Mathematics for Economics and Finance

Transcription

1 Mathematics for Economics and Finance Michael Harrison and Patrick Waldron B Routledge Taylor & Francis Croup LONDON AND NEW YORK

2 Contents List of figures ix List of tables xi Foreword xiii Preface xv Acknowledgements xvii List of abbreviations xviii Notation and preliminaries xix Part I MATHEMATICS Introduction 3 1 Systems of linear equations and matrices Introduction Linear equations and examples Matrix operations Rules of matrix algebra Some special types of matrix and associated rules 15 2 Determinants Introduction Preliminaries Definition and properties Co-factor expansions of determinants Solution of systems of equations 39 3 Eigenvalues and eigenvectors Introduction Definitions and illustration Computation Unit eigenvalues Similar matrices Diagonalization 59

3 vi Contents 4 Conic sections, quadratic forms and definite matrices Introduction Conic sections Quadratic forms Definite matrices 77 5 Vectors and vector spaces 88 f 5.7 Introduction Vectors in 2-space and 3-space n-dimensional Euclidean vector spaces General vector spaces Linear transformations Introduction Definitions and illustrations Properties of linear transformations Linear transformations from W to W" Matrices of linear transformations Foundations for vector calculus Introduction Affine combinations, sets, hulls and functions Convex combinations, sets, hulls and functions Subsets of n-dimensional spaces Basic topology Supporting and separating hyperplane theorems Visualizing functions of several variables Limits and continuity Fundamental theorem of calculus Difference equations Introduction Definitions and classifications Linear, first-order difference equations Linear, autonomous, higher-order difference equations Systems of linear difference equations Vector calculus Introduction Partial and total derivatives Chain rule and product rule Elasticities Directional derivatives and tangent hyperplanes Taylor's theorem: deterministic version Multiple integration Implicit function theorem 236

4 Contents vii 10 Convexity and optimization Introduction Convexity and.concavity Unconstrained optimization Equality-constrained optimization Inequality-constrained optimization Duality 278 Part II APPLICATIONS ~ " v Introduction Macroeconomic applications Introduction Dynamic linear macroeconomic models Input-output analysis Single-period choice under certainty Introduction Definitions Axioms The consumer's problem and its dual General equilibrium theory Welfare theorems Probability theory Introduction Sample spaces and random variables Applications Vector spaces of random variables Random vectors Expectations and moments Multivariate normal distribution Estimation and forecasting Taylor's theorem: stochastic version Jensen's inequality Quadratic programming and econometric applications Introduction Algebra and geometry of ordinary least squares Canonical quadratic programming problem Stochastic difference equations Multi-period choice under certainty Introduction Measuring rates of return 394

5 viii Contents 15.3 Multi-period general equilibrium Term structure of interest rates Single-period choice under uncertainty Introduction Motivation Pricing state-contingent claims The expected-utility paradigm Risk aversion Arbitrage, risk neutrality and the efficient markets hypothesis Uncovered interest rate parity: Siegel's'paradox revisited Mean-variance paradigm Other non-expected-utility approaches Portfolio theory Introduction Preliminaries Single-period portfolio choice problem Mathematics of the portfolio frontier Market equilibrium and the capital asset pricing model Multi-currency considerations 487 Notes ' 493 References 501 Index 505