Mathematics for Economics

by Hoy, Livernois, McKenna, Rees, Stengos

| ISBN: 9780262368780 | Copyright 2022

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An updated edition of a widely used textbook, offering a clear and comprehensive presentation of mathematics for undergraduate economics students.

This text offers a clear and comprehensive presentation of the mathematics required to tackle problems in economic analyses, providing not only straightforward exposition of mathematical methods for economics students at the intermediate and advanced undergraduate levels but also a large collection of problem sets. This updated and expanded fourth edition contains numerous worked examples drawn from a range of important areas, including economic theory, environmental economics, financial economics, public economics, industrial organization, and the history of economic thought. These help students develop modeling skills by showing how the same basic mathematical methods can be applied to a variety of interesting and important issues.

The five parts of the text cover fundamentals, calculus, linear algebra, optimization, and dynamics. The only prerequisite is high school algebra; the book presents all the mathematics needed for undergraduate economics. New to this edition are “Reader Assignments,” short questions designed to test students' understanding before they move on to the next concept. The book's website offers additional material, including more worked examples (as well as examples from the previous edition). Separate solutions manuals for students and instructors are also available.

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Contents (pg. vii)
Preface (pg. xiii)
Glossary of Worked Examples (pg. xv)
Part I. Introduction and Fundamentals (pg. 1)
Chapter 1. Introduction (pg. 3)
1.1 What Is an Economic Model? (pg. 3)
1.2 How to Use This Book (pg. 14)
1.3 Conclusion (pg. 15)
Chapter 2. Review of the Fundamentals (pg. 19)
2.1 Sets and Subsets (pg. 19)
2.2 Numbers (pg. 31)
2.3 Beginning Topology: Point Sets and Distance in Rn (pg. 39)
2.4 Functions (pg. 52)
Chapter 3. Sequences, Series, and Limits (pg. 75)
3.1 Definition of a Sequence (pg. 75)
3.2 Limit of a Sequence (pg. 78)
3.3 Present-Value Calculations (pg. 83)
3.4 Properties of Sequences (pg. 96)
3.5 Series (pg. 100)
Part II. Univariate Calculus and Optimization (pg. 117)
Chapter 4. Continuity of Functions (pg. 119)
4.1 Continuity of a Function of One Variable (pg. 119)
4.2 Economic Applications of Continuous and Discontinuous Functions (pg. 128)
Chapter 5. The Derivative and Differential of Functions of One Variable (pg. 145)
5.1 The Tangent Line and the Derivative (pg. 145)
5.2 Definition of the Derivative and the Differential (pg. 152)
5.3 Conditions for Differentiability (pg. 160)
5.4 Rules of Differentiation (pg. 166)
5.5 Higher Order Derivatives: Concavity and Convexity of a Function (pg. 194)
5.6 Taylor Series Formula, Rolle’s Theorem, and the Mean-Value Theorem (pg. 207)
Chapter 6. Optimization of Functions of One Variable (pg. 225)
6.1 Necessary Conditions for Unconstrained Maxima and Minima (pg. 225)
6.2 Second-Order Conditions for a Local Optimum (pg. 239)
6.3 Optimization over an Interval (pg. 261)
Part III. Linear Algebra (pg. 285)
Chapter 7. Linear Equations and Vector Spaces (pg. 287)
7.1 Solving Systems of Linear Equations (pg. 287)
7.2 Linear Systems in n Variables (pg. 294)
7.3 Vectors in Rn (pg. 305)
Chapter 8. Matrices (pg. 333)
8.1 General Notation (pg. 333)
8.2 Basic Matrix Operations (pg. 338)
8.3 Matrix Transposition (pg. 352)
8.4 Some Special Matrices (pg. 357)
Chapter 9. Determinants and the Inverse Matrix (pg. 365)
9.1 Defining the Inverse (pg. 365)
9.2 Obtaining the Determinant and Inverse of a 3 × 3 Matrix (pg. 379)
9.3 The Inverse of an n ×n Matrix and Its Properties (pg. 384)
9.4 Cramer’s Rule (pg. 390)
9.5 Rank of a Matrix (pg. 400)
Chapter 10. Further Topics in Linear Algebra (pg. 407)
10.1 The Eigenvalue Problem (pg. 407)
10.2 Quadratic Forms (pg. 421)
10.3 Hyperplanes (pg. 430)
Part IV. Multivariate Calculus (pg. 441)
