Kogure, Yosuke and Ikeda, Kiyohiro
(2022):
*Group-theoretic Study of Economic Agglomerations on a Square Lattice.*

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## Abstract

The present paper aims to elucidate the mechanism of economic agglomerations in two-dimensional economic spaces equipped with square road networks, which prosper worldwide (e.g., Chicago and Kyoto). A series of theoretical approaches provided in the present thesis makes it possible to investigate the spatial patterns of economic agglomerations on such spatial platforms systematically. The present paper focuses on square distributions on the square lattice economy, which has not somewhat been given much attention. We apply group-theoretic predictions to the investigation of bifurcation behavior of economic geography models. The present paper provides a systematic analysis procedure that is applicable to a wide range of economic geography models.

Item Type: | MPRA Paper |
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Original Title: | Group-theoretic Study of Economic Agglomerations on a Square Lattice |

Language: | English |

Keywords: | bifurcation, economic agglomeration, economic geography, group-theoretic bifurcation theory, invariant pattern, local-global system, replicator dynamics, square lattice. |

Subjects: | C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods R - Urban, Rural, Regional, Real Estate, and Transportation Economics > R4 - Transportation Economics |

Item ID: | 112842 |

Depositing User: | Prof. Kiyohiro Ikeda |

Date Deposited: | 28 Apr 2022 06:19 |

Last Modified: | 28 Apr 2022 06:21 |

References: | Ago, T., Isono, I., and Tabuchi, T. (2006). Locational disadvantage of the hub. The Annals of Regional Science, 40(4):819–848. Aizawa, H., Ikeda, K., Osawa, M., and Gaspar, J. M. (2020). Breaking and sustaining bifurcations in SN -invariant equidistant economy. International Journal of Bifurcation and Chaos, 30(16):2050240. Akamatsu, T., Mori, T., Osawa, M., and Takayama, Y. (2021). Multimodal agglomeration in economic geography. Unpublished manuscript, arXiv:1912.05113. Akamatsu, T., Takayama, Y., and Ikeda, K. (2012). Spatial discounting, Fourier, and racetrack economy: A recipe for the analysis of spatial agglomeration models. Journal of Economic Dynamics and Control, 36(11):1729–1759. Allen, T. and Arkolakis, C. (2014). Trade and the topography of the spatial economy. The Quarterly Journal of Economics, 129(3):1085–1140. Allgower, E. L., Böhmer, K., and Golubitsky, M. (1992). Bifurcation and symmetry: cross inﬂuence between mathematics and applications. International series of numerical mathematics, volume 104. Birkhäuser. Anas, A. (2004). Vanishing cities: what does the new economic geography imply about the eﬃciency of urbanization? Journal of Economic Geography, 4(2):181–199. Baldwin, R., Forslid, R., Martin, P., Ottaviano, G. I. P., and Robert-Nicoud, F. (2003). Economic geography and public policy. Princeton University Press. Baldwin, R., Forslid, R., Martin, P., Ottaviano, G. I. P., and Robert-Nicoud, F. (2011). Economic Geography and Public Policy. Princeton University Press. Barbero, J. and Zofío, J. L. (2016). The multiregional core–periphery model: The role of the spatial topology. Networks and Spatial Economics, 16(2):469–496. Beavon, K. S. O. (1977). Central place theory: a reinterpretation. Longman. Beckmann, M. J. (1976). Spatial equilibrium in the dispersed city. In Papageorgiou, Y., editor, Mathematical Land Use Theory, pages 117–125. Lexington Books. Bénard, H. (1900). Les tourbillons cellulaires dans une nappe liquide. Revue Générale des Sciences Pures et Appliquées, 11:1261–1271, 1309–1328. Bosker, M., Brakman, S., Garretsen, H., and