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Cyclical Fluctuations in Continuous Time Dynamic Optimization Models: Survey of General Theory and an Application to Dynamic Limit Pricing

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Optimization, Simulation, and Control

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 76))

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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.

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Notes

  1. 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. 2.

    We can introduce other parameters which affect functions F and f, but the formulation in the text is sufficient for our purpose.

  3. 3.

    We assume that the second-order conditions are also satisfied.

  4. 4.

    See mathematical appendix of [2].

  5. 5.

    Theorem 4(1) was referred to by ([15], p. 483) as “Asada-Yoshida Theorem”.

  6. 6.

    We reproduce the proof here. The method of proof is quite simple and straightforward.

  7. 7.

    Gaskins [16] used more general demand function that is not necessarily linear, but we use the linear demand function for simplicity of the analysis following [10] Chap. 10.

  8. 8.

    As for the exhaustive exposition of the theory of differential game, see [12].

  9. 9.

    In the appendix, we reinterpret this equation by means of a continuously distributed lag model of expectation formation.

  10. 10.

    Since \({\partial }^{2}H/\partial {p}^{2} = -2b < 0\) , the second-order condition is always satisfied.

  11. 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. 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

  1. 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)

    Google Scholar 

  2. Asada, T., C. Chiarella, P. Flaschel and R. Franke (2003) : Open Economy Macrodynamics : An Integrated Disequilibrium Approach. Springer, Berlin.

    Book  Google Scholar 

  3. Asada, T. and W. Semmler (1995) : “Growth and Finance : An Intertemporal Model.” Journal of Macroeconomics 17–4, pp. 623–649.

    Article  Google Scholar 

  4. 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.

    Google Scholar 

  5. Asada, T, W. Semmler and A. Novak (1998) : “Endogenous Growth and Balanced Growth Equilibrium.” Research in Economics 52–2, pp. 189–212.

    Article  Google Scholar 

  6. 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.

    Article  MathSciNet  MATH  Google Scholar 

  7. 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.

    Article  MathSciNet  MATH  Google Scholar 

  8. Benhabib, J. and A. Rustichini (1990) : “Equilibrium Cycling with Small Discounting.” Journal of Economic Theory 52, pp. 423–432.

    Article  MATH  Google Scholar 

  9. Chiang, A. (1992) : Elements of Dynamic Optimization. McGraw-Hill, New York.

    Google Scholar 

  10. Dixit, A. K. (1990) : Optimization in Economic Theory(Second Edition). Oxford University Press, Oxford.

    Google Scholar 

  11. Dockner, E. and G. Feichtinger (1991) : “On the Optimality of Limit Cycles in Dynamic Economic Systems.” Journal of Economics 53–1, pp. 31–50.

    Article  MathSciNet  Google Scholar 

  12. Dockner, E., S.Jorgensen, N. Van Long and G. Sorger (2000) : Differential Games in Economics and Management Science. Cambridge University Press, Cambridge.

    Book  MATH  Google Scholar 

  13. Faria, J. R. and J. P. Andrade (1998) : “Investment, Credit, and Endogenous Cycles.” Journal of Economics 67–2, pp. 135–143.

    Article  Google Scholar 

  14. Feichtinger, G., A. Novak and F. Wirl (1994) : “Limit Cycles in Intertemporal Adjustment Models.” Journal of Economic Dynamics and Control 18, pp. 353–380.

    Article  MathSciNet  Google Scholar 

  15. Gandolfo, G. (2009) : Economic Dynamics (Fourth Edition). Springer, Berlin.

    Google Scholar 

  16. Gaskins, D. W. (1971) : “Dynamic Limit Pricing : Optimal Pricing Under Threat of Entry.” Journal of Economic Theory 3, pp. 306–322.

    Article  Google Scholar 

  17. Judd, K. and B. Petersen (1986) : “Dynamic Limit Pricing and Internal Finance.” Journal of Economic Theory 39, pp. 368–399.

    Article  MathSciNet  MATH  Google Scholar 

  18. Liu, W. M. (1994) : “Criterion of Hopf Bifurcation without Using Eigenvalues.” Journal of Mathematical Analysis and Applications 182, pp. 250–256.

    Article  MathSciNet  MATH  Google Scholar 

  19. Romer, P. (1990) : “Endogenous Technological Change.” Journal of Political Economy 98, pp. 71–102.

    Article  Google Scholar 

  20. Shinkai, Y. (1970) : Economic Analysis and Differential-Difference Equations. Toyo Keizai Shinpo-sha, Tokyo. (in Japanese)

    Google Scholar 

  21. 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.

    Article  Google Scholar 

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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.

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Authors and Affiliations

  1. Chuo University, 742-1 Higashinakano, Hachioji, Tokyo, 102-0393, Japan

    Toichiro Asada

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  1. Toichiro Asada
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Correspondence to Toichiro Asada .

Editor information

Editors and Affiliations

  1. , Institute of Mathematics, National University of Mongolia, University Street 1, Ulaanbaatar, 210646, Erdenet, Mongolia

    Altannar Chinchuluun

  2. , Department of Industrial & Systems Engin, University of Florida, Weil Hall 401, Gainesville, 32611, Florida, USA

    Panos M. Pardalos

  3. The School of Economic Studies, Dept.of Mathematics & Computer Science, National University of Mongolia, Ulaanbaatar, 210646, Mongolia

    Rentsen Enkhbat

  4. , Centre for Process Systems Engineering, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom

    E. N. Pistikopoulos

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,

$${p}^{e}(t) ={ \int \limits }_{-\infty }^{t}p(s)\omega (s)\mathrm{d}s, $$
(A1)

where \(\omega (s)\) is a weighting function such that

$$\omega (s) \geqq 0,\quad {\int \limits }_{-\infty }^{t}\omega (s)\mathrm{d}s = 1. $$
(A2)

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

$$\omega (s) = (1/\tau ){\mathrm{e}}^{-(1/\tau )(t-s)} \geqq 0\quad ;\quad \tau > 0. $$
(A3)

Substituting (A3) into (A1), we obtain

$${p}^{e}(t){\mathrm{e}}^{(1/\tau )t} = (1/\tau ){\int \limits }_{-\infty }^{t}p(s){\mathrm{e}}^{(1/\tau )s}\mathrm{d}s. $$
(A4)

Differentiating (A4) with respect to t we obtain

$$\dot{{p}}^{e}(t) = (1/\tau )\{p(t) - {p}^{e}(t)\},$$

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.

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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

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