Mathematical Formulas for Economists PDF
Preview Mathematical Formulas for Economists
MathematicalFormulasforEconomists · Bernd Luderer Volker Nollau Klaus Vetters Mathematical Formulas for Economists Third Edition With62Figuresand6Tables 123 ProfessorDr.BerndLuderer ChemnitzUniversityofTechnology DepartmentofMathematics ReichenhainerStraße41 09126Chemnitz Germany [email protected] ProfessorDr.VolkerNollau Dr.KlausVetters DresdenUniversityofTechnology DepartmentofMathematicsandScience ZellescherWeg12-14 01069Dresden Germany [email protected] [email protected] LibraryofCongressControlNumber:2006934208 ISBN-103-540-46901-XSpringerBerlinHeidelbergNewYork ISBN-13978-3-540-46901-8SpringerBerlinHeidelbergNewYork ISBN3-540-27916-4 2ndEditionSpringerBerlinHeidelbergNewYork Thisworkissubjecttocopyright.Allrightsarereserved,whetherthewholeorpartofthematerialis concerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation,broad- casting,reproductiononmicrofilmorinanyotherway,andstorageindatabanks.Duplicationof thispublicationorpartsthereofispermittedonlyundertheprovisionsoftheGermanCopyright LawofSeptember9,1965,initscurrentversion,andpermissionforusemustalwaysbeobtained fromSpringer.ViolationsareliabletoprosecutionundertheGermanCopyrightLaw. SpringerispartofSpringerScience+BusinessMedia springer.com ©Springer-VerlagBerlinHeidelberg2002,2005,2007 Theuseofgeneraldescriptivenames,registerednames,trademarks,etc.inthispublicationdoes notimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Production:LE-TEXJelonek,Schmidt&V¨ocklerGbR,Leipzig Cover-design:ErichKirchner,Heidelberg SPIN11898412 43/3100YL-543210 Printedonacid-freepaper Preface Thiscollectionofformulasconstitutesacompendiumofmathematicsforeco- nomicsandbusiness.Itcontainsthemostimportantformulas,statementsand algorithms in this significant subfield of modern mathematics and addresses primarilystudentsofeconomicsorbusinessatuniversities,collegesandtrade schools. But people dealing with practical or applied problems will also find this collection to be an efficient and easy-to-use work of reference. First the book treats mathematical symbols and constants, sets and state- ments,number systems andtheir arithmetic aswellasfundamentals ofcom- binatorics.Thechapteronsequencesandseriesisfollowedbymathematicsof finance, the representation of functions of one and several independent vari- ables,theirdifferentialandintegralcalculusandbydifferentialanddifference equations.Ineachcasespecialemphasisisplacedonapplicationsandmodels in economics. The chapteronlinearalgebradealswithmatrices,vectors,determinantsand systems of linear equations. This is followed by the representation of struc- tures and algorithmsof linear programming.Finally, the reader finds formu- las on descriptive statistics (data analysis, ratios, inventory and time series analysis), on probability theory (events, probabilities, random variables and distributions)andoninductivestatistics(pointandintervalestimates,tests). Some important tables complete the work. The present manual arose as a result of many years’ teaching for students of economic faculties at the Institutes of Technology of Dresden and Chem- nitz,Germany.Moreover,theauthorscouldtakeadvantageofexperienceand suggestionsofnumerouscolleagues.Forcriticalreadingofthe manuscriptwe feel obliged to thank Dr M. Richter and Dr K. Eppler. Our special thank is due to M. Schoenherr, Dr U. Wuerker and Dr J. Rudl, who contributed to technical preparationof the book. AftersuccessfulusebyGermanreadersitisagreatpleasureforustopresent thiscollectionofformulastotheEnglishauditorium.Thetranslationisbased on the fifth German edition. We are greatly obliged to Springer-Verlag for giving us the opportunity to publish this book in English. ThesecondEnglisheditionofthisbookwasverypopularbothwithstudents and with practitioners. Thus it was rapidly out of print. So we are very pleased to present this third, carefully checked edition. Finally we would like to emphasize that remarks and criticism are always welcome. Chemnitz/Dresden, Bernd Luderer August 2006 Volker Nollau Klaus Vetters Contents Mathematical Symbols and Constants ........................ 1 Notations and symbols ...................................... 1 Mathematical constants...................................... 2 Sets and Propositions......................................... 3 Notion of a set ............................................. 3 Relations between sets....................................... 3 Operations with sets ........................................ 4 Rules for operations with sets ................................ 5 Product sets and mappings................................... 6 Propositional calculus ....................................... 7 Number Systems and their Arithmetic ....................... 9 Natural, integer, rational, and real numbers .................... 9 Calculation with real numbers ................................ 10 Absolute values............................................. 11 Factorial and binomial coefficients............................. 12 Equations.................................................. 13 Inequalities ................................................ 14 Finite sums ................................................ 15 Powers and roots ........................................... 15 Logarithms................................................. 16 Complex numbers........................................... 17 Combinatorial Analysis ....................................... 19 Permutations............................................... 19 Arrangements .............................................. 19 Combinations .............................................. 20 Sequences and Series ......................................... 21 Sequences of numbers ....................................... 21 Sequences of functions ....................................... 22 Infinite series............................................... 23 Function and power series.................................... 25 Taylor series................................................ 27 Fourier series............................................... 29 VIII Contents Mathematics of Finance ...................................... 31 Simple interest ............................................. 31 Compound interest.......................................... 33 Annuities .................................................. 36 Dynamic annuities .......................................... 38 Amortization calculus ....................................... 39 Price calculus .............................................. 41 Investment analysis ......................................... 42 Depreciations............................................... 43 Numerical methods for the determination of zeros............... 44 Functions of one Independent Variable ....................... 45 Basic notions............................................... 45 Linear functions ............................................ 47 Quadratic functions ......................................... 47 Power functions............................................. 48 Polynomials................................................ 49 Fractional rational functions, partial fraction decomposition ...... 