Optimal sharing rules in repeated partnerships

This paper extends a model of repeated partnerships by Radner et al. (1986) allowing heterogeneous partners to choose their sharing rule. A sharing rule is optimal if the repeated game under the sharing rule has a public strategy equilibrium whose payoff sum is not improved by any public strategy equilibrium under any sharing rule. Two key factors for the analysis are the efficiency loss from allowing only the more productive partner to work and the efficiency loss in any cooperative equilibrium from imperfect observability. If the latter loss is smaller than the former, a threshold discount factor exists below which an asymmetric sharing rule inducing only one partner to work every period is optimal. At the threshold, an optimal sharing rule uniquely exists that is also optimal for any greater discount factor. The latter sharing rule reduces to the equal sharing rule for identical partners. The optimal equilibrium payoff sum as a function of the discount factor is a step function whose jump occurs at the threshold discount factor.