Spatial discounting, Fourier, and racetrack economy: A recipe for the analysis of spatial agglomeration models

Journal of Economic Dynamics & Control 36 (2012) 1729–1759

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Spatial discounting, Fourier, and racetrack economy: A recipe for the analysis of spatial agglomeration models$ Takashi Akamatsu a,n, Yuki Takayama b, Kiyohiro Ikeda c a b c

Graduate School of Information Sciences, Tohoku University, Aramaki-Aoba 6-6-06, Aoba-ku, Sendai, Miyagi 980-8579, Japan Graduate School of Science and Engineering, Ehime University, Japan Graduate School of Engineering, Tohoku University, Japan

a r t i c l e i n f o

abstract

Article history: Received 26 October 2010 Received in revised form 18 April 2012 Accepted 24 April 2012 Available online 13 June 2012

We provide an analytical approach that facilitates understanding the bifurcation mechanism of a wide class of economic models involving spatial agglomeration of economic activities. The proposed method overcomes the limitations of the Turing (1952) approach that has been used to analyze the emergence of agglomeration in the multi-regional core–periphery (CP) model of Krugman (1993, 1996). In other words, the proposed method allows us to examine whether agglomeration of mobile factors emerges from a uniform distribution and to analytically trace the evolution of spatial agglomeration patterns (i.e., bifurcations from various polycentric patterns as well as a uniform pattern) that these models exhibit when the values of some structural parameters change steadily. Applying the proposed method to a multi-regional CP model, we uncover a number of previously unknown properties of the CP model, and notably, the occurrence of ‘‘spatial period doubling bifurcation’’ in the CP model is proved. & 2012 Elsevier B.V. All rights reserved.

JEL classification: R12 R13 F12 F15 F22 C62 C65 Keywords: Economic geography Agglomeration Stability Bifurcation Gravity laws

1. Introduction Most of the world’s population and economic activities are strikingly concentrated in a limited number of areas. Although thousands of articles have been devoted to explain the prevalence of this pattern, the core–periphery (CP) model developed by Krugman (1991) is the first successful attempt to explain this in a full-fledged general equilibrium framework (Fujita and Thisse, 2009). The CP model introduces the Dixit and Stiglitz (1977) model of monopolistic competition and increasing returns into spatial economics and provides a basic framework by which to describe how the interaction among increasing returns, transportation cost, and factor mobility can cause the formation of economic $ We are grateful for comments and suggestions made by Masahisa Fujita, Tomoya Mori, Se-il Mun, Yasusada Murata, Toshimori Otazawa, Daisuke Oyama, Yasuhiro Sato, Takatoshi Tabuchi, Kazuhiro Yamamoto, Dao-Zhi Zeng, as well as participants at the 23rd Applied Regional Science Conference Meeting in Yamagata. Funding from the Japan Society for the Promotion of Science, Grant-in-Aid for Scientific Research (B) 19360227/21360240/ 24360202 is greatly acknowledged. n Corresponding author. E-mail addresses: [email protected] (T. Akamatsu), [email protected] (Y. Takayama), [email protected] (K. Ikeda).

0165-1889/$ - see front matter & 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.jedc.2012.04.010

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agglomeration in geographical space. These new modeling techniques further led to the development of a new branch of spatial economics, known as ‘‘new economic geography (NEG)’’, which has become a fast-growing field. Over the past few decades, numerous extensions of the original CP model have appeared,1 and notably, there has been a proliferation of theoretical and empirical research applying the NEG theory framework to deal with various policy issues, such as trade policy, taxation, or regional redistribution in recent years (Baldwin et al., 2003; Combes et al., 2008). As indicated by its remarkable expansion, the NEG theory has the potential to provide a solid basis for predicting and evaluating the long-run effects of various economic policy proposals. There remain, however, some fundamental limitations that must be addressed before the NEG theory can be confirmed to provide a sound foundation for such applications. One of the most relevant issues is that the theoretical results are mainly restricted to two-region cases. Even though some efforts to overcome this limitation have been made in the early stages of development of the NEG (e.g., Krugman, 1993, 1996; Fujita et al., 1999b), theoretical studies on the NEG models have focused almost exclusively on two regions in the last decade. Although clearly the correct starting point for analysis, two-region models have several limitations. First, these models can neither describe nor explain a rich variety of polycentric patterns that are prevalent in real-world economies. Second, these models cannot provide an adequate framework to consider spatial interactions in a well-defined sense. For example, consider the concept of market accessibility, the differences among regions of which, as evidenced by numerous empirical studies, are major factors that lead to various forms of spatial agglomeration. Note that this concept is defined as the relative position of each region within the entire network of interactions. Due to the existence of such network effects, the impacts of a change in market accessibility in a multi-regional system are quite different from those in a two-region system, the topology of which is too simple to capture the network effects (Behrens and Thisse, 2007). In view of the limitations of two-region models, it remains unclear as to whether we can extrapolate the predictions and implications derived from two-regional analysis to a multi-regional system (Behrens and Thisse, 2007). Therefore, it is fair to say that ‘‘real-world geographical issues cannot be easily mapped into two-regional analysis’’ (Fujita et al., 1999b, p. 79), and the multi-regional framework is a prerequisite for systematic empirical research as well as for practical applications of the NEG theory. In the present paper, an attempt is made to clarify the spatial agglomeration properties of a multi-regional CP model in which regions are located on a circumference. As described in the NEG literature, the two-region CP model exhibits a bifurcation from a symmetric equilibrium (spatially uniform distribution of population) to an asymmetric equilibrium (agglomeration of population), depending on the values of certain structural parameters. In analyzing the multi-regional CP model, the most important, yet most difficult, step is examining the more complex bifurcation (i.e., evolution of spatial agglomeration structure). In the present paper, we investigate the mechanism of the bifurcation through a novel analytical approach synthesizing several techniques each of which is available in economics or in nonlinear mathematics. We shall ¨ explain this approach later in this section. For the analysis, we first present a multi-regional version of Pfluger’s (2004) two-regional CP model, which is a slight modification of Krugman’s original CP model, so that short-run equilibrium is obtained analytically.2 We then show a complete picture of the evolutionary processes of agglomeration patterns in the CP model with K regions equidistantly located on a circumference. The term evolutionary process means that we consider a process in which transportation cost parameter t steadily decreases over time, starting from a very high value of t, at which a uniform distribution of firms/workers is a stable equilibrium. It is first shown that the critical value of t at which the agglomeration structure first emerges (i.e., the first bifurcation occurs) and the associated agglomeration pattern can be obtained analytically. While the first bifurcation is the final and only bifurcation in conventional two-region models, the multi-regional CP model does not end the story. It is indeed proved that the spatial configuration of population evolves in association with further decreases in t, accompanied by the second bifurcation. The analysis of these two bifurcations further reveals that they share a common property, namely the number of core regions (in which firms/workers agglomerate) is reduced by half and the spacing between each pair of neighboring core-regions (agglomeration shadow) doubles after each bifurcation. This suggests a general rule whereby the multi-regional CP economy exhibits the spatial period doubling bifurcation (SPDB). Finally, a proof of this conjecture, including a general analytical formula to predict the occurrence of the SPDB, is presented.3 In addition to these results, we further uncover the effect of the heterogeneity of consumers on agglomeration patterns. Specifically, it is deduced that the CP model may exhibit repeated agglomeration and dispersion with the steady decrease in t, which provides a generalization, as well as a theoretical explanation, of several bell-shaped development results reported in studies on two-region CP models.4

1 For comprehensive reviews of the literature on the NEG/CP models, see the following monographs and handbooks: Fujita et al. (1999b), Papageorgiou and Pines (1998), Fujita and Thisse (2002), Baldwin et al. (2003), Henderson and Thisse (2004), Combes et al. (2008), Glaeser (2008), and Brakman et al. (2009). 2 The approach presented herein, in principle, is applicable to Krugman’s (1991) original CP model and its variants (e.g., Ottaviano et al., 2002; Forslid ¨ and Ottaviano, 2003), but we chose this Pfluger’s model, in favor of its analytical solvability, which is vital in the clarity of exposition. 3 While Krugman (1993, 1996) has presented some numerical results that can be regarded as special examples of the SPDB, no study has presented a rigorous proof of the general SPDB in the CP model. 4 See Tabuchi (1998), Helpman (1998), and Murata (2003).

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While the present paper elucidates the intrinsic properties of a multi-regional CP model, another unique contribution is to provide an analytical approach that facilitates understanding bifurcation properties of economic models (as well as the CP models) involving spatial agglomeration of economic activities. The approach presented in the present paper is characterized by the following three tools: (A) spatial discounting matrix (SDM), (B) discrete Fourier transformation (DFT), and (C) a system of regions on a circumference (racetrack economy (RE)). We outline our approach by briefly reviewing each of these tools/concepts in turn. (A) Geographical location patterns of economic activities are largely shaped by the tension between agglomeration forces and dispersion forces. Take the CP model with two regions, for example. Suppose that a firm moves from one region to another region (say, region 1). This generates an agglomeration force through a circular causation of forward linkages (market access effect on consumers) and backward linkages (market access effect on firms). The former is the incentive for workers to be close to the producers of consumer goods, and the latter is the incentive for producers to concentrate at locations where the market is greater. In opposition to the agglomeration force, a dispersion force (market crowding effect) is also at work. The increased competition in region 1 gives other firms in this region the incentive to move to other regions. If the agglomeration force is strong enough to overcome the dispersion force, then the core–periphery location pattern emerges in the two-regional economy. In a multi-regional economy with more than two regions, it should be emphasized that these agglomeration/dispersion forces not only affect region 1 but also spill over to every other region with a different strength due to the existence of transportation costs and imperfect competition of markets. Furthermore, in the multi-regional CP model, the decay of the agglomeration/dispersion forces with distance (i.e., these forces are spatially discounted) is described by gravity laws.5 Accordingly, the spatial extent and patterns of such spillover effects governed by the gravity laws determine the geographical location patterns of firms and workers. A key component of the multi-regional CP model for systematically dealing with the spatial patterns of the spillover effect is the spatial discounting matrix (SDM), the elements of which represent the spatial discounting factors (SDF) for all pairs of regions. (For the definitions of SDF and SDM, see Section 3.) The SDM encapsulates the essential information required to analyze the multi-regional CP model. In order to determine what types of spatial concentration-dispersion patterns can emerge in the CP model for a racetrack economy, it suffices (a) to obtain the eigenvalues of the SDM and (b) to represent the Jacobian matrix of the indirect utility as a function of the SDM, as shown later. Note that, although this fact holds for a wide variety of models dealing with the formation of spatial patterns in economic activities, we herein restrict ourselves to demonstrating this fact using the CP model. (B) Discrete Fourier transformation (DFT) is a simple and powerful technique for obtaining the essential indices (eigenvalues and eigenvectors) by which to predict the emergence of agglomeration. This may at first appear to be simply a computational issue, but its significance in the analysis of the CP model deserves further explanation. Most economic models involving spatial agglomeration, including the CP model, usually predict the existence of multiple equilibria. As a manner of refining the set of equilibria, it is customary in the NEG literature to assess the stability of equilibria by assuming a particular class of adjustment process (e.g., replicator dynamics) used in evolutionary game theory.6 As demonstrated by Papageorgiou and Smith (1983), Krugman (1996), and Fujita et al. (1999b), combining the evolutionary approach with Turing’s approach (Turing, 1952) provides a very effective method of investigating the emergence of agglomeration. In this approach, the investigation is concerned primarily with the onset of the instability of a uniform equilibrium distribution of mobile workers (which implies the emergence of some agglomeration). The procedure for examining the instability is based on the eigenvalues g ¼ fg k ,k ¼ 0; 1, . . . ,K1g of the Jacobian matrix of the adjustment process evaluated at the symmetric equilibrium (flat earth equilibrium). Accordingly, if the eigenvalues g are represented as functions of the key parameters of the CP model (e.g., transport cost parameter t), we can predict whether a particular agglomeration pattern (bifurcation) will emerge with changes in the parameter values. While Turing’s approach offers a remarkable method for assessing the emergence of agglomeration, it has two important limitations. First, Turing’s approach deals with only the first stage of agglomeration, in which the value of transportation cost steadily decreases, and cannot give a general description of what happens thereafter. Second, despite its conceptual simplicity, performing the eigenvalue analysis in a straightforward manner becomes very complicated, even for a model with only a few cities, and it is, in general, almost impossible to analytically obtain the eigenvalues g evaluated for an arbitrary configuration of mobile workers. This is one of the most difficult obstacles to clarifying the general properties of the multi-regional CP model. As shown in the present paper, however, the DFT coupled with the racetrack economy resolves these two limitations. The eigenvalues g of the Jacobian of the CP model in the racetrack economy can be represented as a simple unimodal

5 The concepts of spatial discounting, accessibility, and gravity laws have a long tradition in fields dealing with spatial aspects of economic activities. Examples include geography, transportation science, location theory, urban economics, regional economics, and international economics. Early empirical studies of gravity laws in the social science literature extend back to Ravenstein (1885), Young (1924), Reilly (1931), Zipf (1946), Voorhees (1955), and Tinbergen (1962). It has been well established in regional science and transportation science that gravity laws can be theoretically derived from the maximum entropy principle (e.g., Wilson, 1967; Smith, 1976; Anas, 1983; Sen and Smith, 1995) or random utility (discrete choice) theory (e.g., Anderson et al., 1992). Since Anderson (1979), it has become increasingly apparent that the gravity models can also be derived from several different models with micro economic underpinning (e.g., Bergstrand, 1989; Evenett and Keller, 2002; Anderson and van Wincoop, 2003). 6 In the present paper, we use perturbed best response dynamics (e.g., Fudenberg and Levine, 1998; Vega-Redondo, 2003; Sandholm, 2010), instead of replicator dynamics, because the former is suitable for dealing with the CP model with heterogeneous agents (see Section 4).

