# Optimal Control Theory with Applications in Economics

Thomas A. Weber
Foreword by A. V. Kryazhimskiy
Pages: 376
https://www.jstor.org/stable/j.ctt5hhgc4

1. Front Matter
(pp. i-vi)
(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)