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.

We use a model of boundedly rational learning to account for the observations of recurrent hyperinflations in the 1980’s. In a standard monetary model we replace the assumption of full rational expectations by a formal definition of quasi-rational learning. The model under learning matches some crucial stylized facts observed during the recurrent hyperinflations experienced by several countries in the 1980’s remarkably well. We argue that, despite being a small departure from rational expectations, quasi-rational learning does not preclude falsifiability of the model, it does not violate reasonable rationality requirements, and it can be used for policy evaluation.

We study the convergence of recursive least-squares learning schemes in economic environments in which there is private information. The presence of private information leads to the presence of hidden state variables from the viewpoint of particular agents. By applying theorems of Ljung, we extend some of our earlier results to characterize conditions under which a system governed by least-squares learning will eventually converge to a rational expectations equilibrium. We apply insights from the learning results to formulate and compute the equilibrium of a version of Townsend's model.

In an economy studied by Lucas and Stokey, tax rates inherit the serial correlation structure of government expenditures, belying Barro's earlier result that taxes should be a random walk for any stochastic process of government expenditures. To recover a version of Barro's random walk tax-smoothing outcome, we modify Lucas and Stokey's economy to permit only risk-free debt. Having only risk-free debt confronts the Ramsey planner with additional constraints on equilibrium allocations beyond one imposed by Lucas and Stokey's assumption of complete markets. The Ramsey outcome blends features of Barro's model with Lucas and Stokey's. In our model, the contemporaneous effects of exogenous government expenditures on the government deficit and taxes resemble those in Lucas and Stokey's model, but incomplete markets put a nearunit root component into government debt and taxes, an outcome like Barro's. However, we show that without ad hoc limits on the government's asset holdings, outcomes can diverge in important ways from Barro's. Our results use and extend recent advances in the consumption-smoothing literature.