Abstract
In this paper, we examine dynamics of a boundedly rational monopoly with continuously distributed time delay. Constructing the gradient dynamic system where the output change is proportional to the expected profit and expected demand is formed based on past data with various shapes of the weighting function, three main results are analytically as well as numerically demonstrated: (1) the stability region depends on the shape of the weighting function and converges to the stability region of the fixed time delay when the shape parameter goes to infinity; (2) delay has a threshold value below which stability is preserved and above which it is lost; (3) the equilibrium point bifurcates to a limit cycle through Hopf bifurcation when it loses stability.
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Notes
- 1.
A dynamic equation with fixed delays can be called a mixed difference-differential equation. However, Gandolfo (2009) points out that such terminology is somewhat dated.
- 2.
See Chap. 5 of Bischi et al. (2010) for stability/instability of economic models with misspecified and uncertain price functions.
- 3.
Monopoly dynamics with multiple continuous delays is considered in Matsumoto and Szidarovszky (2012b).
- 4.
See, for example, Gandolfo (2009) for this theorem.
- 5.
This set of parameters is repeatedly used in the following numerical examples. Notice that \(\gamma =1\) under this set.
- 6.
With tedious calculations, it may be possible to show the similar results in cases for \(m\ge 4.\)
- 7.
Since the first equation of (13) can be rewritten as
$$\begin{aligned} q^{\varepsilon }(t)-q(t)=\omega (q^{e}(t)-q(t)), \end{aligned}$$it can be mentioned that the expected demand is formed in such a way that the expectation error is proportional to the difference between the weighted average level and the current level.
- 8.
Notice that \(f(\tau \gamma )=0\) generates equal roots at point \(C=(8/9,9).\)
- 9.
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Acknowledgments
The authors are particularly grateful to Roberto Dieci and a referee for their comments and suggestions. They highly appreciate financial supports from the MEXT-Supported Programme for the Strategic Research Foundation at Private Universities 2013-2017, the Japan Society for the Promotion of Science (Grant-in-Aid for Scientific Research (C) 24530201 and 25380238) and Chuo University (Joint Research Grant 0981 and Grant for Special Research). The paper was prepared when the second author visited the Department of Economics, Chuo University. He appreciates its hospitality during his stay. The usual disclaimer applies.
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Matsumoto, A., Szidarovszky, F. (2014). Boundedly Rational Monopoly with Single Continuously Distributed Time Delay. In: Dieci, R., He, XZ., Hommes, C. (eds) Nonlinear Economic Dynamics and Financial Modelling. Springer, Cham. https://doi.org/10.1007/978-3-319-07470-2_7
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