Mathematical Analysis On Urban’s Origin

Human activity cultivated on land, city is the product of human economic activity and also the most important places of it. Marx once pointed out that city accompanies with The history of human civilization and has continued to the present. Especially since the industrial civilization, with the progress of technology and the rapid increase of population, the city has been undergoing a change the sharpest ever. More and more problems have appeared in city:traffic jam, increasing shortage of daily water, unreasonable use of city land, overburden of house and public facilities, worsen pollution and so on. These problems have plagued the city urbanization in developed countries, which also appear in the modern China. Since the reform and opening-up in the1980s, China has entered an unprecedented age of urbanization, the problems which developed countries have undergone for hundreds of years concentrate to twenty or thirty years. Such complex problems in short times are testing the ability of city public management and governance. In this article, in order to improve the efficiency of land use and city public governance, we must accurately grasp the historical law city itself in space on the origin, evolution and development. Only have a thorough understanding of the law city itself, can we have a definite object in view in the formulation of public policies on land and city. The purpose of this article is to study on the law city itself, but the developing law of the whole city is a complex and huge task, this article choose the view of city’s origin, tries to answer these questions below:1.How city appears in human activity, that’s to say, What reason prompts city’s appearance;2. What is the core strength that city’s existence and sustainable development based on;3.How to testify the law city itself by using formal mathematical proof.Based on the above goals, this article traced back the whole history about urban research, and pointed out that there are two types of literature, one involved urban’s origin, another involved urban’s location, then summerized the latest research progress on them, one is represented by Krugman’s New Economic Geography urban theory, and the other is New Classical Economics urban theory, which led by Yang Xiaokai. Both of the two theries have been carried out on the mathematical modeling analysis of the city’s origin and development and evolution. Based on the two theories, this article tries to make some improvements and use Matlab to do the numerical analysis of the model simulations, and considers to use the division of labour as the core of model, constructing a general framework, so that the city’s formal analysis of endogenous origin more perfect, in detail, this article makes the imprvements as following:Firstly, this paper based on the general form of the function G-S, constructing a special utility function. This utility function is one of a key innovation in comparison to the past study,which can be able to distinguish different nature between agricultural and non-agricultural products to satisfy the people on the effectiveness. This special utility function has four advantages,1. The elasticity of substitution between agricultural and non-agricultural products increases along with the quantity of agricultural products;2. If the agricultural products is zero, the value of the utility function will be zero, and the quantity of consumption about aricultural has a lower limit;3. When the non-agricultural products is zero, the utility will not be zero;4. Diversity desires are limited by the quantity of agricultural products, the less the quantity of agricultural products, the lower the diversity desires of people, and vice versa.In early human society without city, strived for survival, people would naturally and not particularly care about the consumption of non-agricultural products, even in modern society, agricultural products is still a necessity, only to meet the basic agricultural products necessary for survival, people can consider to consumer non-agricultural products, this is the famous Engel’s Law. C-D function and CES function that previous literature has often been used can not distinguish agricultural and non-agricultural products in the consumption characteristics.Secondly, using the G-S utility function and other improved settings, this article reanalyzes the endogenous origin process of cities in the neoclassical F-K mood in Chapter3and Chapter4. The model of analysis in this article can explain more phenomena of cities than those presented by krugman et.al:1. Differing from the assumption of krugman et.al’s model that the manufacturing industry exists exogenously, the model of this article demonstrates that the manufacturing industry does not emerge abruptly or innately, but appears when there are enough agricultural surplus with the progress of agricultural technology;2. In krugman et.al’s model the relative proportion of number of the manufacturing workers in cities to that of the agricultural labors is given exogenously without the workers flow between the two sections, whereas in the model of this article the proportion is changing endogenously with the exogenous conditions such as the agricultural technology A and the transportation τM+τA;3. In krugman et.al’s model all the agricultural labors are involved in the manufactured products consumption, whereas in the model of this article more and more agricultural labors are participating in the manufactured products consumption with