We obtain a recursive formulation for a general class of optimization problems with forward‐looking constraints which often arise in economic dynamic models, for example, in contracting problems with incentive constraints or in models of optimal policy. In this case, the solution does not satisfy the Bellman equation. Our approach consists of studying a recursive Lagrangian. Under standard general conditions, there is a recursive saddle‐point functional equation (analogous to a Bellman equation) that characterizes a recursive solution to the planner's problem. The recursive formulation is obtained after adding a co‐state variable μ t summarizing previous commitments reflected in past Lagrange multipliers. The continuation problem is obtained with μ t playing the role of weights in the objective function. Our approach is applicable to characterizing and computing solutions to a large class of dynamic contracting problems.

Standard optimal Debt Management (DM) models prescribe a dominant role for long bonds and advocate against issuing short bonds. They require very large positions in order to complete markets and assume each period that governments repurchase all outstanding bonds and reissue (r/r) new ones. These features of DM are inconsistent with U.S. data. We introduce incomplete markets via small transaction costs which serves to make optimal DM more closely resemble the data : r/r are negligible, short bond issuance substantial and persistent and short and long bonds positively co-vary. Intuitively, long bonds help smooth taxes over states and short bonds over time. Solving incomplete market models with multiple assets is challenging so a further contribution of this article is introducing a novel computational method to find global solutions.