Linear Difference Equations
Linear Difference Equations
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线性差分方程
DOI：
10.1142/9789813270886_0007
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发表时间：
2018
期刊：
影响因子：
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通讯作者：
M. Tirelli
中科院分区：
文献类型：
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作者：
M. Tirelli
动态经济模型是研究经济动态并更好地理解增长和经济周期等相关现象的有用工具。均衡条件通常由一组差分方程和一系列边界条件（描述某些变量的极限值）来确定。因此，研究均衡性质需要研究差分方程系统的性质。在许多有趣的情形中，这类差分方程是非线性的；正如你将看到的，即使是教科书式的、确定性的新古典增长模型也是由一个非线性方程组来表示的。尽管非线性使得动态经济系统的研究变得困难，但我们仍然有望得出有用的特征和结果。这可以在局部，即均衡点的邻域内实现，或者对于对数线性化系统在全局实现。在这两种情况下，都使用线性差分方程理论来研究非线性系统。在这些笔记中，我们将总结该理论中的一些有用结果，并将其应用于处理动态经济系统。在这些课堂笔记中，我介绍了一些关于如何求解线性差分方程以及研究解的稳定性的有用材料。这些笔记并不完整；许多重要问题没有解释；但是学生可以在埃拉伊迪（2005年）的《差分方程导论》（施普林格出版社）中找到更多、清晰明了的材料。一本通用的入门参考书籍是西蒙和布卢姆的《经济学家的数学》（诺顿出版社）。对于线性代数问题，我最喜欢的教科书是塞尔日·朗的《线性代数》（施普林格出版社）
Dynamic economic models are a useful tool to study economic dynamics and get a better understanding of relevant phenomena such as growth and business cycle. Equilibrium conditions are normally identified by a system of difference equations and a set of boundary conditions (describing limit values of some variables). Thus, studying equilibrium properties requires studying the properties of a system of difference equations. In many interesting cases such difference equations are nonlinear; as you will see, even the textbook, deterministic, neoclassical, growth model is represented by a system of nonlinear equations. Although nonlinearities make the study of dynamic economic systems difficult, we can still hope to derive useful characterizations and results. This is achieved either locally, in a neighborhood of an equilibrium point, or globally for log-linearized systems. In both cases, non-linear systems are studied using the theory of linear difference equations. In these notes we shall summarize some useful results in this theory and apply them to deal with dynamic economic systems. In these class notes I present some useful material on how to solve linear difference equations and study solution stability. These notes are incomplete; many important questions are left unexplained; but students can find additional, sharp and clear material in Elaydi (2005) ”An Introduction to Difference Equations”, Springer-Verlag. A general, introductory, reference is Simon & Blume, ”Mathematics for Economists”, Norton. For issues on linear algebra, my favorite textbook is Serge Lang ”Linear Algebra”, Springer-Verlag.