THIS HIGHLY specialized paper grew out of a rather general program for analyzing security prices by multiple regression, with the ultimate objective of answering a whole series of questions regarding the supply of and demand for capital. The selection of bank stocks for immediate attention arose out of the unusual condition of bank capital following the World War II inflation, when rapid expansion of deposits was reducing the capital-to-asset ratios of banks to historically low levels. At the same time, the market for bank stocks was somewhat unfavorable, with many issues selling for less than book value, so that bankers often seemed reluctant to raise additional capital by means of new stock issues. Thus the bank stock market commanded the attentions of the bank supervisory officials and bankers alike. Were the common discounts from book value due to unsatisfactory bank earnings? If so, what level of eamings would be required to eliminate these discounts? Or could discounts be substantially reduced, if not entirely eliminated, by merely paying more generous dividends out of existing earnings? Finally, were the discounts in any way affected by capital-to-asset ratios? These questions, whose economic implications are discussed elsewhere,2 can all be attacked by a multiple regression analysis in which the dependent variable consists of bank stock prices and the independent variables consist of such quantities as book value, dividends, and earnings. But although the answers thus obtained are suggestive, and possibly provocative, there remains the haunting doubt that the rigid assumptions of regression analysis do not justify its use with bank stock prices.

IN ALMOST any science, quantitative problems arise for which approximation methods provide the most expedient solution. Refined methods often do not justify the labor they entail, either because the problem in hand does not require great precision, or because errors in the basic data limit the precision attainable. In exterior ballistics, for example, the gravitational attraction of the moon, which affects the motion of a projectile, is not taken into account in the calculation of firing tables because the effects of this attraction are small in comparison with the random errors resulting from unavoidable variations in the most carefully standardized ammunition and the most skillfully manufactured gun barrels. Some time ago in this REVIEW an approximation method for estimating the size distribution of incomes was discussed.' For most years little information is available on the distribution of incomes, but for I935-36 the National Resources Committee has compiled excellent data. Could the N.R.C. distribution be used as the basis for estimating the distribution for other years? If the aggregate income for some other year, say I946, can be determined (or predicted for some future year), and if the inequality of the distribution can be assumed to remain unchanged, the distribution for this other year can be estimated quite readily. This assumption of the same degree of inequality naturally suggests the Lorenz curve as a tool of analysis; however, the real key to the problem is the cumulative frequency curve. In a later article, graphical interpolation with the cumulative frequency curve was proposed as the most direct approach.2 It had the advantage of being simple, easy, and quick. Its accuracy, though not great, appeared adequate. Great precision was almost impossible, regardless of method, because the basic data and the underlying assumptions were approximations themselves. Moreover, most of the problems involving income distributions such as estimates of consumer expenditures or tax receipts do not require great precision. As an example, a problem originally proposed by Ames was discussed: to estimate an income distribution having the same inequality as the N.R.C. distribution, but an average income of $2000 instead of the $I502 for the N.R.C. distribution. cumulative frequency curve was drawn for families (including single persons) from the N.R.C. data -and also the corresponding curve for incomes received and the required distributions of both families and incomes received were read directly from these curves.3 In this example, the curves were drawn on semi-logarithmic paper, and the class intervals chosen were those of the N.R.C. distribution; however, it was stated that many other graphical devices would suffice, and it should have been obvious that any other set of class intervals could have been chosen. Furthermore, the fundamentals of the method are perfectly adaptable to numerical interpolation. In commenting upon this example, Eugene Clark and Leo Fishman might have criticized the arbitrary assumption of an unchanging inequality of income, or pointed out some of the fundamental weaknesses of the underlying data that make precise calculations difficult.4 Instead, they made a major issue out of a very minor point: namely, that the average income in any class interval depends upon the slope of the distribution curve, and that a change in the slope will inevitably affect the average income. 1 Edward Ames, A Method for Estimating the Size Distribution of a Given Aggregate Income, this REVIEW, XXIV (I942), Pp. I84-89. 2David Durand, A Simple Method for Estimating the Size Distribution of a Given Aggregate Income, this REVIEW, XXV ( 943), Pp. 227-30. 3 Ibid., Table I. 4 Eugene Clark and Leo Fishman, Appraisal of Methods for Estimating the Size Distribution of a Given Aggregate Income, this REVIEW, XXIX (I947), PP. 43-46.

NI an article published in this REVIEW,2 Mr. Edward Ames has discussed a method of adjusting the income distribution of families, or other income receiving units, for changes in average income. The essence of Mr. Ames' method is the assumption that the inequalities of the income distribution will not be affected by changes in average income. More technically speaking, the Lorenz curve, which relates the cumulated percentage of families to the cumulated percentage of income received, is assumed to remain unchanged. For example, if with an average income of $I500 the lowest I7 per cent of all families receive only 312 per cent of the total national income (see Table i), then with an average of $2000 the lowest I7 per cent must still receive only 3 Y2 per cent of the income. Mr. Ames derived his method from considerations involving the Lorenz curve, but he could have put up a better theoretical discussion and obtained the same practical results without once mentioning the Lorenz curve. While Mtr. Ames' method is perfectly sound aside from the unrealistic assumptions upon which it is based -it appears to be unduly complex in theory and to require an unnecessary amount of statistical manipulation in practice. This paper presents an alternative method of adjusting an income distribution, which is fundamentally the same as that of Mr. Ames, but which is believed to be both easier to understand and much more economical to operate. If we have a distribution of families by income, and if we wish to estimate how those families will be distributed after an increase of, say, 33 per cent in the average income, we must make certain assumptions about the way in which that increase will affect families at different income levels. One possible though unrealistic assumption, which will permit an easy solution, is that all families on the average enjoy the same percentage increase, that is 33 per cent. This does not necessarily mean that the income of each family will increase exactly 33 per cent; it merely means that the high income families will not receive a greater average percentage increase than the low income families, or vice versa. For the sake of simplifying the discussion, however, we shall pretend that each family does receive an increase of exactly 33 per cent; and we shall give a more rigorous analysis in a mathematical note, following the discussion. necessary result of the assumption of a constant percentage increase for all families is that the inequalities of the income distribution, and hence the Lorenz curve, will remain unchanged. The income of every family can be doubled or halved, increased by 33 per cent or decreased by I5 per cent, without affecting the inequality of the distribution; the lower I7 per cent of families will still receive only 312 per cent of the income. But all this is incidental. The fundamental statistical problem is the adjustment of the frequency distribution so that each family will receive a 33 per cent increase. The simplest method of adjusting an income distribution is by graphic computation from a cumulative frequency curve. There are literally dozens of practical variations by which this method can be employed, but all of them are fundamentally the same. Here we shall illustrate one of t7he variations which appears to be particularly simple and straightforward. Table i presents the National Resources Committee estimate of the distribution of income in the United States for I935-36; column 2 contains the cum lative percentages of families by income level (the distribution used by Mr. Ames in his example), and column 3, the cumulative percentages of income received. These distributions are characterized by an average family income of $I502, or a national income of about 6o billion dollars divided among 39 2million families. The problem at hand is to estimate a new pair of distributions 1 The author is now serving as a Lieutenant with the United States Naval Reserve. The opinions contained in this article are the writer's own, and they are not to be construed as official or as reflecting the viewvs of the Navy Department or the naval service at large. ' Edward Ames, A Method for Estimating the Size Distribution of a Given Aggregate Income, this REVIEW, XXIV (I942), Pp. I84-89.