A Lucas island model to analyse labour movement choice between cities based on personal characteristics

The paper addresses why and how different types of labour choose to reside in different cities and how these choices shape urbanisation and inequality. Against the backdrop of accelerating global urbanisation and pronounced population concentration in certain countries, prior empirical findings disagree on whether inequality in cities follows a U-shaped pattern with city size or increases monotonically. The core question is to understand the drivers of heterogeneous labour movements across cities (modelled as islands) differing in technology and environmental quality. The study motivates a theoretical framework where agents differ by human capital, endowed wealth (capital), and learning ability, and where city technology and personal human capital evolve endogenously. The model incorporates environmental quality into utility to capture moves toward high-amenity cities, aiming to explain observed phenomena such as concentration in superstar cities and high housing price-to-wage ratios in amenity-rich locations. The paper also outlines contributions relative to existing search and matching frameworks by introducing wealth as a determinant of mobility and environmental amenities in preferences.

The paper situates its contribution within several strands:

• City size and inequality: Nord (1980) reports a U-shaped relationship between income inequality and city size, while Madden (2000) and Glaeser et al. (2009) argue inequality rises with population. Subsequent work (Wheeler, 2001; Baum-Snow and Pavan, 2009a) supports a positive effect of city size on wage premia across income quintiles. Behrens and Robert-Nicoud (2014) link larger cities to higher skill premia and competitiveness, reinforcing inequality via selection of productive, exporting firms. Labour mobility spreads knowledge and enhances productivity (Castillo et al., 2020).
• Labour mobility and search: Lucas and Prescott (1978) develop the island model where technology heterogeneity drives wage differentials and unemployment during job search. Coen-Pirani (2010) adapts this framework to inter-city labour flows with land in production.
• Search and heterogeneity models: McCall (1970) models reservation wages; Hopenhayn (1992) studies entry/exit with stochastic productivity; Aiyagari (1994) and Krusell & Smith (1998) incorporate heterogeneity across agents and firms. The current paper differs by adding personal wealth (capital) as a mobility determinant and incorporating environmental quality into utility.
• Prices across cities: Van Nieuwerburgh and Weill (2010) tie productivity-driven migration to housing price dispersion. However, the observed weak or negative wage–price correlations in amenity-rich cities motivate adding non-tradable prices and amenities. The Balassa–Samuelson mechanism (Samuelson, 1964; Kravis & Lipsey, 1983; Samuelson, 1994) and intra-country evidence (Vaona, 2011; Nenna, 2001; Songtao, 2009) rationalise higher non-tradable prices in more productive cities. Environmental amenities raise housing values (Luttik, 2000; Donovan et al., 2019) and wealth of buyers also matters (Catte et al., 2004; Steegmans & Hassink, 2017). Environmental quality affects mobility (Feld et al., 2022; Chen & Oliva, 2022).

Theoretical framework: A Lucas–Prescott style multi-island economy with heterogeneous, infinitely lived agents is developed. Each island is characterised by technology z and environmental quality q. Agents are endowed with human capital h, physical capital k, and learning speed θ. Utility is u(c,q)=ln c + ln q. At the start of each period, after shocks are realised (the steady-state analysis abstracts from uncertainty), each agent decides to stay on the current island or leave to search/move to another island. If staying, the agent earns wage income and capital income, chooses next-period capital k′, and enjoys local environmental quality. If leaving, the agent forgoes current wage income, maintains capital returns, suffers human capital depreciation at rate δ_h during search/move, and arrives at a chosen island next period. Taxes on wage and capital income (τ_w, τ_k) and capital depreciation δ apply. Human capital and technology dynamics follow endogenous growth relationships: technology growth depends on existing technology, research effort, and natural technology growth (Romer, 1986), while human capital growth depends on current human capital and local technology (e.g., Abel et al., 2012). Employment e(h,k,θ; z,q,w,K) maps agent types to employment on islands. Production on each island uses a constant-returns function F(zH,K) hiring effective human capital H and capital K; factor prices satisfy zF_1(zH,K)=w and F_2(zH,K)=r, with free capital mobility equalising returns across islands. The paper defines agents’ Bellman problems for staying and leaving, equilibrium conditions (factor price equalisation, market clearing in labour and capital, and population flows consistent with individual value comparisons), and steady-state conditions. In steady state (with uncertainty suppressed), population on each island is constant, technologies and human capital grow at constant rates, and environmental quality is fixed. Multiple steady-state equilibria can exist given distributions of initial wealth and learning speeds. Analytical results: Comparative statics consider moving from a high-quality/low-technology island (A) to a low-quality/high-technology island (B). The consumption utility gain rises with learning speed θ, while quality utility falls due to lower q in B. Propositions derived:

