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THE PERSONAL DISTRIBUTION OF 
 
 INCOME IN THE UNITED STATES 
 
 FREDERICK ROBERTSON MACAULAY 
 NATIONAL BUREAU OF ECONOMIC RESEARCH. INC. 
 
*ACr, 
 
THE PERSONAL DISTRIBUTION OF 
 
 INCOME IN THE UNITED STATES 
 
 BY 
 
 FREDERICK ROBERTSON MACAULAY 
 NATIONAL BUREAU OF ECONOMIC RESEARCH, INC. 
 
 Submitted in Partial Fulfilment of the Requirements for the Degree 
 
 of Doctor of Philosophy in the Faculty of Political 
 
 Science, Columbia University 
 
 36 
 
 NEW YORK 
 
 HARCOURT, BRACE AND COMPANY 
 1922 
 
/¥/3bci 
 
 COPYBIGHT, 1922, BT 
 NATIONAL BUREAU OP ECONOMIC RESEARCH, INC. 
 
 KXCHANG* 
 
 Printed in the U. £. A. 
 
f cs 
 
 PREFACE 
 
 In the year 1922 the National Bureau of Economic Research, Inc., pub- 
 lished in two volumes the result of an investigation into "Income in the 
 United States." Part III of Volume II of that work consisted of the present 
 study. The author acknowledges with thanks the courtesy of the National 
 Bureau of Economic Research in permitting him to have this reprint made 
 from the original plates. 1 
 
 1 This fact explains the pagination. 
 
 585),, H 
 
TABLE OF CONTENTS 
 
 Chapter Page 
 
 27. The Problem 341 
 
 Practical and theoretical difficulties connected with formulation 
 
 of the problem. Relation of personal distribution to factorial 
 distribution. 
 
 28. Pareto's Law and the Problem of Mathematically Describ- 
 
 ing the Frequency Distribution of Income 344 
 
 Pareto's Law. Improbability that any simple mathematical ex- 
 pression adequately describing the frequency distribution of in- 
 come can ever be formulated. Heterogeneity of the data. 
 
 29. Official Income Censuses 395 
 
 The Australian income census of 1915. 
 
 30. American Income Tax Returns 401 
 
 Pecularities of the tax returns from year to year. Irregularities 
 
 and fluctuations in the distribution of non-reporting and under- 
 statement. The under $5,000 and over $5,000 groups. Wages 
 and total income. 
 
 31. Income Distributions from Other Sources Than Income Tax 
 
 Returns 415 
 
 Purposes for which existing distributions have been collected make 
 them extremely ill adapted to our use — picked data. 
 
 32. Wage Distribution 418 
 
 Relations between rates and earnings, earnings and income. Earn- 
 ings per hour, per day, and per week. Distribution of hours 
 worked in a week, and weeks worked in a year. Federal returns 
 
 of income by sources. The problem of deriving one regression 
 line from the other. 
 
 33. The Construction of a Frequency Curve for All Income 
 
 Recipients 424 
 
 An income census the direct and adequate method of solving the 
 problem. Piecing together the existing data. Checking them 
 for internal consistency and agreement with collateral informa- 
 tion. Conjectural nature of final results. 
 
CHAPTER 27 
 THE PROBLEM 
 
 What is the frequency distribution of annual income among personal 
 income recipients in the United States? Before we can give an intelligent 
 answer to this question, we must formulate it more definitely by indicating 
 certain connotations which logic or expediency leads us to attach to some 
 of its terms. 
 
 By income it seems desirable to mean actual money income, plus the 
 estimated money value of the more important of those items of commodity 
 or service income on which a money value is ordinarily placed. Two of 
 the most important items which are thus included are the annual rental 
 values of owned homes and the value of farm produce consumed by farmers' 
 families. 
 
 In line with the ordinary convention, we have excluded from our defini- 
 tion of income, that income, whether monetary or non-monetary, which a 
 wife receives from her husband or a child from its parents. 1 Not only is 
 such exclusion practically expedient but it is also theoretically defensible 
 and that quite apart from the fact that a money value is not ordinarily 
 placed on the services of wife or child, wages of housekeepers to the con- 
 trary notwithstanding. 
 
 The frequency distribution resulting from the exclusion of such quasi 
 incomes will be less heterogeneous and more significant and interpretable 
 than the distribution which would result from inclusion. For the relation 
 of the incomes of wives and children to the economic struggle is derived 
 and secondary, while that of most other incomes is direct and primary. 
 Now, though the distribution of income among persons is not synonymous 
 with distribution among the factors of production, the two problems are 
 very closely related. An individual's income may be thought of as made 
 up of wages, rent, interest and dividends, profits, and gifts or allowances. 
 If we omit this last type of income, the problem of factorial distribution 
 proposes an investigation of how and why the individual received what 
 remains. Even if gifts and allowances admitted of any such systematic 
 and reasoned explanation as may be given of rent, wages, etc., the ex- 
 planation would be of a totally different kind. Hence, for the purposes of 
 this investigation, it seems undesirable to classify as income, the receipts, 
 
 1 That is, while such income has, of course, been counted in the first instance as income 
 of the husband or parent it has not been re-counted as income of the wife or child. 
 
 341 
 
342 PERSONAL ^StfRIBUTION OF INCOME IN U. S. 
 
 whether monetary or iKHi'-m'oiietary, of those persons receiving merely 
 allowances or gifts. 1 
 
 Similar considerations have led us to think of an income recipient as an 
 individual rather than a family. Just as it is the husband and not the 
 wife, the parent and not the child, so it is the individual and not the family 
 who, as an income receiver, comes into direct economic relationship with 
 the machinery of distribution. 
 
 The chief argument in favor of family rather than individual treatment 
 of the frequency distribution is based upon the idea that, though income 
 accrues to the individual and not the family, the family is a more significant 
 unit of economic need than the individual. But this is a different approach 
 to the question and has, of course, no intimate relation to the problem of 
 factorial distribution. Moreover, we must remember that if we are going 
 to improve appreciably upon the individual, even as a need unit, we can- 
 not stop with actual biological families with their great variation in size 
 and constitution, but must introduce the concept of the theoretical family — 
 father, mother and three children, for example. This last concept is, in 
 its raw form, quite unusable. The population is not made up of such 
 theoretical families. We may discuss what a family of five ought to get 
 to maintain a decent standard of living, but we cannot divide the actual 
 population into families of five and discuss what these non-existent hy- 
 pothetical families actually do get. There remains the alternative of ex- 
 pressing actual families in terms of some need unit such as the "ammain." 2 
 While this last procedure would probably yield an extremely interesting 
 distribution based upon need units, it is impractical to attempt any such 
 solution with the data available. 3 
 
 Though a distribution of income among actual biological families would 
 appear to be somewhat less enlightening and interpretable than a dis- 
 tribution by individuals or by ammains, it would have its own peculiar 
 interest and we would have attempted its construction had the data been 
 adequate for such a purpose. Most of the data bearing on income dis- 
 tribution are in the individual form; wages distributions, for example, are 
 
 1 Of course if the wife or child has "independent" income, that income is no longer of the 
 nature of a gift or allowance even though it may arise from property originally deeded by 
 the husband or father. It is now explainable in terms of rent, interest, etc. 
 
 If income be defined as above, the term personal income recipient will correspond closely 
 to the census expression person gainfully employed. Perhaps the most important difference 
 is that we do not and the Census does include as separate income recipients, farm laborers 
 working on the home farm. 
 
 * Ammain is a word coined by W. I. King and E. Sydenstricker and defined by them, for 
 any given class of people, as "a gross demand for articles of consumption having a total 
 money value equal to that demanded by the average male in that class at the age when his 
 total requirements for expense of maintenance reach a maximum." Measurement of RtHatit* 
 Economic Status of Families. Quarterly Publications of the American Statistical Association, 
 Sept., 1921, p. 852. 
 
 ' It is of course quite possible to estimate the average per ammain income, as has been done 
 by Mr. King; the total income of the people can be divided by the estimated number of 
 ammains in the population. See pages 233 and 234. 
 
THE PROBLEM 343 
 
 almost without exception in that form. Now to estimate the frequency- 
 distribution of income among families from data which, in the first place, 
 are in the individual form and, in the second place, are extremely inade- 
 quate for estimating even the distribution among individuals, could only 
 increase the degree of uncertainty in our results. 
 
 A few words explaining the reason for introducing the next chapter at 
 this point are not out of place here. The data upon which an estimate of 
 even the individual distribution of income in the United States must be 
 based impress one as being in such shape that it is impossible to arrive at 
 more than the roughest sort of approximation by any mere direct adding 
 process. Some more ingenious plan would seem almost necessary. For 
 example, would it not be possible to formulate a general mathematical 
 "law" for the distribution of incomes which law might then be used for 
 "adjusting" the tentative and hypothetical results obtained from piecing 
 together the existing scanty and inadequate material? 
 
 The possibility and desirability of mathematically describing the fre- 
 quency distribution of income would seem intimately tied up with the case 
 for mathematically describing error distributions and statistical distribu- 
 tions in general. The fact that, in our problem, the "law" would be largely 
 derived from the same data as those which were to be "adjusted" need 
 not greatly disturb us. The procedure of adjusting observations in the 
 light of a mathematical expression derived from the same observations is 
 not novel. A number of attempts, one of which has become world-famous, 
 have been made to demonstrate that the distribution of income follows 
 a definite mathematical law. However, the next chapter will show why 
 we fear that no rational and useful mathematical law will soon be formu- 
 lated. 
 
CHAPTER 28 
 
 PARETO'S LAW AND THE GENERAL PROBLEM OF MATHE- 
 MATICALLY DESCRIBING THE FREQUENCY DISTRIBU- 
 TION OF INCOME 
 
 The problem of formulating a mathematical expression which shall de- 
 scribe the frequency distribution of income in all places and at all times, 
 not only closely, but also elegantly, and if possible rationally as opposed 
 to empirically, has had great attractions for the mathematical economist 
 and statistician. The most famous of all attempts at the solution of this 
 fascinating problem are those which have been made by Vilfredo Pareto. 
 Professor Pareto has been intensely interested in this subject for many 
 years and the discussion of it runs through nearly all of his published 
 work. The almost inevitable result is that "Pareto's Law" appears in a 
 number of slightly different forms and Professor Pareto's feelings con- 
 cerning the "law" run all the way from treating it as inevitable and im- 
 mutable to speaking of it as "merely empirical." 
 
 In its best known, most famous, and most dogmatic form, Pareto's Law 
 runs about as follows: 
 
 1. In all countries and at all times the distribution of income is such 
 that the upper (income-tax) ranges of the income frequency distribution 
 curve may be described as follows: If the logarithms of income sizes be 
 charted on a horizontal scale and the logarithms of the numbers of persons 
 having an income of a particular size or over be charted on a vertical 
 scale, then the resulting observational points will lie approximately along 
 a straight line. In other words, if 
 
 x = income size and 
 
 y = number of persons having that income or larger 
 then log y = log b + m log x 
 or y = bx m . 1 
 
 2. In all countries and at all recent times the slope of this straight line 
 fitted to the cumulative distribution, that is, the constant m in the equa- 
 tion y = bx m , will be approximately 1.5. 2 
 
 3. The rigidity and universality of the two preceding conclusions strongly 
 
 1 If the cumulative distribution (cumulating from the higher towards the lower incomes 
 as Pareto does) on a double log scale could be exactly de3cribed by the equation y = bx m , 
 the non-cumulative distribution could be described by the equation Y = — mbx m ~ 1. 
 
 * Strictly, minus 1.5, though Pareto neglects the sign. 
 
 344 
 
PARETO'S LAW 345 
 
 suggest that the shape of the income frequency distribution curve on a 
 double log scale is, for all countries and at all times, inevitably the same 
 not only in the upper (income-tax) range but throughout its entire length. 
 
 4. If then the nature of the whole income frequency distribution is 
 unchanging and unchangeable there is, of course, no possibility of economic 
 welfare being increased through any change in the proportion of the total 
 income going to the relatively poor. Economic welfare can be increased 
 only through increased production. In other words, Pareto's Law in this 
 extreme form constitutes a modern substitute for the Wages Fund Doc- 
 trine. 
 
 This is the most dogmatic form in which the "law" appears. In his 
 later work Professor Pareto drew further and further away from the con- 
 fidence of his first position. He had early stated that the straight line did 
 not seem adequate to describe distributions from all times and places and 
 had proposed more complicated equations. 1 He has held more strongly 
 to the significance of the similarity of slopes but he has wavered in his 
 faith that the lower income portions of the curve (below the income-tax 
 minimum) were necessarily similar for all countries and all times. He has 
 given up the suggestion that existing distributions are inevitable though 
 still speaking of the law as true within certain definite ranges. To translate 
 from his Manuel (p. 391): "Some persons would deduce from it a general 
 law as to the only way in which the inequality of incomes can be dimin- 
 ished. But such a conclusion far transcends anything that can be derived 
 from the premises. Empirical laws, like those with which we are here 
 concerned, have little or no value outside the limits for which they were 
 found experimentally to be true." Indeed Professor Pareto has himself 
 drawn attention to so many difficulties inherent in the crude dogmatic 
 form of the law that this chapter must not be taken as primarily a criticism 
 of his work but rather as a note on the general problem of mathematically 
 describing the frequency distribution of incomes. 
 
 Almost as soon as he had formulated his law Professor Pareto recognized 
 the impossibility of extrapolating the straight line formula into the lower 
 income ranges (outside of the income-tax data which he had been using). 
 The straight line formula involves the absurdity of an infinite number of 
 individuals having approximately zero incomes. Professor Pareto felt 
 that this zero mode with an infinite ordinate was absurd. He believed 
 that the curve must have a definite mode at an income size well above 
 zero 2 and with a finite number of income recipients in the modal group. 
 
 1 The inadequacy of these more complicated equations is discussed later. See pp. 348, 363 
 and 364. 
 
 2 This is, of course, not absolutely necessary. It depends upon our definitions of income 
 and income recipient. If we include the negligible money receipts of young children living 
 at home we might possibly have a mode close to zero. There are few children who do not 
 really earn a few pennies each year. Compare Chart 31A page 416. 
 
346 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 Having come to the conclusion that the income frequency distribution 
 curve must inevitably have a definite mode well above zero income and 
 tail off in both directions from that mode, Professor Pareto was led to 
 think of the possibilities of the simplest of all frequency curves, the normal 
 curve of error. However, after examination and consideration, he felt 
 strongly that the normal curve of error could not possibly be used. He 
 became convinced that the normal curve was not the law of the data for 
 the good and sufficient reason that the part of the data curve given by 
 income-tax returns is of a radically different shape from any part of a 
 normal curve. 1 
 
 Professor Pareto finds a further argument against using the normal curve 
 in the irrationality of such a curve outside the range of the data. 
 The mode of the complete frequency curve for income distribution is at 
 least as low as the minimum taxable income. Income-tax data prove this. 
 However, a normal curve is symmetrical. Hence, if a normal curve could 
 describe the upper ranges of the income curve as given by income-tax data 
 then in the lower ranges it would cut the y axis and pass into the second 
 quadrant, in other words show a large number of negative incomes. 
 
 Now, aside from the fact that this whole argument is unnecessary if 
 the data themselves cannot be described even approximately by a normal 
 curve, Professor Pareto's discussion reveals a curious change in his middle 
 term. If he had said that a symmetrical curve on a natural scale with a 
 mode at least as low as the income-tax minimum would show unbelievably 
 large negative incomes we could follow him but when he states that not 
 only can there be no zero incomes but that there can be no incomes below 
 "the minimum of existence" we realize that he has unconsciously changed 
 the meaning of his middle term. Having examined a mass of income-tax 
 data, all of which were concerned with net money income and from these 
 data having formulated a law, he now apparently without realizing it, 
 changes the meaning of the word income from net money income to money 
 value of commodities consumed, and assumes that those who receive a money 
 income less than a certain minimum must inevitably die of starvation. 
 
 1 Though Pareto seems to have thoroughly understood this fact, his discussion is not al- 
 together satisfactory. He states that the data for the higher incomes show a larger number 
 of such incomes than the normal curve would indicate. This is hardly adequate. To have 
 stated that the upper and lower ranges showed too many incomes as compared with the middle 
 range would have been better. An easy way to realize clearly the impossibility of describing 
 income-tax data by a normal curve is to plot a portion of the non-cumulative data on a natural 
 x log y basis. When so charted the data present a concave shaped curve. However, if the 
 data were describable by any part of a normal curve of error, they would show a convex ap- 
 pearance, or in the limiting case a straight line, as the equation of the normal curve of error 
 — x 1 
 
 \Vx = Voe a ) becomes, on a natural x log y scale, log^/i = logey — 9— , or a second degree 
 
 parabola whose axis is perpendicular to the x axis of coordinates. 
 
 The reader must note that the limiting straight line case mentioned above is on a natural 
 x log y scale and not (as the Pareto straight line) on a log x log y scale. (Note concluded 
 page 347.) 
 
PARETO'S LAW 
 
 347 
 
 Children receive in general negligible money incomes. Many other persons 
 in the community are in the same position. A business man may "lose 
 money" in a given year, in other words he may have a negative money 
 income. There seems no essential absurdity in assuming that a large 
 number of persons receive money incomes much less than necessary to 
 
 (Note 1 page 346 concluded.) 
 
 Chart 28A showing curves fitted to observations on the heights of men illustrates the ap- 
 pearance of the normal curve on a natural scale and on a natural x log y scale. That chart 
 also illustrates another fact of importance in this discussion, namely, that fitting to a different 
 function of the variable gives a different fit. 
 
 -ISO 
 
 -120 
 -90 
 
 DISTRIBUTION OF HEIGHTS OF 1078 MEN. 
 
 B/OMETR/Kfl VOLU P4/5 (FffTHERS) 
 
 1-Normal Curve Fitted to Natural Scale 
 Data by Method of Moments, also 
 Loss of Same. 
 
 2-5econd Degree Parabola Fitted to 
 Natural x Log Y Data by Method of 
 Least Squares, also Antiloss of Same. 
 
 CHART 28A 
 
 msf sss eas els as ess e*s ess eks 675 ess eas x.s 71.5 7&s 7J.5 T*s 7ss a 
 
 HSIGHT l/i /IYCM£S 
 
 SS.S S3S 60S 6'5 CIS 635 6*5 6SS 665 67.5 685 695 ZIS 715 72.5 
 
 HEIGHT IN INCHES 
 
348 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 support existence. When in 1915 Australia took a census of the incomes 
 of all persons "possessed of property, or in receipt of income," over 14 
 per cent of the returns showed incomes "deficit and nil." 1 
 
 Professor Pareto's realization of the impossibility of describing income 
 distributions by means of normal curves led him to the curious conclusion 
 that such distributions were somehow unique and could not be explained 
 upon any "chance" hypothesis. "The shape of the curve which is fur- 
 nished us by statistics, does not correspond at all to the curve of errors, 
 that is to say 2 to the form which the curve would have if the acquisition 
 and conservation of wealth depended only on chance." 3 Moreover, while 
 Professor Pareto's further suggestion of possible heterogeneity in the data 
 corresponds we believe to the facts, his reason for making such a sug- 
 gestion, namely that the data cannot be adequately described by a 
 normal curve, is irrelevant. 4 "Chance" data distributions are no longer 
 thought of as necessarily in any way similar to the normal curve. Even 
 error distributions commonly depart widely from the normal curve. 
 The best known system of mathematical frequency curves, that of 
 Karl Pearson, is intended to describe homogeneous material and is 
 based upon a probability foundation, yet the normal curve is only 
 one of the many and diverse forms yielded by his fundamental 
 
 d log y x + a , 
 
 equation — = ■ -. 5 
 
 dx b -{- bix + b 2 x 2 
 
 While Pareto's Law in its straight line form was at least an interesting 
 suggestion, his efforts to amend the law have not been fruitful. His at- 
 tempts to substitute \og e N = loggA — a log e (x + a) or even loggiV = 
 logeA — a log^x + a) — (3x for the simpler log N = log A — a log £ 
 have not materially advanced the subject. 6 The more complicated curves 
 have the same fundamental drawbacks as the simpler one. Among other 
 peculiarities they involve the same absurdity of an infinite number of 
 persons in the modal interval and none below the mode. Along with the 
 doubling of the number of constants, there comes of course the possibility 
 of improving the fit within the range of the data. Such improvement is, 
 however, purely artificial and empirical and without special significance, 
 as can be easily appreciated by noticing the mathematical characteristics 
 of the equation. 
 
 A number of other statisticians have at various times fitted different 
 types of frequency curves to distributions of income, wages, rents, wealth, 
 
 * Compare Table 29A. 
 
 * My italics. 
 
 » Manuel, p. 385. See also Cours, pp. 416 and 417. 
 
 * Vid. Cours, pp. 416 and 417. 
 
 * Professor A. W. Flux in a review of Pareto's Cours d'Economie Politique (Economic Journal, 
 March, 1897) drew attention to the inadequacy of Pareto's conception of what were and what 
 were not "chance" data. 
 
