Remarks on the first welfare theorem with nonordered preferences

A strong convexity assumption is used to establish the first fundamental theorem of welfare economics using the strong definition of Pareto optimality without assuming that the weak preference relations are complete or transitive. An example demonstrates that Debreu's (Proc. Nat. Acad. Sci., 40 (1954), 588–592) nonsatiation and convexity assumptions are insufficient to establish the result in the absence of transitivity.