Spatial structure of city population growth

The study addresses how domestic migration and natural growth combine to shape spatially heterogeneous population changes within U.S. metropolitan statistical areas (MSAs). Classic city-size regularities (Auerbach, Zipf) and models emphasized random demographic growth, but recent findings indicate that inter-city domestic migration flows, often broadly distributed, significantly affect city trajectories. The research question is how the spatial structure of migration—both within cities (intra-city) and between cities (inter-city)—contributes to heterogeneous population growth at the county level. The authors focus on county-to-county migration to reveal how inflows/outflows vary with urban structure (core vs. external counties), how these flows redistribute population within MSAs, and how they relate to density gradients and international inflows. The purpose is to uncover generalized patterns across diverse cities to inform urban planning and policy.

Prior work established statistical regularities in city sizes and growth (Auerbach, Zipf, Gibrat) and highlighted that inter-city domestic migration, influenced by socioeconomic shocks, can dominate urban growth. Studies link migration-driven growth to externalities like congestion, air pollution, and inequality. Urban spatial heterogeneity is well known, including declining density with distance from cores, fractal morphology, and patterns such as urban heat islets. Research has examined neighborhood inequalities, infrastructure, and migration network topology. However, most analyses operate at the aggregate city level, often neglecting intra-city spatial heterogeneity of flows. This paper builds on these strands by leveraging county-level migration to bridge intra- and inter-city scales.

Data: The analysis covers 2015–2019 using U.S. Census datasets. (1) ACS County-to-County Migration Flows provide 5-year aggregated estimates of annual inflows/outflows between county pairs, derived from approximately 3.54 million sampled housing units per year. (2) County Population Totals 2010–2019 provide population, births, deaths, and migration components; used to compute natural growth (births minus deaths) and domestic netflows. Additional data include Zillow county-level home value indices and IRS year-to-year address-change migration flows (2015–2019) for robustness. Spatial units: 3,141 counties and 384 MSAs. OMB classification distinguishes metropolitan, micropolitan, and non-statistical counties. Core county within an MSA is defined as the county with the highest population density. Key constructs and measures: - Natural growth vs. migration: x = |Inflows − Outflows| / |Births − Deaths| as a county-level ratio to assess migration’s relative importance. - Flow categories: intra-county, intra-state inter-county, inter-state, intra-city (within MSA), inter-city (between MSAs), flows between metro–micro, and metro–non-statistical areas. - Distance from core: county distance to the MSA’s core county used to analyze gradients in inflows/outflows and netflows. - Density gradients: comparison of origin and destination county population densities for netflows to quantify directionality (from high to low density). - Statistical analyses: Pearson correlations (R, p-values) for relationships between flow shares/netflows and distance to core; fraction F of netflows to lower-density destinations; scaling of international inflows Y with city size S via Y = Y0 S^β and residual analysis log(Y/(Y0 S)). - Inter-city flow structure: Define J_ik as aggregate flow from county i to all counties in MSA k, with J_ik ≈ l0 S_i^μ S_k^ν plus noise X_ik. Using flow per capita l_ik = J_ik/S_i and empirical symmetry of l_ik/l_ki yields ν = μ ≈ 0.34 (95% CI [0.33, 0.35]). Define migratory shock variable X_ik = (J_ik − J_ki)/(l0 S_i). The cumulative normalized shock ζ = (1/N) Σ_k X_ik / (l0 N_i S_i) captures net impact of inter-city migratory shocks on county i; its distribution is tested against Gaussian via KS test. Robustness: Analyses replicated for 2005–2009 and 2010–2014 ACS periods and with IRS flows across multiple years; qualitative patterns compared, and differences in intra-/inter-city shares noted.

