Where in cities do "rich" and "poor" people live? The urban economics model revisited

Reproducing the socio-spatial structure of cities is one of the challenges facing the standard urban economics model of Alonso, Muth, Mills (AMM model). In a widely cited paper, Jan K. Brueckner, Jacques-François Thisse and Yves Zenou (1999) asked "Why is central Paris rich and downtown Detroit poor?" and proposed a model with a positive central amenity to account for the structure of European cities, where the city center is usually rich, like in Paris. In this work, we exploit the power of the AMM model and show that various utility functions and plausible conditions offer alternative explanations of households' location by income within a city. We first propose to take into account the empirical fact that the share of income spent for housing decreases when income increases. With a Cobb-Douglas utility function and two income groups, this ingredient yields different social structures than the standard one with low income households in the center and rich ones in the periphery. Depending on the relative values of the city radius and a critical radius related to model parameters, different urban forms appear indeed. These include the existence of a "rich" center and more complex socio-spatial urban forms, for instance alternating a rich center, poor suburbs and a rich outer ring, which have not yet been derived from the AMM model to our knowledge. In this work, we combine analytical ideas and illustrations by the means of an agent-based model. Indeed, starting from a random configuration of the city, a system of agents given relevant behaviour rules can find the equilibrium configuration of the AMM model. This modelling approach is inspired from the Monte Carlo method, and from local search optimization algorithms in computer science. Following Brueckner, Thisse and Zenou (1999), we also study the hypothesis of a central amenity. We show that under certain conditions, a central amenity can also yield a rich city center, and even a U-shaped curve of the income as a function of the distance to the city center. This result can be related in particular to empirical findings in some older North American cities, like New York, Chicago or Philadelphia as well as European cities like Paris. We find with agent-based simulations that these outcomes depend strongly on the respective locations of the employment and the amenity centers. Indeed, considering that the CBD and the amenity do not coincide has an important influence on the socio-spatial structure of the city.