Mathematics and methodology for economics 2016

For a basic course Chaps.1 sets, vectors, trigonometric functions, complexnumbers, 3 mappings and functions, 4 vectors, matrices, systems of linearequations,6functions, limits, derivatio

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Springer Texts in Business and Economics

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Wolfgang Eichhorn • Winfried Gleißner

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ISSN 2192-4333 ISSN 2192-4341 (electronic)

Springer Texts in Business and Economics

ISBN 978-3-319-23352-9 ISBN 978-3-319-23353-6 (eBook)

DOI 10.1007/978-3-319-23353-6

Library of Congress Control Number: 2016932103

Springer Cham Heidelberg New York Dordrecht London

This work is subject to copyright All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, 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 regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

Springer International Publishing AG Switzerland is part of Springer Science+Business Media

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This book about mathematics and methodology for economics is the result of thelifelong teaching experience of the authors It is written for university students

as well as for students of a university of applied sciences It is completely contained and does not assume any previous knowledge of high school mathematics

self-At the end of all chapters and sections, there are exercises such that the readercan test how familiar she or he is with the material of the preceding stuff Aftereach set of exercises, the answers are given to encourage the reader to tackle theproblems

The idea to write such a book was born in 1990 during an internationalmeeting on functional equations which took place at the University of Graz,Austria At this meeting a lot of fascinating applications of functional equations

to solve mathematically formulated economic problems inspired János Aczél,Distinguished Professor of Mathematics, University of Waterloo, Ontario, Canada:

He proposed to one of us (W.E.) to start such an adventure in a form of atextbook for beginners Since then he supported the tentative steps into this direction

by a great wealth of brilliant scientific advices Later on he became for both

of us the lodestar for our endeavour Dear János, we owe you a great debt ofgratitude

For a basic course Chaps.1 (sets, vectors, trigonometric functions, complexnumbers), 3 (mappings and functions), 4 (vectors, matrices, systems of linearequations),6(functions, limits, derivations),7(important nonlinear functions), and

10 (integration) are sufficient If a later course will discuss discrete models ofeconomics, Chap.12(difference equations) should be covered, too For continuousmodels, Chap.11(differential equations) is necessary (However, we decided not to

go very far into details.)

Chapter 2 gives an introduction to linear optimisation and game theory usingproduction systems These ideas are continued in Chaps.5and9, which discussesthe notion of a Nash Equilibrium Chapter 8 deals with nonlinear optimisa-tion

Chapter13, as the conclusion, reflects methodologically most of all that what weoptimistically offered in Chaps.1,2,3,4,5,6,7,8,9,10,11and12

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Many thanks go to Thomas Schlink for typing most of the manuscript in LATEXvery conscientiously and to Dr Roland Peyrer for his inspiring drawings, whichwere transformed to PSTricks, an additional package for graphics in Latex.

Summer, 2015

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1 Sets, Numbers and Vectors 1

1.1 Introduction 1

1.2 Basics 1

1.2.1 Exercises 7

1.2.2 Answers 7

1.3 Subsets, Operations Between Sets 8

1.3.1 Exercises 11

1.3.2 Answers 12

1.4 Cartesian Products of Sets,Rn , Vectors 12

1.4.1 Exercises 18

1.4.2 Answers 19

1.5 Operations for Vectors, Linear Dependence and Independence 19

1.5.1 Sums, Differences, Linear Combinations of Vectors 19

1.5.2 Linear Dependence, Independence 21

1.5.3 Inner Product 24

1.5.4 Exercises 25

1.5.5 Answers 26

1.6 Geometric Interpretations Distance Orthogonal Vectors 26

1.6.1 Exercises 30

1.6.2 Answers 30

1.7 Complex Numbers; the Cosine, Sine, Tangent and Cotangent 31

1.7.1 Multiplication of Complex Numbers 31

1.7.2 Trigonometric Form of Complex Numbers; Sine, Cosine 34

1.7.3 Division of Complex Numbers; Equations 40

1.7.4 Tangent, Cotangent 42

1.7.5 Exercises 43

1.7.6 Answers 43

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2 Production Systems Production Processes, Technologies,

Efficiency, Optimisation 45

2.1 Introduction 45

2.2 Basics 46

2.2.1 Exercises 49

2.2.2 Answers 49

2.3 Linear Production Models, Linear Optimisation Problems 49

2.3.1 Exercises 52

2.3.2 Answers 53

2.4 Simple Approaches to Linear Optimisation Problems 53

2.4.1 Exercises 59

2.4.2 Answers 60

3 Mappings, Functions 61

3.1 Introduction 61

3.2 Basics Domains, Ranges, Images (Codomains) Mappings (Binary Relations), Functions, Injections, Surjections, Bijections Graphs 63

3.2.1 Exercises 72

3.2.2 Answers 72

3.3 Functions of n Variables, n-Dimensional Intervals, Composition of Functions 73

3.3.1 Exercises 77

3.3.2 Answers 78

3.4 Monotonic and Linearly Homogeneous Functions Maxima and Minima 78

3.4.1 Exercises 84

3.4.2 Answers 85

3.5 Convex (Concave) Functions Convex Sets 85

3.5.1 Exercises 92

3.5.2 Answers 92

3.6 Quasi-convex Functions 93

3.6.1 Exercises 99

3.6.2 Answers 100

3.7 Functions in the “Statistical Theory” of Price Indices 100

3.7.1 Exercises 103

3.7.2 Answers 104

4 Affine and Linear Functions and Transformations (Matrices), Linear Economic Models, Systems of Linear Equations and Inequalities 105

