Discounting under disagreement

A group of time consistent agents has access to a common productive resource stock whose output provides their consumption needs. The agents disagree about the appropriate pure rate of time preference to use when choosing a consumption policy, and thus delegate the management of the resource to a social planner who allocates consumption efficiently across individuals and over time. We show that the planner's optimal policy is equivalent to that of a representative agent with a time varying rate of impatience. The representative agent's time preferences depend on the distribution of time preferences in the group, on the agents' tolerance for consumption uctuations, and on the productivity of the resource. The representative agent's rate of impatience coincides with that of the individual with the lowest rate of impatience in the long run, and under plausible conditions, is monotonically declining. In the work-horse case of iso-elastic felicity functions, and Gamma distributed rates of impatience, analytic solutions are possible, and the representative agent has hyperbolic time preferences. We thus provide a normative justification for the use of declining rates of time preference in dynamic welfare analysis.