Abstract
In this chapter, we reconsider the analytical results on the existence of cyclical fluctuations in continuous time dynamic optimization models with two state variables and their applications to dynamic economic theory. In the first part, we survey the useful analytical results which were obtained by Dockner and Feichtinger (J Econom 53–1:31–50, 1991), Liu (J Math Anal Appl 182:250–256, 1994) and Asada and Yoshida (Chaos, Solitons and Fractals 18:525–536, 2003) on the general theory of cyclical fluctuations in continuous time dynamic optimizing and non-optimizing models. In the second part, we provide an application of these analytical results to a particular continuous time dynamic optimizing economic model, that is, a model of dynamic limit pricing with two state variables, which is an extension of Gaskins (J Econom Theor 3:306–322, 1971) prototype model.
Key words
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
This does not necessarily mean that every continuous time dynamic optimization model with two state variables produces cyclical fluctuations. For example, [5] proved analytically that [19] continuous time dynamic optimization model of endogenous growth with two state variables entails only the monotonic convergence to the equilibrium point.
- 2.
We can introduce other parameters which affect functions F and f, but the formulation in the text is sufficient for our purpose.
- 3.
We assume that the second-order conditions are also satisfied.
- 4.
See mathematical appendix of [2].
- 5.
- 6.
We reproduce the proof here. The method of proof is quite simple and straightforward.
- 7.
- 8.
As for the exhaustive exposition of the theory of differential game, see [12].
- 9.
In the appendix, we reinterpret this equation by means of a continuously distributed lag model of expectation formation.
- 10.
Since \({\partial }^{2}H/\partial {p}^{2} = -2b < 0\) , the second-order condition is always satisfied.
- 11.
Note that Eq. (12) means that the initial value of price p(0) is determined if the initial value of a state variable x(0) is given and the initial value of a costate variable μ2(0) is selected.
- 12.
We have \({\int \limits }_{-\infty }^{t}(1/\tau ){\mathrm{e}}^{-(1/\tau )(t-s)}\mathrm{d}s = (1/\tau ){\mathrm{e}}^{-(1/\tau )t}{\int \limits }_{-\infty }^{t}{\mathrm{e}}^{(1/\tau )s}\mathrm{d}s ={ \mathrm{e}}^{-(1/\tau )t}{[{\mathrm{e}}^{(1/\tau )s}]}_{s=-\infty }^{s=t} = 1.\)
References
Asada, T. (2008) : “On the Existence of Cyclical Fluctuations in Continuous Time Dynamic Optimization Models : General Theory and its Application to Economics.” Annals of the Institute of Economic Research, Chuo University 39, pp. 205–222. (in Japanese)
Asada, T., C. Chiarella, P. Flaschel and R. Franke (2003) : Open Economy Macrodynamics : An Integrated Disequilibrium Approach. Springer, Berlin.
Asada, T. and W. Semmler (1995) : “Growth and Finance : An Intertemporal Model.” Journal of Macroeconomics 17–4, pp. 623–649.
Asada, T. and W. Semmler (2004) : “Limit Pricing and Entry Dynamics with Heterogeneous Firms.” M. Gallegati, A. P. Kirman and M. Marsili eds. The Complex Dynamics of Economic Interaction : Essays in Economics and Econophysics, Springer, Berlin, pp. 35–48.
Asada, T, W. Semmler and A. Novak (1998) : “Endogenous Growth and Balanced Growth Equilibrium.” Research in Economics 52–2, pp. 189–212.
Asada, T. and H. Yoshida (2003) : “Coefficient Criterion for Four-dimensional Hopf Bifurcation : A Complete Mathematical Characterization and Applications to Economic Dynamics.” Chaos, Solitons and Fractals 18, pp. 525–536.
Benhabib, J. and K. Nishimura (1979) : “The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth.” Journal of Economic Theory 21. pp. 421–444.
Benhabib, J. and A. Rustichini (1990) : “Equilibrium Cycling with Small Discounting.” Journal of Economic Theory 52, pp. 423–432.
Chiang, A. (1992) : Elements of Dynamic Optimization. McGraw-Hill, New York.