4.1 Introduction 105

4.2 Proportionality, Linear and Affine Functions Additivity, Linear Homogeneity, Linearity 107

4.2.1 Exercises 112

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4.3 Additivity, Linear Homogeneity, Linearity

of Vector-Vector Functions, Matrices 113

4.3.1 Exercises 117

4.3.2 Answers 117

4.4 Matrix Algebra 118

4.4.1 Exercises 124

4.4.2 Answers 125

4.5 Linear Economic Models: Leontief, von Neumann 126

4.5.1 Exercises 133

4.5.2 Answers 134

4.6 Systems of Linear Equations Solution by Elimination Rank Necessary and Sufficient Conditions 135

4.6.1 Exercises 154

4.6.2 Answers 155

4.7 Determinant, Cramer’s Rule, Inverse Matrix 156

4.7.1 Exercises 164

4.7.2 Answers 165

4.8 Applications of Functions of Vector Variables: Aggregation in Economics 165

4.8.1 Exercises 174

4.8.2 Answers 176

5 Linear Optimisation, Duality: Zero-Sum Games 177

5.2 Linear Optimisation Problems 179

5.2.1 Exercises 192

5.2.2 Answers 192

5.3 Duality 194

5.3.1 Exercises 200

5.3.2 Answers 201

5.4 Two-Person Zero-Sum Games 201

5.4.1 Exercises 207

5.4.2 Answers 207

6 Functions, Their Limits and Their Derivatives 209

6.2 Limits, Infinity as Limit, Limit at Infinity, Sequences: Trigonometric Functions, Polynomials, Rational Functions 211

6.2.1 Exercises 220

6.2.2 Answers 221

6.3 Continuity, Sectional Continuity, Left and Right Limits 221

6.3.1 Exercises 226

6.3.2 Answers 227

6.4 Derivative, Derivation 227

6.4.1 Exercises 233

6.4.2 Answers 234

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6.5 Rules Which Make Derivation Easier 234

6.5.1 Exercises 242

6.5.2 Answers 243

6.6 An Application: Price-Elasticity of Demand 243

6.6.1 Exercises 245

6.6.2 Answers 245

6.7 Laws of the Mean, Taylor Series, Bernoulli–L’Hospital Rule 245

6.7.1 Exercises 257

6.7.2 Answers 258

6.8 Monotonicity, Local Maxima, Minima and Convexity of Differentiable Functions 258

6.8.1 Exercises 262

6.8.2 Answers 263

6.9 “Cobweb” Situations in Economics: Points of Intersection of Graphs and Zeros of Functions 263

6.9.1 Exercises 269

6.9.2 Answers 270

6.10 Newton’s Algorithm: Differentials (Linear Approximation) 270

6.10.1 Exercises 275

6.10.2 Answers 275

6.11 Linear Approximation: Differentials and Derivatives of Vector-Vector Functions—Partial Derivatives of Higher Orders 277

6.11.1 Excercises 286

6.11.2 Answers 287

6.12 Chain Rule: Euler’s Partial Differential Equation for Homogeneous Functions 288

6.12.2 Answers 294

6.13 Implicit Functions 294

6.13.2 Answers 299

7 Nonlinear Functions of Interest to Economics Systems of Nonlinear Equations 301

7.2 Exponential and Logarithm Functions Powers with Arbitrary Real Exponents Conditions for Convexity and Applications 302

7.2.1 Exercises 318

7.2.2 Answers 318

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7.3 Applications: “Discrete” and “Continuous”

Compounding, “Effective Interest Rate”, Doubling

Time, Discounting 319

7.3.1 Exercises 324

7.3.2 Answers 324

7.4 Some Interesting Scalar Valued Nonlinear Functions in Several Variables Homothetic Functions 325

7.4.1 Exercises 339

7.4.2 Answers 340

7.5 Fundamental Notions in Production Theory Production Functions Elasticity of Substitution 341

7.5.1 Exercises 356

7.5.2 Answers 356

7.6 Nonlinear Vector-Valued Functions, Systems of Equations Banach’s Fixed Point Theorem 357

7.6.1 Exercises 370

7.6.2 Answers 371

8 Nonlinear Optimisation with One or Several Objectives: Kuhn–Tucker Conditions 373

8.2 Convexity of Differentiable Functions of Several Variables, Matrix–Conditions for Convexity, Eigenvalues, Eigenvectors 375

8.2.1 Exercises 388

8.2.2 Answers 389

8.3 Quadratic Approximation Maxima and Minima of Functions of Several Variables 389

8.3.1 Exercises 405

8.3.2 Answers 406

8.4 Bellman’s Principle of Dynamic Optimisation; Application to a Maximum Problem 407

8.4.1 Exercises 413

8.4.2 Answers 414

8.5 Linear Regression; the “Method of Least Squares” 414

8.5.1 Exercises 420

8.5.2 Answers 421

8.6 Extrema of an Objective Function Under Equality Constraints 422

8.6.1 Exercises 431

8.6.2 Answers 432

8.7 Extrema of an Objective Function Depending on Parameters Envelope Theorems LeChatelier Principle 435

8.7.1 Exercises 447

8.7.2 Answers 448

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8.8 Extrema of an Objective Function Under Inequality

