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Mathematics of Economics and


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)




List of abbreviations


List of notations


1 Introduction


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


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


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


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


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


6.1Preliminaries 230

6.2Operations on vectors 233





Linear dependence and independence 240



Vector spaces 244


7 Matrices and determinants


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


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


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


Eigenvalue problems and quadratic forms




Eigenvalues and eigenvectors 368




Quadratic forms and their sign 376



Functions of several variables


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


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


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






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