Optimal contract design in the joint economic lot size problem with multi-dimensional asymmetric information

Abstract

Previous work has studied the classical joint economic lot size model as an adverse selection problem with asymmetric cost information. Solving this problem is challenging due to the presence of countervailing incentives and two-dimensional information asymmetry, under which the classical single-crossing condition does not need to hold. In the present work we advance the existing knowledge about the problem on hand by conducting its optimality analysis, which leads to a better informed and an easier problem solution: First, we refine the existing closed-form solution, which simplifies problem solving and its analysis. Second, we prove that Karush–Kuhn–Tucker conditions are necessary for optimality, and demonstrate that the problem may, in general, possess non-optimal stationary points due to non-convexity. Third, we prove that certain types of stationary points are always dominated, which eases the analytical solution of the problem. Fourth, we derive a simple optimality condition stating that a weak Pareto efficiency of the buyer’s possible cost structures implies optimality of any stationary point. It simplifies the analytical solution approach and ensures a successful solution of the problem by means of conventional numerical techniques, e.g. with a general-purpose solver. We further establish properties of optimal solutions and indicate how these are related with the classical results on adverse selection.