Constraints 449

8.8.1 Exercises 463

8.8.2 Answers 464

8.9 The Kuhn–Tucker Conditions 465

8.9.1 Exercises 468

8.9.2 Answers 468

8.10 Optimisation with Several Objective Functions 470

8.10.2 Answers 474

9 Set Valued Functions: Equilibria—Games 477

9.2 Set Valued Functions (Correspondences): Shephard’s Axioms 479

9.2.1 Exercises 483

9.2.2 Answers 484

9.3 Competitive Equilibria: Kakutani’s Fixed Point Theorem 485

9.3.1 Exercises 492

9.3.2 Answers 493

9.4 Applications in the Theory of Games: Nash Equilibrium 493

9.4.1 Exercises 505

9.4.2 Answers 506

10 Integrals 509

10.1 Introduction: Definite Integral 509

10.2 Properties of Definite Integrals 512

10.2.2 Answers 513

10.3 Indefinite Integrals (Antiderivatives) 513

10.3.2 Answers 518

10.4 Methods to Calculate Integrals 518

10.4.2 Answers 523

10.5 An Application: Calculating Present Values 524

10.5.2 Answers 529

10.6 Improper Integrals (Integrals on Infinite Intervals or on Intervals Containing Points Where the Function Tends to Infinity) 530

10.6.2 Answers 533

11 Differential Equations 535

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11.2 Basics 539

11.2.2 Answers 542

11.3 Linear Differential Equations of First Order 542

11.3.2 Answers 549

11.4 An Application: Saturation of Markets: “Logistic Growth” 549

11.5 Linear Second Order Differential Equations with Constant Coefficients 552

11.5.2 Answers 559

11.6 The Predator-Prey Model 559

11.6.1 Exercise 562

11.6.2 Answer 563

12 Difference Equations 565

12.1.2 Answers 571

12.2 Linear Difference Equations 571

12.2.2 Answers 582

12.3 Some Applications of Linear Difference Equations 582

12.3.1 The Growth Model of Roy Forbes Harrod (1900–1978) 582

12.3.2 Settlement of Bond Issues 583

12.3.3 Distribution of Wealth 585

12.3.4 The Multi-sector Multiplier Model 586

12.4 Systems of Linear Difference Equations 586

12.5 Nonlinear Difference Equations, Chaos 592

12.5.2 Answers 596

13 Methodology: Models and Theories in Economics 597

13.2 Models in Engineering, Natural Sciences and Mathematics 598

13.3 Models in Economics 600

13.4 Systems of Assumptions 607

13.5 Theories in the Sciences, in Particular in Economics 609

13.6 Why Construct Models and Theories? Types of Models and Theories 616

13.7 Control, Correction and Applicability of Models and Theories 619

13.8 Concluding Remarks 622

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13.9 Exercises 622

13.10 Answers 623

Index 627

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Fig 1.1 Representation of real numbers on the straight line 6

Fig 1.2 Points in the plane 13

Fig 1.3 .x1; x2/ as point and as vector (directed segment) in the plane 15

Fig 1.4 Pythagoras’s theorem 16

Fig 1.5 Addition of vectors 26

Fig 1.6 Multiplication by a scalar 27

Fig 1.7 Construction of x y D x C 1/y 28

Fig 1.8 5; 2/ D 5.1; 0/ C 2.0; 1/ 29

Fig 1.9 Orthogonal vectors 29

Fig 1.10 Trigonometric form of a complex number 34

Fig 1.11 Multiplication of complex numbers 36

Fig 1.12 Cosines and sines 38

Fig 1.13 The inner product x y D jxj jyj cos. / 39

Fig 1.14 Conjugate complex numbers 41

Fig 2.1 A first linear optimisation problem, part 1 54

Fig 3.1 Mapping (multivalued function) 66

Fig 3.2 Single-valued function 66

Fig 3.3 Injection 66

Fig 3.4 Surjection 67

Fig 3.5 Bijection 67

Fig 3.6 Graph 68

Fig 3.7 Graph of the inverse function 68

Fig 3.8 Some graphs 69

Fig 3.9 Cosine function 69

Fig 3.10 Sine function 69

Fig 3.11 Cotangent function 70

Fig 3.12 Tangent function 70

Fig 3.13 Intervals 71

Fig 3.14 A production surface 74

Fig 3.15 Contour-line representation of a real-valued function 74

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Fig 3.16 Extension of a graph 75

