Although the starting point for modern discussions of the optimum division of income is the classic formulation of the problem given by A. P. Lerner, it is difficult to find an unambiguous statement of his analysis in the recent literature of welfare economics. Various interpreters, claiming equally the authority of Lerner, have offered differing and often contradictory statements of Lerner's argument and its implications. Lerner himself has noted this fact, remarking somewhat wistfully, that though he feel[s] very pleased . . . with his argument it is nevertheless . . . the least successful of my inventions.' Our purpose in this paper is to show that the puzzlement and suspicion surrounding the Lerner theorem given in chapter 3 of The Economics of Control probably derives from his ambivalent statement of conclusions that allegedly follow from a single argument. For, as we shall see, there is not one Lerner theorem but two, and Lerner proves his case for only one conclusion (and here only under the most restrictive assumptions) and not the other. However the unsubstantiated Lerner conclusion can still be salvaged through a modified conceptualization of the problem. In Section I, we shall summarize Lerner's statement of his theorem on income distribution under conditions of diminishing marginal utility of income and extend the model to accommodate the possibility of increasing marginal utility as well. In Section II, we shall review a recent revision of the theorem in terms of a voting model and suggest how the unanimity conclusion of this model can be preserved even with the existence of risk takers. In Section III, we shall indicate difficulties encountered when it is realized that Lerner's proof applies only to a mild version of his theorem and not to his bolder conclusion that allegedly follows from his argument. Section IV will be devoted to showing that Lerner's egalitarian conclusion can be rigorously vindicated leaving his basic strong theorem intact.