Lattices and Lotteries

This paper develops techniques for comparative statics in the consumer problem under uncertainty that involve order-based rather than topological assumptions on utility functions and choice sets. We allow for imperfect substitutability and for a broad set of lottery choices. Building on the value lattice approach to the consumer problem developed by Antoniadou (1996), we introduce a partial order that integrates nonlinear pricing. We discuss two extensions to take uncertainty into account, one based on stochastic dominance and the other on lottery pricing. We identify specific difficulties that arise in a general lottery choice setting (ranking budget sets with respect to the strong set order, ensuring antisymmetry). We then provide different ways to adapt our approach, depending on whether choice involves small lotteries or whether partial orders can be refined, or by appealing to a related technique that does not require antisymmetry.