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function of the eigenvalues of either the SDM or its submatrices. More precisely, the kth eigenvalue gk for the flat earth equilibrium reduces to a quadratic function of the kth eigenvalue fk of the SDM (see Section 5), whereas for the equilibrium configurations that emerge at later stages during the course of steadily decreasing transportation cost, a simple technique allows us to obtain each of the eigenvalues g as a function of the eigenvalues of the submatrices of the SDM (see Section 6). This means that, if the eigenvalues of the SDM and its submatrices are obtained analytically, we can completely characterize the intrinsic nature of the CP model without resorting to numerical simulations. The DFT offers a powerful means for obtaining the very eigenvalues, especially when combined with the racetrack economy explained below. (C) The racetrack economy (RE) is an idealized spatial setting in which locations (regions or cities) are spread equidistantly over the circumference of a circle.7 Perhaps, due to its apparently unrealistic setting, the significance of this spatial setting appears to have been neither fully recognized nor exploited in previous studies on the NEG models following the studies of Krugman (1996) and Fujita et al. (1999b).8 In fact, few studies have used this approach to analyze the multi-regional CP model. In our view, however, the analysis of the racetrack economy is, for several reasons, indispensable in fully clarifying the intrinsic self-organizing nature of a wide variety of models dealing with the spatial agglomeration of economic activities. First, the RE is helpful in obtaining a systematic overview of these spatial agglomeration models. Note that these modes have not been systematically compared in a unified spatial setting, despite the fact that the resulting location patterns are significantly affected by spatial boundary conditions or location specific conditions. In the RE, these effects are completely removed, so that we can isolate the self-organizing properties in a pure form. Accordingly, the RE provides a benchmark circumstance or an ideal test-bed designed to compare the intrinsic properties of these models, and such comparisons should facilitate the refinement of a body of knowledge on these models into a coherent and systematic theory. Second, the RE with discrete locations greatly facilitates the eigenvalue analysis of these spatial agglomeration models. More specifically, the SDM for the RE reduces to a circulant matrix, which allows us to obtain the analytical expression of the eigenvalues of the SDM through a simple matrix operation, namely, the DFT of the first row vector of the SDM.9 Furthermore, each row vector of the DFT matrix (i.e., each eigenvector) represents the agglomeration patterns of firms/ workers by the configuration pattern of the entries (see Section 4). Accordingly, the eigenvectors and the associated eigenvalues g thus obtained provide a simple method by which to predict what types of agglomeration patterns emerge for an arbitrary given condition. Finally, note that the results obtained for the RE have a significant degree of generality. From a mathematical point of view, the essence of the RE lies in its periodic boundary condition, which yields structural symmetry of the system. The bifurcation properties of such systems (with symmetric structure) often have some general regularities described by the group-theoretic bifurcation theory (e.g., Golubitsky et al., 1988). As can be inferred from this theoretical link, most of the intrinsic properties of the CP models obtained in the RE are not restricted to a one-dimensional space, but rather hold for a two-dimensional space with periodic boundaries. Therefore, the approach presented in the present paper can be easily extended to a two-dimensional space, although the present paper focuses on the one-dimensional case. Thus, by laying out these three tools/concepts (A), (B), and (C), the present study makes a unique contribution to the understanding of the bifurcation mechanism of a variety of economic agglomeration models, as well as the CP model. Although each of these tools/concepts is rather well established in the literature, the present study is unique in the sense that (1) it sheds new light on the essential role of the spatial discounting factor (SDF) in the context of the bifurcation mechanism/analysis of the economic models involving spatial agglomeration, (2) it provides an analytically tractable method that allows the evolutionary processes of spatial agglomeration, as well as the emergence of agglomeration, to be traced by effectively combining discrete Fourier transformation (DFT) with the circulant properties of the spatial discounting matrix (SDM) in the discrete racetrack economy (RE), and (3) it demonstrates the usefulness of the proposed method by uncovering a number of previously unknown properties of the CP model. The remainder of the present paper is organized as follows. Section 2 discusses related research. Section 3 formulates a multi-regional CP model and defines the stability of the equilibrium of the model. Section 4 defines the SDM in a racetrack economy, the eigenvalues of which are provided by DFT. In Section 5, several types of bifurcations from a uniform distribution are analyzed, which also serves as a preliminary step for the analysis of the SPDB in Section 6. Conclusions are presented in Section 7.

7 RE has reflection and rotation symmetries that are labeled by the dihedral group Dn. Bifurcations for systems with these symmetries have been studied extensively (Sattinger, 1979; Golubitsky et al., 1988), and transformation other than DFT based on this theory is available (Ikeda et al., 2012). Yet, DFT serves as a simple and sufficient means to deal with a wide class of models involving spatial agglomeration of economic activity that is studied in this paper. 8 Before Krugman’s research (1993, 1996), there were few studies using a circular city model (e.g., Salop, 1979; Novshek, 1980; Papageorgiou and Smith, 1983; Eaton and Wooders, 1985). 9 The use of DFT combined with a circulant matrix is not an esoteric technique. Rather, this is an indispensable technique in a wide variety of fields that deal with spatial patterns, including time-series analysis in econometrics, macro economics, and finance; signal processing, image processing, and pattern recognition in computer science; morphogenesis in theoretical biology; and phase transitions in materials in statistical physics. Therefore, it is natural that economic models involving spatial patterns of agglomeration might be analyzed using these tools.

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2. Related research Since the seminal work of Krugman (1991), an enormous number of studies on the CP model and its variants have been published. However, since the work of Krugman (1993, 1996) and Fujita et al. (1999b), theoretical studies on the CP/NEG models have been almost exclusively limited to two-region cases. A few exceptions have dealt with CP models having more than two regions (Mossay, 2003; Tabuchi et al., 2005; Oyama, 2009; Akamatsu and Takayama, 2012; Picard and Tabuchi, 2010; Tabuchi and Thisse, 2011; Ikeda et al., 2012; Castro et al., 2012). Krugman (1993) presented a few numerical simulation runs for the CP model in which 12 regions are equidistantly located on a circumference. The resultant locations of population, in which three types of evenly spaced agglomeration patterns (mono-centric, duo-centric, and tri-centric patterns) emerge, suggest a certain similarity to Christaller’s centralplace patterns (Christaller, 1933). The numerical experiments, however, are limited to a particular set of parameter values, and no theoretical analysis is given. Krugman (1996, Appendix) and Fujita et al. (1999b, Chapter 6) analyzed the CP model with a continuum of locations on a circumference. Using Turing’s approach (Turing, 1952), they succeeded in showing the conditions for the instability of the flat earth equilibrium. Their analysis, however, provides only the initial period of agglomeration (the first bifurcation) when the transportation cost steadily decreases. In order to analyze the possible bifurcations in the later stages, they resort to rather ad hoc numerical simulations. Thus, it is fair to say that their investigations, providing valuable insight, do not systematically show the general properties of the multi-regional CP model. Recently, a few studies have carefully re-examined the robustness of Krugman’s findings in the CP model with a continuous-space racetrack economy. Mossay (2003) theoretically qualified Krugman’s results in the case of workers’ heterogeneous preferences for location. More recently, Picard and Tabuchi (2010) examined the impact of the shape of transport costs on the structure of spatial equilibria. These studies, however, focus only on the first stage of agglomeration as the local instability of a uniform equilibrium distribution. Therefore, the same limitation as Krugman’s approach applies to these studies. Tabuchi et al. (2005) studied the impact of decreasing transport costs on the size and number of cities in a multi-regional model that extend a two-regional CP model by Ottaviano et al. (2002). Oyama (2009) showed that a potential function exists for the multi-regional CP model, and it allowed to identify a stationary state that is uniquely absorbing and globally accessible under the perfect foresight dynamics. These analyses are, however, restricted to a very special class of transport geometry in which regions are pairwise equidistant. Castro et al. (2012) examined the stability of multi-regional CP model, but conducted the stability analysis only on the mono-centric pattern. Akamatsu and Takayama (2012) analyzed a pair of multi-regional models that extend the two-region CP models developed by Forslid and Ottaviano (2003) and Helpman (1998). Although they demonstrated the evolutionary process of spatial agglomeration in these models, their analysis is restricted to the case of four-region models focusing on the comparison between the two models. Only Tabuchi and Thisse (2011) and Ikeda et al. (2012) have discussed the general regularities of bifurcations in the multi-regional CP model. In the former study, which was conducted independently at approximately the same time as the latter study, Tabuchi and Thisse predicted that a CP model could exhibit a spatial period-doubling bifurcation (SPDB) pattern. They also showed (by a numerical example) that a hierarchical urban system structure emerges for the CP model with multiple industries. Although some of their results on the bifurcation properties of the CP model share certain similarity with our findings, the purpose and the emphasis of their study were significantly different from ours. They attempted to derive an urban hierarchical principle from the CP model. Moreover, unlike our study, they appeared not to recognize the essential role of the SDM in the bifurcation analysis and did not provide a general methodology that is applicable to the analysis of a wider class of models other than the CP model. Accordingly, their analysis is specific to the CP model, in which no simple relations between the Jacobian matrix and the SDF are shown. In the latter study, Ikeda et al. presented an approach to understanding the bifurcation behaviors of the multi-regional Krugman model, combining the group-theoretic bifurcation theory with the computational bifurcation theory. They also predicted the possibility of the SPDB based on the former theory and then demonstrated its existence using numerical examples based on the latter theory. Admittedly, the former theory is rather abstract and cannot provide a concrete prediction (e.g., the former theory cannot answer with certainty whether the CP model exhibits the SPDB without eigenvalue analysis), although the former theory provides a very general framework for bifurcation analysis. In contrast, the latter theory allows us to obtain very concrete (numerical) predictions, although the range of predictions is limited to certain specific cases. Thus, the present paper bridges the gap between these two theories, by providing a methodology that allows us to analytically predict the general bifurcation properties of a wide class of economic models expressing spatial agglomeration of economic activities.

3. The core–periphery model In this section, we present the multi-regional CP model. Although the basic assumptions of this model are the same as ¨ those of Pfluger (2004),10 except for the number of regions, for the sake of completeness, we present the basic assumptions 10 The basic assumptions of this model are the same as those of Krugman (1991), except for the functional forms of a utility function and a ¨ production function. Pfluger (2004) replaces the utility function of Krugman with that of the international trade model of Martin and Rogers (1995) and the production function of Krugman with that of Flam and Helpman (1987).

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of this model in Section 3.1. The short-run equilibrium is described in Section 3.2. The long-run equilibrium and its stability are defined in Section 3.3. 3.1. Basic assumptions The economy is composed of K regions (labeled i ¼ 0; 1, . . . ,K1 ), two factors of production (skilled and unskilled labor), and two sectors (agriculture A and manufacturing M). There is a continuum of individuals with mass H þ L, comprising H skilled and L ¼ 1  K unskilled workers. They consume homogeneous A-sector goods and a continuum of varieties of horizontally differentiated M-sector goods, and supply one unit of this type of labor inelastically. Skilled workers are mobile across regions. The number of skilled workers in region i is denoted by hi. Unskilled workers are immobile and equally distributed across all regions (i.e., the number of unskilled workers in each region is one). Preferences U over the M- and A-sector goods are identical across individuals. The utility of an individual in region i is given by A M A UðC M i ,C i Þ ¼ m ln C i þC i ,

ð1Þ C Ai

is the consumption of the A-sector product in region i, and where m 4 0 is constant parameter, aggregate in region i and is defined as 0 1s=ðs1Þ X Z nj M ð s 1Þ= s cji ðkÞ dkA , Ci  @

CM i

is the manufacturing

0

j

where cji(k) is the consumption in region i of a variety k 2 ½0,nj  produced in region j, nj is the number of varieties produced in region j, and s 4 1 is the constant elasticity of substitution between any two varieties. The budget constraint is given by X Z nj pji ðkÞcji ðkÞ dk ¼ Y i , ð2Þ pAi C Ai þ 0

j

where pAi is the price of A-sector goods in region i, pji(k) is the price of a variety k in region i produced in region j and Yi is the income of an individual in region i. An individual in region i maximizes (1) subject to (2). This yields the following demand functions: C Ai ¼

Yi m, pAi

CM i ¼

pAi

ri

m,

cji ðkÞ ¼ mpAi pji ðkÞs rsi 1 , where ri denotes the price index of the differentiated product in region i, which is given by 0 11=ð1sÞ X Z nj 1 s ri ¼ @ pji ðkÞ dkA :

ð3aÞ

ð3bÞ

ð4Þ

0

j

Since the population in region i is hi þ 1, we have the total demand Cji(k) in region i for a variety k produced in region j C ji ðkÞ ¼ mpAi pji ðkÞs ris1 ðhi þ 1Þ:

ð5Þ

The A-sector is perfectly competitive and produces homogeneous goods under constant returns to scale technology, which requires one unit of unskilled labor in order to produce one unit of output. For simplicity, we assume that the A-sector goods are transported freely between regions and are chosen as the nume raire. These assumptions mean that, in equilibrium, the wage of unskilled worker wLi is equal to the price of the A-sector goods in all regions (i.e., pAi ¼ wLi ¼ 1, 8i). The M-sector output is produced under increasing returns to scale technology and Dixit–Stiglitz monopolistic competition. A firm incurs a fixed input requirement of a units of skilled labor and a marginal input requirement of b units of unskilled labor. Given the fixed input requirement a, the skilled labor market clearing implies that, in equilibrium, the number of firms in region i is determined by ni ¼ hi =a. An M-sector firm located in region i maximizes profit with respect to fpij ðkÞ, 8jg X Pi ðkÞ ¼ pij ðkÞC ij ðkÞðawi þ bxi ðkÞÞ, j

where wi is the wage of the skilled worker and xi(k) is the total supply. The transportation costs for M-sector goods are assumed to take on iceberg form. That is, for each unit of the M-sector goods transported from region i to region jai, only a fraction 1=fij o1 arrives. Thus, the total supply xi(k) is given by X xi ðkÞ ¼ fij C ij ðkÞ: ð6Þ j

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Since we have a continuum of firms, each firm is negligible in the sense that its action has no impact on the market (i.e., the price indices). Hence, the first-order condition for profit maximization gives pij ðkÞ ¼

sb f : s1 ij

ð7Þ

This expression implies that the price of the M-sector product does not depend on variety k, so that Cij(k) and xi(k) do not depend on k. Thus, we describe these variables without argument k. Substituting (7) into (4), the price index becomes 0 11=ð1sÞ sb @1 X ri ¼ hd A , ð8Þ s1 a j j ji 1s

where dji ¼ fji is a spatial discounting factor between regions j and i; from (5) and (8), dji is obtained as ðpji C ji Þ=ðpii C ii Þ, which means that dji is the ratio of total expenditure in region i for each M-sector product produced in region j to the expenditure for a domestic product. 3.2. Short-run equilibrium In the short-run, skilled workers are immobile between regions, that is, their spatial distribution (h  ½h0 ,h1 , . . . ,hK1 T ) is assumed to be given. The short-run equilibrium conditions consist of the M-sector goods market clearing condition and the zero-profit condition due to the free entry and exit of firms. The former condition can be written as (6). The latter condition requires that the operating profit of a firm is entirely absorbed by the wage bill of its skilled workers 8 9 = 1
Substituting (5)–(8) into (9), we have the short-run equilibrium wage wi ðhÞ ¼ ms1 fwðHÞ ðhÞ þ wðLÞðhÞ g, i i wiðHÞ ðhÞ 

X dij h, Dj ðhÞ j j

ð10Þ

wðLÞ ðhÞ  i

X dij , Dj ðhÞ j

ð11Þ

P where Dj ðhÞ  k dkj hk denotes the market size of the M-sector in region j. Thus, dij =Dj ðhÞ defines the market share in region j of each M-sector product produced in region i. The indirect utility vi ðhÞ is obtained by substituting (3b), (8), and (10) into (1) 11: vi ðhÞ ¼ Si ðhÞ þ s1 fwðHÞ ðhÞ þwðLÞ ðhÞg, i i

ð12Þ

where Si ðhÞ  ðs1Þ1 ln Di ðhÞ. For the convenience of the analysis in the following sections, we express the indirect utility function vðhÞ in vector form by using the spatial discounting matrix D, the (i,j) entry of which is dij vðhÞ ¼ SðhÞ þ s1 fwðHÞ ðhÞ þwðLÞ ðhÞg, T

where SðhÞ  ½S0 ðhÞ,S1 ðhÞ, . . . ,SK1 ðhÞ , and w

ð13Þ ðHÞ

and w

ðLÞ

are defined as

wðHÞ  Mh,

wðLÞ  M1,

ð14aÞ

M  DD1 ,

D  diag½DT h, 1  ½1; 1, . . . ,1T :

ð14bÞ

3.3. Adjustment process, long-run equilibrium and stability In the long run, the skilled workers are inter-regionally mobile. They are assumed to be heterogeneous in their preferences for location choice as in Tabuchi and Thisse (2002), Murata (2003) and Mossay (2003). That is, the indirect utility for an individual s 2 ð0,hi  in region i is expressed as ðsÞ vðsÞ i ðhÞ ¼ vi ðhÞ þ Ei ,

where EiðsÞ denotes the utility representing the idiosyncratic taste for residential location. We assume that the distribution fEðsÞ ,8sg is (absolutely) continuous. i We present the dynamics of the migration of the skilled workers in order to define the long-run equilibrium and its stability with respect to small perturbations (i.e., local stability). We assume that at each time period t, the opportunity for skilled workers to migrate emerges according to an independent Poisson process with arrival rate l. That is, for each time interval ½t,t þ dtÞ, a fraction l dt of skilled workers have the opportunity to migrate. Given an opportunity at time t, each 11

We ignore the constant terms and the coefficient m, which have no influence on the results in the following sections.