 * Cf. Cours, vol. II, p. 305, note. 
 
PARETO'S LAW 349 
 
 or allied data. 1 However, no one has advanced such claims for a "law" 
 of income 2 distribution as were at one time made by Professor Pareto. 
 When considering the possibility of helpfully describing the distribution 
 of income by any simple mathematical expression, one inevitably begins 
 by examining "Pareto's Law." It is so outstanding. Let us therefore 
 examine Pareto's Law. 
 
 1. Do income distributions, when plotted on a double log scale, 
 approximate straight lines closely enough to give such approxi- 
 mation much significance? 
 
 Before attempting to answer this question it is of course necessary to 
 decide how we shall obtain the straight line with which comparisons are 
 to be made. 
 
 Professor Pareto fitted straight lines directly by the method of least 
 squares to the cumulative distribution plotted on a double log scale. The 
 disadvantage of this procedure is that, though one may obtain the straight 
 line which best fits the cumulative distribution, such a straight line may be 
 anything but an admirable fit to the non-cumulative figures. For example, 
 if a straight line be fitted by the method of least squares to Prussian re- 
 turns for 1886 (as given by Professor Pareto) the total number of income 
 recipients within the range of the data is, according to the fitted straight 
 line, only 5,399,000 while the actual number of returns was 5,557,000, 
 notwithstanding the fact that Prussia, 1886, is a sample which runs much 
 more nearly straight than is usual. How bad the discrepancy may be 
 where the data do not even approximate a straight line is seen in Professor 
 Pareto's Oldenburg material. There the least-squares straight line fitted 
 to the cumulative distribution on a double log scale gives 91,222 persons 
 having incomes over 300 marks per annum while the data give only 54,309. 
 
 1 Among others, Karl Pearson, F. Y. Edgeworth, Henry L. Moore, A. L. Bowley, Lucien 
 March, J. C. Kapteyn, C. Bresciani, C. Gini, F. Savorgnan. 
 
 2 Professor H. L. Moore, in his Laws of Wages, is concerned primarily with wages not 
 income. 
 
 Professor J. C. Kapteyn has presented a pretty but somewhat hypothetical argument sug- 
 gesting that the skewness in the income frequency curve should be such that plotting on a 
 log x basis would eliminate it. 
 
 "In several cases we feel at once that the effect of the causes of deviation cannot be inde- 
 pendent of the dimension of the quantities observed. In such cases we may conclude at once 
 that the frequency curve will be a skew one. To take a single example: 
 
 "Suppose 1000 men to begin trading, each with the same capital; in order to see how their 
 wealth will be distributed after the lapse of 10 years, consider first what will be their condition 
 at some earlier epoch, say at the end of the fifth year. 
 
 "We may admit that a certain trader A will then only possess a capital of £100, while 
 another may possess £100,000. 
 
 "Now if a certain cause of gain or loss comes to operate, what will happen? 
 
 "For instance: Let the price of an article in which both A and B have invested their capital, 
 rise or fall. Then it will be evident that if the gain or loss of A be £10, that of B will not be 
 £10, but £10,000; that is to say, the effect of this cause will not be independent of the capital, 
 but proportional to it." 
 
 J. C. Kapteyn, Skew Frequency Curves in Biology and Statistics, p. 13. 
 
350 PERSONAL DISTRIBUTION OF INCOxME IN U. S. 
 
 The reason for this peculiarity of the fit to the cumulative distribution 
 becomes clear when we remember that the least-squares straight line may 
 easily deviate widely from the first datum point while a straight line giving 
 the same number of income recipients as the data must necessarily pass 
 through the first datum point. 1 
 
 A straight line fitted in such a manner that the total number of per- 
 sons and total amount of income correspond to the data for these items 
 gives what seems a much more intelligible fit. Charts 28B to 28G show 
 cumulative United States frequency distributions from the income-tax 
 returns for the years 1914 to 1919 on a double log scale (Professor Pareto's 
 suggestion). Two straight lines are fitted to each distribution — one a 
 solid least-squares line fitted to the cumulative data points and the 
 other a dotted line so fitted that the total number of persons and total 
 amount of income correspond to the data figures. While the least-squares 
 line may appear much the better fit to these cumulative data, a mere 
 glance at Tables 28B to 28G will reveal the fact that such a line is, to 
 say the least, a less interpretable fit to the non-cumulative distribution. 2 
 It is, of course, evident that neither line is in any year a sufficiently good 
 fit to the actual non-cumulative distribution to have much significance. 
 No mathematics is necessary to demonstrate this. 3 
 
 1 e. g. in the case of Prussia, 1886, the first datum point is x = "over 300M" and y = 54,309 
 persons. 
 
 2 Professor Warren M. Persons discussed the fit of the least-squares straight line to Professor 
 Pareto's Prussian data for 1892 and 1902 in the Quarterly Journal of Economics, May, 1909, 
 and demonstrated the badness of fit of that line to those data. 
 
 5 The income returned for the years 1914 and 1915 was estimated from the number of re- 
 turns. Income is not given in the reports for those years. 
 
 In fitting straight lines to the data of Tables 28B to 28G the lowest income interval (in 
 which married persons making a joint return are exempt) has always been omitted. To have 
 included in our calculations these lowest intervals would have increased still further the bad- 
 ness of the fit in the other intervals. 
 
PARETO'S LAW 
 
 351 
 
 CHART 28 B 
 
 - 1,000,000 
 
 - 100,000 
 
 - 10,000 
 
 -l.ooo S 
 
 -10 
 
 UNITED STATES INCOME TAX RETURNS 
 1314 
 
 CUMULATIVE FREQUENCY DISTRIBUTION 
 
 AND 
 
 FITTED (Least Squares) STRAIGHT LINE 
 
 Scales Logarithmic. 
 
 3 4 5 
 
 I I A. 
 
 INCOME IN THOUSANDS OF DOLLARS 
 10 20 30 40 50 100 200 
 
 I lilt I I 
 
 300 400 500 1,000 2,000 3,000 
 
352 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 
 CHART 28C 
 
 
 UNITED STATES INCOME TAX RETURNS 
 1915 
 
 
 CUMULATIVE FREQUENCY DISTRIBUTION 
 
 
 AMD 
 
 -1,000,000 
 
 FITTED Ueast Squares) STRAIGHT LINE 
 
 
 Scales logarithmic. 
 
 -100,000 
 
 *1||^ 
 
 - 10,000 s 
 
 1 
 
 
 -1,000 ■ 
 
 ^y. 
 
 B 
 
 m 
 
 
 -10Q 
 
 >sr 
 
 -10 
 
 
 2 3 4 5 
 
 INCOME IN THOUSANDS OF DOLLARS 
 10 20 30 40 50 100 200 300 400500 1,000 2.000 
 
PARETO'S LAW 
 
 353 
 
 
 
 CHART 28 D 
 
 
 
 UNITED STATES INCOME TAX RETURNS 
 
 
 
 1316 
 
 
 
 CUMULATIVE FREQUENCY DISTRIBUTION 
 
 
 
 AND 
 
 -1,000,000 
 
 
 TITTED (Least Squares) STRAIGHT LINE 
 Scales Logarithmic 
 
 -100,000 
 
 ^> 
 
 V,. 
 
 URNS 
 
 
 
 
 
 ^v 
 
 Ex. 
 
 
 ^Ov 
 
 O 
 
 
 
 6S 
 
 
 
 - 1,000 B 
 
 
 
 B 
 
 
 ^^^> 
 
 3 
 
 
 
 K 
 
 
 ^sN v 
 
 100 
 
 
 ^^ s 
 
 10 
 
 
 INCOME IN THOUSANDS OF DOLLARS 
 
 2 3 4 5 
 
 10 
 
 20 3,0 40 50 100 200 300 400 500 1,000 2/100 3000 IOO05j00O 
 
354 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 
 
 CHART 28 E 
 
 
 
 UNITED STATES INCOME TAX RETURNS 
 
 
 
 1917 
 
 
 
 CUMULATIVE f REQUENCY DISTRIBUTION 
 
 
 
 AND 
 
 •1,000,000^5. 
 
 
 FITTED (Least Squares) STRAIGHT LINE 
 
 
 . ScalesLogarithmic 
 
 -100,000 
 
 
 
 ■10,000 
 
 
 ^V 
 
 CO 
 
 
 <4NK 
 
 I 
 
 
 
 ■ 
 
 
 >&. 
 
 -1,000 ■ 
 
 1 — 
 
 
 
 
 
 
 
 OS 
 
 
 
 1 
 
 
 
 a 
 
 
 ^f 
 
 5 
 
 
 
 ■ 
 
 
 *^ 
 
 -100 
 
 
 N^^v* 
 
 -10 
 
 
 INCOME IN THOUSANDS OF DOLLARS 
 
 2 345 
 
 10 
 
 20 30 40 50 100 200 300 400500 1,000 2,000 3.0004,000 
 
PARETO'S LAW 
 
 355 
 
 CHART 28 F 
 
 ■ 1,000,000 
 
 100,000 
 
 10,000 
 
 1.000 o 
 
 UNITED STATES INCOME TAX RETURNS 
 1918 
 
 CUMULATIVE FREQUENCY DISTRIBUTION 
 
 AND 
 
 FITTED (Least Souares) STRAIGHT LINE 
 Scales logarithmic. 
 
 2 3 4 5 
 
 INCOME IN THOUSANDS OF DOLLARS 
 10 20 30 40 50 100 200 300 400 500 1,000 2,000 3,000 
 
 I I !■» * . l . 1 1 1 — fci^— —MMi^B^^M 
 
356 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 
 CHART 28 G 
 
 
 UNITED STATES INCOME TAX RETURNS 
 
 
 1919 
 
 
 CUMULATIVE FREQUENCY DISTRIBUTION 
 
 
 AMD 
 
 - 1,000,000 ^^ 
 
 FITTED (Least Squares) STRAIGHT LIKE 
 
 
 Scales Logarithmic 
 
 -100,000 
 
 
 - lO.OOOg 
 
 
 as 
 j 
 
 ^V 
 
 O 
 
 
 o 
 o 
 
 NUMBER 
 
 N^ 
 
 -100 
 
 >£s». 
 
 -10 
 
 *0\ S% s 
 
 2 3 4 5 
 
 INCOME IN THOUSANDS OF DOLLARS 
 
 10 20 30 40 SO 100 200 300 400500 1,000 2,000 3.000 
 
PARETO'S LAW 
 
 357 
 
 TABLE 28B 
 
 UNITED STATES INCOME-TAX RETURNS, 1914 
 
 
 A 
 
 B 
 
 C 
 
 
 
 
 
 
 Straight line 
 
 Per 
 
 Per 
 
 Income class 
 
 U. S. in- 
 come-tax 
 returns 
 
 Least-squares 
 straight line 
 
 giving 
 correct total 
 returns and 
 
 income 
 
 cent 
 A is 
 of B 
 
 cent 
 A is 
 of C 
 
 $ 3,000-$ 4,000 
 
 (82,754) 
 
 
 
 
 
 4,000- 5,000 
 
 66,525 
 
 101,241 
 
 84,683 
 
 65.7 
 
 78.6 
 
 5,000- 10,000 
 
 127,448 
 
 160,545 
 
 115,347 
 
 79.4 
 
 110.5 
 
 10,000- 15,000 
 
 34,141 
 
 38,630 
 
 32,716 
 
 88.4 
 
 104.4 
 
 15,000- 20,000 
 
 15,790 
 
 15,853 
 
 14,102 
 
 99.6 
 
 112.0 
 
 20,000- 25,000 
 
 8,672 
 
 8,230 
 
 7,589 
 
 105.4 
 
 114.3 
 
 25,000- 30,000 
 
 5,483 
 
 4,879 
 
 4,631 
 
 112.4 
 
 118.4 
 
 30,000- 40,000 
 
 6,008 
 
 5,380 
 
 5,267 
 
 111.7 
 
 114.1 
 
 40,000- 50,000 
 
 3,185 
 
 2,793 
 
 2,835 
 
 114.0 
 
 112.3 
 
 50,000- 100,000 
 
 5,161 
 
 4,430 
 
 4,756 
 
 116.5 
 
 108.5 
 
 100,000- 150,000 
 
 1,189 
 
 1,065.5 
 
 1,241 
 
 111.6 
 
 95.8 
 
 150,000- 200,000 
 
 406 
 
 437.3 
 
 535 
 
 92.8 
 
 75.9 
 
 200,000- 250,000 
 
 233 
 
 227.1 
 
 288.1 
 
 102.6 
 
 80.9 
 
 250,000- 300,000 
 
 130 
 
 134.6 
 
 175.5 
 
 96.6 
 
 74.1 
 
 300,000- 400,000 
 
 147 
 
 148.46 
 
 199.9 
 
 99.0 
 
 73.5 
 
 400,000- 500,000 
 
 69 
 
 77.06 
 
 107.6 
 
 89.5 
 
 64.1 
 
 500,000-1,000,000 
 
 114 
 
 122.20 
 
 180.4 
 
 93.3 
 
 63.2 
 
 1,000,000 and over 
 
 60 
 
 62.78 
 
 107.5 
 
 95.6 
 
 55.8 
 
 Total (over $4,000) 
 
 274,761 
 
 344,256.00 
 
 274,761.0 
 
 
 
358 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 TABLE 28C 
 
 UNITED STATES INCOME-TAX RETURNS, 1915 
 
 
 A 
 
 B 
 
 C 
 
 
 
 Income class 
 
 U. S. in- 
 come-tax 
 returns 
 
 Least- 
 squares 
 straight line 
 
 Straight line 
 
 giving 
 
 correct total 
 
 returns and 
 
 income 
 
 Per 
 
 cent 
 "A is 
 of B 
 
 Per 
 
 cent 
 A is 
 of C 
 
 $ 3,000-$ 4,000 
 
 4,000- 5,000 
 
 5,000- 10,000 
 
 10,000- 15,000 
 
 15,000- 20,000 
 
 20.000- 25,000 
 
 25,000- 30,000 
 
 30,000- 40,000 
 
 40,000- 50,000 
 
 50,000- 100,000 
 
 100,000- 150,000 
 
 150,000- 200,000 
 
 200,000- 250,000 
 
 250,000- 300,000 
 
 300,000- 400,000 
 
 400,000- 500,000 
 
 500,000-1,000,000 
 
 1,000,000 and over 
 
 (69,045) 
 
 58,949 
 
 120,402 
 
 34,102 
 
 16,475 
 
 9,707 
 
 6,196 
 
 7,005 
 
 4,100 
 
 6,847 
 
 1,793 
 
 724 
 
 386 
 
 216 
 
 254 
 
 122 
 
 209 
 
 120 
 
 92,064 
 154,507 
 40,358 
 17,406 
 9,372 
 5,716 
 6,508 
 3,503 
 5,880 
 1,536 
 662.5 
 356.6 
 217.5 
 247.7 
 133.3 
 223.8 
 133.6 
 
 68,540 
 119,634 
 33,013 
 14,724 
 8,124 
 5,050 
 5,875 
 3,241 
 5,653 
 1,530 
 695.4 
 383.8 
 238.6 
 277.6 
 153.2 
 267.1 
 177.3 
 
 64.0 
 
 77.9 
 
 84.5 
 
 94.7 
 
 103.6 
 
 108.4 
 
 107.6 
 
 117.0 
 
 116.4 
 
 116.7 
 
 103.3 
 
 108.2 
 
 99.3 
 
 102.5 
 
 91.5 
 
 93.4 
 
 89.8 
 
 86.0 
 
 100.6 
 
 103.3 
 
 111.9 
 
 119.5 
 
 122.7 
 
 119.2 
 
 126.5 
 
 121.1 
 
 114.9 
 
 104.1 
 
 100.6 
 
 90.5 
 
 91.5 
 
 79.6 
 
 78.2 
 
 67.7 
 
 Total (over $4,000) 
 
 287,607 
 
 338,825.0 
 
 267,607.0 
 
 
 
PARETO'S LAW 
 
 359 
 
 TABLE 28D 
 
 UNITED STATES INCOME-TAX RETURNS, 1916 
 
 
 A 
 
 B 
 
 C 
 
 
 
 Income class 
 
 U. S. in- 
 come-tax 
 returns 
 
 Least-squares 
 straight line 
 
 Straight line 
 giving correct 
 total returns 
 
 Per 
 
 cent 
 Ah 
 
 Per 
 
 cent 
 A is 
 
 
 
 and income 
 
 of B 
 
 of C 
 
 $ 3,000-$ 4,000 
 
 (85,122) 
 
 
 
 
 
 4,000- 5,000 
 
 72,027 
 
 139,096 
 
 86,588 
 
 51.8 
 
 83.2 
 
 5,000- 6,000 
 
 52,029 
 
 84,759 
 
 54,221 
 
 61.4 
 
 96.0 
 
 6,000- 7,000 
 
 36,470 
 
 56,533 
 
 36,899 
 
 64.5 
 
 98.8 
 
 7,000- 8,000 
 
 26,444 
 
 39,846 
 
 26,516 
 
 66.4 
 
 99.7 
 
 8,000- 9,000 
 
 19,959 
 
 29,292 
 
 19,801 
 
 68.1 
 
 100.8 
 
 9,000- 10,000 
 
 15,651 
 
 22,529 
 
 15,445 
 
 69.5 
 
 101.3 
 
 10,000- 15,000 
 
 45,309 
 
 60,668 
 
 42,879 
 
 74.7 
 
 105.7 
 
 15,000- 20,000 
 
 22,618 
 
 26,120 
 
 19,311 
 
 86.6 
 
 117.1 
 
 20,000- 25,000 
 
 12,953 
 
 14,044 
 
 10,726 
 
 92.2 
 
 120.8 
 
 25,000- 30,000 
 
 8,055 
 
 8,558 
 
 6,705 
 
 94.1 
 
 120.1 
 
 30,000- 40,000 
 
 10,068 
 
 9,731 
 
 7,854 
 
 103.5 
 
 128.2 
 
 40,000- 50,000 
 
 5,611 
 
 5,232 
 
 4,362 
 
 107.2 
 
 128.6 
 
 50,000- 60,000 
 
 3,621 
 
 3,189 
 
 2,730 
 
 113.5 
 
 132.6 
 
 60,000- 70,000 
 
 2,548 
 
 2,126 
 
 1,857 
 
 119.8 
 
 137.2 
 
 70,000- 80,000 
 
 1,787 
 
 1,499 
 
 1,334.8 
 
 119.2 
 
 133.9 
 
 80,090- 90,000 
 
 1,422 
 
 1,102 
 
 996.8 
 
 129.0 
 
 142.7 
 
 90,000- 100,000 
 
 1,074 
 
 847 
 
 777.5 
 
 123.8 
 
 138.1 
 
 100,000- 150,000 
 
 2,900 
 
 2,282.1 
 
 2,158.4 
 
 127.1 
 
 134.4 
 
 150,000- 200,000 
 
 1,284 
 
 982.6 
 
 972.1 
 
 130.7 
 
 132.1 
 
 200,000- 250,000 
 
 726 
 
 528.2 
 
 539.9 
 
 137.4 
 
 134.5 
 
 250,000- 300,000 
 
 427 
 
 321.9 
 
 337.6 
 
 132.6 
 
 126.5 
 
 300,000- 400,000 
 
 469 
 
 366.1 
 
 395.3 
 
 123.1 
 
 118.6 
 
 400,000- 500,000 
 
 245 
 
 198.8 
 
 219.6 
 
 124.5 
 
 111.6 
 
 500,000-1,000,000 
 
 376 
 
 329.6 
 
 387.4 
 
 114.1 
 
 97.1 
 
 1,000,000-1,500,000 
 
 97 
 
 85.83 
 
 108.7 
 
 113.0 
 
 89.2 
 
 1,500,000-2,000,000 
 
 42 
 
 36.96 
 
 48.88 
 
 113.6 
 
 85.9 
 
 2,000,000-3,000,000 
 
 34 
 
 31.98 
 
 44.19 
 
 106.3 
 
 76.9 
 
 3,000,000-4,000,000 
 
 14 
 
 13.77 
 
 19.91 
 
 101.7 
 
 70.3 
 
 4,000,000-5,000,000 
 
 9 
 
 7.40 
 
 11.05 
 
 121.6 
 
 81.4 
 
 5,000,000 and over 
 
 10 
 
 19.76 
 
 32.87 
 
 50.6 
 
 30.4 
 
 Total (over $4,000) 
 
 344,279 
 
 510,374.00 
 
 344,279.00 
 
 
 
360 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 TABLE 28E 
 
 UNITED STATES INCOME-TAX RETURNS, 1917 
 
 
 A 
 
 B 
 
 C 
 
 
 
 
 U.S. 
 