Fig 3.17 Market share of an improved product 76

Fig 3.18 Total product curve 76

Fig 3.19 Total cost curve 76

Fig 3.20 Composition of mappings 77

Fig 3.21 Unimodal function with maximum 80

Fig 3.22 Unimodal function with minimum 80

Fig 3.23 Extrema at the endpoints of I 80

Fig 3.24 Maximum inside I 80

Fig 3.25 Increasing function onR2 C 81

Fig 3.26 Graph of (part of).x1; x2/ 7! x2 1 x2 2onR2 . 82

Fig 3.27 The ray going through xD x 1; x 2/ 83

Fig 3.28 Concave and convex functions 86

Fig 3.29 The pointu C 1 /v 87

Fig 3.30 Convex hull of six points 88

Fig 3.31 Line of inflection 91

Fig 3.32 Contour-line representation of a function 95

Fig 3.33 Upper level set 96

Fig 3.34 Example 1 97

Fig 4.1 Graph of a linear function 108

Fig 4.2 Graph of an affine function 108

Fig 4.3 A positive homogeneous linear function 110

Fig 5.1 Feasible solutions and contour lines of an optimisation problem 182

Fig 5.2 A problem with no solutions 190

Fig 5.3 Feasible solutions of an optimisation problem 191

Fig 5.4 Expected payoff value 204

Fig 6.1 Production of strawberries 210

Fig 6.2 Neighbourhoods 211

Fig 6.3 Continuity 212

Fig 6.4 f x/ D 2x sin.1 x/ 212

Fig 6.5 g x/ D sin.1=x/ x ¤ 0/ 214

Fig 6.6 f x/ D x2 214

Fig 6.7 Graphs of sin x, cos x 217

Fig 6.8 sin x x tan x (x 0) 218

Fig 6.9 A discontinuous cost function 223

Fig 6.10 Œx for 1 x 4 224

Fig 6.11 Properties of a continuous function on a closed interval 225

Fig 6.12 An unbounded continuous function 225

Fig 6.13 A continuous function with no maximum and no minimum 226

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Fig 6.15 Graph of a function, difference quotient, derivative,

and tangent 228

Fig 6.16 f x/ D jxj is not differentiable at 0 229

Fig 6.17 Properties ofjxj =x D 1 229

Fig 6.18 Germany’s 1998 average tax rate 230

Fig 6.19 Properties of strictly monotone functions 238

Fig 6.20 Sine and Arc sine 240

Fig 6.21 Cosine and Arc cosine 241

Fig 6.22 Tangent and Arc tan 242

Fig 6.23 Law of the mean 246

Fig 6.24 Properties of f1.x/ D jxj 246

Fig 6.25 Properties of f2.x/ D x jxj 246

Fig 6.26 Properties of x 7! x3 260

Fig 6.27 Global and local extrema and horizontal point of inflection 261

Fig 6.28 Supply curve S demand curve D, and equilibrium point p; y/ 264

Fig 6.29 A cobweb 265

Fig 6.30 Both fpng and fyng oscillate between two fixed values 265

Fig 6.31 Both f png and fyng “explode” 265

Fig 6.32 The Newton algorithm 271

Fig 6.33 Newton algorithm oscillates between two points 271

Fig 6.34 Newton algorithm explodes 272

Fig 6.35 Approximation of f at x0; f x0// by the affine function ` 274

Fig 6.36 "-neighbourhood of the point p 277

Fig 6.37 Linear approximation (differentials) of a vector-vector function 279

Fig 6.38 x is in a neighborhood of p on a straight line through p, parallel to ej 281

Fig 6.39 Graphs of two implicit functions 295

Fig 7.1 Decreasing sequence bounded from below 303

Fig 7.2 Exponential functions 305

Fig 7.3 Function f convex from below 305

Fig 7.4 Chord above the graph of a continuous function 306

Fig 7.5 Slopes of chords 308

Fig 7.6 The graph of a tx is a t-fold horizontal contraction of that of a x 309

Fig 7.7 A strictly convex function 313

Fig 7.8 Graphs of growth and decay 324

Fig 7.9 A homogeneous extension 331

Fig 7.10 Bell-shaped curve 332

Fig 7.11 Contour lines of a homothetic production function 339

Fig 7.12 Marginal rate of substitution 343

Fig 7.13 Elasticity of substitution of a production factor 345

Fig 7.14 Examples of equations with two, one or no solution 357

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Fig 7.15 Some curves 363

Fig 7.16 A system of equations with infinitely many solutions 364

Fig 8.1 Open convex sets, interior of a set 375

Fig 8.2 Examples of compact, bounded, closed, and so on sets 391

Fig 8.3 Bounded set S 391

Fig 8.4 Spatial graphs 397

Fig 8.5 A function with no local extremum 398

Fig 8.6 A function with a saddle point 398

Fig 8.7 Saddle point in the origin 399

Fig 8.8 Approximating a cloud of 31 points by a line 415

Fig 8.9 Example of an “envelope” 436

Fig 8.10 Optimisation problem 1 451

Fig 8.11 Optimisation problem 1 under further restrictions 453

Fig 8.12 Optimisation problem 2 455

Fig 8.13 Directional derivative 458

Fig 8.14 Global saddle point 461

Fig 9.1 Cost functions 484

Fig 10.1 Minimum and maximum interest rates 510

Fig 10.2 m b a/ Rb a f x/ dx M.b a/ 515

Fig 10.3 Calculating the difference quotient of F.x/ DRx a f t/ dt 515

Fig 11.1 The solution of the differential equation y0.t/ D y.t/=2 and its vector field 537

Fig 11.2 The logistic curve 551

Fig 12.1 Difference equation for the national income 570

Fig 13.1 Model of simple production of an economy 603

Fig 13.2 The strict law of diminishing returns 614

Fig 13.3 Schneider’s graph 621

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Table 3.1 Values for the function in Fig 3.6 68

