Period Three Implies Heavy Discounting

A condition is derived that must be satisfied by a continuous function h: X → X in order to be the optimal policy function of a strictly concave deterministic dynamic programming problem which is defined on the state space X and which has a given discount factor ρ. We use this condition to show that there is no such dynamic programming problem on a one-dimensional state space that generates optimal solutions which are periodic with minimal period three unless the discount rate exceeds 82%. This bound holds uniformly for all strictly concave problems and all period-three-cycles.