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We develop a two-region economic geography model with vertical innovations that improve the quality of manufactured varieties produced in each region. The chance of innovation depends on the \emph{related variety}, i.e. the importance of interaction between researchers within the same region rather than across different regions. As economic integration increases from a low level, a higher related variety is associated with more agglomerated spatial configurations. However, if the interaction with foreign scientists is relatively more important for innovation, economic activities may (completely) re-disperse after an initial phase of agglomeration due to the increase in the relative importance of a higher chance of innovation in the less industrialized region. This non-monotonic relationship between economic integration and spatial imbalances may exhibit very diverse qualitative properties, not yet described in the literature.

City-size distributions follow an approximate power law in various countries despite high volatility in relative city sizes over time. Our empirical evidence for the United States and Japan indicates that the scaling law stems from a spatial fractal structure owing to the coordination of industrial locations. While the locations of individual industries change considerably over time, there is a persistent pattern in that more localized industries at a given time are found only in larger cities. The spatial organization of cities exhibits a hierarchical structure in which larger cities are spaced apart to serve as centers for surrounding smaller cities, generating a recursive pattern across different spatial scales. In our theoretical replication of the observed regularities, diversity in scale economy among industries induces diversity in their location pattern, which translates into diversity in city size via spatial coordination of industries and population. The city-size power law is a generic feature of Monte-Carlo samples of stationary states resulting from the spontaneous spatial fractal structure in the hypothetical economy. The identified regularities reveal constraints on feasible urban planning at each regional scale. The success or failure of place-based policies designed to take advantage of individual cities' characteristics should depend on their spatial relationships with other cities, subject to the nationwide spatial fractal structure.

Economic activities favor mutual geographical proximity and concentrate spatially to form cities. In a world of diminishing transport costs, however, the advantage of physical proximity is fading, and the role of cities in the economy may be declining. To provide insights into the long-run evolution of cities, we analyzed Japan's census data over the 1970--2015 period. We found that fewer and larger cities thrived at the national scale, suggesting an eventual mono-centric economy with a single megacity; simultaneously, each larger city flattened out at the local scale, suggesting an eventual extinction of cities. We interpret this multi-scale phenomenon as an instance of pattern formation by self-organization, which is widely studied in mathematics and biology. However, cities' dynamics are distinct from mathematical or biological mechanisms because they are governed by economic interactions mediated by transport costs between locations. Our results call for the synthesis of knowledge in mathematics, biology, and economics to open the door for a general pattern formation theory that is applicable to socioeconomic phenomena., Comment: 4 pages, 2 figures

We consider a model of urban spatial structure proposed by Harris and Wilson (Environment and Planning A, 1978). The model consists of fast dynamics, which represent spatial interactions between locations by the entropy-maximizing principle, and slow dynamics, which represent the evolution of the spatial distribution of local factors that facilitate such spatial interactions. One known limitation of the Harris and Wilson model is that it can have multiple locally stable equilibria, leading to a dependence of predictions on the initial state. To overcome this, we employ equilibrium refinement by stochastic stability. We build on the fact that the model is a large-population potential game and that stochastically stable states in a potential game correspond to global potential maximizers. Unlike local stability under deterministic dynamics, the stochastic stability approach allows a unique and unambiguous prediction for urban spatial configurations. We show that, in the most likely spatial configuration, the number of retail agglomerations decreases either when shopping costs for consumers decrease or when the strength of agglomerative effects increases., Comment: 26 pages, 10 figures

We consider an economic geography model with two inter-regional proximity structures: one governing goods trade and the other governing production externalities across regions. We investigate how the introduction of the latter affects the timing of endogenous agglomeration and the spatial distribution of workers across regions. As transportation costs decline, the economy undergoes a progressive dispersion process. Mono-centric agglomeration emerges when inter-regional trade and/or production externalities incur high transportation costs, while uniform dispersion occurs when these costs become negligibly small (i.e., when distance dies). In multi-regional geography, the network structure of production externalities can determine the geographical distribution of workers as economic integration increases. If production externalities are governed solely by geographical distance, a mono-centric spatial distribution emerges in the form of suburbanization. However, if geographically distant pairs of regions are connected through tight production linkages, multi-centric spatial distribution can be sustainable., Comment: 32 pages, 12 figures

Cities have become fewer, bigger, and flatter in many countries. Better transport access has fostered the concentration of economic activities toward fewer, bigger cities, while each city has become more decentralized. We show that two types of dispersion forces against spatial agglomeration explain the dual evolution of cities. Dispersion forces within a city, such as those arising from the consumption of limited land, affect only the city's residents. A decrease in transport costs reduces the relative benefits of agglomeration in a congested central location, making the population distribution in each city flatter and more spread out to achieve a balance between the costs and benefits of agglomeration. Dispersion forces that spill over beyond a city, such as market crowding, make locations near the city unattractive. Lower transport costs promote competition for a wider area around each city, leaving fewer cities with larger distances between them. A model's comparative static results, and thus its policy implications, depend on the type of dispersion forces it incorporates., Comment: 43 pages, 10 figures (main text)