几类具有时滞的动态商业周期模型研究
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摘要
在世界金融危机的今天,经济的完全自由化已经受到质疑,政府政策调节以及监管凸现重要。在政府的政策调节中投资、税收是其中的重要工具,而在实施过程中,任何一项政策工具都会存在时滞。全面地了解那些过程中存在的时滞,时滞如何影响宏观经济运行,如何避免较长的时滞,这些问题无论在理论经济学和实证经济学的研究中都具有重要的理论价值和实际价值。本学位论文在几类商业周期扩展模型的基础上,考虑到商业周期模型中存在时间延滞(资本积累方程中存在时滞或税收收入中存在时滞,时滞有可能是离散的固定时滞也有可能是指数分布时滞),通过运用Routh-Hurwith规则、稳定性切换定理、Hopf分岔理论等相结合的方法探讨时滞对经济系统的影响。全文共分为六章。
在第一章中,我们首先回顾商业周期研究的发展历史、几类主要的动态商业周期模型、再就是对商业周期模型进行分析的动态经济学方法的历史简介。同时,我们也简要地介绍本学位论文的几个重要创新点。
在第二章中,我们考虑资本积累过程中具有固定时滞的动态宏观经济模型,模型建立在Samuleson-Hicks的动态乘数-加速数模型的基础上。通过运用线性稳定性方法及Hopf分岔分析表明时滞能够引起商业周期形成,最后给出数值例子。
第三章,我们在第二章模型的基础上考虑投资过程中存在固定时间延滞及税收收入中存在固定的时间延滞的随机动态经济模型,通过运用Ruan&Wei及Nisbet相关文献的方法证明每个固定的时滞及随机变量都将对经济系统的稳定性产生影响。
在第四章中,我们考虑到更加现实的指数分布税收收入时滞,即大量机构存在不同的税收收入时滞现象,相较于固定税收收入时滞的同质性机构的同一时滞。我们证明指数分布时滞参数对经济系统的稳定性产生影响时,固定时滞将起着更重要的主导作用,数值例子的模拟结果与理论结果一致。
在第五章中,我们在考虑到资本积累方程中存在离散时滞时,在具有政府消费预算约束的动态IS-LM模型基础上通过增加资本积累方程从而形成更广义的含有四个方程的IS-LM的模型,通过运用Hopf分岔定理证明周期解的存在性,唯一性,而且通过规范型理论和中心流形定理的方法得到周期轨道稳定性的充分必要条件。
在第六章中,在Gabisch&Lorenz和Boldrin相关论文探索的扩展IS-LM模型基础上,我们进一步假设投资函数既依赖于过去时间的收入也依赖于过去的资本存量,且具有不同的过去时间,从而形成两个固定时滞的扩展IS-LM模型,在这个模型中我们将时滞对经济系统的影响及周期轨的方向和稳定性进行动态分析。
Today, in the world's financial crisis, laissez-faire of the economy has been questioned. The regulation of government policies and its supervision has become more and more important. Investment and taxes are two kinds of essential tools in the regulation of the government's policy, however, there exists time delay in every implementation process. These questions about the thorough knowledge of those time delay processes, how it affects the macroeconomic operation and how to avoid a long delay, have important theoretical value and practical value both in theoretical economics and positive economics. This paper, based on several business cycle models, considering time delay(wherever in capital accumulation or tax collection, whatever fixed or distributed time delay) into business cycle system, will explore the impacts of time lags on the economic system by employing Routh-Hurwitz criterion, stability switch theorem, Hopf bifurcation theory and so on. The essays is divided into six chapters.
In the first chapter, this paper first reviews the history of researches on business cycles, several types of major dynamic business cycle models, then introduces the history of dynamic analysis methods of several business cycle models. At the same time, the innovations in this thesis are also briefly introduced in this chapter.
In chapter two, by considering the dynamic macroeconomic model with fixed delayed time in capital accumulation, which is based on Samuelson-Hicks's dynamic multiplier-accelerator model, We show that the time lag can cause business cycles by applying linear stability method and Hopf bifurcation, and finally the numerical examples are given.
In chapter three, on the basis of the model in chapter two, we consider the stochastic dynamic economic model with both time lag in capital accumulation and time lag in tax collection, then follow the method of Ruan & Wei and Nisbet's relative works, we can obtain that every fixed time delay and random variables will have an effect on the stability of economic system.
In chapter four, we take into consideration a more realistic exponential distribution in the time delay of tax revenue, we introduce the exponetially distributed time delay, namely, there are different time delays in lots of heterogeneous organizations or companies, comparing with those homogeneous companies which have the same time delay. We prove that the distributed delayed time changes the stability of economic system, the fixed time delay will dominant the stability of the economic system. The numerical simulation is in accordance with theoretical results.
In chapter five, while considering the discrete time lag in capital accumulation, we investigate generalized IS-LM model with four equations after adding the capital accumulation equation on the dynamic IS-LM model with the constraint of government expenditure. Through the application of Hopf bifurcation theorem, normal form method and the center manifold theory, we prove the existence, uniqueness, direction and stability of the business cycle.
