The relationship income / distance to CBD is basically nonlinear and its form varies a lot across the world. We often consider classic forms such as the typical US city one, where rich people live in the suburbs and the European city where they live downtown. Nevertheless, more complex patterns could be observed like in New York City. In this city, the area which is the closest to the city center is very rich; poor people live further away from CBD and the middle class lives in the suburbs. To explain these stylized facts, we use a model inspired by Turnbull (1997). Instead of employing the classical utility functions, this model applies the revealed preference theory. In his original paper, Turnbull established several propositions that made his model consistent with the main results obtained in urban economics (for instance that the prices decrease in relation to the distance from the CBD). Despite its consistence, the model gives different predictions than the main studies regarding the relationship between income and distance to CBD. This model allows us to deal with a great heterogeneity among the households (regarding incomes, tastes, transportation modes, amenities consumption) and thus enables many different potential equilibria. This diversity of equilibria permits to represent in a more accurate way cities and patterns and thus justifies the use of this model. Moreover, we try to take into account every parameter that could influence location choice in a single framework. Some model focuses on access to public transportation (Glaeser et al. (2008), other on amenities (Brueckner et al. (1999)). But no model has ever succeeded in giving a complete framework for studying the income sorting. Utility functions are usually used because the revealed preferences method is highly complex. But our model is quite simple because we demonstrated that the GARP axiom used in the revealed preference theory is equivalent to a simple equation. In our paper, several differences are found when comparing our results to the classic method using utility functions. Firstly, the ratio commuting costs per mile over demand land for rent is found constant between income groups, which is consistent with Wheaton (1977). As this variable is no longer supposed to explain the income sorting, the model focuses on other variables to explain it: income elasticity of housing demand, commuting time or amenity consumption. Moreover, in our model, some of the richer households will locate closer from CBD than a poorer one while other rich households will locate further away. This leads to an overlapping of the different income groups unlike the utility functions models that often predict complete segregation (except in De Bartolome and Ross (2003)).
