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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 Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broadcasting reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationofthispublication orpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyrightLawofSeptember9, 1965,initscurrentversion,andpermissionforusemustalwaysbeobtainedfromSpringer.Violations areliabletoprosecutionundertheGermanCopyrightLaw. Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoesnot imply, even in the absence of a specific statement, that such names are exempt from the relevant protectivelawsandregulationsandthereforefreeforgeneraluse. Coverdesign:WMXDesign,Heidelberg Printedonacid-freepaper SpringerispartofSpringerScience+BusinessMedia(www.springer.com) 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

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