Abstract
The theory of residential location is concerned with the behavior of the consumer who wants to make a choice among various alternate locations within a metropolitan area. It must analyze not only the equilibrium of the consumer, i.e. how much and where the consumer would buy land in a city, given his income, taste, prices and technology of transport and so on, but also the change in equilibrium with respect to change in the above mentioned independent variables, like income, etc.. However, a number of studies, which have examined the equilibrium of the household, apart from being similar in their approach, are also deficient in their analyses. These deficiencies are:
A. They have discussed the necessary conditions of equilibrium, but have not satisfactorily explored the sufficient conditions which are important in order to examine the uniqueness and stability of the equilibrium. For example, Alonso's “Location and Land Use”, though one of the most important ones in this area, is also very erroneous in this point. He discusses and analyzes his theory as if the optimal solution is unique. He does not mention any assumption behind the diagram of the opportunity locus surface. Furthermore, even with the assumptions of well-behaved constraint function, there are possibilities of multiple solutions.
B. Few works have been done to examine the comparative statics of the consumer equilibrium, a very important dimension of the theory of residential location.
C. These studies neglect the possibility of consumer's choice to live at the Central Business District (CBD), because the simple Lagrangian maximization principle, which all of them use, cannot deal with the corner solution. By using the Kuhn-Tucker theorem of nonlinear programming we can consider such a possibility. Besides, use of nonlinear programming has an additional advantage. It brings to light the fundamental problem involved with the non-concavity of the constraint function in the theory of residential location.
D. In Alonso's model, for example, distance from the CBD is thought of as a commodity with negative utility because distance is assumed to represent only accessibility. But distance is also related to environmental quality. In most cities, environmental quality increases as distance increases. Therefore, distance affects utility positively as well as negatively under the assumption of a monocentric city. This means that we need to deal with accessibility and environmental quality separately. This difficulty can be overcome by introducing time element into the model. But to consider this problem we will need a separate paper. Therefore, it is not dealt with in this paper, though we assume that the positive and negative effect of distance on utility cancel out.
The purpose of this paper is to examine the three basic issues raised above. First of all, the necessary and sufficient conditions of the equilibrium is discussed, examining the properties of the constraint function, and then the displacement of equilibrium is analyzed.
