**Title**

On the Principle of Optimality for Nonstationary Deterministic Dynamic Programming

**Abstract**

This note studies a general nonstationary infinite-horizon optimization problem in discrete time. We allow the state space in each period to be an arbitrary set, and the return function in each period to be unbounded. We do not require discounting, and do not require the constraint correspondence in each period to be nonempty-valued. The objective function is defined as the limit superior or inferior of the finite sums of return functions. We show that the sequence of time-indexed value functions satisfies the Bellman equation if and only if its right-hand side is well defined, i.e., it does not involve -∞+∞.

Keywords: Bellman equation, dynamic programming, principle of
optimality, value function

JEL Classification: C61, O41

Takashi KAMIHIGASHI

Research Institute for Economics and Business Administration

Kobe University

Rokkodai-cho, Nada-ku, Kobe

657-8501
Japan

Phone: (81) 78 803 7036

Fax: (81) 78 803 7059