• Proposition 1: High-capital, low-θ agents prefer better-environment islands; low-capital, high-θ agents prefer higher-technology islands.
• Proposition 2: High-capital, high-θ and low-capital, low-θ agents prefer islands with balanced technology and environment. The model implies U-shaped wage inequality across islands as technology rises, due to selective inflows and outflows of the poorest and richest agents. Extended two-goods model: The framework is augmented with two consumption goods: a tradable good A (price normalised to 1) and a non-tradable good B with island-specific price P_B. Utility becomes u(c_A,c_B,q)=ln c_A + ln c_B + ln q. Two sectoral production functions (for A and B) use effective human capital and capital, with factor prices equalised across sectors within an island. In steady state, per the Balassa–Samuelson mechanism, higher-technology islands feature higher wages in tradables, which raise non-tradable prices to equalise factor returns across sectors. This tightens mobility conditions relative to the baseline one-good model and affects the relative attractiveness of high-tech versus high-amenity islands. Empirical strategy: Using IPUMS USA microdata (2016–2021) at the state level, the paper proxies state technology by USPTO patent counts and environmental quality by the EPA Air Quality Index (AQI; higher AQI indicates worse air quality). Binary outcomes indicate whether an individual resides in a high-environment-quality state (AQI below the median) or a high-technology state (patents above the median). Skill is proxied by an 11-level education variable; wealth is proxied by log home value. Controls include sex, age, log(1+income, including rental income), and a farm ownership indicator. Models estimated are Logit and Probit with year, work-industry, and county fixed effects; standard errors are clustered at the family level. Sample size is 734,440.

Theoretical findings:

• Sorting by characteristics: Low-wealth (low k), high-learning-speed (high θ) agents choose high-technology, lower-amenity islands; high-wealth (high k), low-θ agents choose high-amenity, lower-technology islands. Agents with high k and high θ or low k and low θ select balanced islands.
• Inequality pattern: Wage inequality across islands is U-shaped in island technology: both the richest and poorest tend to sort into the most and least technologically advanced islands, while middle-class agents concentrate in intermediate islands.
• Prices of non-tradables: In the two-goods model, higher-technology islands exhibit higher non-tradable prices via the Balassa–Samuelson effect, amplifying sorting effects. High-amenity, lower-wage cities can still have high housing price–to–wage ratios due to wealthy residents’ capital income and amenity demand. Empirical findings (IPUMS USA 2016–2021; n=734,440):
• Living in good-environment states: Education is negatively associated (Logit odds ratio ≈ 0.978, Probit coefficient ≈ -0.013; both p<0.01), wealth is positively associated (Logit odds ratio ≈ 1.015, Probit coefficient ≈ 0.009; both p<0.01).
• Living in high-technology states: Education is positively associated (Logit odds ratio ≈ 1.005, Probit odds ratio ≈ 1.003; both p<0.01), wealth is negatively associated (Logit odds ratio ≈ 0.990, Probit odds ratio ≈ 0.994; both p<0.01).
• Model fit: Pseudo R2 ≈ 0.20–0.22 with year, industry, and county fixed effects; standard errors are robust and clustered at family level. These results support the model’s prediction that high-skill, low-wealth individuals sort into high-technology but poorer-environment states, while low-skill, high-wealth individuals prefer better-environment, lower-technology states.

The model explains heterogeneous labour sorting across cities through the joint roles of endogenous technology and human capital growth, personal wealth, and environmental amenities. High-technology islands attract agents who rely on wage income and benefit from faster human capital accumulation (high θ), even if environmental quality is lower. Conversely, agents with substantial capital income can prioritise environmental amenities and are less inclined to sacrifice quality for higher wages. This sorting generates a U-shaped relationship between island technology and wage inequality: the most technologically advanced islands accumulate both the very rich and the very poor, while intermediate-technology islands attract middle-income agents, reducing inequality there. Incorporating non-tradable goods prices clarifies why some amenity-rich cities exhibit high housing prices despite lower local wages: the Balassa–Samuelson channel raises non-tradable prices in productive cities, and wealthy households’ capital income sustains demand in amenity-dominant locales. The empirical state-level evidence from the US aligns with these mechanisms, showing that skill and wealth have opposite associations with choosing high-tech versus high-environment states. These findings shed light on population concentration in superstar cities and the observed high housing price–to–wage ratios in scenic cities.

This paper develops a Lucas–Prescott style island model with endogenous technology and human capital growth to study heterogeneous agents’ location choices across cities differing in technology and environmental quality. The framework yields a balanced growth steady state and multiple equilibria. The baseline model implies that ideal cities lie on a technology–amenity frontier; low-wealth, high-learning-speed agents prefer high-technology cities, while high-wealth agents prefer high-amenity cities. Wage inequality displays a U-shaped relation with city technology, while total income inequality is ambiguous. Extending the model to include tradable and non-tradable goods shows that higher-technology cities have higher non-tradable prices (Balassa–Samuelson effect), which amplifies sorting and helps explain high housing price–to–wage ratios in amenity-rich cities. Empirical evidence from US state-level data supports the theoretical predictions: higher education (skill) and lower wealth are associated with residence in high-technology, poorer-environment states, whereas higher wealth is associated with choosing better-environment, lower-technology states. Future work will incorporate government policy, calibrate and simulate the model with US data, and test implications at more granular county/city levels, including joint dynamics of migration and non-tradable prices.

• Theoretical simplifications: The steady-state analysis abstracts from stochastic shocks to ensure balanced growth, which limits realism relative to persistent uncertainty in actual city dynamics. Multiple steady-state equilibria depend on assumed distributions of initial wealth and learning speeds, leaving some predictions context-specific.
• Omitted policy environment: Government roles in attracting workers, supporting R&D, and improving environmental quality are not modelled; policy implications are therefore incomplete.
• Empirical scope: Evidence is limited to state-level US data (2016–2021). Finer spatial resolution (county/city) and richer controls for cost-of-living and housing markets would improve identification. Wealth is proxied by home value, which may imperfectly capture liquid wealth; technology is proxied by patents, which may not fully reflect productivity across sectors.