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- (a) Urban agglomrations in 1970 (b) Urban agglomrations in 2015 (c) Highway and high-speed railway network in 1970 (d) Highway and high-speed railway network in 2015 Fukuoka Tokyo Nagoya Osaka Sapporo Fukuoka Tokyo Nagoya Osaka Sapporo Tokyo Nagoya Osaka Tokyo Nagoya Osaka Fukuoka Sapporo Highway High-speed railway Highway High-speed railway Figure 14: UAs and transport network in Japan (a) Population share growth (b) Area growth (c) Population density growth Average = 0.21 Standard deviation= 0.75 Average = 0.94 Standard dveiation = 1.05 Average = -0.22 Standad dveiation = 0.22 Figure 15: Growth rates of the sizes of UAs in Japan B Racetrack economy and simplification of the stability analysis The friction matrix D for a racetrack economy is a circulant matrix. In this appendix, we first review some useful properties of circulant matrices. Then, we see how the stability analysis of the flat-earth pattern in a symmetric racetrack economy is simplified by these properties.
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- < 0. (D.2.14) By employing these formulae, we can show that whenever the equilibrium is unique (Ã(1 − ) > 1), we have GH( f ) < 0 and thus ak ≥ 0 for all k. It also follows that dak/dr < 0 for all k ≥ 1. We also note that GH( f ) < 0 implies the stability of h̄. Further, if there are no exogenous heterogeneities in A, the model is isomorphic to Redding and Sturm (2008) and Allen and Arkolakis (2014) regarding the second-nature mechanism.
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- 59Population count data are obtained from Statistics Bureau, Ministry of Internal Affairs and Communications of Japan (1970, 2015). 60UA i in year s is said to be associated with UA j in year t (, s) if the intersection of the spatial coverage of i and that of j accounts for the largest population of i among all the UAs in year t. For years s < t, if i and j are associated with each other, they are considered to be the same UA. If i is associated with j but not vice versa, then i is considered to have been absorbed into j, while if j is associated with i but not vice versa, then j is considered to have separated from i. If i is not associated with any UA in year t, then i is considered to have disappeared by year t, while if j is not associated with any UA in year s, then j is considered to have newly emerged by year t.
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- 61Interested readers should consult Chapter 8 of Sandholm (2010) for local stability analysis via the linearization of evolutionary dynamics in population games as well as the consequences of assuming random utility models on the Jacobian matrix of the dynamic J at an equilibrium. C Analyses of economic geography models In this appendix, we derive the Jacobian matrix of the payoff function at the flat-earth equilibrium, ∇v(h̄), for the models included in Table 1. As discussed in the main text and as in Appendix B above, this suffices for our purpose. Table 2 at the end of this appendix summarizes the exact mappings from each model to the coefficients of a model-dependent function G( f ) c0 + c1 f + c2 f 2 . We note that as soon as one has an analytical expression of G( f ), one can derive the break points with respect to the relevant parameters and study the implications of the model.
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- − θ̂ Ã − 1 Ã . (C.8.19) Remark C.6. We must require that 0 ≤ θ̂ < 1/{2(K − 1)} < 1 to ensure that Si is positive for all region i. In particular, when H 1 and K 2, meaning that h 1/2 as in the original study, we have θ̂ θ/{2(1 − θ)} and θ̂ ∈ (0, 1/2). Moreover, by letting γ ≡ θ̂ and ≡ 1 − θ̂, the model is isomorphic to Helpman (1998)’s model with LL, albeit there is a restriction on γ.
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- Remark C.4. The regional model formulated in 3 of Redding and Rossi-Hansberg (2017) is an enhanced version of the Hm model with LL, in which the variable input of skilled labor is allowed to depend on region i (i.e., productivity differs across regions). That is, the cost function of firms in region i becomes Ci(xi(ξ)) wi(α + βi xi(ξ)). (C.4.16) This then implies that the short-run equilibrium price and price index in region i become pij(ξ) Ãβi à − 1 Äij wi , (C.4.17) Pi à à − 1 ( 1 αà ∑ j∈K hj(βj wj Äji)1−à )1/(1−Ã) , (C.4.18) respectively. As the model assumes LL, the wage equation for the model is wi hi ∑ j∈K hiAi w1−à i dij ∑ k∈K hkAk w1−à k dk j wj hj , (C.4.19) where Ai ≡ β1−à i . Thus, by abstracting from first natures by setting Ai Ā, the model reduces to the Hm model under LL.
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- The first and perhaps simplest example is a location-fixed factor in the payoff function: vi(h, Ai) v̂i(h) + Ai , (D.1.1) where v̂i(h) is the A-independent component of vi(h, Ai), which we term local heterogeneity. The specification (D.1.1) includes many models with location-fixed factors that directly affect the (indirect) utility of mobile workers. For instance, by taking the logarithm, the indirect utility function of Allen and Arkolakis (2014)’s model that incorporates location-fixed amenities reduces to (D.1.1). Such effects also arise from local non-tradable goods, with a representative example being Helpman (1998). As is evident from (C.4.11), when we let Ai : (1 − ) log[Ai], the model reduces to (D.1.1).
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