Emergence and Evolution of Heterogeneous Spatial Patterns

We live in a quite heterogeneous space. There are cities and rural areas, and population density varies a lot across space. People migrate and commute to the places of their work. The goal of this article is to clarify the mechanism of commuting as an equilibrium in heterogeneous space with different technologies. It is well known that agricultural production requires substantial amount of land per unit of labour, while most industrial production and services require much lower land input. We assume that all industrial production and service sector is located in urban areas, while all agriculture is in rural area. Historically, the share of labour in agriculture was declining due to more rapid growth of productivity there in comparison to service sector. At the same time, people change the location of their residence much slower. That is why at some point in time we face the situation, when rural area has excessive labour (not enough work for all in agriculture), while urban areas create an increasing number of jobs. A relatively simple mathematical model is proposed to explain the emergence of spatial pattern with heterogeneous density and phase transition between urban and rural areas. There are three types of agents: workers who live in a city, farmers who live in a rural area and workers-commuters from rural area to a city. In an equilibrium they are indifferent between occupation and residence. An indifference across locations for a priori identical agents implies the shape of land rent. If some parameters of the model change, they imply the change of the whole spatial pattern. In particular, split of rural residents into commuters and farmers depends on road infrastructure development through transport cost. Two types of shocks (decline in commuting transport cost by construction of fast roads and the relative decline in agricultural price) can perturb agricultural zone. Some former farmers start commuting to city while keeping residence in rural area. This is how a functional area of a city with integrated labour market emerges.