Breaking and sustaining bifurcations in SN-Invariant equidistant economy

This paper elucidates the bifurcation mechanism of an equidistant economy in spatial economics. To this end, we derive the rules of secondary and further bifurcations as a major theoretical contribution of this paper. Then we combine them with pre-existing results of direct bifurcation of the symmetric group SN [Elmhirst, 2004]. Particular attention is devoted to the existence of invariant solutions which retain their spatial distributions when the value of the bifurcation parameter changes. Invariant patterns of an equidistant economy under the replicator dynamics are obtained. The mechanism of bifurcations from these patterns is elucidated. The stability of bifurcating branches is analyzed to demonstrate that most of them are unstable immediately after bifurcation. Numerical analysis of spatial economic models confirms that almost all bifurcating branches are unstable. Direct bifurcating curves connect the curves of invariant solutions, thereby creating a mesh-like network, which appears as threads of warp and weft. The theoretical bifurcation mechanism and numerical examples of networks advanced herein might be of great assistance in the study of spatial economics.info:eu-repo/semantics/publishedVersio

商業集積を促進する地域差別政策の厚生分析：小売企業と居住者の空間分布と市場の失敗

Tohoku University博士（学術）thesi

Bifurcation theory of a square lattice economy: Racetrack economy analogy in an economic geography model

Bifurcation theory for an economic agglomeration in a square lattice economy is presented in comparison with that in a racetrack economy. The existence of a series of equilibria with characteristic agglomeration patterns is elucidated. A spatial period doubling bifurcation cascade between these equilibria is advanced as a common mechanism to engender fewer and larger agglomerations in both economies. Analytical formulas for a break point, at which the uniformity is broken under reduced transport costs, are proposed for an economic geography model by synthetically encompassing both economies

Group-theoretic Study of Economic Agglomerations on a Square Lattice

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

正方形格子における経済集積の群論的分岐メカニズム

Tohoku University風間基樹課

Break and sustain bifurcations of S_N-invariant equidistant economy

This paper aims at the elucidation of the bifurcation mechanism of an equidistant economy in Economic Geography. An attention is paid to the existence of invariant solutions that retain their spatial patterns when the bifurcation parameter changes. Theoretical results on symmetrybreaking bifurcation of the symmetric group SN, which describes the symmetry of this economy, is combined with the mechanism of sustain bifurcation of invariant patterns that is inherent to the economy. The stability of bifurcating branches is investigated theoretically to demonstrate that most of them are asymptotically unstable. Among a plethora of theoretically possible spatial patterns, those which actually become stable for spatial economic models are investigated numerically. The solution curves of the economy are shown to display a complicated mesh-like structure, which looks like threads of warp and weft

Break and sustain bifurcations of S_N-invariant equidistant economy

This paper aims at the elucidation of the bifurcation mechanism of an equidistant economy in Economic Geography. An attention is paid to the existence of invariant solutions that retain their spatial patterns when the bifurcation parameter changes. Theoretical results on symmetrybreaking bifurcation of the symmetric group SN, which describes the symmetry of this economy, is combined with the mechanism of sustain bifurcation of invariant patterns that is inherent to the economy. The stability of bifurcating branches is investigated theoretically to demonstrate that most of them are asymptotically unstable. Among a plethora of theoretically possible spatial patterns, those which actually become stable for spatial economic models are investigated numerically. The solution curves of the economy are shown to display a complicated mesh-like structure, which looks like threads of warp and weft

Break and sustain bifurcations of S_N-invariant equidistant economy

This paper aims at the elucidation of the bifurcation mechanism of an equidistant economy in Economic Geography. An attention is paid to the existence of invariant solutions that retain their spatial patterns when the bifurcation parameter changes. Theoretical results on symmetrybreaking bifurcation of the symmetric group SN, which describes the symmetry of this economy, is combined with the mechanism of sustain bifurcation of invariant patterns that is inherent to the economy. The stability of bifurcating branches is investigated theoretically to demonstrate that most of them are asymptotically unstable. Among a plethora of theoretically possible spatial patterns, those which actually become stable for spatial economic models are investigated numerically. The solution curves of the economy are shown to display a complicated mesh-like structure, which looks like threads of warp and weft

Group-theoretic analysis of a scalar field on a square lattice

In this paper, we offer group-theoretic bifurcation theory to elucidate the mechanism of the self-organization of square patterns in economic agglomerations. First, we consider a scalar field on a square lattice that has the symmetry described by the group $\textrm{D}_{4} \ltimes \mathbb{Z}_{n} \times \mathbb{Z}_{n})$ and investigate steady-state bifurcation of the spatially uniform equilibrium to steady planforms periodic on the square lattice. To be specific, we derive the irreducible representations of the group $\textrm{D}_{4} \ltimes (\mathbb{Z}_{n} \times \mathbb{Z}_{n})$ and show the existence of bifurcating solutions expressing square patterns by two different mathematical ways: (i) using the equivariant branching lemma and (ii) solving the bifurcation equation. Second, we apply such a group-theoretic methodology to a spatial economic model with the replicator dynamics on the square lattice and demonstrate the emergence of the square patterns. We furthermore focus on a special feature of the replicator dynamics: the existence of invariant patterns that retain their spatial distribution when the value of the bifurcation parameter changes. We numerically show the connectivity between the uniform equilibrium and invariant patterns through the bifurcation. The square lattice is one of the promising spatial platforms for spatial economic models in new economic geography. A knowledge elucidated in this paper would contribute to theoretical investigation and practical applications of economic agglomerations

Bifurcation theory of a racetrack economy in a spatial economy model

Racetrack economy is a conventional spatial platform for economic agglomeration in spatial economy models. Studies of this economy up to now have been conducted mostly on $2^k$ cities, for which agglomerations proceed via so-called spatial period doubling bifurcation cascade. This paper aims at the elucidation of agglomeration mechanisms of the racetrack economy in a general setting of an arbitrary number of cities. First, an attention was paid to the existence of invariant solutions that retain their spatial distributions when the transport cost parameter is changed. A complete list of possible invariant solutions, which are inherent for replicator dynamics and are dependent on the number of cities, is presented. Next, group-theoretic bifurcation theory is used to describe bifurcation from the uniform state, thereby presenting an insightful information on spatial agglomerations. Among a plethora of theoretically possible invariant solutions, those which actually become stable for spatial economy models are obtained numerically. Asymptotic agglomeration behavior when the number of cities become very large is studied