 income-tax 
 returns 
 
 
 Straight line 
 
 Per 
 
 Per 
 
 Income class 
 
 Least-squares 
 straight line 
 
 giving correct 
 total returns 
 
 cent 
 A is 
 
 cent 
 A is 
 
 
 
 and income 
 
 of B 
 
 of C 
 
 $ 1,000-$ 2,000 
 
 (1,640,758) 
 
 
 
 
 
 2,000- 2,500 
 
 480,486 
 
 618,069 
 
 517,512 
 
 77.7 
 
 92.8 
 
 2,500- 3,000 
 
 358,221 
 
 367,835 
 
 284,620 
 
 97.4 
 
 125.9 
 
 3,000- 4,000 
 
 374,958 
 
 407,366 
 
 376,117 
 
 92.0 
 
 99.7 
 
 4,000- 5,000 
 
 185,805 
 
 212,569 
 
 184,854 
 
 87.4 
 
 100.5 
 
 5,000- 6,000 
 
 105,988 
 
 126,507 
 
 111,097 
 
 83.8 
 
 95.4 
 
 6,000- 7,000 
 
 64,010 
 
 82,746 
 
 73,355 
 
 77.4 
 
 87.3 
 
 7,000- 8,000 
 
 44,363 
 
 57,357 
 
 51,285 
 
 77.3 
 
 86.5 
 
 8,000- 9,000 
 
 31,769 
 
 41,556 
 
 37,362 
 
 76.4 
 
 85.0 
 
 9,000- 10,000 
 
 24,536 
 
 31,551 
 
 28,551 
 
 77.8 
 
 85.9 
 
 10,000- 11,000 
 
 19,221 
 
 24,097 
 
 21,900 
 
 79.8 
 
 87.8 
 
 11,000- 12,000 
 
 15,035 
 
 19,412 
 
 17,747 
 
 77.5 
 
 84.7 
 
 12,000- 13,000 
 
 12,328 
 
 15,707 
 
 14,440 
 
 78.5 
 
 85.4 
 
 13,000- 14,000 
 
 10,427 
 
 12,751 
 
 11,761 
 
 81.8 
 
 88.7 
 
 14,000- 15,000 
 
 8,789 
 
 10,709 
 
 9,909 
 
 82.1 
 
 88.7 
 
 15,000- 20,000 
 
 29,896 
 
 34,161 
 
 31,891 
 
 87.5 
 
 93.7 
 
 20,000- 25,000 
 
 16,806 
 
 17,825 
 
 16,876 
 
 94.3 
 
 99.6 
 
 25,000- 30,000 
 
 10,571 
 
 10,609 
 
 10,159 
 
 99.6 
 
 104.1 
 
 30,000- 40,000 
 
 12,733 
 
 11,749 
 
 11,385 
 
 108.4 
 
 111.8 
 
 40,000- 50,000 
 
 7,087 
 
 6,130 
 
 6,021 
 
 115.6 
 
 117.7 
 
 50,000- 60,000 
 
 4,541 
 
 3,649 
 
 3,622 
 
 124.4 
 
 125.4 
 
 60,000- 70,000 
 
 2,954 
 
 2,387 
 
 2,391 
 
 123.8 
 
 123.5 
 
 70,000- 80,000 
 
 2,222 
 
 1,653.5 
 
 1,672 
 
 134.4 
 
 132.9 
 
 80,000- 90,000 
 
 1,539 
 
 1,198.5 
 
 1,217.9 
 
 128.4 
 
 126.4 
 
 90,000- 100,000 
 
 1,183 
 
 910.0 
 
 930.8 
 
 130.0 
 
 127.1 
 
 100,000- 150,000 
 
 3,302 
 
 2,384.4 
 
 2,469.5 
 
 138.5 
 
 133.7 
 
 150,000- 200,000 
 
 1,302 
 
 985.2 
 
 1,039.6 
 
 132.2 
 
 125.2 
 
 200,000- 250,000 
 
 703 
 
 514.1 
 
 550.5 
 
 136.7 
 
 127.7 
 
 250,000- 300,000 
 
 342 
 
 305.9 
 
 330.8 
 
 111.8 
 
 103.4 
 
 300,000- 400,000 
 
 380 
 
 338.9 
 
 371.2 
 
 112.1 
 
 102.4 
 
 400,000- 500,000 
 
 179 
 
 176.8 
 
 196.3 
 
 101.2 
 
 91.2 
 
 500,000- 750,000 
 
 225 
 
 199.96 
 
 225.56 
 
 112.5 
 
 99.8 
 
 750,000-1,000,000 
 
 90 
 
 82.61 
 
 94.97 
 
 108.9 
 
 94.8 
 
 1,000,000-1,500,000 
 
 67 
 
 68.77 
 
 80.51 
 
 97.4 
 
 83.2 
 
 1,500,000-2,000,000 
 
 33 
 
 28.42 
 
 ' 33.90 
 
 116.1 
 
 97.3 
 
 2,000,000-3,000,000 
 
 24 
 
 23.65 
 
 28.71 
 
 101.5 
 
 83.6 
 
 3,000,000-4,000,000 
 
 5 
 
 9.77 
 
 12.10 
 
 51.2 
 
 41.3 
 
 4,000,000-5,000,000 
 
 8 
 
 5.10 
 
 6.40 
 
 156.9 
 
 125.0 
 
 5,000,000 and over 
 
 4 
 
 12.42 
 
 16.25 
 
 32.2 
 
 24.6 
 
 Total (over $2,000) 
 
 1,832,132 
 
 2,123,640.00 
 
 1,832,132.00 
 
 
 
PARETO'S LAW 
 
 331 
 
 TABLE 28F 
 
 UNITED STATES INCOME-TAX RETURNS, 1918 
 
 Income class 
 
 $ 1,000-$ 2,000 
 
 2,000- 3,000 
 
 3,000- 4,000 
 
 4,000- 5,000 
 
 5,000- 6,000 
 
 6,000- 7,000 
 
 7,000- 8,000 
 
 8,000- 9,000 
 
 9,000- 10,000 
 
 10,000- 11,000 
 
 11,000- 12,000 
 
 12,000- 13,000 
 
 13,000- 14,000 
 
 14,000- 15,000 
 
 15,000- 20,000 
 
 20,000- 25,000 
 
 25,000- 30,000 
 
 30,000- 40,000 
 
 40,000- 50,000 
 
 50,000- 60,000 
 
 60,000- 70,000 
 
 70,000- 80,000 
 
 80,000- 90,000 
 
 90,000- 100,000 
 
 100,000- 150,000 
 
 150,000- 200,000 
 
 200,000- 250,000 
 
 250,000- 300,000 
 
 300,000- 400,000 
 
 400,000- 500,000 
 
 500,000- 750,000 
 
 750,000-1,000,000 
 
 1,000,000-1,500,000 
 
 1,500,000-2,000,000 
 
 2,000,000-3,000,000 
 
 3,000,000-4,000,000 
 
 4,000,000-5,000,000 
 
 5,000,000 and over 
 
 U.S. 
 
 income-tax 
 
 returns 
 
 (1,516,938) 
 
 1,496,878 
 
 610,095 
 
 322,241 
 
 126,554 
 
 79,152 
 
 51,381 
 
 35,117 
 
 27,152 
 
 20,414 
 
 16,371 
 
 13,202 
 
 10,882 
 
 9,123 
 
 30,227 
 
 16,350 
 
 10,206 
 
 11,887 
 
 6,449 
 
 3,720 
 
 2,441 
 
 1,691 
 
 1,210 
 
 934 
 
 2,358 
 
 866 
 
 401 
 
 247 
 
 260 
 
 122 
 
 132 
 
 46 
 
 33 
 
 16 
 
 11 
 
 4 
 
 2 
 
 1 
 
 Least-squares 
 straight line 
 
 Straight line 
 giving correct 
 total returns 
 
 and income 
 
 Per 
 
 cent 
 A i3 
 of B 
 
 1,375,372 
 537,892 
 269,674 
 155,513 
 99,102 
 67,184 
 47,740 
 35,628 
 26,793 
 21,283 
 16,999 
 13,638 
 11,328 
 35,214 
 17,654 
 10,181 
 10,886 
 5,458 
 3,147 
 2,006 
 1,359.5 
 966.2 
 721.0 
 1,822.3 
 712.7 
 357.3 
 208.0 
 220.3 
 110.5 
 119.28 
 46.66 
 36.88 
 14.42 
 11.40 
 4.46 
 2.24 
 4.86 
 
 1,470,366 
 566,044 
 280,477 
 160,366 
 101,389 
 68,258 
 48,266 
 35,795 
 26,832 
 21,231 
 16,873 
 13,515 
 11,165 
 34,486 
 17,097 
 9,762 
 10,336 
 5,121 
 2,928 
 1,852 
 1,246 
 881.4 
 653.7 
 1,636.3 
 629.8 
 312.1 
 178.3 
 188.7 
 93.55 
 99.70 
 38.36 
 29.88 
 11.50 
 8.96 
 3.44 
 • 1.71 
 3.60 
 
 108.8 
 
 113.4 
 
 119.5 
 
 81.4 
 
 79.9 
 
 76.5 
 
 73.6 
 
 76.2 
 
 76.2 
 
 76.9 
 
 77.7 
 
 79.8 
 
 80.5 
 
 85.8 
 
 92.6 
 
 100.2 
 
 109.2 
 
 118.2 
 
 118.2 
 
 121.7 
 
 124.4 
 
 125.2 
 
 129.5 
 
 129.4 
 
 121.5 
 
 112.2 
 
 119.9 
 
 118.0 
 
 110.4 
 
 110.7 
 
 98.6 
 
 89.5 
 
 111.0 
 
 96.5 
 
 89.7 
 
 89.3 
 
 20.6 
 
 Per 
 
 cent 
 A is 
 of C 
 
 101.8 
 
 107.8 
 
 114.9 
 
 78.9 
 
 78.1 
 
 75.3 
 
 72.8 
 
 75.9 
 
 76.1 
 
 77.1 
 
 78.2 
 
 80.5 
 
 81.7 
 
 87.7 
 
 95.6 
 
 104.5 
 
 115.0 
 
 125.9 
 
 127.0 
 
 131.8 
 
 135.7 
 
 137.3 
 
 142.9 
 
 144.1 
 
 137.5 
 
 128.5 
 
 138.5 
 
 137.8 
 
 130.4 
 
 132.4 
 
 119.9 
 
 110.4 
 
 139.1 
 
 122.8 
 
 116.3 
 
 117.0 
 
 27.8 
 
 Total (over $2,000) 
 
 2,908,176 
 
 2,769,408.00 
 
 2,908,176.00 
 
362 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 TABLE 28G 
 
 UNITED STATES INCOME-TAX RETURNS, 1919 
 
 
 A 
 
 B 
 
 C 
 
 
 
 
 U. S. 
 
 
 Straight line 
 
 Per 
 
 Per 
 
 Income class 
 
 income-tax 
 returns 
 
 Least-squares 
 straight line 
 
 giving correct 
 total returns 
 
 cent 
 A is 
 
 cent 
 A is 
 
 
 
 and income 
 
 of B 
 
 of C 
 
 $ 1,000-$ 2,000 
 
 (1,924,872) 
 
 
 
 
 
 2,000- 3,000 
 
 1,569,741 
 
 1,984,285 
 
 1,673,688 
 
 79.1 
 
 93.8 
 
 3,000- 4,000 
 
 742,334 
 
 764,739 
 
 660,950 
 
 97.1 
 
 112.3 
 
 4,000- 5,000 
 
 438,154 
 
 379,330 
 
 333,645 
 
 115.5 
 
 131.3 
 
 5,000- 6,000 
 
 167,005 
 
 216,921 
 
 193,470 
 
 77.0 
 
 86.3 
 
 6,000- 7,000 
 
 109,674 
 
 137,278 
 
 123,953 
 
 79.9 
 
 88.5 
 
 7,000- 8,000 
 
 73,719 
 
 92,511 
 
 84,273 
 
 79.7 
 
 87.5 
 
 8,000- 9,000 
 
 50,486 
 
 65,403 
 
 60,066 
 
 77.2 
 
 84.1 
 
 9,000- 10,000 
 
 37,967 
 
 48,583 
 
 44,980 
 
 78 1 
 
 84.4 
 
 10,000- 11,000 
 
 28,499 
 
 36,386 
 
 33,887 
 
 78.3 
 
 84.1 
 
 11,000- 12,000 
 
 22,841 
 
 28,796 
 
 27,027 
 
 79.3 
 
 84.5 
 
 12,000- 13,000 
 
 18,423 
 
 22,921 
 
 21,600 
 
 80.4 
 
 85.3 
 
 13,000- 14,000 
 
 15,248 
 
 18,329 
 
 17,395 
 
 83.2 
 
 87.7 
 
 14,000- 15,000 
 
 12,841 
 
 15,181 
 
 14,459 
 
 84.6 
 
 88.8 
 
 15,000- 20,000 
 
 42,028 
 
 46,868 
 
 45,162 
 
 89.7 
 
 93.1 
 
 20,000- 25,000 
 
 22,605 
 
 23,249 
 
 22,797 
 
 97.2 
 
 99.2 
 
 25,000- 30,000 
 
 13,769 
 
 13,294 
 
 13,228 
 
 103.6 
 
 104.1 
 
 30,000- 40,000 
 
 15,410 
 
 14,084 
 
 14,219 
 
 109.4 
 
 108.4 
 
 40,000- 50,000 
 
 8,298 
 
 6,986 
 
 7,178 
 
 118.8 
 
 115.6 
 
 50,000- 60,000 
 
 5,213 
 
 3,994 
 
 4,162 
 
 130.5 
 
 125.3 
 
 60,000- 70,000 
 
 3,196 
 
 2,528 
 
 2,665 
 
 126.4 
 
 119.9 
 
 70,000- 80,000 
 
 2,237 
 
 1,704 
 
 1,813 
 
 131.3 
 
 123.4 
 
 80,000- 90,000 
 
 1,561 
 
 1,205 
 
 1,292 
 
 129.5 
 
 120.8 
 
 90,000- 100,000 
 
 1,113 
 
 894 
 
 968.3 
 
 124.5 
 
 114.9 
 
 100,000- 150,000 
 
 2,983 
 
 2,240 
 
 2,461.5 
 
 133.2 
 
 121.2 
 
 150,000- 200,000 
 
 1,092 
 
 863.2 
 
 971.6 
 
 126.5 
 
 112.4 
 
 200,000- 250,000 
 
 522 
 
 428.1 
 
 490.4 
 
 121.9 
 
 106.4 
 
 250,000- 300,000 
 
 250 
 
 245.0 
 
 284.4 
 
 102.0 
 
 87.9 
 
 300,000- 400,000 
 
 285 
 
 259.2 
 
 306.0 
 
 110.0 
 
 93.1 
 
 400,000- 500,000 
 
 140 
 
 128.6 
 
 154.4 
 
 108.9 
 
 90.7 
 
 500,000- 750,000 
 
 129 
 
 137.32 
 
 168.2 
 
 93.9 
 
 76.7 
 
 750,000-1,000,000 
 
 60 
 
 52.89 
 
 66.4 
 
 113.4 
 
 90.4 
 
 1,000,000-1,500,000 
 
 34 
 
 41.25 
 
 52.95 
 
 82.4 
 
 64.2 
 
 1,500,000-2,000,000 
 
 13 
 
 15.89 
 
 20.90 
 
 81.8 
 
 62.2 
 
 2,000,000-3,000,000 
 
 7 
 
 12.40 
 
 16.68 
 
 56.5 
 
 42.0 
 
 3,000,000 and over 
 
 11 
 
 12.15 
 
 17.27 
 
 90.5 
 
 63.7 
 
 Total (over $2,000) 
 
 3,407,888 
 
 3,929,905.00 
 
 3,407,888.00 
 
 
 
PARETO'S «LAW 
 
 363 
 
 Why do the least-squares straight lines appear graphically such good 
 fits to the cumulative distributions (for at least the later years) when a 
 merely arithmetic analysis shows even this fit to the cumulative data to 
 be so illusory? Because the percentage range in the number of persons is so 
 extremely wide. The deviations of the cumulative data on a double log 
 scale from the least-squares straight line are minute when compared with 
 the percentage changes in the data from the smallest to the largest incomes. 
 But this is not helpful. The fact that there are 100,000 times as many 
 persons having incomes over $2,000 per annum as there are persons 
 having incomes over $5,000,000 per annum, does not make a theoretical 
 reading for a particular income interval of twenty or thirty per cent over 
 or under the data reading an unimportant deviation. Charting data on 
 a double log scale may thus become a fertile source of error unless ac- 
 companied by careful interpretation. 1 This fact has long been recognized 
 by engineers and others who have had much experience with similar prob- 
 lems in curve fitting. 
 
 Another matter of some importance must be noted here. The devia- 
 tions of the data from the straight lines might be much less than they are 
 and yet constitute extremely bad fits. The data points (even on a non- 
 cumulative basis) do not flutter erratically from side to side of the fitted lines; 
 they run smoothly, passing through the fitted line at small angles in the way 
 that one curve cuts another. Now, in curve fitting, such a condition always 
 strongly suggests that the particular mathematical curve used is not in 
 any sense the "law" of the data. 
 
 2. Are the slopes of the straight lines fitted to income data 
 from different times and places similar in any significant degree? 
 
 1 The dangers of fitting curves with such a combination as a cumulative distribution and 
 a double log scale, without further analysis, is well illustrated by the results Professor Pareto 
 obtained for Oldenburg. To the Oldenburg data he fitted the rather complicated equation 
 log N = log A — a log (x + a) — Bx and obtained the following results. (The value Pareto 
 gives for 8, namely .0000631, does not check with his calculated figures given below. B = 
 .0000274 is evidently what he intended.) 
 
 Income in 
 marks (over) 
 
 N 
 
 Logarithms of N 
 
 
 Observed 
 
 Calculated 
 
 A 
 
 300 
 
 600 
 
 900 
 
 1,500 
 
 3,000 
 
 6,000 
 
 9,000 
 
 15,300 
 
 30,000 
 
 54,309 
 
 24,043 
 
 16,660 
 
 9,631 
 
 3,502 
 
 994 
 
 445 
 
 140 
 
 25 
 
 4.7349 
 4.3810 
 4.2217 
 3.9837 
 3.5443 
 2.9974 
 2.6484 
 2.1461 
 1.3979 
 
 4.7349 
 4.4368 
 4.2304 
 3.9409 
 3.5008 
 2.9997 
 2.6671 
 2.1838 
 1 . 3364 
 
 — .0558 
 
 — .0086 
 + . 0428 
 + .0435 
 
 — .0023 
 
 — .0187 
 
 — .0377 
 + .0615 
 
 (From Cours d' Economie Politique, vol. II, p. 307.) 
 
 The above table may give the reader a vague idea that the fit is rather good, 
 from the above table the following table may be directly derived : 
 (Note concluded page 364.) 
 
 However, 
 
364 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 If income distributions charted on a double log scale not only cannot 
 be approximately represented by straight lines, but also differ radically 
 
 (Note 1 page 363 concluded.) 
 
 
 Number of persons 
 
 
 Income in marks 
 
 Actual 
 
 Computed 
 
 of computed 
 
 300- 600 
 
 600- 900 
 
 900- 1,500 
 
 1,500- 3,000 
 
 3,000- 6,000 
 
 6,000- 9,000 
 
 9,000-15,300 
 
 15,300-30,000 
 
 Over 30,000 
 
 30,266 
 
 7,383 
 
 7,029 
 
 6,129 
 
 2,508 
 
 549 
 
 305 
 
 115 
 
 25 
 
 26,969 
 
 10,342 
 
 8,270 
 
 5,560 
 
 2,169 
 
 534 
 
 312 
 
 131 
 
 22 
 
 112.2 
 
 71.4 
 
 85.0 
 
 110.2 
 
 115.6 
 
 102.8 
 
 97.8 
 
 87.8 
 
 113.6 
 
 Total 
 
 54,309 
 
 54,309 
 
 100.0 
 
 The fit no longer impresses one as quite so good. See Chart 28H below. 
 
 ■/O0,t 
 
 CHART 28 H 
 
 OLDENBURG INCOME TAX RETURNS 
 1890 
 CUMULATIVE FREQUENCY DISTRIBUTION 
 WITH TWO riTTED CURVES 
 
 (f) log y*9.0077-/.632l/ogz (least smarts straigM line) 
 (Z)loj y ° d.72M4 -1.465 /og (x+2Z0) - .0000274 x 
 
 (Parefo's second 'approximation) 
 
 Scales Logarithmic 
 
 70,000 
 
 7.000 
 
 I 
 I 
 
 I 
 
 •100 
 
 //iCOME /TV HUNDREDS OF/YWRKS 
 9 7S 30 60 90 
 
 ISO 
 
PARETO'S LAW 
 
 365 
 
 in shape, it is of course not of great importance whether the straight lines 
 fitted to such data from different times and places have or have not ap- 
 proximately constant slopes. For example, a comparison of Chart 28C 
 showing the cumulative distribution of United States income-tax returns 
 for 1915 on a double log scale and Chart 28F showing similar data for 
 1918, makes it plain that, even were the slopes of the fitted straight lines 
 for the two years identical, the data curves would still be so different as 
 to make the similarity of slope of the fitted lines of almost no significance. 1 
 In considering slopes, let us examine further both the data and the 
 fitted lines for these two years 1915 and 1918. Tables 281 and 28J give 
 some numerical illustrations of the differences between the distributions 
 for the two years. Table 281 gives the number of returns in each income 
 interval each year and the percentages that the 1918 figures are of the 
 1915 figures. 
 
