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Mathematics for Economics
Michael Hoy, John Livernois, Chris McKenna, Ray Rees, Anthanassios Stengos
The MIT Press
This paperback edition is not available in the U.S. and Canada.
This book offers a comprehensive presentation of the mathematicsrequired to tackle problems in economic analysis. To give a betterunderstanding of the mathematical concepts, the text follows the logicof the development of mathematics rather than that of an economicscourse. After a review of the fundamentals of sets, numbers, andfunctions, the book covers limits and continuity, the calculus offunctions of one variable, linear algebra, multivariate calculus, anddynamics. To develop the student's problem-solving skills, the bookworks through a large number of examples and economic applications. Thesecond edition includes simple game theory, l'Hôpital's rule, Leibniz'srule, and a more intuitive development of the Hamiltonian. Aninstructor's manual is available.
About the Authors
Michael Hoy is a faculty member in the Economics Department at the University of Guelph, Ontario.
John Livernois is a faculty member in the Economics Department at the University of Guelph, Ontario.
Chris McKenna is a faculty member in the Economics Department at the University of Guelph, Ontario.
Ray Rees is a faculty member at the Ludwig Maximilians University, Munich.
Anthanassios Stengos is a faculty member in the Economics Department at the University of Guelph, Ontario.
Preface xiii
I.Introduction and Fundamentals
1.Introduction3
1.1What Is an Economic Model?3
1.2How to Use This Book8
1.3Conclusion9
2.Review of Fundamentals11
2.1Sets and Subsets11
2.2Numbers23
2.3Some Properties of Point Sets in Rn33
2.4Functions43
2.5Proof, Necessary and Sufficient Conditions *60
3.Sequences, Series, and Limits67
3.1Definition of a Sequence67
3.2Limit of a Sequence70
3.3Present-Value Calculations75
3.4Properties of Sequences84
3.5Series89
II.Univariate Calculus and Optimization
4.Continuity of Functions115
4.1Continuity of a Function of One Variable115
4.2Economic Applications of Continuous and Discontinuous
Functions125
4.3Intermediate-Value Theorem143
5.The Derivative and Differential for Functions of One
Variable155
5.1Definition of a Tangent Line155
5.2Definition of the Derivative and the Differential162
5.3Conditions for Differentiability169
5.4Rules of Differentiation175
5.5Higher-Order Derivatives: Concavity and Convexity of a
Function208
5.6Taylor Series Formula and the Mean-Value Theorem218
6.Optimization of Functions of One Variable227
6.1Necessary Conditions for Unconstrained Maxima and
Minima227
6.2Second-Order Conditions253
6.3Optimization over an Interval265
III.Linear Algebra
7.Systems of Linear Equations279
7.1Solving Systems of Linear Equations279
7.2Linear Systems in n-Variables293
8.Matrices317
8.1General Notation317
8.2Basic Matrix Operations324
8.3Matrix Transposition340
8.4Some Special Matrices345
9.Determinants and the Inverse Matrix353
9.1Defining the Inverse353
9.2Obtaining the Determinant and Inverse of a 3 x 3
Matrix370
9.3The Inverse of an n x n Matrix and Its Properties376
9.4Cramer's Rule386
10.Some Advanced Topics in Linear Algebra *405
10.1Vector Spaces405
10.2The Eigenvalue Problem421
10.3Quadratic Forms436
IV.Multivariate Calculus
11.Calculus for Functions of n-Variables455
11.1Partial Differentiation455
11.2Second-Order Partial Derivatives469
11.3The First-Order Total Differential477
11.4Curvature Properties: Concavity and Convexity498
11.5More Properties of Functions with Economic
Applications513
11.6Taylor Series Expansion *534
12.Optimization of Functions of n-Variables545
12.1First-Order Conditions545
12.2Second-Order Conditions560
12.3Direct Restrictions on Variables569
13.Constrained Optimization585
13.1Constrained Problems and Approaches to Solutions585
13.2Second-Order Conditions for Constrained Optimization616
13.3Existence, Uniqueness, and Characterization of
Solutions622
14.Comparative Statics631
14.1Introduction to Comparative Statics631
14.2General Comparative-Statics Analysis643
14.3The Envelope Theorem660
15.Concave Programming and the Kuhn-Tucker Conditions677
15.1The Concave-Programming Problem677
15.2Many Variables and Constraints686
V.Integration and Dynamic Methods
16.Integration701
16.1The Indefinite Integral701
16.2The Riemann (Definite) Integral709
16.3Properties of Integrals721
16.4Improper Integrals733
16.5Techniques of Integration742
17.An Introduction to Mathematics for Economic Dynamics753
17.1Modeling Time754
18.Linear, First-Order Difference Equations763
18.1Linear, First-Order, Autonomous Difference Equations763
18.2The General, Linear, First-Order Difference Equation780
19.Nonlinear, First-Order Difference Equations789
19.1The Phase Diagram and Qualitative Analysis789
19.2Cycles and Chaos799
20.Linear, Second-Order Difference Equations811
20.1The Linear, Autonomous, Second-Order Difference
Equation811
20.2The Linear, Second-Order Difference Equation with a
Variable Term838
21.Linear, First-Order Differential Equations849
21.1Autonomous Equations849
21.2Nonautonomous Equations870
22.Nonlinear, First-Order Differential Equations879
22.1Autonomous Equations and Qualitative Analysis879
22.2Two Special Forms of Nonlinear, First-Order Differential
Equations888
23.Linear, Second-Order Differential Equations897
23.1The Linear, Autonomous, Second-Order Differential
Equation897
23.2The Linear, Second-Order Differential Equation with a
Variable Term919
24.Simultaneous Systems of Differential and Difference
Equations929
24.1Linear Differential Equation Systems929
24.2Stability Analysis and Linear Phase Diagrams951
24.3Systems of Linear Difference Equations976
25.Optimal Control Theory999
25.1The Maximum Principle1002
25.2Optimization Problems Involving Discounting1014
25.3Alternative Boundary Conditions on x(T)1026
25.4Infinite-Time Horizon Problems1040
25.5Constraints on the Control Variable1053
25.6Free-Terminal-Time Problems (T Free)1063
Appendix: Complex Numbers and Circular Functions1081
Answers1091
Index1123
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