THE C-K MODEL WITH SOLVABLE ENDOGENOUS FERTILITY

§ 1　 IntroductionThe relationship of population and economic growth is one of the most importantproblems in modern economic growth theory.Establishing a mathematical model theoreti-cal to study the interrelation of population and economic growth becomes a fundamentalmethod in this field[1～ 7] .In the papers[8,9] ,we obtain the especial C-K models with solv-able endogenous fertility under the specified strongly additive utility function and Cobb-Douglas production function.In the models,th…     

可解内生生育率的C-K模型

Abstract. In this paper,a C-K model with solvable endogenous fertility under the strongly addi-tive utility function is presented. The discrimination conditions of the existence of the nonzerosteady states are given. Under a kind of utility function and production function,we prove thatthese conditions are satisfied and the economy at least has an optimal growth path. The position-al relationship of the multiple steady states on the plane is discussed when multiple steady statesand multiple growth paths exist. By numerical analysis ,the fertility decreses with the per capitacapital and per capita consumption increasing and increases with the per capita capital and percapita consumption decreasing on the economic growth path are obtained.     

含内生生育率的RCK模型

本文在经典的Ｒａｍｓｅｙ－Ｃａｓｓ－Ｋｏｏｐｍａｎｓ（ＲＣＫ）经济增长模型中 内生生育率、建立了一个能刻划经济增长和人口增长相互作用的ＲＣＫ模型,证明此模型在生产函数满足一定条件下存在最优增长路径和稳态解。     

一般Rebelo增长模型的动态分析

在本文中，我们考虑一个一般Rebelo增长模型：虽然在这一般模型中要求均衡增长轨道与均衡值是更困难的，但我们通过精巧的计算和动态分析，引出了唯一可行轨道及均衡值，且分析了参数对它的影响，当时间偏好率（折现因子增大时，分配在物资生产部门的物资资本与人力资本的比重会增大。而物资资本，人力资本及总生产量的增长率都会相应减少。     

具有人力资本和科技进步的经济增长模型研究

通过引入新的变量,将二维非自治动力系统转化为二维自治的动力系统,得到包含教育,实物资本,人口和技术进步的经济增长模型.证明二维自治动力系统存在唯一的平衡点且为结点.对于给定的初始状态,模型描述的经济存在一条增长路径.通过对模型动态性质的讨论,得出维持经济长期增长的关键因素是技术进步.最后,讨论了死亡率、折旧率和对教育的投资对经济增长的影响,得出在相同的生育率的情况下,死亡率越低,经济增长越快;低折旧率对应于高经济增长,加大教育的投入也可有效地促进经济增长.     

存在人口增长极限下的改进Ramsey模型

在存在人口增长极限的假设下,对标准的Ramsey模型进行了修正,结果表明:首先,与常数人口增长率的Ramsey模型不同,模型的人均消费和人均资本并不一定随时间的推移而递增;其次,人均消费和人均资本不存在平衡增长路径均衡,但是存在渐进平衡增长路径均衡;再次,在外生参数满足一定条件的前提下,人均消费和人均资本渐近于均衡的过程是单调的.     

Economic growth and fertility rate cycles

This paper presents the global characterization of paths of optimal capital accumulation and fertility rate in a model of economic growth with endogenous fertility. The conditions for per capita capital to oscillate, to move monotonically, or to jump to steady state in the second period depend on the income elasticity of demand for children and the income elasticity of marginal utility of income. Movements in the optimal fertility rates are found to follow the movements in optimal capital to a certain extent.     

含公共开支的两部门内生经济增长模型

本文研究了一个含公共开支的两部门的内生经济增长模型 ,其生产函数具有最一般的形式 .用于两部分的三要素—物质资本 ,人力资本 ,有效劳动的份额正相关 ,产出用于公共开支的份额与用于生产部门的三要素的份额负相关 ,经济增长极大地依赖于生产弹性 .并给出均衡值为正的条件和经济系统沿二维稳定流形收敛于稳定点的条件     

THE DYNAMICS OF FERTILITY IN C-K MODEL WITH ENDOGENOUS FERTILITY

lhtroduCtlonThe relatloDshlp bet W6W6en the economic gr OWOWth and population growth Is the one of mostimportant problems In the economic growth theorem．In the empirical analyses,the level offertility and the level ofincomeper caplta Is one of*the strongest observably negative correl砒ions,In cross－country dataIS]．Constructing a mathematical model to confirm the empirical indingsIs a m叼or method In thetheoretical study、BeckerI6];GalorI7]and TaTamuraW use OLG modelto discuss this…     

我国省际经济发展水平的拐点及转换速度分析——基于平滑转换回归模型

本文利用LSTR模型对我国31个省1978-2008年的人均GDP分别进行了模型拟合,根据模型参数确定了各省人均GDP的拐点及转换速度,并将31个省分成发展历程接近的5个组。结果显示,经济发达的东部省份的人均GDP转换速度最快,中、西部省份的人均GDP的转换速度没有显著差别、它们的分组是混合在一起的;1994-1995年是东部发达地区人均GDP拐点出现的时间,2003-2005年是中、西部欠发达地区人均GDP拐点出现的时间。     

具有非线性传染率和时滞的随机SIQR计算机病毒模型的渐近行为

研究了一类具有非线性发生率和时滞的随机SIQR计算机病毒模型.首先证明了该系统具有唯一的全局正解,然后通过构造适当的Lyapunov函数并利用伊藤公式,分析了该模型的解在无病平衡点附近及地方病平衡点附近的渐近行为,最后通过数值模拟对随机系统解的渐近行为做了进一步的分析并给出了结论.     