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worker chooses the region that provides the highest indirect utility viðsÞ ðhðtÞÞ, which depends on the current distribution hðtÞ  ½h0 ðtÞ,h1 ðtÞ, . . . ,hK1 ðtÞT . The fraction P i ðvðhðtÞÞÞ of skilled workers who choose region i at time t is given by P i ðvðhðtÞÞÞ ¼ Pr½viðsÞ ðhðtÞÞ 4vðsÞ ðhðtÞÞ8jai. Thus, we have j hi ðt þ dtÞ ¼ ð1l dtÞhi ðtÞ þ l dt HP i ðvðhðtÞÞÞ: By normalizing the unit of time so that l ¼ 1, we obtain the following adjustment process: _ ¼ FðhðtÞÞ  HPðvðhðtÞÞÞhðtÞ, hðtÞ

ð15Þ

_ where hðtÞ denotes the time derivative of hðtÞ, and PðvðhÞÞ  ½P0 ðvðhÞÞ, P 1 ðvðhÞÞ, . . . ,P K1 ðvðhÞÞT , and H is the population of the skilled worker. For the specific functional form of P i ðvðhðtÞÞÞ, we use the logit choice function exp½yvi ðhÞ , P i ðvðhÞÞ  P j exp½yvj ðhÞ

ð16Þ

where y 2 ð0,1Þ is the parameter denoting the inverse of variance of the idiosyncratic tastes. This implies the assumption that the distributions of fEiðsÞ ,8sg are Gumbel distributions, which are identical and independent across regions (e.g, McFadden, 1974; Anderson et al., 1992). The adjustment process described by (15) and (16) is the logit dynamics, which has been studied in evolutionary game theory (e.g., Fudenberg and Levine, 1998; Hofbauer and Sandholm, 2002, 2007; Sandholm, 2010). Next, we define the long-run equilibria and its stability. The long-run equilibria are stationary points of the adjustment process of (15). n

Definition 1. The long-run equilibria are defined as distributions h that satisfy n

n

n

Fðh Þ  HPðvðh ÞÞh ¼ 0:

ð17Þ

In the limit of y-1, vðsÞ ðhÞ ¼ vi ðhÞ holds for all i (i.e., EðsÞ -0), and the condition given in (17) reduces to a conventional case i i ( n n n V vi ðh Þ ¼ 0 if hi 40, ð18Þ n n V n vi ðh Þ Z 0 if hi ¼ 0, where Vn denotes the equilibrium utility. This limit case does not have idiosyncratic taste and is called homogeneous consumer case. n n We restrict our concern to the neighborhood of h , and define the stability of h in the sense of linear asymptotic n stability. In dynamic system theory, h is linearly stable if all the eigenvalues of the Jacobian matrix rFðhÞ  ½@F i ðhÞ=@hj  of n n the adjustment process of (15) have negative real part; h is critical if at least one eigenvalue is on the imaginary axis; h is linearly unstable if at least one eigenvalue has positive real part (Hirsch and Smale, 1974). These facts mean that the linear stability can be assessed by examining the following Jacobian matrix:

rFðhÞ ¼ HJðhÞrvðhÞI,

ð19Þ

where JðhÞ and rvðhÞ are K-by-K matrices, the (i,j) entries of which are @P i ðvðhÞÞ=@vj and @vi ðhÞ=@hj , respectively. For the logit choice function of (16), it is easily verified that the former Jacobian matrix JðhÞ is expressed as JðhÞ ¼ yfdiag½PðvðhÞÞPðvðhÞÞPðvðhÞÞT g:

ð20Þ

The latter Jacobian matrix rvðhÞ is given by

rvðhÞ ¼ rSðhÞ þ s1 frwðHÞ ðhÞ þ rwðLÞ ðhÞg,

ð21Þ

where rSðhÞ ¼ ðs1Þ1 M T , rwðHÞ ¼ MMHM T , rwðLÞ ¼ MM T and H  diag½h. 4. Spatial discounting matrix and racetrack economy In order to understand the bifurcation mechanism of the CP model, we need to know how the eigenvalues of the Jacobian matrix of the adjustment process of (15) depend on the bifurcation parameters (e.g., transportation cost t). A combination of the racetrack economy (RE) defined in Section 4.1 and the resultant circulant properties of the spatial discounting matrix (SDM) shown in Section 4.2 greatly facilitates the analysis. Section 4.3 shows that the eigenvalues fg k g of the Jacobian matrix of the adjustment process can be expressed as simple functions of the eigenvalues ff k g of the SDM. 4.1. Racetrack economy The so-called racetrack economy is considered as a geometrical distribution of regions. To be precise, K regions f0; 1,2, . . . ,K1g are assumed to be equidistantly located on a circle of radius 1, and the distance tði,jÞ between two regions i and j is defined as the minimum path length, i.e. tði,jÞ ¼ ð2p=KÞmði,jÞ,

ð22Þ

T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

1737

where mði,jÞ  minf9ij9,K9ij9g. The set ftði,jÞ,i,j ¼ 0; 1,2, . . . ,K1g of the distances can be represented by the spatial discounting matrix (SDM) D ¼ ðdij Þ. We define this matrix by dij  exp½ðs1Þttði,jÞ:

ð23Þ

Defining the spatial discount factor (SDF) by r  exp½ðs1Þtð2p=KÞ,

ð24Þ

mði,jÞ

. It follows from the definition that the SDF r is a monotonically decreasing function of the we can represent dij as r transportation cost parameter t, and hence the feasible range of the SDF (corresponding to 0 r t o1) is given by ð0; 1 : t ¼ 03r ¼ 1, and t-13r-0. As shown by Definitions (22) and (23), the SDM D in a racetrack economy is a circulant, which is constructed from the vector d0  ½1,r,r 2 , . . . ,r M , . . . ,r 2 ,rT (see, for example, Gray, 2006, for the definition of circulants). This circulant property of the matrix D offers a key to understanding the essential properties of the CP model. 4.2. Eigenvalue of the spatial discounting matrix By exploiting the fact that the SDM D is a circulant matrix, we can easily obtain the eigenvalues f  ½f 0 ,f 1 , . . . ,f K1 T of the SDM by discrete Fourier transformation (DFT). More specifically, the eigenvalues f are given by the following similarity transformation of D: Z n DZ ¼ diag½f ,

ð25Þ jk

n

where Z is a DFT matrix, element (j,k) of which is zjk  o ðj,k ¼ 0; 1, . . . ,K1Þ, o  exp½ið2p=KÞ, and Z denotes the conjugate transpose of Z. Furthermore, (25) and the properties of a circulant (see, for example, Gray, 2006) show that DFT of the vector d0 (the first row vector of D) directly yields the eigenvalues f f ¼ Zd0 :

ð26Þ

Since the entries of d0 satisfy d0,m ¼ d0,Km (m ¼ 1; 2, . . . ,M1), where M  K=2, we see that the eigenvalues f are composed of M-1 roots with multiplicity 2 and two simple roots (k ¼ 0,M). This together with the straightforward calculation of (26) yields the following lemma (proofs of lemmas are hereinafter given in the Appendices). Lemma 4.1. The eigenvalues f of the SDM on the RE with K regions are given by f m ¼ cm ðrÞRm ðrÞ f m ¼ f Km

ðm ¼ 0; 1, . . . ,MÞ,

ð27aÞ

ðm ¼ 0; 1, . . . ,MÞ,

ð27bÞ

where

cm ðrÞ 

1r 2 , 12r cosð2pm=KÞ þ r 2

Rm ðrÞ  1ð1Þm r M

ðm ¼ 0; 1, . . . ,MÞ:

As shown in Section 4.3, the eigenvalues f ðrÞ of matrix D=d, which can be obtained by normalizing the eigenvalues f ðrÞ by f 0 ¼ d0  1  d, play a key role in the analysis of the bifurcation from a uniform distribution h (flat earth equilibrium). It follows from Lemma 4.1 that the eigenvalues f ðrÞ have the following properties. Lemma 4.2. The normalized SDM D=d on the RE with K regions has the following eigenvector and the associated eigenvalues: (i) the kth eigenvector (k ¼ 0; 1,2, . . . ,K ¼ 1) is given by the kth row vector, zk , of the discrete Fourier transformation (DFT) matrix Z, that is, zk  ½1, ok , o2k , . . . , okðK1Þ ,

where o  exp½ið2p=KÞ:

ð28Þ

(ii) the kth eigenvalue f k (k ¼ 0; 1,2, . . . ,K1) is given by f 0 ¼ 1,

f M ¼ cM ðrÞ2 , (

f m ¼ f Km ¼

cm ðrÞcM ðrÞ ðm : evenÞ cm ðrÞcM ðrÞEM ðrÞ ðm : oddÞ

ð29aÞ ðm ¼ 1; 2, . . . ,M1Þ,

where cM ðrÞ ¼ ð1rÞ=ð1 þrÞ, EM ðrÞ  ð1 þr M Þ=ð1r M Þ. (iii) f m (m ¼ 1; 2, . . . ,M) is a monotonically decreasing function of r for 8r 2 ½0; 1Þ. (iv) 8 > ff m ðrÞg ¼ f 0 ðrÞ ¼ 1 < max m 8r 2 ½0; 1Þ: 2 > : minff m ðrÞg ¼ f M ðrÞ ¼ cM ðrÞ m

ð29bÞ

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(v)

T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

(

1 ¼ f 0 4 f 2 4    4f 2k 4    4f M ¼ cM ðrÞ2 1 Zf 1 4f 3 4    4 f 2k þ 1 4    4 f M1 ¼ cM ðrÞ2 EM

8r 2 ½0; 1Þ:

Fig. 1 illustrates the properties of the eigenvalues f in Lemma 4.2 for the case of K ¼16 (M ¼ K=2 ¼ 8). For convenience in the bifurcation analysis in later sections, we depict the eigenvalues as functions of the transportation cost parameter t. It follows from Definition (24) that the SDF r 2 ½0; 1Þ is a monotonically decreasing function of t (see Definition (24)), and that t ¼ 03r ¼ 1 and t-13r-0. This, together with (iii) in the lemma, indicate that f m is a monotonically increasing function of t and that t ¼ 03f m ¼ 0 and t-13f m -1. 4.3. Eigenvalues of the Jacobi matrices The eigenvalues g of the Jacobian matrix rFðhÞ at an arbitrary distribution h of the skilled labor cannot be obtained without resorting to numerical techniques. It is, however, possible at some symmetric distributions h to obtain analytical expressions for the eigenvalues g of the Jacobian. In particular, a uniform distribution of skilled workers (flat earth equilibrium), h  ½h,h, . . . ,h,hT and h  H=K, which has intrinsic significance in examining the emergence of agglomeration, gives us the simplest example for illustrating this fact. At the uniform configuration h, the Jacobian matrix of the adjustment process

rFðhÞ ¼ HJðhÞrvðhÞI

ð30Þ

reduces to a circulant, as shown in the following lemmas. Lemma 4.3. The Jacobian matrices JðhÞ and rvðhÞ at h are given by JðhÞ ¼ yK 1 ðIK 1 1 1T Þ,

ð31Þ

rvðhÞ ¼ h1 fb½D=da½D=d2 g,

ð32Þ

1

where a  s1 ð1þ h

Þ, b  ðs1Þ1 þ s1 , 1  ½1; 1, . . . ,1T .

Lemma 4.4. The Jacobian matrices JðhÞ, rvðhÞ, and rFðhÞ at h are circulants. The fact that rFðhÞ is a circulant allows us to obtain the eigenvalues g of rFðhÞ using the DFT matrix Z. Specifically, both sides of (30) can be diagonalized by the similarity transformation based on Z diag½g ¼ H diag½d diag½ediag½1,

ð33Þ

where d and e are the eigenvalues of JðhÞ and rvðhÞ, respectively. The two eigenvalues, d and e, in the right-hand side of (33) are easily obtained as follows. The former eigenvalues d are obtained by DFT of the first row vector of JðhÞ in (32)

d ¼ yK 1 ½0; 1,1, . . . ,1T : As for the latter eigenvalues e, note that rvðhÞ in (31) consists of additions and multiplications of the circulant D=d. This implies that the eigenvalues e are represented as functions of the eigenvalues f of D=d n o 1 b½f a½f 2 , e¼h

Fig. 1. Eigenvalues f m (m ¼ 0; 1,: :,8) of D=d for K¼16 as functions of transportation cost parameter t. (A) k: even. (B) k: odd.

T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

1739

where ½x2  ½x  ½x, and ½x  ½y denotes the component-wise products of vectors x and y. Thus, we arrive at the following lemma for the eigenvalues of rFðhÞ. Lemma 4.5. Consider a uniform distribution h of skilled workers in the RE with K regions. The Jacobian matrix rFðhÞ of the adjustment process of (15) at h has the following eigenvector and associated eigenvalues: (i) the kth eigenvector (k ¼ 0; 1, . . . ,K1) is given by the kth row vector, zk , of the discrete Fourier transformation (DFT) matrix Z. (ii) the kth eigenvalue gk (k ¼ 1; 2, . . . ,K1) is given by a quadratic function of the kth eigenvalue f k of the spatial discounting matrix D=d g k ¼ yGðf k Þ

ðk ¼ 1; 2, . . . ,K1Þ,

g 0 ¼ 1,

1

GðxÞ  bxax2 y where a  s

1

1

ð1 þh

ð34aÞ ð34bÞ

1

Þ, b  ðs1Þ

þs

1

.

