Isotone Recursive Methods: The Case of Homogeneous Agents

Abstract

Standard-Nutzungsbedingungen:Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden.Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen.Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in AbstractOver the last decade, isotone recursive methods have provided uni…ed catalog of results on existence, characterization, and computation of Markovian Equilibrium Decision Processes (MEDPs) in in…-nite horizon economies where the second welfare theorem fails. Such economies include models with production nonconvexities, taxes, valued …at money, models with monopolistic competition, behavioral heterogeneity, and incomplete markets. In this paper, we survey this emerging class of methods. Our methods use a qualitative approach to economic equilibria …rst introduced in the work in operations research by Veinott and Topkis. As the methods emphasize the role of order, they are amenable for obtaining conditions for monotone comparison theorems on the space of economies. We are also able to describe monotone iterative procedures that provide the needed foundations for a theory of numerical solutions for MEDPs and stationary Markov equilibrium (SME). One interesting additional result of independent interest is we construct su¢ cient conditions for the existence of a new class of envelope theorems for nonconcave programming problems.Email addresses: Manjira.Datta@asu.edu, Kevin.Re¤ett@asu.edu. We are deeply indebted to Len Mirman and Olivier Morand for numerous lengthy discussions concerning many issues discussed in this survey. Many of the results presented in this paper were developed originally in some form during our joint work with Len and Olivier over the last …ve years. We dedicate this paper to Len Mirman on the occasion of his sixty-…fth birthday. Indeed, this paper would not have been written without Len's ongoing pioneering work on equilibrium growth under uncertainty. We also thank Elena Antoniadou, Hector Chade, John Coleman, Jeremy Greenwood, Seppo Heikkila, Ken Judd, Tom Krebs, Cuong Le Van, Robert Lucas, Jr., Jianjun Miao, Chris Shannon, John Stachurski, Yiannis Vailakis, Charles Van Marrewijk, Jean-Marie Viaene, Itzhak Zilcha, and especially Robert Becker and Manuel Santos for many helpful conversations over the past years. All mistakes remain our own.

Recursive Utility and Thompson Aggregators, Ii: Uniqueness of the Recursive Utility Representation

“…They recast the result in a more general form than the nonlinear eigenvalue problem motivating Krasnosel'ski¼ ¬'s seminal work. 9 Datta and Re¤ett [18] survey monotone operator methods in recursive models and include uniqueness arguments for Coleman's model based on u 0 concave operator methods. Le Van, Morhaim, and Vailakis [26] prove a version of Krasnosel'ski¼ ¬'s Theorem that also includes a monotone operator existence result.…”

“…Condition (T 6) turns out to be an important joint restriction on the preferences embodied in the aggregator function as well as on the underlying commodity space, as might arise from properties of technologies in production economies and/or endowments in exchange economies. 18 In our setting the commodity space excludes sustainable growth and the parameter = 1.…”

Intergenerational Interactions in Human Capital Accumulation

“…One important caveat is that in our current case, the appropriate operator whose fixed points are the Markov perfect equilibria of the considered economy turns out to be decreasing while the relevant theorems summarized by Datta and Reffett (2006) deal with increasing operators only. Hence, we cannot apply the established methods directly: the appropriate theorems must be generalized first.…”

“…Technically, to show existence and uniqueness of Markov perfect equilibrium (MPE) in an economy with strategic interactions we use the method of monotone operators pioneered by Coleman (1991) and developed further by Datta and Reffett (see a survey of these methods in Datta and Reffett (2006)). One important caveat is that in our current case, the appropriate operator whose fixed points are the Markov perfect equilibria of the considered economy turns out to be decreasing while the relevant theorems summarized by Datta and Reffett (2006) deal with increasing operators only.…”