Distributional Equality and Aggregate Utility: Further Comment
对Abba Lerner的最优分配定理给出N人情形下的形式化证明，利用信息论中的无知概念，并澄清了平等假设的模糊性。
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