Linear Sharing Rule Sample Clauses
Linear Sharing Rule. As a general assumption throughout Rayo [2007]’s model, linear sharing is the most commonly seen sharing in real life because of its simplicity. However, whether linear sharing can be optimal is not clear. The restriction makes our search for optimality much easier but the comfort might be on the cost of efficiency. Thus in this section we would try to disclose whether anything is lost by assuming linearity in the sharing rule. From now on we would assume linear sharing and search for the optimal linear sharing to see if there is any efficiency loss by putting linearity restriction on the sharing rule. For each i, we assume si(F) = αiF + βi and Σ αi = 1, Σ βi = 0. In general sharing, we need to specify shares for each agent and each output level. However for each individual we only need to solve two parameters. With linear sharing constraint 4.7 would be affected by both parameter α and β, the utility among agents are no-longer transferable. Any constant transfers of output will affect individual IC constraints. Thus, the maximization of total team surplus can only support certain distribution outcomes under linear sharing. In this sense if we restrict to linear sharing rules, maximizing team surplus is merely a typ- ical case with a specific social welfare function. While there might exist other outcomes that some specific agents might prefer while don’t maximize the team surplus. Which social welfare function the team would choose depends on the organization structure, it might be principal-agent structure where there is one agent who dominates and put his own welfare as the priority, or it could be a ▇▇▇▇ bargaining process. We raise this issue here to make critical argument about the surplus maximization in Rayo [2007]. When individuals’ surpluses enter their IC constraints endogenously, utilities are no longer transferable. Which specific target effort will be preferable depends on the social welfare function of the team, thus we will leave the efficiency issue to the future study and focus on solving the optimal contract first. Due to the budget balancing constraint, it’s impossible to increase every agent’s share to 1. The problem is then how to derive an optimal sharing rule to create the maximum surplus possible. We start by simplifying our IC constraints
Linear Sharing Rule. Here we would restrict our attention to a special kind of sharing rule which is most commonly seem in real life - linear sharing rule.
3.1. A sharing rule S : R → Rn, where n = 2, 3, . . . is said to be linear if:
1. For any Y ∈ R, the share to each team member i has the following struc- ture: si(Y ) = αiY + βi, where αi and βi are constants or functions that are independent of Y , where the share is non-negative: αi ≥ 0.
2. The budget is balanced such that Σ αi = 1 and Σ βi = 0. The definition says, a linear sharing rule contains two parts, for each team member i it specifies a portion αi of the output which must sum up to 1 and a constant income/payment βi which must sum up to 0. The simple structure of linear sharing rules makes them easily enforced by courts. Moreover, by using a linear sharing rule the team members are pretty sure that the shares they get si(Y ) is monotone increasing with the output Y , since Sir(Y ) = αi ≥ 0. However, the incentive provided by linear sharing rule is not strong enough to produce the first best outcome ( ▇▇▇▇▇▇▇▇▇ [1982]). This is due to the fact that any agent’s ▇▇▇▇▇ will save his/her effort cost fully while the negative impact on the output will be shared with others. If we stick to balancing the budget, the punishment from a linear sharing rule can never be strong enough to achieve an efficient outcome. In the static model in ▇▇▇▇▇▇▇▇▇ [1982], the action vector of the team aˆ constitutes a ▇▇▇▇ equilibrium if and only if for each i, aˆi solves ▇▇▇ ▇▇(f (ai, aˆ−i)) − ci(ai), subject to for each i, individual rationality is satisfied that is πi ≥ π¯i.