Stabilizing the Hamiltonian system for constructing optimal trajectories

An infinite-horizon optimal control problem based on an economic growth model is studied. The goal in the problem is to optimize the mechanisms of investment in basic production assets in order to increase the growth rate of the consumption level. The main output variable-the gross domestic product (GDP)-depends on three production factors: capital stock, human capital, and useful work. The first two factors are endogenous variables of the model, and the useful work is an exogenous factor. The dependence of the GDP on the production factors is described by the Cobb-Douglas power-type production function. The economic system under consideration is assumed to be closed, so the GDP is distributed between consumption and investment in the capital stock and human capital. The optimal control problem consists in determining optimal investment strategies that maximize the integral discounted relative consumption index on an infinite time interval. A solution to the problem is constructed on the basis of the Pontryagin maximum principle adapted to infinite-horizon problems. We examine the questions of existence and uniqueness of a solution, verify necessary and sufficient optimality conditions, and perform a qualitative analysis of Hamiltonian systems on the basis of which we propose an algorithm for constructing optimal trajectories. This algorithm uses information on solutions obtained by means of a nonlinear regulator. Finally, we estimate the accuracy of the algorithm with respect to the integral cost functional of the control process.     

A positional game for an overlapping generation economy

We develop a model with intra-generational consumption externalities, based on the overlapping generation model by Diamond (1965). More specifically, we consider a two-period lived overlapping generation economy, assuming that the utility of each consumer depends also on the average consumption level by the consumers in the same generation. We suppose that such level is not taken as a parameter by agents, who behave strategically. We characterize the consumption and saving choices for the two periods in the Nash equilibrium path and we determine a dynamic equation for capital accumulation. For the associated dynamical system, we find a unique positive steady state for capital and we investigate how its position, as well as that of the steady states for consumption in both periods, change with respect to variations in the degree of interaction in the two periods. We finally compare the steady states for capital with and without social interaction.     

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.     

Modeling of optimal investment in science and technology

The latest achievements in science and technology lead to the development of new and more productive capital that can essentially increase a company’s profit. On the other hand, companies should invest not only in the productive capital, but also in science and technology.

The optimal control of an economic system that divides its output among the production of consumption goods, the accumulation of new capital, and the contribution to science and technology is considered. The model is expressed as nonlinear integral equations with unknowns in the integrands and lower limits of integration. An optimization problem for the profit maximization is suggested. The necessary condition for an extremum and the second variation of the functional are derived. The structure of optimal solutions is analyzed. Interpretation of all results is provided.     

A stochastic control model of investment,production, and consumption on a finite horizon

In this paper, we consider a stochastic control problem on a finite time horizon. The unit price of capital obeys a logarithmic Brownian motion, and the income from production is also subject to the random Brownian fluctuations. The goal is to choose optimal investment and consumption policies to maximize the finite horizon expected discounted hyperbolic absolute risk aversion utility of consumption. A dynamic programming principle is used to derive a time‐dependent Hamilton–Jacobi–Bellman equation. The Leray–Schauder fixed point theorem is used to obtain existence of solution of the HJB equation. At last, we derive the optimal investment and consumption policies by the verification theorem. The main contribution in this paper is the use of PDE technique to the finite time problem for obtaining optimal polices. Copyright © 2014 John Wiley & Sons, Ltd.     

人力资本、自然资本与最优经济增长要素分析

通过把人力资本、自然资本与社会生产总量联系起来 ,由消费物资多少、获得知识多少及自然资本存量大小这三因素来确定效用 ,提供了物质、人力和自然资本三种重要资产增长的模型框架 ,得到了经济的消费增长率、均衡时各种资本的增长率之间的关系和实现最优经济增长的控制策略 .     

Productive public expenditures, expectation formations and nonlinear dynamics

In this paper, we investigate the dynamic properties of an overlapping generations’ model with capital accumulation and publicly funded inventions under three different expectations: perfect foresight, myopic expectations and adaptive expectations. We show that considering productive public expenditures in the model will increase the dimension of the dynamical system. To study the dynamic behavior of a high-dimensional dynamical system, we focus on the case when the elasticity of publicly funded invention to output is small and approximate the system by using a one-dimensional dynamical system. This approximation method provides an efficient way to rigorously prove the existence of chaos in high-dimensional dynamical systems. We show that when agents are perfectly foresighted, there exists a unique, nontrivial steady state which is a global attractor. Cycles or even chaos may occur under myopic and adaptive expectations when the inter-temporal elasticity of substitution of consumption is large enough. Furthermore, we find that the impact of fiscal policy is sensible to the expectation formation.     

Optimal control and long-run dynamics for a spatial economic growth model with physical capital accumulation and pollution diffusion

In this work we analyze the large-time behavior of a spatially structured economic growth model coupling physical capital accumulation and pollution diffusion. This model extends other results in the literature along different directions. Alongside the classical Cobb–Douglas production function, a convex–concave production function is considered. We add a negative feedback to the production function in order to describe the (negative) influence of pollution on output, and therefore on capital accumulation. We also present an optimal control problem for the above model.     

