##
**Recursive methods in economic dynamics. With the collaboration of Edward C. Prescott.**
*(English)*
Zbl 0774.90018

Cambridge, MA: Harvard University Press. xviii, 588 p. (1989).

This is a graduate-level text that provides a comprehensive and rigorous treatment of dynamic programming for both deterministic and stochastic models, with a view towards applications in economics. The emphasis is on discrete-time time-invariant models, and in the stochastic case on systems with a Markov process (or chain) description, where the instrument (control) variables are allowed to depend on the current values of the state variables. Hence, using the terminology of stochastic control theory, problems with imperfect (noisy) state measurements are not covered.

The book is, for the most part, self-contained, as most of the mathematical background needed is developed in parallel and with considerable attention given to detail. It is organized into four parts, comprising a total of 18 chapters. The first part (comprising Chapters 1 and 2) provides a general introduction to the models as well as the solution techniques to be studied in the remainder of the book, and motivates the need for dynamic models (and thereby recursive methods) for decision-making in economics. Parts II and III are devoted to deterministic and stochastic models, respectively, and study in each case the dynamic optimization problems faced by an individual decision maker (which could be a firm, a consumer or a social planner). Part IV, on the other hand, analyses the existence and characterization of competitive equilibria in such systems, be they Pareto-optimal or not.

Part II, which covers deterministic systems, opens with a discussion of mathematical preliminaries, such as metric spaces, completeness, contraction mappings and fixed points, use of contraction mappings in the Bellman equation for discounted cost problems, and the theory of (static) optimization – all these constituting Chapter 3. Chapter 4 is devoted to a detailed study of the dynamic programming (Bellman) equation for infinite-horizon discounted cost problems, first discussing the principle of optimality, followed by discussion of the existence of a value function and an optimum, the method of successive approximations, differentiability of the value function, Euler equations, and dynamic optimization with unbounded returns. This theory is then applied in Chapter 5 to various dynamic models that arise in economics, among which are the one-sector model of economic growth, human capital accumulation and growth model, the inventory control problem, and a consumption- savings problem. In the final chapter of Part II, the steady-state behavior of dynamic models under optimal or non-optimal plans are studied, by focusing attention on the Lyapunov (global) stability theorem for nonlinear systems, and on linear approximations and the stability of linear systems (with particular application to the Euler equation).

Part III, which is devoted to stochastic models, follows the general pattern of Part II. The first two chapters (7 and 8) introduce the requires mathematical framework and tools for the modeling, analysis and optimization of dynamic stochastic models. The first of these covers measure theory (specifically probability measures), and the second one discusses stochastic processes in general and Markov processes in particular (with emphasis placed on transition functions defined by stochastic difference equations). Chapter 9 utilizes this framework in the derivation of the stochastic Bellman equation and the corresponding Euler equations. This development is then applied in Chapter 10 to several stochastic dynamic models, such as multi-sector models of economic growth, search models, inventory accumulation, and job matching, to name a few, all of which include an element of uncertainty. Toward a study of the stability of the resulting optimal (or nonoptimal) systems, the next two chapters discuss convergence concepts in stochastic (particularly, Markov) processes, with Chapter 11 devoted to strong convergence (of measures), and Chapter 12 to weak convergence. The latter also discusses the dependence of the limiting invariant measure on parameters that characterize the transition probabilities, and proves Helly’s theorem, which is used to prove convergence results for two classes of monotone Markov processes. Chapter 13 discusses the application of these tools to some of the models of Chapter 10, and Part III ends with a discussion of the law of large numbers for Markov processes in Chapter 14. This last chapter of Part III seems to be somewhat out of place, since the ergodic theory discussed there is not used anywhere else in the book.

Part IV comprises four chapters, the first of which discusses the connections between Pareto-optimality and competitive equilibrium. Here some tools of functional analysis are introduced, such as dual spaces and the Hahn-Banach theorem, which leads to a geometric interpretation for prices. Also discussed here are the two fundamental theorems of welfare economics. Chapter 16 is devoted to different applications of equilibrium theory, which illustrate how solutions to various planning problems can be interpreted, under suitable prices, as market equilibria. Finally, to handle problems where such an equivalence (between a social planner’s optimization problem and market equilibrium) does not hold, the next (and last) two chapters introduce some fixed point theorems (Brouwer’s and Schauder’s), and illustrate their application.