Dixit, A. K. (1990) : Optimization in Economic Theory(Second Edition). Oxford University Press, Oxford.
Dockner, E. and G. Feichtinger (1991) : “On the Optimality of Limit Cycles in Dynamic Economic Systems.” Journal of Economics 53–1, pp. 31–50.
Dockner, E., S.Jorgensen, N. Van Long and G. Sorger (2000) : Differential Games in Economics and Management Science. Cambridge University Press, Cambridge.
Faria, J. R. and J. P. Andrade (1998) : “Investment, Credit, and Endogenous Cycles.” Journal of Economics 67–2, pp. 135–143.
Feichtinger, G., A. Novak and F. Wirl (1994) : “Limit Cycles in Intertemporal Adjustment Models.” Journal of Economic Dynamics and Control 18, pp. 353–380.
Gandolfo, G. (2009) : Economic Dynamics (Fourth Edition). Springer, Berlin.
Gaskins, D. W. (1971) : “Dynamic Limit Pricing : Optimal Pricing Under Threat of Entry.” Journal of Economic Theory 3, pp. 306–322.
Judd, K. and B. Petersen (1986) : “Dynamic Limit Pricing and Internal Finance.” Journal of Economic Theory 39, pp. 368–399.
Liu, W. M. (1994) : “Criterion of Hopf Bifurcation without Using Eigenvalues.” Journal of Mathematical Analysis and Applications 182, pp. 250–256.
Romer, P. (1990) : “Endogenous Technological Change.” Journal of Political Economy 98, pp. 71–102.
Shinkai, Y. (1970) : Economic Analysis and Differential-Difference Equations. Toyo Keizai Shinpo-sha, Tokyo. (in Japanese)
Yoshida, H. and T. Asada (2007) : “Dynamic Analysis of Policy Lag in a Keynes-Goodwin Model : Stability, Instability, Cycles and Chaos.” Journal of Economic Behavior and Organization 62, pp. 441–469.
Acknowledgements
This chapter is based on the paper that was written in March 2010 while the author was staying at School of Finance and Economics, University of Technology Sydney (UTS) as a visiting professor under the “Chuo University Leave Program for Special Research Project”, and an earlier version of this chapter was tentatively published as Discussion Paper Series No. 139 of the Institute of Economic Research, Chuo University, Tokyo, Japan (April 2010). This research was financially supported by the Japan Society for the promotion of Science (Grant-in-Aid (C) 20530160) and Chuo University. Grant for Special Research Section 2 of this chapter is based on [1], although Sects. 3 and 4 and Appendix are not based on [1]. Needless to say, only the author is responsible for possible remaining errors. The author is grateful to Dr. Masahiro Ouchi of Nihon University, Tokyo, Japan for preparing LATEX version of this chapter.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Appendix
Appendix
In this appendix, we reinterpret Eq. (8) in the text by means of a continuously distributed lag model of expectation formation following the procedure that was adopted by [20, 21]. Let us assume that the expected price is the weighted average of actual past prices, that is,
where \(\omega (s)\) is a weighting function such that
In particular, we assume that our model is described by means of the following “simple exponential distributed lag” (cf. [20] Chap. 6 and [21]).Footnote 12
Substituting (A3) into (A1), we obtain
Differentiating (A4) with respect to t we obtain
which is equivalent to Eq. (8) in the text if we write \(\beta = 1/\tau.\) We can interpret τ as the average time lag of expectation adaptation.
Rights and permissions
Copyright information
© 2013 Springer Science+Business Media New York
About this chapter
Cite this chapter
Asada, T. (2013). Cyclical Fluctuations in Continuous Time Dynamic Optimization Models: Survey of General Theory and an Application to Dynamic Limit Pricing. In: Chinchuluun, A., Pardalos, P., Enkhbat, R., Pistikopoulos, E. (eds) Optimization, Simulation, and Control. Springer Optimization and Its Applications, vol 76. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-5131-0_13
Download citation
DOI: https://doi.org/10.1007/978-1-4614-5131-0_13
Published:
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4614-5130-3
Online ISBN: 978-1-4614-5131-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)