On Time-Consistent Collective Choice with Heterogeneous Quasi- Hyperbolic Discounting

A general setup is considered where agents are characterised by quasi-hyperbolic discounting and by heterogeneous bias for the present and heterogenous discounting parameters. Consumptions are moreover subject to a standard feasibility constraint. A collective utility function is defined as a function of the intertemporal utilities of the selves of the different agents, the elementary unit being thus the self of a given period for a given agent. The analysis is further specialized to time-independent collective utility functions. Such a framework generating a tension between Pareto-optimality and time-consistency for the optimal allocations, two approaches are suggested in order to tackle this issue. The first one imposes restrictions on the collective utility function that ensure the timeconsistency of the optimal decisions. The second one builds from an a priori time-inconsistent collective utility function. The benevolent planner is then to be considered as a sequence of successive incarnations, any of these incarnations being endowed with its own objective. The associated optimal policy is the equilibrium of a game between the successive incarnations of the planner when the players follow Markovian strategies. The results obtained for both solution concepts are compared through an example that also shows how they can be recovered through a competitive equilibrium.