Mathematical Formulas for Economists

Transcription

1 Mathematical Formulas for Economists

2 Springer-Verlag Berlin Heidelberg GmbH

3 Bernd Luderer. Volker Nollau Klaus Vetters Mathematical Formulas for Economists With 58 Figures and 6 Tables, Springer

4 Professor Dr. Bernd Luderer Chemnitz University of Technology Faculty of Mathematics StraBe der Nationen Chemnitz Germany Professor Dr. Volker Nollau Dr. Klaus Vetters Dresden University of Technology Faculty of Mathematics and Natural Science MommsenstraBe Dresden Germany ISBN Library of Congress Cataloging-in-Publication Data applied for Die Deutsche Bibliothek - CIP-Einheitsaufnahme Luderer, Bernd: Mathematical formulas for economists: with 6 tables I Bernd Luderer; Volker Nollau; Klaus Vetters. ISBN ISBN (ebook) DOI / This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH. Violations are liable for prosecution under the German Copyright Law. Springer-Verlag Berlin Heidelberg 2002 Originally pnblished by Springer-Verlag Berlin Heidelberg New York in 2002 The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regnlations and therefore free for general use. Softcover-Design: Erich Kirchner, Heidelberg SPIN / Printed on acid-free paper

5 Preface This collection of formulas constitutes a compendium of mathematics for economics and business. It contains the most important formulas, statements and algorithms in this significant subfield of modern mathematics and addresses primarily students of economics or business at universities, colleges and trade 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 statements, number systems and their arithmetic as well as fundamentals of combinatorics. The chapter on sequences and series is followed by mathematics of finance, the representation of functions of one and several independent variables, their differential and integral calculus and by differential and difference equations. In each case special emphasis is placed on applications and models in economics. The chapter on linear algebra deals with matrices, vectors, determinants and systems of linear equations. This is followed by the representation of structures and algorithms of linear programming. Finally, the reader finds formulas on descriptive statistics (data analysis, ratios, inventory and time series analysis), on probability theory (events, probabilities, random variables and distributions) and on inductive statistics (point and interval estimates, tests). Some important tables complete the work. The present manual arised as a result of many years' teaching for students of economic faculties at the Institutes of Technology of Dresden and Chemnitz, Germany. Moreover, the authors could take advantage of experience and suggestions of numerous colleagues. For critical reading of the manuscript we feel obliged to thank Dipl.-Math. 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 preparation of the book. After successful use by German readers it is a great pleasure for us to present this collection of formulas to the English auditorium. The translation is based on the third, revised German edition. We are greatly obliged to Springer Verlag for giving us the opportunity to publish this book in English. Finally we would like to emphasize that remarks and criticism are always welcome. Chemnitz / Dresden, July 2001 Bernd Luderer Volker Nollau Klaus Vetters

6 Contents Mathematical Symbols and Constants... 1 Notations and symbols... 1 Mathematical constants Sets and Propositions Notion of a set... 3 Relations between sets Operations with sets... 4 Rules for operations with sets... 5 Product sets and mappings Propositional calculus Number Systems and their Arithmetic... 9 Natural, integer, rational, and real numbers Calculation with real numbers Absolute values Factorial and binomial coefficients Equations....'. 13 Inequalities Finite sums Powers and roots Logarithms Complex numbers... " Combinatorial Analysis Permutations Arrangements Combinations Sequences and Series Sequences of numbers Sequences of functions Infinite series Function and power series Taylor series Fourier series

7 VIII Contents Mathematics of Finance Simple interest Compound interest Annuities Dynamic annuities Amortization calculus Price calculus Investment analysis Depreciations Numerical methods for the determination of zeros Functions of one Independent Variable Basic notions Linear functions Quadratic functions Polynomials Fractional rational functions, partial fraction decomposition Exponential functions Logarithmic functions Trigonometric functions Inverse trigonometric functions Hyperbolic functions Area-hyperbolic functions Some economic functions Differential Calculus for Functions of one Variable Limit of a function Continuity..., 60 Differentiation Economic interpretation of the first derivative Rates of change and elasticities..., 66 Mean value theorems Higher derivatives and Taylor expansion Description of function features by means of derivatives Investigation of economic functions, profit maximization Integral Calculus for Functions of one Variable Indefinite integral Definite integral Tables of indefinite integrals Improper integrals Parameter integrals Economic applications of integral calculus

