Underlying much modern economic theory and econometric methodology is a function space setting. Motivated by this perspective, this dissertation provides a set of tools to construct and solve dynamic economic models with functions as state variables and apply them in models of inequality over space and across individuals. I show how to characterize the solution to function-valued models by linearization in function space, provide a set of algorithms to compute this solution numerically in both regular and ill-posed models, and prove that the algorithms are consistent. The power and efficacy of the methods are illustrated in several examples including a dynamic stochastic model of trade, migration, and economic geography.
Portions of this PhD thesis have been adapted into the paper Solution of Rational Expectations Models with Function Valued States. The thesis contains additional results including a very different algorithm for the non-compact case, expanded analysis of the economic geography model, and additional numerical applications.