The eigenvectors fzk ðk ¼ 0; 1, . . . ,K1Þg in the first part of Lemma 4.5 represent the agglomeration patterns of skilled workers by the configuration pattern of the entries. For example, all entries of z0 are equal to 1, and the entry pattern of z0 ¼ ½1; 1, . . . ,1 corresponds to the state in which skilled workers are uniformly distributed among K regions, whereas zM ¼ ½1,1; 1,1, . . . ,1,1, where the alternating sequence of 1 and - 1, represents an M-centric agglomerate pattern in which skilled workers reside in M ¼ K=2 regions alternately. Similarly, zM=2 correspond to an M/2-centric pattern. The eigenvalues gk in Lemma 4.5(ii) can be interpreted as the strength of net agglomeration force that leads the uniform distribution to the direction of the kth agglomeration pattern (i.e., zk ). The term net agglomeration force refers to the net effect of the agglomeration force minus the dispersion force. Specifically, these forces correspond to bx and ax2, respectively, in (34b). As is clear from the derivation of the eigenvalues g, the term bx stems from rS and the first term of rwðHÞ , which represent forward linkage (or price-index effect) and backward linkage (or market access effect of firms), respectively. Thus, the term bx represents the agglomeration force induced by the increase in the variety of products that would be realized when the uniform distribution h deviates to agglomeration pattern zk . The term ax2, which stems from rwðLÞ and the second term of rwðHÞ , represents the dispersion force due to increased market competition (market crowding effect) in the agglomerated pattern zk . 5. Uniform distribution and agglomeration 5.1. The first bifurcation—emergence of agglomeration We are now ready to examine the emergence of agglomeration in the CP model. In this and subsequent sections, we consider the bifurcation process of the equilibrium, where the value of the transportation cost t steadily decreases from a very high value.12 In order for the bifurcation from the flat earth equilibrium to occur with the changes in t, either of the eigenvalues fg k , 8ka0g must changes sign. Since the eigenvalues gk are given by Gðf k Þ defined in (34b), changes in sign indicate that there should exist real solutions for the following quadratic equation with respect to f k : Gðf k Þ ¼ 0:

ð35Þ

Moreover, the solutions must lie in the interval ½0; 1Þ, which is the possible range of the eigenvalue f k . These conditions lead to the following proposition. Proposition 5.1. Starting from a very high value of the transport cost t, we steadily decrease the value of t. In order for a bifurcation from a uniform equilibrium to some agglomeration to occur in the RE, the parameters of the CP model should satisfy pffiffiffiffiffi Y  b2 4ay1 Z 0 and b þ Y r2a: ð36Þ The first inequality in Proposition 5.1 is the condition for the existence of real solutions of (35) with respect to f k . This condition is not necessarily satisfied for cases in which the heterogeneity of consumers is very large (i.e., y is very small), which implies that no agglomeration occurs in such cases. The second inequality, which stems from the requirement that solutions of (35) be less than 1, corresponds to the no-black-hole condition, as described in the literature dealing with the two-region CP model. For cases in which the no-black-hole condition is not satisfied, the eigenvalues fg k , 8ka0g are positive even when the transportation cost t is very high, which means that the uniform distribution h cannot be a stable equilibrium. 12 In this paper, the evolution from the trivial solution (i.e., flat earth equilibrium) is investigated. As shown in Ikeda et al. (2012), the core–periphery models have multiple stable equilibria. A full analysis/proof of the bifurcation diagram is an important issue, but is beyond the scope of this paper. See Golubitsky et al. (1988) for general theory on the bifurcation of symmetric system and Ikeda et al. (2012) for the application of this theory to RE. For an asymmetric equilibrium, the SDM approach, however, loses its simple and systematic structure expounded in Sections 4.2 and 4.3.

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Fig. 2. First break point in the course of decreasing t.

In the following analyses, we assume that the parameters ðh, s, yÞ of the CP model satisfy (36) in Proposition 5.1. We then have two real solutions for (35) with respect to f k : pffiffiffiffiffi pffiffiffiffiffi bþ Y b Y and xn  : ð37Þ xnþ  2a 2a Each of the solutions xn7 represents a critical value (break point) at which a bifurcation from the flat earth equilibrium h occurs if we regard the eigenvalue f k as a bifurcation parameter. Since we are interested in the process of decreasing the value of t, it is more meaningful to express the break point in terms of t (rather than in terms of f k ). For this purpose, we define the inverse function, tk ðÞ, of the eigenvalue f k ðtÞ as a function of t

tn ¼ tk ðxn Þ3xn ¼ f k ðtn Þ:

ð38Þ

The inverse tk ðÞ is a monotonically increasing function because f k ðÞ is a monotonically increasing function of t (see Fig. 1). We now investigate bifurcation behavioral characteristics at the critical values xnþ and xn in this sequence. Suppose that the transportation cost is initially large enough for the condition t 4 tk ðxnþ Þ to hold. From Definition (38) of the inverse function tk ðÞ, this is equivalent to f k ðtÞ 4xnþ

8ka0:

ð39Þ

Since the eigenvalue gk is a concave function G of f k (see Lemma 4.5), the condition given in (39) implies that g k o0 8ka0 holds, and hence the flat earth equilibrium is stable. We then consider what happens when the value of t steadily decreases. The decrease in t leads to a state in which t r tk ðxnþ Þ holds for some k. This can be equivalently represented as f k ðtÞ rxnþ

(k,

ð40Þ

which means that g k Z0 (k, and, therefore, the flat earth equilibrium is unstable. While (40) does not explicitly show which of the eigenvalues ff k ðtÞg first satisfies this condition, Lemma 4.2 shows that the Mth eigenvalue first reaches the critical value xnþ because f M ðtÞ ¼ minff k g k

8t 2 ½0,1Þ:

Accordingly, the critical value of t at which the first bifurcation occurs is given by

tnþ  maxftk ðxnþ Þg ¼ tM ðxnþ Þ:

ð41Þ

k

The above procedure for obtaining the break point13 can also be illustrated graphically, as shown in Fig. 2. In the course of steadily decreasing the value of t, the bifurcation from the flat earth equilibrium occurs when either of f k ðtÞ first takes a value less than xnþ . Fig. 2 shows that this occurs at the point of intersection C þ of the horizontal line ‘ þ (which represents f k ¼ xnþ ) and the curve f M ðtÞ (which is the minimum of ff k ðtÞg). This figure also shows that C þ is characterized as the point at which the maximum of ftk ðxnþ Þg is attained. This yields (41) for the critical value of t. The fact that the Mth eigenvalue first reaches the critical value xnþ also enables us to identify the associated agglomeration pattern that emerges at the first bifurcation. Recall here that the equilibrium moves in the direction of the eigenvector for which the associated eigenvalue reaches the critical value. As stated in Lemma 4.5, this direction is the Mth eigenvector zM ¼ ½1,1, . . . ,1,1. Therefore, the pattern of agglomeration that first emerges is h ¼ h þ dzTM ¼ ½h þ d,hd, . . . ,h þ d,hdT

ð0 r d rhÞ,

in which skilled workers agglomerate in alternate regions.

ð42Þ 14

13 As stated in Fujita et al. (1999b), ‘break point’ is a bifurcation point at which the symmetry of the system is broken and agglomeration patterns emerge when the transportation cost t steadily decreases. 14 The form of (42) has a symmetry labeled by a group Dn=2 , which has a partial symmetry of Dn that labels the symmetry of the RE. The state of Dn=2 -symmetry emerges by a first bifurcation and is preserved until undergoing another bifurcation. The stability of this state is to be determined by a

T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

1741

Fig. 3. Stability and eigenvalue gk. (A) Homogeneous consumer case. (B) Heterogeneous consumer case.

Finally, note that (41) implicitly indicates how the changes in the values of the CP model parameters affect the critical value tnþ . More specifically, (41) is equivalent to xnþ ¼ f M ðrðtnþ ÞÞ, which yields the critical value in terms of the SDF: pffiffiffiffiffiffiffi 1 xnþ pffiffiffiffiffiffiffi : ð43Þ rðtnþ Þ ¼ 1þ xnþ This yields the relationship between tnþ and the CP model parameters ðh, s, yÞ because xnþ is explicitly represented as a function of ðh, s, yÞ in (37) and the SDF r is a simple exponential function of t, as defined in (24). Thus, we can characterize the bifurcation from the uniform distribution as follows. Proposition 5.2. Suppose that the CP model satisfies the conditions (36) described in Proposition 5.1. Starting from a state in which the value of the transportation cost t is large enough for the uniform distribution h to be a stable equilibrium, we consider the process in which the value of t steadily decreases. (i) The net agglomeration force (i.e., the eigenvalue) gk for each agglomeration pattern (i.e., the eigenvector) zk increases with the decrease in t, and the uniform distribution becomes unstable at the break point t ¼ tnþ given by (37) and (41). (ii) The critical value tnþ increases as (a) the heterogeneity of skilled workers (in location choice) becomes smaller (i.e., y is large), (b) the number of skilled workers relative to that of the unskilled workers becomes larger (i.e., h is large), and (c) the elasticity of substitution between two varieties becomes smaller (i.e., s is small). (iii) The pattern of agglomeration that first emerges is h ¼ ½hþ d,hd, . . . ,h þ d,hdT ð0 r d r 1Þ, in which skilled workers agglomerate in alternate regions. 5.2. The last bifurcation—collapse of agglomeration In Section 5.1, we have derived two critical values (with respect to the eigenvalue f k ), xnþ and xn , at each of which a bifurcation from the uniform distribution h occurs, and only the properties of the bifurcation at the former critical value (xnþ ) have been shown. We shall examine the bifurcation at the latter critical value (xn ). We discuss two cases, A and B. In Case A, consumers are homogeneous with respect to location choice (i.e., y-1), and in Case B, consumers are inhomogeneous (i.e., y is finite). As shown below, these two cases exhibit significantly different properties of the bifurcation at the critical value xn . 2 (A) For the homogeneous consumer case, Y ¼ b and the critical value given by (37) reduces to the following simpler expression: lim xn7 ðyÞ ¼

y-1

b a

or 0:

Substituting this into the definition of tk ðÞ, we see that tk ðxn Þ ¼ tk ð0Þ ¼ 0 8a0. Therefore, the critical value tn for the bifurcation from some agglomeration to the flat earth equilibrium is zero lim tn ¼ minftk ð0Þg ¼ 0:

y-1

k

This means that at least one of the eigenvalues fg k g is always positive for the interval ð0, tM ðxnþ ÞÞ of t (see Fig. 3). That is, no matter how small the transportation cost, agglomeration never collapses, except for the minimum limit of t ¼ 0. (footnote continued) coefficient containing up to the third-order derivatives of the governing equation (Thompson and Hunt, 1973, p. 96). This is an important issue, but is out of the scope of this paper, which focuses on the analysis of the Jacobian matrix of (19) that demands its second-order derivatives.

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Fig. 4. Repetitions of agglomeration and dispersion.

(B) For the case in which consumers are heterogeneous, we see from (37) that the critical value xn is always positive regardless of the values of the CP model parameters, and thus tk ðxn Þ 4 0 8k. This means that the net agglomeration force gk can be negative on the interval ð0, tk ðxn Þ of t (see Fig. 3). Accordingly, in the course of decreasing t, a bifurcation from some agglomeration to the uniform distribution (re-dispersion) h necessarily occurs at the critical value

tn ¼ minftk ðxn Þg 4 0: k

ð44Þ

That is, the CP economy with heterogeneous consumers necessarily moves from agglomeration to dispersion when the transportation cost t decreases to the critical value. This is a generalized result of the conventional bell-shaped bifurcation in the two-region CP model with idiosyncratic differences in location choice (e.g., Tabuchi, 1998; Helpman, 1998; Murata, 2003). It is worthwhile to examine the properties of the re-dispersion more closely. First, it can be deduced that, even if the consumers are homogeneous, the re-dispersion occurs when some dispersion forces (e.g., land rent) that increase with agglomeration are introduced into the CP model. This is because the necessary and sufficient condition for the occurrence of the re-dispersion is that the parameters of the CP model satisfy xn 40. This requirement can be satisfied by either adding an extra constant term in the function G in (34) or introducing heterogeneity in location choice (i.e., y is finite). Second, the re-dispersion can be observed not only at the critical value tn but also in the course of evolving agglomeration. The re-dispersion at tn is the last among multiple re-dispersions in the course of decreasing t. The mechanism of how such repetitions of agglomeration and dispersion may occur can best be explained by the example in Fig. 4.15 Each curve in this figure represents the eigenvalue f k at the uniform distribution h as a function of t. Recall that a bifurcation from agglomeration to dispersion occurs when all the eigenvalues fg k g at h become negative. We can see from the figure that this actually occurs twice during the process of decreasing t. Specifically, the first re-dispersion is observed at the boundary tnn  between ranges A and B. Some agglomeration is present in range A, because either of ff k ,8ka0; 1g is in ½xn ,xnþ , and hence either of the eigenvalues fg k g at h is positive, but h become stable below the boundary tnn  (range B) because all the eigenvalues fg k g at h are negative. (Note that f 1 4 xnþ and f k o xn 8ka0; 1, which means that none of ff k g is in ½xn ,xnþ .) Note here that some agglomeration again emerges in range C because f 1 ¼ f K1 are in ½xn ,xnþ , and hence the eigenvalues g 1 ¼ g K1 at h are positive. This leads to the second re-dispersion at the boundary tn between ranges C and D, and h again becomes stable below the boundary tn (range D) because all the eigenvalues fg k g at h are negative. (Note that f k o xn 8ka0, which means that none of ff k g is in ½xn ,xnþ .) The discussions in (A) and (B) can be summarized as the following proposition: Proposition 5.3. Starting with an agglomeration state, we consider the process in which the value of the transportation cost t decreases: (i) The net agglomeration force gk for each zk monotonically decreases with the decrease in t after a monotonic increase (i.e., gk is a unimodal function of tÞ. (ii) For the CP model with homogeneous consumers (i.e., y-1), all the net agglomeration forces reach zero only at the limit of t ¼ 0. That is, the agglomeration equilibrium never reverts to the uniform distribution equilibrium during the course of decreasing t. (iii) For the CP model with heterogeneous consumers (i.e., y is finite), all the net agglomeration forces reach zero at a strictly positive value of t. That is, a bifurcation from some agglomeration to the uniform distribution equilibrium (re-dispersion) 15 Detailed discussions on the economic implications/interpretations of this phenomenon are relegated to Akamatsu et al. (2009) because the purpose of the present paper is to demonstrate a method by which to analyze the CP models.

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occurs at t ¼ tn , where tn is given by (44) and (37). Furthermore, repetitions of agglomeration and dispersion may be observed (see Fig. 4).

6. Evolution of agglomeration patterns In conventional two-region CP models, steady decreases in transportation cost lead to the occurrence of a bifurcation from the uniform distribution h to a mono-centric agglomeration. In the multi-regional CP model, however, the first bifurcation shown in Section 5 does not directly branch to the mono-centric agglomeration. Instead, further bifurcations leading to a more concentrated pattern can repeatedly occur. In this section, we examine such evolution of agglomeration after the first bifurcation, restricting ourselves to the homogeneous consumer case (i.e., y-1). First, in Section 6.1, it is shown that a second bifurcation from h indeed occurs, which suggests a general rule of the agglomeration process whereby the multi-regional CP economy exhibits the spatial period doubling bifurcation (SPDB). In Section 6.2, a proof of this conjecture, including a general analytical formula to predict the occurrence of the SPDB, is presented. In order to illustrate the theoretical results, a number of numerical examples are presented in Section 6.3.

