turkmath.org

Discrete time optimal growth model is one of the important models of the theory of economics. This simple and elegant model is described by the presence of a social planner who maximizes the infinite sum of discounted utilities of consumption subject to a convex one sector production set. One useful approach to solve this problem is dynamic programming which is in fact the study of a dynamic optimization problem through the analysis of value functional equations. In finite horizon, these types of problems can be solved by means of Lagrange multipliers method. In infinite horizon optimal growth models, these multipliers will typically belong to an infinite dimensional decision space. Therefore, the questions whether the Lagrange multipliers exist and whether they can be represented by a summable sequence arise. These problems have been overcome by extending the Lagrangean to infinite dimensional spaces and sufficient conditions for Lagrangean to be represented by a summable sequence of multipliers are then provided. In this talk, I will present the problem in general form and give the extension of the Kuhn-Tucker theorem where the multipliers are represented in $(1^\infty)'$, the dual space of $1^\infty$, and then give sufficient conditions for having an $1^1$ representation of the multipliers.