Preface

Today, a firm understanding of mathematics is essential for any serious student of economics. Students of economics need nowadays several important mathematical tools. These include calculus for functions of one or several variables as well as a basic understanding of optimization with and without constraints, e.g. linear programming plays an important role in optimizing production programs. Linear algebra is used in economic theory and econometrics. Students in other areas of economics can benefit for instance from some knowledge about differential and difference equations or mathematical problems arising in finance. The more complex economics becomes, the more deep mathematics is required and used. Today economists consider mathematics as the most important tool of economics and business. This book is not a book on mathematical economics, but a book on higher-level mathematics for economists.

Experience shows that students who enter a university and specialize in economics vary enormously in the range of their mathematical skills and aptitudes. Since mathematics is often a requirement for specialist studies in economics, we felt a need to provide as much elementary material as possible in order to give those students with weaker mathematical backgrounds the chance to get started. Using this book may depend on the skills of readers and their purposes. The book starts with very basic mathematical principles. Therefore, we have included some material that covers several topics of mathematics in school (e.g. fractions, powers, roots and logarithms in Chapter 1 or functions of a real variable in Chapter 3). So the reader can judge whether or not he (she) is sufficiently familiar with mathematics to be able to skip some of the sections or chapters.

Studying mathematics is very difficult for most students of economics and business. However, nowadays it is indeed necessary to know a lot of results of higher mathematics to understand the current economic literature and to use modern economic tools in practical economics and business. With this in mind, we wrote the book as simply as possible. On the other hand, we have presented the mathematical results strongly correct and complete, as is necessary in mathematics. The material is appropriately ordered according to mathematical requirements (while courses, e.g. in macroeconomics, often start with advanced topics such as constrained optimization for functions of several variables). On the one hand, previous results are used by later results in the text. On the other hand, current results in a chapter make it clear why previous results were included in the book.

The book is written for non-mathematicians (or rather, for those people who only want to use mathematical tools in their practice). It intends to support students in learning the basic mathematical methods that have become indispensable for a proper understanding of the current economic literature. Therefore, the book contains a lot of worked examples and economic applications. It also contains many illustrations and figures to simplify the

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mathematical techniques used and show how mathematical results may be used in economics and business. Some of these examples have been taken from former examinations (at the Otto-von-Guericke University of Magdeburg), and many of the exercises given at the end of each chapter have been used in the tutorials for a long time. In this book, we do not show how the mathematical results have been obtained and proved, but we show how they may be used in real-life economics and business. Therefore, proofs of theorems have been skipped (with a few exceptions) so that the volume of the book does not substantially exceed 500 pages, but in spite of the relatively short length the book includes the main mathematical subjects useful for practical economics and an efficient business.

The book should serve not only as a textbook for a course on mathematical methods for students, but also as a reference book for business people who need to use higher-level mathematics to increase profits. (Of course, one will not increase profit by solving e.g. a differential equation, but one can understand why somebody has increased profits after modelling a real process and finding a solution for it.) One of the purposes of this book is to introduce the reader to the most important mathematical methods used in current economic literature. We also provide an introduction to the close relationship between mathematical methods and problems arising in the economy. However, we have included only such economic applications as do not require an advanced knowledge of economic disciplines, since mathematics is usually taught in the first year of studies at university.

The reader needs only knowledge of elementary mathematics from secondary school to understand and use the results of the book, i.e. the content is self-sufficient for understanding. For a deeper understanding of higher mathematics used in economics, we also suggest a small selection of German and English textbooks and lecture materials listed in the literature section at the end of the book. Some of these books have been written at a comparable mathematical level (e.g. Opitz, Mathematik; Simon and Blume, Mathematics for Economists; Sydsaeter and Hammond, Mathematics for Economic Analysis) while others are more elementary in style (e.g. Misrahi and Sullivan, Mathematics and Finite Mathematics; Ohse, Mathematik für Wirtschaftswissenschaftler; Rosser, Basic Mathematics for Economists). The booklets (Schulz, Mathematik für wirtschaftswissenchaftliche Studiengänge; Werner, Mathematics for Students of Economics and Management) contain most important definitions, theorems of a one-year lecture course in mathematics for economists in a compact form and they sketch some basic algorithms taught in the mathematics classes for economists at the Otto-von-Guericke University of Magdeburg during recent decades. Bronstein and Semandjajew, Taschenbuch der Mathematik, and Eichholz and Vilkner, Taschenbuch der Wirtschaftsmathematik, are well-known handbooks of mathematics for students. Varian, Intermediate Microeconomics, is a standard textbook of intermediate microeconomics, where various economic applications of mathematics can be found.

The book is based on a two-semester course with four hours of lectures per week which the first author has given at the Otto-von-Guericke University of Magdeburg within the last ten years. The authors are indebted to many people in the writing of the book. First of all, the authors would like to thank Dr Iris Paasche, who was responsible for the tutorials from the beginning of this course in Magdeburg. She contributed many suggestions for including exercises and for improvements of the contents and, last but not least, she prepared the answers to the exercises. Moreover, the authors are grateful to Dr Günther Schulz for his support and valuable suggestions which were based on his wealth of experience in teaching students of economics and management at the Otto-von-Guericke University of Magdeburg for more than twenty years. The authors are grateful to both colleagues for their contributions.

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The authors also thank Ms Natalja Leshchenko of the United Institute of Informatics Problems of the National Academy of Sciences of Belarus for reading the whole manuscript (and carefully checking the examples) and Mr Georgij Andreev of the same institute for preparing a substantial number of the figures. Moreover, many students of the International Study Programme of Economics and Management at the Otto-von-Guericke University of Magdeburg have read single chapters and contributed useful suggestions, particularly the students from the course starting in October 2002. In addition, the authors would like to express their gratitude to the former Ph.D. students Dr Nadezhda Sotskova and Dr Volker Lauff, who carefully prepared in the early stages a part of the book formerly used as printed manuscript in LATEX and who made a lot of constructive suggestions for improvements.

Although both authors have taught in English at universities for many years and during that time have published more than 100 research papers in English, we are nevertheless not native speakers. So we apologize for all the linguistic weaknesses (and hope there are not too many). Of course, for all remaining mathematical and stylistic mistakes we take full responsibility, and we will be grateful for any further comments and suggestions for improvements by readers for inclusion in future editions (e-mail address for correspondence: frank.werner@mathematik.uni-magdeburg.de). Furthermore, we are grateful to Routledge for their pleasant cooperation during the preparation of the book. The authors wish all readers success in studying mathematics.

We dedicate the book to our parents Hannelore Werner, Willi Werner, Maja N. Sotskova and Nazar F. Sotskov.

F.W.

Y.N.S.