6.1. The second bifurcation As we decrease the value of t from the break point t ¼ tnþ (at which the first bifurcation occurs), the deviation d from the uniform distribution h increases monotonically (for the proof, see Appendix C). This eventually leads to an M-centric n pattern, h ¼ ½2h,0; 2h,0, . . . ,2h,0T , where skilled workers equally exist only in the alternate two regions. In other words, skilled workers/firms in half of the regions are completely absorbed into the neighboring core regions (in which skilled n n workers agglomerate) in h . The fact that the M-centric pattern h may exist as an equilibrium of the CP model can be n confirmed by examining the sustain point for h . The sustain point is the value of t below which the equilibrium condition n for h n

n

n

v0 ðh Þ ¼ v2 ðh Þ ¼ maxfvk ðh Þg k

ð45Þ

is satisfied. As shown in Appendix D, the condition of (45) indeed holds for any t smaller than tn01 , which is the sustain n point for h . n After the emergence of the M-centric pattern h ¼ ½2h,0; 2h,0, . . . ,2h,0T , further decreases in t below the sustain point n can lead to further bifurcations (i.e., h become unstable). In order to investigate such a possibility, we need to obtain the n eigenvectors and the associated eigenvalues for the Jacobian matrix of the adjustment process at h

rFðhn Þ ¼ HJðhn Þrvðhn ÞI:

ð46Þ n

A seeming difficulty encountered in obtaining the eigenvalues is that, unlike the Jacobian rFðhÞ at h, the Jacobian rFðh Þ n n at h is no longer a circulant matrix. This is due to the loss of symmetry in the configuration h of skilled workers, which n n leads to Jðh Þ and rvðh Þ not being circulant matrices. However, it is still possible to find a closed-form expression for the n n n n eigenvalues of rFðh Þ based on the fact that h has partial symmetry and the submatrices of Jðh Þ and rvðh Þ are circulants. n In order to exploit the symmetry remaining in the M-centric pattern h , we begin by dividing the set C ¼ f0; 1,2, . . . ,K1g of regions into two subsets, namely the subset C 0 ¼ 0; 2, . . . ,K2 of regions with skilled workers and the subset C 1 ¼ f1; 3, . . . ,K1g of regions without skilled workers. Corresponding to this division of the set of regions, we consider the following permutation s of the set C:

so that the first half elements fsð0Þ, sð1Þ, . . . , sðM1Þg and the second half elements fsðMÞ, sðM þ1Þ, . . . , sðK1Þg of the set C P ¼ fsð0Þ, sð1Þ, . . . , sðK1Þg correspond to the subsets C0 and C1, respectively. For this permutation, we define the associated K-by-K permutation matrix P ð1Þ ð47Þ where Peð1Þ and P oð1Þ are M-by-K matrices. The (i,j) element of P eð1Þ is 1 if j ¼ 2i (i ¼ 0; 1,2, . . . ,M1), and is otherwise 0, whereas that of Poð1Þ is 1 if j ¼ 2i þ1, and is otherwise 0. It is readily verified that, for a K-by-K matrix A, the (i,j) element of which is aij, P ð1Þ AP Tð1Þ yields a matrix for which the (i,j) element is asðiÞsðjÞ , and that P ð1Þ P Tð1Þ ¼ P Tð1Þ P ð1Þ ¼ I. That is, the similarity transformation P ð1Þ AP Tð1Þ gives a consistent renumbering of the rows and columns of A by the permutation s.

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The permutation s (or the permutation matrix Pð1Þ in (47)) constitutes a new coordinate system for analyzing the Jacobian matrix of the adjustment process. Under the new coordinate system, the SDM D can be represented as ð48Þ ð0Þ

where each of the submatrices Dð0Þ and Dð1Þ is a M-by-M circulant generated from vectors d0  ½1,r 2 ,r 4 , . . . ,r M , . . . ,r 4 ,r 2 T ð1Þ n and d0  ½r,r 3 ,r 5 , . . . ,r M1 , r M1 , . . . ,r 3 ,rT , respectively. Similarly, the Jacobian matrix rFðh Þ of the adjustment process is transformed into

r Fðhn Þ  P ð1Þ rFðhn ÞP Tð1Þ ¼ HJ  ðhn Þr vðhn ÞI,

ð49Þ

where the Jacobian matrices in the right-hand side are, respectively, defined as ð50aÞ

ð50bÞ For these Jacobian matrices under the new coordinate system, we have Lemma 6.1. All the submatrices J ðijÞ and V ðijÞ (i,j ¼0,1) are circulants. This leads to the following lemma, which significantly facilitates obtaining the eigenvalues g n ¼ fg nk g of the Jacobian n matrix rFðh Þ. n

Lemma 6.2. The eigenvalues g nk (k ¼ 1; 2, . . . ,K1) of the Jacobian matrix rFðh Þ of the adjustment process at n h ¼ ½2h,0; 2h,0, . . . ,2h,0T are represented as ( ð00Þ yek=2 1 ðk : evenÞ, g nk ¼ ð51Þ 1 ðk : oddÞ, ð00Þ T  denotes the eigenvalues of the Jacobian V ð00Þ . where eð00Þ  ½e0ð00Þ ,e1ð00Þ , . . . ,eM1

This lemma implies that knowing only V ð00Þ is sufficient to obtain the eigenvalues g n of the Jacobian matrix rFðh Þ. Furthermore, the submatrix V ð00Þ is a circulant consisting only of submatrices Dð0Þ , Dð1Þ , and Dð1ÞT of the SDM D. Since they ð0Þ ð1Þ ð1TÞ ð0Þ ð1Þ ð1TÞ are circulants, for which DFT of vectors d0 , d0 , and d0 yields the associated eigenvalues f , f , and f , we can obtain ð00Þ ð00Þ the eigenvalues e of the matrix V . n

Lemma 6.3. The Jacobian V ð00Þ and the associated eigenvalues eð00Þ are given by 2

V ð00Þ ¼ b½Dð0Þ =dð0Þ a1 fð1 þ 2hÞ½Dð0Þ =dð0Þ 2 þ½Dð1Þ Dð1ÞT =dð1Þ g, eð00Þ ¼ b½f

ð0Þ

where b  ðs1Þ

=dð0Þ a1 fð1 þ2hÞ½f

1

ð0Þ

=dð0Þ 2 þ½f 1

þ s1 and a1  s1 ð2hÞ

;

ð1Þ

  ½f

ðiÞ dðiÞ  d0

ð1TÞ

ð52aÞ

2

=dð1Þ g,

ð52bÞ

 1 (i¼ 0,1).

Combining this with (51) in Lemma 6.2, we have n

n

Lemma 6.4. The largest eigenvalue of the Jacobian matrix rFðh Þ of the adjustment process of (15) at h is given by n

maxfg nk g ¼ g nM ¼ yGn ðf M=2 ðrÞÞ k

8r 2 ð0; 1,

ð53Þ

and the associated eigenvector is the M  K=2th row vector znM of Z n  P Tð1Þ Z  P ð1Þ , where Z   diag½Z ½M ,Z ½M , Z ½M is an M-by-M n DFT matrix; f M=2  cðr 2 Þ2 1

Gn ðxÞ  bxan x2 y

,

an  a1 ð1 þ 2hÞ ¼ s1 ð1 þ ð2hÞ1 Þ,

ð54aÞ b  ðs1Þ1 þ s1 :

ð54bÞ

This lemma allows us to determine the critical value at which a bifurcation from the M-centric pattern n h ¼ ½2h,0; 2h,0, . . . ,2h,0T occurs. In a similar manner to the discussions for the first bifurcation from the uniform n distribution h, we see that the second bifurcation from the M-centric pattern h occurs when the maximum eigenvalue n g nM ðxÞ changes sign. Accordingly, the critical values of f M=2  cðr 2 Þ2 at which the second bifurcation occurs are the solutions

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of a quadratic equation Gn ðxÞ ¼ 0. For the homogeneous consumer case (i.e., y-1), the quadratic equation reduces to lim Gn ðxÞ ¼ bxan x2 ¼ 0,

ð55Þ

y-1

and hence, the critical values (i.e., the solutions of (55)) are given by xnn þ ¼

b bs ¼ an 1 þ ð2hÞ1

and

xnn  ¼ 0:

ð56Þ

Since the possible range of f M=2 is ½0; 1Þ, the critical value xnn þ must be less than 1 for the occurrence of the second bifurcation. This condition is equivalent to ho

1s1 : 2

ð57Þ

n

Note here that f M=2 is a monotonically increasing function of t. This implies that, in the course of decreasing t, the second nn nn 2 bifurcation occurs when t first reaches the critical value tnn þ that satisfies x þ ¼ cðrðt þ ÞÞ . That is, the critical value of the second bifurcation in terms of the SDF is given by sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffi 1 xnn þ pffiffiffiffiffiffi ffi: ð58Þ Þ ¼ rðtnn þ 1 þ xnn þ n

n It is easily verified that rðtnn þ Þ is larger than rðt þ Þ, which means that the bifurcation at the M-centric pattern h occurs after that of the uniform distribution h in the course of decreasing t. We can also identify the associated agglomeration pattern that emerges at this bifurcation. The direction of movement n away from h at this bifurcation is the Mth eigenvector of Z n : znM ¼ ½1; 0,1; 0, . . . ,1; 0,1; 0. Accordingly, the pattern of agglomeration is given by

h ¼ h þ dznMT ¼ ½2h þ d,0; 2hd,0, . . . T n

ð0 r d r 2hÞ:

Thus, the properties of the second bifurcation can be summarized as follows: Proposition 6.1. Suppose that the M-centric pattern h ¼ ½2h,0; 2h,0, . . . T is a stable equilibrium (i.e., the condition (57) is n satisfied) for the CP model with homogeneous consumers. With the decreases in t, the M-centric pattern h becomes unstable at nn the second break point t ¼ t þ given by (56) and (58), and a more concentrated pattern h ¼ ½2h þ d,0; 2hd,0, . . . T then emerges. n

n

After the second bifurcation, the deviation d from the M-centric pattern h monotonically increases with the decrease in nn the transportation cost t, which leads to an ðM=2Þ-centric pattern, h ¼ ½4h,0; 0,0, . . . ,4h,0; 0,0T . That is, half of the core n nn regions in the M-centric pattern h are completely taken over by the neighboring new core regions in h . This fact can be confirmed by examining a sustain point for the ðM=2Þ-centric pattern. As shown in Appendix D, the equilibrium condition nn for h nn

nn

v0 ðh Þ ¼ maxfvk ðh Þg k

ð59Þ

is satisfied for any t smaller than some critical value. 6.2. Recursive bifurcations—spatial period doubling bifurcations The analysis of the first and second bifurcations thus far reveals that the first and second bifurcations share a common property whereby the number of core regions is reduced by half and the spacing between each pair of neighboring coren regions doubles after each bifurcation. An intuitive explanation for this striking regularity is that the K/2-centric pattern h in an RE with K regions and a uniform pattern in an RE with K/2 regions can be regarded as approximately equivalent. As n shown in Lemma 6.2, the occurrence of the bifurcation from h is determined by only the marginal utilities V ð00Þ of skilled n workers in core regions. Therefore, the bifurcation at h may be qualitatively similar to that of a uniform pattern in an RE with K/2 regions, in which the K/4-centric pattern should emerge at the first bifurcation. (For clarity of exposition, we assume below that the number of regions K is a power of 2.)16 Following this reasoning, we conjecture that, in the course of decreasing t, the multi-regional CP economy exhibits the spatial period doubling bifurcation (SPDB), whereby a third nn nnn nn bifurcation at the K=ð22 Þ-centric pattern h leads to a K=ð23 Þ-centric pattern h (in which alternate core regions of h are 4 extinct), at which a fourth bifurcation leads to a K=ð2 Þ-centric pattern, and such recursive bifurcations continue until a mono-centric agglomeration is obtained (see Fig. 5). To a certain extent, the above conjecture is true, but the dispersion force due to unskilled workers living in periphery regions should not be overlooked. Since their import of M-sector goods affects the wages of skilled workers in the core 16 Note that the analytical approach presented in this paper is applicable to cases other than K ¼ 2k . The detailed discussion on general cases is given in Ikeda et al. (2012).

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 Fig. 5. A series of agglomeration patterns that emerge in the course of decreasing t.

regions, the marginal utilities V ð00Þ of the core regions in the K regional system are not exactly the same as those in a K/2 regional system. Thus, the occurrence of the SPDB is not necessarily obvious and must be proven. In order to examine the possibility of the SPDB, we must obtain the largest eigenvalue g ðnÞ of the Jacobian matrix k rFðhðnÞ Þ at the K=2n -centric configuration 2 3T ðnÞ

h

6 7 6 7  62n h,0; 0, . . . ,0 ,2n h,0, . . . ,0 , . . . ,2n h,0, . . . ,0 7 , |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 5 4|fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} 2n

2n

2n

which would emerge after the nth bifurcation from the flat earth equilibrium for an arbitrary natural number n. This eigenvalue is obtained in a similar manner to the case of the second bifurcation. ðnÞ

Lemma 6.5. The largest eigenvalue of the Jacobian matrix rFðh Þ of the adjustment process of (15) at h ðnÞ g ¼ gM maxfg ðnÞ k k

ðnÞ

¼ yG ðrÞ

8r 2 ð0; 1,

ðnÞ

is given by ð60Þ

ðnÞ  of Z ðnÞ  P TðnÞ Z  and the associated eigenvector is the Mth row vector zM ðnÞ P ðnÞ , where Z ðnÞ  diag½Z ½LðnÞ , . . . ,Z ½LðnÞ , Z ½LðnÞ is an L(n)by-L(n) DFT matrix, where LðnÞ  K=2n ðnÞ 1 GðnÞ ðrÞ  c0 ðr,nÞ  G^ ðrÞy ,

ð61aÞ

8 9 n = 2X 1 ðnÞ 1 < n ^ ð1 þ2 hÞc0 ðr,nÞ þ cj ðr,nÞ , G ðrÞ  b n ; 2 hs : j¼1 (

cj ðr,nÞ 

n

1r 2 2j n 1 þ r 2 2j

( pðj,n,KÞ ¼

ð61bÞ

)pðj,n,KÞ ðj ¼ 0; 1,2, . . . ,2n 1Þ,

1

if j ¼ 0 and MðnÞ  LðnÞ=2 ¼ 1,

2

otherwise:

ð61cÞ

ð61dÞ

The n þ1th bifurcation for the homogeneous consumer case (i.e., y-1) occurs at the critical value r ðnÞ of the SDF at ðnÞ which the largest eigenvalue g M changes sign, that is, r ðnÞ is the solution of lim GðnÞ ðrÞ ¼ 0

y-1

ð3G^

ðnÞ

ðrÞ ¼ 0Þ:

ð62Þ

In order for the SPDB to occur in the course of decreasing t (i.e., increasing r), the critical value r ðnÞ must be an increasing function of n r ðnÞ o r ðn þ 1Þ

for n ¼ 1; 2, . . . ,log2 K1:

ð63Þ

The following lemma helps to determine whether the condition of (63) holds. ðnÞ Lemma 6.6. The function G^ ðrÞ of the SDF r has the following properties:

(i) G^ ðnÞ ðrÞ is a monotonically increasing function of r 2 ½0; 1. (ii) G^ ðnÞ ð0Þ ¼ bs1 f1 þh1 ð2n hÞ1 g, G^ ðnÞ ð1Þ ¼ b 4 0. (iii) For any n Z1, G^ ðnÞ ðrÞ Z G^ ðn þ 1Þ ðrÞ 8r 2 ½0; 1. (The equality holds only if r¼ 1.) ðnÞ ðnÞ From properties (i) and (ii) of G^ ðrÞ in Lemma 6.6, (62) has a unique solution r ðnÞ 2 ½0; 1 if and only if G^ ð0Þ o 0, which is equivalently written as

h oð1s1 Þð1ð2n Þ1 Þ:

ð64Þ

ðnÞ Furthermore, if this condition holds, then it can be concluded from property (iii) of G^ ðrÞ that the inequality of (63) holds, i.e., the SPDB occurs in the CP model. Note that the condition of (64) for the occurrence of the SPDB is very similar to the

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no-black-hole condition (for the homogeneous consumer case), h o1s1 , obtained in Proposition 5.1. The right-hand side of (64) increases with n (i.e., the requirement in (64) becomes ‘‘weak’’ with the increases of n), and the condition of (64) reduces to the no-black-hole condition as n becomes large. Therefore, we can predict that, if the no-black-hole condition is satisfied and the first bifurcation occurs, then the succeeding SPDB necessarily occurs in the CP model. Finally, it is easily verified that the associated agglomeration pattern that emerges at the n þ 1 th bifurcation is ðnÞ consistent with the SPDB. The direction of movement away from h at this bifurcation is the M  K=2 th eigenvector zðnÞ M of Z ðnÞ . Accordingly, the pattern of agglomeration is given by ðnÞ

h¼h

þ dzðnÞ M

ð0 r d r 2n hÞ,

2

3

6 7 6 7 zðnÞ 1; 0,0, . . . ,0 ,1; 0,0, . . . ,0 , . . . . . . ,1; 0,0, . . . ,0 ,1; 0,0, . . . ,0 7, M ¼6 |fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} 5 4|fflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflffl} |fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl} 2n