Global bifurcation of positive equilibria in nonlinear population models

Existence of nontrivial nonnegative equilibrium solutions for age-structured population models with nonlinear diffusion is investigated. Introducing a parameter measuring the intensity of the fertility, global bifurcation is shown of a branch of positive equilibrium solutions emanating from the trivial equilibrium. Moreover, for the parameter-independent model we establish existence of positive equilibria by means of a fixed point theorem for conical shells.     

Construction of a regulator for the hamiltonian system in a two-sector economic growth model

We consider an optimal control problem of investment in the capital stock of a country and in the labor efficiency. We start from a model constructed within the classical approaches of economic growth theory and based on three production factors: capital stock, human capital, and useful work. It is assumed that the levels of investment in the capital stock and human capital are endogenous control parameters of the model, while the useful work is an exogenous parameter subject to logistic-type dynamics. The gross domestic product (GDP) of a country is described by a Cobb-Douglas production function. As a utility function, we take the integral consumption index discounted on an infinite time interval. To solve the resulting optimal control problem, we apply dynamic programming methods. We study optimal control regimes and examine the existence of an equilibrium state in each regime. On the boundaries between domains of different control regimes, we check the smoothness and strict concavity of the maximized Hamiltonian. Special focus is placed on a regime of variable control actions. The novelty of the solution proposed consists in constructing a nonlinear stabilizer based on the feedback principle. The properties of the stabilizer allow one to find an approximate solution to the original problem in the neighborhood of an equilibrium state. Solving numerically the stabilized Hamiltonian system, we find the trajectories of the capital of a country and labor efficiency. The solutions obtained allow one to assess the growth rates of the GDP of the country and the level of consumption in the neighborhood of an equilibrium position.     

An Improved Solow—Swan Model

§1.　IntroductionThetheoryofeconomicgrowthisanimportanttopicinMacroeconomics.Afterador-mancynearlytwodecades,thestudyofeconomicgrowthbecamevigorousagaininthemid-1980s.TheSolow-Swanmodel[1]isoneofthemostusedinthisfield.Thismodelhasarousedmanyresearchersinterestintherecentyears.N.G.Manki,etal(1992)[2]appliedthismodeltoexaminewhetheritisconsistentwiththeinternationalvariationinthestandardofliving.W.Easterly(1993)[3]gavearevisedSolow-SwanmodelbyintroducingtheCESproductionfunction.Inthepaper…     

An equilibrium model of resource scarcity

A new equilibrium model of resource scarcity is discussed, whose solution is the intersection point of two maps: the gradient (subgradient) of the sensitivity function of a convex programming problem and the gradient (subgradient) of an a priori given resource cost function. In fact, we compute the equilibrium consumption volumes of resources with equal internal and external marginal costs. The proposed method of computing the equilibrium solution of the model reduces to solving a saddle-point problem. Convergence of the proposed method is proved.     

具有固定时滞的经济周期模型的动态分析

在经济活动中,投资行为和资本存量存在一定的时滞效应,这会影响经济周期模型的动态行为,进而使得投资政策对经济的稳定调整复杂化.考虑到资本存量的预期时间以及投资时滞对经济活动的影响,采用Hopf分岔理论,研究具有固定时滞的经济周期模型的均衡点的稳定性以及形成经济周期的条件.研究发现,投资过程中的投资时滞,以及投资决策中对于资本存量的预测时间构成经济周期产生的诱因;同时可通过政府投资政策调整达到预期均衡目标,这对保持经济周期稳定及经济政策制定有一定的指导作用.     

Oligopoly and stability

In this paper, the so called Theocharis–Cournot problem is reconsidered. It concerns the relation between oligopoly and perfect competition, in particular the destabilization of Cournot equilibrium when the number of competitors increases. Using a CES production function where one input, capital, is fixed during periods of investment, a mixed short/long run market dynamics is set up. In the short run, with capital fixed, there is a capacity limit for production possibilities, whereas, at moments of capital renewal there are constant returns to scale. In this setting the local stability of Cournot equilibrium is reconsidered. It is demonstrated that if no more than two firms reinvest in the same time period, and the wage rate is not too high, then the Cournot equilibrium is stable.     

高薪养廉的效用模型分析

针对我国近年来提出的"高薪养廉"观点,采用VNM效用函数建立了相应的数学模型.通过分析,找到了一个满足一定约束的均衡点(y_1~*,y_2~*),此点即达到了高薪养廉的目的.但在进一步的讨论中,发现均衡点(y_1~*,y_2~*)有很大的局限性,且模型显示出加强监督与惩罚力度的重要性.     

Saddle point equilibrium under uncertain environment

The saddle point equilibrium problem is of great importance in differential game theory. In this paper, an optimistic value model for the saddle point equilibrium problem under uncertain environment is investigated. The equilibrium equation for the proposed model is presented. Then a linear quadratic model is discussed. Finally, a counter terror problem is analyzed by the results obtained in the paper.     

具有垂直传染和接触传染的传染病模型的稳定性研究

本文研究了一类具有垂直传染和接触传染的传染病模型.利用常微分方程定性与稳定性方法,分析了该模型非负平衡点的存在性及其局部稳定性.同时,利用LaSalle不变性原理和通过构造适当的Lyapunov函数,获得了平凡平衡点、无病平衡点和地方病平衡点全局渐近稳定的充分条件.结果表明当基本再生数小于等于1时,所有种群趋于灭绝;当基本再生数大于1和病毒主导再生数小于1时,病毒很快被清除;当基本再生数大于1和病毒主导再生数大于1以及满足一定条件时,病毒持续流行并将成为一种地方病.