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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 theorem . . . . . . . . . . . . . . . 78 Revised: December 2, 1998 CONTENTS iii 4.7.3 Existence of equilibrium . . . . . . . . . . . . . . . . . . 78 4.8 The Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . . 78 4.8.1 The Edgeworth box . . . . . . . . . . . . . . . . . . . . . 78 4.8.2 Pareto efficiency . . . . . . . . . . . . . . . . . . . . . . 78 4.8.3 The First Welfare Theorem . . . . . . . . . . . . . . . . . 79 4.8.4 The Separating Hyperplane Theorem . . . . . . . . . . . 80 4.8.5 The Second Welfare Theorem . . . . . . . . . . . . . . . 80 4.8.6 Complete markets . . . . . . . . . . . . . . . . . . . . . 82 4.8.7 Other characterizations of Pareto efficient allocations . . . 82 4.9 Multi-period General Equilibrium . . . . . . . . . . . . . . . . . 84 5 CHOICE UNDER UNCERTAINTY 85 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.2 Review of Basic Probability . . . . . . . . . . . . . . . . . . . . 85 5.3 Taylor’s Theorem: Stochastic Version . . . . . . . . . . . . . . . 88 5.4 Pricing State-Contingent Claims . . . . . . . . . . . . . . . . . . 88 5.4.1 Completion of markets using options . . . . . . . . . . . 90 5.4.2 Restrictions on security values implied by allocational efficiency and covariance with aggregate consumption . . . 91 5.4.3 Completing markets with options on aggregate consumption 92 5.4.4 Replicating elementary claims with a butterfly spread . . . 93 5.5 The Expected Utility Paradigm . . . . . . . . . . . . . . . . . . . 93 5.5.1 Further axioms . . . . . . . . . . . . . . . . . . . . . . . 93 5.5.2 Existence of expected utility functions . . . . . . . . . . . 95 5.6 Jensen’s Inequality and Siegel’s Paradox . . . . . . . . . . . . . . 97 5.7 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.8 The Mean-Variance Paradigm . . . . . . . . . . . . . . . . . . . 102 5.9 The Kelly Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.10 Alternative Non-Expected Utility Approaches . . . . . . . . . . . 104 6 PORTFOLIO THEORY 105 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.2 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . 105 6.2.1 Measuring rates of return . . . . . . . . . . . . . . . . . . 105 6.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 108 6.3 The Single-period Portfolio Choice Problem . . . . . . . . . . . . 110 6.3.1 The canonical portfolio problem . . . . . . . . . . . . . . 110 6.3.2 Risk aversion and portfolio composition . . . . . . . . . . 112 6.3.3 Mutual fund separation . . . . . . . . . . . . . . . . . . . 114 6.4 Mathematics of the Portfolio Frontier . . . . . . . . . . . . . . . 116 Revised: December 2, 1998 iv CONTENTS 6.4.1 The portfolio frontier in <N: risky assets only . . . . . . . . . . . . . . . . . . . . . . 116 6.4.2 The portfolio frontier in mean-variance space: risky assets only . . . . . . . . . . . . . . . . . . . . . . 124 6.4.3 The portfolio frontier in <N: riskfree and risky assets . . . . . . . . . . . . . . . . . . 129 6.4.4 The portfolio frontier in mean-variance space: riskfree and risky assets . . . . . . . . . . . . . . . . . . 129 6.5 Market Equilibrium and the CAPM . . . . . . . . . . . . . . . . 130 6.5.1 Pricing assets and predicting security returns . . . . . . . 130 6.5.2 Properties of the market portfolio . . . . . . . . . . . . . 131 6.5.3 The zero-beta CAPM . . . . . . . . . . . . . . . . . . . . 131 6.5.4 The traditional CAPM . . . . . . . . . . . . . . . . . . . 132 7 INVESTMENT ANALYSIS 137 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 7.2 Arbitrage and Pricing Derivative Securities . . . . . . . . . . . . 137 7.2.1 The binomial option pricing model . . . . . . . . . . . . 137 7.2.2 The Black-Scholes option pricing model . . . . . . . . . . 137 7.3 Multi-period Investment Problems . . . . . . . . . . . . . . . . . 140 7.4 Continuous Time Investment Problems . . . . . . . . . . . . . . . 140

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