幾類具有時滯的動態商業周期模型研究

中文摘要

在世界金融危機的今天，經濟的完全自由化已經受到質疑，政府政策調節以及監管凸現重要。在政府的政策調節中投資、稅收是其中的重要工具，而在實施過程中，任何一項政策工具都會存在時滯。

全面地了解那些過程中存在的時滯，時滯如何影響巨觀經濟運行，如何避免較長的時滯，這些間題無論在理論經濟學和實證經濟學的研究中都具有重要的理論價值和實際價值。

本學位論文在幾類商業周期擴展模型的基礎上，考慮到商業周期模型中存在時間延滯(資本積累方程中存在時滯或稅收收入中存在時滯，時滯有可能是離散的固定時滯也有可能是指數分布時滯)，通過運用Routh-Hurwith規則、穩定性切換定理、Hopf分岔理論等相結合的方法探討時滯對經濟系統的影響。

全文共分為六章。在第一章中，我們首先回顧商業周期研究的發展歷史、幾類主要的動態商業周期模型、再就是對商業周期模型進行分析的動態經濟學方法的歷史簡介。同時，我們也簡要地介紹本學位論文的幾個重要創新點。在第二章中，我們考慮資本積累過程中具有固定時滯的動態巨觀經濟模型，模型建立在Samuleson-Hicks的動態乘數-加速數模型的基礎上。通過運用線性穩定性方法及Hopf分岔分析表明時滯能夠引起商業周期形成，最後給出數值例子。第三章，我們在第二章模型的基礎上考慮投資過程中存在固定時間延滯及稅收收入中存在固定的時間延滯的隨機動態經濟模型，通過運用Ruan&Wei及Nisbet相關文獻的方法證明每個固定的時滯及隨機變數都將對經濟系統的穩定性產生影響。

在第四章中，我們考慮到更加現實的指數分布稅收收入時滯，即大量機構存在不同的稅收收入時滯現象，相較於固定稅收收入時滯的同質性機構的同一時滯。我們證明指數分布時滯參數對經濟系統的穩定性產生影響時，固定時滯將起著更重要的主導作用，數值例子的模擬結果與理論結果一致。

在第五章中，我們在考慮到資本積累方程中存在離散時滯時，在具有政府消費預算約束的動態IS-LM模型基礎上通過增加資本積累方程從而形成更廣義的含有四個方程的IS-LM的模型，通過運用Hopf分岔定理證明周期解的存在性，唯一性，而且通過規範型理論和中心流形定理的方法得到周期軌道穩定性的充分必要條件。

在第六章中，在Gabisch&Lorenz和Boldrin相關論文探索的擴展IS-LM模型基礎上，我們進一步假設投資函式既依賴於過去時間的收入也依賴於過去的資本存量，且具有不同的過去時間，從而形成兩個固定時滯的擴展IS-LM模型，在這個模型中我們將時滯對經濟系統的影響及周期軌的方向和穩定性進行動態分析。關鍵字：商業周期；時滯；Hopf分歧；穩定性切換；稅收收人；白噪聲收起

外文摘要

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 multiplieraccelerator 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. Keywords: business cycle; time delay; Hopf bifurcation; stability switch; tax collection; white noise

幾類具有時滯的動態商業周期模型研究

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