50 Exponential functions ....................................... 51 Logarithmic functions ....................................... 52 Trigonometric functions...................................... 53 Inverse trigonometric functions ............................... 55 Hyperbolic functions ........................................ 56 Area-hyperbolic functions .................................... 56 Some economic functions..................................... 57 Functions of one Variable: Differential Calculus............... 61 Limit of a function .......................................... 61 Continuity ................................................. 62 Differentiation.............................................. 63 Economic interpretation of the first derivative .................. 66 Rates of change and elasticities ............................... 68 Mean value theorems ........................................ 70 Higher derivatives and Taylor expansion ....................... 70 Description of function features by means of derivatives.......... 72 Investigation of economic functions, profit maximization ......... 75 Functions of one Variable: Integral Calculus .................. 79 Indefinite integral ........................................... 79 Definite integral ............................................ 80 Tables of indefinite integrals.................................. 81 Improper integrals .......................................... 88 Parameter integrals ......................................... 88 Economic applications of integral calculus...................... 89 Contents IX Differential Equations ........................................ 91 First-order differential equations .............................. 91 Linear differential equations of n-th order ...................... 92 First-order linear systems with constant coefficients ............. 95 Difference Equations.......................................... 97 First-order linear difference equations.......................... 97 Economic models ........................................... 98 Linear second-order difference equations ....................... 99 Economic models ........................................... 101 Linear difference equations of n-th order with constant coefficients 102 Differential Calculus for Functions of Several Variables ....... 103 Basic notions............................................... 103 Point sets of the space IRn ................................... 103 Limit and continuity ........................................ 104 Differentiation of functions of several variables.................. 105 Total differential............................................ 108 Unconstrained extreme value problems......................... 109 Constrained extreme value problems........................... 110 Least squares method ....................................... 112 Propagationof errors........................................ 113 Economic applications ....................................... 114 Linear Algebra ............................................... 115 Vectors .................................................... 115 Equations of straight lines and planes ......................... 117 Matrices ................................................... 119 Determinants............................................... 121 Systems of linear equations................................... 122 Gaussian elimination ........................................ 123 Cramer’s rule............................................... 125 Exchange method........................................... 125 Inverse matrix.............................................. 126 Eigenvalue problems for matrices ............................. 126 Matrix models.............................................. 127 Linear Programming. Transportation Problem................ 129 Normal form of a linear programming problem.................. 129 Simplex method ............................................ 130 Dual simplex method........................................ 132 Generation of an initial simplex table.......................... 133 Duality .................................................... 135 Transportation problem...................................... 136 X Contents Descriptive Statistics ......................................... 139 Basic notions............................................... 139 Univariate data analysis ..................................... 139 Statistical parameters ....................................... 140 Bivariate data analysis ...................................... 141 Ratios ..................................................... 144 Inventory analysis........................................... 145 Time series analysis ......................................... 147 Calculus of Probability ....................................... 149 Random events and their probabilities ......................... 149 Conditional probabilities..................................... 151 Random variables and their distributions ...................... 153 Discrete distributions........................................ 153 Continuous distributions ..................................... 155 Special continuous distributions............................... 156 Random vectors ............................................ 159 Inductive Statistics ........................................... 163 Sample .................................................... 163 Point estimates ............................................. 163 Confidence interval estimates ................................. 165 Statistical tests ............................................. 167 Significance tests under normal distribution .................... 168 Tables..................................................... 170 References.................................................... 181 Index......................................................... 183 Mathematical Symbols and Constants Notations and symbols IN – set of natural numbers IN0 – set of natural numbers inclusively zero ZZ – set of integer numbers Q – set of rational numbers IR – set of real numbers + IR – set of nonnegative real numbers IRn – set of n-tuples of real numbers (n-dimensional vectors) C – set of complex numbers √ x – nonnegative number y (square root) such that y2 =x, x≥0 √ nx – nonnegative number y (n-th root) such yn=x, x≥0 (cid:1)n xi – sum of the numbers xi: x1+x2+...+xn i=1 (cid:2)n xi – product of the numbers xi: x1·x2·...·xn i=1 n! – 1·2·...·n (n factorial) min{a,b} – minimum of the numbers a and b: a for a≤b, b for a≥b max{a,b} – maximum of the numbers a and b: a for a≥b, b for a≤b (cid:4)x(cid:5) – smallest integer y such that y ≥x (rounding up) (cid:6)x(cid:7) – greatest integer y such that y ≤x (rounding down) sgn x – signum: 1 for x>0, 0 for x=0, −1 for x<0 |x| – absolute value of the real number x: x for x≥0 and −x for x<0 (a,b) – open interval, i.e. a<x<b [a,b] – closed interval, i.e. a≤x≤b (a,b] – half-open interval closed from the right, i.e. a<x≤b [a,b) – half-open interval open at the right, i.e. a≤x<b ≤, ≥ – less or equal; greater or equal def = – equality by definition := – the left-hand side is defined by the right-hand side
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