Limit of the Solutions for the Finite Horizon Problems as the Optimal Solution to the Infinite Horizon Optimization Problems

TL;DR

This paper extends previous work by analyzing the limits of finite horizon solutions as optimal solutions for infinite horizon problems, considering time-dependent utility and production functions, and examining the effects of discounting.

Contribution

It generalizes existing results by incorporating time-dependent functions and additional criteria, clarifying conditions for optimality and uniqueness of solutions in infinite horizon models.

Findings

Limit of finite horizon solutions can be optimal for infinite horizon under certain conditions.

Conditions for the limit to be the unique optimum are established.

Application to a growth model shows how discounting affects optimal paths.

Abstract

We aim to generalize the results of Cai and Nitta (2007) by allowing both the utility and production function to depend on time. We also consider an additional intertemporal optimality criterion. We clarify the conditions under which the limit of the solutions for the finite horizon problems is optimal among all attainable paths for the infinite horizon problems under the overtaking criterion, as well as the conditions under which such a limit is the unique optimum under the sum-of-utilities criterion. The results are applied to a parametric example of the one-sector growth model to examine the impacts of discounting on optimal paths.