Abba Lerner's formulation of the optimum division of remains interesting although in some ways ambiguous. The comment by William Breit and William Culbertson in this Review is evidence of both points. There is still no formal proof of the theorem for an arbitrary number of persons, even if its outlines are clear enough. The equal assumption remains unclear. This note contains a proof of the weak' theorem for N persons. It uses a concept of ignorance derived from information theory.2 Throughout this paper the term income is used in a somewhat special sense, following Lerner et al.; a sense somewhat akin to Gary Becker's full (p. 497).3 It is evaluated on the assumption that the individual works the maximum feasible hours at maximum effort; purchases of (quantitative or qualitative) leisure are treated like any other purchases.4 Moreover, it is assumed that we may discuss redistribution without considering the effect of resulting price changes on the utility-of-income schedules, (see Lerner (1944) pp. 23-24). Suppose that the distribution of by individual shares is given, that is, when Yi is the of individual i, and Y is total Yi/ Y = yi is a given constant for i = 1, ... ,N. The yi may be regarded as the probabilities that an incremental dollar of purchasing power goes to individual i. Now, suppose it is known that each individual's utility function is drawn from a finite family fj(Yi), for j = 1, . . . , J; i = 1, N. The probability that individual i's utility function is function j is pij. The expected utility of a dollar of purchasing power allocated to individual i is

In the process of cultivation of taste, tastes are changed by the experience of consumption. To model this we might assume that cultivation of taste for a particular good involves a change in only one parameter of the utility function, and that the other parameters, that is, the underlying utility function, remains unchanged. For example, assume that there are just two goods, x and z, of which x is subject to cultivation of taste, and z is not. Then