Maths forEconomicsSecond Edition

Geoff Renshawwith contributions from Norman Ireland

OXPORDUNIVERSITY PRESS

Brief contents

Detailed contents ix

About the author xiv

About the book xv

How to use the book xvii

Chapter map xviii

Guided tour of the textbook features xx

Guided tour of the Online Resource Centre xxii

Acknowledgements xxiv

Part One Foundations

1 Arithmetic 3

2 Algebra 43

3 Linear equations 63

4 Quadratic equations 109

5 Some further equations and techniques 134

Part Two Optimization with one independent variable

Part Four Optimization with two or more independent variables

6 Derivatives and differentiation 165

7 Derivatives in action 184

8 Economic applications of functions and derivatives 213

9 Elasticity 256

Part Three Mathematics of finance and growth i

10 Compound growth and present discounted value 297

11 The exponential function and logarithms 328

12 Continuous growth and the natural exponential function 342

13 Derivatives of exponential and logarithmic functions and theirapplications 368

14 Functions of two or more independent variables 389

15 Maximum and minimum values, the total differential, and applications 441

16 Constrained maximum and minimum values 479

17 Returns to scale and homogeneous functions; partial elasticities;growth accounting; logarithmic scales 519

CO

oo

en

Pa it Five Some further topics

18 Integration

19 Matrix algebra

20 Difference and differential equations

W21 Extensions and future directions (on the Online Resource Centre)

551

577

597

Appendix: Answers to chapter 1 self-test

Glossary

Index

623

624

632

O

Detailed contents

About the authorAbout the book

How to use the book

Chapter map

Guided tour of the textbook features

Guided tour of the Online Resource Centre

Acknowledgements

ijjgfl Part One Foundations

1 Arithmetic1.1 Introduction

xivXV

xvii

xviii

XX

xxii

xxiv

3

3

1.2 Addition and subtraction with positiveand negative numbers

1.3 Multiplication and division with positive

3 Linear equations 63

3.1 Introduction 63

3.2 How we can manipulate equations 64

3.3 Variables and parameters 69

3.4 Linear and non-linear equations 69

3.5 Linear functions 72

3.6 Graphs of linear functions 73

3.7 The slope and intercept of a linear function 75

3.8 Graphical solution of linear equations 80

3.9 Simultaneous linear equations 81

3.10 Graphical solution of simultaneous linearequations 84

3.11 Existence of a solution to a pair of linearsimultaneous equations 87

3.12 Three linear equations with three unknowns 90

1.4

1.5

1.6

1.7

1.8

1.9

1.10

1.11

1.12

1.13

1.14

1.15

1.16

2

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

2.13

and negative numbersBrackets and when we need them

Factorization

Fractions

Addition and subtraction of fractions

Multiplication and division of fractions

Decimal numbers

Adding, subtracting, multiplying, anddividing decimal numbers

Fractions, proportions, and ratios

Percentages

Index numbers

Powers and roots

Standard index form

Some additional symbols

SELF-TEST EXERCISE

AlgebraIntroduction

Rules of algebraAddition and subtraction of algebraic

expressionsMultiplication and division of algebraic

expressionsBrackets and when we need themFractions

Addition and subtraction of fractions

Multiplication and division of fractions

Powers and roots

Extending the idea of powers

Negative and fractional powers

The sign of a"

