Equilibrium growth when the planner is time inconsistent

This paper takes a step toward defining a continuous time methodology to analyze a class of time inconsistent capital accumulation games when the discount rate is non-constant and commitment is not possible. The subgame perfect and history independent equilibrium consumption policies are characterized with an equation that must be satisfied by the value function. The equation is reminiscent of the classical Hamilton-Jacobi-Bellman equation of optimal control, but with a non-local term that is relevant for the qualitative behavior of the solution. The methodology is applied to an overlapping generations growth model with a utilitarian social planner. The social planner problem is shown to be isomorphic to a time inconsistent capital capital accumulation problem with a non constant discount rate. Existence of multiple continuously differentiable subgame perfect equilibrium consumption policies leading to multiple steady state is established. Decentralization of the equilibrium is possible but requires a distorting positive or negative tax on capital in the long term.