Limitations in Direct and Indirect Methods for Solving Optimal Control Problems in Growth Theory
Abstract
The focus of this paper is on a comprehensive analysis of different methods and mathematical techniques used for solving optimal control problems (OCP) in growth theory. Most important methods for solving dynamic non-linear infinite-horizon growth models using optimal control theory are presented and a critical view of the limitations of different methods is given. The main problem is to determine the optimal rate of growth over time in a way that maximizes the welfare function over an infinite horizon. The welfare function depends on capital-labor ratio, the state variable, and the per-capita consumption, the control variable. Numerical methods for solving OCP are divided into two classes: direct and indirect approach. How the indirect approach can be used is given in the example of the neo-classical growth model. In order to present the indirect and the direct approach simultaneously, two endogenous growth models, one written by Romer and another by Lucas and Uzawa, are studied. Advantages and efficiency of these different approaches will be discussed. Although the indirect methods for solving OCP are still the most expanded in growth theory, it will be seen that using direct methods can also be very efficient and help to overcome problems that can occur by using the indirect approach.
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