An urban economic model over a continuous plane with spatial characteristic vector field : consideration of heterogeneous geographical conditions

Among others Beckmann (1952) firstly introduced the concept of a two dimensional continuous space into economics. This great step had unfortunately not shown further expansion in economics. Through several papers related to Beckmann's initiation, Beckmann and Puu (1985) at last reached a systematic treatment of the continuous spatial economics. Although their achievement is fascinated by employing a partial differential equations approach, Beckmann's original philosophy, that is, the gradient law has still been inherited. Following their achievement, Puu (2003) alone developed their theory by using many computer simulations to visually show the significance of their theory. Beckmann and Puu's book (1985) aims to study formation of urban configuration in a two dimensional continuous space, focusing on flows of commodities. However, consideration of households and firms location is not necessarily sufficient, resulting in reconsideration from a new urban economics point of view. As another exceptional urban economic study of a plane city, Lucus and Rossi-Hansberg (2002) is pointed out. They were inspired by Fujita and Ogawa (1982), and indicate endogenous land use pattern over a plane city. However they neglect commodity market to simplify the analysis. Well discussion about formation of urban configuration is summarized in Anas, Arnott, and Small (1998). Differing from Beckmann and Puu's studies, Miyata (2010) introduces bid rent functions (Fujita (1989)), which are familiar in the new urban economics, for land of households and firms, and then it studies how the results of Beckmann and Puu are rigorously modified by using the theory of partial differential equations (Courant and Hilbert (1953, 1962). However Miyata (2010) deals with a symmetric equilibrium which seems to be a little unrealistic. This article extends the author's previous study introducing spatial characteristic vector field in the model which stands for heterogeneity in geographical conditions in a city, and try to show asymmetry in land use pattern and endogenous formation of transport networks in a two dimensional continuous space.