Harrison,_Michael_&_Waldron,_Patrick_-_Mathematical_Economics_and_Finance

Harrison,_Michael_&_Waldron,_Patrick_-_Mathematical_Economics_and_Finance

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Unformatted text preview: 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 Taylors 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 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ....
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This note was uploaded on 04/07/2008 for the course ECON dont know taught by Professor Dontknow during the Spring '08 term at Case Western.

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Harrison,_Michael_&_Waldron,_Patrick_-_Mathematical_Economics_and_Finance

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