Recursive contracts in non-convex environments

TL;DR

This paper investigates non-convex dynamic optimization with forward-looking constraints, demonstrating that recursive multiplier methods yield optimal solutions under certain conditions, and explores the role of lotteries in economic models like the Ramsey problem.

Contribution

It establishes conditions under which recursive multiplier formulations are optimal in non-convex environments and analyzes the importance of lotteries in economic decision-making.

Findings

Recursive multiplier formulation yields optimal value with access to a public randomization device.

The formulation as sup-inf or inf-sup affects the timing of lotteries and constraint expectations.

Lotteries are essential for optimal solutions in certain economic problems like the Ramsey policy.

Abstract

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.