Table 4.1 Input–output table of an economy 127

Table 4.2 Aggregating recommendations by m decision makers

on allocating the amount s among n projects 166

Table 4.3 Aggregation of input or purchase quantities which

establish output value or utility 171

Table 5.1 Slack variables and function values at the vertices in Fig 5.1 183

Table 5.2 Simplex tableau for a zero-sum game 188

Table 5.3 Simplex tableaus: the tableau format and its use for

solving the linear optimisation problem (5.21), (5.22),

(5.23), (5.24), and (5.25) 188

Table 5.4 Matrix of payoffs ajk for the player P The payoffs for

the player Q are ajk 202

Table 5.5 Example of a payoff matrix of a deterministic game 202

Table 5.6 The payoff matrix of a non-deterministic game 203

Table 7.1 Effective interest corresponding to different stated

rates of interest (first line) The first column is the

number of payments per year The last row shows the

continuous compounded interest 321

Table 9.1 Payoff matrices (payoff functions) in a duopoly 478

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of relations in economics and to mathematical notions and methods which will

be the subject of this book The belief that mathematics and its applications toeconomics are just about calculations is mistaken Mathematics and mathematiciansare needed to discover or create and analyse structures in a logically sound way.Chapter13at the end of the book will deal, among other things, with the basics ofmathematical–logical reasoning

In the present chapter we not only summarise basic knowledge about natural numbers, integers, rational and real numbers but define also complex numbers as a

particular case of vectors They will make, among others, the derivation of important

trigonometric formulas easier than usual Vectors and sets, to be introduced in this

chapter, form the basis of much that will follow

Most of the contents of this section just restates the obvious or the well known

It may, however, be useful to remind the reader of these building stones in whatfollows

A set is a collection of distinct objects (this is really just paraphrasing not

defining; we do not define such apparently simple things in this book) The objects,

of which it consists, are the elements of the set For instance you are an element of

W Eichhorn, W Gleißner, Mathematics and Methodology for Economics,

1

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(or: belong to) the set of all people who are reading this sentence (“belongs to” is

a synonym of “element of”) A set is usually given by enumerating all its elements(if there are only finitely many of them) or by giving a procedure (often called

“algorithm”) enabling us to determine all its elements

For instance, the set S consisting of the elements A ; B; C is usually written as

S D fA; B; Cg or S D fA; C; Bg or S D fB; A; Cg or

S D fB; C; Ag or S D fC; A; Bg or S D fC; B; Ag:

The order of the elements is irrelevant (unless told otherwise; if the order is ofpartial or total relevance then we speak of partially or totally ordered sets; to thelatter belong the sequences with which we will deal in detail in Sect.5.4; comparealso Sects.1.5and3.7)

The set of all positive integers, in other words the set of all natural numbers

1; 2; 3; : : : is written as

N D f1; 2; 3; : : :g:

We also mention the notation

N D fn j n is a natural numberg:

After n follows the condition imposed on n separated form n by j.

The symbol 2 reads “element of”, while … means “is not among the elements of”(or “does not belong to”) For instance,

that is, the set of fractions with integer numerator and positive integer denominator,

whose greatest common divisor (gcd) is 1 We assume also that the rules foraddition, subtraction, multiplication and division of rational numbers are known

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As is also known, rational numbers can be represented as finite or periodic infinite decimal fractions Confining ourselves, for simplicity, to positive rational numbers, a finite decimal fraction can be written as

8 .) Take

xD 5:4181818 : : : :

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1000x D 5418:1818 : : : 10x D 54:1818 : : :

and, by subtraction (really multiplication and subtraction of infinite decimal fractionhave to be justified but they are quite intuitive here),

990x D 5364; so x D 5364

990 D 29855:There is an obvious way to make a (periodic) infinite decimal fraction out of afinite one:

31:46 D 31:460000 : : :but we agree that, if in a decimal fraction (finite or infinite) there are only 0’s from

a place on (after the decimal point), then we omit them There is also a less obviousway:

31:46 D 31:459999 : : : :Indeed, using the above procedure for

xD 31:45999 : : :

we get

1000x D 31459:999 : : :

100x D 3145:999 : : : 900x D 28314

xD 28314900 D 3146100 D 31:46:

Actually, those ending with 999 and those ending with 000 are the only infinite decimal fractions which equal finite ones and they are the only pairs of infinite decimal fractions which are equal without all their digits being equal (in the same

order)

Clearly there are also non-periodic decimal fractions; for instance

111:1010010001000010 : : : :(While only 1’s and 0’s figure in it, there is no finite segment which keeps exactlyrepeating.) These (and their products by .1/) are the irrational numbers The

numbers 2 (the length of the circumference of the unit circle) andp2 (the number

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whose square is 2) are also irrational Actually, in a certain sense, which can bemade precise, there are “many more” irrational than rational numbers This is quiteintuitive: we would be rather surprised if the same numbers in the same order keptrepeating as winners in a lottery every fixed (albeit possibly large) number of weeks.The rational and irrational numbers together form the setR of real numbers It follows from the above that every real number can be represented as a finite or infinite decimal fraction—multiplied by.1/ if the real number was negative.There is a pretty proof showing thatp

2 is indeed irrational, that is, it cannot be

a rational number We prove this by contradiction (see Appendix): Suppose

p

2 D m

n (n 2 N, m 2 N sincep2 is positive) We may choose m and n so that not both are even (either just one or neither of them is even; an even number is an integer

divisible by 2; an integer which is not even, is odd) because, if both the numeratorand the denominator were even, then we could cancel the highest power of 2 bywhich both would be divisible (for instance 1624 D 23)

