Managerial Limited Commitment : A New Class of Stochastic Control Problems

The theory of investment and growth of firms has been an important source of stochastic control problems in economics and management science. The issue of optimal management contract under the constraint that a CEO can depart to pursue an outside option ("managerial limited commitment") has been addressed more recently, in particular in a seminal paper by Ai and Rui (2015). This problem leads to a new class of stochastic control problems, with a stochastic constraint capturing the CEO's limited commitment. Ai and Rui (2015) introduced a dynamic programming equation which has unusual boundary conditions. Many formal arguments are used in the proof, although the amount of intuition is impressive. Our objective in this paper is to address the new technical challenges arising in this class of problems and to rigorously solve these, also studying a related dual problem