tailieunhanh - Mathematical Economics and Finance

Why are we interested in solving simultaneous equations? We often have to find a point which satisfies more than one equation simultaneously, for example when finding equilibrium price and quantity given supply and demand functions. To be an equilibrium, the point (Q; P) must lie on both the supply and demand curves. Now both supply and demand curves can be plotted on the same diagram and the point(s) of intersection will be the equilibrium (equilibria) | 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 Introduction. 3 Systems of Linear Equations and Matrices. 3 Matrix Operations. 7 Matrix Arithmetic. 7 Vectors and Vector Spaces . 11 Linear Independence. 12 Bases and Dimension . 12 Rank. 13 Eigenvalues and Eigenvectors. 14 Quadratic Forms . 15 Symmetric Matrices . 15 Definite Matrices. 15 2 VECTOR CALCULUS 17 Introduction . 17 Basic Topology . 17 Vector-valued Functions and Functions of Several Variables . 18 Revised December 2 1998 ii CONTENTS Partial and Total Derivatives. 20 The Chain Rule and Product Rule . 21 The Implicit Function Theorem. 23 Directional Derivatives. 24 Taylor s Theorem Deterministic Version . 25 The Fundamental Theorem of Calculus . 26 3 CONVEXITY AND OPTIMISATION 27 Introduction . 27 Convexity and Concavity . 27 Definitions. 27 Properties of concave functions . 29 Convexity and differentiability. 30 Variations on the convexity theme. 34 Unconstrained Optimisation . 39 Equality Constrained Optimisation The Lagrange Multiplier Theorems . 43 Inequality Constrained Optimisation The Kuhn-Tucker Theorems . 50 Duality . 58 II APPLICATIONS 61 4 CHOICE UNDER CERTAINTY 63 Introduction . 63 Definitions . 63 Axioms . 66 Optimal Response Functions Marshallian and Hicksian Demand . 69 The consumer s problem. 69 The No Arbitrage Principle . 70 Other Properties of Marshallian demand. 71 The dual problem . 72 Properties of Hicksian demands . 73 Envelope Functions Indirect Utility and Expenditure . 73 Further Results in Demand Theory . 75 General Equilibrium Theory . 78 Walras law . 78 Brouwer s fixed point .