Perturbation theory with heterogeneous agents: Theory and applications

This dissertation studies the application of perturbation theory to dynamic problems in macroeco- nomics with heterogenous agents. The first chapter develops an algorithm to apply perturbation theory to models heterogeneous agents, exploiting and underlying symmetry of the problem to make it computationally tractable. The theory is then applied to a standard multi-country model. Accuracy of the approximation is then compared to standard uses of perturbation theory and more conventional algorithms. The second chapter applies the theory to an optimal capital taxation problem with ex-post heterogenous agents and idiosyncratic investment risk.;In chapter 1, I develop an algorithm that allows for the tractable application of perturbation theory to economic models with large numbers of heterogeneous agents. This method exploits a previously unused symmetry in such models, which allows for the complexity of the problem to grow linearly with the number of agents, even with higher order approximations. A key feature of the algorithm is that the solution is simulated using a sequence of local approximations around the current distribution of individual state variables. As an application, I use this method to ap- proximate the decision rules in a multi-country model and compare accuracy to traditional solution methods. When the number of agents exceeds 4 the accuracy of my new algorithm exceeds that of standard approaches for both Euler equation and resource constraint errors. It improves upon the accuracy standard perturbation theory (where the approximation is taken around the degenerate vi steady state) by 2 to 3 orders of magnitude.;In the final chapter, I study the role of capital taxation in a general equilibrium heterogeneous agent economy with uninsurable investment risk characterized by persistent productivity shocks to the firms privately owned by entrepreneurs. In contrast to models with i.i.d. investment risk, the inclusion of persistent investment risk does not allow for easy aggregation. Instead, in solving this model I use a novel application of perturbation theory to economies with heterogeneous agents developed in the previous chapter. The presence of idiosyncratic investment risk leads to an under- accumulation of capital, which leads the Ramsey planner to have two conflicting motives in setting the capital tax. On the one hand, the planner wishes to correct the under-accumulation of capital by subsidizing saving. On the other hand, the planner wants to use capital taxes to redistribute from agents whose investments paid off to those whose investments did not perform well. In the presence of i.i.d. productivity shocks the first motive dominates and the planner moves to correct the savings decisions of the agents, but I find that with persistent shocks the motive for redistribution dominates, leading the government to tax capital. This result is robust to the inclusion of financial frictions. In addition, I study the response of the Ramsey plan to aggregate shocks when agents either have access to a complete set of arrow securities with respect to the aggregate shock or are restricted to trade only a risk free bond. I find that, in contrast to the results with a representative agent, the presence of incomplete markets reduces volatility of capital taxes in the optimal Ramsey allocation.