Schramm, M. (2010). Adding geography to the new economic geography: bridging the gap between theory and empirics. Journal of Economic Geography, 10(6):793–823. Brakman, S., Garretsen, H., Marrewijk, C. V., and Berg, M. V. D. (1999). The return of Zipf: Towards a further understanding of the rank-size distribution. Journal of Regional Science, 39(1):183–213. Brakman, S., Garretsen, H., and van Marrewijk, C. (2001). The New Introduction to Geographical Economics. Cambridge University Press; 2nd edition, 2009. Busse, F. H. (1978). Non-linear properties of thermal convection. Reports on Progress in Physics, 41(12):1929–1967. Castro, S. B. S. D., Correia-da Silva, J., and Mossay, P. (2012). The core–periphery model with three regions and more. Papers in Regional Science, 91(2):401–418. Ceccherini-Silberstein, T., Scarabotti, F., and Tolli, F. (2010). Representation theory of the symmetric groups: the Okounkov-Vershik approach, character formulas, and partition algebras. Cambridge studies in advanced mathematics. Cambridge University Press. Chow, S. N. and Hale, J. K. (1982). Methods of bifurcation theory. A series of comprehensive studies in mathematics. Springer. Christaller, W. (1933). Die zentralen Orte in Süddeutschland. Gustav Fischer. English translation: Central places in southern Germany. Prentice Hall (1966). Cicogna, G. (1981). Symmetry breakdown from bifurcation. Lettere al Nuovo Cimento, 31:600–602. Clarke, M. and Wilson, A. G. (1983). The dynamics of urban spatial structure: Progress and problems. Journal of Regional Science, 23(1):1–18. Clarke, M. and Wilson, A. G. (1985). The dynamics of urban spatial structure: The progress of a research programme. Transactions of the Institute of British Geographers, 10(4):427–451. Combes, P. P., Mayer, T., and Thisse, J.-F. (2008). Economic geography: the integration of regions and nations. Princeton University Press. Coriell, S. R. and McFadden, G. B. (1993). Morphological stability. In Hurle, D. T. J., editor, Handbook of crystal growth, volume 1, part B, pages 785–857. Elsevier. Curtis, C. W. and Reiner, I. (1962). Representation theory of ﬁnite groups and associative algebras, volume 45 of Wiley classic library. Wiley (Interscience). Dellnitz, M. and Werner, B. (1989). Computational methods for bifurcation problems with symmetries–with special attention to steady state and hopf bifurcation points. Journal of Computational and Applied Mathematics, 26(1):97–123. Dicken, P. and Lloyd, P. E. (1990). Location in space: theoretical perspectives in economic geography, 3rd edition. Prentice Hall. Dionne, B., Silber, M., and Skeldon, A. C. (1997). Stability results for steady, spatially periodic planforms. Nonlinearity, 10(2):321–353. Eaton, B. C. and Lipsey, R. G. (1975). The principle of minimum diﬀerentiation reconsidered: Some new developments in the theory of spatial competition. Review of Economic Studies, 42:27–49. Eaton, B. C. and Lipsey, R. G. (1982). An economic theory of central places. The Economic Journal, 92(365):56–72. Elmihirst, T. (2004). SN -equivariant symmetry-breaking bifurcations. International Journal of Bifurcation and Chaos, 14(3):1017–1036. Forslid, R. and Ottaviano, G. I. P. (2003). An analytically solvable core–periphery model. Journal of Economic Geography, 3(3):229–240. Fujita, M. and Krugman, P. (1995). When is the economy monocentric?: von Thünen and Chamberlin uniﬁed. Regional Science and Urban Economics, 25(4):505–528. Fujita, M., Krugman, P., and Mori, T. (1999a). On the evolution of hierarchical urban systems. European Economic Review, 43(2):209–251. Fujita, M., Krugman, P., and Venables, A. (1999b). The spatial economy: cities, regions, and international trade. The MIT Press. Fujita, M. and Mori, T. (1997). Structural stability and evolution of urban systems. Regional Science and Urban Economics, 27(4):399–442. Fujita, M. and Thisse, J.