2n

2n n

2n ðnÞ

which means that alternate halves of K=2 core regions in h absorb skilled workers from the neighboring core regions. ðnÞ Further decreases in the transportation cost t increase the deviation d from the K=2n -centric pattern h and eventually ðn þ 1Þ lead to a K=2n þ 1 -centric agglomeration pattern h . This is consistent with the SPDB rule. Thus, we can confirm the occurrence of the SPDB in the CP model. Proposition 6.2. Consider the bifurcation process of the CP model with homogeneous consumers in the RE with K regions in the course of decreasing transportation cost t: (i) The K=2n -centric pattern h becomes unstable at the n þ1th break point (in terms of the SDF) r ¼ r ðnÞ given by the solution of (62). (ii) If the CP model satisfies (64), then the CP model exhibits a series of spatial period doubling bifurcations, in which the number of core regions is reduced by half and the spacing between each pair of neighboring core-regions doubles after each bifurcation, and the recursive bifurcations continue until a mono-centric agglomeration is reached. ðnÞ

6.3. Numerical examples We present a number of numerical examples to illustrate the theoretical results. We show the cases of (i) homogeneous consumers and (ii) heterogeneous consumers. In (ii), we demonstrate two examples, (ii-1) and (ii-2), which correspond to the cases with single re-dispersion and multiple re-dispersions, respectively. For the computation of these examples, we use the algorithm developed by Ikeda et al. (2012), which applies the computational bifurcation theory to the multiregional CP model. This algorithm allows us to obtain a complete picture of the evolutionary process of the agglomeration patterns, which includes not only the theoretical results but also transition processes of (i) and agglomeration patterns of (ii). The results of the numerical analysis are shown in Fig. 6, in which the horizontal axis denotes the SDF r and the curve represents the fraction hi =H of the skilled workers in the most populous region. Case (i): The parameter values in this case are s ¼ 2:1, y ¼ 3000, H¼1.6, and K ¼16 (y is sufficiently large, so that the equilibrium configurations can be regarded as those of the homogeneous consumer case). The evolutionary process of the agglomeration pattern in the course of increasing r is illustrated in Fig. 6(a). This figure shows the occurrence of the SPDB. Specifically, the recursive bifurcations lead, in turn, to the 8-centric, 4-centric, duo-centric and mono-centric patterns. Furthermore, it is confirmed that each break point r ðnÞ derived from the numerical analysis coincides with the theoretical prediction (i.e., the solution of (62)). These results are consistent with Proposition 6.2. Case (ii-1): The parameter values are s ¼ 2:1, y ¼ 10, H¼3.2, and K ¼16. The evolutionary process of this case, which is illustrated in Fig. 6 (b), is qualitatively the same as that of the case (i). The agglomeration patterns of both cases evolve from a uniform distribution toward the 8-centric, 4-centric, duo-centric, and mono-centric patterns in turn as r increases. However, there are two main differences between this and case (i). As shown in Proposition 5.3(iii), the re-dispersion from the mono-centric pattern occurs at rðtn Þ, and the population of any region never becomes zero. Case (ii-2): The parameter values are identical to those of case (ii-1), except for s ¼ 2:5. As illustrated in Fig. 6(c), multiple re-dispersions occur in the course of increasing r. The first re-dispersion is observed at the boundary rðtnn Þ between range A and B. That is, unlike case (ii-1), the increase in r from range A to range B causes the collapse of the duocentric agglomeration, which leads to the uniform distribution. After the emergence of a mono-centric pattern in range C, the second re-dispersion occurs at the boundary rðtn Þ between range C and range D. These results are consistent with Proposition 5.3(iii). 7. Concluding remarks We have presented an approach to capture the bifurcation properties of the multi-regional core–periphery (CP) model. The presented method is characterized by three key tools: a spatial discounting matrix (SDM), discrete Fourier transformation (DFT), and a discrete racetrack economy (RE). An effective combination of these tools overcomes the limitations of the Turing

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Fig. 6. Evolution of the agglomeration patterns. (a) Case (i): homogeneous consumer. (b) Case (ii-1): Heterogeneous consumer. (c) Case (ii-2): heterogeneous consumer case with multiple re-dispersions.

(1952) approach, enabling an examination of whether agglomeration of mobile factors emerges from a uniform distribution and allowing the evolution of spatial agglomeration patterns exhibited by the CP model when the transportation cost steadily decreases to be traced analytically. Through the analysis, we revealed a number of interesting behaviors of the CP model, and notably, we proved analytically the occurrence of spatial period doubling bifurcation, in which the number of core regions reduces by half and the spacing between each pair of neighboring core-regions doubles after each bifurcation, and the recursive bifurcations continue until a mono-centric agglomeration is reached.

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Although we restricted ourselves to the analysis of the CP model, the approach presented herein can be used for a wide class of models involving spatial agglomeration of economic activities. For instance, analyzing the other variants of the CP model, such as that by Ottaviano et al. (2002), is straightforward. Note that the application area of the proposed approach is not restricted to inter-regional migration models but also covers the analysis of a variety of urban land-use (internal structure of cities) models, in which the formation of urban centers is endogenized (see, for example, Fujita and Thisse, 2002, Chapters 6 and 7). A typical example of such an application can be found in Akamatsu and Takayama (2010). In order to deal with agglomeration models having more complex structure/circumstances, the proposed method may be extended in several ways. A straightforward extension would be to apply the method to a two-dimensional rectangular domain with periodic boundaries. This may be achieved in a relatively straightforward manner by replacing the one-dimensional DFT in the present method with the two-dimensional DFT (See, e.g., Davis, 1979). Another interesting extension would be to allow for multiple types of mobile agents. There are a number of spatial economic models of this type. Examples include NEG models with multiple industry sectors, models of the emergence of new cities (Fujita et al., 1999a), and self-organizing urban structure models in which the spatial interactions between mobile firms and consumers result in the formation of CBDs (e.g., Fujita and Ogawa, 1982; Fujita, 1988; Berliant et al., 2002; Lucas and Rossi-Hansberg, 2002; Berliant and Wang, 2008). This extension would be challenging, but would greatly enhance our understanding of the nature of agglomeration economies. Appendix A. Proofs of Lemmas 4.1 and 4.2 Proof of Lemma 4.1. A calculation of (26) yields that f m ¼ 1 þð1Þm r M þ

M 1 X

ðomk þ omk Þr k ¼

k¼1

f m ¼ f Km

ð1r 2 Þð1ð1Þm r M Þ 1rðom þ om Þ þ r 2

ðm ¼ 0; 1, . . . ,MÞ,

ðm ¼ 1; 2, . . . ,M1Þ:

Substituting o  exp½ið2p=KÞ into (A.1a), we obtain (27a).

ðA:1aÞ ðA:1bÞ

&

Proof of Lemma 4.2. (i) Since D is diagonalized by the similarity transformation shown in (25), the kth eigenvector is given by (28) in Lemma 4.2. (ii) Using f ¼ Zd0 =d and Lemma 4.1, we can obtain (29). (iii) If m is even, differentiating f m with respect to r yields df m 2ð1r 2 Þð1cosm Þ ¼ , dr ð12r cosm þr 2 Þ2 where cosm  cosð2pm=KÞ. Since 1 rcosm r 1, df m =dr o 0. If m is odd, we can rewrite the condition df m =dr o 0 as cosm o1 þ

MrM ð2yÞ x2MrM ð1rÞ2

,

ðA:2Þ

where x  ð1r 2 Þð1r 2M Þ, y  rð1r 2 Þ2 ð1þ r 2 Þ o 2. Since the second term of the right-hand side of (A.2) is positive, this inequality is always satisfied. Therefore, f m is a monotonically decreasing function of r for 8r 2 ½0; 1Þ. (iv) Since cm ðrÞ is a increasing function of cosm , f m is also a increasing function of cosm . Thus, we obtain maxm ff m g ¼ f 0 , minm ff m g ¼ f M . 2 (v) From the facts that f m is a increasing function of cosm , f 1 r1 and f M1 4 cM EM , we obtain the inequalities in (v). &

Appendix B. Proofs of Lemmas 4.3 and 4.4 Proof of Lemma 4.3. From the definition of M, M  DD1 , we can reduce M at h to MðhÞ ¼ ðhdÞ1 D. Substituting this into (21) yields (32). Substituting h ¼ h into (20), we obtain (31). & Proof of Lemma 4.4. We examine each of rvðhÞ and JðhÞ in turn. The right-hand side of (32) consists only of additions and multiplications of the circulant matrix D. It follows from this that rvðhÞ is a circulant. Eq. (31) clearly shows that JðhÞ is a circulant because I and 1 1T in the right-hand side are obviously circulants. Thus, both rvðhÞ and JðhÞ are circulants, and this concludes that the Jacobian matrix rFðhÞ at the configuration h is a circulant. & Appendix C. Proof of dd=dr 40 For the homogeneous consumer case, the equilibrium condition (18) for h ¼ ½hþ d,hd, . . . ,hþ d,hdT is rewritten as     1 b þ ðd,rÞ ð1rÞ2 4rh 1 1 ln þ  ¼ 0, ðC:1Þ b þ ðd,rÞ b ðd,rÞ s1 b ðd,rÞ s n

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b þ ðd,rÞ  ð1 þrÞ2 hþ ð1rÞ2 d 4 0,

ðC:2Þ

b ðd,rÞ  ð1þ rÞ2 hð1rÞ2 d 40:

ðC:3Þ

Differentiating both sides of (C.1) with respect to r, we obtain

s1 1 1 2ð1 þ rÞdY þ f2ð1rÞ þ4hgðb b þ Þ2ð1rÞdX dd s ¼ , dr ð1rÞ2 X X



ðC:4Þ

! 1 1 s1 1 1 þ fð1rÞ2 4rhg 2 þ 2 ,  bþ b s bþ b

ðC:5Þ

! 1 1 s1 1 1 2  fð1rÞ 4rhg 2  2 :  Y b þ b s b þ b 

ðC:6Þ

Substituting (C.1) into (C.5), (C.6) yields (   2 !  ) 2 b bþ bþ bþ o 0, X¼ ln 1 þ 1þ b þ ðb b þ Þ b b b

Y¼

1 bþ



1

ðC:7Þ

   bþ bþ bþ þ 1þ ln 4 0: b b b

ðC:8Þ

It follows from (C.4), (C.7) and (C.8) that d is a monotonically increasing (decreasing) function of r (t).

n

&

nn

Appendix D. Sustain points for h and h

n

nn

We will show the derivation of sustain points for h and h , in turn. n (1) For the M-centric pattern h ¼ ½2h,0; 2h,0, . . . ,2h,0T , we can easily obtain the indirect utility for each city by n substituting h ¼ h into (12) 8 < s1 ½1 þ h1  þ ðs1Þ1 lnð2hdð0Þ Þ ði mod 2 ¼ 0Þ, n vi ðh Þ ¼ : s1 ½ð2hÞ1 ðx1 þ ð1 þ 2hÞxÞ þ ðs1Þ1 lnð2hdð1Þ Þ ðotherwiseÞ, ðiÞ

n

where x  dð1Þ =dð0Þ , dðiÞ  d0  1. To obtain the sustain point for h , we represent the utility difference between the ‘‘core’’ cities and the ‘‘periphery’’ cities as a function of the SDF n

n

v01 ðrÞ  v0 ðh Þv1 ðh Þ ¼ ð1 þ2hÞx2 þ 2ð1 þ hÞx12hsðs1Þ1 ðx ln xÞ: Differentiating v01 ðrÞ with respect to r, we have dv01 ðrÞ dx ¼ aðrÞ, dr dr

aðrÞ  2ð1 þ 2hÞx

2hs 2h ln x þ 2 , s1 s1

ðD:1Þ

dx 2ð1r 2 Þ ¼ 4 0: dr ð1 þ r 2 Þ2

ðD:2Þ

It follows from (D.1) and (D.2) that aðrÞ is a monotonically decreasing function of r, að0Þ 4 0 and að1Þ o0. This implies that dv01 ðrÞ=dr changes sign only once. That is, v01 ðrÞ is a unimodal function of r. From this fact, we see that v01 ðrÞ takes zero value at r ¼1 and r ¼ r n01 4 0 (i.e., the equation v01 ðrÞ ¼ 0 has two positive solutions 1 and r n01 ) and that ( o 0 for 0 o r o r n01 , v01 ðrÞ Z 0 for r n01 o¼ r o1: n

n

n

This means that the equilibrium condition for h , v0 ðh Þ ¼ maxk fvk ðh Þg, is satisfied for any r larger than r n01 ; that is, r ¼ r n01 n is the sustain point for h ¼ ½2h,0; 2h,0, . . . ,2h,0T .

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(2) The sustain point for the M=2-centric pattern h ¼ ½4h,0; 0,0, . . . ,4h,0; 0,0T can be obtained by the similar manner. nn The indirect utility at h is given by 8 s1 ½1 þ h1  þ ðs1Þ1 lnð4hdð0Þ Þ ði mod 22 ¼ 0Þ, > > < 2 nn 1 1 1 vi ðh Þ ¼ s ½ð4hÞ ðx01 þ ð1 þ 4hÞx10 þ x12 þx21 Þ þðs1Þ lnð4hdð1Þ Þ ði mod 2 ¼ 1; 3Þ, > > : s1 ½ð4hÞ1 ð2 þx þð1 þ4hÞx Þ þðs1Þ1 lnð4hd Þ ði mod 22 ¼ 2Þ, nn

