The PREFACE of the book is given here. The complete Ebook can be now FREELY DOWNLOADED (open archive), e.g. from the publisher's website (Taylor & Francis Group): https://www.taylorfrancis.com/books/9781134319312 . Now the Ebook can also be FREELY DOWNLOADED AT ALL KINDLE STORES OF AMAZON (in all countries), e.g. under : https://www.amazon.de/Mathematics-Economics-Business-English-Werner-ebook/dp/B000Q7ZFKW/ref=tmm_kin_swatch_0?_encoding=UTF8&qid=&sr= Since it is still the first edition from 2006 without any changes, a list of identified misprints can be found at my homepage under : https://www.math.uni-magdeburg.de/~werner/misprints.pdf . I hope this list is rather complete since the book was intensively used in the tutorials of my lectures on this subject since the publication of the book in 2006. The book is based on a two-semester course which the first author taught at the Otto-von-Guericke University Magdeburg for students of the Faculty of Economics and Management since 1997. Of course, any further feedback is appreciated.

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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 opti-

mization with and without constraints, e.g. linear programming plays an important role in

optimizing production programs. Linear algebra is used in economic theory and economet-

rics. 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. To-

day 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, have

an enormous range of 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 might 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 this book as simple as possible. On the other hand,

we presented the mathematical results strongly correct and complete as it 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 clear why

previous results were included into the book.

The book is written for non-mathematicians (or better to say, for those people who only want

to use mathematical tools in their practice). It intends to support the 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

used mathematical techniques 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

1

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 exceed 500 pages, but in

spite of the rather small volume, 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 their profit (of course, one will not increase the profit by solving e.g.

a differential equation, but one can understand why somebody has increased the profit 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 economy. However, we have included only such economic

applications which do not required an advanced knowledge in economic disciplines since

mathematics is usually taught in the first year of the studies at the university.

The reader of this book needs only knowledge of elementary mathematics from the secondary

school to understand and use the results of the book, i.e. the material of the book is self-

sufficient for understanding. For a deeper understanding of higher mathematics used in

economics, we also propose a small selection of some German and English textbooks and

lecture materials [1 - 20] listed in the literature section at the end of the book. Some of

these books have been written in a comparable mathematical level (e.g. references [12,

16, 17]) while others have been written in a more elementary style (e.g. [8, 9, 11, 14]).

The booklets [18, 20] contain most important definitions, theorems of a one-year lecture 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 the last decades. References [2] and [5] are well-known handbooks of mathematics

for students. Reference [19] is a standard textbook for 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 with many suggestions for

including exercises and for improvements of the contents of the book and last but not least,

she prepared the answers to the exercises. Moreover, the authors are grateful to Dr. G¨unther

Schulz for his support and valuable suggestions which were based on his rich experience with

teaching students of economics and management at the Otto-von-Guericke-University of

Magdeburg for more than 20 years. The authors are grateful to both colleagues for their

contributions.

The authors also thank Ms. Natalja Leshchenko from the United Institute of Informat-

ics Problems of the National Academy of Sciences of Belarus for proofreading the whole

manuscript (and checking carefully the examples) and Mr. Georgij Andreev from the same

institute for preparing a substantial part of the figures. Moreover, many students of the inter-

national study program of Economics and Management at the Otto-von-Guericke-University

2

Magdeburg have read single chapters and contributed with 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 early stages a part of the book formerly used as printed

manuscript in L

A

T

E

X and who gave a lot of constructive suggestions for improvements.

Although both authors have to teach in English at the universities for many years and

although we have meanwhile more than 100 research papers written in English, we are nev-

ertheless not native speakers. So we apologize for all linguistic weaknesses (and hope that

there are not too many). Of course, for all remaining mathematical and stylistic mistakes

we overtake responsibility, and we will be grateful for all further comments and suggestions

for improvements by the readership to be included into future editions (email address for

correspondence: frank.werner@mathematik.uni-magdeburg.de). Furthermore, we are grate-

ful to Routledge for the pleasant cooperation during the preparation of this textbook. The

authors wish all readers success in studying mathematics.

We dedicate this book to our parents Hannelore Werner, Willi Werner, Maja N. Sotskova

and Nazar F. Sotskov.

Frank Werner and Yuri Sotskov

Magdeburg and Minsk, January 2005

3

... (see Peasnell, 1982, p. 108). The above relation coincides with the recursion formula used in financial and actuarial mathematics for computing the balance (residual debt) in a loan contract (Promislow, 2006;Werner and Sotskov, 2006;Kellison, 2009), where b 0 is the amount borrowed, a t b t−1 represents interest and f t is the installment. This fact enables one to interpret f as a loan contract whereby shareholders lend the firm the amount b 0 and receive the installment f t at time t. ...

