Influence of production function parameters on the solution and value function in optimal control problem

The paper deals with economical growth models [4, 19] and corresponding optimal control problems with infinite time horizon [2, 14, 17]. The production function for the model of economic growth is selected from the class of exponential functions of the Cobb–Douglas type (a nonlinear model). For this class, solutions of the corresponding optimal control problems are constructed within the framework of the Pontryagin maximum principle and their asymptotic behavior is investigated. Namely, trends of value functions are analyzed when the elasticity coefficient of the Cobb–Douglas production function tends to unit. In the limit case when the elasticity coefficient equals to unit the optimal control problem has several specific features: (1) the production function transforms to a linear function; (2) the property of strictly concavity of the production function is disappeared; (3) optimal trajectories and the value function for the linear optimal control problem can be constructed analytically. It is shown that optimal trajectories and value functions of non-linear models converge to corresponding solutions of the linear model when the elasticity coefficient grows up to unit.
It is investigated changes of qualitative properties of Hamiltonian systems and the series of value functions when the elasticity parameter grows. The paper is completed by results of numerical experiments.