A note on optimal contracting with public ex post information under limited liability

Riordan and Sappington (J Econ Theory 45:189–199, 1988) show that in an agency relationship in which the agent’s type is correlated with a public ex post signal, the principal may attain first best (full surplus extraction and efficient output levels) if the agent is faced with a lottery such that each type is rewarded for one signal realization and punished equally for all the others. Gary-Bobo and Spiegel (RAND J Econ 37:431–448, 2006) show that there exist locally incentive compatible lotteries such that limited liability constraints are satisfied, only if these constraints are satisfied when that kind of lottery is used. We explore how lotteries should be designed to attain not only local but also global incentive compatibility when the agent is protected by limited liability. The main issue with global incentive compatibility rests with intermediate types being potentially attractive reports to both lower and higher types. A lottery including three levels of profit (rather than only two) is found to be optimal under limited liability, in that it is most likely to be globally incentive compatible, if local incentive constraints are strictly satisfied. Conditions for first-best implementation are identified.