Squaring the above equation, we get

numbers In fact, in their geometric representation on the straight line they are quite

indistinguishable from the rational numbers: If one chooses (Fig.1.1) a point 0 and

a point 1 on the line then every point represents a real number (either rational

or irrational) and, conversely, every real number is represented by a point of that

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Fig 1.1 Representation of real numbers on the straight line The rational numbers12 D 0:5 and

11

3 D 3:66 are represented by the points between 0 and 1, and 3 and 4, respectively The irrational numbers

p

2 D 1:41 , D 3:14 , and e D 2:71 are represented by

the points between 1 and 2, 3 and 4, and 3 and 2, respectively

line We will identify that point with the real number which it represents (use them

interchangeably) and call this line the “real line” or the “number line” We note that any real number can be approximated both by rational and by irrational numbers as closely as one wants, that is, the distance from the real number to an appropriately

chosen rational resp irrational number can be made as small as one wishes The

distance of two (real or rational or integer or positive) numbers x and y is defined by

is, either positive or 0)

We denote the set of nonnegative real numbers by RC, that of positive realnumbers byRCC:

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(b) Let a be a rational number and be an irrational number Are a C , a ,

a , a= irrational numbers?

(c) Is for any pair, of distinct irrational numbers C , , = irrational?

4 The expressions (a), (d), (e), (f) are sets The expression (b) means the distance

(number) 5, not the set consisting of the single element 5 (that would be f5g).

The expression (c) is no set, since not all numbers (elements) are distinct

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5 (a) Yes, (b) Yes,

(c) No For D 1 Cp2, D 1 p2 we get C D 1, for D p2,

Dp1=2 we get D 1 and = D 2

A set T is a subset of a set S if every element of T is also element of S (while elements

of S may or may not be elements of T) This is written as

T S or, what is the same, S T;

and is sometimes verbalised as “S contains T” For instance,

N Z; Z Q; Q R(which also can be written asN Z Q R ),

R R;

f3; 5g f8; 5; 3g; f8g f3; 5; 8g:

Note from the last example that there are sets having only one element It is

often convenient to speak also about a set with no element, the empty set which is denoted by ; This is not to be confused with the set f0g which has one element: the

number 0

Clearly, if T S and S T then S D T, that is, S and T are the same set (because every element of T belongs also to S and every element of S is also element of T) The set T needs not be a subset of S in order to define

S n T D fx j x 2 S but x … Tg (which may be empty) as the complement of T with respect to S But S n T is a subset

of S Examples:

f3; 4; 6g n f3; 6g D f4g; f3; 4; 6g n f1; 2; 3g D f4; 6g; RCn RCCD f0g:

The union of the sets S and T (neither of which needs to be a subset of the other)

is the set V which contains those elements which belong either to S or to T (or to

both) In symbols:

V D S [ T WD fx j x 2 S or x 2 Tg:

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We use this occasion to call attention to a fine point Let the sets A, B and C

consist of the employees (“elements”; of course, a company consists of more

than its employees but we will ignore this here) a1; a2; : : : ; a10, b1; b2; : : : ; b90,

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which has three elements (A, B, and C) while the union

Sk D fx j x 2 S1and x 2 S2and: : : and x 2 Sng:

and verify for any sets S, T, V

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(why?), while

.S \ T/ \ V D S \ T \ V/ and S [ T/ [ V D S [ T [ V/

is called the associativity of \ and [, respectively, and the first and second

part of (1.1) is the distributivity of \ over [ and of [ over \, respectively.

While these “identities” are quite important, one can construct manyothers

The symbols 8 (“for all”) and 9 (“there exists”) help express somemathematical facts For instance,

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(c) S \ T [ V/ D S \ T/ [ S \ V/ (distributivity of \ over [),

(d) S [ T \ V/ D S [ T/ \ S [ V/ (distributivity of [ over \),

5 Verify for arbitrary sets S, T, V

(a) S T and T V imply S V,

(e) and (f) are the sets ; and f0g, respectively

2 The statements (c), (e), (g) are correct

S T WD f.s; t/ j s 2 S; t 2 Tg:

A few remarks may be useful here: This is a “set of sets” as discussed in the previous

section on the example of a “set of companies”: The elements of ST are the ordered pairs s; t/ just as the elements of the Cartesian product of n sets (the notations on the left and in the middle can be used interchangeably):

@n kD1Sk WD S1 S2 : : : Sn

WD f.s1; s2; : : : ; sn / j s12 S1; s22 S2; : : : ; sn 2 Sng

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-4 -3 -2 -1 0

1

3 2 1

-1 -2 -3 -4

Vertical axis Y-axis

Horizontal axis X-axis (-4,1)

(1,-4)

(2,3) (3,2)

Fig 1.2 The points in the plane are represented by pairs of real numbers If the numbers of such

a pair are written in different order, we usually get different points

are ordered n-tuples s1; s2; : : : ; sn/ “Ordered”, because their order is of importance

(at the beginning of Sect.1.2we have already indicated that later some sets may

be ordered or, at least, partially ordered) The importance of ordering is seen onthe example in Fig.1.2: As usual (see also below), a point in the Cartesian plane

is represented by its “x and y coordinates”, that is, its distances from the “vertical

axis” f.0; y/ j y 2 Rg and from the “horizontal axis” f.x; 0/ j x 2 Rg, respectively.

