Treating the Future Equally: Solving Undiscounted Infinite Horizon Optimization Problems

TL;DR

This paper addresses the challenges of solving infinite horizon optimization problems without discounting, proposing conditions under which finite horizon solutions converge to optimal infinite horizon paths, with applications to economic models.

Contribution

It introduces general conditions ensuring finite horizon solutions approximate infinite horizon optima, and applies these to a Ramsey model demonstrating unique path properties.

Findings

Finite horizon solutions converge to infinite horizon optima under certain conditions.

The derived paths in the Ramsey model exhibit intriguing properties.

Legitimacy of solutions is justified through non-standard arguments.

Abstract

Infinite horizon optimization problems accompany two perplexities. First, the infinite series of utility sequences may diverge. Second, boundary conditions at the infinite terminal time may not be rigorously expressed. In this paper, we show that under two fairly general conditions, the limit of the solution to the undiscounted finite horizon problem is optimal among feasible paths for the undiscounted infinite horizon problem, in the sense of the overtaking criterion. Applied to a simple Ramsey model, we show that the derived path contains intriguing properties. We also comprehend the legitimacy of the derived paths by addressing the perplexities with non-standard arguments.