##
**Economic dynamics in discrete time.**
*(English)*
Zbl 1369.91001

Cambridge, MA: MIT Press (ISBN 978-0-262-02761-8/hbk; 978-0-262-32558-5/ebook). xxii, 710 p. (2014).

This book

Dynamic general equilibrium theory has become the standard framework in economics to describe the formation and dynamic evolution of prices and allocations on markets with rational agents whose behavior is derived from an underlying microeconomic decision problem. Almost all theoretical and applied work based on this framework takes place in discrete time and draws on methods from a variety of mathematical areas including dynamical systems theory, probability and measure theory, functional analysis, and numerical mathematics. This well-structured and clearly written book aims to provide a comprehensive, self-contained review of these methods and illustrate their application in various branches of economics including decision theory and dynamic equilibrium analysis.

With the general objective set, the contents of the book unfolds along two stages. The first one is the methodological stage which reviews the general mathematical concepts and results. The second one is the application stage which applies these methods to specific economic problems and models. In the book, the methodological stage consists mostly of Parts I and II while the application stage is represented mainly by Parts III and IV. To make the book self-contained and open to self-study, a mathematical appendix reviews basic results from real, functional, and convex analysis and measure and probability theory. While the latter presentation is confined to concepts and results needed in the main text, the book also provides references to their general mathematical foundation and proofs.

Methodological stage

Part I starts by reviewing basic concepts from dynamical systems theory to solve deterministic and stochastic difference equations. The focus then specializes to a particular class of stochastic difference equations, the so-called rational expectations models. Many dynamic macroeconomic models yield equilibrium conditions which can be cast in this form. The book reviews and compares various theoretical results how to obtain and approximate solutions to these conditions and provides program code to compute them numerically using the DYNARE software package.

The following chapters provide a general discussion on the existence of stationary solutions corresponding to invariant probability measures for a general class of stochastic processes including Markov processes with finitely or infinitely many states. Necessary and sufficient conditions under which these solutions exist are provided and various notions of convergence are discussed to describe their asymptotic stability. In a final chapter, basic results from ergodic theory including Birkhoff’s theorem are proved and applied to obtain several laws of large numbers for a general class of stationary stochastic processes including stationary Markov processes. All these results are presented in a clear and rigorous fashion providing the reader with a valuable survey of existing results which are applicable to a large class of problems in dynamic economic theory.

Part II centers around dynamic optimization theory and optimal control problems which constitute an integral part of dynamic economic models. The book first formalizes a class of stochastic decision problems which possess a recursive structure on a suitably defined state space. This permits the application of dynamic programming techniques to solve these problems which are discussed for both finite and infinite time horizon. The recursive approach is contrasted with and compared to the more general characterization of solutions by Euler equations and transversality conditions which can also be applied to non-recursive problems. Several economic applications such as inventory problems and optimal consumption-savings and investment decisions illustrate these general results.

The next chapters focus on problems of optimal control for linear-quadratic problems and under partial information and review basic numerical approximation techniques which are applied to specific problems from dynamic programming. The methodological part is completed by presenting selected econometric methods such as the ‘Generalized Method of Moments’ and the ‘Maximum Likelihood’ approach. The application of these and related methods to estimate parameters of fully specified dynamic stochastic general equilibrium models is a particularly intriguing field in the economics science that has seen rapid progress being made in recent years.

Application stage

The application of the theoretical results starts with Part III which analyzes dynamic equilibria in selected macroeconomic models. The first two chapters focus on frictionless models such as pure exchange economies with complete markets and stochastic production economies derived from the neoclassical growth model. These models are widely used in the literature to study problems of asset pricing and business cycles. An additional third chapter outlines Bayesian methods permitting to estimate the parameters of these models using empirical data.

The remaining chapters of this part extend the analysis to models with incomplete markets and frictions such as the overlapping generations model which is used to study asset bubbles, search and matching models of the labor market to study unemployment, and the New Keynesian model to analyze monetary policy. The latter has become the standard workhorse at central banks to provide guidance to monetary policy.

All these models are presented in a clear and rigorous fashion with a strong focus on methodical aspects in line with the general objective of the book. References to simulation scripts are again provided to replicate all quantitative results.

The final Part IV extends the previous methods to more specialized applications which somewhat deviate from the standard framework. One such application is recursive utility theory which overcomes several problems with the traditional expected utility approach to study models of asset pricing. Further topics in this part comprise dynamic games to model strategic interaction in a dynamic environment and recursive contract theory to study credibility, commitment, and reputation effects. All the previous applications are formulated in consistent continuation of the previously developed formal apparatus with all adaptions presented in a clear and consistent fashion.

Summary

In summary, this book provides a comprehensive toolkit of methods which are indispensable to any researcher working in theoretical macroeconomics and related fields. A major strength is the nexus between the thoroughly presented mathematical theory and its application to solve specific economic problems. For the latter purpose, the numerical simulation scripts provided are extremely helpful to illustrate and quantify the theoretical results.

With these features, the book appeals to a broad audience ranging from researchers with a strong theoretical interest to those who mainly seek to apply the results in quantitative and empirical studies. It has the potential to become a standard reference in the economics profession and provides a valuable teaching resource for any advanced course in economic dynamics.

