A Note on Dynamic Processes

Abstract

This dissertation is written in two parts. In the first part, we overview and extend a novel, robust and computationally efficient method come to be known as an auxiliary function method for long-time averages, which is capable of computing sharp upper or lower bounds on time averaged quantities in underlying dynamical variables via convex optimization and semidefinite programming techniques. We then turn to studying the validity of asymptotic methods for computing long-time statistics in non- linear or nonautonomous dynamical systems. Asymptotic methods, such as Fourier expansions, the Galerkin method, and harmonic balance methods, are ubiquitous in the literature when studying dynamical systems, but as these methods produce only approximate solutions, it is natural to ask how well these approximate solutions agree with a system’s true solutions. We show for the Duffing equation and the nonlinear, damped driven pendulum that the mean squared amplitude as produced by the harmonic balance method agrees quite well with the system’s true solution. However, asymptotic methods fail to accurately predict the regions of stability for a parametrically driven, coupled oscillator system. We show that the regions of stability are particularly sensitive to the coupling effects across a broad range of modulation frequencies, and hence show the auxiliary function method as a more robust means of determining stability regions. In the second part of this work, we first overview dynamic choice in the presence of uncertainty while discussing the classical paradigms of Von Neumann-Morgenstern expected utility and discounted expected utility. We then discuss the ethical theory of utilitarianism from the perspective of Jeremy Bentham and discuss its connections to decision theory; in particular, social choice theory. We briefly overview the social choice literature by reviewing the seminal work of Kenneth Arrow and John Harsanyi and subsequent results. Then we present a novel extension of Harsanyi’s theorem to an infinite time horizon, multi-generation setting. Under some additional assumptions, a Pareto condition is equivalent to utilitarian aggregation and the utilitarian weights are unique. We analyze the properties of utilitarian weights, such as the limiting behavior of utilitarian weights for distant future generations, and the comparative statics of utilitarian weights as the social discount factor or the social risk attitude changes. Among other findings, we show that a higher social discount rate is associated with a more unequal assignment of utilitarian weights across generations.