Monotone second-best optimal contracts

Abstract

We review the two stage nonlinear programming approach to the principal-agent problem proposed by Grossman and Hart (G-H) in 1983. The objective of the principal is to determine an optimal payment schedule to an agent that depends only on observable outcomes when (random) outcomes are influenced by unobservable actions of the agent. We establish that these ‘second-best’ optimal contracts are monotonically relted to observable outputs under fairly mild conditions. Two methods of proof are employed to establish that the monotone likelihood ratio condition (MLRC) on the probabilities of outcomes is sufficient for monotonicity of second-best optimal contracts in a three-state model. The first uses ideas related to the variation-diminishing property of totally positive matrices. The second approach uses a theorem of the alternative from linear programming theory to establish a condition that must hold under MLRC. The paper concludes with a counter-example that is a perturbed version of an example in G-H that shows that MLRC is not sufficient for monotonicity of second-best optimal contracts when there are four or more states.