Using lattice programming methods and order theoretic fixpoint theory, we develop a powerful class of monotone iterative methods on a lattice that is able to provide the first step toward a qualitative theory of Markovian equilibrium in a broad class of infinite horizon economies with capital under uncertainty. The class of economies we consider is very general, and includes those where the second welfare theorem fails. Using generalizations of a famous theorem of Tarski [51], we prove

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... of decentralized Markovian equilibrium for each element of this class. This class of economies is very large, and includes many models with distortionary monetary and fiscal policies, market imperfections (e.g., monopolistic competition), production externalities and nonconvexities in production where these frictions create value functions for agent decision problems that cannot be studied using concave programming methods. Unlike existing Euler equation methods, the key step in our work is constructing an infinite-dimensional single crossing property, which allows us to obtain monotone operator on a complete lattice of functions. We next consider the question of existence of monotone stationary Markovian equilibrium. For this class of economies, we develop a class of monotone iterative methods that can be used to prove both existence and allows for the study issues relating to monotone comparative dynamics. Finally, we conclude by relating our results to those in the existing literature, and including a discussion of robustness of existing and the possibility for new necessary conditions under which these procedures will work. For this latter consideration, we call upon the work of LiCalzi and Veinott (1991) and Milgrom and Shannon (1994) on ordinal conditions, and apply these methods to characterizing conditions under which monotone controls exist for our choice of commodity lattices. The results allow us to characterize all of the existing literature as special cases of our monotone methods on lattice.