Necessary and sufficient conditions

APPENDIX: The Greek alphabet

710

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3.13

3.14

3.15

3.16

3.17

4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

4.8

4.9

4.10

4.11

4.12

4.13

4.14

5

5.1

5.2

5.3

5.4

5.5

5.6

Economic applications

Demand and supply for a good

The inverse demand and supply functions

Comparative statics

Macroeconomic equilibrium

Quadratic equationsIntroduction

Quadratic expressions

Factorizing quadratic expressions

Quadratic equations

The formula for solving any quadraticequation

Cases where a quadratic expressioncannot be factorized

The case of the perfect square

Quadratic functions

The inverse quadratic function

Graphical solution of quadratic equations

Simultaneous quadratic equations

Graphical solution of simultaneousquadratic equations

Economic application 1: supply and demand

Economic application 2: costs and revenue

Some further equations andtechniquesIntroduction

The cubic function

Graphical solution of cubic equations

Application of the cubic function ineconomics

The rectangular hyperbola

Limits and continuity

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102

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aLU

iiLUO

5.7 Application of the rectangular hyperbola

in economics 146

5.8 The circle and the ellipse 149

5.9 Application of circle and ellipse in

economics 151

5.10 Inequalities 152

5.11 Examples of inequality problems 156

5.12 Applications of inequalities in economics 159

Part Two Optimization with oneindependent variable

6 Derivatives and differentiation 165

6.1 Introduction 165

6.2 The difference quotient 166

6.3 Calculating the difference quotient 167

6.4 The slope of a curved line 168

6.5 Finding the slope of the tangent 170

6.6 Generalization to any function of x 172

6.7 Rules for evaluating the derivative ofa function 173

6.8 Summary of rules of differentiation 182

7 Derivatives in action 184

7.1 Introduction 184

7.2 Increasing and decreasing functions 185

7.3 Optimization: finding maximum and

minimum values 187

7.4 A maximum value of a function 187

7.5 The derivative as a function of x 188

7.6 A minimum value of a function 189

7.7 The second derivative 191

7.8 A rule for maximum and minimumvalues 191

7.9 Worked examples of maximum and

minimum values 192

7.10 Points of inflection 195

7.11 A rule for points of inflection 198

7.12 More about points of inflection 199

7.13 Convex and concave functions 206

7.14 An alternative notation for derivatives 209

7.15 The differential and linear approximation 210

8 Economic applications of functions

and derivatives 213

8.1 Introduction 213

8.2 The firm's total cost function 214

8.3 The firm's average cost function 216

8.4 Marginal cost 218

8.5 The relationship between marginal and

average cost 220

8.6 Worked examples of cost functions 222

8.7 Demand, total revenue, and marginalrevenue 229

8.8 The market demand function 229

8.9 Total revenue with monopoly 231

8.10 Marginal revenue with monopoly 232

8.11 Demand, total and marginal revenuefunctions with monopoly 234

8.12 Demand, total and marginal revenuewith perfect competition 235

8.13 Worked examples on demand, marginal

and total revenue 236

8.14 Profit maximization 239

8.15 Profit maximization with monopoly 240

8.16 Profit maximization using marginal costand marginal revenue 242

8.17 Profit maximization with perfectcompetition 244

8.18 Comparing the equilibria under monopolyand perfect competition 246

8.19 Two common fallacies concerning profitmaximization 248

8.20 The second order condition for profitmaximization 248APPENDIX 8.1: The relationship between

total cost, average cost, and marginal cost 253

APPENDIX 8.2: The relationship betweenprice, total revenue, and marginal revenue 254

