Implicit Programming and the Stable Manifold for Optimal Growth Problems

The stable manifold for the discrete-time, one-sector optimal growth model is shown to be constructed by solving a functional equation for an implicit programming problem based on the planner's Euler equation. The implicit programming problem is described by a minimum-gain operator, and its unique fixed-point solution is the optimal policy function. This operator was introduced by Becker and Foias (1998). The term "implicit programming" comes from the observation that the unknown optimal policy function appears in the constraints of the optimization problem as well as in the problem's value function. The implicit programming problem defines an algorithm for computing the optimal policy function. The proposed paper numerically illustrates convergence of the algorithm -- in particular, that its iterates converge to the optimal policy function. We apply the algorithm to several different economic growth models. To assess the efficiency and accuracy of the algorithm, we also solve the example models with alternative methods, such as projection and perturbation methods.