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EnjoyMathematical Economics and FinanceMichael Harrison Patrick WaldronDecember 2, 1998ContentsList of Tables iiiList of Figures vPREFACE viiWhat Is Economics? . . . . . . . . . . . . . . . . . . . . . . . . . . . viiWhat Is Mathematics? . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiNOTATION ixI MATHEMATICS 11 LINEAR ALGEBRA 31.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Systems of Linear Equations and Matrices . . . . . . . . . . . . . 31.3 Matrix Operations . . . . . . . . . . . . . . . . . . . . . . . . . . 71.4 Matrix Arithmetic . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5 Vectors and Vector Spaces . . . . . . . . . . . . . . . . . . . . . 111.6 Linear Independence . . . . . . . . . . . . . . . . . . . . . . . . 121.7 Bases and Dimension . . . . . . . . . . . . . . . . . . . . . . . . 121.8 Rank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131.9 Eigenvalues and Eigenvectors . . . . . . . . . . . . . . . . . . . . 141.10 Quadratic Forms . . . . . . . . . . . . . . . . . . . . . . . . . . 151.11 Symmetric Matrices . . . . . . . . . . . . . . . . . . . . . . . . . 151.12 Definite Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . 152 VECTOR CALCULUS 172.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.2 Basic Topology . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.3 Vector-valued Functions and Functions of Several Variables . . . 18Revised: December 2, 1998ii CONTENTS2.4 Partial and Total Derivatives . . . . . . . . . . . . . . . . . . . . 202.5 The Chain Rule and Product Rule . . . . . . . . . . . . . . . . . 212.6 The Implicit Function Theorem . . . . . . . . . . . . . . . . . . . 232.7 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . . 242.8 Taylor’s Theorem: Deterministic Version . . . . . . . . . . . . . 252.9 The Fundamental Theorem of Calculus . . . . . . . . . . . . . . 263 CONVEXITY AND OPTIMISATION 273.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.2 Convexity and Concavity . . . . . . . . . . . . . . . . . . . . . . 273.2.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . 273.2.2 Properties of concave functions . . . . . . . . . . . . . . 293.2.3 Convexity and differentiability . . . . . . . . . . . . . . . 303.2.4 Variations on the convexity theme . . . . . . . . . . . . . 343.3 Unconstrained Optimisation . . . . . . . . . . . . . . . . . . . . 393.4 Equality Constrained Optimisation:The Lagrange Multiplier Theorems . . . . . . . . . . . . . . . . . 433.5 Inequality Constrained Optimisation:The Kuhn-Tucker Theorems . . . . . . . . . . . . . . . . . . . . 503.6 Duality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58II APPLICATIONS 614 CHOICE UNDER CERTAINTY 634.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.2 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.3 Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 664.4 Optimal Response Functions:Marshallian and Hicksian Demand . . . . . . . . . . . . . . . . . 694.4.1 The consumer’s problem . . . . . . . . . . . . . . . . . . 694.4.2 The No Arbitrage Principle . . . . . . . . . . . . . . . . . 704.4.3 Other Properties of Marshallian demand . . . . . . . . . . 714.4.4 The dual problem . . . . . . . . . . . . . . . . . . . . . . 724.4.5 Properties of Hicksian demands . . . . . . . . . . . . . . 734.5 Envelope Functions:Indirect Utility and Expenditure . . . . . . . . . . . . . . . . . . 734.6 Further Results in Demand Theory . . . . . . . . . . . . . . . . . 754.7 General Equilibrium Theory . . . . . . . . . . . . . . . . . . . . 784.7.1 Walras’ law . . . . . . . . . . . . . . . . . . . . . . . . . 784.7.2 Brouwer’s fixed point theorem . . . . . . . . . . . . . . . 78Revised: December 2, 1998CONTENTS iii4.7.3 Existence of equilibrium . . . . . . . . . . . . . . . . . . 