-F. (2002). Economics of Agglomeration: Cities, Industrial Location, and Regional Growth. Cambridge University Press. Gaspar, J. M., Castro, S. B. S. D., and Correia-da Silva, J. (2018). Agglomeration patterns in a multi-regional economy without income eﬀects. Economic Theory, 66:863–899. Gaspar, J. M., Castro, S. B. S. D., and Correia-da Silva, J. (2020). The footloose entrepreneur model with a ﬁnite number of equidistant regions. International Journal of Economic Theory, 16(4):420–446. Golubitsky, M. and Schaeﬀer, D. G. (1985). Singularities and Groups in Bifurcation Theory, volume 1. Applied mathematical sciences, volume 51. Springer. Golubitsky, M., Schaeﬀer, D. G., and Stewart, I. (1988). Singularities and Groups in Bifurcation Theory, volume 2. Applied mathematical sciences, volume 69. Springer. Golubitsky, M. and Stewart, I. (2002). The symmetry perspective: from equilibrium to chaos in phase space and physical space. Progress in mathematics. Birkhäuser. Harris, B. and Wilson, A. G. (1978). Equilibrium values and dynamics of attractiveness terms in production-constrained spatial-interaction models. Environment and Planning A: Economy and Space, 10(4):371–388. Healey, T. J. (1988). A group-theoretic approach to computational bifurcation problems with symmetry. Computer Methods in Applied Mechanics and Engineering, 67(3):257–295. Helpman, E. (1998). The size of regions. In Pines, D., Sadka, E., and Zilcha, I., editors, Topics in public economies: theoretical and applied analysis, pages 33–54. Cambrigde University Press. Hoyle, R. B. (2006). Pattern formation: an introduction to methods. Cambridge texts in applied mathematics. Cambridge University Press. Ikeda, K., Aizawa, H., Kogure, Y., and Takayama, Y. (2018a). Stability of bifurcating patterns of spatial economy models on a hexagonal lattice. International Journal of Bifurcation and Chaos, 28(11):1850138. Ikeda, K., Akamatsu, T., and Kono, T. (2012a). Spatial period-doubling agglomeration of a core–periphery model with a system of cities. Journal of Economic Dynamics and Control, 36(5):754–778. Ikeda, K., Kogure, Y., Aizawa, H., and Takayama, Y. (2019a). Invariant patterns for replicator dynamics on a hexagonal lattice. International Journal of Bifurcation and Chaos, 29(06):1930014. Ikeda, K. and Murota, K. (2014). Bifurcation theory for hexagonal agglomeration in economic geography. Springer. Ikeda, K. and Murota, K. (2019). Imperfect bifurcation in structures and materials, 3rd edition. Springer. Ikeda, K., Murota, K., and Akamatsu, T. (2012b). Self-organization of Lösch’s hexagons in economic agglomeration for core–periphery models. International Journal of Bifurcation and Chaos, 22(08):1230026. Ikeda, K., Murota, K., Akamatsu, T., Kono, T., and Takayama, Y. (2014). Self-organization of hexagonal agglomeration patterns in new economic geography models. Journal of Economic Behavior and Organization, 99:32–52. Ikeda, K., Murota, K., Akamatsu, T., and Takayama, Y. (2017a). Agglomeration patterns in a long narrow economy of a new economic geography model: Analogy to a racetrack economy. International Journal of Economic Theory, 13(1):113–145. Ikeda, K., Murota, K., and Takayama, Y. (2017b). Stable economic agglomeration patterns in two dimensions: Beyond the scope of central place theory. Journal of Regional Science, 57(1):132–172. Ikeda, K., Onda, M., and Takayama, Y. (2018b). Spatial period doubling, invariant pattern, and break point in economic agglomeration in two dimensions. Journal of Economic Dynamics and