Convexities of Consumption Preferences

The concept of convexity plays an important role in the studies of economics in general and consumer theory in particular. By removing the unrealistic assumption that consumer preferences are complete on the set X of consumption choices, this chapter, which is mainly based on Forrest et al. (Studia Universitatis “Vasile Goldis” Arad—Economics Series, 2023), examines how the various concepts of convex preferences and relevant properties can be reestablished. Different from the literature, this chapter derives a series of formal propositions and constructs counterexamples. Among others, it shows the following: (1) The weighted combination of two possible consumptions is not necessarily comparable with any of the possibilities; (2) not every convergent sequence of a consumer’s preferable consumptions asymptotically preserves his preference preordering; (3) not all preferences satisfy either positive multiplicativity or additive conservation; (4) all three types of preference convexities—weak convexity, convexity, and strong convexity—can be introduced to general convex spaces. In the conclusion section, some research topics of expected significance are suggested for future works.