.1 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 150
7.3 OrthogonalMatrices . . . . . . . . . . . . . . . . . . . . . . 150
7.3.1 OrthogonalProjection . . . . . . . . . . . . . . . . . 150
7.3.2 OrthogonalComplementofaMatrix . . . . . . . . . 152
7.3.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 153
7.4 EigenvaluesandEigenvectors . . . . . . . . . . . . . . . . . . 153
7.4.1 ComplexEigenvalues . . . . . . . . . . . . . . . . . 159
7.4.2 RepeatedEigenvalues . . . . . . . . . . . . . . . . . 160
7.4.3 EigenvaluesandtheDeterminantandTraceof
aMatrix . . . . . . . . . . . . . . . . . . . . . . . . 164
7.4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 166
7.5 FactorizationofSymmetricMatrices . . . . . . . . . . . . . . 167
7.5.1 SomeInterestingPropertiesofSymmetricMatrices . 167
7.5.2 FactorizationofMatrixwithRealDistinctRoots . . . 170
7.5.3 FactorizationofaPositiveDefiniteMatrix . . . . . . 172
7.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 176
7.6 LUFactorizationofaSquareMatrix . . . . . . . . . . . . . . 176
7.6.1 CholeskyFactorization . . . . . . . . . . . . . . . . 181
7.6.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 182
7.7 KroneckerProductandVecOperator . . . . . . . . . . . . . . 183
7.7.1 VectorizationofaMatrix . . . . . . . . . . . . . . . 185
7.7.2 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 185
PartIII Calculus
8 Differentiation:FunctionsofOneVariable . . . . . . . . . . . . . 189
8.1 MarginalAnalysisinEconomics . . . . . . . . . . . . . . . . 189
8.1.1 MarginalConceptsandDerivatives . . . . . . . . . . 190
8.1.2 ComparativeStaticAnalysis . . . . . . . . . . . . . . 192
8.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 193
8.2 LimitandContinuity . . . . . . . . . . . . . . . . . . . . . . 194
8.2.1 Limit . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.2.2 Continuity . . . . . . . . . . . . . . . . . . . . . . . 196
8.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . 198