Mathematics for Economists. with Applications. James Bergin. Routledge R Taylor & Francis Group LONDON AND NEW YORK

Transcription

1 Mathematics for Economists with Applications James Bergin Routledge R Taylor & Francis Group LONDON AND NEW YORK

2 NjjjNjN Contenl List offigures Preface xv xxi 1 Introduction Introduction A brief summary by chapter 1 2 Matrices and systems of equations Introduction Some motivating examples Vectors Matrices Matrix addition and subtraction Matrix multiplication Matrix transpose Matrix inversion Additional remarks on determinants Cramer's rule Gaussian elimination 2.4 Linear dependence and independence Matrix rank and linear independence 2.5 Solutions of systems of equations v

3 2.5.1 Solving systems of equations: a geometric view Solutions of equation systems: rank and linear independence Special matrices 64 Exercises for Chapter Linear algebra: applications Introduction The market model Adjacency matrices Input-output analysis Inflation and unemployment dynamics Stationary (invariant) distributions Convergence to a stationary distribution Computation of the invariant distribution Econometrics Derivation of the least squares estimator 97 Exercises for Chapter Linear programming Introduction Formulation The feasible region Finding an optimal Solution Dual prices and slack variables Some examples Duality reconsidered Basic solutions Equality-constrained programs Definition and identification ofbasic solutions Duality principles Duality and dual prices 134 Exercises for Chapter Functions of one variable 5.1 Introduction 5.2 Some examples

4 vii CONTENTS Demand functions Present value and the interest rate Taxation Options Continuity and differentiability Preliminaries Inequalities and absolute value Continuity Differentiability Types of function Plots of various functions The log and exponential functions Growth rates Additivity of continuous growth rates The functions lnx (logg x) and e kx Some rules of calculus l'hospital's rule Higher derivatives 170 Exercises for Chapter Functions of one variable: applications Introduction Optimization The first-order condition: issues The second-order condition Applications: profit maximization and related Problems Concavity and optimization Summary Externalities The free-rider problem Comparative statics Elasticity of demand Elasticity, revenue and profit maximization Taxation and monopoly Taxation incidence in supply and demand 205

5 viii Aunittax An ad-valorem tax Elasticities and taxation Ramsey pricing 211 Exercises for Chapter Systems of equations, differentials and derivatives Introduction Partial derivatives Level contours The elasticity of substitution Market equilibrium: an application Oligopoly: linear demand Oligopoly: non-linear demand Industry concentration indices Total differentials The impact of parameter changes Solutions to systems of equations Existence and properties of solutions The implicit function theorem General systems of equations 253 Exercises for Chapter Taylor series Introduction Taylor series approximations: an example Taylor series expansions Approximations and accuracy Applications of Taylor series expansions Concavity and the second derivative Roots of a function Numerical optimization Expected utility theory and behavior towards risk Expected utility theory Risk-averse preferences and welfare loss Risk aversion and the cost of small risk Diversification and portfolio selection 293

6 ix CONTENTS Diversification Portfolio selection The multivariate version oftaylor's theorem Proof oftaylor's theorem 297 Exercises for Chapter Vectors Introduction Vectors: length and distance Vectors: direction and angles Hyperplanes and direction of increase Parkas' lemma 316 Exercises for Chapter Quadratic forms Introduction Quadratic forms Positive and negative definite matrices Symmetrie matrices Criteria for positive and negative definiteness Positive and negative semi-definiteness Definiteness with equality constraints 330 Exercises for Chapter Multivariate optimization Introduction The two-variable case Motivation for the second-order conditions Failure of the second-order conditions The envelope theorem Optimization: n variables Motivation for the Hessian conditions Definiteness and second-order conditions Concavity, convexity and the Hessian matrix 367 Exercises for Chapter Equality-constrained optimization 12.1 Introduction

7 A motivating example The two-variable case Motivation for first- and second-order conditions The first-order condition The second-order condition A geometric perspective The n-variable case Quadratic forms and bordered matrices Bordered Hessians and optimization The second-order condition reconsidered Interpretation of the Lagrange multiplier Optimization: concavity and convexity Quasiconcavity and bordered Hessians Optimization with many constraints 411 Exercises for Chapter Inequality-constrained optimization Introduction A motivating example Overview Optimization The constraint qualification Complementary slackness Minimization and maximization Global and local optima Non-negativity constraints Necessary conditions and non-convexities The Lagrange multiplier Equality and inequality constraints 444 Exercises for Chapter Integration Introduction The integral Common integrals and rules of Integration Measuring welfare: surplus Consumer surplus 452

8 Producer surplus Consumer surplus and welfare Cost, supply and profit Average and marginal cost Taxation and consumer surplus Present value Leibnitz's rule Inequality measures Ramsey pricing Welfare measures Consumer surplus Compensating Variation Equivalent Variation Comparison of welfare measures 485 Exercises for Chapter Eigenvalues and eigenvectors Introduction Definitions and basic properties An application: dynamic models Supply and demand dynamics A model of population growth The Perron Frobenius theorem Some applications Webpage ranking Leslie matrices Eigenvalues and eigenvectors of Symmetrie matrices Real and complex eigenvalues and eigenvectors Diagonalization Properties of eigenvalues and eigenvectors Largest eigenvalues and eigenvalue multiplicity 534 Exercises for Chapter Differential equations Introduction Preliminary discussion First-order linear differential equations 544

9 xii Constant coefficients Variable coefficients Some non-linear first-order differential equations The logistic model The Bernoulli equation First-order differential equations: existence of solutions Second- and higher-order differential equations Constant coefficients Derivation of Solution Systems of differential equations Stability 566 Exercises for Chapter Linear difference equations Introduction Motivating examples First-order linear difference equations Solving first-order linear difference equations The constant forcing function The cobweb model Equations with variable forcing function Second-order linear difference equations The inhomogeneous equation The homogeneous equation The general Solution Stability Examples The Samuelson multiplier-accelerator model Vector difference equations The particular Solution The complementary Solution The»-variable case Miscellaneous calculations Various forcing functions: solutions Roots of the characteristic equation Stability ggg

10 xiii CONTENTS Exercises for Chapter Probability and distributions Introduction Random variables The distribution function and density The conditional distribution Joint distributions Discrete distributions The Bernoulli distribution The binomial distribution The Poisson distribution The Poisson process Continuous distributions The uniform distribution The exponential distribution The normal distribution Moment-generating functions The chi-square distribution The t distribution The F distribution 631 Exercises for Chapter Estimation and hypothesis testing Introduction Estimation Unbiasedness and efficiency Large-sample behavior Law of large numbers The central limit theorem Hypothesis testing A hypothesis test Types of error Simple null and alternative hypotheses Uniformly most powerful tests Two-sided tests Unbiased tests 662

11 Use of chi-square and F tests Econometric applications Matrix computations Ordinary least squares Distributions and hypothesis testing Test statistics Hypothesis testing for the linear model 672 Exercises for Chapter Bibliography 681 Index 683