Common Shocks and Relative Compensation Schemes

This paper studies qualitative properties of an optimal contract in a multi-agent setting in which agents are subject to a common shock. We derive a necessary and sufficient condition for the optimal reward of an agent producing an output level y to be a decreasing (increasing) function of the outputs of the other agents, under the assumption that the agents’ outputs are informative signals of the value of the common shock. The condition is that the likelihood ratio p(y, e, u)/p(y, e', u), where e is a higher effort level than e', and u is the value of the common shock, be a decreasing (increasing) function of u. We derive conditions on the way the common shock affects the marginal product of effort under which the likelihood ratio is decreasing for all output levels, or increasing for some output levels and decreasing for others.