Time consistent Markov policies in dynamic economi

Authors: Łukasz Balbus Kevin Reffett Łukasz Woźny

Publish Date: 2014/04/04

Volume: 44, Issue: 1, Pages: 83-112

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Abstract

We study the question of existence and computation of timeconsistent Markov policies of quasihyperbolic consumers under a stochastic transition technology in a general class of economies with multidimensional action spaces and uncountable state spaces Under standard complementarity assumptions on preferences as well as a mild geometric condition on transition probabilities we prove existence of timeconsistent solutions in Markovian policies and provide conditions for the existence of continuous and monotone equilibria We present applications of our methods to habit formation models environmental policies and models of consumption under borrowing constraints and hence show how our methods extend the results obtained by Harris and Laibson Econometrica 69935–957 2001 to a broad class of dynamic economies We also present a simple successive approximation scheme for computing extremal equilibrium and provide some results on the existence of monotone equilibrium comparative statics in the model’s deep parametersWe thank Robert Becker Madhav Chandrasekher Manjira Datta Paweł Dziewulski Amanda Friedenberg Ed Green Seppo Heikkilä Len Mirman Peter Streufert and especially Ed Prescott as well participants of our SAET 2011 session for helpful conversations on the topics of this paper We especially thank two anonymous referees and the associate editor for their excellent comments on an earlier draft of this paper Balbus and Woźny reserach has been supported by NCN Grant No UMO2012/07/D/HS4/01393 All the usual caveats applyWe begin with some useful definitions not provided earlier in the paper but used in proofs of the proposition An arbitrary set Xge is partially ordered set or poset if X is equipped with an order relation ge Xtimes Xrightarrow X that is reflexive antisymmetric and transitive If every element of a poset X is comparable then X is chain If X is a chain and countable X is a countable chain An upper respectively lower bound for a set Bsubset X is an element xurespectively xlin X such that for any other element xin B xle xu respectively xlle x If there is a point xu respectively xl such that xu is the least element in the subset of upper bounds of Bsubset X respectively the greatest element in the subset of lower bounds of Bsubset X we say xu respectively xl is the supremum respectively infimum of B Clearly if the supremum or infimum of a set X exists it must be uniqueWe say a set Lsubset X is a lattice if for any two elements say x and xprime in L L is closed under the operation of infimum denoted by xwedge xprime and supremum denoted xvee xprime The former is referred to as “the meet” of the two points while the latter is “the join” A subset L 1 of L is a sublattice of L if it contains the sup and the inf with respect to L of any pair of points in L 1 A lattice is complete if any L 1subset L the least upper bound denoted vee L 1 and the greatest lower bound denoted wedge L 1 are both in L If this completeness property only holds for countable subsets L c the lattice is sigma complete In a poset X if every subchain Csubset X is complete then X is referred to as a chain complete poset or equivalent a complete partially ordered set or CPO A set C is countable if it is either finite or there is a bijection from the natural numbers onto C If every countable chain Csubset X is complete then X is referred to as a countably chain complete posetLet X 1ge X 1 and X 2ge X 2 be posets A function or equivalently operator fX 1rightarrow X 2 is monotone or orderpreserving or isotone if fxprime ge X 2fx when xprime ge X 1x for xxprime in X 1 A sequence h n in H is order convergent if there exists two monotonic sequences of elements from H one decreasing h downarrow n and one increasing h uparrow n such that h=inf h downarrow n=sup h uparrow n and h uparrow nle h nle h downarrow n A necessary and sufficient condition for an increasing sequence h nrightarrow h to be order convergent is h=sup h nLet X be a countably chain complete poset with the greatest and least elements and T a poset If FXtimes Trightarrow X is increasing and monotonicallysupinf preserving on X then trightarrow overlinePhi t and trightarrow underlinePhi t are isotoneLet t 1le t 2 From Theorem 7 we know that m i= overlinePhi t i=vee Gamma i=vee xFxt ile x Note that by isotonicity of Fxcdot we obtain m 1=Fm 1t 1le Fm 1t 2 Hence m 1in Gamma 2 Since m 2 is the greatest element of Gamma 2 hence m 1le m 2 square

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