Differential equations, stability and chaos in dynamic economics.

*(English)*Zbl 0693.90001
Advanced Textbooks in Economics, 27. Amsterdam etc.: North-Holland. xvi, 389 p. $ 49.00; Dfl 150.00 (1989).

The theme of the book is to show that the stability properties of an economic model must be investigated and become understood before such a model is used to supply insights into the working of the actual economic system. The book introduces the reader to three advanced mathematical topics: ordinary differential equations, stability techniques and chaotic dynamics.

Chapter 1 provides a general treatment of ordinary differential equations, emphasizing topics such as existence, continuation of solutions, uniqueness, successive approximations and dependence on initial data and parameters. Chapter 2 discusses linear differential equations with a balanced approach between their properties and solutions. Chapter 3 gives numerous definitions and examples of stability notions, with emphasis on linear systems, the linearization of nonlinear systems, the counting and examination of roots of characteristic equations, and a comprehensive presentation of two-dimensional systems and their phase diagrams. Chapter 4 continues on the topic of stability at a more advanced level, with an emphasis on Liapunov theory for local stability and global asymptotic stability. Chapter 5 surveys the stability contributions of mathematical economists with emphasis on methods of stability analysis of optimal control problems. Chapter 5 is used as a foundation for later chapters. Chapter 10 removes the emphasis from stability by stressing instabilities; it discusses chaos in macroeconomics and statistical theory for nonlinear dynamics.

The applications selected in chapters 6 through 10 include microeconomic dynamics, investment theory, macroeconomic policies, capital theory, business cycles, financial economics and many others.

All chapters conclude with two sections on miscellaneous applications and exercises and further remarks and references. In total the reader will find a valuable guide to over 500 selected references that use differential equations, stability analysis and chaotic dynamics.

The book is well written, and fairly self-contained. The audience for this book will include Ph D students in economics with a special interest in economic theory, economic researchers and applied mathematicians.

Chapter 1 provides a general treatment of ordinary differential equations, emphasizing topics such as existence, continuation of solutions, uniqueness, successive approximations and dependence on initial data and parameters. Chapter 2 discusses linear differential equations with a balanced approach between their properties and solutions. Chapter 3 gives numerous definitions and examples of stability notions, with emphasis on linear systems, the linearization of nonlinear systems, the counting and examination of roots of characteristic equations, and a comprehensive presentation of two-dimensional systems and their phase diagrams. Chapter 4 continues on the topic of stability at a more advanced level, with an emphasis on Liapunov theory for local stability and global asymptotic stability. Chapter 5 surveys the stability contributions of mathematical economists with emphasis on methods of stability analysis of optimal control problems. Chapter 5 is used as a foundation for later chapters. Chapter 10 removes the emphasis from stability by stressing instabilities; it discusses chaos in macroeconomics and statistical theory for nonlinear dynamics.

The applications selected in chapters 6 through 10 include microeconomic dynamics, investment theory, macroeconomic policies, capital theory, business cycles, financial economics and many others.

All chapters conclude with two sections on miscellaneous applications and exercises and further remarks and references. In total the reader will find a valuable guide to over 500 selected references that use differential equations, stability analysis and chaotic dynamics.

The book is well written, and fairly self-contained. The audience for this book will include Ph D students in economics with a special interest in economic theory, economic researchers and applied mathematicians.

Reviewer: Y.M.El-Fattah

##### MSC:

90-02 | Research exposition (monographs, survey articles) pertaining to operations research and mathematical programming |

93-02 | Research exposition (monographs, survey articles) pertaining to systems and control theory |

93D05 | Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory |

93C15 | Control/observation systems governed by ordinary differential equations |

91B28 | Finance etc. (MSC2000) |

91B62 | Economic growth models |

34D20 | Stability of solutions to ordinary differential equations |

93C05 | Linear systems in control theory |

93C10 | Nonlinear systems in control theory |

93D20 | Asymptotic stability in control theory |

93C95 | Application models in control theory |