• 论文 •

### 效用最大化、logit变换和城市地理学的数量分析模型

1. 1. 北京大学城市与环境学系 北京 100871;
2. 东北师范大学地理系 吉林 长春 130024
• 收稿日期:2002-03-10 修回日期:2002-04-27 出版日期:2002-09-20 发布日期:2002-09-20
• 基金资助:
国家自然科学基金资助项目的部分内容(项目编号:40071035)。

### Utility-Maximization, Logit Transformation and the Basic Mathematical Models for Analytical Urban Geography

CHEN Yan-guang1, LIU Ji-sheng2, FANG Yan-gang2

1. 1. Department of Geography, Peking University, Beijing 100871;
2. Department of Geography, Northeast Normal University, Changchun, Jilin 130024
• Received:2002-03-10 Revised:2002-04-27 Online:2002-09-20 Published:2002-09-20

Abstract: Many basic mathematical models as scale laws can be derived from a pair of assumptions, i.e. ① The utility in some sense is proportional to some kinds of variables such as time, space, and class; ② The respondent variables of the utility such as population size and city number is the exponential function of the utility. The basic models include the allometric relationship of urban and rural population, logistical model of urbanization process, the relationship between level of urbanization and that of economic development, i.e. Zhou’s model, urban population density model, i.e. Clark’s model, the generalized Beckmann-Davis model, and the three-parameter Zipf model, etc. It is proved that the exponential relationships between the utility and its respondency come from utility-maximization that is consistent with entropy-maximization, and utility-maximization and entropy-maximization are actually principles of duality. Ont the other hand, the basic models for analytical urban geography can be changed into logit models, for example, the urban and rural population allometric model is linked to binary logit model (BLM), and the Clark model can be connected with multinomial logit model (MNL). This means that we can simulate the urban dynamics based on utility-maximization principle by means of logit transformation that can simplify the researched objects to a great extent. The logit transformation will become an important part of geocomputation and the simulation based on the logit models consociated with utility-maximization principle will contribute to the studies of spatial complexity. Utility-maximization is not only one of the general principles of urban geography but the foundation of urban maximization from which we can develop new methods for urban planning. Perhaps the consistence of utility-maximization and entropy-maximization is the key to solving the problems of spatial complexity. Unlike the concepts such as "the edge of chaos" and "self-organized criticality", the duality of utility-and entropy-maximization can answer many theoretical questions without using one concept to replace another.

• P290