A Qualitative Theory of Markovian Equilibrium in Infinite Horizon Economies with Capital

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Abstract
Using lattice programming methods and order-theoretic fixpoint theory, we are able to provide a first step in describing an ordinal (or qualitative) theory of equilibrium growth under uncertainty for a broad class of accumulation problems. The setting is one where in general the second welfare theorem fails, and equilibrium distortions can be modeled as elements of a partially ordered set. By a qualitative theory, we refer to the fact that all comparative statements in a parameter made by this class of models is closed under a well defined class of strictly increasing transformations. We characterize optimal planning problems as well as prove the existence of decentralized Markovian equilibrium. The class of environments considered include models with distortionary monetary and fiscal policy, market imperfections (e.g., monopolistic competition), production externalities, and models with incomplete markets with a continuum of income heterogeneity (e.g, Bewley models). For the optimal growth problem, we show that all existing optimal growth models exhibiting monotone controls in the (modified) planner’s problem are special cases of the class of larger class of superextremal models on a lattice. We next provide monotonicity results on this larger class, which includes models with serially correlated shocks. We then prove existence of decentralized Markovian equilibrium within a class of monotone functions, and conduct monotone or 'robust' comparative analysis in important parameters of the underlying primitive data of the economy. We show how one can sharpen the comparative analysis (in the sense of set orders) by restricting the class of primitive ecomomic data for the models, obtaining the existing cases in the literature as special cases. As our methods are constructive, computational issues are easily discussed. We conclude by showing how to use the methods in this paper to study models such as those in Krusell and Smith [39].
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