Equilibrium theory in infinite dimensional spaces

This dissertation aims to investigate economic ‘negligibility’ in economies which have infinite dimensional spaces as their components, and to prove the existence of equilibrium in such frameworks.; The first chapter provides purpose, background, analytical ideas and overview of the other chapters.; In chapter 2, we prove the existence of a competitive equilibrium for an economy with a measure space of agents and with an infinite dimensional commodity space which has interior points of its positive cone. We dispense with convexity and completeness assumptions on preferences. We utilize the convexifying effect on aggregation without further assumptions on the atomless measure space of agents.; In the third chapter, we consider an exchange economy with a continuum of agents and infinite dimensional commodity space which lacks interior points of its positive cone. To be concrete, we take l ₂ for our commodity space. The fact that l₂ is a Hilbert space makes it natural to take advantage of the projection. McKenzie (1959) used the projection in his proof of a competitive equilibrium with a finite dimensional commodity space. We take McKenzie's (1959) approach as repackaged by Khan (1993) to derive the existence of a competitive equilibrium directly without relying on the existence results in finite dimensional commodity spaces.; The fourth chapter turns to a large sequential game. Despite the widespread use of dynamic economies with ‘many’ negligible stochastic entities and global stability in the literature, a consistent theoretical model for these situations is still unavailable. This is due to the lack of a law of large numbers for a continuum of independent random variables. We consider this problem in the framework of a large sequential game and provide a solution. We depart from the unit Lebesgue interval to a new set of agents modelled by an atomless Loeb space. Then the proper law of large numbers for ‘many’ independent random variables, proved by Sun (1998), can be used to cancel out a continuum of negligible random shocks, and thus leads to period-wise no aggregate uncertainty. We also obtain the existence of a Nash equilibrium strategy.