A Divide and Conquer Algorithm for Exploiting Policy Function Monotonicity

Abstract

A divide and conquer algorithm for exploiting policy function monotonicity is proposed and analyzed. To solve a discrete problem with n states and n choices, the algorithm requires at most nlog2(n)+5n objective function evaluations. In contrast, existing methods for nonconcave problems require n2 evaluations in the worst case. For concave problems, the solution technique can be combined with a method exploiting concavity to reduce evaluations to 14n+2log2(n). A version of the algorithm exploiting monotonicity in two‐state variables allows for even more efficient solutions. The algorithm can also be efficiently employed in a common class of problems that do not have monotone policies, including problems with many state and choice variables. In the sovereign default model of Arellano (2008) and in the real business cycle model, the algorithm reduces run times by an order of magnitude for moderate grid sizes and orders of magnitude for larger ones. Sufficient conditions for monotonicity and code are provided.