the improvement of exogenous conditions. In other words, in the early stages of the manufacturing industry development, the agricultural labors that live in the rural areas far away from the cities probably have no access to consuming the manufacturing products.4. More to the point is that, in krugman et.al’s model the nominal wages of city workers together with the consumption shares are given exogenously due to the exogenously existing of the manufacturing industry, therefore, cities are pseudo endogenous, while in the model of this article cities are real endogenous on account of the endogenous manufacturing industry, the endogenous nominal wages of city workers and the endogenous consumption shares.Thirdly, although this article improves the F-K model by using the G-S utility function settings to make the city origin endogenous in the analyses of Chapter3and Chapter4, many basic phenomena of city origin remain unexplainable because of the inherent defects of the neoclassical frame. Hence, after the comparative study, this article points out in Chapter5that the division of labor is the core economic power for the origin and development of cities. Furthermore, Chapter5and Chapter6reconstruct the model of city origin core on the basis of the division of labor according to the new classical city theory. Judging from the details, the model in this article differs from that presented by Yang Xiaokai in the following points:1. This article redefines and distinguishes the endogenous transaction costs and the exogenous transaction costs. In a fixed time node, all the transaction costs produced by the spatial distance between the two parties can be referred to as the exogenous transaction costs; while the endogenous transaction costs refers to the transaction costs produced by the parties themselves, that is, the transaction costs derived from the elements such as the public security, the information asymmetry, the ethical risk, the uncertainty and so on in the transactional interaction process between the two parties. In addition, what is different from that of Chen Junchang(2009) is that, this article sets the transaction efficiency as k=k1k2, instead of k=k1+k2, and points out that under the condition of the division of labor, the exogenous transaction costs results in the gathering and therefore the formation of cities, whereas the endogenous transaction costs endows the government with the rationality and the economic necessity. In other words, the government should bear the responsibility of reducing the endogenous transaction costs. After the cities come into being in the history of mankind, in the evolution and deepening process of labor division the interaction between the exogenous transaction costs and the endogenous transaction costs makes the cities keeping their development and evolution.2. The utility function. With regards to the agricultural products and the non-agricultural products, the settings of using the G-S function in Chapter3and Chapter4are continued and the CES function is adopted as the subfunction.3. The settings of agricultural production function. The production function form in the new classical model is suitable for any product, nonetheless, this article adds the land element into the agricultural production to make it conform to the characteristics of agriculture further and besides, the adoption of land element can substantialize the space so as to make the model possess the character of spatiality.4. By setting the study costs of each job and the minimum survival number of agricultural products, the number of items that the individuals can produce and consume under the limited condition is assumed endogenously instead of exogenously.5. The linear space is adopted. In the city model put forward by Yang Xiaokai, the space problem is abstract without being substantialized. The linear space possesses the actual spatiality although the two-dimensional plane and the three-dimensional space in reality are simplified. Settings of the linear space and the agricultural production function concretize the locations of cities making them not imaginary any more. Generally speaking, this article incorporates the essential elements of forming a city such as the professional workers, the professional traders, the councilors and the markets into the same analysis frame, and on that basis, the process of city origin is explained more distinctly.Besides the above, as a by-product the analysis process, this article reconstructs labor theory of value and Land rent theory, and points out that the defect of classical labor theory of value, which dates from Adam Smith and end at Marx, is taht they neglected ownership. Under the condition of ownership, this article classifies the valve into two categories, one is exogenous value, which is created by nature, another is endogenous value, which is created by people’s labour, so that value, use value and exchange value of the product can unify into one single meaning. With regard to land rent theory, if the land is Homogeneous, only the rent from locational differences can be regarded as reasonable rent, this kind of rent derives from the saving of transaction costs, and the rent should be used to improve the endogenous transactional efficiency, in other words, it should be considered as a revenue of government, and expenditure As a public service, benefits all the the people in society. In a contrary, monopoly rent,which derives from by the private ownership, is the root of the unfair distribution of income in society.