Fundamental Methods of Mathematical Economics

Citation

Chiang, A. C., & Wainwright, K. (2005). Fundamental Methods of Mathematical Economics (4th ed.). McGraw-Hill.

Chapter Summary

The book has twenty chapters in five parts, and runs to roughly 700 pages. McGraw-Hill's description says that twenty years separated this edition from the previous one, that Kevin Wainwright, a long-time user of the text, updated the examples, applications and theory, and that Alpha Chiang's presentation style was kept. Two chapters, 13 and 20, are marked as new.

Chapters 1–2: The Nature of Mathematical Economics; Economic Models

Part 1, "Introduction", sets the scene: what mathematical economics is, and what an economic model is. Both chapters come before any technique is taught.

Chapter 3: Equilibrium Analysis in Economics

Part 2, "Static (or Equilibrium) Analysis", opens with a chapter on equilibrium analysis in economics, the basic analytical idea on which the later parts build.

Chapters 4–5: Linear Models and Matrix Algebra

Two chapters, the second labelled a continuation of the first, take up linear models and the matrix algebra used to handle them. The title shows that matrices are taught inside the book rather than assumed.

Chapters 6–8: Comparative Statics and the Concept of the Derivative; Rules of Differentiation and Their Use in Comparative Statics; Comparative-Static Analysis of General-Function Models

Part 3, "Comparative-Static Analysis", pairs comparative statics with the concept of the derivative, then covers the rules of differentiation and their use in comparative statics, and finally applies comparative statics to models stated with general functions.

Chapters 9–13: Optimization: A Special Variety of Equilibrium Analysis; Exponential and Logarithmic Functions; The Case of More Than One Choice Variable; Optimization with Equality Constraints; Further Topics in Optimization

Part 4, "Optimization Problems", treats optimisation as a special case of equilibrium analysis. It proceeds from one choice variable to several, adds a chapter on exponential and logarithmic functions, takes up optimisation under equality constraints, and ends with a chapter new to this edition that includes the envelope theorem and duality.

Chapters 14–20: Economic Analysis and Integral Calculus; Continuous Time: First-Order Differential Equations; Higher-Order Differential Equations; Discrete Time: First-Order Difference Equations; Higher-Order Difference Equations; Simultaneous Differential Equations and Difference Equations; Introduction to Optimal Control Theory

Part 5, "Dynamic Analysis", begins with integral calculus, treats continuous-time models through differential equations and discrete-time models through difference equations, first at first order and then at higher order, and combines them in simultaneous systems. The final chapter, new to this edition, introduces optimal control theory.

Key Concepts

Parts 1 and 2: Introduction and Static Analysis (Chapters 1–5)

• Economic model: a simplified mathematical representation of an economic situation.
• Equilibrium analysis: finding the values of the variables at which a model is in balance.
• Matrix algebra: the notation and operations used to handle systems of linear equations.

Part 3: Comparative-Static Analysis (Chapters 6–8)

• Comparative statics: comparing equilibria to see how they respond to a change in an exogenous parameter.
• Derivative: the measure of a rate of change used to express such responses.
• General-function models: models stated with general rather than specific functional forms.

Part 4: Optimization Problems (Chapters 9–13)

• Optimization: choosing the value of one or more variables to make an objective as large or as small as possible.
• Equality constraints: restrictions that the choice variables must satisfy exactly.
• Envelope theorem and duality: results on how optimal values respond to parameters and on paired formulations of a problem, covered in the chapter new to this edition.

Part 5: Dynamic Analysis (Chapters 14–20)

• Differential equations: equations describing change in continuous time.
• Difference equations: equations describing change from one discrete period to the next.
• Optimal control theory: the choice of a path over time to optimise an objective, introduced in the last chapter.

Critical Analysis

Strengths

• A progression that follows how economists use mathematics. The five parts move from equilibrium to comparative statics, then optimisation, then dynamics, and the part titles name kinds of economic analysis (static, comparative-static, optimisation, dynamic) rather than only mathematical topics, which ties each technique to a use.
• Tools are taught inside the book. Matrix algebra begins at Chapter 4 and the concept of the derivative at Chapter 6, so the text does not depend entirely on separate courses having been completed.
• Breadth in dynamics. Continuous-time and discrete-time methods each have chapters at first and higher order, followed by simultaneous systems and a closing introduction to optimal control.
• A deliberately revised edition. The publisher says examples, applications and theory were updated, and the contents list shows two new chapters, one on further optimisation topics and one on optimal control, that extend the book's reach from comparative statics into duality and optimal control.

Limitations

• Dated by edition. The most recent edition is from 2005, after a gap of twenty years since the previous one; examples and applications reflect that date.
• Optimal control is only introduced. The final chapter is titled as an introduction, so readers needing dynamic optimisation in depth will need a more specialised text.
• A methods book. The chapters are organised by mathematical technique rather than by economic topic, so it works as a companion to economics courses rather than a replacement for them.

Where It Fits in the Course

Chiang and Wainwright sits between the gentle introductions and the specialised treatments in this list: it is more systematic than the former and broader than the latter. Lis and Rosser's Basic Mathematics for Economists is aimed at students whose school mathematics is limited or rusty, rebuilding arithmetic and algebra before introducing calculus and optimisation, so it suits readers who are not yet ready for Chapter 3 onward. Sydsæter, Hammond, Strøm and Carvajal's Essential Mathematics for Economic Analysis runs from elementary algebra through single- and multi-variable calculus, optimisation and basic linear algebra with many worked problems; it covers some of the same ground as Parts 2 to 4 and can serve as a parallel or alternative source. Ian Jacques's Mathematics for Economics and Business is another general mathematics text for economics and business students.

Brock and Malliaris's Differential Equations, Stability, and Chaos in Dynamic Economics is described as a rigorous graduate-level treatment covering stability, optimal control, nonlinear dynamics and chaos. It is the natural step after Chiang and Wainwright's Part 5 for readers who want dynamics at greater depth.

How to Use This Book

The book suits economics students who need the mathematics of intermediate and later courses: comparative statics, constrained optimisation and basic dynamics. The prerequisites are school-level algebra, some exposure to introductory calculus and a principles-level economics course. Because matrices and the derivative are introduced from scratch, it can also serve as a first systematic course for a motivated reader.

A practical plan:

• Core: Chapters 1 to 3, then the derivative and comparative statics in Chapters 6 to 8, and the optimisation chapters on one choice variable, several choice variables and equality constraints (Chapters 9, 11 and 12). These are the chapters to prioritise if time is short.
• As needed: Chapters 4 and 5 on matrices if linear algebra has not been studied; Chapter 10 on exponential and logarithmic functions if they are unfamiliar; Chapter 13 for envelope and duality results needed in advanced microeconomic theory.
• Dynamics: Chapters 14, 15 and 17 give the basic dynamic tools: integral calculus, first-order differential equations and first-order difference equations. Chapters 16, 18 and 19 extend these, and Chapter 20 is an introduction for readers moving towards dynamic macroeconomics or graduate study.

Work the problems by hand as the chapters proceed and check each technique against an economic model, since the book is built around applying mathematics to models. Read the titles of the parts as a map: if a course covers only static analysis and optimisation, Part 5 can be left for later.