A Nonlinear Dynamic Model of Short Run Fluctuations
A Nonlinear Dynamic Model of Short Run Fluctuations
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短期波动的非线性动态模型
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发表时间：
1981
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通讯作者：
Garry J. Schinasi
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作者：
Garry J. Schinasi
最近对“商业周期”建模的大多数尝试(见Lucas(1975)、Sargent-Wallace(1975)、Taylor(1979)的工作)采用了Frisch-Slutsky框架，在该框架中，宏观经济活动中波动的持续性通过冲击其他稳定的(通常是线性的)系统成分的随机冲击的持续性来解释。卡尔多(1940)和古德温(1951)以及希克斯(1950)(以及最近的一村(1954)、克莱因和普雷斯顿(1969)、Change和Smyth(1971)、托尔(1977)和瓦里安(1979))的工作试图通过使用非线性行为函数，用系统性经济行为来模拟波动的持续性(或当时被认为是“内生周期”)。非线性动力学方法要么用Poincare-Bendixon定理的变种来证明闭轨的存在，要么用关于Leinard方程(一个二阶非线性微分方程)的定理来证明极限环的存在、唯一性和稳定性。本文证明了传统的动态IS-LM宏观模型的改进形式的输出动态可归结为Leinard方程。这里考虑的模型是对传统IS-LM模型的两个重要方面的修改。首先，我们在模型中加入了允许货币和债券融资的政府预算约束；其次，也是更重要的是，我们将投资行为建模为产出的Kaldor型非线性函数。结果表明，对于一般平衡点局部不稳定的情况，模型仍然是全局稳定的，即所有局部不稳定点都收敛到一个唯一稳定的极限环。本文的研究内容如下。第二节建立了一般模型，并讨论了其几何性质。第三节推导了Leinard方程，并证明了模型各子结构极限环的存在唯一性和稳定性。第四节得出了一些启示。附录陈述了正文中使用的关于Leinard方程的一个定理。
Most recent attempts to model the "business cycle" (see the work of Lucas (1975), Sargent-Wallace (1975), Taylor (1979)) adopt a Frisch-Slutsky framework, in which the persistence of fluctuations in macroeconomic activity is explained by the persistence of random shocks impinging on an otherwise stable (and usually linear) systematic component. Work by Kaldor (1940) and Goodwin (1951), as well as Hicks (1950) (and more recently Ichimura (1954), Klein and Preston (1969), Change and Smyth (1971), Torre (1977) and Varian (1979)) attempted to model the persistence of fluctuations (or what was then believed to be an "endogenous cycle") with systematic economic behaviour by employing nonlinear behavioural functions. The nonlinear dynamic approach to modelling cycles has used either a variant of the Poincare-Bendixon theorem, to prove the existence of closed orbits, or a theorem on Leinard's equation (a second order non-linear differential equation) to prove the existence, uniqueness and stability of limit cycles. This paper shows that the output dynamic of a modified version of a traditional, dynamic IS-LM macro-model is reducible to Leinard's equation. The models considered here are modifications of a traditional IS-LM model in two important ways. First, we augment the model with a government budget constraint allowing for money and bond financing; and secondly and more importantly, we model investment behaviour as a Kaldor-type nonlinear function of output. It is shown that for cases in which the general equilibrium of the model is locally unstable, the model is nevertheless globally stable in the sense that all locally unstable points converge to a unique and stable limit cycle. The paper proceeds as follows. Section 2 sets up the general model, and discusses its geometric properties. Section 3 derives Leinard's equation and proves the existence, uniqueness and stability of a limit cycle for various substructures of the model. Section 4 draws some implications. The Appendix states a theorem on Leinard's equation used in the text.