Applications Of Stochastic Optimal Methods To Resources Allocation Models

The paper discusses the optimal allocation of human and physical capital in production sectors, risky investment decisions and economic growth, by using methods of stochastic optimal, stochastic differential equation and numerical analysis.Firstly, two models are considered. one is an endogenous economic growth model about allocation of physical and human capital in three sectors. Optimal allocation of physical capital and human capital in three sectors is derived, the fractions of physical and human capital in each sector are positively related. The allocation of human resources between applied technology and basic scientific research sectors are considered. By the complicate mathematic deduction, the paper reaches the following conclusion: the fraction of human resources used in applied technologies sector is higher than one in the basic scientific sector. The fraction used in the basic research sectors,are positively related to the shares of the two factors in the three sectors, the fraction used in the applied research sectors,are positively related to the shares of applied technology in the three sectors.Secondly, an economic growth model of physical and human capital accumulation under a continue-time stochastic framework, are developed. When allocation is determined enogenously, the optimal allocation decision is independent of the volatility of the stochastic variables. Volatility has a positive effects on the expected growth rates of physical and human capital, however, the effect of volatility on social well-being is negative, which is exactly unanimous to Hu's results of human capital and education model. Finally, the stability for two-dimensional stochastic system are discussed, the conditions of the stability of zero solution are given. When allocation is fixed, we investigate the stability of a two-sector economic growth model under stochastic case. A two-dimensional stochastic differential system is deduced by Ito's formula, by using Liapunov function methods, whether the growth rates of physical capital and human capital are exponentially stable or unstable depends on the values for parameters.Thirdly, the allocation of resources in R&D sector and and good sector and its economic growth, are examined. Economic growth, consumption, welfare and relations between them and volatility are discussed when allocation is determined exogenously and engenously. When allocation fixed, if relatively risky everasion coefficient is more one, the economic growth rates are related negatively to productivity disturbances, and social welfare is related negatively to productivity disturbances. When allocation is determined enogenousely, the economic growth rates are related positively to productivity disturbances, and social welfare is related negatively to productivity disturbances, which is animous to Hu's. Finally, when labor increases stochasticly, the conditions that the expected growth rates of physical capital and technology increase or reduce at exponentially rate, are given.Finally, a risky investment decision model with learning-by-doing production function under two different utility functions are discussed. Under expected utility function,by using stochastic optimal methods, we obtain expected growth rate, risk-free rate and propensity to consume at the steady state. The impact of taxation and technology parameter and technology volatility on growth rates, risk-free and consume-wealth ratio, are analyzed elaborately. Under recursive utility function which disentangles risk aversion from intertemporal elasticity of substitution, when riskless rate is exogenous variable, the optimization solution for the problem are obtained, comparing with the optimization consumption and savings under the two utility functions. The impact of the intertemporal elasticity of substitution, the coefficient of relative risk aversion, technology volatility and technology parameter on risk portfolio, propensity to consume and the expected growth rate of wealth, are analyzed elaborately, when riskless rate is enogenous variable, the fraction of the risk investment and consumption-wealth ratio increase with government expenditure and decrease with income taxation, and the expected growth rate decreases with them. The impact of the intertemporal elasticity of substitution and the risk aversion on the risk portfolio, propensity to consume and the expected growth rate, are analyzed elaborately. Compare with the effect of the increase of taxation and expenditure on economic variables, and point out the errors committed by using the constant elasticity utility function.