Recursive contracts in non-convex environments

Abstract

Many dynamic contracting problems in economics are non-convex, and deterministic optima can be dominated by randomized solutions. We develop a recursive method to compute optimal lotteries in the presence of forward-looking constraints, extending the approach of \cite{marcet2019recursive} to non-convex settings. We show that an inf-sup functional equation recovers the correct value when lotteries are conducted after agents commit to actions (``ex-ante'' lotteries), though the resulting solution often lacks economic meaning. Replacing the inf-sup with a sup-inf operator delivers the correct value of an economically more relevant ``ex-post'' lottery problem, where agents act after observing the lottery's outcome. We also develop an algorithm to recover the optimal lottery policies from the solution of the functional equation.