Both “Cartesian product” and “Cartesian plane” refer to the name of the French

mathematician René Descartes (1596–1650) We emphasise that the couples and tuples are ordered: As we see in the Fig.1.2, (2,3) and (3,2) are two different points.

n-(Actually.s; t/ and t; s/ give the same points only in the obvious case t D s).

Example The Cartesian product of the sets

S1D fa; b; cg; S2D fx; yg; S3D fzg and S4D fwg

(continued)

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is given by

S1 S2 S3 S4

D f.a; x; z; w/; a; y; z; w/; b; x; z; w/; b; y; z; w/; c; x; z; w/; c; y; z; w/g:

This is a set of six elements.a; x; z; w/; : : : ; c; y; z; w/ and not of seven elements

a ; b; c; x; y; z; w: the ordered sets a; x; z; w/; a; y; z; w/; : : : themselves are the elements of S1 S2 S3 S4

By the way, the S1 S2 : : : Sn notation is legitimate because the Cartesian product is associative:

.S1 S2/ S3D S1 S2 S3/ D S1 S2 S3

D f.s1; s2; s3/ j s12 S1; s22 S2; s3 2 S3g:

But the Cartesian product is not commutative:

S1 S2D f.s; t/ j s 2 S1; t 2 S2g ¤ f.s; t/ j s 2 S2; t 2 S1g D S2 S1;for instance

fa; b; cg fx; yg D f.a; x/; a; y/; b; x/; b; y/; c; x/; c; y/g

and

fx; yg fa; b; cg D f.x; a/; x; b/; x; c/; y; a/; y; b/; y; c/g:

While the latter equals f.x; a/; y; a/; x; b/; y; b/; x; c/; y; c/g (compare the duction of sets at the beginning of Sect.1.2), this is still not the same as fa; b; cg fx; yg above, because x; a/ is not the same ordered pair as a; x/, and y; a/ not the

intro-same as.a; y/, and so on.

If all sets S1; S2; : : : ; Snare the same

S1D S2D : : : D Sn D S then their Cartesian product is the n-th Cartesian power

S n WD f.s1; s2; : : : ; sn/ j s12 S; s22 S; : : : ; sn 2 Sg:

In particular, for S DR, we get

Rn D f.x1; x2; : : : ; xn/ j xk 2 R k D 1; 2; : : : ; n/g:

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In other words, the elements of Rn

are the vectors with n real components or

“n-component real vectors” Similarly, the elements of S n are “vectors with n components in S” For instance, the elements of Rn

CC are the vectors with n

positive components, those ofNn

are the vectors whose all n components are natural

numbers, similarly forNn

0, whereN0 D N [ f0g is the set of nonnegative integers,and so on

There are many examples of such vectors in economics and other social sciences,

for instance the price vector p1; : : : ; pn/ 2 R n

CCof the present prices and the vector

could be, say, the number of unemployed in n different job categories or the number

of students enrolled in n faculties of a university, and so on.

As mentioned (compare Figs.1.2and1.3), for n D 2, every element x1; x2/ of

R2can be identified with the point in the (Cartesian) plane, whose coordinates are x

1

and x2 We identify.x1; x2/ 2 R2also with the directed segment of the straight lineconnecting the origin (the point (0,0)) with the point.x1; x2/ (Fig.1.3) That directedsegment is the arrow usually associated with the word “fig1.3”, in this case a “2-

component real vector” (x1; x2 are its components) Similarly a 3-component realvector can be identified with a point in the three-dimensional (Euclidean) space andalso with a directed segment from the origin (0,0,0) to that point As a generalisation

-1

-2 -3

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we call the n-component real vector x1; x2; : : : ; xn/ 2 R n

(x1; x2; : : : ; xn are its

components) also a point in the n-dimensional (Cartesian) space.

We will write bold face letters for vectors, in particular for real vectors:

A :

At present we treat these interchangeably: we will not distinguish them till Chap.4,

where they will turn out to be two different special cases of matrices.

For n D 2 the length of the vector (directed segment) x D x1; x2/ is jjxjj D

.x2

1C x2

2/1=2by the theorem of Pythagoras While the reader is surely familiar with

this theorem, the simple proof in Fig.1.4 may not be so well known Actually,Pythagoras’s theorem proves

jjxjj D x2

1C x2

2/1=2

only for positive x1, x2but it implies the same expression for the length of all x D

.x1; x2/ 2 R2and we accept as definition of jjxjj the similar formula

Fig 1.4 jjxjj2 D x2 C x2: Pythagoras’s theorem proved by taking away four equal rectilinear

triangles each from the two equal (big) squares

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for all x D.x1; : : : ; xn/ 2 R n

(n D 1; 2; 3; : : :; note that, for n D 1; jjxjj D jxj) and

call it the Euclidean norm (though “Pythagorean” may be appropriate) For n D3

it still has the geometric meaning of length of x Vectors e with norm 1 (jjejj D1)

are called unit vectors.

