Optimizing Contracts in Principal-Agent Team Production

I study a principal-agent team production model. The principal hires a team of agents to participate in a common production task. The exact effort of each agent is unobservable and unverifiable, but the total production outcome (e.g. the total revenue) can be observed. The principal incentivizes the agents to exert effort through contracts. Specifically, the principal promises that each agent receives a pre-specified amount of share of the total production output. The principal is interested in finding the optimal profit-sharing rule that maximizes her own utility. I identify a condition under which the principal's optimization problem can be reformulated as solving a family of convex programs, thereby showing the optimal contract can be found efficiently.