Urban hierarchy system’s three elements

Population size, populationdensity, and innovation are key elements for urban hierarchy upgrading. Existing studies often examine each dimension in isolation, lacking an integrated analytical framework. This paper constructs a spatial innovation theoretical model that incorporates scale and density constraints, and conducts empirical testing using panel data of 297 prefecture-level and above cities in China from 2010 to 2020. The findings are as follows: First, urban scale has a significantly positive correlation with innovation performance, but this relationship is moderated by spatial density. Second, there is an inverted U-shaped relationship between innovation performance and population density, with a specific optimal density threshold. Third, the optimal density threshold increases with the expansion of urban scale. Fourth, digitalization promotes virtual agglomeration, thereby reducing spatial communication costs, effectively raising the optimal density threshold, and amplifying the innovation spillover effects of scale and density. The calculation results show that most Chinese cities, including megacities and super-large cities, are still below the optimal density threshold and have significant room for density improvement. This study reveals the nonlinear interaction mechanism among the three elements of urban scale, density, and innovation, and proposes a three-element theory of the urban hierarchy system from the perspective of spatial economics, which has implications for the high-quality development of Chinese cities.