We emphasised that the n-tuples of components are ordered In another sense, the

setR of real numbers is ordered (“totally ordered”, to be exact): for any a; b 2 R either a < b or a D b or a > b (one and only one of these can hold) “Greater” (or “smaller” and, of course, “equal”) can be usefully defined also for n-component real vectors with n> 1, even in two, in general different, ways One is

x> y the same as y < x/ if x1 > y1; x2> y2; : : : ; xn > ynI

Of course,

xD y means x1D y1; x2D y2; : : : ; xn D yn:

If this does not hold (that is, x and y are not the same vector) then we write x ¤ y.

Knowing that xk yk for real numbers means that xk is either greater or equal yk, we define for n-component real vectors the second “greater” (or “smaller”) relation by

x y the same as y x/ if x1 y1; x2 y2; : : : ; xn yn but x¤ y;

that is xk yk for all k.D 1; 2; : : : ; n/ but, at least for one `, “sharply” x` > y`(` 2 f1; 2; : : : ; ng) This is not the same as

x > D y or y <D x/ which means that x1 y1; x2 y2; : : : ; xn yn but no x`needs to be really greater than y` In other words, x > D y contains x D y as particular case, but x y does not Strictly speaking, inR1(=R, that is, for reals),

we should write x > D y if x can be either greater or equal y but it is traditional to use the simpler x y notation in this (exceptional) n D 1 case (where the “” in the

above sense is not needed, because it means the same as “ >” for n D 1, which is not the case if n> 1)

Under either of these “greater” relations (there are also others, these are the mostuseful ones),Rn

is not totally ordered, it is only partially ordered, meaning that,

while for some pairs of vectors x 2Rn

for which neither x > y nor x < y nor x D y (neither x y nor x y nor

x D y) holds For instance, of the two vectors 3; 2/ and 2; 3/ in Fig.1.2neither is greater (either in the sense > or ) than the other (Their norms happen to be equal,

both arep

13, but they are not equal according to the above definition, since alreadytheir first components are different.) Another example is given by the three vectors

of goods

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(the first components being, say, pounds of butter, the second pounds of honey).Clearly

a < b because 3 < 4; 2 < 5/ and a < c since 3 < 6; 2 < 3/ but neither b < c nor b D c, not even b c or b c (since 4 < 6 but 5 > 3) This

is not only of theoretical importance: because of this it is not clear which of the two

vectors of quantities of goods, b or c is of more economic utility This is what makes

synthesising (merging, index) methods necessary

We note that there does exist a total order on Rn

, the lexicographical order.

In this order, the point with the greater first component is considered greater; incase of equal first components that with greater second component, and so on.The ordering is called “lexicographic” because that is how “lexicons” (dictionaries,phone directories, etc.) are ordered: in the alphabetical order of the first letter; ifthat is the same in two words then by the second letter, and so on The words canconsist of differently many letters Any word W stands in front of every longerword starting with W Applying this rule accordingly we can establish a complete(lexicographical) order for all vectors ofR2,R3,R4, The lexicographical order

is, however, not practical for most applications in economics

1.4.1 Exercises

1 For the sets S1D fa; bg, S2D fc; d; e; f g, S3D fxg determine

(a) S1 S2,

(b) S2 S1,

(c) the Cartesian product S1 S2 S3,

(d) the fourth Cartesian power of S1

2 Calculate the length of the vectors

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(a) Which of these vectors are comparable with respect to<, , <D ?

(b) Order them in the lexicographical order (Start with a which has the smallest

first component.)

1.4.2 Answers

1 (a) f.a; c/; a; d/; a; e/; a; f /; b; c/; b; d/; b; e/; b; f /g;

(b) f.c; a/; c; b/; d; a/; d; b/; e; a/; e; b/; f ; a/; f ; b/g;

(c) f.a; c; x/; a; d; x/; a; e; x/; a; f ; x/; b; c; x/; b; d; x/; b; e; x/;

.b; f ; x/g:

(d) S41 D f.a; a; a; a/; a; a; a; b/; a; a; b; a/; a; b; a; a/; b; a; a; a/;

.a; a; b; b/; a; b; a; b/; b; a; a; b/; a; b; b; a/; b; a; b; a/;

.b; b; a; a/; a; b; b; b/; b; a; b; b/; b; b; a; b/; b; b; b; a/; b; b; b; b/g:

While not any two vectors could be compared in the sense of the above “>” or

“” order, any two (n-component real) vectors can be added, subtracted, any vector

can be multiplied by a real number (“scalar” in this context) and even any two

n-component vectors can be multiplied in a sense (giving a “scalar product”, not an n-component vector as product).

1.5.1 Sums, Differences, Linear Combinations of Vectors

If the prices p01; p0

2; : : : ; p0

n of n goods in a “basket of goods” in the base year are

considered to be the components of a vector

p0D p0

1; : : : ; p0n/ 2 R n

CC

and during a certain time-interval the prices increase by d1; : : : ; dn, which we collect

again into a vector

D d ; : : : ; dn/ 2 R n ;

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then the new prices will be p0C d1; p0C d2; : : : ; p0

n C dn, forming the new price

because the addition of real numbers has these properties (write the above equation

in components) The sum of more than three vectors can be defined similarly

As motivation for the rule on multiplication of vectors by scalars, consider a bank

which pays on 90-day term deposit 4 % (nominal) yearly interest, that is 1 % for the

90 day period Denote the amounts of n term deposits by t01; t0

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