• Scale of movement: During 2015–2019, about 45.6 million people per year moved in the U.S. (14.2% of population), with 43.5 million domestic moves and 2.1 million international inflows. - Composition of domestic moves: ~59% intra-county; ~24% intra-state inter-county; ~17% inter-state. - Migration vs. natural growth: For most counties, domestic migration dominates growth. The ratio x = |Inflows−Outflows|/|Births−Deaths| follows a lognormal distribution and is ≤1 for 76.5% of counties, indicating migration often outweighs natural growth in magnitude. - Heterogeneity across and within MSAs: Relative dispersion of county growth due to netflows exceeds 1 for ~85% of MSAs, indicating strong intra-city heterogeneity. - Inter- vs. intra-city shares: Inter-city flows are the largest component (~55%), intra-city ~25%; flows between metro–micro (~13%) and metro–non-statistical (~7%) are smaller. Across states (excluding single-MSA states and single-county MSAs), inter- and intra-city shares within states are on average comparable (mean ~0.5, SD ~0.2). - Spatial structure within MSAs: • Core counties are more likely to have negative netflows (net out-migration), with intra-city netflows oriented radially outward from higher- to lower-density counties, contributing to suburban expansion. • Inflows from other MSAs concentrate in core counties and decay with distance from the core; inflows from within the same MSA increase slightly with distance (Fig. 3A,B). • Outflows from central regions tend to go to other MSAs (Fig. 3D) and to other counties within the same MSA (Fig. 3C), indicating central areas are highly dynamic. • Intra-city flows are the main driver of population increases in external counties; relative growth due to intra-city flows increases with distance (Fig. 3E; positive correlation), whereas inter-city contributions show no clear trend and can be negative at the periphery (Fig. 3F). - Density gradient: Within MSAs, netflows predominantly go from higher- to lower-density counties. For the 46 MSAs with >5 counties, more than 93% have a fraction F > 0.5 of intra-city netflows to lower-density counties; F increases with city population. - Inter-city density patterns: About 57% of inter-city netflows go to lower-density counties overall. F decreases with destination city size (inflows to large cities tend to originate from lower-density counties) and increases with origin city size (outflows from large cities tend toward lower-density destinations). - Housing and employment: No clear national pattern linking intra-city netflows to lower house prices; results vary by city. However, in about two-thirds of cities, intra-city netflows favor counties with lower unemployment rates. - Migratory shocks at county level: The distribution of ζ (sum of normalized netflows per county) is approximately Gaussian, indicating absence of heavy-tailed shocks at the county level. This suggests spill-over redistribution of city-level shocks across counties, smoothing extremes. - International inflows: Concentrated in large MSAs; top 10 MSAs receive ~40% of international inflows (e.g., New York 8.5%, Los Angeles 5.4%, Miami 5.0%). International inflows scale superlinearly with city size with β ≈ 1.19 (95% CI [1.13, 1.24]). Residual analysis shows college towns receive more international inflows than expected, while some largest cities receive less than expected. Within MSAs, international inflows are concentrated in core counties and decline sharply with distance from the core. - Robustness over time and data sources: Similar spatial trends observed in 2005–2009 and 2010–2014, though the dominance of moves to lower-density counties decreased from ~95.7% (2005–2009) to ~76.1% (2010–2014). IRS migration data (2015–2019) corroborate patterns and indicate stronger intra-city correlations with distance, with intra-city flows comprising ~80% of metro inflows/outflows (except cores), compared to ACS’s roughly equal intra-/inter-city split.

The findings show that county-level population growth within U.S. MSAs is driven by a structured interaction of intra- and inter-city migration. Intra-city flows redistribute residents outward along negative density gradients, fueling suburban expansion and contributing to heterogeneous growth across a city’s counties. Inter-city flows predominantly impact core counties and shape overall city-level population change. Although cities experience heavy-tailed migratory shocks, these are dissipated at the county scale as inflows are spread across multiple counties, yielding a Gaussian distribution of normalized shocks. This spill-over effect explains smoother county-level dynamics relative to city-level volatility. The concentration of international inflows in large-city cores further accentuates central dynamism, while outward domestic movements reflect household trade-offs (e.g., housing, employment, amenities) and life-course dynamics. These spatial patterns have implications for planning: anticipating asymmetric urban sprawl, managing infrastructure demands, and addressing distributional impacts across metropolitan cores and peripheries. The results underscore the importance of distinguishing flow components (core vs. external, inter- vs. intra-city) when analyzing urban growth and inequality dynamics.

This study introduces a county-level framework to disentangle intra- and inter-city migration contributions to heterogeneous urban population growth. Key contributions include: (1) documenting outward intra-city redistribution along density gradients; (2) showing inter-city flows concentrate in cores; (3) demonstrating that county-level migratory shocks are tempered (Gaussian ζ) compared to city-level heavy tails; and (4) revealing superlinear scaling of international inflows with city size and their concentration in core counties. The framework and empirical patterns offer actionable insights for urban planners regarding sprawl, infrastructure, and equity. Future research avenues include integrating intra-county flows (e.g., neighborhood-scale), extending analyses to other countries and income contexts, constructing typologies of flows and drivers (housing, employment, amenities), and combining multiple data sources (e.g., mobile, administrative) to model polycentric dynamics and forecast expansion/densification.

• Data granularity: Intra-county (sub-county) moves, which comprise ~60% of domestic moves, are not analyzed due to lack of sub-county data. - Scope of international flows: ACS flow files provide international inflows but not outflows; international components are thus partially observed. - Structural constraints: States with a single MSA have no within-state inter-city flows; MSAs with a single county preclude intra-city flow estimation. - Temporal variability: Migration patterns evolve over decades; results reflect 2015–2019 with historical comparisons indicating changing intensities. - Generalizability: Patterns may not hold in low- and middle-income countries or regions with lower urbanization and dominant rural-to-urban flows. - Measurement and heterogeneity: Housing price effects show city-specific heterogeneity; IRS vs. ACS indicate different intra-/inter-city shares. - Classification dynamics: OMB reclassification of fringe rural counties into metro may obscure rural populations in proximate areas, complicating interpretation of external county trends.