 TABLE 281 
 
 COMPARISON OF UNITED STATES INCOME-TAX RETURNS FOR 
 
 1915 AND 1918 
 
 Income class 
 
 $ 4,000 «»-$. 5,000. 
 5,000- 10,000. 
 
 10,000- 15,000 . 
 
 15,000- 20,000. 
 
 20,000- 25,000 . 
 
 25,000- 30,000. 
 
 30,000- 40,000. 
 
 40,000- 50,000. 
 
 50,000- 100,000. 
 100,000- 150,000. 
 150,000- 200,000. 
 200,000- 250,000. 
 250,000- 300,000. 
 300,000- 400,000. 
 400,000- 500,000. 
 500,000-1,000,000. 
 1,000,000 and over. . 
 
 in umuer < 
 
 ji returns 
 
 Ratio of 1918 
 
 1915 
 
 1918 
 
 to 1915 
 
 58,949 
 
 322,241 
 
 5.4664 
 
 120,402 
 
 319,356 
 
 2.6524 
 
 34,102 
 
 69,992 
 
 2.0524 
 
 16,475 
 
 30,227 
 
 1.8347 
 
 -9,707 
 
 16,350 
 
 1.6844 
 
 • 6,196 
 
 10,206 
 
 1.6472 
 
 7,005 
 
 11,887 
 
 1.6969 
 
 ,-4,100 
 
 6,449 
 
 1.5729 
 
 6,847 
 
 9,996 
 
 1.4599 
 
 1,793 
 
 2,358 
 
 1.3151 
 
 724 
 
 866 
 
 1.1961 
 
 386 
 
 401 
 
 1.0389 
 
 216 
 
 247 
 
 1 . 1435 
 
 254 
 
 260 
 
 1.0236 
 
 122 
 
 122 
 
 1.0000 
 
 209 
 
 178 
 
 .8517 
 
 120 
 
 67 
 
 .5583 
 
 a The $3,000-$4,000 class is not included, as in 1915 married persons in that class were 
 exempted while in 1918 they were not. 
 
 The change as we pass from the $4,000-$5,000 interval, where the 1918 
 figures are nearly five-and-a-half times the 1915 figures, to the intervals 
 above $500,000, where the 1918 figures are actually less than the 1915 
 figures, illustrates the great and fundamental difference between the slopes 
 of the two distributions. However, such a comparison of unadjusted 
 
 1 Compare also the deviations from the fitted lines as given in Tables 28C and 28F. 
 
366 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 money intervals, while it throws into relief the differences in slope of the 
 two distributions, is by no means as enlightening for purposes of exhibiting 
 their other essential dissimilarities as a comparison of the two sets of data 
 after they have been adjusted for changes in average (per capita) income 
 and changes in population. Table 28 J gives some comparisons between the 
 data for the two years and between the fitted lines for the two years on 
 such an adjusted basis. Two intervals, one in the relatively low income 
 range and the other in the high income range, are used to illustrate the 
 essentially different character of the distributions for the two years. 
 
 TABLE 28J 
 
 COMPARISONS OF UNITED STATES INCOME-TAX RETURNS FOR THE YEARS 1915 AND 
 1918 ADJUSTED FOR CHANGES IN AVERAGE (PER CAPITA) INCOME AND CHANGES 
 IN POPULATION 
 
 ACTUAL INCOME-TAX DATA 
 
 Income intervals 
 
 Number of returns 
 (1) (2) 
 
 Fraction of population 
 (3) (4) 
 
 Ratio of 
 
 Column (4) 
 
 to Column (3) 
 
 
 1915 
 
 1918 
 
 1915 
 
 1918 
 
 
 Between 12 and 13 
 times average income 
 
 21,190 
 
 31,197 
 
 .00021099 
 
 . 00029945 
 
 1.4193 
 
 Between 1,200 and 1,300 
 times average income 
 
 43.85 
 
 20.37 
 
 .0000004366 
 
 .0000001955 
 
 .4478 
 
 Over 12 times average 
 income 
 
 248,600 
 
 271,452 
 
 .00247536 
 
 .00260561 
 
 1.0526 
 
 
 Amount in dollars 
 
 Per cent of total income 
 
 
 Over 12 times average 
 income 
 
 1915 
 $4,283,010,735 
 
 1918 
 $5,312,832,516 
 
 1915 
 11.9% 
 
 1918 
 
 8.7% 
 
 . 7311 
 
 LEAST-SQUARES STRAIGHT LINES 
 
 Income intervals 
 
 Number of returns 
 (1) (2) 
 
 Fraction of population 
 (3) (4) 
 
 Ratio of 
 Column (4) 
 to Column (3) 
 
 
 1915 
 
 1918 
 
 1915 
 
 1918 
 
 
 Between 12 and 13 
 times average income 
 
 32,886 
 
 41,730 
 
 .00032745 
 
 .00040056 
 
 1.2233 
 
 Between 1,200 and 1,300 
 times average income 
 
 47.63 
 
 17.10 
 
 .0000004743 
 
 .0000001641 
 
 .3460 
 
 STRAIGHT LINES FITTED TO GIVE THE SAME TOTAL NUMBER OF RETURNS AND THE 
 SAME TOTAL INCOME AS THE INCOME-TAX DATA 
 
 Income intervals 
 
 Number of returns 
 (1) (2) 
 
 Fraction of population 
 (3) (4) 
 
 Ratio of 
 
 Column (4) 
 
 to Column (3) 
 
 
 1915 
 
 1918 
 
 1915 
 
 1918 
 
 
 Between 12 and 13 
 times average income 
 
 24,510 
 
 42,460 
 
 .00024405 
 
 .00040756 
 
 1.6700 
 
 Between 1,200 and 1 300 
 times average income 
 
 54.73 
 
 14.15 
 
 .0000005450 
 
 .0000001358 
 
 .2492 
 
PARETO'S LAW 
 
 367 
 
 
 NOTES TO TABLE 28J 
 "Average Income" Intervals 
 
 
 
 
 1915 
 
 1918 
 
 
 $ 358 
 
 4,296 
 
 4,654 
 
 429,600 
 
 465,400 
 
 $ 586 
 
 7,032 
 
 7,618 
 
 703,200 
 
 761,800 
 
 
 
 
 13 " " " 
 
 
 1,200 " " " 
 
 
 1,300 " " " 
 
 
 
 
 Equations of Fitted Straight Lines on a Cumulative Double Log Basis 
 
 
 Least- squares lines 
 
 Lines giving correct total 
 
 number of returns and 
 
 total income 
 
 1914 
 
 y = 11 . 153322 — 1 . 559256 x 
 y = 10.643299 — 1.419579 x 
 y = 10.839435—1.424638 x 
 y = 11.410606— 1.539996 x 
 y = 12 . 033697 — 1 . 693823 x 
 y = 12 . 320963 — 1 . 734802 x 
 
 y = 10 . 557242 — 1 . 420936 x 
 
 1915 
 
 y = 10.202382 — 1 .325598 x 
 
 1916 
 
 y = 10.212702 — 1.298088 x 
 
 1917 
 
 y = 11 . 170980 — 1 .486817 x 
 
 1918 
 
 y = 12 . 202452 — 1 . 738497 x 
 
 1919 
 
 y = 12.036155— 1.667258 x 
 
 
 
 Table 28J needs little discussion. In the section treating actual income- 
 tax data we notice that while the adjusted number of returns in the lower 
 income interval 1 increased 41.93 per cent from 1915 to 1918, the adjusted 
 number of returns in the upper income interval 2 decreased 55.22 per cent. 
 Moreover, while the adjusted total number of returns above the " 12-times- 
 average-income" point increased 5.26 per cent, the adjusted amount of 
 income reported in these returns decreased 26.89 per cent. 
 
 Such figures suggest a rather radical change in the distribution of in- 
 come during this short three-year period. Similar conclusions may be 
 drawn from the figures for the two pairs of fitted lines, though we must 
 of course remember that these lines describe only very inadequately the 
 actual data. The lines so fitted as to give each year the same total number 
 of returns and total amount of income as the data for that year yield 
 sensational results. While the adjusted number of returns in the lower 
 income-interval increased 67 per cent, the adjusted number of returns 
 in the upper income-interval decreased 75.08 per cent. 
 
 Finally, it has been suggested that changes in the characteristics of the 
 tax-income-distribution in the United States from 1915 to 1918 may be 
 accounted for as the results of the increase in the surtax rates with 1917. 
 We do not believe any large part of these changes can be so accounted 
 for. Notwithstanding the fact that the country entered the European 
 war during the interval, the difference between the 1915 distribution and 
 the 1918 distribution in the United States, extreme as it is, cannot be said 
 to be unreasonably or unbelievably great. Even the changes in the slope 
 of the least-squares line are not phenomenal. Pareto's Prussian figures 
 contain fluctuations in slope from — 1.60 to — 1.89 while the slope of the 
 least-squares straight line fitted to his Basle data is only — 1.25. The 
 
 1 Between 12 and 13 times the average income (per capita) each year. 
 
 1 Between 1,200 and 1,300 times the average income (per capita) each year. 
 
368 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 slopes of the least-squares straight lines fitted to the American data are 
 —1.42 for 1915 and —1.69 for 1918. 
 
 3. If the upper income ranges (or "tails") of income distributions 
 were, when charted on a double log scale, closely similar in shape, 
 would that fact justify the assumption that the lower income ranges 
 were likewise closely similar? 
 
 Before attempting to answer the above question, let us summarize the 
 case we have just made against believing the "tails" significantly similar. 
 We can then discuss how much importance such similarity would have 
 did it exist. 
 
 We have found upon examination that the approximation to straight 
 lines of the tails of income distributions plotted on double log scales is 
 specious; that the slopes of the fitted straight lines differ sufficiently to 
 produce extreme variations in the relative number of income recipients 
 in the upper as compared with the lower income ranges of the tails; 
 that the upper and lower income ranges of the actual data for different 
 times or places tell a similar story of extreme variation; and that the 
 irregularities in shape of the tails of the actual data, entirely aside 
 from any question of approximating or not approximating straight lines 
 of constant slope, vary greatly from year to year and from country to 
 country, ranging all the way from the irregularities of such distributions 
 as the Oldenburg data, through the American data for 1914, 1915 and 1916 
 to such an entirely different set of irregularities as those seen in the Amer- 
 ican data for 1918 1 . 
 
 At this stage of the discussion the reader may ask whether a general 
 appearance of approximating straight lines on a double log scale, poor as the 
 actual fit may be found to be under analysis, has not some meaning, some 
 significance. The answer to this question must be that, if we were not deal- 
 ing with a frequency distribution but with a correlation table showing a 
 relationship between two variables, an approximation of the regression lines 
 to linearity when charted on a double log scale might easily be the clue 
 to a first approximation to a rational law; but that, on the other hand, ap- 
 proximate linearity in the tail of a frequency distribution charted on a double 
 log scale signifies relatively little because it is such a common charac- 
 teristic of frequency distributions of many and varied types. 
 
 The straight line on a double log scale or, in other words, the equation 
 y = bx m , when used to express a relationship between two variables, is, to 
 quote a well-known text on engineering mathematics, "one of the most 
 useful classes of curves in engineering." 2 In deciding what type of equa- 
 tion to use in fitting curves by the method of least squares to data con- 
 
 1 Compare Charts 28H, 28B, 28C, 28D and 28F. 
 * P. Steinmetz, Engineering Mathematics, p. 216. 
 
PARETO'S LAW 369 
 
 cerning two variables the texts usually mention y = bx m as "a quite com- 
 mon case." ! A recent author writes, "simple curves which approximate 
 a large number of empirical data are the parabolic and hyperbolic curves. 
 The equation of such a curve is y = ax b [y = bx m ], parabolic for 6 positive 
 and hyperbolic for 6 negative." 2 A widely used text on elementary 
 mathematics speaks of the equation y = bx m as one of "the three funda- 
 mental functions" in practical mathematics. 3 The market for "logarith- 
 mic paper" shows what a large number of two-variable relationships may 
 be approximated by this equation. Moreover this equation is often a 
 close first approximation to a rational law. Witness "Boyle's Law." In- 
 deed, sufficient use has not been made of this curve in economic discus- 
 sions of two-variable problems. 
 
 The primary reason why approximation to linearity on a double log 
 scale has no such significance in the case of the tail of a frequency distribu- 
 tion as it often has in the case of a two-variable problem is because of 
 the very fact that we are considering the tail of the distribution, in other 
 words, a mere fraction of the data. While frequency distributions which 
 can be described throughout their length by a curve of the type y = bx m are 
 extremely rare, a large percentage of all frequency distributions have tails 
 approximating straight lines on a double log scale. 4 It is astonishing how 
 many homogeneous frequency distributions of all kinds may be described 
 with a fair degree of adequacy by means of hyperbolas 5 fitted to the data 
 on a double log scale. Along with this characteristic goes, of course, the 
 possibility of fitting to the tails of such distributions straight lines approxi- 
 mately parallel to the asymptotes of the fitted hyperbola. However we 
 have by no means adequately described an hyperbola when we have 
 stated the fact that one of its asymptotes is (of course) a straight line and 
 that its slope is such and such. Had we even similar information con- 
 cerning the other asymptote also, we should know little about the hyper- 
 bola or the frequency distribution which it would describe on a double 
 log scale. The hyperbola might coincide with its asymptotes and hence 
 have an angle at the mode or it might have a very much rounded "top." 
 Such a variation in the shape of the top of the hyperbola 6 would generally 
 correspond to a very great variation in the scatter or "inequality" of the 
 distribution as well as many other characteristics. 
 
 1 D. P. Bartlett, Method of Least Squares, p. 33. 
 
 2 J. Lipka, Graphical and Mechanical Computation, p. 128. 
 
 3 C. S. Slichter, Elementary Mathematical Analysis, preface. 
 
 4 A very large percentage of the remainder have tails approximating straight lines on a 
 natural x log y basis. 
 
 5 N. B. Not a straight line on the double log scale, which is a so-called hyperbola on the 
 natural scale, but a true conic section hyperbola on the double log scale. 
 
 Charts 28K and 28L (Earnings per Hour of 318,946 Male Employees in 1919) illustrate 
 how excellent a fit may often be obtained by means of an hyperbola even though fitted only 
 by selected points. A comparison of the least-squares parabola and the selected-points 
 hyperbola on Chart 28K illustrates also the straight-tail effect. 
 
 » Compare Karl Pearson's concept of "kurtosis." 
 
370 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 
 
 CHART 28 K 
 
 
 
 EARNINGS PER HOUR 
 
 
 
 OF 
 
 
 
 318,946 MALE EMPLOYEES 
 
 
 
 IN 1919 
 
 
 
 MONTHL Y LHBO/f REV/EIY, SEPT. /3I9 
 
 
 
 SCfULES-JLOG/f/f/THM/C 
 
 g 
 
 
 
 > 
 
 
 
 ■ 
 
 
 
 •100.0O0W 
 
 
 
 1 
 
 
 
 •< 
 
 
 
 * 
 
 
 ^^* 
 
 * 
 
 PLOYEES PER TEA 
 
 
 
 § 
 
 
 
 M 
 
 
 
 3 
 
 
 
 1,000 < 
 
 
 
 b. 
 
 
 
 © 
 
 
 
 i 
 
 
 
 § 
 
 
 
 g 
 
 
 EARNINGS PER HOUR IN DOLLARS \| 
 
 
 
 .30 
 
 .40 .50 .60 -.70 .80 .90 1.00 1.25 1.50 1.75 \l 
 
PARETO'S LAW 
 
 371 
 
 Rough similarity in the tails of two distributions on a double log scale 
 by no means proves even rough similarity in the remainder of the dis- 
 tributions. Charts 28M, 28N, 280 and 28P illustrate both cumulatively 
 
 
 
 
 CHART 28 M 
 
 -100,000 
 
 
 
 CUMULATIVE FREQUENCY DISTRIBUTION 
 RATES OF WAGES PER HOUR 
 
 FOR 
 
 72,231 MALE EMPLOYEES 
 
 SLAUGHTERING & M EAT- PACKING INDUSTRY 
 
 in the u.s. in i9i7. 
 sounasuftis/ofuiiiaismisriaMifrmisi. 
 
 
 
 EMPLOYEES 
 
 
 
 Scales Logarithmic 
 
 ■ 
 
 S 
 
 a 
 s 
 
 
 
 
 u. 
 -1,000 o 
 
 
 
 
 ■ 
 ■ 
 1 
 := 
 z 
 
 
 
 
 -100 
 
 
 
 
 
 10 
 
 WAGES IN CENTS PER HOUR 
 15 20 25 30 40 
 
 50 60 
 
372 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 
 
 CHART 28 N 
 
 
 
 FREQUENCY DISTRIBUTION 
 
 
 
 
 
 
 RATES OF WAGES PER HOUR 
 
 
 
 FOR 
 
 
 
 7?,29l MALE EMPLOYEES 
 
 
 
 in the 
 
 
 
 SLAUGHTERING & MEAT-PACKING INDUSTRY 
 
 
 
 IN THE U.S. Itl 1917. 
 
 
 
 source aatm/annBatsmis7KS. Auiumtu 
 
 
 
 Scales Logarithmic 
 
 -10,000 
 
 
 
 B 
 
 ■ 
 >* 
 
 
 / N^ 
 
 s 
 
 E 
 
 
 / v? 
 
 -1,000 s 
 
 
 
 M 
 
 
 
 a 
 
 
 
 < 
 
 
 
 n 
 
 
 
 b. 
 
 
 
 © 
 
 
 
 M 
 
 
 
 -100 9 
 
 ■ 
 
 
 
 1 
 
 
 
 g 
 
 
 
 -10 
 
 
 
 -1 
 
 
 WAGES IN CENTS PER HOUR 
 
 
 10 
 
 15 20 25 30 40 50 60 
 
 and non-cumulatively on a double log scale two wages distributions whose 
 extreme tails appear roughly to approximate straight lines of about equal 
 slope. 1 Charts 28M and 28N are from data concerning wages per hour 
 of 72,291 male employees in the slaughtering and meat-packing industry 
 in 1917; 2 Charts 280 and 28P are from data concerning wages per hour 
 of 180,096 male employees in 32 manufacturing industries in the United 
 States in 1900. 3 A mere glance at the two non-cumulative distributions 
 will bring home the fact that while they show considerable similarity in 
 the upper income range tails, they are quite dissimilar in the remainder 
 
 'The illustration shows only "rough similarity" in the extreme tails. However, there 
 seems no good reason for believing that even great similarity in the tails proves similarity 
 in the rest of the distribution. It certainly cannot do so in the case of essentially hetero- 
 geneous distributions, such as income distributions. 
 
 » Bureau of Labor Statistics, Bulletin No. 252. 
 
 •Twelfth Census of the United States (1900), Special Report on Employees and Wages, 
 Davis R. Dewey. 
 
PARETO'S LAW 
 
 373 
 
 -100,000 
 
 -10,000 
 
 1,000 £ 
 
 CHART 280 
 
 CUMULATIVE FREOUfNCY DISTRIBUTION 
 RATES OF WAGES PER HOUR 
 
 FOR 
 
 130,096 MALE EMPLOYEES 
 
 32 MANUFACTURING INDUSTRIES 
 IN THE U.S. IN I90O. 
 
 aoosa ■DctveYii"Cfnsus 
 Scales Logarithmic 
 
 WAGES IN CENTS PER HOUR 
 20 25 30 40 50 
 
 60 70 
 
 80 90 100 
 
 1 ' I 
 
 of the curves. Moreover, in spite of this similarity of tails, the slaughtering 
 and meat-packing distribution has a coefficient of variation of 30.5 while 
 the manufacturing distribution has a coefficient of 47.7. In other words, 
 the relative scatter or "inequality of distribution" is more than one-and-a- 
 half times as great in the manufacturing data as it is in the slaughtering 
 and meat-packing data. Furthermore, no discussion and explanation of 
 greater essential heterogeneity in the one distribution than in the other 
 will offset the fact that the tails are similar but the distributions are dif- 
 ferent. There seems indeed to be almost no correlation between the slope 
 of the upper-range tail and the degree of scatter in wages distributions. 
 Some distributions showing extremely great scatter have very steep tails, 
 some have not. 1 The frequency curve for the distribution of income in 
 Australia in 1915 is radically different from either the curve for the United 
 States in 1910 constructed by Mr. W. I. King or the curve for the United 
 States in 1918 constructed by the National Bureau of Economic Research. 
 
