Which recursive equilibrium?

Since the seminal work of Kydland and Prescott and Abreu, Pearce, and Stacchetti, researchers have sought to develop correspondence-based monotone continuation methods for constructing pure strategy sequential (subgame perfect) equilibrium, and/or pure strategy Markov perfect Nash equilibrium in classes of dynamic games. These are "strategic dynamic programming" methods (or, the so-called "APS approach") for mapping between spaces of correspondences to (i) verify the existence of subgame perfect Nash equilibrium, as well as (ii) suggesting explicit methods for computing approximate solutions. In the last decade, economists have attempted to extend these APS methods to study competitive equilibrium in dynamic general equilibrium models. In this line of work, emphasis has both been on analyzing sequential competitive equilibrium, as well as Markovian or recursive equilibrium. In this paper, we reconsider recent results reported in this emerging literature using "APS methods" for the existence and computation of recursive/Markov competitive equilibrium in nonoptimal competitive economies using function-based APS methods. And, to keep things simple, we consider these questions in the setting of a simple one-sector nonoptimal growth model with a state-contingent tax. We find several interesting results. First, we extend the uniqueness result for continuous Markov equilibrium for the policy iteration method proposed by Coleman to a larger class of functions (i.e., spaces of bounded functions). However, despite this generalization, there exist other fixed point procedures that potentially construct continuous Markov equilibrium that exist outside this set. This result shows the delicate nature of existing uniqueness results in the literature even for the simplest nonoptimal models. Next, we extend Coleman's policy iteration approach to prove existence of (locally Lipschitz) continuous recursive equilibrium in economies previously thought not to possess them. Specifically, we show the delicate nature of the existing correspondence-based continuation APS methods. In general, these APS methods do not verify the existence of recursive equilibrium (even for simple one-dimensional cases). Also, using constructive arguments, we show that even when existence of Markov equilibrium is known, the solutions to the abstract functional equations considered in the APS methods of Miao and Santos admit solutions or selections that are not necessarily Markov equilibrium. This is a serious problem for numerical work. In particular, even when existence of Markov equilibrium selections exist, our results show that current APS procedures for competitive economies do not, in general, provide a rigorous method for constructing or approximating a recursive equilibrium selection from the limiting (greatest fixed point) "equilibrium" correspondence even for very simple economies. To remedy this situation, we propose a new APS method with correspondences valued in function spaces which succeeds in verifying the existence of a recursive equilibrium. This method defines an interval approximation method (valued in function spaces) that provide, in principle, an explicit method for computing and characterizing continuous Markov equilibrium.