A Deterministic and Linear Model of Dynamic Optimization

Somdeb Lahiri

Papers from arXiv.org

Abstract: We introduce a model of infinite horizon linear dynamic optimization and obtain results concerning existence of solution and satisfaction of the competitive condition and transversality condition being unconditionally sufficient for optimality of a trajectory. We also show that under some mild restrictions the optimal trajectory satisfies the Euler condition and a related transversality condition. We show that the optimal value function is concave and continuous and the optimal trajectory satisfies the functional equation of dynamic programming. Linearity bites when it comes to the definition of optimal decision rules which can no longer be guaranteed to be single-valued. We show that the optimal decision rule is an upper semi-continuous correspondence. For linear cake-eating problems, we obtain monotonicity results for the optimal value function and a conditional monotonicity result for optimal decision rules. We also introduce the concept of a two-phase linear cake eating problem and obtain a necessary condition that must be satisfied by all solutions of such problems. We show that for a class of linear dynamic optimization problems, known as interlinked linear dynamic optimization problems, a slightly modified version of the functional equation of dynamic programming is satisfied.

Date: 2025-02, Revised 2025-03
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