Abstract

In this paper, we analyze a procurement problem between a principal and an agent with discrete types, where contracts must be drawn from finite grids of quantities and transfers. Contract discreteness does not preclude the usual constraint simplifications, but calculus methods must be replaced with discrete methods and optimal transfers may involve “rounding up” costs—which can impact optimal quantities.

When transfers need no rounding up, we take a discrete first-order approach. We show that optimal quantities need not be unique—even under strict discrete concavity—but they cannot take more than two adjacent values for each type. Also, ensuring strict monotonicity entails limiting the “degree of concavity” of the principal’s utility. When rounding up is needed, the screening problem becomes a finite-horizon, discrete dynamic programming problem. Now, virtual costs being strictly increasing cannot even guarantee monotonicity of the quantities, and we can have distortions at the top—both downwards and upwards.

Alejandro Francetich
Ph.D. in Economic Analysis and Policy, Stanford GSB, Stanford University. Associate Professor, School of Business, University of Washington Bothell. His research interest is Microeconomic Theory, in particular Mechanism Design.