Preface ...................................................... xiii
Part I Introduction and Fundamentals
Chapter 1 Introduction ......................................... 3
1.1 What Is an Economic Model? ................................. 3
1.2 How to Use This Book ....................................... 8
1.3 Conclusion ................................................. 9
Chapter 2 Review of Fundamentals .............................. 11
2.1 Sets and Subsets .......................................... 11
2.2 Numbers ................................................... 23
2.3 Some Properties of Point Sets in  n ....................... 31
2.4 Functions ................................................. 41
Chapter 3 Sequences, Series, and Limits ....................... 61
3.1 Definition of a Sequence .................................. 61
3.2 Limit of a Sequence ....................................... 65
3.3 Present-Value Calculations ................................ 69
3.4 Properties of Sequences ................................... 79
3.5 Series .................................................... 84
Part II Univariate Calculus and Optimization
Chapter 4 Continuity of Functions ............................ 103
4.1 Continuity of a Function of One Variable ................. 103
4.2 Economic Applications of Continuous and Discontinuous
Functions
Chapter 5 The Derivative and Differential for Functions of
One Variable .................................................. 127
5.1 Definition of a Tangent Line ............................. 127
5.2 Definition of the Derivative and the Differential ........ 134
5.3 Conditions for Differentiability ......................... 141
5.4 Rules of Differentiation ................................. 147
5.5 Higher Order Derivatives: Concavity and Convexity of
a Function ............................................... 175
5.6 Taylor Series Formula and the Mean-Value Theorem ......... 185
Chapter 6 Optimization of Functions of One Variable .......... 195
6.1 Necessary Conditions for Unconstrained Maxima and
Minima ................................................... 196
6.2 Second-Order Conditions .................................. 211
6.3 Optimization over an Interval ............................ 220
Part III Linear Algebra
Chapter 7 Systems of Linear Equations ........................ 235
7.1 Solving Systems of Linear Equations ...................... 236
7.2 Linear Systems in n-Variables ............................ 250
Chapter 8 Matrices ........................................... 267
8.1 General Notation ......................................... 267
8.2 Basic Matrix Operations .................................. 273
8.3 Matrix Transposition ..................................... 288
8.4 Some Special Matrices .................................... 293
Chapter 9 Determinants and the Inverse Matrix ................ 301
9.1 Defining the Inverse ..................................... 301
9.2 Obtaining the Determinant and Inverse of a 3 × 3
Matrix ................................................... 318
9.3 The Inverse of an n × n Matrix and Its Properties ........ 324
9.4 Cramer's Rule ............................................ 329
Chapter 10 Some Advanced Topics in Linear Algebra ............. 347
10.1 Vector Spaces ............................................ 347
10.2 The Eigenvalue Problem ................................... 363
10.3 Quadratic Forms .......................................... 378
Part IV Multivariate Calculus
Chapter 11 Calculus for Functions of n-Variables .............. 393
11.1 Partial Differentiation .................................. 393
11.2 Second-Order Partial Derivatives ......................... 407
11.3 The First-Order Total Differential ....................... 415
11.4 Curvature Properties: Concavity and Convexity ............ 436
11.5 More Properties of Functions with Economic
Applications ............................................. 451
11.6 Taylor Series Expansion .................................. 464
Chapter 12 Optimization of Functions of n-Variables ........... 473
12.1 First-Order Conditions ................................... 474
12.2 Second-Order Conditions .................................. 484
12.3 Direct Restrictions on Variables ......................... 491
Chapter 13 Constrained Optimization ........................... 503
13.1 Constrained Problems and Approaches to Solutions ......... 504
13.2 Second-Order Conditions for Constrained Optimization ..... 516
13.3 Existence, Uniqueness, and Characterization of
Solutions ................................................ 520
Chapter 14 Comparative Statics ................................ 529
14.1 Introduction to Comparative Statics ...................... 529
14.2 General Comparative-Statics Analysis ..................... 540
14.3 The Envelope Theorem ..................................... 554
Chapter 15 Concave Programming and the Kuhn-Tucker
Conditions .................................................... 567
15.1 The Concave-Programming Problem .......................... 567
15.2 Many Variables and Constraints ........................... 575
Part V Integration and Dynamic Methods
Chapter 16 Integration ........................................ 585
16.1 The Indefinite Integral .................................. 585
16.2 The Riemann (Definite) Integral .......................... 593
16.3 Properties of Integrals .................................. 605
16.4 Improper Integrals ....................................... 613
16.5 Techniques of Integration ................................ 623
Chapter 17 An Introduction to Mathematics for Economic
Dynamics ...................................................... 633
17.1 Modeling Time ............................................ 634
Chapter 18 Linear, First-Order Difference Equations ........... 643
18.1 Linear, First-Order, Autonomous Difference Equations ..... 643
18.2 The General, Linear, First-Order Difference Equation ..... 656
Chapter 19 Nonlinear, First-Order Difference Equations ........ 665
19.1 The Phase Diagram and Qualitative Analysis ............... 665
19.2 Cycles and Chaos ......................................... 673
Chapter 20 Linear, Second-Order Difference Equations .......... 681
20.1 The Linear, Autonomous, Second-Order Difference
Equation ................................................. 681
20.2 The Linear, Second-Order Difference Equation with
a Variable Term .......................................... 708
Chapter 21 Linear, First-Order Differential Equations ......... 715
21.1 Autonomous Equations ..................................... 715
21.2 Nonautonomous Equations .................................. 731
Chapter 22 Nonlinear, First-Order Differential Equations ...... 739
22.1 Autonomous Equations and Qualitative Analysis ............ 739
22.2 Two Special Forms of Nonlinear, First-Order
Differential Equations ................................... 748
Chapter 23 Linear, Second-Order Differential Equations ........ 753
23.1 The Linear, Autonomous, Second-Order Differential
Equation ................................................. 753
23.2 The Linear, Second-Order Differential Equation with
a Variable Term .......................................... 772
Chapter 24 Simultaneous Systems of Differential and
Difference Equations .......................................... 781
24.1 Linear Differential Equation Systems ..................... 781
24.2 Stability Analysis and Linear Phase Diagrams ............. 803
24.3 Systems of Linear Difference Equations ................... 825
Chapter 25 Optimal Control Theory ............................. 845
25.1 The Maximum Principle .................................... 848
25.2 Optimization Problems Involving Discounting .............. 860
25.3 Alternative Boundary Conditions on x(T) .................. 872
25.4 Infinite-Time Horizon Problems ........................... 886
25.5 Constraints on the Control Variable ...................... 899
25.6 Free-Terminal-Time Problems (T Free) ..................... 909
Answers ....................................................... 921
Index ......................................................... 953
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