Mathematics for Economics and Business An

This text offers the ideal approach for economics and business students seeking to understand the mathematics relevant to them. Each chapter demonstrates basic mathematical techniques, while also explaining the economic analysis and business context where each is used.Now in its second edition, the text features expanded summaries of economic analysis, new sections on matrix algebra and linear programming, and additional demonstrations of economics applications. Along with these new features, the book continues to separate mathematical methods and economics applications into discrete sections, allowing the student to learn the mathematics needed, or to proceed immediately to the economics examples.

Jean Soper is Lecturer in Economics at the University of Leicester and is involved in the Executive Committee responsible for the continuing development of WinEcon.

Preface

ix

Features of the Book

xii

List of MathEcon Screens

xvii

List of Excel Worksheets

xxiii

1 Functions in Economics

1

(60)

1.1 Introduction

1

(1)

1.2 Coordinates and Graphs

2

(3)

1.3 Variables and Functions

5

(7)

1.4 Basic Rules of Algebra

12

(10)

1.5 Fractions and Sharing

22

(7)

1.6 Powers and Indices

29

(4)

1.7 Functions of More Than One Variable

33

(1)

1.8 Economic Variables and Functions

34

(1)

1.9 Total and Average Revenue

35

(3)

1.10 Total and Average Cost

38

(3)

1.11 Profit

41

(1)

1.12 Production Functions, Isoquants and the Average Product of Labour

42

(3)

1.13 Functions in Excel

45

(8)

Chapter 1: Answers to Practice Problems

53

(8)

2 Equations in Economics

61

(60)

2.1 Introduction

61

(1)

2.2 Rewriting and Solving Equations

62

(5)

2.3 Substitution

67

(1)

2.4 Demand and Supply

68

(8)

2.5 Cost-Volume-Profit Analysis

76

(2)

2.6 Linear Equations

78

(6)

2.7 Budget Lines

84

(3)

2.8 Constant Substitution Along a Line

87

(1)

2.9 Quadratic Equations

88

(3)

2.10 Intersection of MC with MR or AVC

91

(3)

2.11 Simultaneous Equations

94

(2)

2.12 Simultaneous Equilibrium in Related Markets

96

(3)

2.13 Exponential Functions

99

(4)

2.14 Logarithmic Functions

103

(3)

2.15 Plotting and Solving Equations in Excel

106

(6)

Chapter 2: Answers to Practice Problems

112

(9)

3 Macroeconomic Models

121

(26)

3.1 Introduction

121

(1)

3.2 Notation and Model Specification

122

(8)

3.3 Identifying Equilibrium Income

130

(1)

3.4 Keynesian Cross Model

131

(5)

3.5 IS-LM Analysis

136

(6)

Chapter 3: Answers to Practice Problems

142

(5)

4 Changes, Rates, Finance and Series

147

(40)

4.1 Introduction

147

(2)

4.2 Percentage Changes

149

(3)

4.3 Simple and Compound Interest

152

(5)

4.4 Annual Equivalent Rate

157

(3)

4.5 Nominal and Real Interest Rates

160

(2)

4.6 Growth Rates and Logarithms

162

(3)

4.7 Depreciation

165

(2)

4.8 Net Present Value

167

(4)

4.9 Series

171

(2)

4.10 Savings and Loans with Regular Payments

173

(5)

4.11 Prices of Bills and Bonds and the Rate of Interest

178

(2)

4.12 Financial Calculations with Excel

180

(4)

Chapter 4: Answers to Practice Problems

184

(3)

5 Differentiation in Economics

187

(40)

5.1 Introduction

187

(1)

5.2 Using Differentiation in Economics

188

(2)

5.3 The Basic Rules of Differentiation

190

(11)

5.4 Application to Economic Analysis

201

(1)

5.5 Utility Functions

201

(3)

5.6 Revenue Functions

204

(4)

5.7 Short-run Production Functions

208

(2)

5.8 Cost Functions

210

(3)

5.9 Point Elasticity of Demand and of Supply

213

(8)

5.10 Investment Multiplier

221

(2)

Chapter 5: Answers to Practice Problems

223

(4)

6 Maximum and Minimum Values

227

(22)

6.1 Introduction

227

(1)

6.2 Identifying Maximum and Minimum Turning Points

228

(4)

6.3 Maximum Total Revenue

232

(2)

6.4 Maximum Profit

234

(5)

6.5 Minimum Average Cost

239

(2)

6.6 Average and Marginal Product of Labour

241

(2)

6.7 Tax Rate which Maximizes Tax Revenue

243

(1)

6.8 Minimizing Total Inventory Costs

244

(2)

Chapter 6: Answers to Practice Problems

246

(3)

7 Further Rules of Differentiation

249

(21)

7.1 Introduction

249

(1)

7.2 The Chain Rule

250

(1)

7.3 The Product Rule

251

(2)

7.4 The Quotient Rule

253

(1)

7.5 Marginal Revenue, Price Elasticity and Maximum Total Revenue

254

(3)

7.6 Optimal Production and Cost Relationships

257

(5)

7.7 Exponential and Natural Logarithmic Functions

262

(2)

7.8 Applications to Economic Analysis

264

(1)

Chapter 7: Answers to Practice Problems

265

(5)

8 Partial Differentiation in Economics

270

(37)

8.1 Introduction

270

(1)

8.2 Partial Derivatives

271

(2)

8.3 Small Increments Formula

273

(2)

8.4 Multipliers

275

(7)

8.5 Differentials and Implicit Differentiation

282

(2)

8.6 Slope of an Indifference Curve

284

(3)

8.7 Elasticities

287

(2)

8.8 Logarithmic Demand Functions

289

(1)

8.9 Long-run Production Functions

290

(5)

8.10 Second Order Partial Derivatives

295

(2)

8.11 Optimization of Multivariate Functions

297

(2)

8.12 Price Discriminating Monopoly

299

(3)

Chapter 8: Answers to Practice Problems

302

(5)

9 Constrained Maxima and Minima

307

(13)

9.1 Introduction

307

(1)

9.2 Substitution Method

308

(3)

9.3 Lagrange Multiplier Method

311

(4)

Chapter 9: Answers to Practice Problems

315

(5)

10 Integration in Economics

320

(16)

10.1 Introduction

320

(1)

10.2 Rules of Integration

321

(4)

10.3 Finding Total Functions from Marginal Functions

325

(3)

10.4 Area Under a Curve

328

(3)

10.5 Definite Integrals in Economics

331

(2)

Chapter 10: Answers to Practice Problems

333

(3)

11 Linear Programming

336

(21)

11.1 Introduction

336

(1)

11.2 Formulating Linear Programming Problems

337

(2)

11.3 Graphical Solution

339

(4)

11.4 The Optimal Solution and the Feasible Set

343

(1)

11.5 Using Excel Solver

344

(4)

11.6 Sensitivity Analysis

348

(1)

11.7 Minimization Subject to Constraints

349

(4)

Chapter 11: Answers to Practice Problems

353

(4)

12 Matrices in Economics

357

(42)

12.1 Introduction

357

(1)

12.2 Matrices and Vectors

358

(3)

12.3 Matrix Addition and Subtraction

361

(1)

12.4 Multiplication of Matrices

362

(11)

12.5 Handling Equation Systems

373

(4)

12.6 Determinants

377

(5)

12.7 Matrix Inversion

382

(4)

12.8 Systems of Equations in Economics

386

(2)

12.9 Matrix Operations in Excel

388

(5)

Chapter 12: Answers to Practice Problems

393

(6)

Index

399

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Mathematics for Economics and Business An Interactive Introduction

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