 1 The tails of wage distributions have in general much greater slopes than those of the 
 upper (i. e., income-tax) range rj income distributions. This is an outstanding difference 
 between the two distributions Pareto's conclusions with respect to the convex appearance 
 of the curve for wages are consistent with curves showing number of dollars per income-tax 
 interval traceable to wa^es but not with actual wage distributions showing number of 
 recipients per wage intel ^al. Distributions based upon income from effort and distributions 
 based upon income from such sources (mostly profits and income from property) as yield the 
 higher incomes seem to have tails the one as roughly straight as the other. Indeed many 
 wage distributions have tails more closely approximating straight lines than do income-tax 
 data. 
 
374 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 
 
 
 CHART 28 P 
 
 
 
 % 
 
 FREQUENCY DISTRIBUTION 
 
 RATES OF WAGES PER HOUR 
 
 180,096 MALE EMPLOYEES 
 
 IN 
 
 3E MANUFACTURING INDUSTRIES 
 
 IN THE 0. S. IN 1900 
 
 3ou#c£ oiMm* census 
 
 
 
 Scales Logarithmic 
 
 -10,000 
 
 
 
 
 •1,000 
 
 
 
 
 B 
 S 
 
 E 
 -100 £ 
 
 1 
 
 O 
 
 
 
 
 H 
 H 
 
 -io g 
 5 
 z 
 
 
 
 
 i 
 
 
 
 
 
 10 
 
 15 
 
 WAGES IN CENTS PER HOUR 
 
 20 25 30 40 50 60 70 80 90 100 
 
 Yet all three curves have tails on a double log scale quite as similar as is 
 common with income-tax returns. 1 
 
 From this discussion we may draw the corollary that it is futile to at- 
 tempt to measure changes in the inequality of distribution of income 
 throughout its range by any function of the mere tail of the income fre- 
 quency distribution. It seems unnecessary therefore to discuss Pareto's 
 suggestions on this subject. 
 
 4. Is it probable that the distribution of income is similar enough 
 from year to year in the same country to make the formulation 
 of any useful general "law" possible? 
 
 1 As will be seen in Chapter 29, there seems reason for believing that the extreme difference 
 between the distribution of incomes obtained by the Australian Census and the estimate 
 made by the National Bureau of Economic Research is due largely to difference in definition 
 of income and income recipient. However, this does not alter the fact that we have here 
 again two distributions with tails as similar as is usual with income-tax distributions and 
 lower ranges about as different as it is possible to imagine. 
 
PARETO'S LAW 375 
 
 Before answering this question we must decide what we should mean 
 by the word similar. If income distributions for two years in the same 
 country were such that each distribution included the same individ- 
 uals and each individual's income was twice as large in the second year 
 as it had been in the first year, it would seem reasonable to speak of the 
 distributions as strictly similar. If in a third year (because of a doubling 
 of population due to some hypothetical immigration) the number of per- 
 sons receiving each specified income size was exactly twice what it was 
 in the second year, it would still seem reasonable to speak of the distribu- 
 tions as strictly similar. Tested by any statistical criterion of dispersion 
 which takes account of relative size (such as the coefficient of variation), 
 the dispersion is precisely the same in each of the three years. Moreover 
 the three distributions mentioned above x must necessarily have identically 
 the same shape on a double log scale, and furthermore any two distribu- 
 tions which have identically the same shape on a double log scale 2 must 
 necessarily have the same relative dispersion as measured by such indices 
 as the coefficient of variation, interquartile range divided by median, etc. 
 Approximation to identity of shape on a double log scale seems then a 
 useful concept of "similarity." It is the concept implicit in Pareto's work. 3 
 
 Now we have already found considerable evidence that income dis- 
 tributions are not, to a significant degree, similar in shape on a double log 
 scale. The income-tax tails of income distributions for different times and 
 places neither approximate straight lines of constant slope nor approxi- 
 mate one another; they are of distinctly different shapes. Moreover, such 
 tails do not show in respect of their numbers of income recipients and 
 
 1 Or, any distributions whose equations may be reduced to one another by substituting 
 kix for x and hiy for y. 
 
 2 The curve may be thought of as consisting of two parts, which before reduction to log- 
 arithms, would be (1) the positive income section and (2) the negative income section with 
 positive signs. 
 
 3 While approximate identity of shape on a natural scale, a natural x and log y scale, or 
 any other similar criterion would constitute a "law," no such approximate identity of shape 
 on such scales has yet been discovered and it seems difficult to advance any very cogent 
 a priori reasons for expecting it. 
 
 In this connection we must remember that had we the exact figures for the entire frequency 
 curves of the distribution of income in the United States from year to year, if moreover we 
 could imagine definitions of income and income recipient which would be philosophically 
 satisfactory and statistically usable — and if further we managed year by year to describe 
 our data curves adequately by generalized mathematical frequency curves of more or less 
 complicated variety we should not necessarily have arrived at any particularly valuable re- 
 sults. Any series of data may be described to any specified degree of approximation by a 
 power series of the type y = A -f Bx + Cz 2 -{- Dx 3 + • : but such fit is purely em- 
 pirical and absolutely meaningless except as an illustration of MacLaurin's theorem in the 
 differential calculus. We might be able to describe each year's data rather well by one of 
 Karl Pearson's generalized frequency curves, but if the essential characteristics of the curve — 
 skewness, kurtosis, etc., changed radically from year to year, description of the data by such 
 a curve might well give no clue whatever as to any "law." Not only might the years be dif- 
 ferent but the fits might be empirical. Professor Edgeworth has well said that "a close fit 
 of a curve to given statistics is not, per se and apart from a priori reasons, a proof that the 
 curve in question is the form proper to the matter in hand. The curve may be adapted to the 
 phenomena merely as the empirically justified system of cycles and epicycles to the planetary 
 movements, not like the ellipse, in favor of which there is the Newtonian demonstration, as 
 well as the Keplerian observations." Journal of the Royal Statistical Society, vol. 59, p. 533. 
 
376 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 total amounts of income any uniformity of relation to the total number 
 of income recipients and total amount of income in the country, even 
 after adjustments have been made for variations in population and average 
 income. 1 Considerations such as these, reenforce the conclusion which 
 we arrived at from an examination of wage distributions, namely, that 
 there is little necessary relation between the shape of the tail and the shape 
 of the body of a frequency distribution, and have led us to suspect that, 
 even if the tails of income distributions were practically identical in shape, 
 it would be extremely dangerous to conclude therefore that the lower 
 income ranges of the curves were in any way similar. 
 
 A most important matter remains to be discussed. What right have 
 we to assume that the heterogeneity necessarily inherent in all income 
 distribution data is not such as inevitably to preclude not only uniformity 
 of shape of the frequency curve from year to year and country to country 
 but also the very possibility of rational mathematical description of any 
 kind unless based upon parts rather than the whole? What evidence have 
 we as to the extent and nature of heterogeneity in income distribution 
 data? 
 
 In the first place we must remember that lower range incomes are pre- 
 dominantly from wages and salaries, while upper range incomes are pre- 
 dominantly from rent, interest, dividends and profits. 2 While 74.67 per 
 cent of the total income reported in the United States in the $l,000-$2,000 
 income interval in 1918 was traceable to wages and salaries, only 33.10 
 per cent of the income in the $10,000-$20,000 interval was from those 
 sources, and only 15.92 per cent of the income in the $100,000-$150,000 
 interval and 3.27 per cent of the income in the over-$500,000 intervals. 
 On the other hand, while only 1.93 per cent of the total income reported 
 in the $l,000-$2,000 interval in 1918 was traceable to dividends, 23.73 
 per cent was so traceable in the $10,000-$20,000 interval, 43.18 per cent 
 in the $100,000-$150,000 interval, and 59.44 per cent in the over-$500,000 
 intervals. 3 The difference in constitution of the income at the upper and 
 
 1 Estimated per cent of total income received by highest 5% of income receivers in United 
 States: 
 
 1913 33 
 
 1914 32 
 
 1915 32 
 
 1916 34 
 
 1917 29 
 
 1918 26 
 
 1919 24 
 
 National Bureau of Economic Research, Income in the United States, vol. 1, p. 116. 
 » Compare Professor A. L. Bowley's paper on "The British Super-Tax and the Distribution 
 of Income," Quarterly Journal of Economics, February, 1914. 
 » Statistics of Income 1918, pp. 10 and 44. 
 
 While the reporting of dividends was almost certainly less complete in the lower than in 
 the upper income classes, the difference could not be sufficient to invalidate the general con- 
 clusion. Lower range incomes are predominantly wage and salary incomes; upper range in- 
 comes are not. 
 
PARETO'S LAW 377 
 
 lower ends of the distribution is sufficient to justify the statement that 
 most of the individuals going to make up the lower income range of the 
 frequency curve are wage earners, while the individuals going to make up 
 the upper income range are capitalists and entrepreneurs. 1 What do we 
 know about the shapes of these component distributions? Is the funda- 
 mental difference in their relative positions on the income scale their only 
 dissimilarity? 
 
 In any particular year the upper income tail of the frequency distribu- 
 tion of income among capitalists and entrepreneurs seems not greatly dif- 
 ferent from the extreme upper income tail of the frequency distribution 
 of income among all classes. This is what we might expect. Not only is 
 the percentage of the total income in the extreme upper income ranges 
 reported as coming from wages and salaries small but much of this so- 
 called wages and salaries income must be merely technical. For example, 
 it is often highly " convenient" to pay "salary" rather than dividends. 
 Furthermore, in so far as the tail of the curve of distribution of income 
 among capitalists and entrepreneurs is not identical with the tail of the 
 general curve, it will show a smaller rather than a larger slope, because the 
 percentage of the number of persons in each income interval who are 
 capitalists and entrepreneurs increases as we pass from lower to higher 
 incomes. 2 Now the slopes of the straight lines fitted to the extreme tails 
 of non-cumulative income distributions on a double log scale fluctuate 
 within a range of about 2.4 to 3.0. 
 
 The upper range tails of wages distributions tell an entirely different 
 story. Aside from surface irregularities often quite evidently traceable to 
 concentration on certain round numbers, the majority of wages distribu- 
 tions have tails which, on a double log scale, are roughly linear. 3 How- 
 ever the slopes of straight lines fitted to these tails are much greater than 
 the slopes of corresponding straight lines fitted to income distribution 
 tails. 4 While the slopes of income distribution tails range from about 2.4 
 
 1 Many individuals in the middle income ranges must necessarily be difficult to classify. 
 This does not mean that the concept of heterogeneity is inapplicable. There are countries 
 in which the population is a mixture of Spanish, American Indian, and Negro blood. Now 
 such a population must, for many statistical purposes, be considered extremely heterogeneous 
 even though the percentage of the population which is of any pure blood be quite negligible. 
 
 2 In 1917, the only year in which returns are classified according to "principal source of 
 income" (wages and salaries, income from business, income from investment) the difference 
 in slope, in the income range $100,000 to $2,000,000, between the distribution for all returns 
 and the distribution for those returns which did not report wages and salaries as their prin- 
 cipal source of income was less than .05. The slope in this range of the line fitted to all re- 
 turns was about 2.64; the business and investment line was about 2.59 and the wages line 
 about 3.21. In 1916, the only year in which returns are classified according to occupations, 
 the distribution of income among capitalists shows a slope of only 2.08 while public service 
 employees (civil) show a slope of 2.70 and skilled and unskilled laborers a slope of 2.74. 
 
 3 Attention has already been drawn to the fact that this is a characteristic of many fre- 
 quency distributions of various kinds. 
 
 4 A furthee difference between the upper range income distribution among capitalists and 
 entrepreneurs and the upper range of the distribution among all persons seems to be, from 
 the 1916 occupation distributions, that the distribution among all persons shows less of a roll, 
 i. e., is straighter. 
 
378 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 to 3.0, the slopes of wages distributions tails commonly range between 
 4.0 and 6.0. They seldom run below about 4.5; they sometimes run as 
 high as 10.0 and 11.0. 
 
 A distribution of wages per hour for 26,183 male employees in iron and 
 steel mills in the United States in 1900 ! shows a tail with a slope of about 
 3.35. However, the total of which this is a part, the distribution of wages 
 per hour among 180,096 male employees in 32 manufacturing industries 
 in 1900, shows a tail-slope of about 4.8. The estimated distribution of 
 weekly earnings of 5,470,321 wage earners in the United States in 1905 2 
 shows a tail-slope of about 5.0. The distribution of earnings per hour 
 among 318,946 male employees in 29 different industries in the United 
 States in 1919 3 shows a tail-slope of about 5.86. The distribution of 
 wages per month among 1,939,399 railroad employees in the United States 
 in 1917 4 shows a tail-slope of about 6.25. The distribution of wages per 
 hour among 43,343 male employees in the foundries and metal working 
 industry of the United States in 1900 5 shows a tail-slope of about 7.8. 
 The distribution of earnings in a week among 9,633 male employees in the 
 woodworking industry — agricultural implements — in the United States in 
 1900 6 shows a tail-slope of over 11.0. At the other extreme was the case 
 of the wages-per-hour distribution among 26,183 male employees in Amer- 
 ican iron and steel mills in 1900 with a slope of 3.35. Both 11.0 and 3.35 
 are exceptional, but the available data make it clear that wages distribu- 
 tions of either earnings or rates have tail-slopes which are always much 
 greater than the maximum tail-slope of income distributions. 
 
 The illustrations in the preceding paragraph are illustrations of the tail- 
 slopes of wages distributions among wage earners. However all the evi- 
 dence points to frequency distributions of income among wage earners 
 having tail-slopes only very slightly less steep than the tail-slopes of wages 
 distributions. We have almost no usable data concerning the relation 
 between individual wage distributions and income distributions for the 
 same individuals, but we have a few samples showing the relation between 
 family earnings distributions and family income distributions. 7 More- 
 over, we can without great risk base certain extremely general conclusions 
 
 •Twelfth Census of the United States (1900), Special Report on Employees and Wages, 
 Davis R. Dewey. 
 
 2 1905 Census of Manufacturers, Part IV, p. 647. 
 
 • Monthly Labor Review, Sept., 1919. 
 
 1 Report of the Railroad Wage Commission to the Director General of Railroads, 1919, p. 96. 
 
 'Twelfth Census of the United States (1900), Special Report on Employees and Wages, 
 Davis R. Dewey. 
 
 •Twelfth Census of the United States (1900), Special Report on Employees and Wages, 
 Davis R. Dewey. 
 
 7 The reader must not confuse the percentage of the income not derived from wages going 
 to wage-earners in any particular income class with the percentage of the income not derived 
 from wages going to all income recipients in any particular income class. Soirtte of these last 
 recipients are not wage earners at all, they receive no wages. Information concerning the 
 second of these relations but not the first is given in the income tax reports. 
 
PARETO'S LAW 379 
 
 concerning individual wage-earners' income distributions on these family- 
 data. The upper tails of the family-wage distributions are the tails of the 
 wage distributions for the individuals who are the heads of the families. 
 This is apparent from an analysis of the samples. Now income from rents 
 and investments belongs almost totally to heads of families. Such income 
 is however so small in amount that it cannot alter appreciably the slope 
 of the tail. 1 While income from other sources than rents and investments 
 (lodgers, garden and poultry, gifts and miscellaneous) may not be so con- 
 fidently placed to the credit of the head of the family, this item changes 
 its percentage relation to the total income so slowly as to be negligible in 
 its effect upon the tail-slope of the distribution. 2 Notwithstanding the 
 danger of reasoning too assuredly about individuals from these picked 
 family distributions, we seem justified in believing that the tail-slopes of 
 income distributions among individual wage earners are not very different 
 from the tail-slopes of wage distributions among the same individuals. 3 
 The upper tail-slopes of income distributions among typical wage earners 
 
 1 For example, in the report on the incomes of 12,096 white families published in the Monthly 
 Labor Review for December, 1919, we find the income from rents and investments less than 
 one per cent of the total family income for each of the income intervals. 
 
 Percentage income from 
 Income group rents and investments 
 
 is of total income 
 Under $900 .079 
 
 $ 900-$l,200 .176 
 
 1,200- 1,500 .410 
 
 1,500- 1,800 .551 
 
 1,800- 2,100 .606 
 
 2,100- 2,500 .998 
 
 2,500 and over .778 
 
 2 As a somewhat extreme example, the Bureau of Labor investigation mentioned in the 
 preceding note shows the following relations between total family earnings and total family 
 income (including income from rents and investments, lodgers, garden and poultry, gifts and 
 miscellaneous). 
 
 Income group Percentage that total 
 
 earnings are of total income 
 
 Under $900 96.2 
 
 $ 900-$l,200 96.5 
 
 1,200- 1,500 96.3 
 
 1,500- 1,800 96.0 
 
 1,800- 2,100 96.3 
 
 2,100- 2,500 95.1 
 
 2,500 and over 96.2 
 
 3 Further corroboratory evidence, of some slight importance, that the tail-slopes of wage 
 distributions among wage earners are not very different from the tail-slopes of income dis- 
 tributions among wage earners is yielded by the fact that the tail-slopes of income distribu- 
 tions among families (which are virtually identical with the tail-slopes of both income and 
 wage distributions among the heads of these families) have roughly the same range as the 
 tail-slopes of wage distributions among individuals. The British investigation into the in- 
 comes of 7,616 workingmen's families in the United States in 1909 shows a tail-slope of about 
 3.5. (Report of the British Board of Trade on Cost of Living in American Towns, 1911. [Cd. 
 5609], p. XLIV.) The Bureau of Labor's investigation into the income of 12,096 white fam- 
 ilies in 1919 shows a tail-slope of about 4.0. Mr. Arthur T. Emery's extremely careful in- 
 vestigation into the incomes of 2,000 Chicago households in 1918 shows a tail-slope of 
 about 4.4. At the other extreme we find that the Bureau of Labor's investigation into the 
 income of 11,156 families in 1903 (Eighteenth Annual Report of the Commissioner of Labor, 
 1903, p. 558) shows a tail-slope of about 10.0, and that Mr. R. C. Chapin's investigation into 
 the income of 391 workingmen's families in New York City (Standard of Living Among Work- 
 ingmen's Families in New York City, p. 44) also shows a slope of about 10.0. The tails of 
 these last two cases are very irregular so that the slope itself is not determinable with much 
 precision. 
 
380 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 may then be assumed to have much greater slopes than the upper tail- 
 slopes of income distributions among capitalists and entrepreneurs. It 
 does not seem possible to make any very definite statement concerning 
 the body and lower tail of the capitalist and entrepreneurial distribution — 
 even in so far as that term is a significant one. 1 All the evidence suggests 
 that the mode of what we have termed the capitalist-entrepreneurial dis- 
 tribution is consistently higher than the wage-earners' mode. 2 Its lower 
 income tail undoubtedly reaches out into the negative income range, which 
 the tail of the wage-earners' distribution may, both a priori and from evi- 
 dence, be assumed not to do. It seems a not irrational conclusion then to 
 speak of the capitalist-entrepreneurial distribution as having a lesser tail- 
 slope than the wage- earners' distribution on the lower income side as well 
 as on the upper income side, 3 and as a corollary almost certainly a much 
 greater dispersion both actual and relative than the wage-earners' dis- 
 tribution. 
 
 Though the above generalizations concerning differences between the 
 wage-earners' income distribution and the capitalist-entrepreneurial in- 
 come distribution seem sound, they tell but a fraction of the story. Aside 
 from the difficulty of classifying all income recipients in one or the other 
 of these two classes, we are faced with the further fact that investigation 
 suggests that our two component distributions are themselves exceedingly 
 heterogeneous. 4 We have already noted that wage distributions for dif- 
 ferent occupations and times are extremely dissimilar in shape and we 
 suspect that the same applies to capitalist-entrepreneurial distributions. 
 For example, what little data we possess suggest that the distribution of 
 income among farmers has little in common with other entrepreneurial 
 distributions. 
 
 Moreover, the component distributions, into which it would seem nec- 
 essary to break up the complete income distribution before any rational 
 description would be possible, not only have different shapes and different 
 positions on the income scale (i. e., different modes, arithmetic averages, 
 etc.), but the relative position with respect to one another on the income scale 
 of these different component distributions changes from year to year. 5 
 
 1 In the total income curve there is a broad twilight zone where individuals are often both 
 wage or salary earners and capitalists or even entrepreneurs. 
 
 2 In the 1916 occupation distributions the only occupations showing more returns for the 
 $4,000-$5,000 interval than the $3,0O0-$4,000 (that is the only occupations showing any 
 suggestion of a mode) are of a capitalistic or entrepreneurial description — bankers; stock- 
 brokers; insurance brokers; other brokers; hotel proprietors and restaurateurs; manufacturers; 
 merchants; storekeepers; jobbers; commission merchants, etc.; mine owners and mine op- 
 erators; saloon keepers; sportsmen and turfmen. 
 
