Lattices and Lotteries

We consider the consumer problem under uncertainty when the consumer can choose the quantity of a risk-free good and the lottery, or distribution, of a risky good from a set of distributions. These goods are imperfect substitutes in the consumer preferences, with additive preferences a special case. We develop sufficient conditions for the choice of the risky good to be monotone with respect to income, exploring different notions of monotonicity. The sufficient conditions are ordinal, independent of concavity, and do not require differentiability or continuity. Cardinal conditions and conditions from the single good case are not necessary and are not always sufficient. The sufficient conditions are formulated in appropriate value lattices. The framework is flexible and adaptable to handle different uncertainty applications. Examples demonstrate the sufficient conditions and different applications where available lotteries may be finite in number, may have discrete support, or may form a chain or a lattice.