MATHEMATICS FOR ECONOMISTS An Introductory Textbook Third Edition Malcolm Pemberton and Nicholas Rau UNIVERSITY OF TORONTO PRESS Toronto Buffalo
Contents Preface Dependence of Chapters Answers and Solutions The Greek Alphabet xi xvi xvii xviii 1 LINEAR EQUATIONS 1 1.1 Straight line graphs 1 1.2 An economic application: supply and demand 9 1.3 Simultaneous equations 11 1.4 Input-output analysis 18 Problems on Chapter 1 20 2 LINEAR INEQUALITIES 22 2.1 Inequalities 22 2.2 Economic applications 27 2.3 Linear programming 31 Problems on Chapter 2 36 3 SETS AND FUNCTIONS 38 3.1 Sets and numbers 38 3.2 Functions 45 3.3 Mappings 54 Problems on Chapter 3 56 Appendix to Chapter 3 58 4 QUADRATICS, INDICES AND LOGARITHMS 60 4.1 Quadratic functions and equations 60 4.2 Maximising and minimising quadratic functions 67 4.3 Indices 69 4.4 Logarithms 75 Problems on Chapter 4 78 v
vi 5 SEQUENCES AND SERIES ÖU 5.1 Sequences ^ 5.2 Serie: 85 5.3 Geometrie progressions in economics 89 Problems on Chapter 5 ^ 6 INTRODUCTION TO DIFFERENTIATION 97 6.1 The derivative 98 6.2 Linear approximations and differentiability 106 6.3 Two useful rules Hl 6.4 Derivatives in economics 114 Problems on Chapter 6 117 Appendix to Chapter 6 118 7 METHODS OF DIFFERENTIATION 121 7.1 The produet and quotient rules 121 7.2 The composite funetion rule 123 7.3 Monotonie funetions 127 7.4 Inverse funetions 133 Problems on Chapter 7 136 Appendix to Chapter 7 138 8 MAXIMA AND MINIMA 140 8.1 Critical points 140 8.2 The second derivative 145 8.3 Optimisation 148 8.4 Convexity and concavity 157 Problems on Chapter 8 165 9 EXPONENTIAL AND LOGARITHMIC FUNCTIONS 166 9.1 The exponential funetion 166 9.2 Natural logarithms 172 9.3 Time in economics Problems on Chapter 9 Appendix to Chapter 9 ^82 10 APPROXIMATIONS 184 10.1 Linear approximations and Newton's method 185 10.2 The mean value theorem 2gg 10.3 Quadratic approximations and Taylor's theorem 193 10.4 Taylor series ^97 Problems on Chapter 10 200 Appendix to Chapter 10 9f, 9
vii 11 MATRIX ALGEBRA 205 11.1 Vectors 206 11.2 Matrices 211 11.3 Matrix multiplication 216 11.4 Square matrices 220 Problems on Chapter 11 222 12 SYSTEMS OF LINEAR EQUATIONS 225 12.1 Echelon matrices 226 12.2 More on Gaussian elimination 229 12.3 Inverting a matrix 235 12.4 Linear dependence and rank 241 Problems on Chapter 12 244 13 DETERMINANTS AND QUADRATIC FORMS 246 13.1 Determinants 247 13.2 Transposition 252 13.3 Inner products 256 13.4 Quadratic forms and Symmetrie matrices 259 Problems on Chapter 13 265 Appendix to Chapter 13 268 14 FUNCTIONS OF SEVERAL VARIABLES 269 14.1 Partial derivatives 269 14.2 Approximations and the chain rule 276 14.3 Production functions 282 14.4 Homogeneous functions 285 Problems on Chapter 14 290 Appendix to Chapter 14 292 15 IMPLICIT RELATIONS 295 15.1 Implicit differentiation 295 15.2 Comparative statics 303 15.3 Generalising to higher dimensions 308 Problems on Chapter 15 313 Appendix to Chapter 15 316 16 OPTIMISATION WITH SEVERAL VARIABLES 318 16.1 Critical points and their Classification 318 16.2 Global optima, concavity and convexity 327 16.3 Non-negativity constraints 335 Problems on Chapter 16 338 Appendix to Chapter 16 340