 » Of course the very word slope is an ambiguous term to use concerning the tail of a curve 
 which enters the second quadrant. 
 
 * Evidence suggesting definite heterogeneity in the "wage and salary" figures of the income- 
 tax returns is presented in Chapter 30. 
 
 • This fact is one of the simpler pieces of evidence against the existence of a "law." Of 
 course, even though the income distribution were made up of heterogeneous material, if the 
 
PARETO'S LAW 
 
 381 
 
 Table 28Q l is interesting as showing the changes in the relative positions 
 of the arithmetic averages of different wage distributions in 1909, 1913 
 and 1918. 
 
 TABLE 280 
 
 CHANGES IN THE RELATIVE POSITIONS OF THE AVERAGE ANNUAL 
 EARNINGS OF EMPLOYEES ENGAGED IN VARIOUS INDUSTRIES 
 
 Industry 
 
 All Industries 
 
 Agriculture 
 
 Production of Minerals 
 
 Manufacturing: 
 
 Factories 
 
 Hand Trades 
 
 All Transportation 
 
 Railway, Express, Pullman, Switching and 
 Terminal Cos 
 
 Street Railway, Electric Light and Power, 
 Telegraph and Telephone Cos 
 
 Transportation by Water 
 
 Banking 
 
 Government 
 
 Unclassified Industries 
 
 1909 
 
 1913 
 
 1918 
 
 100.0 
 
 48.2 
 95.7 
 
 91.2 
 111.7 
 104.9 
 
 104.0 
 
 99.5 
 123.5 
 123.0 
 118.1 
 114.4 
 
 100.0 
 
 45.4 
 
 104.4 
 
 97.5 
 103.5 
 105.4 
 
 108.2 
 
 93.8 
 114.1 
 128.6 
 113.8 
 107.7 
 
 100.0 
 
 54.7 
 
 119.0 
 
 105.5 
 110.8 
 119.3 
 
 129.3 
 
 81.4 
 
 147.5 
 
 135.5 
 
 83.0 
 
 97.8 
 
 The data are so inadequate that the construction of a similar table for 
 capitalist-entrepreneurial distributions is not feasible. However, there are 
 comparatively good figures for total income of farmers and total number 
 of farmers year by year. 2 The average incomes of farmers, year by year, 
 were the following percentages of the estimated average incomes of all 
 persons gainfully employed in the country. 
 
 Percentages 
 
 1910 75.19 
 
 1911 69.13 
 
 1912 72.41 
 
 1913 74.88 
 
 1914 76.33 
 
 1915 80.45 
 
 1916 82.85 
 
 1917 104.51 ♦ 
 
 1918 109.68 
 
 1919 103.95 
 
 1920 63.88 
 
 This is a wide range. 
 
 Exactly what effects have such internal movements of the component 
 distributions upon the total income frequency distribution curve? This 
 is a difficult question to answer as we have not sufficient data to break 
 
 component parts remained constant in shape and in their relative positions with respect to one 
 another on the income scale, these relations would of themselves constitute a s 'law" 
 
 1 Based upon Income in the United States, Vol. I, pp. 102 and 103. 
 
 2 See Income in the United States, Vol. I, p. 112. 
 
382 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 down the total, composite, curve into its component parts with any de- 
 gree of confidence. 1 However, the movements of wages in recent years 
 would appear to give us a clue to the sort of phenomena we might expect 
 to find if we had complete and adequate data. 
 
 The slopes of the upper income tails of wages distributions are great, 
 4 to 5 or more. 2 Now the wage curve moved up strongly from 1917 to 
 1918 if we may judge by averages. The average wage of all wage earners 
 in the United States 3 increased 15.6 per cent 4 from 1917 to 1918. During 
 the same period the average income of farmers increased 19. 1 per cent 5 
 and the average income of persons other than wage earners and farmers 
 remained nearly constant. Total amounts of income by sources in millions 
 of dollars were: 
 
 1917 
 
 1918 
 
 Percentage 1918 
 was of 1917 
 
 Total Wages a 
 
 Total Farmers' Income . 
 All other Income 
 
 $27,795 
 
 8,800 
 
 17,265 
 
 $32,575 
 10,500 
 17,291 
 
 117.20 
 119.32 
 100.15 
 
 Total Income. 
 
 $53,860 
 
 $60,366 
 
 112.08 
 
 a Includes pensions, etc., and includes soldiers, sailors, and marines. 
 
 Stockholders in corporations saw income from that source actually decline 
 from 1917 to 19 18. 6 What happened to American income-tax returns 
 during this time? 
 
 1 The processes by which the income distribution curve published in Income in the United 
 States, Vol. I, pp. 132-135 was arrived at were such that to use that material here would 
 practically amount to circular reasoning. ■ The conclusions arrived at here were used in build- 
 ing up that curve. 
 
 1 The slope of the tail of the wage and salary curve in the 1917 income tax returns is only 
 about 3.21 (compare, note 2, p. 377). However we must remember that the individuals there 
 classified are largely of an entirely different type of "wage-earner" from those in the lower 
 groups. In this upper group occur the salaried entrepreneurs, professional men, etc., and 
 those whose "salaries" are really profits or dividends. The evidence points to a rather dis- 
 tinct and significant heterogeneity along this division in the wage and salary distribution. 
 See Chapter 30. i .,„.«,., 
 
 » Excluding soldiers, sailors, and marines, and professional classes but including officials 
 and "salaried entrepreneurs." 
 
 « From $945 per annum in 1917 to $1,092 per annum in 1918. 
 
 * From $1,370 per annum in 1917 to $1,632 per annum in 1918. 
 
 • CORPORATION DIVIDENDS, SURPLUS AND EARNINGS 
 
 (In millions of dollars) 
 
 
 Dividends 
 
 Surplus 
 
 Net earnings 
 
 1917. . 
 
 3,995 
 2,568 
 
 3,963 
 1,945 
 
 7,958 
 
 
 4,513 
 
 See page 324. 
 
PARETO'S LAW 
 
 383 
 
 TOTAL AMOUNT OF NET INCOME RETURNED BY SOURCES (RETURNS 
 REPORTING OVER $2,000 PER ANNUM NET INCOME) a 
 
 (Millions of dollars) 
 
 Income class 
 
 Wages and salaries 
 
 All other sources b 
 
 1917 
 
 1918 
 
 1917 
 
 1918 
 
 $3,648 
 
 $6,493 
 
 $7,543 
 
 $7,198 
 
 1,553 
 
 3,687 
 
 1,799 
 
 2,036 
 
 301 
 
 703 
 
 528 
 
 736 
 
 661 
 
 849 
 
 1,167 
 
 1,296 
 
 1,133 
 
 1,254 
 
 4,049 
 
 3,130 
 
 Over $2,000 . . 
 
 2,000- 4,000. 
 
 4,000- 5,000. 
 
 5,000-10,000. 
 Over 10,000. 
 
 ° Wages income from returns reporting between $1,000 and $2,000 per annum is not avail- 
 able for 1917. 
 
 b "Other sources" are total net income minus wages and salaries, i. e., total general deduc- 
 tions have been assumed as deductible from other sources (gross). All things considered, 
 this seems proper here though it may easily be criticised. In connection with changes in the 
 relation between net and gross income from 1917 to 1918 see Chapter 30, pp. 401 and 402. 
 
 While reported income from all other sources than wages and salaries 
 declined 4.6 per cent, 1 reported income from wages and salaries increased 
 78.0 per cent. 2 Moreover, the great increases in wages and salaries were 
 in the lowest intervals. The wage curve with its steep tail-slope was 
 moving over into the income tax ranges. 3 The effect upon the total curve 
 is very pronounced, as may be seen from Table 28R. 
 
 TABLE 28R 
 
 AMERICAN INCOME TAX RETURNS IN 1917 AND 1918 
 
 Total Number of Returns 
 (In thousands) 
 
 1917 
 
 1918 
 
 Percentage 1918 
 was of 1917 
 
 $2,000-$4,000 
 4,000- 5,000 
 5,000-10,000 
 
 Over 10,000, 
 
 1,214 
 186 
 271 
 162 
 
 2,107 
 322 
 319 
 160 
 
 173.56 
 173.12 
 117.71 
 
 98.77 
 
 On a double log scale we see the curve changing its shape radically. While 
 the 1917 curve is comparatively smooth and regular, the 1918 curve 
 develops a distinct "bulge" in the lower ranges. 4 
 
 The preceding discussion has been concerned with equal dollar-income 
 
 1 Had "other sources" been taken gross instead of net, that item would have shown an 
 increase of 5.3 per cent instead of a decrease of 4.6 per cent. 
 
 2 The actual spread is still greater than the figures show. Income from professions, which 
 in 1917 was classed under wages, in 1918 and 1919 was classed under business. 
 
 3 This seems to be a fact though it is not the whole story. The "intensive drive" of 1919 
 may easily account for some of the increase. See Chapter 30 for a discussion of the probable 
 extent of this influence. 
 
 * See Income in the United States, Vol. I, Charts 28 and 30. 
 
384 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 intervals. However, $2,000 income in 1918 was relatively less than $2,000 
 income in 1917. The average (per capita) income of the country was 
 $523 in 1917 and $586 in 1918. 1 The adjustment is theoretically crude, 
 but $2,241 2 in 1918 might be considered as in one sense equivalent to 
 $2,000 in 1917. The results of comparisons of the two years upon this 
 basis are given in Table 28S. 3 
 
 TABLE 28S 
 
 INCOME RETURNED— BY SOURCES 
 
 (Millions of dollars) 
 1917 
 
 Income class 
 
 Wages and 
 salaries 
 
 Total net 
 income 
 
 Total net 
 
 income 
 
 minus 
 
 wages and 
 
 salaries 
 
 Total gross 
 income 
 
 Total gross 
 
 income 
 
 minus 
 
 wages and 
 
 salaries 
 
 $2,000-$4,000 
 
 4,000- 5,000 
 
 5,000-10,000 
 
 Over 10,000 
 
 $1,553 
 
 301 
 
 661 
 
 1,133 
 
 $3,352 
 
 829 
 1,828 
 5,182 
 
 $1,799 
 
 528 
 
 1,167 
 
 4,049 
 
 $3,713 
 
 895 
 
 1,951 
 
 5,518 
 
 $2,161 
 
 594 
 
 1,290 
 
 4,384 
 
 1918 
 
 $2,241-$4,482 
 
 $3,236 
 
 $5,359 
 
 $2,123 
 
 $5,766 
 
 $2,530 
 
 4,482- 5,602. ... 
 
 498 
 
 1,111 
 
 613 
 
 1,247 
 
 749 
 
 5,602-11,205 
 
 773 
 
 1,960 
 
 1,187 
 
 2,315 
 
 1,542 
 
 Over 11,205 
 
 1,153 
 
 4,129 
 
 2,976 
 
 4,842 
 
 3,689 
 
 (Multiplied by — , that is reduced to " 1917 dollars") 
 586 
 
 $2,241-$4,482. ... 
 
 $2,888 
 
 $4,783 
 
 $1,895 
 
 $5,146 
 
 $2,258 
 
 4.482- 5,602. . . . 
 
 445 
 
 992 
 
 547 
 
 1,113 
 
 668 
 
 5,602-11,205. ... 
 
 690 
 
 1,749 
 
 1,059 
 
 2,066 
 
 1,376 
 
 Over 11,205.... 
 
 1,029 
 
 3,685 
 
 2,656 
 
 4,321 
 
 3,292 
 
 (Percentages of Total Income of Country) 
 IM7 
 
 $2,000-$4,000 
 
 2.88 
 
 6.22 
 
 3.34 
 
 6.89 
 
 4.01 
 
 4,000- 5,000. ... 
 
 .56 
 
 1.54 
 
 .98 
 
 1.66 
 
 1.10 
 
 5,000-10,000 
 
 1.23 
 
 3.39 
 
 2.16 
 
 3.62 
 
 2.39 
 
 Over 10,00 
 
 2.10 
 
 9.61 
 
 7.51 
 
 10.24 
 
 8.14 
 
 1918 
 
 $2,241-$4,482 
 
 5.30 
 
 8.78 
 
 3.48 
 
 9.45 
 
 4.15 
 
 4,482- 5,602. ... 
 
 .82 
 
 1.82 
 
 1.00 
 
 2.05 
 
 1.23 
 
 5,602-11,205. ... 
 
 1.27 
 
 3.21 
 
 1.94 
 
 3.80 
 
 2.53 
 
 Over 11,205... 
 
 1.89 
 
 6.77 
 
 4.88 
 
 7.94 
 
 6.05 
 
 1 Income in the United States, Vol. I, p. 76. 
 
 * $2,000 X H|- 
 
 * The figures for the amounts of income in the irregular 1918 income intervals of that table 
 ($2,241-$4,482, etc.) were calculated by straight line interpolation on a double log scale ap- 
 plied to the even thousand dollar intervals of the income-tax returns. Though the total 
 income curve does not approximate linearity it may be assumed linear within the small 
 range of one income tax interval without serious error. 
 
PARETO'S LAW 
 
 385 
 
 (Table 28S concluded.) 
 
 NUMBER OF RETURNS 
 
 (Thousands) 
 
 Income class 
 
 1917 
 
 Income class 
 
 1918 
 
 Percentage 1918 
 was of 1917 
 
 $2,000-$4,000 
 
 4,000- 5,000 
 
 5,000-10,000 
 
 Over 10,000 
 
 1,214 
 186 
 271 
 162 
 
 $2,241-84,482 
 
 4,482- 5,602 
 
 5,602-11,205 
 
 Over 11,205 
 
 1,758 
 220 
 260 
 136 
 
 144.81 
 
 118.28 
 
 95.94 
 
 83.95 
 
 It is from this table once again apparent that the wage distribution moved 
 independently up on the income scale and that the effect of this movement 
 was confined to the lowest income intervals. Charts 28T, 28U, 28V, 28W, 
 28X, 28Y, 28Z, and 28AA which show the number of dollars income per 
 dollar-income interval, by sources, are enlightening as illustrating in still 
 
 CHART 28 T 
 
 100,000 
 
 U.S. INCOME TAX RETURNS 
 1916 
 
 NUMBER OF DOLLARS !N EACH INCOME 
 INTERVAL BY SOURCES 
 
 Scales Logarithmic. 
 
 t TOTAL INCOME. 
 Z. WAGES. 
 
 3. BUSINESS 
 
 4. OTHER INCOME 
 
 INCOME IN THOUSANDS OP DOLLARS 
 20 30 40 SO 100 200 
 
 i i i J ■ i i T i 
 
 300 400 500 
 
 2,000 
 
386 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 CHART 28 tf 
 
 U. S. INCOME TAX RETURNS 
 
 1916 
 
 NUMBER OF DOLLARS IN EACH 
 
 INCOME INTERVAL BY SOURCES. 
 
 Scales Logarithmic. 
 
 A INCOME OTHER THAN WAGES 
 
 OR BUSINESS. 
 5. RENTS. 
 
 INCOME IN THOUSANDS OF DOLLARS 
 20 30 40 50 100 200 
 
 400 500 
 I I 
 
 2.00(1 
 
PARETO'S LAW 
 
 387 
 
 CHART 28V 
 
 U.S. INCOME TAX RETURNS. 
 1317 
 
 NUMBER OF DOLLARS IN EACH 
 INCOME INTERVAL BY SOURCES. 
 
 Scales Logarithmic. 
 
 I. TOTAL INCOME. 
 8 WAGES. 
 
 3. BUSINESS. 
 
 4. OTHER INCOME. 
 
 3 4 5 
 
 INCOME IN THOUSANDS OF DOLLARS 
 10 20 30 40 50 100 200 300 400 500 
 
 I I I I I I I I I I 
 
 1.000 
 
388 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 CHART 28W 
 
 U 5 INCOME TAX RETURNS 
 
 1917 
 
 NUMBER OF DOLLARS IN EACH 
 
 INCOME INTERVAL BY SOURCES 
 
 Scales Logarithmic. 
 
 4 INCOME OTHER THAN WAGES OR BUSINESS. 
 
 5 RENTS 
 
 6 INTEREST 
 7. DIVIDENDS 
 
 INCOME IN THOUSANDS OF DOLLARS 
 20 30 40 50 100 200 
 
 I I I I i I I 
 
PARETO'S LAW 
 
 389 
 
 ^^^ 
 
 
 
 CHART 28 X 
 
 1 o*"""\..--V 
 
 z«-' 
 
 \ ^^» 
 
 
 
 -1,000,000 
 
 V N \ 
 
 
 US. INCOME TAX RETURNS 
 
 ,.**• 
 
 ^<A Nl 
 
 
 1918 
 
 
 -^ A X 
 
 
 NUMBER Or DOLLARS IN EACH 
 
 3 
 
 
 INCOME INTERVAL BY SOURCES 
 
 
 ^Sv 
 
 
 Scales Logarithmic 
 
 -100,000 
 
 
 IV Nv 
 
 1 TOTAL INCOME- 
 
 2 WAGES. 
 
 3 BUSINESS 
 
 
 
 ■"•S^ - ""-^* >» 
 
 4. OTHER INCOME. 
 
 
 
 
 
 -10,000 
 
 
 
 i 
 
 l.OOOg 
 
 
 
 i 
 
 
 \ 
 
 
 I 
 
 
 
 
 8 
 
 z 
 
 
 
 -100 as 
 
 
 
 \ \ Vs. 
 
 1 
 
 8 
 
 
 
 \\X 
 
 1 
 
 Ok 
 
 
 
 V ^~-\ Ai 
 
 -io a 
 
 
 
 ~X \ 
 
 1 
 
 
 
 \ \ 
 
 s 
 
 
 
 \ \ 
 
 -1 
 
 
 
 \ ' 
 
 \ 
 
 \ 
 \ 
 
 I 
 
 2 
 
 3 4 5 
 
 INCOME IN THOUSANDS OF DOLLARS 
 10 20 30 40 SO 100 
 
 \ 
 \ 
 
 \ 
 
 \ 
 200 300 400 500 1,000 2,000 3.000'' 
 
390 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 U. 3. INCOME TAX RETURNS 
 1316 
 
 NUMBER OF DOLLARS IN EACH 
 INCOME INTERVAL BY SOURCES 
 
 .Scales Logarithmic. 
 4. INCOME OTHER THAN WAGES 
 
 OR BUSINESS. 
 5 RENTS. 
 e. IHTERE3T. 
 7 DIVIDEND5. 
 
 INCOME IN THOUSANDS OF DOLLARS 
 
 20 30 40 50 100 200 
 
 1.000 
 
PARETO'S LAW 
 
 391 
 
 U. S. INCOME TAX RETURNS 
 
 1913 
 
 NUMBER OF DOLLARS IN EACH 
 
 INCOME INTERVAL EST SOURCES 
 
 Scales Logarithmic 
 
 1. TOTAL INCOME. 
 
 2. WAGES. 
 
 3. BUSINESS. 
 
 4. OTHER INCOME. 
 
 V_— a 
 
 INCOME IN THOUSANDS OF DOLLARS 
 
392 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 U a INCOME TAX RETURNS 
 
 1919 
 
 NUMBER OK DOLLARS IN EACH 
 
 INCOME INTERVAL BY SOURCES 
 
 Scales Logarithmic 
 A INCOME OTHER THAN WA6E5 
 OR BUSINESS 
 
 5. RENTS. 
 
 6. INTEREST. 
 7 DIVIDENDS. 
 
 \ f \ 
 
 INCOME IN THOUSANDS OF DOLLARS 
 20 30 40 SO 100 
 
 ■■;■■ 
 
PARETO'S LAW 393 
 
 greater detail the changes in the constitution of the returns from year to 
 year. 
 
 Such material and the appearance of the "bulge" on the income-tax 
 curve in the lowest income ranges 1 in the years 1918 and 1919 when wages 
 and salaries were high and average (per capita) incomes also high 2 strongly 
 suggest that the income curve, in so far as it shows any similarity from 
 year to year, changes its general appearance and turns up (on a double 
 log scale) as it approaches those ranges where wages and salaries are of 
 predominant influence. 3 The great slopes of wage distributions are on 
 this hypothesis not inconsistent with the smaller slope of the general 
 income curve in its higher (income-tax) ranges. 4 
 
 Conclusions: 
 
 (1) Pareto's Law is quite inadequate as a mathematical generalization, 
 for the following reasons: 
 
 (a) The tails of the distributions on a double log scale are not, 
 in a significant degree, linear; 
 
 (b) They could be much more nearly linear than they are without 
 that condition being especially significant, as so many dis- 
 tributions of various kinds have tails roughly approaching 
 linearity; 
 
 (c) The straight lines fitted to the tails do not show even approxi- 
 mately constant slopes from year to year or between country 
 and country; 
 
 (d) The tails are not only not straight lines of constant slope but 
 are not of the same shape from year to year or between 
 country and country. 
 
