Differentiability of Preferences without Derivatives

A new definition of differentiable convex preferences is proposed for spaces without requiring any algebraic structure. Given a set of primitive orderings Λ, preferences are Λ-convex-differentiable if, at each alternative, there is a unique ordering in Λ whose advancement is necessary for preference improvement and whose reduction is sufficient for preference deterioration. As an application, it is shown that differentiability of preferences guarantees the secondwelfare theoremin a two-agent exchange economy with a finite set of distinct indivisible goods.