Two theorems are derived about social choice functions, which are defined on comprehensive convex subsets of utility allocation space. Theorem 1 asserts that a linearity condition, together with Pareto optimality, implies that a social choice function must be utilitarian. Theorem 2 asserts that a concavity condition, together with Pareto optimality and independence of irrelevant alternatives, implies that a social choice function must be either utilitarian or egalitarian. These linearity and concavity conditions have natural interpretations in terms of the timing of social welfare analysis (before or after the resolution of uncertainties) and its impact on social choices.