Control, 92:129–152. Ikeda, K., Onda, M., and Takayama, Y. (2019b). Bifurcation theory of a racetrack economy in a spatial economy model. Networks and Spatial Economics, 19(1):57–82. Ikeda, K., Osawa, M., and Takayama, Y. (2022). Time evolution of city distributions in Germany. Networks and Spatial Economics. DOI: 10.1007/s11067-021-09557-2 Ikeda, K., Takayama, Y., Onda, M., and Murakami, D. (2018c). Group-theoretic spectrum analysis of population distribution in Southern Germany and Eastern Usa. International Journal of Bifurcation and Chaos, 28(14):1830045. Isard, W. (1975). Introduction to regional science. Prentice-Hall. Kettle, S. F. A. (2007). Symmetry and structure: readable group theory for chemists, 3rd edition. Wiley. Kim, S. K. (1999). Group theoretical methods and applications to molecules and crystals. Cambridge University Press. Kirchgässner, K. (1979). Exotische lösungen des bénardschen problems. Mathematical Methods in the Applied Sciences, 1(4):453–467. Kochendörfer, R. (1970). Group Theory. McGraw-Hill. Kogure, Y., Ikeda, K., and Aizawa, H. (2021). Group-theoretic bifurcation mechanism of economic agglomerations on a square lattice. International Journal of Bifurcation and Chaos, 31(13):2130040. Koschmieder, E. L. (1993). Bénard cells and Taylor vortices. Cambridge University Press. Krugman, P. (1991). Increasing returns and economic geography. Journal of Political Economy, 99(3):483–499. Krugman, P. (1993). On the number and location of cities. European Economic Review, 37(2):293–298. Krugman, P. (1996). The Self-organizing Economy. Blackwell. Lloyd, P. and Dicken, P. (1972). Location in space: a theoretical approach to economic geography. Harper & Row. Lösch, A. (1940). Die räumliche Ordnung der Wirtschaft. Gustav Fischer. English translation: The economics of location. Yale University Press (1954). Marsden, J. and Ratiu, T. (1999). Introduction to mechanics and symmetry, 2nd edition. Texts in applied mathematics, volume 17. Springer. Melbourne, I. (1999). Steady-state bifurcation with Euclidean symmetry. Transactions of the American Mathematical Society, 351(4):1575–1603. Mitropolsky, Y. and Lopatin, A. (1988). Nonlinear mechanics, groups and symmetry. Mathematics and its applications. Kluwer. Mori, T. (1997). A modeling of megalopolis formation: The maturing of city systems. Journal of Urban Economics, 42(1):133–157. Mossay, P. and Picard, P. M. (2011). On spatial equilibria in a social interaction model. Journal of Economic Theory, 146(6):2455–2477. Munz, M. and Weidlich, W. (1990). Settlement formation, part II: numerical simulation. The Annals of Regional Science, 24(3):177–196. Murata, Y. and Thisse, J.-F. (2005). A simple model of economic geography á la Helpman–Tabuchi. Journal of Urban Economics, 58(1):137–155. Olver, P. J. (1995). Equivalence, invariants, and symmetry. Cambridge University Press. Osawa, M., Akamatsu, T., and Kogure, Y. (2020). Stochastic stability of agglomeration patterns in an urban retail model. Unpublished manuscript, arXiv:2011.06778. Osawa, M., Akamatsu, T., and Takayama, Y. (2017). Harris and wilson (1978) model revisited: The spatial period-doubling cascade in an urban retail model. Journal of Regional Science, 57(3):442–466. Pﬂüger, M. (2004). A simple, analytically solvable, Chamberlinian agglomeration model. Regional Science and Urban Economics, 34(5):565–573. Pﬂüger, M. and Südekum, J. (2008). Integration, agglomeration and welfare. Journal of Urban Economics, 63(2):544–566. Picard, P. M. and Tabuchi, T. (2010). Self-organized agglomerations and transport costs. Economic Theory, 42(3):565–589. Puga, D. (1999). The rise and fall of regional inequalities. European Economic Review, 43(2):303–334. Redding, S. J. and