Dynamic general equilibrium theory has become the standard framework in economics to describe the formation and dynamic evolution of prices and allocations on markets with rational agents whose behavior is derived from an underlying microeconomic decision problem. Almost all theoretical and applied work based on this framework takes place in discrete time and draws on methods from a variety of mathematical areas including dynamical systems theory, probability and measure theory, functional analysis, and numerical mathematics. This well-structured and clearly written book aims to provide a comprehensive, self-contained review of these methods and illustrate their application in various branches of economics including decision theory and dynamic equilibrium analysis.

With the general objective set, the contents of the book unfolds along two stages. The first one is the methodological stage which reviews the general mathematical concepts and results. The second one is the application stage which applies these methods to specific economic problems and models. In the book, the methodological stage consists mostly of Parts I and II while the application stage is represented mainly by Parts III and IV. To make the book self-contained and open to self-study, a mathematical appendix reviews basic results from real, functional, and convex analysis and measure and probability theory. While the latter presentation is confined to concepts and results needed in the main text, the book also provides references to their general mathematical foundation and proofs.

Methodological stage

Part I starts by reviewing basic concepts from dynamical systems theory to solve deterministic and stochastic difference equations. The focus then specializes to a particular class of stochastic difference equations, the so-called rational expectations models. Many dynamic macroeconomic models yield equilibrium conditions which can be cast in this form. The book reviews and compares various theoretical results how to obtain and approximate solutions to these conditions and provides program code to compute them numerically using the DYNARE software package.

The following chapters provide a general discussion on the existence of stationary solutions corresponding to invariant probability measures for a general class of stochastic processes including Markov processes with finitely or infinitely many states. Necessary and sufficient conditions under which these solutions exist are provided and various notions of convergence are discussed to describe their asymptotic stability. In a final chapter, basic results from ergodic theory including Birkhoff’s theorem are proved and applied to obtain several laws of large numbers for a general class of stationary stochastic processes including stationary Markov processes. All these results are presented in a clear and rigorous fashion providing the reader with a valuable survey of existing results which are applicable to a large class of problems in dynamic economic theory.

Part II centers around dynamic optimization theory and optimal control problems which constitute an integral part of dynamic economic models. The book first formalizes a class of stochastic decision problems which possess a recursive structure on a suitably defined state space. This permits the application of dynamic programming techniques to solve these problems which are discussed for both finite and infinite time horizon. The recursive approach is contrasted with and compared to the more general characterization of solutions by Euler equations and transversality conditions which can also be applied to non-recursive problems. Several economic applications such as inventory problems and optimal consumption-savings and investment decisions illustrate these general results.

The next chapters focus on problems of optimal control for linear-quadratic problems and under partial information and review basic numerical approximation techniques which are applied to specific problems from dynamic programming. The methodological part is completed by presenting selected econometric methods such as the ‘Generalized Method of Moments’ and the ‘Maximum Likelihood’ approach. The application of these and related methods to estimate parameters of fully specified dynamic stochastic general equilibrium models is a particularly intriguing field in the economics science that has seen rapid progress being made in recent years.

Application stage

The application of the theoretical results starts with Part III which analyzes dynamic equilibria in selected macroeconomic models. The first two chapters focus on frictionless models such as pure exchange economies with complete markets and stochastic production economies derived from the neoclassical growth model. These models are widely used in the literature to study problems of asset pricing and business cycles. An additional third chapter outlines Bayesian methods permitting to estimate the parameters of these models using empirical data.

The remaining chapters of this part extend the analysis to models with incomplete markets and frictions such as the overlapping generations model which is used to study asset bubbles, search and matching models of the labor market to study unemployment, and the New Keynesian model to analyze monetary policy. The latter has become the standard workhorse at central banks to provide guidance to monetary policy.

All these models are presented in a clear and rigorous fashion with a strong focus on methodical aspects in line with the general objective of the book. References to simulation scripts are again provided to replicate all quantitative results.

The final Part IV extends the previous methods to more specialized applications which somewhat deviate from the standard framework. One such application is recursive utility theory which overcomes several problems with the traditional expected utility approach to study models of asset pricing. Further topics in this part comprise dynamic games to model strategic interaction in a dynamic environment and recursive contract theory to study credibility, commitment, and reputation effects. All the previous applications are formulated in consistent continuation of the previously developed formal apparatus with all adaptions presented in a clear and consistent fashion.

Summary

In summary, this book provides a comprehensive toolkit of methods which are indispensable to any researcher working in theoretical macroeconomics and related fields. A major strength is the nexus between the thoroughly presented mathematical theory and its application to solve specific economic problems. For the latter purpose, the numerical simulation scripts provided are extremely helpful to illustrate and quantify the theoretical results.

With these features, the book appeals to a broad audience ranging from researchers with a strong theoretical interest to those who mainly seek to apply the results in quantitative and empirical studies. It has the potential to become a standard reference in the economics profession and provides a valuable teaching resource for any advanced course in economic dynamics.

Reviewer: Marten Hillebrand (Konstanz)

### MSC:

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

91B55 | Economic dynamics |

91B51 | Dynamic stochastic general equilibrium theory |

91B64 | Macroeconomic theory (monetary models, models of taxation) |

91B62 | Economic growth models |

91B06 | Decision theory |

37N40 | Dynamical systems in optimization and economics |

37Mxx | Approximation methods and numerical treatment of dynamical systems |

91-04 | Software, source code, etc. for problems pertaining to game theory, economics, and finance |