A Method for Obtaining Closed-Form Policy Solutions to Dynamic Optimization Problems

Abstract

This project develops a “choose and verify” method for obtaining closed-form policy functions in dynamic optimization environments. In so doing, it purports to enlarge the class of problems with analytic solutions. The two-step method entails specifying various components of the return and restrictions in a manner that respects the problem’s overall choice structure and then deliberately winnowing the pool of candidate specifications. To demonstrate its applicability, the paper formulates a simple health-capital model with two salient features: the stock variable is an economic bad, and leisure is productive. This conceptual inversion prompts two important questions: How long can we maintain unhealthy lifestyles before crossing a point-of-no-return on the path to persistent unwellness? And, do some people have a diminished or nonexistent capacity to stave off certain illnesses? En route to answering, the application shows that parsimonious models can exhibit considerable explanatory power in the presence of choose-and-verify-based structure. In particular, one obtains an analytic policy function capable of generating one, two, or three steady states. High levels of personal discounting and risk preference reduce individuals’ resilience and increase the likelihood of health-related poverty traps.