Geometric Intersection Method For Solving Environmental Economic Equilibrium

The main opinion of classical economics, Adam Smith built, is that the market economy is the most efficient system to maximize the efficiency in the process of optimizing allocation of resource distribution. Rational economic man is a base of classical economics theory, supposed an economic individual would benefit from the maximization of his economical utility, furthermore realization of social benefit.Modern Western economists make an assumption of New Economical Man to try to balance relations between individual and society, and microeconomics and macroeconomics. However, the assumption only considers the benefit of modern man, not one of the offspring. Also, the theory of market economy overemphasizesfree competition, but underestimates to make use of natural resource and ecology in reason, which results in a series of bad influences, such as wildcatting,wasting resources and environment pollutions, and being unfair to modern men, moreover, hazarding the long-term development of human beings. All these are contrary to the aim of economy, raising the level of a social welfare. So the externality of economy causes the above.Economical growth is the main aim pursued by the macroeconomicsl policy, and one of problems economists concerned about most. A base of GDP growth theory is the input-output model, where capitals, labors and technique are main input factors. Stable environmental quality is supposed in this kind of economical model, that is, the ecology factor are considered as an external variable. But this theory can't represent the efficient production technology by using environmental resource reasonably, and gradually go far away from the real economic activity. In the frame of the above economical growth theory, popular system of GDP Accounts and statistic methods neglect some costs of social development, such as the resource depletion and environment degeneration. So economical data by their computation are local, and can't represent the changes of net welfare at the present and in the future. Hence, behind the speedy growth of GDP, great deficits of resource stocks and ecology environment have always been swept under the rug, serious unreasonable economical structures and the amazing unfairness in economical social are hidden.So the sustainable development idea, balancing benefits between modern men and the offspring, start to come out for the purpose of resolving the above limitation in current economical structure and economical growth theory and accounts methods. After the United Nations Congress on environment and development in Rio, the sustainable development became the common point. Chinese government emphasis the scientific development concept "everything basing the humanity" and the absolute new concept on constructing the socialist harmony society in recent years. Making the harmonic development between human beings and the nature into consideration, and bettering the relationship among economy construction, population growth and resource exploitation, and ecology protection promote our society development and pave the way for the civilized development of better ecology and life.We should develop and build a system of new national performance evaluation in place of the current national accounts around the GDP, so that make the sustainable from the theoretic philosophy to the real public. It is the strategic step for measuring the socialist harmony society and sustainability to take the resource, environment and non-market act into the national statistic and accounts and build GGDP account. Green GDP minus the environmental and ecological cost, adjust the classical statistic item from the angle of natural and humanity,which reflect the true development level, and provide the more scientific and sound standard for decision. GGDP theory and practice reflect the more rational choice of in macroeconomic policy. Since the last 70's, all experts, from worldwide and some organizations of UN, have always been explored in building a theory of the system of national accounts around Green GDP. Their works on GGDP are foundations of the evaluation of sustainable development in real life, but their works are most from eyes of statistics, and seldom analysis the economic internal principle."Demand, supply, equilibrium and growth" are the core and eternal theme of economics. After Walras built the theory of general equilibrium, this theme of economics has always been researched deeply in different research ways by experts, such as Arrow,Debru, Scarf, Samuelsson, von Neumann and etc.. All these research ways are based by mathematical models in economics, and mainly use analysis techniques and methods so their works are entirely new and impressive. Also their works always could be present by geometrical graphs, which would give readers more intuitive understanding and be convenient to the latter analysis.So straightly using geometry to describe and research economical equilibrium is an important characteristics, the theory of mathematical economics is differentfrom other economical research methods, and become a direction quantities economists concerned most. Here, our works resolve some problems in economical equilibrium and applications by virtue of some ideas from geometry.In the first chapter, we analysis the economics principle of sustainable developmentafter introducing its definition, and show that the base of measure theory, used in sustainable development, is a reasonable price of the externality of environment, that is, the shadow price which is the premise of GGDP accounts theory.Then, in the second chapter, we first introduce some classical economical problems, included input-output, linear programming and stable general equilibrium.Then we show the common property of theirs, that is, they can be integrated into the theory of geometry intersection. Furthermore, we give some properties of Bernstein parameter model, owning property of keeping shape in computational geometry, and classical methods in solving stable general equilibrium.These would pave the way of constructing related theory of geometry intersection in the sequence. In this chapter, we mainly discuss the method for solving stable general equilibrium. We divided the stable general equilibrium problem into two kind of mathematical models, the first is of the explicit functionsof supply and demand, and the second is only of the discrete data.We give the following General skeleton of Geometric Intersection Theory:to solve the distinct root of the nonlinear polynomial system by using pure geometric method.1. To construct the equation of nonlinear polynomial system by fitting and approximation;2. Consider the root of the nonlinear polynomial system the same as the super surface in the space of one more dimension;3. Change the above super surface into Bezier surface by basis transformation and normalization;4. Turn the solving the nonlinear system into the geometric intersection of some Bezier surface and one super plane by using the linear precision property of Bernstein Model;5. To get the distinct root of nonlinear system by computing the convex hull, subdivision, degree elevation and other pure geometric method.The above algorithm can be applied in the case of continuous function and the discrete data.We give a precise error estimation in the case of approaching the continuous function by Bernstein Model, the corresponding approximation Theorem is:Theorem 1. Suppose that the intersection of y = f₁(t) and y = f₂(t) is x_I = (t_I,f_i(t_I)), and denote f = f₁-f₂, the corresponding Bernstein approachingpolynomials are B_n1(f₁;t) and B_n2(f₂;t), the intersection of them is x_B = (t_B, B_ni(f_i;t_B), we can assume n₁ = n₂ and denote n = n₁= n₂ by degree elevation. Thus B_n(f;t) = B_n(f_i;t) - B_n(f₂;t). We can obtainSimilarly , we can adopt the fitting method to change the general equilibrium into the geometric intersection of Bernstein Model by dealing with the discrete data. Then we introduce the PP algorithm in geometric intersection, the algorithmis of geometric intuition, good stability and without the need of exact initial value. Combing it with the classical method for general equilibrium, can adopt the virtue of the Global method and Local method, no need of exact initial value and a fast computational complexity.Also, we obtain a amasing fact by approaching the production function with Bernstein Model:Theorem 2. If the effect function is of the formThen the following optimization with constraintscan obtain the linear expenditure systemwhere I is identity operator, E_i are moving operator of the ith direction with controlled vertex b_0…0 andΔ_i is the difference operator of the ith direction with controlled vertex b_0…0. In the Chapter 3, we discuss the computation of shadow price of environmentalexternality. After the introduction, we give our own computational formulathe formula is the application of geometric intersection theory in the measure theory.In Chapter 4, we review the environmental input and output and computationalgeneral equilibrium and then combine the environmental factor with multi-department and dynamic economic system, some new algorithm such as super plane algorithm is obtained in this chapter.