“This book is immensely valuable, especially for students coming to this material for the first time. Professor Chiang has a singular talent for clear exposition of complex mathematical concepts. This text is simply the best introduction to dynamic optimization I have ever seen.” — John McDermott, University of South Carolina

“Chiang has done it again.” — Henry Thompson, Auburn University

“A brilliant, highly readable book. Bringing together tractable dynamics and a rich array of applications, it covers in depth some major analytical developments in dynamic macroeconomics. Dynamic macroeconomics in the 1990s was about introducing various kinds of market imperfections and heterogeneity in the models available before. This book teaches, in a comprehensive and understandable way, how to use and formulate these models. Chiang makes it insightful and natural for the reader, using the tools he has laid out, to go on to attack substantive and original research in dynamic macroeconomics. Invaluable for teachers and students alike.” — Zuhair Al-Fakhouri, Wayne State University

“This is the most understandable text I have come across on topics of optimization. The author discusses the formal elements of problems in an informal way to facilitate an easy grasp of the crucial points. I have learned and re-learned control theory from this book better than any other text.” — Abdul Qayum, Portland State University

Part I. INTRODUCTION

1. The Nature of Dynamic Optimization
Salient Features of Dynamic Optimization Problems / Variable Endpoints and Transversality Conditions / The Objective Functional / Alternative Approaches to Dynamic Optimization

Part II. THE CALCULUS OF VARIATIONS

2. The Fundamental Problem of the Calculus of Variations
The Euler Equation / Some Special Cases / Two Generalizations of the Euler Equation / Dynamic Optimization of a Monopolist / Trading Off Inflation and Unemployment

3. Transversality Conditions for Variable-Endpoint Problems
The General Transversality Condition / Specialized Transversality Conditions / Three Generalizations / The Optimal Adjustment of Labor Demand

4. Second-Order Conditions
The Concavity/Convexity Sufficient Condition / The Legendre Necessary Condition / First and Second Variations

5. Infinite Planning Horizon
Methodological Issues of Infinite Horizon / The Optimal Investment Path of a Firm / The Optimal Social Saving Behavior / Phase-Diagram Analysis / The Concavity/Convexity Sufficient Condition Again

6. Constrained Problems
Four Basic Types of Constraints / Some Economic Applications Reformulated / The Economics of Exhaustible Resources

Part III. OPTIMAL CONTROL THEORY

7. Optimal Control: The Maximum Principle
The Simplest Problem of Optimal Control / The Maximum Principle / The Rationale of the Maximum Principle / Alternative Terminal Conditions / The Calculus of Variations and Optimal Control Theory Compared / The Political Business Cycle / Energy Use and Environmental Quality

8. More on Optimal Control
An Economic Interpretation of the Maximum Principle / The Current-Value Hamiltonian / Sufficient Conditions / Problems with Several State and Control Variables / Antipollution Policy

9. Infinite-Horizon Problems
Transversality Conditions / Some Counterexamples Reexamined / The Neoclassical Theory of Optimal Growth / Exogenous and Endogenous Technological Progress

10. Optimal Control with Constraints
Constraints Involving Control Variables / The Dynamics of a Revenue-Maximizing Firm / State-Space Constraints / Economic Examples of State-Space Constraints / Limitations of Dynamic Optimization

Answers to Selected Exercise Problems