02

20

ð2Þ

nn

where xij ¼ dðiÞ =dðjÞ . Define the following utility difference functions at h : v01  v0 ðh Þv1 ðh Þ ¼ ð4h þ 1Þðx10 Þ2 þ 4ð1 þhÞx10 1x20 x12 x10 4hsð1sÞ1 ðx10 ln x10 Þ, nn

nn

v02  v0 ðh Þv2 ðh Þ ¼ ð4h þ 1Þðx20 Þ2 þ 2ð1 þ2hÞx20 14hsð1sÞ1 ðx20 ln x20 Þ: nn

nn

Differentiating v0i ðrÞ with respect to r, we have dv0i ðrÞ dx ¼ i0 a0i ðrÞ, dr dr

a01 ðrÞ  2ð1 þ4hÞx10 

4hs 4h ln x10 bðrÞ þ4 , s1 s1

a02 ðrÞ  2ð1 þ4hÞx20 

4hs 4h ln x20 þ 4 , s1 s1

bðrÞ 

dðx20 þ x12 x10 Þ=dr 6rð1r 4 Þ ¼ , dx10 =dr 1 þ3r 2 3r 4 r 6

dx10 ð1r 6 Þ þ 3r 2 ð1r 2 Þ ¼ 40, dr ð1 þ r 4 Þ2

dx20 4rð1r 4 Þ ¼ 4 0: dr ð1 þ r 4 Þ2

From the fact that a0i ðrÞ is a monotonically decreasing function of r, a0i ð0Þ 4 0 and a0i ð1Þ o 0, v0i ðrÞ is a unimodal function of r. Therefore, we can show that ( o 0 for 0 o r or nn 0i nn v0i ðrÞ ðD:3Þ ði ¼ 1; 2Þ and 0 or nn 01 o r 02 Z 0 for r nn 0i rr o 1 nn

nn

nn

where r nn 0i is the solution of v0i ðrÞ ¼ 0 ði ¼ 1; 2Þ. That is, the equilibrium condition for h , v0 ðh Þ ¼ maxk fvk ðh Þg, is satisfied nn nn for any r larger than r nn 02 , and r ¼ r 02 is the sustain point for h . Appendix E. Proofs of Lemmas 6.1 and 6.2 Proof of Lemma 6.1. We prove that each of the submatrices J ðijÞ and V ðijÞ is a circulant in turn. (1) Let each of pð0Þ and pð1Þ denotes the location choice probability of the subset of cities C 0 ði ¼ 0Þ and C 1 ði ¼ 1Þ at the n agglomeration pattern h , respectively ( n ð1EÞM 1 ði ¼ 0Þ, exp½yvi ðh Þ ¼ pðiÞ  ðE:1Þ n n EM1 ði ¼ 1Þ: Mfexp½yv0 ðh Þ þexp½yv1 ðh Þg A straightforward calculation of the definition (50a) of J   P ð1Þ Jðh ÞP Tð1Þ reveals that the submatrices J ðijÞ (i,j¼0,1) are circulants whose first row vector is given by ( ypðiÞ ½1pðjÞ ,pðjÞ , . . . ,pðjÞ  ði ¼ jÞ, J 0ðijÞ ¼ ðE:2Þ ypðiÞ ½pðjÞ ,pðjÞ , . . . ,pðjÞ  ðiajÞ: n

n

n

(2) As is shown in (21), the Jacobian matrix rvðh Þ consists of additions and multiplications of Mðh Þ  DfDðhÞg1 . It n n n follows from this that the Jacobian matrix r vðh Þ  P ð1Þ rvðh ÞP Tð1Þ defined in (50b) consists of those of P ð1Þ Mðh ÞP Tð1Þ , ðijÞ which in turn is composed of submatrices M (i,j¼0,1)

Therefore, in order to prove that V ðijÞ (i,j ¼0,1) are circulants, it suffices to show that the submatrices M ðijÞ (i,j¼ 0,1) are n circulants. Note here that P ð1Þ Mðh ÞP Tð1Þ can be represented as P ð1Þ Mðh ÞP Tð1Þ ¼ ½P ð1Þ DP Tð1Þ ½P ð1Þ fDðh Þg1 P Tð1Þ : n

n

ðE:3Þ

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The first bracket of the right-hand side of (E.3) is given in (48), and a simple calculation of the second bracket yields

ð0Þ

ð1Þ

where dð0Þ  d0  1 ¼ ð1r M Þð1 þr 2 Þ=ð1r 2 Þ and dð1Þ  d0  1 ¼ ð1r M Þ2r=ð1r 2 Þ. Thus, we have ðE:4Þ which shows that the submatrices M ðijÞ (i,j¼0,1) are circulants. 

&

n

Proof of Lemma 6.2. Consider the Jacobian matrix r Fðh Þ defined in (49)

r Fðhn Þ  P ð1Þ =Fðhn ÞP Tð1Þ X

F ðijÞ 

J ðikÞ V ðkjÞ

ðE:5aÞ

ði,j ¼ 0; 1Þ:

ðE:5bÞ

k ¼ 0;1

Note here that the submatrices F ðijÞ of the Jacobian matrix r Fðh Þ are circulants because all submatrices J ðijÞ and V ðijÞ (i,j¼0,1) are circulants. This enables us to diagonalize each of the submatrices F ðijÞ by using a M-by-M DFT matrix Z ½M n

ðE:6Þ ðijÞ

is the eigenvalues of F ðijÞ , and Z   diag½Z ½M ,Z ½M . It also follows that applying the similarity transformation where f based on Z ½M to both sides of (E.5b) yields X ðikÞ ðijÞ f ¼ ½d   ½eðkjÞ  ði,j ¼ 0; 1Þ, ðE:7Þ k ¼ 0;1 ðijÞ

ðijÞ

ðijÞ

ðijÞ ðijÞ ðijÞ T where d  ½d0 , d1 , . . . , dM1 T and eðijÞ  ½eðijÞ and V ðijÞ 0 ,e1 , . . . ,eM1  denote the eigenvalues of the Jacobian matrices J (i,j¼0,1), respectively. The former eigenvalues dðijÞ can be given analytically by the DFT of the first row vector J ðijÞ 0 of the submatrix J ðijÞ ðijÞ

dðijÞ ¼ Z ½M J 0ðijÞ

ði,j ¼ 0; 1Þ:

ðE:8Þ

Substituting (E.1) into (E.8), we have

dð00Þ ¼ ðyM1 Þð1EÞ½E,1, . . . ,1T , dð11Þ ¼ ðyM1 ÞE½1E,1, . . . ,1T , dð01Þ ¼ ðyM1 Þð1EÞ½E,0, . . . ,0T , dð10Þ ¼ ðyM1 ÞE½E1; 0, . . . ,0T : It follows from this that

(

lim :dðijÞ ¼ limdðijÞ ¼ E-0

pð0Þ -1=2

ðijÞ

Substituting these d ( lim :f

ðijÞ

pð0Þ -1=2

¼

ðyM 1 Þ½0; 1,1, . . . ,1T

if i ¼ j ¼ 0,

0

otherwise:

(i,j ¼0,1) into (E.7) yields

ð0jÞ T ðyM 1 Þ½0,e1ð0jÞ ,e2ð0jÞ , . . . ,eM1 

if i ¼ 0

0

if i ¼ 1

ðj ¼ 0; 1Þ:

Thus, when pð0Þ -1=2, (E.6) reduces to ðZ  Þ1 r FZ  Converting this into the original coordinate system, we obtain P T ½ðZ  Þ1 r FZ  P ¼ ðZ n Þ1 rFZ n ¼ ydiag½e^ 0 , e^ 1 , . . . , e^ M1 I, where n

T



Z  P Z P;

e^ 0 



0

0

0

0



" and

e^ k 

ekð00Þ

eð01Þ k

0

0

# ðk ¼ 1; 2, . . . ,M1Þ:

T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

1753

Since eigenvalues of an upper-triangular matrix (e^ k ) are given by the diagonal entries, we can conclude that the n eigenvalues of the Jacobian rFðh Þ are given by ( ð00Þ yek=2 1 ðk : evenÞ, gk ¼ & ðE:9Þ 1 ðk : oddÞ:

Appendix F. Proofs of Lemmas 6.3 and 6.4 

n

Proof of Lemma 6.3. Substituting (21) and (E.4) into the definition of r vðh Þ in (50), we obtain (52a) in lemma 6.3. Since the right-hand side of (52a) consisting only of circulants Dð0Þ , Dð1Þ and Dð1ÞT , we have the associated eigenvalues eð00Þ in (52b). & ðiÞ

Proof of Lemma 6.4. Each of the eigenvalues f ðiÞ (i¼0,1) of DðiÞ =dðiÞ is obtained by DFT of vectors d0 ð0Þ

f ð0Þ ¼ Z ½M d0 =dð0Þ ,

ð1Þ

f ð1Þ ¼ Z ½M d0 =dð1Þ ,

ð1TÞ

f ð1TÞ ¼ Z ½M d0 =dð1Þ :

ðF:1Þ

A straight-forward calculation of (F.1) yields f M=2 ¼ cðr 2 Þ2 , ð0Þ

ð1Þ

ð1TÞ

f M=2 ¼ f M=2 ¼ 0,

ð1Þ ð1TÞ

f m f m 40

ðm ¼ 0; 1,2, . . . ,M1Þ:

ðF:2Þ

Substituting these into (52b) yields e0ð00Þ ¼ Gð00Þ ð1; 1Þ ¼ const:, ð0Þ

ð00Þ eM=2 ¼ Gð00Þ ðcðr 2 Þ2 ,0Þ,

ð1Þ ð1TÞ

ð00Þ ð00Þ ¼ eMm ¼ Gð00Þ ðf m ,f m f m Þ em

ðF:3aÞ

ðm ¼ 0; 1,2, . . . ,M1Þ,

ðF:3bÞ

where Gð00Þ ðx,yÞ  bxa1 ½ð1 þ 2hÞx2 þ y. It follows from (F.3) that maxfenm ðxÞg ¼ Gð00Þ ðcðr 2 Þ2 ,0Þ: m

Combining this with (E.9) in Lemma 6.2, we obtain (53).

&

ðnÞ

Appendix G. Eigenvalues of submatrices of = Fðh Þ ðnÞ

We introduce a particular division of the set of C by which we exploit the symmetry of the configuration h so that the ðnÞ n resultant submatrices of the Jacobian rFðh Þ reduce to circulants. First, recall that in obtaining the eigenvalues of rFðh Þ n at a K/2-centric configuration h that emerges after the first bifurcation, we have divided the set C of regions into two subsets, namely, the subset C 0 ¼ f0; 2, . . . ,K2g of regions with skilled workers and the subset C 1 ¼ f1; 3, . . . ,K1g of nn regions without skilled workers. In a similar vein, for a K/4-centric configuration h that would emerge after the second n bifurcation from the configuration h , it is natural to divide each Ci (i¼0,1) into two subsets, Ci0 and Ci1, the elements of which are the even-numbered regions in Ci and the odd-numbered regions in Ci, respectively. It is also natural in the nn division to suppose that only the regions in subset C00 have skilled workers in configuration h . This division procedure ðnÞ can be repeated recursively. For configuration h after n bifurcations, each of the 2n1 subsets C i0 i1 in2 (i0 ,i1 , . . . ,in2 ¼ 0; 1) of regions generated at the preceding (n1 th) bifurcation is divided into two subsets, C i0 i1 in1 (in1 ¼ 0; 1). This division procedure naturally leads to the following definition of the permutation snK of the set C:

where L  K=2n . Note here that the division and the associated permutation are the same as those used in the fast Fourier transform (FFT) algorithm (see, for example Van Loan, 1992). This implies that the kth element snK ðkÞ (k ¼ 0; 1, . . . ,K1) of the permutation snK is given by a bit reversal operation used in the FFT technique whereby an index k, written in binary with digits bK1 bK2    b1 b0 , is transferred to the index with reversed digits b0 b1    bK2 bK1 , which yields snK ðkÞ. It also follows that the first component of the subset C i0 i1 ,in1 is given by the bit reversal of i0 i1 i2    in1 : an  i0 þ i1 2þ    þ in1 2n1 (for a concrete illustration of the division of the set C and the associated permutation snK , see Appendix H). The permutation snK defined above can be expressed in terms of the associated K  K permutation matrix P ðnÞ

ðG:1Þ

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T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

where P eðnÞ and P oðnÞ are K=2n þ 1 by K=2n matrices. The (i,j) element of P eðnÞ is 1 if j ¼ 2i (i ¼ 0; 1,2, . . . ,K=2n þ 1 1), and is otherwise 0, whereas that of P oðnÞ is 1 if j ¼ 2i þ1, and is otherwise 0. Under the new coordinate system transformed by the permutation snK , the SDM D can be represented as T D ðnÞ  P ðnÞ DP ðnÞ :

ðG:2Þ

n 2 n Clearly, the transformed SDM, D ðnÞ , consists of ð2 Þ blocks (submatrices), in which each of the 2 block divisions in each of the row and column directions corresponds to the subset C i0 i1 i2 in1 (i0 ,i1 , . . . ,in1 ¼ 0; 1) of C. As it turns out, we do not have to deal with all of the submatrices. For our purpose, it is sufficient to focus on 2n submatrices of the first block-row, in which each submatrix, denoted as Dði0 i1 i2 in1 Þ , represents the spatial discounting factors between the core regions of set C 0000 and the periphery regions of set C i0 i1 i2 in1 . It is readily shown that the submatrix Dði0 i1 i2 in1 Þ is a circulant generated ði i i i Þ ðaðnÞÞ ðaðnÞÞ from a vector d0 0 1 2 n1  d0 with LðnÞ  K=2n elements, the kth element d0,k of which is given by ( n if 0 r k rMðnÞ, r aðnÞ þ k2 ðaðnÞÞ d0,k  ðG:3Þ n r bðnÞ þ ðLðnÞkÞ2 if MðnÞ ok rLðnÞ,

where aðnÞ  i0 þi1 2þ    þ in1 2n1 (i.e., the bit reversal of i0 i1 i2    in1 ðmod 2Þ ¼ in1 þ in2 2 þ    þi0 2n1 ), bðnÞ  2n aðnÞ, and MðnÞ  LðnÞ=2. ðnÞ Similarly, applying this transformation to the Jacobian matrix rFðh Þ of the adjustment process, we have

r FðhðnÞ Þ  P ðnÞ rFðhðnÞ ÞP TðnÞ ¼ HJ  ðhðnÞ Þr vðhðnÞ ÞI,

ðG:4Þ

where the Jacobian matrices in the right-hand side are, respectively, defined as J  ðh Þ  P ðnÞ Jðh ÞP TðnÞ , ðnÞ

ðnÞ

r vðhðnÞ Þ  P ðnÞ rvðhðnÞ ÞP TðnÞ : ðnÞ



ðG:5Þ n 2

ðnÞ

Each of the Jacobian matrices J  ðh Þ and r vðh Þ has a block structure with ð2 Þ submatrices, just like the transformed SDM. These submatrices under the new coordinate system have the following desirable properties. Lemma G.1. The submatrices J ð0,aðnÞÞ of J  ðh Þ and V ð0,aðnÞÞ of r vðh Þ, (aðnÞ  i0 þ i1 2þ    þ in1 2n1 ¼ 0; 1,2, . . . ,2n 1), are circulants, where the superscript ð0,aðnÞÞ of each submatrix denotes that the block-row and the block-column of the submatrix correspond to set C 0000 and set C i0 i1 i2 in1 , respectively. ðnÞ



ðnÞ

This leads to the following result (parallel to Lemma 6.2 in the second bifurcation), which is very useful for obtaining ðnÞ the eigenvalues g ðnÞ ¼ fg kðnÞ g of rFðh Þ. ðnÞ

ðnÞ

Lemma G.2. The eigenvalues g ðnÞ (k ¼ 0; 1,2, . . . ,K1) of the Jacobian matrix rFðh Þ of the adjustment process at h are k represented as 8 < yeð0;0Þn 1 ðk mod 2n ¼ 0Þ, ðnÞ k=2 ðG:6Þ gk ¼ : 1 ðotherwiseÞ, ð0;0Þ T ,eð0;0Þ , . . . ,eð0;0Þ . where eð0;0Þ  ½eð0;0Þ 0 1 LðnÞ1  are the eigenvalues of the submatrix V

This lemma indicates that the eigenvalues eð0;0Þ of the submatrix V ð0;0Þ immediately yield the eigenvalues g ðnÞ . Since the V ð0;0Þ is a circulant consisting only of the sub-matrices DðaðnÞÞ  Dði0 i1 i2 in1 Þ of the SDM, we can obtain the eigenvalues eð0;0Þ . Lemma G.3. The Jacobian V ð0;0Þ and the associated eigenvalues eð0;0Þ are given by 8 " # " #2 " #" #9 n < 2X 1 Dð0Þ Dð0Þ DðaðnÞÞ DðaðnÞÞT = n ð0;0Þ , ¼b þ V an ð1 þ 2 hÞ : dð0Þ dð0Þ dðaðnÞÞ dðaðnÞÞ ; aðnÞ ¼ 1

ðG:7aÞ

8 # " # " # " #9 n ð0Þ ð0Þ 2 ðaðnÞÞ ðaðnÞTÞ = 2X 1 < f f f f n ¼b þ an ð1þ 2 hÞ  , : dð0Þ dð0Þ dðaðnÞÞ dðaðnÞÞ ; aðnÞ ¼ 1 "

ð0;0Þ

e

where an  s1 ð2n hÞ1 , dðaðnÞÞ  d0

ðaðnÞÞ

 1, f

ðaðnÞÞ

ðaðnÞÞ

¼ Z ½LðnÞ d0

, and f

ðaðnÞTÞ

ðG:7bÞ ðaðnÞTÞ

¼ Z ½LðnÞ d0

.

