An Appraisal of the Errors Involved in Estimating the Size Distribution of a Given Aggregate Income

Abstract

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.