Recursive contracts in non-convex environments

In this paper we examine non-convex dynamic optimization problems with forward looking constraints. We prove that the recursive multiplier formulation in \cite{marcet2019recursive} gives the optimal value if one assumes that the planner has access to a public randomization device and forward looking constraints only have to hold in expectations. Whether one formulates the functional equation as a sup-inf problem or as an inf-sup problem is essential for the timing of the optimal lottery and for determining which constraints have to hold in expectations. We discuss for which economic problems the use of lotteries can be considered a reasonable assumption. We provide a general method to recover the optimal policy from a solution of the functional equation. As an application of our results, we consider the Ramsey problem of optimal government policy and give examples where lotteries are essential for the optimal solution.