  • Carlo Alberto Magni Carlo Alberto Magni

This paper presents a theoretical framework for valuation, investment decisions, and performance measurement based on a nonstandard theory of residual income. It is derived from the notion of "unrecovered" capital, which is here named "lost" capital because it represents the capital foregone by the investors. Its theoretical strength and meaningfulness is shown by deriving it from four main perspectives: financial, microeconomic, axiomatic, accounting. Implications for asset valuation, capital budgeting and performance measurement are investigated. In particular: an aggregation property is shown, which makes the simple average residual income play a major role in valuation; a dual relation between the standard theory and the lost-capital theory is proved, clarifying the way periodic performance is computed in the two paradigms and the rationale for measuring performance with either paradigm; the average accounting rate of return is shown to be more reliable than the internal rate of return as a capital budgeting criterion, and maximization of the average residual income is shown to be equivalent to maximization of Net Present Value (NPV). Two metrics are also presented: one enjoys the nice property of robust goal congruence irrespective of the sign of the cash flows; the other one enjoys periodic consistency in the sense of Egginton (1995). The results obtained suggest that this theory might prove useful for real-life applications in firm valuation, capital budgeting decision-making, ex ante and ex post performance measurement, incentive compensation. A numerical example illustrates the implementation of the paradigm to the EVA model and the Edwards-Bell-Ohlson model.

... Let frictional unemployment resulting from supply > demand in preliminary diagram be μ Then, μ = μ -a L D – L S ………… (2) L D Interpretation: (i) At excess demand = 0, unemployment =μ (ii) At excess demand > 0, unemployment <μ (iii) At excess demand < 0, unemployment >μ We can combine equations (1) and (2) above. ϖ/b = 1/a ( μ -μ) (Ekanem 2000) ϖ = b/a μ -b/a μ ……………………… (3) where a, b, and μ are constants Hence ϖ = α -âμ, where: b/a μ = α(intersect), b/a = â (slope). (Ojameruaye et al. 2001; Iyoha et al. 2002). ...

  • Igberaese Francis Ilenloa Igberaese Francis Ilenloa
  • Raphael Igbinosa Adeghe

Algebraic models are more endowed to economists than the traditional theories and diagrams from which they are derived, being more precise and clearer. Using algebra, there exist equilibrium wage and unemployment in labour market, through the age long A.W. Philips curve. Then, the question of sensitivity is the extent of unemployment that must be tolerated to ensure price stability. The objective of the paper is to show that algebra is effective in determining equilibrium wage and unemployment, and it recommends among others, a policy of price control, so that the negative effects unemployment and inflation have on wage can be neutralized in the different periods.

  • Chenmao Wu
  • Zeren Wang

Aiming at the problem that existing fuzzy clustering with quadratic polynomial prototype cannot effectively deal with the uneven illumination images, a robust kernel-based fuzzy local neighborhood clustering-related algorithm based on quadratic polynomial is proposed. In this paper, the samples and cluster centers of fuzzy clustering with quadratic polynomial prototype are firstly mapped into the high-dimensional feature space. Furthermore, an optimization model of kernel-based fuzzy local neighborhood information clustering with quadratic polynomial prototype is constructed by the induced kernel distance metric. Then the iterative algorithm of membership and coefficient matrix of the quadratic polynomial prototype is proposed by Lagrange multiplier method and numerical algebra. In the end, the convergence of the proposed is strictly proved by using the combination of Zangwill theorem and bordered Hessian matrix. Experimental results show that the proposed algorithm outperforms existing fuzzy clustering with quadratic polynomial prototype, and it can effectively resolve the segmentation problem of inhomogeneous image and has very high accuracy in segmenting MRI brain images with a certain intensity of noise or bias field.

Wind turbine tower (WTT) is a vital part of wind energy development, which gains huge momentum globally over the past decades. In this study, an ingenious solution is provided with the proposed performance-based design framework for the WTTs. To represent the performance of the wind projects, an intuitive variable, i.e., the expected lifetime profit, is considered as the performance index. The fragility functions were used to probabilistically incorporate the uncertainties from the characteristics of natural hazards, the structural capacity estimations, the variations in geometric irregularities of towers, etc. The lifetime profit were roughly estimated based on the structural performance associated with the existing economic data collected from the current WTT projects. Furthermore, with a hypothetical case study, this paper presents a solution to improve the structural performance using the proposed methodology. The influences of the initial costs as well as the operation and maintenance costs on the profit were also investigated. The final profit and the profit-operating time curves were provided, which are valuable references to the designers, the owners and the insurers of the wind farm projects.

  • Shou-Jen Chang-Chien
  • Yessica Nataliani
  • Miin-Shen Yang Miin-Shen Yang

Partitional clustering is the most used in cluster analysis. In partitional clustering, hard c-means (HCM) (or called k-means) and fuzzy c-means (FCM) are the most known clustering algorithms. However, these HCM and FCM algorithms work worse for data sets in a noisy environment and get inaccuracy when the data set has different shape clusters. For solving these drawbacks in HCM and FCM, Wu and Yang (Pattern Recognit 35:2267–2278, 2002) proposed the alternative c-means clustering with an exponential-type distance that extends HCM and FCM into alternative HCM (AHCM) and alternative FCM (AFCM). In this paper, we construct a more generalization of AHCM and AFCM with Gaussian-kernel c-means clustering, called GK-HCM and GK-FCM. For theoretical behaviors of GK-FCM, we analyze the bordered Hessian matrix and then give the theoretical properties of the GK-FCM algorithm. Some numerical and real data sets are used to compare the proposed GK-HCM and GK-FCM with AHCM and AFCM methods. Experimental results and comparisons actually demonstrate these good aspects of the proposed GK-HCM and GK-FCM algorithms with its effectiveness and usefulness. Finally, we apply the GK-FCM algorithm to MRI segmentation.