Straightforward calculation of (G.7b) together with Lemma G.2 leads to Lemma 6.5. Appendix H. An example of the division of the set C We show an example of the division of the set C and the associated permutation snK for the case of K ¼16. For an 8n centric configuration h that emerges after a first bifurcation from the flat earth equilibrium (n ¼1), the set C with 16 elements is divided into two subsets, C0 and C1, each of which is a set of core regions and that of periphery regions, n nn respectively. After a second bifurcation from the 8-centric configuration h (n¼2), a 4-centric configuration h emerges,

T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

1755

and then each of the subsets Ci (i¼0,1) is divided into two subsets i ¼ 0 : C 00 ¼ f0; 4,8; 12g

and

C 01 ¼ f2; 6,10; 14g,

i ¼ 1 : C 10 ¼ f1; 5,9; 13g

and

C 11 ¼ f3; 7,11; 15g:

where C00 is the set of core regions, and all other subsets are the sets of periphery regions in the 4-centric configuration nn ð3Þ h . For a duo-centric configuration h , which emerges after a third bifurcation (n¼3), each of the subsets fC i0 i1 ,ði0 ,i1 ¼ 0; 1Þg is further divided into two subsets i0 i1 ¼ 00 : C 000 ¼ f0; 8g i0 i1 ¼ 01 : C 010 ¼ f2; 10g i0 i1 ¼ 10 : C 100 ¼ f1; 9g i0 i1 ¼ 11 : C 110 ¼ f3; 11g

and and and and

C 001 ¼ f4; 12g, C 011 ¼ f6; 14g, C 101 ¼ f5; 13g, C 111 ¼ f7; 15g,

where only the two regions in the subset C000 has skilled workers. Thus, after the third bifurcation, C is divided into 23 ¼ 8 subsets fC i0 i1 i2 ,ði0 ,i1 ,i2 ¼ 0; 1Þg. Accordingly, the permutation s316 corresponding to this division of C is given by ðH:1Þ It is readily verified that snK ðkÞ is given by the bit-reversal operation; for example, k¼12 is written as ‘‘1100’’ (mod 2), whose bit-reversal yields ‘‘0011’’ (mod 2), and hence, s316 ð12Þ ¼ 0011ðmod 2Þ ¼ 3. This also shows that the first component of C110 is given by the bit reversal of ‘‘110’’ (mod 2). Appendix I. Proofs of Lemmas G.1, G.2 and G.3 Proof of Lemma G.1. We prove that the submatrices J ð0,aðnÞÞ and V ð0,aðnÞÞ are circulants in turn. ðnÞ (1) Let pðaðnÞÞ denotes the location choice probability of the subset of cities C i0 i1 i2 in1 at h ¼ h ( 2n =K if aðnÞ ¼ 0, exp½yvaðnÞ  ¼ pðaðnÞÞ ¼ n P K=2 f k exp½yvk g 0 otherwise: Substituting this into (G.5), we have ( ðy2n K 1 ÞðI1 1T Þ if i ¼ j ¼ 0, J ðijÞ ¼ 0 otherwise:

ðI:1Þ

This shows that the submatrices J ð0,aðnÞÞ are circulants. ðnÞ ðnÞ ðnÞ ðnÞ (2) Since rvðh Þ consists of additions and multiplications of Mðh Þ  DfDðh Þg1 , r vðh Þ defined in (G.5) also ðnÞ consists of those of P ðnÞ Mðh ÞP TðnÞ . From this, in order to prove that V ð0,aðnÞÞ are circulants, it is suffices to show that the ðnÞ ðnÞ submatrices M ðijÞ of P ðnÞ Mðh ÞP TðnÞ are circulants. P ðnÞ Mðh ÞP TðnÞ is expressed as P ðnÞ Mðh ÞP TðnÞ ¼ ½P ðnÞ DP TðnÞ ½P ðnÞ fDðh Þg1 P TðnÞ : ðnÞ

ðnÞ

ðI:2Þ

ðijÞ

The submatrices D of the second bracket of (I.2) is given by ( n ð2 hÞ1 diag½dðaðnÞÞ 1 if i ¼ j ¼ aðnÞ, DðijÞ ¼ 0 otherwise, where n

ðaðnÞÞ

dðaðnÞÞ  d0

1¼

r aðnÞ þ r 2 aðnÞ ð1r K=2 Þ, n 1r 2

and this shows that DðijÞ are circulants. As is shown in Section 6.2, the submatrices of the first bracket of the right-hand side of (I.2) defined in (G.2) are circulants. Thus, M ðijÞ are circulants. & ðnÞ

Proof of Lemma G.2. The Jacobian matrix r Fðh Þ defined in (G.4) is given by

r FðhðnÞ Þ ¼ HJ  ðhðnÞ Þr vðhðnÞ Þ

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T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

where F ðijÞ  matrix Z ½LðnÞ

P

kJ

ðikÞ

V ðkjÞ , LðnÞ  K=2n . Since F ðijÞ are circulants, these submatrices are diagonalized by L(n)-by-L(n) DFT

ðnÞ



1 ðZ  r Fðh ÞZ ðnÞ ðnÞ Þ

ðI:3Þ

where Z  ðnÞ  diag½Z ½LðnÞ ,Z ½LðnÞ , . . . ,Z ½LðnÞ , and f X ðikÞ ðijÞ f ¼ ½d   ½eðkjÞ ,

ðijÞ

is the eigenvalues of F ðijÞ , which is given by ðI:4Þ

k ðijÞ

ðijÞ

ðijÞ

ðijÞ ðijÞ T ðijÞ dðijÞ ¼ ½d0 , d1 , . . . , dLðnÞ T and eðijÞ ¼ ½eðijÞ and V ðijÞ , respectively. Applying the 0 ,e1 , . . . ,eLðnÞ  denote the eigenvalues of J similarity transformation based on Z ½LðnÞ to both sides of (I.1), we have ( ðy=LðnÞÞ½0; 1,1, . . . ,1T if i ¼ j ¼ 0, ðijÞ d ¼ 0 otherwise:

Substituting this into (I.4) yields ( ð0jÞ T ðy=LðnÞÞ½0,e1ð0jÞ ,e2ð0jÞ , . . . ,eLðnÞ  ðijÞ f ¼ 0

if i ¼ j ¼ 0, otherwise:

Thus, converting (I.3) into the original coordinate system, we obtain ðnÞ

ðnÞ

1 r Fðh ÞZ ðnÞ P ðnÞ ¼ ðZ nðnÞ Þ1 r Fðh ÞZ nðnÞ ¼ ydiag½0, e^ 1 , . . . , e^ LðnÞ I, P TðnÞ ðZ  ðnÞ Þ ðnÞ



where Z nðnÞ  P TðnÞ Z  ðnÞ P ðnÞ , and 2 ð00Þ e eð01Þ  k 6 k 6 0 ðnÞ 0   e^ k  6 6 ^ & 4 ^ 0

0

ðnÞ



n1

ekð02

Þ

3 7 7 7 7 5

0 ^

ðk ¼ 1; 2, . . . ,LðnÞÞ:

0



ðnÞ

Since eigenvalues of an upper-triangular matrix (e^ k ) are given by the diagonal entries, we can conclude that the eigenvalues of the Jacobian are given by (G.6). & 

ðnÞ

Proof of Lemma G.3. Substituting (21) and (I.2) into the definition of r vðh Þ in (G.5), we obtain (G.7a) in lemma G.3. Since the right-hand side of (G.7a) consisting only of circulants Dð0Þ , DðaðnÞÞ and fDðaðnÞÞ gT , we have the associated eigenvalues eð00Þ by (G.7b). & Appendix J. Proof of Lemma 6.5 We see from Lemmas G.2 and G.3 that Lemma 6.5 is equivalent to ð00Þ arg maxfem  E0m E1m g ¼ MðnÞ, m

ð0Þ

f MðnÞ =dð0Þ ¼ c0 ðr,nÞ,

n 2X 1

ðaðnÞÞ

MðnÞ  LðnÞ=2  K=2n þ 1 , ðaðnÞÞT

2

f MðnÞ  f MðnÞ =dðaðnÞÞ ¼ c0 ðr,nÞ

aðnÞ ¼ 1

n 2X 1

ðJ:1Þ

cj ðr,nÞ,

ðJ:2Þ

j¼1

P2n 1

ðaðnÞÞ ðaðnÞÞT 2  fm =dðaðnÞÞ (m ¼ 0; 1, . . . ,Lðn1Þ). To prove these, we aðnÞ ¼ 1 f m ðaðnÞÞ ðaðnÞÞT first derive analytical expressions of the eigenvalues f m and f m (m ¼ 0; 1, . . . ,LðnÞ1). Since DðaðnÞÞ is a circulant, the ðaðnÞÞ ðaðnÞÞ ðaðnÞÞ ðaðnÞÞ eigenvalues ff m g are obtained by DFT of d0 :f ¼ Z ½LðnÞ d0 . Substituting (G.3) into this and some calculation yields

where E0m  ðf m =dð0Þ Þfban ð1þ 2n hÞðf m =dð0Þ Þg, E1m  an ð0Þ

ðaðnÞÞ

fm

ð0Þ

¼ ½am ðnÞ þ ibm ðnÞ  gm ðnÞ,

ðJ:3Þ

where cosm  cosð2pðm=LðnÞÞÞ, n

Rn  r 2 ,

An  r aðnÞ ,

sinm  sinð2pðm=LðnÞÞÞ,

1 2 am ðnÞ  An ð1Rn cosm ÞA1 n ð1Rn cosm ÞRn ,

bm ðnÞ  ðAn A1 gm ðnÞ  n ÞðRn sinm Þ,

1ð1Þm RnMðnÞ 12Rn cosm þR2n

4 0:

T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

1757

In a similar manner, we obtain the eigenvalues of DðaðnÞÞT as ðaðnÞÞT

fm

¼ ½am ðnÞibm ðnÞgm ðnÞ:

ðJ:4Þ

From (J.3) and (J.4), we have ðaðnÞÞ ðaðnÞÞT fm

fm

¼ ½aðnÞ þ bðnÞcosm gm ðnÞ2 ,

ðJ:5Þ

1 2 2 2 2 2 where aðnÞ  ð1 þR2n ÞðAn Rn A1 n Þ þ ð1 þRn Þ 2Rn Z 0, bðnÞ  2Rn An ðAn 1ÞðRn An Þ Z0. To prove (J.1), it suffices to show that

arg maxfE0m g ¼ MðnÞ m

arg minfE1m g ¼ MðnÞ:

and

m

The former is true since it is readily verified from (J.3), (J.4) that ð0Þ

arg minff m g ¼ MðnÞ: m

The latter is also evident from the fact that ðaðnÞ ¼ 1; 2, . . . ,2n 1Þ,

ðaðnÞÞ ðaðnÞÞT fm g ¼ MðnÞ

arg minff m m

which is obvious from (J.5) since aðnÞ Z 0, bðnÞ Z 0, gm ðnÞ Z0. Thus, (J.1) has been proved. (J.2) can be easily proved by ðaðnÞÞ ðaðnÞÞT calculating dðaðnÞÞ and f MðnÞ f MðnÞ ðaðnÞÞ

dðaðnÞÞ ¼ d0

 1 ¼ ðr aðnÞ þ r 2

n

aðnÞ

Þ

1r K=2 n , 1þ r 2

8 K=2 2 > < ðr aðnÞ r 2n aðnÞ Þ2 ð1r n Þ ðaðnÞÞ ðaðnÞÞT ð1 þ r 2 Þ2 fm fm ¼ > : aðnÞ 2n aðnÞ 2 ðr r Þ

ðJ:6Þ

if MðnÞ Z2, ðJ:7Þ if MðnÞ ¼ 1:

Setting aðnÞ ¼ 0 in (J.3), (J.4) and (J.6) yields the first equality of (J.2). The second equality of (J.2) also can be readily verified by substituting (J.6) and (J.7) into the LHS. &

Appendix K. Proof of Lemma 6.6 (i) Defining

En ðrÞ  C 2n =2 ðrÞ þ E^ n ðrÞ, E^ n ðrÞ 

ð2nX =2Þ1

C k ðrÞ

k¼1

and C k ðrÞ 



1r 2k 1 þ r 2k

2

,

we have ðnÞ G^ ðrÞ ¼ s1 fc0 ðrÞ þ ð2n hÞ1 En ðrÞg:

ðK:1Þ

Since dc0 ðr,nÞ=dr o 0 and dEn ðrÞ=dr o0 8r 2 ð0; 1, we can conclude that the first derivative of the RHS of (K.1) is ðnÞ negative, that is, dG^ ðrÞ=dr o0 8r 2 ð0; 1. (ii) Substituting C k ð0Þ ¼ 1 and En ð0Þ ¼ 1 þ 2fð2n =2Þ1g ¼ 2n 1 into (K.1), we obtain (62). Substituting C k ð1Þ ¼ 0 and ðnÞ En ð1Þ ¼ 0 into (K.1) yields G^ ð1Þ ¼ b 4 0. (iii) It follows from (K.1) that ðnÞ ðn þ 1Þ ðrÞ ¼ s1 ½fc0 ðr,n þ 1Þc0 ðr,nÞg þ ð2n þ 1 hÞ1 fEn þ 1 ðrÞ2En ðrÞg: G^ ðrÞG^ ðnÞ

ðK:2Þ

ðn þ 1Þ

ðrÞ Z 0, it suffice to show that En þ 1 2En Z0 since it is obvious that c0 ðr,n þ 1Þ Z c0 ðr,nÞ (the To prove G^ ðrÞG^ equality holds only if r ¼1). From the definition of En ðrÞ

En þ 1 2En ¼ C 2n þ 1 =2 2C 2n þ 1 =2 þ2ðE^ n þ 1 2E^ n Þ:

ðK:3Þ

It also follows from the definitions of E^ n ðrÞ and Ck(r) that

E^ n þ 1 2E^ n ¼ ðE^ n þ 1 E^ n ÞE^ n ¼

1 2n þX =21

k ¼ 2n =2

Ck

2nX =21 k¼1

C k ¼ C 2n =2 þ

2nX =21

ðC k þ ð2n =2Þ C k Þ:

k¼1

ðK:4Þ

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T. Akamatsu et al. / Journal of Economic Dynamics & Control 36 (2012) 1729–1759

Substituting (K.4) into (K.3), we have

En þ 1 2En ¼ C 2n þ 1 =2 þ 2

2nX =21

ðC k þ ð2n =2Þ C k Þ:

k¼1

Since C k þ ‘ ðrÞ ZC k ðrÞ 8‘,k 4 0, for any 0 rr r 1,n Z1, we conclude that En þ 1 2En Z0 (the equality holds only if r ¼1). &

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