784.8 The Welfare Theorems . . . . . . . . . . . . . . . . . . . . . . . 784.8.1 The Edgeworth box . . . . . . . . . . . . . . . . . . . . . 784.8.2 Pareto efficiency . . . . . . . . . . . . . . . . . . . . . . 784.8.3 The First Welfare Theorem . . . . . . . . . . . . . . . . . 794.8.4 The Separating Hyperplane Theorem . . . . . . . . . . . 804.8.5 The Second Welfare Theorem . . . . . . . . . . . . . . . 804.8.6 Complete markets . . . . . . . . . . . . . . . . . . . . . 824.8.7 Other characterizations of Pareto efficient allocations . . . 824.9 Multi-period General Equilibrium . . . . . . . . . . . . . . . . . 845 CHOICE UNDER UNCERTAINTY 855.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 855.2 Review of Basic Probability . . . . . . . . . . . . . . . . . . . . 855.3 Taylor’s Theorem: Stochastic Version . . . . . . . . . . . . . . . 885.4 Pricing State-Contingent Claims . . . . . . . . . . . . . . . . . . 885.4.1 Completion of markets using options . . . . . . . . . . . 905.4.2 Restrictions on security values implied by allocational ef-ficiency and covariance with aggregate consumption . . . 915.4.3 Completing markets with options on aggregate consumption 925.4.4 Replicating elementary claims with a butterfly spread . . . 935.5 The Expected Utility Paradigm . . . . . . . . . . . . . . . . . . . 935.5.1 Further axioms . . . . . . . . . . . . . . . . . . . . . . . 935.5.2 Existence of expected utility functions . . . . . . . . . . . 955.6 Jensen’s Inequality and Siegel’s Paradox . . . . . . . . . . . . . . 975.7 Risk Aversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 995.8 The Mean-Variance Paradigm . . . . . . . . . . . . . . . . . . . 1025.9 The Kelly Strategy . . . . . . . . . . . . . . . . . . . . . . . . . 1035.10 Alternative Non-Expected Utility Approaches . . . . . . . . . . . 1046 PORTFOLIO THEORY 1056.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.2 Notation and preliminaries . . . . . . . . . . . . . . . . . . . . . 1056.2.1 Measuring rates of return . . . . . . . . . . . . . . . . . . 1056.2.2 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . 1086.3 The Single-period Portfolio Choice Problem . . . . . . . . . . . . 1106.3.1 The canonical portfolio problem . . . . . . . . . . . . . . 1106.3.2 Risk aversion and portfolio composition . . . . . . . . . . 1126.3.3 Mutual fund separation . . . . . . . . . . . . . . . . . . . 1146.4 Mathematics of the Portfolio Frontier . . . . . . . . . . . . . . . 116Revised: December 2, 1998iv CONTENTS6.4.1 The portfolio frontier in <N:risky assets only . . . . . . . . . . . . . . . . . . . . . . 1166.4.2 The portfolio frontier in mean-variance space:risky assets only . . . . . . . . . . . . . . . . . . . . . . 1246.4.3 The portfolio frontier in <N:riskfree and risky assets . . . . . . . . . . . . . . . . . . 1296.4.4 The portfolio frontier in mean-variance space:riskfree and risky assets . . . . . . . . . . . . . . . . . . 1296.5 Market Equilibrium and the CAPM . . . . . . . . . . . . . . . . 1306.5.1 Pricing assets and predicting security returns . . . . . . . 1306.5.2 Properties of the market portfolio . . . . . . . . . . . . . 1316.5.3 The zero-beta CAPM . . . . . . . . . . . . . . . . . . . . 1316.5.4 The traditional CAPM . . . . . . . . . . . . . . . . . . . 1327 INVESTMENT ANALYSIS 1377.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1377.2 Arbitrage and Pricing Derivative Securities . . . . . . . . . . . . 1377.2.1 The binomial option pricing model . . . . . . . . . . . . 1377.2.2 The Black-Scholes option pricing model . . . . . . . . . . 1377.3 Multi-period Investment Problems . . . . . . . . . . . . . . . . . 1407.4 Continuous Time Investment Problems . . . . . . . . . . . . . . . 140Revised: December 2, 1998