Optimal Control Theory with Applications in Economics

Optimal Control Theory with Applications in Economics

Thomas A. Weber
Foreword by A. V. Kryazhimskiy
Copyright Date: 2011
Published by: MIT Press
Pages: 376
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  • Book Info
    Optimal Control Theory with Applications in Economics
    Book Description:

    This book bridges optimal control theory and economics, discussing ordinary differential equations, optimal control, game theory, and mechanism design in one volume. Technically rigorous and largely self-contained, it provides an introduction to the use of optimal control theory for deterministic continuous-time systems in economics. The theory of ordinary differential equations (ODEs) is the backbone of the theory developed in the book, and chapter 2 offers a detailed review of basic concepts in the theory of ODEs, including the solution of systems of linear ODEs, state-space analysis, potential functions, and stability analysis. Following this, the book covers the main results of optimal control theory, in particular necessary and sufficient optimality conditions; game theory, with an emphasis on differential games; and the application of control-theoretic concepts to the design of economic mechanisms. Appendixes provide a mathematical review and full solutions to all end-of-chapter problems. The material is presented at three levels: single-person decision making; games, in which a group of decision makers interact strategically; and mechanism design, which is concerned with a designer's creation of an environment in which players interact to maximize the designer's objective. The book focuses on applications; the problems are an integral part of the text. It is intended for use as a textbook or reference for graduate students, teachers, and researchers interested in applications of control theory beyond its classical use in economic growth. The book will also appeal to readers interested in a modeling approach to certain practical problems involving dynamic continuous-time models.

    eISBN: 978-0-262-29848-3
    Subjects: Business

Table of Contents

  1. Front Matter
    (pp. i-vi)
  2. Table of Contents
    (pp. vii-viii)
  3. Foreword
    (pp. ix-x)
    A. V. Kryazhimskiy

    Since the discovery, by L. S. Pontryagin, of the necessary optimality conditions for the control of dynamic systems in the 1950s, mathematical control theory has found numerous applications in engineering and in the social sciences. T. A. Weber has dedicated his book to optimal control theory and its applications in economics. Readers can find here a succinct introduction to the basic control-theoretic methods, and also clear and meaningful examples illustrating the theory.

    Remarkable features of this text are rigor, scope, and brevity, combined with a well-structured hierarchical approach. The author starts with a general view on dynamical systems from the...

  4. Acknowledgments
    (pp. xi-xii)
  5. 1 Introduction
    (pp. 1-16)

    Change is all around us. Dynamic strategies seek to both anticipate and effect such change in a given system so as to accomplish objectives of an individual, a group of agents, or a social planner. This book offers an introduction to continuous-time systems and methods for solving dynamic optimization problems at three different levels: single-person decision making, games, and mechanism design. The theory is illustrated with examples from economics. Figure 1.1 provides an overview of the book’s hierarchical approach.

    The first and lowest level, single-person decision making, concerns the choices made by an individual decision maker who takes the evolution...

  6. 2 Ordinary Differential Equations
    (pp. 17-80)

    An ordinary differential equation (ODE) describes the evolution of a variable$x(t)$as a function of timet. The solution of such an equation depends on the initialstate${x_0}$at a given time${t_0}$. For example,$x(t)$might denote the number of people using a certain product at time$t \ge {t_0}$(e.g., a mobile phone). An ordinary differential equation describes how the (dependent)variable$x(t)$changes as a function of time and its own current value. The change of state from$x(t)$to$x(t + \delta )$between the time instantstand$t + \delta $as the increment$\delta $tends to zero defines the...

  7. 3 Optimal Control Theory
    (pp. 81-148)

    Chapter 2 discussed the evolution of a system from a known initial state$x({t_0}) = {x_0}$as a function of time$t \ge {t_0}$, described by the differential equation$\dot x(t) = f(t,x(t))$. For well-posed systems the initial data$({t_0},{x_0})$uniquely determine the state trajectory$x(t)$for all$t \ge {t_0}$. In actual economic systems, such as the product-diffusion process in example 2.1, the state trajectory may be influenced by a decision maker’s actions, in which case the decision maker exerts control over the dynamic process. For example, product diffusion might depend on the price, or the higher the marketing effort, the faster one expects the product’s consumer base...

  8. 4 Game Theory
    (pp. 149-206)

    The strategic interaction generated by the choices available to different agents is modeled in the form of agame. A game that evolves over several time periods is called adynamic game, whereas a game that takes place in one single period is termed astatic game. Depending on the information available to each agent, a game may be either ofcompleteorincomplete information. Figure 4.1 provides an overview of these main types of games, which are employed for the exposition of the fundamental concepts of game theory in section 4.2.

    Every game features a set of players, together...

  9. 5 Mechanism Design
    (pp. 207-230)

    This chapter reviews the basics of static mechanism design in settings where a principal faces a single agent of uncertain type. The aim of the resulting screening contract is for the principal to obtain the agent’s type information in order to avert adverse selection (see example 4.19), maximizing her payoffs. Nonlinear pricing is discussed as an application of optimal control theory.

    A decision maker may face a situation in which payoff-relevant information is held privately by another economic agent. For instance, suppose the decision maker is a sales manager. In a discussion with a potential buyer, she is thinking about...

  10. Appendix A: Mathematical Review
    (pp. 231-252)
  11. Appendix B: Solutions to Exercises
    (pp. 253-332)
  12. Appendix C: Intellectual Heritage
    (pp. 333-334)
  13. References
    (pp. 335-348)
  14. Index
    (pp. 349-360)