Abstract
The spatial agglomeration is investigated for a racetrack economy, comprising a number of cities equally spreading on the circumference of a circle, with micromechanism by Krugman’s core–periphery model. The group-theoretic bifurcation analysis procedure presented in Chap. 2 is applied to a problem with the dihedral group, expressing the symmetry of the racetrack economy. The theoretically possible agglomeration (bifurcation) patterns of this economy are predicted by using block-diagonalization, bifurcation equation, and equivariant branching lemma. Spatial period doubling bifurcation cascade is highlighted as the most characteristic progress of agglomeration. This chapter, as a whole, serves as an introduction to the methodology for a more general analysis in Chaps. 5–9 in Part II of an economy on a hexagonal lattice with a larger and more complicated symmetry group.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
Bifurcation behavior of systems with dihedral group symmetry has been studied fully in the literature of applied mathematics: e.g., Sattinger, 1979 [14], 1983 [15]; Healey, 1988 [7]; Golubitsky, Stewart, and Schaeffer, 1988 [6]; Dellnitz and Werner, 1989 [4]; and Ikeda and Murota, 2010 [8]. Sections 3.2–3.4 are adaptations of these results to the racetrack economy.
- 2.
See Sect. 1.5.3 for the definition of interior and corner equilibria.
- 3.
See Remark 3.3 for the proof.
- 4.
The transformation matrix Q used here is an orthogonal matrix.
- 5.
This corresponds to (2.98) with \(\boldsymbol{u} = \boldsymbol{\lambda }\), \(\boldsymbol{u}_{\mathrm{c}} = \boldsymbol{\lambda }_{0}\), \(\boldsymbol{w} = {w}^{(-,+)}\boldsymbol{{q}}^{(-,+)}\), and \(\overline{\boldsymbol{w}} =\sum _{ j=1}^{n/2-1}({w}^{(j),1}\boldsymbol{{q}}^{(j),1} + {w}^{(j),2}\boldsymbol{{q}}^{(j),2})\). The term \({w}^{(+,+)}\boldsymbol{{q}}^{(+,+)}\) is lacking due to the condition of no population growth in (3.4).
- 6.
By (2.76) of Ikeda and Murota, 2010 [8], a simple critical point is classified as a bifurcation point if \(\boldsymbol{\xi }_{1}^{\,\top }(\partial \boldsymbol{F}/\partial \tau )_{\mathrm{c}} = 0\), and a limit point of τ if \(\boldsymbol{\xi }_{1}^{\,\top }(\partial \boldsymbol{F}/\partial \tau )_{\mathrm{c}}\neq 0\).
- 7.
- 8.
When a stable equilibrium path becomes unstable at a critical point where a stable bifurcated path does not exist, the stable path often shifts dynamically to another stable path. This is called dynamical shift in this book.
- 9.
References
Akamatsu T, Takayama Y, Ikeda K (2012) Spatial discounting, Fourier, and racetrack economy: a recipe for the analysis of spatial agglomeration models. J Econ Dynam Contr 36(11): 1729–1759
Behrens K, Thisse J-F (2007) Regional economics: a new economic geography perspective. Reg Sci Urban Econ 37(4):457–465
Bosker M, Brakman S, Garretsen H, Schramm M (2010) Adding geography to the new economic geography: bridging the gap between theory and empirics. J Econ Geogr 10: 793–823
Dellnitz M, Werner B (1989) Computational methods for bifurcation problems with symmetries—with special attention to steady state and Hopf bifurcation points. J Comput Appl Math 26:97–123
Fujita M, Krugman P, Venables AJ (1999) The spatial economy: cities, regions, and international trade. MIT, Cambridge
Golubitsky M, Stewart I, Schaeffer DG (1988) Singularities and groups in bifurcation theory, vol 2. Applied mathematical sciences, vol 69. Springer, New York
Healey TJ (1988) A group theoretic approach to computational bifurcation problems with symmetry. Comp Meth Appl Mech Eng 67:257–295
Ikeda K, Murota K (2010) Imperfect bifurcation in structures and materials: engineering use of group-theoretic bifurcation theory, 2nd edn. Applied mathematical sciences, vol 149. Springer, New York
Ikeda K, Akamatsu T, Kono T (2012) Spatial period-doubling agglomeration of a core–periphery model with a system of places. J Econ Dynam Contr 36(5):754–778
Kettle SFA (1995) Symmetry and structure, 2nd edn. Wiley, Chichester
Kim SK (1999) Group theoretical methods and applications to molecules and crystals. Cambridge University Press, Cambridge
Krugman P (1993) On the number and location of places. Eur Econ Rev 37:293–298
Picard PM, Tabuchi T (2010) Self-organized agglomerations and transport costs. Econ Theor 42:565–589
Sattinger DH (1979) Group theoretic methods in bifurcation theory. Lecture notes in mathematics, vol 762. Springer, Berlin
Sattinger DH (1983) Branching in the presence of symmetry. Regional conference series in applied mathematics, vol 40. SIAM, Philadelphia
Tabuchi T, Thisse J-F (2011) A new economic geography model of central places. J Urban Econ 69:240–252
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer Japan
About this chapter
Cite this chapter
Ikeda, K., Murota, K. (2014). Agglomeration in Racetrack Economy. In: Bifurcation Theory for Hexagonal Agglomeration in Economic Geography. Springer, Tokyo. https://doi.org/10.1007/978-4-431-54258-2_3
Download citation
DOI: https://doi.org/10.1007/978-4-431-54258-2_3
Published:
Publisher Name: Springer, Tokyo
Print ISBN: 978-4-431-54257-5
Online ISBN: 978-4-431-54258-2
eBook Packages: EngineeringEngineering (R0)