Group for Research in Decision Analysis

Lattices and Lotteries

Leonard J. Mirman

This paper addresses comparative statics in the consumer problem under uncertainty. Unlike the majority of research in this area we do not assume univariate preferences. We allow a second good to enter preferences independently of the risky good with possibly imperfect substitutability/ complementarity. Also in contrast to standard analysis, we allow in the consumption set the presence of different lotteries from which the consumer may choose subject to budgetary constraints. The methodology we follow is based on the lattice programming approach to the deterministic consumer problem developed by Antoniadou (1996) and Mirman and Ruble (2008), which we extend to take account of the presence of uncertainty. We discuss choice and comparative statics as income increases when desired monotonicity is with respect to expenditure on the risky good, as well as when monotonicity is with respect to First or Second Order Stochastic dominance. We derive sufficient conditions in terms of superextremal variant properties of the expected utility function in the constructed lattices for such monotone comparative statics. Our model encompasses the classical portfolio problem under uncertainty which in our context corresponds to perfect substitutability between the two goods. However, our analysis not only extends the existing univariate analysis to the multivariate setting but it can do so in the absence of differentiability of the utility function, or divisibility of the risky good.