Chapter 11. Calculus for Functions of n Variables (pg. 443)
11.1 Partial Differentiation (pg. 443)
11.2 Second-Order Partial Derivatives (pg. 456)
11.3 The First-Order Total Differential (pg. 462)
11.4 Implicit Differentiation (pg. 465)
11.5 Curvature Properties: Concavity and Convexity (pg. 481)
11.6 Quasiconcavity and Quasiconvexity (pg. 496)
11.7 More Properties of Functions with Economic Applications (pg. 501)
11.8 Taylor Series Expansion (pg. 509)
Chapter 12. Optimization of Functions of n Variables (pg. 519)
12.1 First-Order Conditions (pg. 520)
12.2 Second-Order Conditions (pg. 530)
12.3 Direct Restrictions on Variables (pg. 541)
Chapter 13. Constrained Optimization (pg. 551)
13.1 Constrained Problems and Approaches to Solutions (pg. 551)
13.2 Second-Order Conditions for Constrained Optimization (pg. 573)
13.3 Existence, Uniqueness, and Characterization of Solutions (pg. 577)
13.4 Problems, Problems (pg. 586)
Chapter 14. Comparative Statics (pg. 597)
14.1 Introduction to Comparative Statics (pg. 597)
14.2 General Comparative Statics Analysis (pg. 603)
14.3 The Envelope Theorem (pg. 623)
Chapter 15. Nonlinear Programming and the Kuhn-Tucker Conditions (pg. 635)
15.1 The Kuhn-Tucker Conditions (pg. 636)
15.2 Hyperplane Theorems and Quasiconcavity (pg. 655)
Part V. Integration and Dynamic Methods (pg. 679)
Chapter 16. Integration (pg. 681)
16.1 The Indefinite Integral (pg. 681)
16.2 The Riemann (Definite) Integral (pg. 689)
16.3 Properties of Integrals (pg. 702)
16.4 Improper Integrals (pg. 710)
16.5 Techniques of Integration (pg. 720)
Chapter 17. An Introduction to Mathematics for Economic Dynamics (pg. 731)
17.1 Modeling Time (pg. 732)
Chapter 18. Linear, First-Order Difference Equations (pg. 743)
18.1 Linear, First-Order, Autonomous Difference Equations (pg. 743)
18.2 The General, Linear, First-Order Difference Equation (pg. 756)
Chapter 19. Nonlinear, First-Order Difference Equations (pg. 767)
19.1 The Phase Diagram and Qualitative Analysis (pg. 767)
19.2 Cycles and Chaos (pg. 774)
Chapter 20. Linear, Second-Order Difference Equations (pg. 783)
20.1 The Linear, Autonomous, Second-Order Difference Equation (pg. 783)
20.2 The Linear, Second-Order Difference Equation with a Variable Term (pg. 810)
Chapter 21. Linear, First-Order Differential Equations (pg. 817)
21.1 Autonomous Equations (pg. 817)
21.2 Nonautonomous Equations (pg. 833)
Chapter 22. Nonlinear, First-Order Differential Equations (pg. 843)
22.1 Autonomous Equations and Qualitative Analysis (pg. 843)
22.2 Two Special Forms of Nonlinear, First-Order Differential Equations (pg. 852)
Chapter 23. Linear, Second-Order Differential Equations (pg. 857)
23.1 The Linear, Autonomous, Second-Order Differential Equation (pg. 857)
23.2 The Linear, Second-Order Differential Equation with a Variable Term (pg. 876)
Chapter 24. Simultaneous Systems of Differential and Difference Equations (pg. 885)
24.1 Linear Differential Equation Systems (pg. 885)
24.2 Stability Analysis and Linear Phase Diagrams (pg. 907)
24.3 Systems of Linear Difference Equations (pg. 930)
Chapter 25. Optimal Control Theory (pg. 949)
25.1 The Maximum Principle (pg. 952)
25.2 Optimization Problems Involving Discounting (pg. 964)
25.3 Alternative Boundary Conditions on x(T) (pg. 975)
25.4 Infinite-Time-Horizon Problems (pg. 990)
25.5 Constraints on the Control Variable (pg. 1003)
25.6 Free-Terminal-Time Problems (T Free) (pg. 1013)
References and Further Reading (pg. 1025)
Answers (pg. 1027)
Index (pg. 1061)

Michael Hoy

Michael Hoy is Professor in the Department of Economics and Finance at the University of Guelph.

John Livernois

John Livernois is Professor in the Department of Economics at the University of Guelph

Chris McKenna

Chris McKenna is formerly a Professor in the Department of Economics and Finance at the University of Guelph.

Ray Rees

.Ray Rees is Professor of Economics Emeritus at the Center for Economic Studies (CES) at the University of Munich.

Thanasis Stengos

Thanasis Stengos is Professor in the Department of Economics and Finance at the University of Guelph.

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