Pareto optimal levels of effort
There is a team technology in which two workers must work together in order to produce. The revenue from their efforts is given by R = (e1e2)^1/2 where ei is the effort that worker ∈ {1, 2} puts into production. Each worker's utility is given by ui = wi − ei^2 where wi is worker i's income.
(a) What are the Pareto optimal levels of effort (this maximizes total surplus from production minus the costs of effort)?
(b) If the sharing rule is wi =R/2 , for both agents. How much effort will each worker put in a Nash equilibrium if they cannot observe each other's effort level?
(c) Try and obtain the sharing rule wi = bR, where b is a scalar, such that in a Nash equilibrium the workers will have an incentive to put in the optimal effort obtained in part a). Is this sharing rule reasonable? How could this sharing rule be implemented?