Delay Differential Nonlinear Economic Models
The asymptotical behavior of dynamic economic systems has been the focus of a large number of studies with both discrete and continuous time scales. They are based on the qualitative theory of difference or ordinary differential equations (Bellman, 1969; Goldberg, 1958). It has been shown by many authors that the introduction of information delay into the dynamic models significantly changes their asymptotical properties. For example, Chiarella and Szidarovszky (2004) consider dynamic oligopolies with partial information on the price function and Huang (2008) examines the role of information lag in economic dynamics, to name a few.
There is a significant difference between models with fixed time lags and models with continuously distributed delays. In the first case there is an infinite spectrum, and in the second case with gamma-function type kernel functions, the spectrum is finite. An important special case of continuously distributed time lags is given by exponentially decreasing kernel functions.
In this paper we compare dynamics generated by fixed time lags and continuously distributed delay with exponential kernel function. We will first show that these two types of models generate the same local dynamics if the delay is sufficiently small. This is, however, not true if the delay becomes large.
KeywordsHopf Bifurcation Price Function Cournot Model Business Cycle Model Investment Function
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This paper was done when the first author visited the Department of Systems and Industrial Engineering of the University of Arizona. He appreciated its hospitality over his stay. The authors wish to express their gratitude to an anonymous referee for useful comments and also want to acknowledge the encouragement and support by Kei Matsumoto for the research leading to this paper. They appreciate financial support from Chuo University (Joint Research Project 0981) and the Japan Ministry of Education, Culture, Sports, Science and Technology (Grant-in-Aid for Scientific Research (C) 21530172). The usual disclaimer applies.
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