Equilibrium policies when preferences are time inconsistent
  This paper characterizes differentiable and subgame Markov perfect equilibria in a continuous time intertemporal decision problem with non-constant discounting. Capturing the idea of non commitment by letting the commitment period being infinitesimally small, we characterize the equilibrium strategies by a value function, which must satisfy a certain equation. The equilibrium equation is reminiscent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term leading to differences in qualitative behavior. As an application, we formulate an overlapping generations Ramsey model where the government maximizes a utilitarian welfare function defined as the discounted sum of successive generations' lifetime utilities. When the social discount rate is different from the private discount rate, the optimal command allocation is time inconsistent and we retain subgame perfection as a principle of intergenerational equity. Existence of multiple subgame perfect equilibria is established. The multiplicity is due to the successive governments' inability to coordinate their beliefs and we single out one of them as (locally) renegotiation-proof. Decentralization can be achieved with both age and time dependent lump sum transfers and, long term distorting capital interest income taxes/subsidy. 
Ivar Ekeland and Ali Lazrak