Mathematics for Economics - 2nd Edition

by Michael Hoy, John Livernois, Chris McKenna, Ray Rees and Thanasis Stengos

• ISBN13: 9780262082945
• ISBN10: 0262082942
• Edition: 2nd
• Format: Hardcover
• Copyright: 2001-06-11
• Publisher: Mit Pr

Summary

This book offers a comprehensive presentation of the mathematics required to tackle problems in economic analysis. To give a better understanding of the mathematical concepts, the text follows the logic of the development of mathematics rather than that of an economics course. After a review of the fundamentals of sets, numbers, and functions, the book covers limits and continuity, the calculus of functions of one variable, linear algebra, multivariate calculus, and dynamics. To develop the student's problem-solving skills, the book works through a large number of examples and economic applications. The second edition includes simple game theory, l'Hocirc;pital's rule, Leibniz's rule, and a more intuitive development of the Hamiltonian. An instructor's manual is available.

Table of Contents

Preface	p. xiii
Introduction and Fundamentals
Introduction	p. 3
What Is an Economic Model?	p. 3
How to Use This Book	p. 8
Conclusion	p. 9
Review of Fundamentals	p. 11
Sets and Subsets	p. 11
Numbers	p. 23
Some Properties of Point Sets in Rn	p. 33
Functions	p. 43
Proof, Necessary and Sufficient Conditions *	p. 60
Sequences, Series, and Limits	p. 67
Definition of a Sequence	p. 67
Limit of a Sequence	p. 70
Present-Value Calculations	p. 75
Properties of Sequences	p. 84
Series	p. 89
Univariate Calculus and Optimization
Continuity of Functions	p. 115
Continuity of a Function of One Variable	p. 115
Economic Applications of Continuous and Discontinuous Functions	p. 125
Intermediate-Value Theorem	p. 143
The Derivative and Differential for Functions of One Variable	p. 155
Definition of a Tangent Line	p. 155
Definition of the Derivative and the Differential	p. 162
Conditions for Differentiability	p. 169
Rules of Differentiation	p. 175
Higher-Order Derivatives: Concavity and Convexity of a Function	p. 208
Taylor Series Formula and the Mean-Value Theorem	p. 218
Optimization of Functions of One Variable	p. 227
Necessary Conditions for Unconstrained Maxima and Minima	p. 227
Second-Order Conditions	p. 253
Optimization over an Interval	p. 265
Linear Algebra
Systems of Linear Equations	p. 279
Solving Systems of Linear Equations	p. 279
Linear Systems in n-Variables	p. 293
Matrices	p. 317
General Notation	p. 317
Basic Matrix Operations	p. 324
Matrix Transposition	p. 340
Some Special Matrices	p. 345
Determinants and the Inverse Matrix	p. 353
Defining the Inverse	p. 353
Obtaining the Determinant and Inverse of a 3 x 3 Matrix	p. 370
The Inverse of an n x n Matrix and Its Properties	p. 376
Cramer's Rule	p. 386
Some Advanced Topics in Linear Algebra *	p. 405
Vector Spaces	p. 405
The Eigenvalue Problem	p. 421
Quadratic Forms	p. 436
Multivariate Calculus
Calculus for Functions of n-Variables	p. 455
Partial Differentiation	p. 455
Second-Order Partial Derivatives	p. 469
The First-Order Total Differential	p. 477
Curvature Properties: Concavity and Convexity	p. 498
More Properties of Functions with Economic Applications	p. 513
Taylor Series Expansion *	p. 534
Optimization of Functions of n-Variables	p. 545
First-Order Conditions	p. 545
Second-Order Conditions	p. 560
Direct Restrictions on Variables	p. 569
Constrained Optimization	p. 585
Constrained Problems and Approaches to Solutions	p. 585
Second-Order Conditions for Constrained Optimization	p. 616
Existence, Uniqueness, and Characterization of Solutions	p. 622
Comparative Statics	p. 631
Introduction to Comparative Statics	p. 631
General Comparative-Statics Analysis	p. 643
The Envelope Theorem	p. 660
Concave Programming and the Kuhn-Tucker Conditions	p. 677
The Concave-Programming Problem	p. 677
Many Variables and Constraints	p. 686
Integration and Dynamic Methods
Integration	p. 701
The Indefinite Integral	p. 701
The Riemann (Definite) Integral	p. 709
Properties of Integrals	p. 721
Improper Integrals	p. 733
Techniques of Integration	p. 742
An Introduction to Mathematics for Economic Dynamics	p. 753
Modeling Time	p. 754
Linear, First-Order Difference Equations	p. 763
Linear, First-Order, Autonomous Difference Equations	p. 763
The General, Linear, First-Order Difference Equation	p. 780
Nonlinear, First-Order Difference Equations	p. 789
The Phase Diagram and Qualitative Analysis	p. 789
Cycles and Chaos	p. 799
Linear, Second-Order Difference Equations	p. 811
The Linear, Autonomous, Second-Order Difference Equation	p. 811
The Linear, Second-Order Difference Equation with a Variable Term	p. 838
Linear, First-Order Differential Equations	p. 849
Autonomous Equations	p. 849
Nonautonomous Equations	p. 870
Nonlinear, First-Order Differential Equations	p. 879
Autonomous Equations and Qualitative Analysis	p. 879
Two Special Forms of Nonlinear, First-Order Differential Equations	p. 888
Linear, Second-Order Differential Equations	p. 897
The Linear, Autonomous, Second-Order Differential Equation	p. 897
The Linear, Second-Order Differential Equation with a Variable Term	p. 919
Simultaneous Systems of Differential and Difference Equations	p. 929
Linear Differential Equation Systems	p. 929
Stability Analysis and Linear Phase Diagrams	p. 951
Systems of Linear Difference Equations	p. 976
Optimal Control Theory	p. 999
The Maximum Principle	p. 1002
Optimization Problems Involving Discounting	p. 1014
Alternative Boundary Conditions on x(T)	p. 1026
Infinite-Time Horizon Problems	p. 1040
Constraints on the Control Variable	p. 1053
Free-Terminal-Time Problems (T Free)	p. 1063
Appendix: Complex Numbers and Circular Functions	p. 1081
Answers	p. 1091
Index	p. 1123
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