Characterizing Properties of Stochastic Objective Functions

This paper develops tools for analyzing properties of stochastic objective functions that take the form ( , ) ( , ) ( ; ) V u dF ≡ ∫s x x s s θ θ . The paper analyzes the relationship between properties of the primitive functions, such as the utility functions u and probability distributions F, and properties of the stochastic objective. The methods are designed to address problems where the utility function is restricted to lie in a set of functions which is a “closed convex cone” (examples of such sets include increasing functions, concave functions, or supermodular functions). It is shown that approaches previously applied to characterize monotonicity of V (that is, stochastic dominance theorems) can be used to establish other properties of V as well. The first part of the paper establishes necessary and sufficient conditions for V to satisfy “closed convex cone properties” such as monotonicity, supermodularity, and concavity, in the parameter θ. Then, we consider necessary and sufficient conditions for monotone comparative statics predictions, building on the results of Milgrom and Shannon (1994). A new property of payoff functions is introduced, called lsupermodularity, which is shown to be necessary and sufficient for V(x,θ) to be quasisupermodular in x (a property which is, in turn, necessary for comparative statics predictions). The results are illustrated with applications.