viii 17 PRINCIPLES OF CONSTRAINED OPTIMISATION 343 17.1 Lagrange multipliers 17.2 Extensions and warnings 350 17.3 Economic applications 354 17.4 Quasi-concave funetions 363 Problems on Chapter 17 370 18 FURTHER TOPICS IN CONSTRAINED OPTIMISATION 373 18.1 The meaning of the multipliers 373 18.2 Envelope theorems 376 18.3 Non-negativity constraints again 383 18.4 Inequality constraints 387 Problems on Chapter 18 394 19 INTEGRATION 397 19.1 Areas and integrale 397 19.2 Rules of Integration 404 19.3 Integration in economics 410 19.4 Numerical Integration 413 Problems on Chapter 19 420 Appendix to Chapter 19 422 20 ASPECTS OF INTEGRAL CALCULUS 424 20.1 Methods of Integration 424 20.2 Infinite Integrals 429 20.3 Differentiation under the integral sign 433 20.4 Double integrals 438 Problems on Chapter 20 445 21 INTRODUCTION TO DYNAMICS 448 21.1 Differential equations 449 21.2 Linear equations with constant coefrcients 453 21.3 Härder first-order equations 459 21.4 Difference equations Problems on Chapter 21 22 THE CIRCULAR FUNCTIONS 475 22.1 Cycles, circles and trigonometry 475 22.2 Extending the definitions 4^2 22.3 Calculus with circular funetions 22.4 Polar coordinates Problems on Chapter 22, Q7
ix 23 COMPLEX NUMBERS 499 23.1 The complex number system 499 23.2 The trigonometric form 504 23.3 Complex exponentials and polynomials 508 Problems on Chapter 23 514 24 FURTHER DYNAMICS 515 24.1 Second-order differential equations 515 24.2 Qualitative behaviour 524 24.3 Second-order difference equations 531 Problems on Chapter 24 539 Appendix to Chapter 24 541 25 EIGENVALUES AND EIGENVECTORS 543 25.1 Diagonalisable matrices 543 25.2 The characteristic polynomial 548 25.3 Eigenvalues of Symmetrie matrices 555 Problems on Chapter 25 559 Appendix to Chapter 25 562 26 DYNAMIC SYSTEMS 564 26.1 Systems of difference equations 564 26.2 Systems of differential equations 572 26.3 Qualitative behaviour 576 26.4 Nonlinear systems 588 Problems on Chapter 26 595 Appendix to Chapter 26 598 27 DYNAMIC OPTIMISATION IN DISCRETE TIME 599 27.1 The basic problem 599 27.2 Variants of the basic problem 605 27.3 Dynamic programming 609 Problems on Chapter 27 615 Appendix to Chapter 27 617 28 DYNAMIC OPTIMISATION IN CONTINUOUS TIME 619 28.1 The basic problem and its variants 619 28.2 The maximum principle 624 28.3 Two problems in resource economics 629 28.4 Problems with an infinite horizon 636 Problems on Chapter 28 640 Appendix to Chapter 28 643
X 29 INTRODUCTION TO ANALYSIS 648 29.1 Rigour 648 29.2 More on the real number system 652 29.3 Sequences of real numbers 656 29.4 Continuity 661 Problems on Chapter 29 664 30 METRIC SPACES AND EXISTENCE THEOREMS 666 30.1 Metrie spaces 667 30.2 Open, closed and compact sets 672 30.3 Continuous mappings 677 30.4 Fixed point theorems 680 Problems on Chapter 30 685 Appendix to Chapter 30 687 Notes on Further Reading 691 Index 693