Where Alonso meets Sierpinski: an urban economic model of a fractal metropolitan area

The coexistence of residential and agricultural activities within 'periurban belts' characterises many modern metropolitan areas. Unfortunately, few theoretical works in standard urban economics take this type of mixed space into account. This paper is an attempt to fill this gap: we present a residential location model (standard in urban economics) that is based on a support provided by fractal geography. More precisely, on the one hand, a Sierpinski carpet is used to render the nested hierarchies of the rural and urban sites within a metropolitan area. On the other hand, a household maximises, subject to a budget constraint, a Cobb - Douglas/constant elasticity of substitution (CES) utility function, wherein sub-CES functions portray the consumer's taste for variety in urban and rural amenities according to their hierarchical rank. As the household's optimum depends on accessibility to these various amenities, we propose a coding scheme of the sites on the Sierpinski carpet and a procedure for computing the distances between each site and each amenity. The urban equilibrium solution in the case of an open city is analytically determined. Numerical simulations are performed. They reveal the link between the rent gradient and the accessibility to the rural and urban amenities. In particular, we show that the rent is not always monotonous in distance from the origin (central business district) and that the structure of the fractal city is very different from that of the Thünian city when (1) commuting costs are low, (2) preferences for rural amenities are high, or (3) the substitutability of urban (rural) amenities is low.