In chapter six, based on Gabisch & Lorenz and Boldrin's related essays of augmented IS-LM model, we then assume that the investment function will depend on the income in the past and also the capital stock in the past but with different past time, we construct the augmented IS-LM model with two fixed different time lags, dynamically analyze the impacts of the time lags on the dynamic model, in which we will analyze the direction and stability of the business cycle.
在第一章中,我们首先回顾商业周期研究的发展历史、几类主要的动态商业周期模型、再就是对商业周期模型进行分析的动态经济学方法的历史简介。同时,我们也简要地介绍本学位论文的几个重要创新点。
在第二章中,我们考虑资本积累过程中具有固定时滞的动态宏观经济模型,模型建立在Samuleson-Hicks的动态乘数-加速数模型的基础上。通过运用线性稳定性方法及Hopf分岔分析表明时滞能够引起商业周期形成,最后给出数值例子。
第三章,我们在第二章模型的基础上考虑投资过程中存在固定时间延滞及税收收入中存在固定的时间延滞的随机动态经济模型,通过运用Ruan&Wei及Nisbet相关文献的方法证明每个固定的时滞及随机变量都将对经济系统的稳定性产生影响。
在第四章中,我们考虑到更加现实的指数分布税收收入时滞,即大量机构存在不同的税收收入时滞现象,相较于固定税收收入时滞的同质性机构的同一时滞。我们证明指数分布时滞参数对经济系统的稳定性产生影响时,固定时滞将起着更重要的主导作用,数值例子的模拟结果与理论结果一致。
在第五章中,我们在考虑到资本积累方程中存在离散时滞时,在具有政府消费预算约束的动态IS-LM模型基础上通过增加资本积累方程从而形成更广义的含有四个方程的IS-LM的模型,通过运用Hopf分岔定理证明周期解的存在性,唯一性,而且通过规范型理论和中心流形定理的方法得到周期轨道稳定性的充分必要条件。
在第六章中,在Gabisch&Lorenz和Boldrin相关论文探索的扩展IS-LM模型基础上,我们进一步假设投资函数既依赖于过去时间的收入也依赖于过去的资本存量,且具有不同的过去时间,从而形成两个固定时滞的扩展IS-LM模型,在这个模型中我们将时滞对经济系统的影响及周期轨的方向和稳定性进行动态分析。
Today, in the world's financial crisis, laissez-faire of the economy has been questioned. The regulation of government policies and its supervision has become more and more important. Investment and taxes are two kinds of essential tools in the regulation of the government's policy, however, there exists time delay in every implementation process. These questions about the thorough knowledge of those time delay processes, how it affects the macroeconomic operation and how to avoid a long delay, have important theoretical value and practical value both in theoretical economics and positive economics. This paper, based on several business cycle models, considering time delay(wherever in capital accumulation or tax collection, whatever fixed or distributed time delay) into business cycle system, will explore the impacts of time lags on the economic system by employing Routh-Hurwitz criterion, stability switch theorem, Hopf bifurcation theory and so on. The essays is divided into six chapters.
In the first chapter, this paper first reviews the history of researches on business cycles, several types of major dynamic business cycle models, then introduces the history of dynamic analysis methods of several business cycle models. At the same time, the innovations in this thesis are also briefly introduced in this chapter.
In chapter two, by considering the dynamic macroeconomic model with fixed delayed time in capital accumulation, which is based on Samuelson-Hicks's dynamic multiplier-accelerator model, We show that the time lag can cause business cycles by applying linear stability method and Hopf bifurcation, and finally the numerical examples are given.
In chapter three, on the basis of the model in chapter two, we consider the stochastic dynamic economic model with both time lag in capital accumulation and time lag in tax collection, then follow the method of Ruan & Wei and Nisbet's relative works, we can obtain that every fixed time delay and random variables will have an effect on the stability of economic system.
In chapter four, we take into consideration a more realistic exponential distribution in the time delay of tax revenue, we introduce the exponetially distributed time delay, namely, there are different time delays in lots of heterogeneous organizations or companies, comparing with those homogeneous companies which have the same time delay. We prove that the distributed delayed time changes the stability of economic system, the fixed time delay will dominant the stability of the economic system. The numerical simulation is in accordance with theoretical results.
In chapter five, while considering the discrete time lag in capital accumulation, we investigate generalized IS-LM model with four equations after adding the capital accumulation equation on the dynamic IS-LM model with the constraint of government expenditure. Through the application of Hopf bifurcation theorem, normal form method and the center manifold theory, we prove the existence, uniqueness, direction and stability of the business cycle.
In chapter six, based on Gabisch & Lorenz and Boldrin's related essays of augmented IS-LM model, we then assume that the investment function will depend on the income in the past and also the capital stock in the past but with different past time, we construct the augmented IS-LM model with two fixed different time lags, dynamically analyze the impacts of the time lags on the dynamic model, in which we will analyze the direction and stability of the business cycle.
引文
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