This chapter examines the problem of maximizing total utility subject to an upper bound on an unnormalized Gini coefficient. It shows that, remarkably, the solution consists of only two or three distinct utility levels, suggesting that the occurrence of two or three major socioeconomic classes in historical societies may be related to the underlying mathematics of the distribution problem. The chapter also shows by counterexample that the problem is not regionally decomposable and conducts an extensive marginal analysis of stakeholder incentives to improve efficiency. In particular, it shows that efficiency improvements in the upper tier of stakeholders (those receiving the largest utility allotments) benefit stakeholders in the middle tier, but benefits flow in the opposite direction only when the inequality bound is relatively tight.