Efficient Computations

Abstract

This chapter describes fast algorithms for computing the value function and optimal decision rule for the type of social planning problem to be described in Chapter 5. The chapter is organized as follows. First, it displays a transformation that removes both discounting and cross-products between states and controls. It then describes invariant subspace methods for solving an optimal linear regulator problem. Next, it describes a closely related method called the doubling algorithm that effectively skips steps in iterating on the Bellman equation. The calculations can be further accelerated by partitioning the state vector to take advantage of patterns of zeros in various matrices that define the problem. This is followed by discussions of fast methods for computing equilibria for periodic economies; the rapid solution of a periodic optimal linear regulator problem; and how the calculations can be adapted to handle Hansen and Sargent's (1995) recursive formulation of Jacobson's and Whittle's risk-sensitive preferences.