Rossi-Hansberg, E. (2017). Quantitative spatial economics. Annual Review of Economics, 9(1):21–58. Redding, S. J. and Sturm, D. M. (2008). The costs of remoteness: Evidence from german division and reuniﬁcation. American Economic Review, 98(5):1766–1797. Robert-Nicoud, F. (2005). The structure of simple ‘new economic geography’ models (or, on identical twins). Journal of Economic Geography, 5(2):201–234. Saiki, I., Ikeda, K., and Murota, K. (2005). Flower patterns appearing on a honeycomb structure and their bifurcation mechanism. International Journal of Bifurcation and Chaos, 15(2):497–515. Sandholm, W. H. (2010). Population Games and Evolutionary Dynamics. The MIT press. Sattinger, D. H. (1979). Group Theoretic Methods in Bifurcation Theory. Lecture notes in mathematics, volume 762. Springer. Sattinger, D. H. (1983). Branching in the presence of symmetry. Regional conference series in applied mathematics, volume 40. Society for Industrial and Applied Mathematics. Schrijver, A. (1998). Theory of linear and integer programming. Wiley-interscience series in discrete mathematics and optimization. Wiley. Serre, J. P. (1977). Linear representations of ﬁnite groups. Graduate texts in mathematics, volume 42. Springer. Sheard, N. (2021). The network of us airports and its eﬀects on employment. Journal of Regional Science, 61(3):623–648. Silber, M. and Proctor, M. R. E. (1998). Nonlinear competition between small and large hexagonal patterns. Physical Review Letter, 81:2450–2453. Sivashinsky, G. I. (1983). Instabilities, pattern formation, and turbulence in ﬂames. Annual Review of Fluid Mechanics, 15(1):179–199. Skeldon, A. C. and Silber, M. (1998). New stability results for patterns in a model of long-wavelength convection. Physica D: Nonlinear Phenomena, 122(1):117–133. Stelder, D. (2005). Where do cities form? A geographical agglomeration model for Europe. Journal of Regional Science, 45(4):657–679. Tabuchi, T. (1998). Urban agglomeration and dispersion: A synthesis of Alonso and Krugman. Journal of Urban Economics, 44(3):333–351. Tabuchi, T. and Thisse, J.-F. (2011). A new economic geography model of central places. Journal of Urban Economics, 69(2):240–252. Tabuchi, T., Thisse, J.-F., and Zeng, D.-Z. (2005). On the number and size of cities. Journal of Economic Geography, 5(4):423–448. Takayama, Y. and Akamatsu, T. (2011). Emergence of polycentric urban conﬁgurations from combination of commu-nication externality and spatial competition. Journal of Japan Society of Civil Engineers, Series D3 (Infrastructure Planning and Management), 67(1):1–20. Takayama, Y., Ikeda, K., and Thisse, J.-F. (2020). Stability and sustainability of urban systems under commuting and transportation costs. Regional Science and Urban Economics, 84:103553. Tanaka, R., Saiki, I., and Ikeda, K. (2002). Group-theoretic bifurcation mechanism of pattern formation in three-dimensional uniform materials. International Journal of Bifurcation and Chaos, 12(12):2767–2797. Taylor, G. I. (1923). Stability of a viscous liquid contained between two rotating cylinders. Philosophical Transactions of the Royal Society of London, Series A (Containing Papers of a Mathematical or Physical Character), 223:289–343. Turing, A. M. (1952). The chemical basis of morphogenesis. Philosophical Transactions of the Royal Society of London, Series B (Biological Sciences), 237:37–72. Vanderbauwhede, A. L. (1982). Local bifurcation and symmetry. Research notes in mathematics, volume 75. Pitman. Weidlich, W. and Haag, G. (1987). A dynamic phase transition model for spatial agglomeration processes. Journal of regional science, 27(4):529–569. |

URI: | https://mpra.ub.uni-muenchen.de/id/eprint/112842 |