A Deterministic and Linear Model of Dynamic Optimization

TL;DR

This paper develops a comprehensive linear dynamic optimization model for infinite horizon problems, establishing conditions for optimality, solution existence, and properties of optimal trajectories and decision rules, with applications to cake-eating problems.

Contribution

It introduces a deterministic linear model for infinite horizon dynamic optimization, analyzing solution conditions, optimality criteria, and decision rule properties, including new results for cake-eating and interlinked problems.

Findings

Optimal trajectories satisfy the Euler and transversality conditions.

Under convexity, the value function is concave and continuous.

Optimal decision rules form an upper semi-continuous correspondence.

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. The optimal trajectory satisfies the functional equation of dynamic programming. Under an additional convexity assumption for the two-period constraint sets, we show that the optimal value function is concave and continuous. 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 if all the two-period constraint sets are convex, then 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.