A Static Analysis of Residential Location: A Unified Approach toward the Models of New Urban Economics

Abstract

The prime object of this study is to propose an analytical framework which enables us to synthesize the static analyses of residential location. The setting for the study is the one typical in the literature of the new urban economics. Namely, the city is assumed to be monocentric, linear (or fan-shaped), and located on a featureless plane.
As in the other fields of economics, there are two possible ways to deal with these problems. One is the positive approach based on individual utility maximization, typically represented by the Alonso-Muth type models, and the other is the normative approach to seek some kind of social optimum. However, when the problem is properly formulated in the latter approach, it is possible to describe the competitive equilibrium in terms of the mathematical programming.
Meanwhile, the linear programming model known as the Herbert-Stevens model is proved to be able to reproduce, as its optimum, the equilibrium of the Alonso-Muth model, when the distance is discretized to a finite number of districts in this model. This conversely suggests that when the Herbert-Stevens model is reformulated to accommodate the continuous distance, this new model should have a close relationship with the original Alonso-Muth model.
In the light of this observation, the Herbert-Stevens model is first reformulated as an optimal control problem, and the properties of its solution are examined from its optimality condition. In particular, it is important to learn that the optimality condition is sufficient for optimality, and the existence of such an optimum is guaranteed. The Alonso-Muth type model, which is derived under the same circumstances as the optimization model, is then formulated, and the solutions from these two models are compared.
From this comparison, it is demonstrated that the continuous reformulation of the Herbert-Stevens model is theoretically equivalent to the equilibrium model of the Alonso-Muth type in the absence of externalities. Thus the existence of equilibrium can be concluded from that of the corresponding optimal control, and the Pareto optimality of the equilibrium can also be proved by the similar reasoning.
These observations suggest that this Herbert-Stevens type model could provide a powerful analytical framework for the study on residential location, which maintains the theoretical correspondence with the competitive equilibrium. To demonstrate the versatility of our framework to the various urban problems, two types of urban externalities, transport congestion and neighborhood externalities, are chosen to illustrate how these externalities can be incorporated in our framework.