MATHEMATICS FOR ECONOMISTS. An Introductory Textbook. Third Edition. Malcolm Pemberton and Nicholas Rau. UNIVERSITY OF TORONTO PRESS Toronto Buffalo

Transcription

1 MATHEMATICS FOR ECONOMISTS An Introductory Textbook Third Edition Malcolm Pemberton and Nicholas Rau UNIVERSITY OF TORONTO PRESS Toronto Buffalo

2 Contents Preface Dependence of Chapters Answers and Solutions The Greek Alphabet xi xvi xvii xviii 1 LINEAR EQUATIONS Straight line graphs An economic application: supply and demand Simultaneous equations Input-output analysis 18 Problems on Chapter LINEAR INEQUALITIES Inequalities Economic applications Linear programming 31 Problems on Chapter SETS AND FUNCTIONS Sets and numbers Functions Mappings 54 Problems on Chapter 3 56 Appendix to Chapter QUADRATICS, INDICES AND LOGARITHMS Quadratic functions and equations Maximising and minimising quadratic functions Indices Logarithms 75 Problems on Chapter 4 78 v

3 vi 5 SEQUENCES AND SERIES ÖU 5.1 Sequences ^ 5.2 Serie: Geometrie progressions in economics 89 Problems on Chapter 5 ^ 6 INTRODUCTION TO DIFFERENTIATION The derivative Linear approximations and differentiability Two useful rules Hl 6.4 Derivatives in economics 114 Problems on Chapter Appendix to Chapter METHODS OF DIFFERENTIATION The produet and quotient rules The composite funetion rule Monotonie funetions Inverse funetions 133 Problems on Chapter Appendix to Chapter MAXIMA AND MINIMA Critical points The second derivative Optimisation Convexity and concavity 157 Problems on Chapter EXPONENTIAL AND LOGARITHMIC FUNCTIONS The exponential funetion Natural logarithms Time in economics Problems on Chapter 9 Appendix to Chapter 9 ^82 10 APPROXIMATIONS Linear approximations and Newton's method The mean value theorem 2gg 10.3 Quadratic approximations and Taylor's theorem Taylor series ^97 Problems on Chapter Appendix to Chapter 10 9f, 9

4 vii 11 MATRIX ALGEBRA Vectors Matrices Matrix multiplication Square matrices 220 Problems on Chapter SYSTEMS OF LINEAR EQUATIONS Echelon matrices More on Gaussian elimination Inverting a matrix Linear dependence and rank 241 Problems on Chapter DETERMINANTS AND QUADRATIC FORMS Determinants Transposition Inner products Quadratic forms and Symmetrie matrices 259 Problems on Chapter Appendix to Chapter FUNCTIONS OF SEVERAL VARIABLES Partial derivatives Approximations and the chain rule Production functions Homogeneous functions 285 Problems on Chapter Appendix to Chapter IMPLICIT RELATIONS Implicit differentiation Comparative statics Generalising to higher dimensions 308 Problems on Chapter Appendix to Chapter OPTIMISATION WITH SEVERAL VARIABLES Critical points and their Classification Global optima, concavity and convexity Non-negativity constraints 335 Problems on Chapter Appendix to Chapter

5 viii 17 PRINCIPLES OF CONSTRAINED OPTIMISATION Lagrange multipliers 17.2 Extensions and warnings Economic applications Quasi-concave funetions 363 Problems on Chapter FURTHER TOPICS IN CONSTRAINED OPTIMISATION The meaning of the multipliers Envelope theorems Non-negativity constraints again Inequality constraints 387 Problems on Chapter INTEGRATION Areas and integrale Rules of Integration Integration in economics Numerical Integration 413 Problems on Chapter Appendix to Chapter ASPECTS OF INTEGRAL CALCULUS Methods of Integration Infinite Integrals Differentiation under the integral sign Double integrals 438 Problems on Chapter INTRODUCTION TO DYNAMICS Differential equations Linear equations with constant coefrcients Härder first-order equations Difference equations Problems on Chapter THE CIRCULAR FUNCTIONS Cycles, circles and trigonometry Extending the definitions 4^ Calculus with circular funetions 22.4 Polar coordinates Problems on Chapter 22, Q7

6 ix 23 COMPLEX NUMBERS The complex number system The trigonometric form Complex exponentials and polynomials 508 Problems on Chapter FURTHER DYNAMICS Second-order differential equations Qualitative behaviour Second-order difference equations 531 Problems on Chapter Appendix to Chapter EIGENVALUES AND EIGENVECTORS Diagonalisable matrices The characteristic polynomial Eigenvalues of Symmetrie matrices 555 Problems on Chapter Appendix to Chapter DYNAMIC SYSTEMS Systems of difference equations Systems of differential equations Qualitative behaviour Nonlinear systems 588 Problems on Chapter Appendix to Chapter DYNAMIC OPTIMISATION IN DISCRETE TIME The basic problem Variants of the basic problem Dynamic programming 609 Problems on Chapter Appendix to Chapter DYNAMIC OPTIMISATION IN CONTINUOUS TIME The basic problem and its variants The maximum principle Two problems in resource economics Problems with an infinite horizon 636 Problems on Chapter Appendix to Chapter

7 X 29 INTRODUCTION TO ANALYSIS Rigour More on the real number system Sequences of real numbers Continuity 661 Problems on Chapter METRIC SPACES AND EXISTENCE THEOREMS Metrie spaces Open, closed and compact sets Continuous mappings Fixed point theorems 680 Problems on Chapter Appendix to Chapter Notes on Further Reading 691 Index 693