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

Urbanization, a global trend accelerated in recent decades, has led to significant population concentration in major cities. This concentration, however, often results in income inequality. Existing empirical research offers conflicting perspectives on the relationship between city size and inequality. While some studies suggest a U-shaped relationship, others find a consistent positive correlation. The core of understanding this phenomenon lies in the driving forces behind labor movements between cities. This paper utilizes the Lucas-Prescott island model, modified to incorporate skill heterogeneity and environmental factors, to explore the underlying mechanisms that influence individual labor choices and their impact on income distribution and urban development. Previous models often treat labor as homogeneous, neglecting the crucial roles of individual skills and wealth in location decisions. By considering these elements, alongside the quality of the environment as a significant non-wage factor affecting job choice, the model strives to provide a more comprehensive understanding of urban labor dynamics and regional inequality.

Existing literature on labor mobility and urban inequality reveals contrasting views. Nord (1980) argues that income inequality follows a U-shaped pattern with increasing city size due to changes in the occupational and wage structure. Conversely, Madden (2000) and Glaeser et al. (2009) find that inequality consistently increases with population size, attributing this to the expansion of high-skilled job markets and increased competition for low-skilled workers. Behrens and Robert-Nicoud (2014) develop a model linking city size to inequality and productivity, emphasizing the skill premium and effects of urbanization on competitiveness. Castillo et al. (2020) highlight the role of knowledge diffusion through labor mobility. The Lucas and Prescott (1978) island model emphasizes the trade-off between higher wages and job search costs, explaining unemployment. Coen-Pirani (2010) extends this model by including land in the production function, but still overlooks skill heterogeneity and environmental considerations. This paper builds on these studies by incorporating skill heterogeneity, endogenous technology growth, and environmental preferences into a Lucas-Prescott framework to offer a richer and more nuanced explanation of urban labor dynamics.

The paper develops a two-part methodological approach. First, it presents a baseline Lucas-Prescott style island model with heterogeneous agents. Agents' income stems from wages and capital rent. They choose between staying on their current island or moving to another, trading off higher potential wages against the forgone income during the search period. The model introduces skill heterogeneity and endogenous technological growth, where technology growth is a function of existing technology, the number of researchers, and a constant 'natural technology' growth rate. Human capital growth depends on current human capital and city technology level. Agents maximize utility, a function of consumption and environmental quality. The model's equilibrium conditions define the optimal location choices of agents with different skill levels and wealth, given technological and environmental characteristics of each island. Steady-state conditions are derived, and the paper analyzes the equilibrium location of agents with varying characteristics. Specifically, propositions are developed predicting location choices based on capital and learning speed. Second, the paper extends this model to incorporate a two-goods setting with tradable and non-tradable goods. This extension introduces the price of non-tradable goods as an additional factor influencing location decisions. The Balassa-Samuelson effect is integrated to model the relationship between technology levels, wages, and non-tradable goods prices. This expanded model is used to analyze the implications of both wage differences and non-wage factors on location choices and urban inequality. The model's predictions are then tested empirically using US census data from 2016-2021, using Logit and Probit regression models to investigate how wealth, education, and other variables affect location choices between cities with differing technology and environmental levels.

The baseline model shows that agents’ location choices depend on their capital and speed of learning. Those with high capital and low learning speed tend to favor locations with better environments, while those with low capital and high learning speed gravitate towards high-technology areas. The model demonstrates a U-shaped relationship between wage income inequality and technology level across islands, aligning with Nord's (1980) findings. However, the relationship between total income inequality and technology is ambiguous. The two-goods model enhances the understanding of labor mobility by highlighting the role of non-tradable goods prices. Higher technology levels are associated with higher wages for tradable goods, and to ensure wage parity between tradable and non-tradable sectors, the price of non-tradable goods rises in high-technology areas (Balassa-Samuelson effect). This effect influences the magnitude of labor mobility implications. The empirical analysis using US data confirms the model's predictions. Individuals with high skills and low wealth are more likely to live in high-technology states, even with poorer environments. Conversely, less-skilled but wealthier individuals prefer locations with better environments. The results highlight the interplay between individual characteristics, city attributes (technology and environment), and the decision-making process behind labor mobility.

The findings contribute to the literature in several ways. First, the model offers a theoretical framework for understanding the heterogeneous location choices of workers with diverse skills and wealth, moving beyond simplistic homogeneous labor assumptions. Second, the model provides a theoretical basis for empirical research on wage differences across cities, especially relevant for developing countries where technology disparities are more pronounced. Third, the model explains price heterogeneity across cities, addressing situations where the typical positive relationship between wages and housing prices does not hold. The incorporation of environmental preferences and non-tradable goods clarifies the influence of non-wage factors on urban development patterns. The model helps explain the concentration of highly skilled individuals in major cities and the tendency of wealthier individuals to reside in environmentally appealing areas, even if those areas have lower wages. The empirical results support the model's predictions. This suggests that individual heterogeneity plays a crucial role in shaping labor migration patterns and urban inequality.

This study contributes a novel Lucas-Prescott style island model to the literature on labor mobility and urban inequality, incorporating endogenous technology growth, skill heterogeneity, and environmental considerations. The model generates a nuanced understanding of location choices and reveals the U-shaped relationship between wage inequality and technological advancement. Future research will extend the model to include government interventions, calibrate it with US microdata, and conduct a more thorough analysis of city-level data, examining the reciprocal effects between migration patterns and non-tradable goods prices.

The model has some limitations. The assumption of a steady-state equilibrium simplifies the dynamic processes of urban development, potentially underestimating the role of short-term fluctuations. The empirical analysis focuses on US data, limiting the generalizability of the findings to other contexts. The model's complexity might hinder the intuitive understanding of the implications. Future research should address these limitations by incorporating more realistic assumptions and exploring the model's applicability across diverse socio-economic settings.