8 Contents IX Differential Equations First-order differential equations Linear differential equations of n-th order First-order linear systems with constant coefficients Difference Equations First-order linear difference equations Economic models Linear second-order difference equations Economic models Linear difference equations of n-th order with constant coefficients 100 Differential Calculus for Functions of Several Variables Basic notions Point sets of the space lr n Limit and continuity Differentiation of functions of several variables Total differential Unconstrained extreme value problems Constrained extreme value problems Least squares method Propagation of errors Economic applications Linear Algebra Vectors Equations of straight lines and planes Matrices Determinants... : Systems of linear equations Gaussian elimination Cramer's rule Exchange method Inverse matrix Eigenvalue problems for matrices Matrix models Linear Programming and Transportation Problem Normal form of a linear programming problem Simplex method Dual simplex method..., 130 Generation of an initial simplex table..., 131 Duality Transportation problem..., 134

9 X Contents Descriptive Statistics Basic notions Univariate data analysis Statistical parameters Bivariate data analysis Ratios Inventory analysis Time series analysis Calculus of Probability Random events and their probabilities Conditional probabilities Random variables and their distributions Discrete distributions Continuous distributions Special continuous distributions Random vectors Inductive Statistics Sample Point estimates Confidence interval estimates Statistical tests Significance tests under normal distribution Tables References Index

10 Mathematical Symbols and Constants n LXi i=l n I1 Xi i=l n! - set of natural numbers - set of natural numbers inclusively zero - set of integer numbers - set of rational numbers - set of real numbers - set of nonnegative real numbers - set of n-tuples of real numbers (n-dimensional vectors) - set of complex numbers - nonnegative number y (square root) such that y2 = X, X ~ 0 - nonnegative number y (n-th root) such yn=x, X ~ 0 - sum of the numbers Xi: Xl + X Xn - product of the numbers Xi: Xl' X2... xn 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 r X 1 - smallest integer y such that y ~ X (rounding up) LxJ - greatest integer y such that y ~ x (rounding down) sgn X Ixl (a, b) [a,b] (a, b] [a, b) ~, > ±, =F def - signum: 1 for X > 0, 0 for X = 0, -1 for X < 0 - absolute value of the real number x: X for x ~ 0 and -x for x < 0 - open interval, i. e. a < x < b - closed interval, i. e. a ~ x ~ b - half-open interval closed from the right, i. e. a < x ~ b - half-open interval open at the right, i. e. a ~ x < b - less or equal; greater or equal - first plus, then minus; first minus, then plus - equality by definition - the left-hand side is defined by the right-hand side

11 2 Mathematical symbols and constants 'V - for all; for any there exists... ; there is (at least one)... P 1\ q - conjunction; p and q p V q - disjunction; p or q p =:::} q - implication; from p it follows q p{::::=}q...,p aem a~m - equivalence; p is equivalent to q - negation; not p n! = n - factorial (~) AcB o rang (A) det A, IAI 8ij lim an n-too lim f(x) x-txo lim f(x) x.j.xo - a is an element of the set M - a is not an element of the set M - binomial coefficient - A subset of B - empty set - norm (of a vector, matrix,... - rank of the matrix A - determinant of the matrix A - Kronecker's symbol: 1 for i = j and 0 for i =f. j - limit of the sequence {an} for n tending to infinity - limit of the function f at the point Xo - limit from the right (right-hand limit) at the point Xo lim f(x) - limit from the left (left-hand limit) at the point Xo xtxo Ue(x*) - c-neighbourhood of the point x* f(x)l~ = [f(x)]~ = feb) - f(a) 7r = e = r 1 = = 180 I' = /1 =