A Qualitative Approach to Markovian Equilibrium in Infinite Horizon Economies with Capital

Abstract

Using lattice programming and order theoretic fixpoint theory, we develop a powerful class of monotone iterative methods that provide a qualitative theory of Markovian equilibrium for a large class of infinite horizon economies with capital. The class of economies is large and includes situations where the second welfare theorem fails as in models with public policy, valued fiat money, various forms of market imperfections (e.g., monopolistic competition), production externalities, and various other nonconvexities in the production sets. The methods can be easily adapted to construct symmetric Markov equilibrium in models with many agents and market incompleteness. As our methods are constructive, we prove they have important implications for characterizing the structure of numerical approximations to extremal Markovian equilibrium within the class. Of independent interest is our new approach to characterizing dynamic complementarities. We apply recent generalized envelope theorems found in the literature on nonsmooth analysis to characterize equilibrium value functions in our dynamic programming setting. Our fixed point algorithms are sharp, and are able to distinguish sufficient conditions under which Markovian equilibrium form a complete lattice of Lipschitz continuous, uniformily continuous and semi-continuous monotone functions as well as unique differentiable equilibrium. We develop a new collection of partial orders that allow us to conduct extensive monotone comparative dynamics on the space of economies. We conclude with a discussion of how the methods can be extended to economies with ordinal (as opposed to cardinal) forms of complementary.