Elements of Mathematics for Economics and Finance (Classroom Companion: Economics)

This book equips undergraduates with the mathematical skills required for degree courses in economics, finance, management, and business studies. The fundamental ideas are described in the simplest mathematical terms, highlighting threads of common mathematical theory in the various topics. Coverage helps readers become confident and competent in the use of mathematical tools and techniques that can be applied to a range of problems. Preface Preface to the Second Edition Contents 1 Essential Skills 1.1 Introduction 1.2 Numbers 1.2.1 Addition and Subtraction 1.2.2 Multiplication and Division 1.2.3 Evaluation of Arithmetical Expressions 1.3 Fractions 1.3.1 Multiplication and Division 1.4 Decimal Representation of Numbers 1.4.1 Standard Form 1.5 Percentages 1.6 Powers and Indices 1.7 Simplifying Algebraic Expressions 1.7.1 Multiplying Brackets 1.7.2 Factorization 2 Linear Equations 2.1 Introduction 2.2 Solution of Linear Equations 2.3 Solution of Simultaneous Linear Equations 2.4 Graphs of Linear Equations 2.4.1 Slope of a Straight Line 2.5 Budget Lines 2.6 Supply and Demand Analysis 2.6.1 Multicommodity Markets 3 Quadratic Equations 3.1 Introduction 3.2 Graphs of Quadratic Functions 3.3 Quadratic Equations 3.4 Applications to Economics 4 Functions of a Single Variable 4.1 Introduction 4.2 Limits 4.3 Polynomial Functions 4.4 Continuity 4.5 Reciprocal Functions 4.6 Inverse Functions 5 The Exponential and Logarithmic Functions 5.1 Introduction 5.2 Exponential Functions 5.3 Logarithmic Functions 5.4 Returns to Scale of Production Functions 5.4.1 Cobb-Douglas Production Functions 5.5 Compounding of Interest 5.6 Applications of the Exponential Function in EconomicModelling 6 Differentiation 6.1 Introduction 6.2 Rules of Differentiation 6.2.1 Constant Functions 6.2.2 Linear Functions 6.2.3 Power Functions 6.2.4 Sums and Differences of Functions 6.2.5 Product of Functions 6.2.6 Quotient of Functions 6.2.7 The Chain Rule 6.3 Exponential and Logarithmic Functions 6.4 Marginal Functions in Economics 6.4.1 Marginal Revenue and Marginal Cost 6.4.2 Marginal Propensities 6.5 Approximation to Marginal Functions 6.6 Higher Order Derivatives 6.7 Production Functions 7 Maxima and Minima 7.1 Introduction 7.2 Local Properties of Functions 7.2.1 Increasing and Decreasing Functions 7.2.2 Concave and Convex Functions 7.3 Local or Relative Extrema 7.4 Global or Absolute Extrema 7.5 Points of Inflection 7.6 Optimization of Production Functions 7.7 Optimization of Profit Functions 7.8 Other Examples 8 Partial Differentiation 8.1 Introduction 8.2 Functions of Two or More Variables 8.3 Partial Derivatives 8.4 Higher Order Partial Derivatives 8.5 Partial Rate of Change 8.6 The Chain Rule and Total Derivatives 8.7 Some Applications of Partial Derivatives 8.7.1 Implicit Differentiation 8.7.2 Elasticity of Demand 8.7.3 Utility 8.7.4 Production 8.7.5 Graphical Representations 9 Optimization 9.1 Introduction 9.2 Unconstrained Optimization 9.3 Constrained Optimization 9.3.1 Substitution Method 9.3.2 Lagrange Multipliers 9.3.3 The Lagrange Multiplier λ: An Interpretation 9.4 Iso Curves 10 Matrices and Determinants 10.1 Introduction 10.2 Matrix Operations 10.2.1 Scalar Multiplication 10.2.2 Matrix Addition 10.2.3 Matrix Multiplication 10.3 Solutions of Linear Systems of Equations 10.4 Cramer's Rule 10.5 More Determinants 10.6 Special Cases 10.7 Eigenvalues and Eigenvectors 11 Integration 11.1 Introduction 11.2 Rules of Integration 11.3 Definite Integrals 11.4 Definite Integration: Area and Summation 11.5 Producer's Surplus 11.6 Consumer's Surplus 11.7 Integration by Substitution 11.8 Partial Fractions in Integration 11.9 Integration by Parts 12 Linear Difference Equations 12.1 Introduction 12.2 Difference Equations 12.3 First Order Linear Difference Equations 12.4 Stability 12.5 The Cobweb Model 12.6 Second Order Linear Difference Equations 12.6.1 Complementary Solutions 12.6.2 Particular Solutions 12.6.3 Stability 13 Differential Equations 13.1 Introduction 13.2 First Order Linear Differential Equations 13.2.1 Stability 13.3 Nonlinear First Order Differential Equations 13.3.1 Separation of Variables 13.3.2 Integrating Factors 13.4 Solow Differential Equation 13.5 Second Order Linear Differential Equations 13.5.1 The Homogeneous Case 13.5.2 The General Case 13.5.3 Stability A Differentials B Answers to Self-Assessment Questions Index Based on over 15 years'experience in the design and delivery of successful first-year courses, this book equips undergraduates with the mathematical skills required for degree courses in economics, finance, management, and business studies. The book starts with a summary of basic skills and takes its readers as far as constrained optimisation helping them to become confident and competent in the use of mathematical tools and techniques that can be applied to a range of problems in economics and finance.Designed as both a course text and a handbook, the book assumes little prior mathematical knowledge beyond elementary algebra and is therefore suitable for students returning to mathematics after a long break. The fundamental ideas are described in the simplest mathematical terms, highlighting threads of common mathematical theory in the various topics. Features of the book include: a systematic approach: ideas are touched upon, introduced gradually and thenconsolidated through the use of illustrative examples;several entry points to accommodate differing mathematical backgrounds; numerous problems, with full solutions, and exercises to illustrate the theory and applications; key learning objectives and self-assessment questions provided for each chapter; full solutions to exercises, available to lecturers via the Springer Nature Extramaterial - Lecturer Material website. Vass Mavron is Emeritus Professor of Mathematics at Aberystwyth University. Tim Phillips is Professor of Applied Mathematics in the School of Mathematics at Cardiff University.