9 Elasticity 256

9.1 Introduction 256

9.2 Absolute, proportionate, and percentagechanges 257

9.3 The arc elasticity of supply 259

9.4 Elastic and inelastic supply 260

9.5 Elasticity as a rate of proportionate change 260

9.6 Diagrammatic treatment 261

9.7 Shortcomings of arc elasticity 263

9.8 The point elasticity of supply 263

9.9 Reconciling the arc and point supplyelasticities 265

9.10 Worked examples on supply elasticity 265

9.11 The arc elasticity of demand 268

9.12 Elastic and inelastic demand 270

9.13 An alternative definition of demand elasticity 272

9.14 The point elasticity of demand 273

9.15 Reconciling the arc and point demandelasticities 274

9.16 Worked examples on demand elasticity 275

9.17 Marginal revenue and the elasticity ofdemand 279

9.18 The elasticity of demand under perfectcompetition 282

9.19 Worked examples on demand elasticity

and marginal revenue 284

9.20 Other elasticities in economics 288

9.21 The firm's total cost function 288

9.22 The aggregate consumption function 290

9.23 Generalizing the concept of elasticity 292

PartThree Mathematics of financeand growth

10 Compound growth and presentdiscounted value 297

10.1 Introduction 297

10.2 Arithmetic and geometric series 298

10.3 An economic application 300

10.4 Simple and compound interest 304

10.5 Applications of the compound growth

formula 307

10.6 Discrete versus continuous growth 309

10.7 When interest is added more than once

per year 309

10.8 Present discounted value 314

10.9 Present value and economic behaviour 316

10.10 Present value of a series of future receipts 316

10.11 Present value of an infinite series 319

10.12 Market value of a perpetual bond 320

10.13 Calculating loan repayments 322

11 The exponential function andlogarithms 328

11.1 Introduction 328

11.2 The exponential function y= 10* 330

11.3 The function inverse t o / = 10x 331

11.4 Properties of logarithms 333

11.5 Using your calculator to find commonlogarithms 333

11.6 The graph o f / = log10x 334

11.7 Rules for manipulating logs 335

11.8 Using logs to solve problems 337

11.9 Some more exponential functions 338

12 Continuous growth and the natural

exponential function 342

12.1 Introduction 342

12.2 Limitations of discrete compound growth 343

12.3 Continuous growth: the simplest case 343

12.4 Continuous growth: the general case 346

12.5 The graph of y = aere 347

12.6 Natural logarithms 349

12.7 Rules for manipulating natural logs 351

12.8 Natural exponentials and logs on yourcalculator 351

12.9 Continuous growth applications 353

12.10 Continuous discounting and present value 358

12.11 Graphs with semi-log scale 361

13 Derivatives of exponential and

logarithmic functions and their

applications 368

13.1 Introduction 368

13.2 The derivative of the natural exponentialfunction 369

13.3 The derivative of the natural logarithmicfunction 370

13.4 The rate of proportionate change, or rateof growth

13.5 Discrete growth

13.6 Continuous growth

13.7 Instantaneous and nominal growth ratescompared

13.8 Semi-log graphs and the growth rate again

13.9 An important special case

13.10 Logarithmic scales and elasticity

Part Four Optimization with two ormore independent variables

14 Functions of two or moreindependent variables 389

14.1 Introduction 389

14.2 Functions with two independent variables 390

14.3 Examples of functions with twoindependent variables 393

14.4 Partial derivatives 398

14.5 Evaluation of first order partial derivatives 401

14.6 Second order partial derivatives 403

14.7 Economic applications 1: the productionfunction 411

14.8 The shape of the production function 411

14.9 The Cobb-Douglas production function 420

14.10 Alternatives to the Cobb-Douglas form 425

14.11 Economic applications 2: the utility function 428

14.12 The shape of the utility function 429

14.13 The Cobb-Douglas utility function 434

APPENDIX 14.1: A variant of the partial

derivatives of the Cobb-Douglas function 439

15 M a x i m u m and minimum values, thetotal differential, and applications 441

15.1 Introduction 441

15.2 Maximum and minimum values 442

15.3 Saddle points 448

15.4 The total differential of z = f(x, y) 452

15.5 Differentiating a function of a function 457

15.6 Marginal revenue as a total derivative 458

15.7 Differentiating an implicit function 460

15.8 Finding the slope of an iso-zsection 463

15.9 A shift from one iso-z section to another 463

15.10 Economic applications 1: the productionfunction 465

15.11 Isoquants of the Cobb-Douglas productionfunction 468

15.12 Economic applications 2: the utility function 470

15.13 The Cobb-Douglas utility function 472

15.14 Economic application 3: macroeconomicequilibrium 473

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374

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16.1

16.2

16.3

16.4

16.5

16.6

16.7

16.8

16.9

16.10

16.11

15.15 The Keynesian multiplier 473

15.16 The IS curve and its slope 474

15.17 Comparative statics: shifts in the IS curve 475

16 Constrained maximum andminimum valuesIntroduction

The problem, with a graphical solution

Solution by implicit differentiation

Solution by direct substitution

The Lagrange multiplier method

Economic applications 1: cost minimization

Economic applications 2: profit maximization

A worked example

Some problems with profit maximization

Profit maximization by a monopolist

Economic applications 3: utility maximization

by the consumer

16.12 -Deriving the consumer's demand functions

17 Returns to scale and homogeneousfunctions; partial elasticities; growthaccounting; logarithmic scales

17.1 Introduction

17.2 The production function and returns to scale

17.3 Homogeneous functions

17.4 Properties of homogeneous functions

17.5 Partial elasticities

17.6 Partial elasticities of demand

17.7 The proportionate differential of a function

17.8 Growth accounting

17.9 Elasticity and logs

17.10 Partial elasticities and logarithmic scales

17.11 The proportionate differential and logs

17.12 Log linearity with several variables

Part Five Some further topics

18 Integration18.1 Introduction

18.2 The definite integral 552

18.3 The indefinite integral 554

18.4 Rules for finding the indefinite integral 555

18.5 Finding a definite integral 562

18.6 Economic applications 1: deriving the479479

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519

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551

18.7

18.8

18.9

18.10

19

19.1

19.2

19.3

19.4

19.5

19.6

19.7

19.8

19.9

19.10

19.11

19.12

19.13

19.14

20

20.1

20.2

20.3

20.4

20.5

20.6

20.7

20.8

20.9

total cost function from the marginalcost function

Economic applications 2: deriving totalrevenue from the marginal revenue function

Economic applications 3: consumers' surplus

Economic applications 4: producers' surplus

Economic applications 5: present value of acontinuous stream of income

Matrix algebraIntroduction

Definitions and notation

Transpose of a matrix

Addition/subtraction of two matrices

Multiplication of two matrices

Vector multiplication

Scalar multiplication

Matrix algebra as a compact notation

The determinant of a square matrix

The inverse of a square matrix

Using matrix inversion to solve linearsimultaneous equations

Cramer's rule

A macroeconomic application

Conclusions

Difference and differential equationsIntroduction

Difference equations

Qualitative analysis

The cobweb model of supply and demand

Conclusions on the cobweb model

Differential equations

Qualitative analysis

Dynamic stability of a market

Conclusions on market stabilitv

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610

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615

616

620

W21 Extensions and future directions APPENDIX 21.3: The firm's maximum profit(on the Online Resource Centre) function with two products

21.1 Introduction APPENDIX 21.4: Removing the imaginary number

21.2 Functions and analysis

21.3 Comparative statics

21.4 Second order difference equations

APPENDIX 21.1: Proof of Taylor's theorem Appendix: Answers to chapter 1 self-test 623

APPENDIX 21.2: Using Taylor's formula to relate Glossary 624production function forms Index 632