Dr. Rhodes' Analysis of the distribution of Single Incomes in the United States

THE purpose of this note is to raise some questions concerning the method suggested by Dr. Rhodes in his paper The Distribution of Incomes in the United States , ptublished in the August issue of ECONOMICA. Dr. Rhodes' method implies that the number of incomes in each income bracket, with the exception of the highest and the lowest brackets, is determined by two different laws : one described by a formula derived for a different income range and extrapolated for this particular bracket, and another approximated by a curve fitted to residual frequencies including the bracket considered. In other words, the proportion of incomes falling into a particular bracket is the sum of two frequencies. To disenitangle them, Dr. Rhodes fitted a curve to the section of the empirical data where frequencies are supposed to be not composite (namely, the part of the Tail to which the original curve was fitted), and from the formula obtained he estimated what part of total frequencies in the composite part of the distribution is ruled by this first distribution law. A second formula was derived for residual frequencies of a certain number of income brackets, and a new set of residuals obtained by extrapolating the second law of distribuition (the ) beyond the lower limit of the income range for which it was derived. Though no third curve was fitted to these residuals as the number of known values of x is small (p. 231), nothing in Dr. Rhodes' method prevents us from refining the anatomic approach. In another distribution, we could obtain a Neck in addition to a Tail and a . The decomposition of the total frequencies thus is guided not by economic considerations but merely by the endeavour to obtain several partial distributions that can be described by a simple formula and to add up as nearly as possible to the original frequencies. We venture to challenge the rationale of this method for economic analysis. It is sometimes very important to derive, from empirical income data, particular distributions for component parts of the population. One might be interested in analysing separately farm and non-farm incomes, or incomes derived mainly from property and those earned in gainful pursuits; in other words, to obtain partial distributions for more homogeneous components of the empirical distribution. Such an endeavour can obviously be successful only if considerable external information concerning the empirical data is available (if, for instance, the assumption can be made that practically all incomes below $3,000 are essentially earned incomes); but the inherent difficulties can hardly be solved by the proposed method of successive fitting of curves, a technique which is possibly attractive for the mathematician but has scarcely any economic meaning. This leads us to the core of the problem: Why do we try to describe an empirical income distribution by a mathematical formuLla ? Probably mainly for the following three reasons: 1. In order to compare distributions for different populations. 2. In order to compare changes over time in the same population. 3. In order to interpolate (or to rearrange in convenient brackets) a given distribution. Dr. Rhodes' method has obviously no particular advantages for No. 3. In the first two cases also, we can make little use of it, for after we have torn and Tail apart (and are left, in addition, vith the Beak which in the example studied includes not less than a fourth of the total population) in one distribution, we are not sure that when we repeat the same operation with another distribution (either for another population or for the same population at a different time point) we shall obtain a comparable Head and Tail . If we decompose a distribution, making certain assumptions

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