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Mathematics for Economics and Finance

The aim of this book is to bring students of economics and finance who have only an introductory background in mathematics up to a quite advanced level in the subject, thus preparing them for the core mathematical demands of econometrics, economic theory, quantitative finance and mathematical economics, which they are likely to encounter in their final-year courses and beyond. The level of the book will also be useful for those embarking on the first year of their graduate studies in Business, Economics or Finance. The book also serves as an introduction to quantitative economics and finance for mathematics students at. | Mathematical Economics and Finance Michael Harrison Patrick Waldron December 2 1998 CONTENTS i Contents List of Tables iii List of Figures v PREFACE vii What Is Economics . vii What Is Mathematics .viii NOTATION ix I MATHEMATICS 1 1 LINEAR ALGEBRA 3 1.1 Introduction. 3 1.2 Systems of Linear Equations and Matrices. 3 1.3 Matrix Operations. 7 1.4 Matrix Arithmetic. 7 1.5 Vectors and Vector Spaces . 11 1.6 Linear Independence. 12 1.7 Bases and Dimension . 12 1.8 Rank. 13 1.9 Eigenvalues and Eigenvectors. 14 1.10 Quadratic Forms . 15 1.11 Symmetric Matrices . 15 1.12 Definite Matrices. 15 2 VECTOR CALCULUS 17 2.1 Introduction . 17 2.2 Basic Topology . 17 2.3 Vector-valued Functions and Functions of Several Variables . 18 Revised December 2 1998 ii CONTENTS 2.4 Partial and Total Derivatives. 20 2.5 The Chain Rule and Product Rule . 21 2.6 The Implicit Function Theorem. 23 2.7 Directional Derivatives. 24 2.8 Taylor s Theorem Deterministic Version . 25 2.9 The Fundamental Theorem of Calculus . 26 3 CONVEXITY AND OPTIMISATION 27 3.1 Introduction . 27 3.2 Convexity and Concavity . 27 3.2.1 Definitions. 27 3.2.2 Properties of concave functions . 29 3.2.3 Convexity and differentiability. 30 3.2.4 Variations on the convexity theme. 34 3.3 Unconstrained Optimisation . 39 3.4 Equality Constrained Optimisation The Lagrange Multiplier Theorems . 43 3.5 Inequality Constrained Optimisation The Kuhn-Tucker Theorems . 50 3.6 Duality . 58 II APPLICATIONS 61 4 CHOICE UNDER CERTAINTY 63 4.1 Introduction . 63 4.2 Definitions . 63 4.3 Axioms . 66 4.4 Optimal Response Functions Marshallian and Hicksian Demand . 69 4.4.1 The consumer s problem. 69 4.4.2 The No Arbitrage Principle . 70 4.4.3 Other Properties of Marshallian demand. 71 4.4.4 The dual problem . 72 4.4.5 Properties of Hicksian demands . 73 4.5 Envelope Functions Indirect Utility and Expenditure . 73 4.6 Further Results in Demand Theory . 75 4.7 General Equilibrium Theory . 78 4.7.1 Walras law . 78 4.7.2 Brouwer s fixed point .