 (2) It seems unlikely that any useful mathematical law describing the 
 entire distribution can ever be formulated, because: 
 
 (a) Changes in the shape of the income curve from year to year 
 seem traceable in considerable measure to the evident hetero- 
 geneity of the data; 
 
 (b) Because of such heterogeneity it seems useless to attempt to 
 
 1 See Chapter 30 for further discussion of this "bulge" in connection with an examination 
 of how far it may be the result of irregularity in reporting. 
 
 2 Average (per capita) incomes being high means that a definite money income (such as 
 $2,000) takes us relatively further down the income curve than if average incomes were low. 
 
 3 It is difficult to say just where the "bulge" might have appeared in the 1917 distribution 
 if as great efforts had been made to obtain correct returns in that year as were made under 
 the "intensive drive" for 1918 returns. The wages line on the 1917 number of dollars income 
 per dollar-income interval chart (Chart 28V) shows signs of turning up somewhere between 
 $4,000 and $5,000 and the business line somewhere in the $5,000-$10,000 interval. However 
 neither movement is large nor can their positions be accurately determined on account of the 
 size of the reporting intervals. See also Chapter 30, p. 412. 
 
 4 The " bulge " on the income from wages and salaries curve itself, as seen in the income- 
 tax returns for 1918 and 1919 (see Charts 28X and28Z), seems the result of heterogeneity in 
 these wage and salary data themselves. This hypothesis is considered in Chapter 30. 
 
394 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 describe the whole distribution by any mathematical curve 
 designed to describe homogeneous distributions (as any simple 
 mathematical expression must almost necessarily be designed 
 to do) ; 
 
 (c) Furthermore, the existing data are not adequate to break up 
 the income curve into its constituent elements; 
 
 (d) If the data were complete and adequate we might still remain 
 in our present position of knowing next to nothing of the 
 nature of any "laws" describing the elements. 1 
 
 (3) Pareto's conclusion that economic welfare can be increased only 
 through increased production is based upon erroneous premises. 
 The income curve is not constant in shape. The internal movements 
 of its elements strongly suggest the possibility of important changes 
 in distribution. The radically different mortality curves for Roman 
 Egypt and modern England, 2 and the decrease in infant mortality 
 in the last fifty years illustrate well what may happen to heteroge- 
 neous distributions. 
 
 The next four chapters review the data from which any income frequency 
 distribution for the United States must be constructed. 
 
 1 Though all the evidence points to hope of further progress lying in the analysis of the 
 parts rather than in any direct attack upon the unbroken heterogeneous whole. 
 « See Biometrika, Vol. I, pp. 261-264. 
 
CHAPTER 29 
 OFFICIAL INCOME CENSUSES 
 
 There has never been a complete income census of the American people. 
 The Federal income-tax data cannot take the place of such a census. Re- 
 specting the distribution of income among persons having incomes of less 
 than $1,000 Federal income-tax data give us no information whatsoever." 
 Furthermore, on account of the exemption of married persons, compara- 
 tively little use can be made of the $1,000 to $2,000 interval. The number 
 of persons reporting incomes over $2,000 in our best year, 1918, was only 
 7.3 per cent of the estimated total number of income-recipients in the 
 country. Moreover, not only because of direct evasion and illegal non- 
 reporting, but also because of "legal evasion" and the large amount of 
 tax-exempt income which need not be reported at all, these income-tax 
 data cannot give an approximately correct picture of even that part of 
 the frequency curve which lies above $2,000. The adjustments of the 
 income-tax data necessary to obtain such a picture are extremely large, 
 as we shall presently see. 
 
 Only one country in the world has ever taken an official income census 
 which made any pretense of completeness. Under the War Census Act 
 the Commonwealth of Australia took an official income census of incomes 
 received during the year ended June 30, 1915, by everyone, man, woman, 
 or child, who was "possessed of property, or in receipt of income." * The 
 results of that census are summarized by G. H. Knibbs, the Commonwealth 
 Statistician, in The Private Wealth of Australia and its Growth. A Re- 
 port of the War Census of 1915. (See Table 29A and Charts 29A, 29B 
 and 29C.) 
 
 Now while it would naturally be impossible to construct a complete 
 frequency distribution for American incomes from Australian data, 2 we 
 might perhaps hope to discover some characteristics of income-distribution 
 
 1 While the first clause of the Australian "Wealth and Income Card" stated merely that 
 it was "to be filled in by all persons aged 18 or upwards possessed of property, or holding 
 property on trust, or in receipt of income," etc. (p. 9), "a special instruction was issued that 
 in the case of all persons under the age of 18, possessed of property, or in receipt of income, 
 a return must be furnished by the parent or guardian in respect of such property or income." 
 (p. 10.) The income from such trust funds was not all, but only "in the main," allocated to 
 individual beneficiaries, (p. 22.) 
 
 G. H. Knibbs, The Private Wealth of Australia and its Growth. A Report of the War Census 
 of 1915. 
 
 2 Aside from the questionableness of such a procedure, the large size of the low income 
 intervals in the Australian distribution and the lack of information concerning the amount 
 of negative income make that distribution a difficult one to work with. A classification by 
 such large intervals tells very little. 
 
 395 
 
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 ■ 
 
AUSTRALIAN CENSUS OF INCOMES - 1915. 
 
 r~ 
 
 — i 
 
 4 
 
 CHART 29 A 
 
 NON-CUMULATIVE FREQUENCY DISTRIBUTION 
 
 SC#L£S NffTURfiL 
 
 ■Males and Females 
 Males 
 ■Females 
 
 , , 
 
 SCALE 
 
 100.000 
 PERSONS 
 
 I 
 
 lob" 
 
 "A 
 
 i 
 
 300 4(lo 500 600 
 
 INCOME IN POUNDS STERLING 
 
 200 
 
 700 
 
 ■1,000,000 
 
 100,000 g 
 
 E 
 
 CO 
 
 ■ 10,000 § 
 
 »i n — L*_ 
 
 CHART 29 B 
 
 IN-CUMULATIVE FREQUENCY DISTRIBUTION 
 
 SCALES LOGARITHMIC 
 
 I 
 
 I 
 
 I 
 
 •1,000 
 ■100 
 
 I 
 
 I— , 
 
 n 
 
 50 
 
 "1,000,000 
 
 100 
 
 INCOME IN POUNDS STERLING 
 
 29O 300 500 1,000 
 
 "L- 
 
 2,000 3,000 
 
 5,000 
 
 CHART 29 C 
 
 CUMULATIVE FREQUENCY DISTRIBUTION 
 
 SCALES LOGARITHMIC 
 
 X 
 
 . Income in pounds sterling 
 tqa 290 390 590 l.Qoo 
 
 2,000 MM , 5,000 
 
398 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 curves in general from this, the only actual census ever taken. A knowl- 
 edge of such general characteristics might then, quite imaginably, be a 
 little helpful in the problem of describing the American or any other 
 income distribution. 
 
 However, when we come to examine the Australian figures, we find that 
 they have certain pronounced peculiarities which would be extremely diffi- 
 cult to read into the American material. For example, the Australian dis- 
 tribution shows a flatness and lack of pronounced mode totally unlike the 
 results we have built up from an analysis of American data. In the Aus- 
 tralian distribution there are nearly the same number of persons having 
 incomes between and £50, £50 and £100, and £100 and £150. 1 
 
 What are the causes of this rather startling peculiarity of the Australian 
 frequency curve? 2 In the first place let us suggest a possibly minor but 
 by no means necessarily negligible factor. We know little about the good- 
 ness of the Australian reporting in this census. Income is, from its nature, 
 a difficult subject to investigate. When the material is collected by means 
 of schedules to be filled in by the informants, as was the case in the Aus- 
 tralian census, the returns may easily be full of errors. The average in- 
 dividual is surprisingly ignorant concerning the amount of his total income. 
 The further fact that the census was taken in order to estimate possi- 
 bilities of future taxation may well have been a powerful incentive towards 
 great irregularities all along the line, but especially in the lower income 
 groups. Persons whose income brought them distinctly into the upper 
 groups (over £156) were, at the time of the income census, about to make 
 returns under oath for income-tax purposes and would hardly care to 
 show a radical discrepancy between the two returns. On the other hand, 
 many persons, whose true incomes were around £156 and the modal income, 
 might easily have "underestimated" with the idea of evading if possible 
 future taxation based upon a lowering of the exemption limit. The result 
 of such practices would tend to show up graphically in a flattening of the 
 curve in the vicinity of the mode of the distribution and a raising of the 
 numbers in the lowest groups. 3 
 
 However, poor reporting is probably only a secondary element ac- 
 counting for the peculiarities of the Australian curve. It is most of all the 
 
 • See Table 29A and Chart 29A. 
 
 1 Notwithstanding the fact that distributions for different times and for different countries 
 probably vary greatly (see Chapter 28), the difference between the Australian curve and 
 the Bureau's American estimate seems too radical to explain upon this basis. 
 
 • It is difficult to determine the extent of actual non-reporting. The number of males 
 filling out income cards was 2,527,831. All males "possessed of property, or in receipt of 
 income" are supposed to be included in this number. It amounted, however, to only 54.60 
 per cent of the total male population. Males "possessed of property, or in receipt of income" 
 necessarily constitute a larger percentage of the total male population than do male "bread- 
 winners," yet in the Australian census of 1911 male breadwinners constituted 69.4 per cent 
 of the total male population, and male breadwinners 20 years of age or older 58.9 per cent. 
 Even if we assume that the number of income returns for males under 18 was negligible we 
 still are faced with a discrepancy difficult to account for. 
 
OFFICIAL INCOME CENSUSES 399 
 
 concentration of female returns in the lowest income groups which gives 
 the flat and modeless appearance to the total curve. The Australian fre- 
 quency distribution among males only, is much more like our estimated 
 American distribution ! than is the Australian distribution among males 
 and females together. Now the concentration of female returns in the 
 lower income intervals would seem to be the result of a large number of 
 returns made by women and female children receiving petty incomes from 
 property who would be classified, in the Australian Census of Population, 
 as "dependents" and not as "breadwinners." 2 
 
 Of the total female population in 1915, 33.46 per cent made out income 
 cards and 23.18 per cent reported positive incomes (10.28 per cent re- 
 ported zero or negative incomes) . But according to the Australian census 
 of 1911, only 18.6 per cent of the total female population were classified 
 as "breadwinners." Thus the women reporting positive incomes in 1915 
 constituted a much larger percentage of the total female population than 
 did female "breadwinners" in 1911 of the total female population in that 
 year. The discrepancy seems too great to be accounted for by the in- 
 crease in the number of women "breadwinners" caused by the war. More 
 than half of the 23.18 per cent of the female population reporting positive 
 incomes in 1915 reported incomes under £50 per annum. Moreover, the 
 average income of this group was only £22 per annum — under the arith- 
 metic average of the interval. This strongly suggests petty incomes from 
 property, and part time occupations such as keeping boarders, lodgers, 
 chickens, etc., rather than any great increase in the number of female 
 "breadwinners." The fact that over 30 per cent of the returns made by 
 females reported zero or negative incomes is further evidence that the 
 large number of extremely small incomes reported was largely the result 
 of the schedule calling for income returns from all persons "possessed of 
 property." 
 
 Negative incomes arise in general from business or speculative losses. 
 Bad as may be the condition of any laboring class, its members are seldom 
 faced with negative incomes. It is unlikely that many of the 249,476 
 females reporting "deficit and nil" were wage-earners. They were in 
 general the owners of small investments which showed losses, such as 
 town lots upon which taxes had been paid. 3 
 
 1 See Income in the United States, Vol. I, pp. 128, 129, 132-135. 
 
 2 All persons are classified as "breadwinners" or as "dependents" by the Australian census. 
 Male "breadwinners" in Australia constituted in 1911, according to the census of that year, 
 69.4 per cent of the total male population, female "breadwinners" 18.6 per cent of the total 
 female population, and total "breadwinners" 45.0 per cent of the total population. These 
 figures compare with American census figures for 1910 showing males "gainfully employed" 
 to constitute 63.6 per cent of total males, females "gainfully employed" 18.1 per cent of 
 total females, and total "gainfully employed" 41.5 per cent of the total population. 
 
 3 It is worth noting that in the Australian schedule "rates and taxes paid" could be de- 
 ducted before making an income return. This consideration may be of some importance in 
 explaining the very large number of small, zero, and negative incomes. 
 
400 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 While the frequency curve for Australian males is much more like the 
 American distribution than the curve representing both male and female 
 Australian income recipients, even it shows a much greater concentration 
 in the lowest income intervals than does the American distribution. This 
 can probably be accounted for to some extent by a large number of income 
 returns for young male "dependents" "possessed of property." 
 
 The essential difference in appearance between the American income- 
 distribution curve which we presented in Volume I and the Australian 
 curve of 1915 is, then, probably traceable to (1) Australian underreporting 
 and (2) Australian inclusion of a large number of "dependents" who re- 
 ceived petty incomes from property and who were in no important sense 
 "breadwinners" or "gainfully employed." 
 
 What shall we say about the desirability or undesirability of including 
 in an income frequency distribution dependents receiving petty incomes 
 from property? While it is true that their incomes, positive or negative, 
 are in a way as real as any other incomes, we must remember that probably 
 almost all individuals over six years of age not only receive but earn some 
 money income during each year. Shall we then include the entire popu- 
 lation over six years old in our distribution? As we approach this theo- 
 retical limit it is seen that the concept becomes less and less practically or 
 even theoretically interesting. Both practically and theoretically we are 
 interested in the incomes of persons who, though they be minors, have 
 "economically come of age" and have entered into certain definite rela- 
 tions to the machinery of factorial distribution. They are "breadwinners" 
 or "persons gainfully employed," and the concept back of such expres- 
 sions, though like many economic concepts somewhat of a compromise, 
 seems a good compromise for our purposes. 
 
 Defining income recipient as we have, we cannot use the Australian 
 material as an aid to the graduation or adjustment of the American income- 
 distribution curve in its lower ranges. In the upper income ranges, the 
 Australian distribution offers, as we shall see, an interesting illustration 
 of the same double swing (letter S) appearance of the curve seen in some 
 of the more recent American data. 1 
 
 1 When charted on a double log scale. 
 
CHAPTER 30 
 AMERICAN INCOME TAX RETURNS 
 
 At the beginning of the preceding chapter attention was drawn to some 
 reasons why income-tax returns cannot take the place of an adequate 
 income census. Nevertheless tax returns are in many respects the most 
 important single source of information we have for estimating the fre- 
 quency distribution of incomes. Were there neither tax returns nor in- 
 come censuses for any country, it is difficult to see how we could make 
 even an interesting guess as to the distribution of income in the upper 
 ranges. 
 
 American income-tax data go back to 1913. We have now at our dis- 
 posal returns for the seven years, 1913 to 1919, inclusive. 1 However, the 
 amount of information given in the official reports for the earlier years 
 1913, 1914 and 1915 is not great. Little is shown beyond the number 
 of returns classified by large income intervals and the same returns classi- 
 fied by districts. The 1916 tax report is the most voluminous and in one 
 respect the most adequate report which has yet appeared. 2 It contains 
 a set of tables which we are sorry to miss in the later reports, showing 
 the frequency distribution of incomes by separate occupations. Other 
 features of this report which have been retained in later years are tables 
 showing both number of returns and amount of net income for each income 
 class for the country as a whole, and the same by States; tables showing 
 the sources of the income returned in each income interval, that is the 
 amount from wages, business, property; distribution tables arranged by 
 sex and conjugal condition; amounts of tax collected from each income 
 class, etc. 
 
 Changes in the Federal Income Tax Law during the period have not 
 been such as greatly to affect any conclusions which we have drawn from 
 the data. From the standpoint of this investigation, probably the most 
 important changes in the law relate to general deductions, professions, and 
 minimum taxable income. 
 
 In the 1916 returns all deductions were classified as general deductions. 
 
 1 The Annual Reports of the Commissioner of Internal Revenue are the sources for American 
 income-tax data for the years 1913 to 1915. Since 1915 the data have appeared annually 
 as a separate Treasury Department publication entitled Statistics of Income. 
 
 2 A peculiarity of the 1916 data is that the returns are tabulated as family rather than in- 
 dividual returns. "The net incomes reported on separate returns made by husband and wife 
 in 1916 are combined and included as one return in the figures for the several classes." Statis- 
 tics of Income, 1917, p. 22. 
 
 401 
 
402 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 In the 1917 returns the types of deductions classified as general deductions 
 were greatly reduced; not even contributions were included. In 1918 the 
 category was enlarged; contributions, for example, were again placed in 
 the general deductions class. Now these changes affect greatly the rela- 
 tions between net and total income from year to year. Reported net income 
 was in 1916 only 75.43 per cent of reported total income, in 1917 it was 
 92.67 per cent, in 1918 89.74 per cent, and in 1919 88.51 per cent. As 
 it is the total and not the net income which in the Statistics of Income, is 
 divided up according to source, such fluctuations as the above interfere 
 with comparisons of different years. 
 
 While income from professions was tabulated separately in 1916, in 1917 
 it was included in wages and salaries, and in 1918 and 1919 in business. 
 
 In the 1913 to 1916 returns exemptions were $3,000 per annum for an 
 unmarried person, or a married person not living with his wife (or her 
 husband), and $4,000 per annum aggregate exemption for married persons 
 living together. 1 In the 1917 and later returns these minima were reduced 
 to $1,000 and $2,000 respectively. However, the increase in usefulness for 
 our purposes of the 1917 and later returns was even greater than the 
 lowered minima would suggest. Not only was the minimum taxable 
 income lowered from $3,000 to $1,000, but this reduction occurred in the 
 face of a rapidly rising general level of incomes. With the rise in incomes, 
 $3,000 in 1918 or 1919 was relatively a much smaller income than $3,000 
 in 1913. In other words, we might logically expect $3,000 to be relatively 
 further down the income distribution curve in 1918 than in 1916 or 
 1917. 
 
 The accuracy of the reporting is, of course, a matter of great importance 
 for this investigation. Now, while it does not seem possible to measure 
 directly from the data changes in accuracy of reporting during the period, 
 the rapid expansion of the income-tax organization and its increasing 
 attention to the investigation and checking of returns establish the pre- 
 sumption of greater statistical value in the reports for the later years. 
 Offsetting this to an unknown degree is the apparently increasing amount 
 of "legal evasion" in the higher income classes. The reporting for the 
 years 1913, 1914, 1915 and 1916 appears to have been peculiarly bad in 
 the lower income ranges. The distinct improvement in 1917 (compare 
 the 1917 returns with those for earlier years in Tables 28B, 28C, 28D, 28E, 
 and Charts 27 and 28 of Volume I) seems associated with the patriotic 
 enthusiasm engendered by the war. Upon our entry into the war, not 
 only did the Bureau of Internal Revenue make an increased effort to ob- 
 
 1 As the returns for 1913 were for income received for the ten months March 1 to December 
 31, 1913, the actual minima used for reporting purposes were $2,500 and $3,333.33 (i. e., |8 
 of $3,000 and $4,000 respectively). 
 
AMERICAN INCOME TAX RETURNS 403 
 
 tain correct returns but individuals, under the spur of patriotism, seem to 
 have made less effort to evade. 1 
 
 The remainder of this chapter is concerned largely with a discussion 
 of possible irregularities in the distribution of non-reporting and under- 
 statement in the later years. While the total amount of non-reporting 
 and understatement was almost certainly greater in the returns for 1917 
 than in those for 1918 and 1919, are we sure that the non-reporting and 
 understatement of these later years are not possibly more irregularly dis- 
 tributed along the frequency curve than was the case in 1917? Is it 
 possible that the improvement in the accuracy of the published returns 
 for 1918, as compared with those for 1917, was so much greater in the 
 income intervals under $5,000 that the resulting change in the shape of 
 the frequency curve may amount to something almost akin to an "over- 
 adjustment"? 
 
 Income returns by individuals are made on two types of blanks, a blank 
 to be filled in by persons reporting incomes under $5,000 and another 
 blank to be filled in by persons reporting incomes over that figure. Now, 
 while the returns of incomes under $5,000 and made on "under $5,000" 
 blanks are examined, investigated and audited in the field soon after 
 their receipt, the investigation and audit of the returns for incomes over 
 $5,000 are handled in Washington. If an individual has an actual income 
 of $8,000 but reports $4,600 (on an "under $5,000" blank), as soon as a 
 Field Collector discovers this discrepancy, he passes the matter over to 
 the Revenue Agent in charge of the District for Field Investigation. The 
 return, accompanied by the Agent's report, is forwarded to Washington 
 for final audit. Thus the Field Collectors audit only returns that are (a) 
 made on "under $5,000" blanks and (b) believed, after investigation, to be 
 for incomes which are actually under $5,000. 
 
