Dynamic Optimization Methods For A Kind Of Continuous-time Economic Growth Models

The world is constantly changing in the eyes of economists.and the economic system,which is studied extensively,is also in the dynamic changes. Since Evans, Ramsey and Hotelling has analysed economic problems dynamically in 1920's, authors are always very interested in the study of this field. Economist Paul Samuel-son has said that economic problems can dynamically handle has been well known. While analyzing the economic issues, we often encounter many dynamic optimiza-tion problems,and the method of dynamic optimization can be widely used in the field of analyzing the stability of equilibrium and resolving the dynamic optimiza-tion problems. With the development of the world economy, this approach is also constantly developing,and in analyzing economic problems it has become increas-ingly important.Ramsey model which was proposed by the famous economist Frank Ramsey first has been regarded as one of the classical economic growth model.Just for the framework of macroeconomic,many authors believe that the dynamic equilibrium model which was based on Ramsey model has been become a major research and analysis paradigm of modern macroeconomics.The dynamic optimization methods for a kind of continuous-time economic growth models are described in the paper. And we state the issue in the framework of continuous-time and take the Ramsey model as a example.We introduce three methods of dynamic optimization and mainly introduce the third method. The pa-per can be divided into four parts. In the first part, we introduce the development process of the dynamic optimization theory and the methods of dynamic optimiza-tion. In the second part,the Ramsey (Ramsey-Cass-Koopmans) model,which is a classical model,is stated. In the third part,we introduce the general methods: optimal control and dynamic programming, which are used to solve the dynamic op-timization problems for the continuous-time.Then they are applied to the Ramsey model. In the fourth part, on the basis of the third part, we introduce a asymptotic method for solving dynamic equilibrium. And the basic idea of this method is stated through Ramsey model. Besides, we compare the results of this asymptotic method to the approximation results of the optimal control using some datum. At last, the error analysis between the Taylor series and Pade expansions can be stated as follows:(1) The higher-order approximating results by means of this method is better than the first-order ones.(2) The error valid range of the Pade expansions approximation is larger than the Taylor series',so the Pade approximations'effects are better than the Taylor ones. (3) The first-order approximating result around the steady state of the asymp-totic method is the same as the approximating result of the optimal control.And by the conclusion (1) we know the asymptotic method can get better approximate effect.Therefore, the validity of this algorithm can be verified.