Policy with stochastic hysteresis

TL;DR

This paper introduces a comprehensive methodology for analyzing path-dependent policies in stochastic models with forward-looking agents, applying it to a macro-climate model to derive explicit dynamics of optimal taxes and damages.

Contribution

It develops a new theoretical framework for stochastic hysteresis analysis, deriving closed-form dynamics of optimal policies and introducing a total derivative formula for conditional expectations.

Findings

Derived closed-form dynamics of the optimal Pigouvian tax.

Proposed a tractable class of hysteresis functionals.

Characterized the stochastic hysteresis elasticity and optimal policy process.

Abstract

The paper develops a general methodology for analyzing policies with path-dependency (hysteresis) in stochastic models with forward looking optimizing agents. Our main application is a macro-climate model with a path-dependent climate externality. We derive in closed form the dynamics of the optimal Pigouvian tax, that is, its drift and diffusion coefficients. The dynamics of the present marginal damages is given by the recently developed functional It\^o formula. The dynamics of the conditional expectation process of the future marginal damages is given by a new total derivative formula that we prove. The total derivative formula represents the evolution of the conditional expectation process as a sum of the expected dynamics of hysteresis with respect to time, a form of a time derivative, and the expected dynamics of hysteresis with the shocks to the trajectory of the stochastic process, a form of a stochastic derivative. We then generalize the results. First, we propose a general class of hysteresis functionals that permits significant tractability. Second, we characterize in closed form the dynamics of the stochastic hysteresis elasticity that represents the change in the whole optimal policy process with an introduction of small hysteresis effects. Third, we determine the optimal policy process.