Chapter 5 Nonlinear pricing and mechanism design

This chapter describes methods for solving simple versions that arise in nonlinear pricing and mechanism design. Nonlinear pricing is one instance of a wide variety of problems derived from the principalagent paradigm and its extensions to mechanism design. The key ingredient of the standard formulation is the representation of an agent's private information as a point in a Euclidean space. Characterization of a solution relies on the necessary conditions derived from the seller's incomplete problem. Computational procedures naturally divide into two phases. In Phase 1, the differential equations are solved with fixed (Dirichlet) boundary values. Straightforward approaches to solving these equations based on discrete approximations encounter inconsistencies. In Phase 2, the boundary values are adjusted to improve the approximation of the boundary conditions imposed by the transversality condition. For both algorithms, the differential condition derived from the integrability condition provides an adequate von Neumann condition on the upper boundary. In the pure relaxation algorithm, the conditions on the lower boundaries can be derived from the welfareoptimality condition.