- •Contents
- •Preface
- •Abbreviations
- •Notations
- •1 Introduction
- •2 Sequences; series; finance
- •3 Relations; mappings; functions of a real variable
- •4 Differentiation
- •5 Integration
- •6 Vectors
- •7 Matrices and determinants
- •8 Linear equations and inequalities
- •9 Linear programming
- •10 Eigenvalue problems and quadratic forms
- •11 Functions of several variables
- •12 Differential equations and difference equations
- •Selected solutions
- •Literature
- •Index
Mathematics of Economics and
Business
Knowledge of mathematical methods has become a prerequisite for all students who wish to understand current economic and business literature. This book covers all the major topics required to gain a firm grounding in the subject, such as sequences, series, application in finance, functions, differentiations, differential and difference equations, optimizations with and without constraints, integrations and much more.
Written in an easy and accessible style with precise definitions and theorems, Mathematics of Economics and Business contains exercises and worked examples, as well as economic applications. This book will provide the reader with a comprehensive understanding of the mathematical models and tools used in both economics and business.
Frank Werner is Extraordinary Professor of Mathematics at Otto-von-Guericke University in Magdeburg, Germany.
Yuri N. Sotskov is Professor at the United Institute of Informatics Problems, National Academy of Science of Belarus, Minsk.
Mathematics of Economics
and Business
Frank Werner and Yuri N. Sotskov
First published 2006 by Routledge
2 Park Square, Milton Park, Abingdon, Oxon OX14 4RN
Simultaneously published in the USA and Canada by Routledge
270 Madison Ave, New York, NY10016
Routledge is an imprint of the Taylor & Francis Group
© 2006 Frank Werner and Yuri N. Sotskov
This edition published in the Taylor & Francis e-Library, 2006.
“To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk.”
All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library
Library of Congress Cataloging in Publication Data
A catalog record for this title has been requested
ISBN10: 0–415–33280–X (hbk)
ISBN10: 0–415–33281–8 (pbk)
ISBN13: 9–78–0–415–33280–4 (hbk)
ISBN13: 9–78–0–415–33281–1 (pbk)
Contents
Preface |
ix |
List of abbreviations |
xiii |
List of notations |
xv |
1 Introduction |
1 |
1.1Logic and propositional calculus 1
1.1.1Propositions and their composition 1
1.1.2Universal and existential propositions 7
1.1.3Types of mathematical proof 9
1.2Sets and operations on sets 15
1.2.1Basic definitions 15
1.2.2Operations on sets 16
1.3Combinatorics 26
1.4Real numbers and complex numbers 32
1.4.1Real numbers 32
1.4.2Complex numbers 47
2 Sequences; series; finance |
61 |
2.1Sequences 61
2.1.1Basic definitions 61
2.1.2Limit of a sequence 65
2.2Series 71
2.2.1Partial sums 71
2.2.2Series and convergence of series 73
2.3Finance 80
2.3.1Simple interest and compound interest 80
2.3.2Periodic payments 85
2.3.3Loan repayments, redemption tables 90
2.3.4Investment projects 97
2.3.5Depreciation 101
vi Contents
3 Relations; mappings; functions of a real variable |
107 |
3.1Relations 107
3.2Mappings 110
3.3Functions of a real variable 116
3.3.1Basic notions 117
3.3.2Properties of functions 121
3.3.3Elementary types of functions 126
4 Differentiation |
148 |
4.1Limit and continuity 148
4.1.1Limit of a function 148
4.1.2Continuity of a function 151
4.2Difference quotient and the derivative 155
4.3Derivatives of elementary functions; differentiation rules 158
4.4Differential; rate of change and elasticity 164
4.5Graphing functions 168
4.5.1Monotonicity 168
4.5.2Extreme points 169
4.5.3Convexity and concavity 175
4.5.4Limits 178
4.5.5Further examples 181
4.6Mean-value theorem 184
4.7Taylor polynomials 186
4.8Approximate determination of zeroes 189
5 Integration |
197 |
5.1Indefinite integrals 197
5.2Integration formulas and methods 198
5.2.1Basic indefinite integrals and rules 198
5.2.2Integration by substitution 200
5.2.3Integration by parts 204
5.3The definite integral 209
5.4Approximation of definite integrals 215
5.5Improper integrals 219
5.5.1Infinite limits of integration 219
5.5.2Unbounded integrands 220
5.6Some applications of integration 222
5.6.1Present value of a continuous future income flow 222
5.6.2Lorenz curves 224
5.6.3Consumer and producer surplus 225
6 Vectors |
230 |
6.1Preliminaries 230
6.2Operations on vectors 233
|
Contents |
vii |
6.3 |
Linear dependence and independence 240 |
|
6.4 |
Vector spaces 244 |
|
7 Matrices and determinants |
253 |
7.1Matrices 253
7.2Matrix operations 258
7.3Determinants 263
7.4Linear mappings 271
7.5The inverse matrix 273
7.6An economic application: input–output model 277
8 Linear equations and inequalities |
287 |
8.1Systems of linear equations 287
8.1.1Preliminaries 287
8.1.2Existence and uniqueness of a solution 290
8.1.3Elementary transformation; solution procedures 292
8.1.4General solution 302
8.1.5Matrix inversion 306
8.2Systems of linear inequalities 308
8.2.1Preliminaries 308
8.2.2Properties of feasible solutions 309
8.2.3A solution procedure 315
9 Linear programming |
328 |
9.1Preliminaries 328
9.2Graphical solution 330
9.3Properties of a linear programming problem; standard form 334
9.4Simplex algorithm 339
9.5Two-phase simplex algorithm 350
9.6Duality; complementary slackness 357
9.7Dual simplex algorithm 363
10 |
Eigenvalue problems and quadratic forms |
368 |
|
|
10.1 |
Eigenvalues and eigenvectors 368 |
|
|
10.2 |
Quadratic forms and their sign 376 |
|
11 |
Functions of several variables |
383 |
11.1Preliminaries 383
11.2Partial derivatives; gradient 387
11.3Total differential 394
11.4Generalized chain rule; directional derivatives 397
11.5Partial rate of change and elasticity; homogeneous functions 402
11.6Implicit functions 405
viiiContents
11.7Unconstrained optimization 409
11.7.1Optimality conditions 409
11.7.2Method of least squares 419
11.7.3Extreme points of implicit functions 423
11.8Constrained optimization 424
11.8.1Local optimality conditions 424
11.8.2Global optimality conditions 434
11.9Double integrals 436
12 Differential equations and difference equations |
444 |
12.1Differential equations of the first order 445
12.1.1Graphical solution 445
12.1.2Separable differential equations 447
12.2Linear differential equations of order n 451
12.2.1Properties of solutions 451
12.2.2Differential equations with constant coefficients 454
12.3Systems of linear differential equations of the first order 461
12.4Linear difference equations 472
12.4.1Definitions and properties of solutions 472
12.4.2Linear difference equations of the first order 474
12.4.3Linear difference equations of the second order 478
Selected solutions |
486 |
Literature |
511 |
Index |
513 |