商业信用额度下易腐品最优订货策略

运作实践中,供应商不仅会给予零售商延期付款待遇以刺激其订购量,同时也会为信用期内零售商的应付账款设置上限值(即商业信用额度),从而规避零售商的应付账款对其造成的资金机会成本和违约风险。鉴于此,本文基于商业信用额度构建了零售商关于易腐品的订购决策模型。通过分析零售商资金机会成本的不同形式,可以得到零售商关于易腐品五种不同的成本函数表达式。结合数学证明可以得到零售商五种目标函数的性质,以及在商业信用额度条款下零售商最优的订购决策。算例表明供应商可以通过设置一个合理的商业信用额度条款将零售商的订购量和信用期内的应付账款控制在一个合适范围。同时商业信用额度条款有助于缓解供应商的营运资金压力,但会在一定程度上增加零售商的运营成本。     

Product innovation with lumpy investment

The paper provides a framework that enables us to analyze the important topic of capital accumulation under technological progress. We describe an algorithm to solve Impulse Control problems, based on a (multipoint) boundary value problem approach. Investment takes place in lumps and we determine the optimal timing of technology adoptions as well as the size of the corresponding investments. Our numerical approach led to some guidelines for new technology investments. First, we find that investments are larger and occur in a later stadium when more of the old capital stock needs to be scrapped. Moreover, we obtain that the size of the firm’s investments increase when the technology produces more profitable products. We see that the firm in the beginning of the planning period adopts new technologies faster as time proceeds, but later on the opposite happens. Furthermore, we find that the firm does not invest such that marginal profit is zero, but instead marginal profit is negative.     

Growth-security profiles in capital accumulation under risk

This paper considers the tradeoff between growth and security in the problem of capital accumulation under risk. It is shown how growth can be continuously traded for security with simple deterministic strategies generated from the optimal growth and optimal security problems. A lower bound is derived for the error resulting from the use of such strategies.     

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.     

Optimal Consumption and Portfolio with Ambiguity to Markovian Switching

This paper considers the problem of maximizing expected utility from consumption and terminal wealth under model uncertainty for a general semimartingale market, where the agent with an initial capital and a random endowment can invest. To find a solution to the investment problem we use the martingale method. We first prove that under appropriate assumptions a unique solution to the investment problem exists. Then we deduce that the value functions of primal problem and dual problem are convex conjugate functions. Furthermore we consider a diffusion-jump-model where the coefficients depend on the state of a Markov chain and the investor is ambiguity to the intensity of the underlying Poisson process. Finally, for an agent with the logarithmic utility function, we use the stochastic control method to derive the Hamilton-Jacobi-Bellmann (HJB) equation. And the solution to this HJB equation can be determined numerically. We also show how thereby the optimal investment strategy can be computed.     

基于营运资金信息匹配平台的供应链金融动态折扣决策研究

当前逐渐受到实业界关注的一种新型供应链金融模式是基于营运资金信息匹配平台的动态折扣。本文针对两级供应链的动态折扣决策问题进行建模,揭示了动态折扣的应用对于改善供应链参与方现金流的影响机理。首先通过考虑供需双方动态折扣的独立决策,得出日折扣率的边界条件、双方效用最大时的日折扣率、买方混合还款方式下营运资金的最优准备方案和最低边界值;其次考虑供需双方动态折扣的最优决策,推导出在不同折扣率和利率关系下的最优还款策略,研究表明动态折扣可以明显提升供需双方的利润情况。     

The Tradeoff between Consumption and Investment in Incomplete Financial Markets

In this paper we are concerned with the tradeoff between long term growth of the expected utility of wealth and consumption. The goal is to find a consumption policy for which the optimal rate of capital growth is zero, i.e. a policy for which balance between consumption and investment is reached. The asymptotic limit of this investment problem when the HARA parameter γ → -∝ is also studied.     

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.     

Optimal control of a big financial company with debt liability under bankrupt probability constraints

This paper considers an optimal control of a big financial company with debt liability under bankrupt probability constraints. The company, which faces constant liability payments and has choices to choose various production/business policies from an available set of control policies with different expected profits and risks, controls the business policy and dividend payout process to maximize the expected present value of the dividends until the time of bankruptcy. However, if the dividend payout barrier is too low to be acceptable, it may result in the company’s bankruptcy soon. In order to protect the shareholders’ profits, the managements of the company impose a reasonable and normal constraint on their dividend strategy, that is, the bankrupt probability associated with the optimal dividend payout barrier should be smaller than a given risk level within a fixed time horizon. This paper aims at working out the optimal control policy as well as optimal return function for the company under bankrupt probability constraint by stochastic analysis, partial differential equation and variational inequality approach. Moreover, we establish a riskbased capital standard to ensure the capital requirement can cover the total given risk by numerical analysis, and give reasonable economic interpretation for the results.     

Irreversible capital accumulation under interest rate uncertainty

We consider the optimal sequential irreversible investment policy of a value maximizing firm facing decreasing returns to scale and interest rate uncertainty. We characterize the optimal accumulation policy and its value for a broad class of diffusion models of the short interest rate by focusing on the marginal investment decision and deriving the marginal value of capital explicitly. We also state a set of conditions under which there is a maximal capital stock above which the option to expand productive capacity further in the future becomes valueless. Hence, our results indicate that interest rate uncertainty may limit the size of an optimally investing firm.     

Change of economic structure-Capital, knowledge, population and pollution

This paper proposes a nonlinear dynamic economic model with endogenous technological change, population, capital accumulation and pollutant accumulation. The system consists of two, industrial and environmental, sectors. The industrial sector produces goods with knowledge, labor, and capital as inputs under perfect competitive conditions, subject to the government's tax policy. The tax income is used by the environmental sector to employ labor and capital. We examine the effects of the environmental policy, knowledge accumulation efficiency and preference change on the environment. We show that the simple system has multiple equilibria and may exhibit permanent oscillations.