Non-causal models in long term planning via set contractive optimal control methods

The notion of consistency for optimal plans introduced in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491] is studied in relation to dynamic programming and to multi-objective optimal control. It is demonstrated that the consistency condition for an optimal sequence of n policies for periods from 1 to n corresponding to a sequence of n economic agents’ decisions that together constitute an optimal plan for an agreed-upon social objective function is equivalent to a multi-objective problem with n + 1 criteria. An optimal plan is consistent if and only if the corresponding ( n + 1 ) -criteria problem is balanced, and consistent optimal plans do exist in such economic situations. Otherwise, the consistency requirement generates the balance set and a set of Pareto solutions, and both can be computed as illustrated in the inflation–unemployment example extended for two periods with three objectives to optimize. The procedure of “consistent planning for the infinite horizon” considered in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491] is shown to concur with Bellman’s principle of optimality that may be invalid in many practical cases including the case considered in [F.E. Kydland, E.C. Prescott, Rules rather than decisions: The inconsistency of optimal plans, J. Polit. Econ. 85 (3) (1977) 473–491]. Finitely causal and non-causal models are introduced for long term planning in a multi-objective (Pareto) framework, and it is demonstrated how to deal with non-causal models using new set-contractive methods for multi-objective global optimization. The results are illustrated by examples.