  • Miin-Shen Yang Miin-Shen Yang
  • Yessica Nataliani

Fuzzy clustering algorithms generally treat data points with feature components under equal importance. However, there are various data sets with irrelevant features involved in clustering process that may cause bad performance for fuzzy clustering algorithms. That is, different feature components should take different importance. In this paper, we present a novel method for improving fuzzy clustering algorithms that can automatically compute individual feature weight, and simultaneously reduce these irrelevant feature components. In fuzzy clustering, the fuzzy c-means (FCM) algorithm is the best known. We first consider the FCM objective function with feature-weighted entropy, and construct a learning schema for parameters, and then reduce these irrelevant feature components. We call it a feature-reduction FCM (FRFCM). During FRFCM processes, a new procedure for eliminating irrelevant feature(s) with small weight(s) is created for feature reduction. The computational complexity of FRFCM is also analyzed. Some numerical and real data sets are used to compare FRFCM with various feature-weighted FCM methods in the literature. Experimental results and comparisons actually demonstrate these good aspects of FRFCM with its effectiveness and usefulness in practice.

  • Dan Greenwood Dan Greenwood

In the longstanding debate in political economy about the feasibility of socialism, the Austrian School of Economists have argued that markets are an indispensable means of evaluating goods, hence a prerequisite for productive efficiency. Socialist models for non-market economic calculation have been strongly influenced by the equilibrium model of neoclassical economics. The Austrians contend that these models overlook the essence of the calculation problem by assuming the availability of knowledge that can be acquired only through the market process itself. But the debate in political economy has not yet considered the recent emergence of agent-based systems and their applications to resource allocation problems. Agent-based simulations of market exchange offer a promising approach to fulfilling the dynamic functions of knowledge encapsulation and discovery that the Austrians show to be performed by markets. Further research is needed in order to develop an agent-based approach to the calculation problem, as it is formulated by the Austrians. Given that the macro-level objectives of agent-based systems can be easily engineered, they could even become a desirable alternative to the real markets that the Austrians favour.

We analyze annual revenues and earnings data for the 500 largest-revenue U.S. companies during the period 1954–2007. We find that mean year profits are proportional to mean year revenues, exception made for few anomalous years, from which we postulate a linear relation between company expected mean profit and revenue. Mean annual revenues are used to scale both company profits and revenues. Annual profit fluctuations are obtained as difference between actual annual profit and its expected mean value, scaled by a power of the revenue to get a stationary behavior as a function of revenue. We find that profit fluctuations are broadly distributed having approximate power law tails with a Lévy-type exponent α1.7, from which we derive the associated break-even probability distribution. The predictions are compared with empirical data.

  • Cornelia Codruła
  • Ilie Dura
  • Imola Mitran
  • Driga

Equilibrium can be described as an ideal market situation, in which the interests of economic agents are best served, and the resources are allocated and used based on certain criteria and at a normal level of efficiency for every stage. Market equilibrium is derived from a problem specific to non-cooperative game theory with zero-sum and two players. In order to analyze market equilibrium, the main issue is to determine the equilibrium price in different situations: knowing the functions of supply and demand, knowing the elasticity of these functions, taking into account the existence of income tax etc. The results presented in this paper are broad and dwell on the well-known results for Cobweb and Laffer models. An important advantage of these findings is that they are convenient in terms of calculations and they have interesting economic interpretations. Key-Words: equilibrium price, supply-demand relation, the elasticity of the supply function, the elasticity of the demand function, the equation of price dynamics, the dynamic index of prices.

This paper presents an axiomatization of residual income, aka excess profit, and illustrates how it may univocally engenders fixed-income or variable-income assets. In the first part it is shown that, depending on the relations between excess profit and the investor's excess wealth, a well-specified theory of residual income is generated: one is the standard theory, which historically traces back to Hamilton (1777) and Marshall (1890) and is a deep-rooted notion in economic theory, finance, and accounting. Another one is the systemic value added or lost-capital paradigm: introduced in Magni (2000, 2003), the theory is enfolded in Keynes's (1936) notion of user cost and is naturally generated by an arbitrage-theory perspective. In the second part, the paper reverts the usual analysis: instead of computing residual incomes profits from a pattern of cash flows, residual incomes are fixed first to derive vectors of cash flows. It is shown that variable- or fixed-income assets may be constructed on the basis of either theory starting from pre-determined growth rates for excess profit. In particular, zero-coupon bonds and coupon bonds traded in a capital market are shown to be deducted as equilibrium vectors of residual-income-based assets.

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