The book is written very competently, by the authors who are acknowledged authorities on the use of dynamic programming techniques in economics. It will undoubtedly remain as a standard reference book in the field for years to come.

The book is, for the most part, self-contained, as most of the mathematical background needed is developed in parallel and with considerable attention given to detail. It is organized into four parts, comprising a total of 18 chapters. The first part (comprising Chapters 1 and 2) provides a general introduction to the models as well as the solution techniques to be studied in the remainder of the book, and motivates the need for dynamic models (and thereby recursive methods) for decision-making in economics. Parts II and III are devoted to deterministic and stochastic models, respectively, and study in each case the dynamic optimization problems faced by an individual decision maker (which could be a firm, a consumer or a social planner). Part IV, on the other hand, analyses the existence and characterization of competitive equilibria in such systems, be they Pareto-optimal or not.

Part II, which covers deterministic systems, opens with a discussion of mathematical preliminaries, such as metric spaces, completeness, contraction mappings and fixed points, use of contraction mappings in the Bellman equation for discounted cost problems, and the theory of (static) optimization – all these constituting Chapter 3. Chapter 4 is devoted to a detailed study of the dynamic programming (Bellman) equation for infinite-horizon discounted cost problems, first discussing the principle of optimality, followed by discussion of the existence of a value function and an optimum, the method of successive approximations, differentiability of the value function, Euler equations, and dynamic optimization with unbounded returns. This theory is then applied in Chapter 5 to various dynamic models that arise in economics, among which are the one-sector model of economic growth, human capital accumulation and growth model, the inventory control problem, and a consumption- savings problem. In the final chapter of Part II, the steady-state behavior of dynamic models under optimal or non-optimal plans are studied, by focusing attention on the Lyapunov (global) stability theorem for nonlinear systems, and on linear approximations and the stability of linear systems (with particular application to the Euler equation).

Part III, which is devoted to stochastic models, follows the general pattern of Part II. The first two chapters (7 and 8) introduce the requires mathematical framework and tools for the modeling, analysis and optimization of dynamic stochastic models. The first of these covers measure theory (specifically probability measures), and the second one discusses stochastic processes in general and Markov processes in particular (with emphasis placed on transition functions defined by stochastic difference equations). Chapter 9 utilizes this framework in the derivation of the stochastic Bellman equation and the corresponding Euler equations. This development is then applied in Chapter 10 to several stochastic dynamic models, such as multi-sector models of economic growth, search models, inventory accumulation, and job matching, to name a few, all of which include an element of uncertainty. Toward a study of the stability of the resulting optimal (or nonoptimal) systems, the next two chapters discuss convergence concepts in stochastic (particularly, Markov) processes, with Chapter 11 devoted to strong convergence (of measures), and Chapter 12 to weak convergence. The latter also discusses the dependence of the limiting invariant measure on parameters that characterize the transition probabilities, and proves Helly’s theorem, which is used to prove convergence results for two classes of monotone Markov processes. Chapter 13 discusses the application of these tools to some of the models of Chapter 10, and Part III ends with a discussion of the law of large numbers for Markov processes in Chapter 14. This last chapter of Part III seems to be somewhat out of place, since the ergodic theory discussed there is not used anywhere else in the book.

Part IV comprises four chapters, the first of which discusses the connections between Pareto-optimality and competitive equilibrium. Here some tools of functional analysis are introduced, such as dual spaces and the Hahn-Banach theorem, which leads to a geometric interpretation for prices. Also discussed here are the two fundamental theorems of welfare economics. Chapter 16 is devoted to different applications of equilibrium theory, which illustrate how solutions to various planning problems can be interpreted, under suitable prices, as market equilibria. Finally, to handle problems where such an equivalence (between a social planner’s optimization problem and market equilibrium) does not hold, the next (and last) two chapters introduce some fixed point theorems (Brouwer’s and Schauder’s), and illustrate their application.

The book is written very competently, by the authors who are acknowledged authorities on the use of dynamic programming techniques in economics. It will undoubtedly remain as a standard reference book in the field for years to come.

Reviewer: T.Başar (Urbana, IL) (MR 92h:90027)

### MSC:

91B62 | Economic growth models |

91-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to game theory, economics, and finance |

93E03 | Stochastic systems in control theory (general) |

90C39 | Dynamic programming |