Being serious about non-commitment: subgame perfect equilibrium in continuous time

TL;DR

This paper develops a framework for differentiable subgame perfect equilibria in continuous time with non-constant discounting, revealing indeterminacy in steady states but robustness to small deviations.

Contribution

It introduces a novel equilibrium equation with a non-local term and analyzes its implications for growth models with non-constant discount rates.

Findings

Equilibrium characterized by a modified Hamilton-Jacobi-Bellman equation

Non-constant discounting causes indeterminacy in steady states

Steady state levels are robust to small changes in discount rates

Abstract

This paper characterizes differentiable subgame perfect equilibria in a continuous time intertemporal decision optimization problem with non-constant discounting. The equilibrium equation takes two different forms, one of which is reminescent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term. We give a local existence result, and several examples in the consumption saving problem. The analysis is then applied to suggest that non constant discount rates generate an indeterminacy of the steady state in the Ramsey growth model. Despite its indeterminacy, the steady state level is robust to small deviations from constant discount rates.