Abstract: |
One important goal in the study of economic models is to make optimal decision on a time series of consumptions that can bring forth a maximal expected level of satisfaction brought subject to the constraint of available resources which themselves are determined by the current saving and the production. The optimization problem, referred to as the consumer’s problem, involves infinitely many decisions simultaneously, which makes the solution extremely difficult to find. One ingenious idea, known as Bellman’s principle of optimality, asserts that an optimal policy should have the property that the subsequent decisions from any given initial state and decision remain optimal with regard to the state resulting from the first decision. This principle effectively reformulates the consumer’s problem into a recursive form known as the Bellman equation in the field of dynamical programming. Quite a few numerical methods have already been proposed in the literature for solving the Bellman equation. One possible approach is to tackle the first order optimality condition of the Bellman equation via the notion of Lagrange multipliers, which gives rise to the Euler equation. Under the setting of economic dynamics, the Euler equation in general is a three-term finite difference equation F(k_t; k_{t+1}; k_{t+2}) = 0 where F : R^3 \to R is some known nonlinear function. For decision-making, it is sometimes desirable to acquire the dynamics in the sequential decisions k_{t+1} = p(k_t); known the policy function, from any initial value k0. Finding the policy function for the consumer’s problem therefore means reducing the three-term recurrence relation to a two-term relation. Can this be done and how can this be accomplished? In this talk, we present a preliminary investigation of this term reduction problem in general and propose a rather simple, yet highly precise, scheme for computation in particular. |