 While the Field Audit of returns of these incomes is well under way 
 before the preparation of the statistical tables in the Statistics of Income 
 and hence appears in that tabulation to an unknown extent, the Washing- 
 ton audit of incomes over $5,000 has hardly begun and hence the amended 
 figures for these higher incomes do not appear in the Statistics of Income. 
 It is impossible to say exactly how much of the "bulge" 2 which appears 
 in the $1,000 to $5,000 interval on the double log charts of the 1918 and 
 1919 tax income distributions is caused by a difference in the accuracy 
 of the published figures for returns of incomes under and over $5,000. 
 However, the Treasury Department states that "the Statistics of Income 
 
 1 It must not, of course, be assumed that the increase in the number of returns in 1917 is 
 traceable solely to increased goodness of reporting. 
 
 2 Described in Chapter 28. At many points in the following discussion the reader should 
 refer back to the presentation of the case for heterogeneity in the income-tax data contained 
 in Chapter 28. 
 
404 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 are compiled almost entirely from unaudited returns whether they be for 
 'under $5,000' or 'over $5,000.'" It seems probable therefore that the 
 sudden change in slope of the 1918 curve (on a double log scale) at about 
 $5,000 can be explained only partially by a change in accuracy of the 
 published returns at that point. 
 
 Moreover, a considerable amount of evidence, some of which has already 
 been presented in Chapter 28, suggests that the "bulge" on the income 
 curves for the later years corresponds to a reality on the actual income 
 curves. While it may be somewhat over-accented in the published figures 
 for 1918 and 1919, and while the figures for 1917 might have shown more 
 of such a "bulge" * had the reporting been better, we must not assume 
 that the published figures for either 1917 or 1918 give a radically incorrect 
 picture of the facts merely because the income curves for the two years 
 are so different. The dogma of the similarity of the income curve from 
 year to year has little evidence to support it. 
 
 It is by no means certain that even the apparently definite and sharp 
 angles on the curves in this $4,000 to $6,000 region give an unreal picture. 
 While it is true that we find the same angles on the wages and salaries 
 curve, that curve itself seems heterogeneous. An income distribution 
 curve composed of wage and salary earners (in the ordinary sense of the 
 terms) may well cut an income distribution curve composed of "salaried 
 entrepreneurs," and business and financial experts somewhere in the lower 
 income ranges. The angle on the composite curve may give a decidedly 
 accurate picture of the facts. 2 
 
 Let us see what light the data throw on some of these problems. 
 Table 30A showing the number of returns for the lower income intervals 
 in 1917, 1918, and 1919 and the percentage movements from year to year 
 illustrates the great increase in the number of returns in the under-$5,000 
 intervals between 1917 and the later years. 
 
 Chart No. 28 of Volume I, on which are drawn the frequency distributions 
 for each year from 1916 to 1919 on a double log scale, shows the difference 
 in the appearance of the income curves for the three years. Examining 
 that chart we notice that the 1918 data-points, which in the upper income 
 ranges run nearly as smoothly as the 1917 points, in the $4,000 to $5,000 
 interval move abruptly upwards and from there on into the lowest income 
 ranges are well above the 1917 points, showing on the chart an irregular, 
 plateau-like effect in these lowest income ranges. No such "plateau" 
 is apparent on the 1917 line. The year 1919 presents in that chart a 
 
 1 While the 1917 curve runs much more smoothly in the $3,000 to $6,000 range than either 
 the 1918 or 1919 curves, it is not without the hint of a bulge beginning at about $4,500. See 
 p. 412. 
 
 * In constructing the complete income distribution curve for 1918, published in Volume I. 
 the influence of changes in the accuracy of reporting around $5,000 income was probably 
 overestimated. 
 
AMERICAN INCOME TAX RETURNS 
 
 405 
 
 TABLE 30A 
 
 
 Number of returns 
 
 Percentage increases 
 
 Income intervals 
 
 1917 
 
 1918 
 
 1919 
 
 1918 
 over 
 1917 
 
 1919 
 over 
 1918 
 
 1919 
 over 
 1917 
 
 $2,000-13,000 
 3,000- 4,000 
 4,000- 5,000 
 
 838,707 
 374,958 
 185,805 
 
 1,496,878 
 610,095 
 322,241 
 
 1,569,741 
 742,334 
 438,154 
 
 78.47 
 62.71 
 73.43 
 
 4.87 
 21.68 
 35.97 
 
 87.16 
 
 97.98 
 
 135.81 
 
 5,000- 6,000 
 6,000- 7,000 
 7,000- 8,000 
 8,000- 9,000 
 9,000-10,000 
 
 105,988 
 64,010 
 44,363 
 31,769 
 24,536 
 
 126,554 
 79,152 
 51,381 
 35,117 
 27,152 
 
 167,005 
 
 109,674 
 
 73,719 
 
 50,486 
 
 37,967 
 
 19.40 
 23.66 
 15.82 
 10.54 
 10.66 
 
 31.96 
 38.56 
 43.48 
 43.77 
 39.83 
 
 57.57 
 71.34 
 66.17 
 58.92 
 54.74 
 
 similar appearance to 1918 though the absence of small intervals in the 
 range immediately above $5,000 disguises the characteristics of the curve 
 materially. 1 
 
 The change in the contour of the lower range of the tax income frequency 
 curve from 1917 to 1918 and 1919, is, as we have mentioned, associated 
 with a large increase in the relative amount of income from wages and 
 salaries in the lower intervals. Tables 30B and 30C are interesting in 
 this connection. 2 
 
 The 1916 figures in Table 30B are introduced simply because they 
 are computable. 3 However, too much weight must not be attached to 
 them. The 1916 returns are undoubtedly extremely inadequate. The 
 high percentages that year from $3,000 income (the 1916 minimum) up 
 to about $10,000 may possibly be the result of the ease with which salary 
 returns (as opposed to wage, business, or other returns) are obtainable. 
 The $4,000 to $5,000 interval is the lowest comparable interval for the 
 four years. 4 In that interval the numbers of returns by years were: 
 
 1916- 72,027 
 1917-185,805 
 1918-322,241 
 1919-438,154 
 
 iWhen chart 28 was drawn for Volume I, only "preliminary" large interval data were 
 available. Final small interval data show a "bulge" very similar to that seen in the" 1918 line. 
 
 2 The 1917 official wages figures include income from professions. The 1918 and 1919 wages 
 figures do not. This makes the increase in the percentages in 1918 still more striking. In- 
 come from professions was tabulated separately in 1916, but was included in the wages figures 
 for that year in order that 1916 and 1917 might be comparable. 
 
 3 No data are available from which corresponding figures for 1913, 1914 or 1915 might 
 be calculated. 
 
 4 The $3,000-$4,000 interval did not in 1916, include married persons making a joint return. 
 
406 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 TABLE 30B 
 
 PER CENT THAT INCOxME FROM WAGES AND SALARIES IN EACH NET 
 INCOME CLASS WAS OF TOTAL NET INCOME IN THAT CLASS 
 
 Income class 
 
 1916 
 
 1917 
 
 1918 
 
 1919 
 
 $ 1,000-$ 2,000 
 
 
 
 79.45 
 
 83.49 
 
 2,000- 3,000 
 
 
 
 69.75 
 
 74.53 
 
 3,000- 4,000 
 
 76.98 
 
 
 55.21 
 
 61.86 
 
 2,000- 4,000 
 
 
 46.32 
 
 (64.42) 
 
 (69.45) 
 
 4,000- 5,000 
 
 66.86 
 
 36.30 
 
 48.85 
 
 52.48 
 
 5,000- 10,000 
 
 53.31 
 
 36.16 
 
 39.59 
 
 43.24 
 
 10,000- 20,000 
 
 36.38 
 
 32.94 
 
 38.60 
 
 38.11 
 
 20,000- 40,000 
 
 24.60 
 
 26.82 
 
 33.16 
 
 33.38 
 
 40,000- 60,000 
 
 17.23 
 
 22.74 
 
 27.88 
 
 27.57 
 
 60,000- 80,000 
 
 16.20 
 
 19.67 
 
 25.36 
 
 24.01 
 
 80,000- 100,000 
 
 13.37 
 
 18.51 
 
 22.16 
 
 22.70 
 
 100,000- 150,000 
 
 13.34 
 
 15.75 
 
 18.44 
 
 18.75 
 
 150,000- 200,000 
 
 9.39 
 
 12.65 
 
 13.16 
 
 15.42 
 
 200,000- 250,000 
 
 9.14 
 
 12.30 
 
 13.07 
 
 13.62 
 
 250.000- 300,000 
 
 7.87 
 
 9.36 
 
 12.57 
 
 11.92 
 
 300,000- 500,000 
 
 6.59 
 
 10.17 
 
 11.27 
 
 10.18 
 
 500,000-1,000,000 
 
 5.21 
 
 6.39 
 
 5.42 
 
 6.80 
 
 1,000,000-1,500,000 
 
 4.84 
 
 2.83 
 
 7.54 
 
 1.60 
 
 1,500,000-2,000,000 
 
 3.23 
 
 3.76 
 
 2.21 
 
 10 00 
 
 2,000,000 and over 
 
 .51 
 
 2.39 
 
 .85 
 
 4.02 
 
 The amounts of income from wages and salaries and from other net income 
 in the $4,000-$5,000 interval were year by year in millions of dollars: 
 
 
 1916 
 
 1917 
 
 1918 
 
 1919 
 
 Wages and salaries ° 
 
 216 
 107 
 
 301 
 
 528 
 
 703 
 736 
 
 1,029 
 
 Other net income 
 
 931 
 
 
 
 ° Income from professions is included in the 1916 and 1917 wages and salaries figures. 
 
 The percentage changes in these items from one year to the next were: 
 
 1917 1918 1919 
 
 1916 1917 1918 
 
 Wages and salaries 139 . 3 233 . 7 146 . 4 
 
 Other Net Income 493 . 139 . 4 126 . 6 
 
 It is plain that the great increase in the $4,000-^5,000 interval 1 in 1917 
 was in income from other sources than wages and salaries. 
 
 Table 30C shows the wage and salary figures compared with total income 
 instead of net income as in Table 30B. It was, of course, necessary to re- 
 tain the net income intervals as the data are not classified in total income 
 
 1 As may be seen from Tables 30B and 30C, the increase from 1916 to 1917 in income from 
 other sources than wages and salaries was greater than the increase in income from wages 
 and salaries not only in the $4,000-$5,000 interval but also in the $5,000-$ 10,000 interval. 
 
AMERICAN INCOME TAX RETURNS 
 
 407 
 
 intervals. Though the relations between years are different in this table 
 from what they are in the net income table, 1 the distribution of the per- 
 centages in each individual year shows much the same characteristics in 
 both tables. 
 
 TABLE 30C 
 
 FER CENT THAT INCOME FROM WAGES AND SALARIES IN EACH NET 
 INCOME CLASS WAS OF TOTAL INCOME IN THAT CLASS 
 
 Income class 
 (Net) 
 
 1916 
 
 1917 
 
 1918 
 
 1919 
 
 $ 1,000- $2,000 
 
 
 
 74.67 
 
 77.25 
 
 2,000- 3,000 
 
 
 
 65.42 
 
 69.14 
 
 3,000- 4,000 
 
 47.74 
 
 
 51.14 
 
 56.71 
 
 2,000- 4,000 
 
 
 41.82 
 
 (60.15) 
 
 (64.12) 
 
 4,000- 5,000 
 
 45.96 
 
 33.60 . 
 
 44.82 
 
 47.12 
 
 5,000- 10,000 
 
 36.38 
 
 33.87 
 
 33.55 
 
 36.60 
 
 10,000- 20,000 
 
 25.76 
 
 30.89 
 
 33.10 
 
 32.70 
 
 20,000- 40.000 
 
 18.81 
 
 25.20 
 
 28.76 
 
 28.36 
 
 40,000- 60,000 
 
 13.75 
 
 21.23 
 
 23.79 
 
 23.39 
 
 60,000- 80,000 
 
 12.76 
 
 18.56 
 
 21.51 
 
 20.33 
 
 80,000- 100,000 
 
 10.74 
 
 17.61 
 
 19.00 
 
 19.25 
 
 100,000- 150,000 
 
 11.06 
 
 15.05 
 
 15.92 
 
 15.40 
 
 150,000- 200,000 
 
 7.68 
 
 12.01 
 
 13.10 
 
 12.41 
 
 200,000- 250,000 
 
 7.83 
 
 11.75 
 
 11.22 
 
 11.26 
 
 250,000- 300,000 
 
 6.64 
 
 8.71 
 
 10.73 
 
 9.80 
 
 300,000- 500,000 
 
 5.50 
 
 9.59 
 
 9.62 
 
 8.19 
 
 500,000-1,000,000 
 
 4.35 
 
 5.88 
 
 4.37 
 
 5.38 
 
 1,000,000-1,500,000 
 
 4.12 
 
 2.62 
 
 6.29 
 
 1.34 
 
 1,500,000 2,000.000 
 
 2.82 
 
 3.54 
 
 1.81 
 
 8.54 
 
 2,000,000 anH. over 
 
 .47 
 
 2.18 
 
 .63 
 
 .32 
 
 The percentages in Tables 30B and 30C show each year a sudden increase 
 (as we approach the lower income intervals) somewhere in the $4,000 to 
 $5,000 or the $5,000 to $10,000 interval. At exactly what point each year 
 do these sudden increases seem to occur? Charts 30D, 30E and 30F pre- 
 sent the material in a slightly different form. They illustrate the relation- 
 ship between the average income from wages and salaries in each net 
 income interval and the average total income in the same net income in- 
 terval for the years 1917, 1918 and 1919 on a double log scale. The 1918 
 and 1919 charts immediately suggest the improbability of being able to 
 describe the data by a single simple mathematical expression. To the 
 1918 data-points have been applied two distinct mathematical curves, 
 which fit the data remarkably well and intersect at about $6,700 total 
 income. The curve fitted to the upper income ranges is a parabola, while 
 that fitted to the lower income ranges is an hyperbola, one of whose asymp- 
 totes is the 45° line which divides the chart into a "possible" and an "im- 
 
 1 Some reasons for the changes in relation of net to total income from year to year are 
 mentioned en pages 401 and 402. 
 
 fr 
 
 <<& 
 
408 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S 
 
 
 
 
 CHART 30 D 
 
 
 US. INCOME TAX RETURNS / 
 
 
 
 
 1317. / 
 
 
 
 
 AVERAGE INCOME / 
 
 
 
 
 FROM / 
 
 
 
 
 WA6ES AND SALARIES / 
 
 
 
 B 
 
 < 
 
 AND / 
 
 
 
 AVERA6E TOTAL INCOME / 
 
 
 
 o 
 
 IN / 
 
 EACH NET INCOME INTERVAL. / 
 
 
 
 S 
 
 b. 
 
 Scales logarithmic. / 
 
 
 
 o 
 
 
 
 
 -lOOg 
 
 . / 
 
 
 
 
 z 
 
 
 
 
 < 
 
 
 
 
 W 
 
 
 
 O^^" - " - ""^ 
 
 3 
 
 f 
 
 
 
 -50 £ 
 -40 B 
 
 
 
 
 
 -30 a 
 
 
 
 
 ■ 
 
 / ^^* 
 
 
 
 -20 j 
 
 s ^^ 
 
 
 
 < 
 
 
 
 
 m 
 
 
 
 
 a 
 
 
 
 
 z 
 
 
 
 
 -10 n 
 
 ■ 
 
 
 
 
 9 
 
 
 
 
 ■< 
 
 
 
 
 » 
 
 
 
 
 -5 S 
 
 
 
 
 O 
 
 
 
 
 -4 H 
 
 
 
 
 u. 
 
 
 
 
 -3 N 
 
 
 
 
 8 
 
 
 
 
 " 2 2 / 
 
 
 
 
 4 
 
 TOTAL INCOME IN THOUSANDS OF DOLLARS 
 
 
 
 2 3 4 5 10 20 30 40 50 100 200 800 400 500 
 
 1,000 
 
 2,000 3,000 4,000 
 
 / 
 
AMERICAN INCOME TAX RETURNS 
 
 409 
 
 
 
 / CHART 30E 
 
 
 U.5. INCOME TAX RETURNS / 
 
 
 
 I9IS / 
 
 
 
 AVERAGE INCOME / 
 
 
 
 FROM S 
 
 
 
 WAGES AMD SALARIES / 
 
 
 B 
 
 AND / 
 
 
 AVERAGE TOTAL INCOME / 
 
 
 < 
 
 IN / 
 
 
 g 
 
 EACH NET INCOME INTERVAL / 
 
 
 o 
 
 
 
 a 
 
 Scales Logarithmic / 
 
 
 O 
 
 
 
 100m 
 
 B 
 
 
 • 
 
 Z 
 
 
 
 «5 
 
 
 
 ■ 
 
 
 
 
 3 
 
 
 
 -50 | 
 
 -40 H 
 
 / '* 
 
 6 
 
 """—- 
 
 •30 J 
 
 
 e 
 
 2 
 
 
 
 -203 
 
 
 
 < 
 
 
 
 m 
 
 
 
 
 
 
 
 -10 < 
 
 
 « 
 
 B 
 
 
 
 a 
 
 
 
 -< 
 
 
 
 . is 
 
 
 
 " 5 s 
 
 
 
 -4 5 
 
 
 
 * « 
 
 
 
 E 
 
 
 
 •3 w 
 
 
 
 ■ 
 
 
 
 - 2 8/ 
 
 
 
 2/ 
 
 
 
 
 
 ■ 
 
 / 
 
 TOTAL INCOME IN THOUSANDS OF DOLLARS 
 
 
 % 
 
 3 4 5 10 20 30 40 50 100 200 300 400 500 
 
 1,000 2,000 3,0004,000 
 
410 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 CHART 30F 
 
 3 
 h 
 -100© 
 
 ■50 o 
 -40g 
 -30 
 
 U. 3. INCOME TAX RETURNS 
 1919 
 AVERAGE INCOME 
 
 FROM 
 
 WA6E5 AMD SALARIES 
 AND 
 
 AVERAGE TOTAL INCOME 
 
 EACH NET INCOME INTERVAL 
 Scales Logarithmic 
 
 TOTAL INCOME IN THOUSANDS OF DOLLARS 
 20 30 40 50 100 200 300 400 500 
 
 1.000 2.000 3.000 4,000 
 
AMERICAN INCOME TAX RETURNS 
 
 411 
 
 possible" area. The equations of the two (1918) curves on a double log 
 scale are (I) y + 3.92945 — 2.744 x + -22 x 2 =0 (parabola) 
 
 (II) y 2 — 3.981909 y — .867246 xy + 3.981909 x — .132754 x 2 
 — .060262 = (hyperbola) 
 As it is difficult to estimate accurately by eye the goodness of fit of a curve 
 to data when charted on a log scale, Table 30E is introduced: 
 
 TABLE 30E 
 
 WAGES AND INCOME IN THE 1918 INCOME TAX RETURNS 
 
 
 
 Average income from 
 
 
 
 
 wages and salaries 
 
 Percentages 
 
 Net income 
 
 Average 
 total income 
 
 
 
 that data are of 
 
 intervals (1918) 
 
 Data 
 
 Mathematical 
 
 mathematical 
 curves 
 
 
 
 
 curves 
 
 
 $ 1,000-$ 2,000... 
 
 $ 1,566 
 
 $ 1,169 
 
 $ 1,178 
 
 99.2 
 
 2,000- 3,000... 
 
 2,583 
 
 1,690 
 
 1,652 
 
 102.3 
 
 3,000- 4,000... 
 
 3,710 
 
 1,897 
 
 1,955 
 
 97.0 
 
 4,000- 5,000... 
 
 4,866 
 
 2,181 
 
 2,117 
 
 103.0 
 
 5,000- 6,000... 
 
 6,388 
 
 2,192 
 
 2,216 
 
 98.9 
 
 6,000- 7,000... 
 
 7,620 
 
 2,537 
 
 2,555 
 
 99.3 
 
 7,000- 8,000... 
 
 8,952 
 
 2,963 
 
 3,012 
 
 98.4 
 
 8,000- 9,000 . . . 
 
 10,148 
 
 3,341 
 
 3,407 
 
 98.1 
 
 9,000- 10,000... 
 
 11,214 
 
 3,747 
 
 3,760 
 
 99.7 
 
 10,000- 11,000... 
 
 12,207 
 
 4,171 
 
 4,078 
 
 102.3 
 
 11,000- 12,000... 
 
 13,707 
 
 4,555 
 
 4,542 
 
 100.3 
 
 12,000- 13,000 . . . 
 
 14,263 
 
 4,806 
 
 4,709 
 
 102.1 
 
 13,000- 14,000... 
 
 15,922 
 
 5,529 
 
 5,204 
 
 106.2 
 
 14,000- 15,000... 
 
 16,778 
 
 5,801 
 
 5,455 
 
 106.3 
 
 15.000- 20,000... 
 
 20,167 
 
 6,375 
 
 6,400 
 
 99.6 
 
 20,000- 25,000 . . . 
 
 25,859 
 
 7,891 
 
 7,860 
 
 100.4 
 
 25,000- 30,000 . . . 
 
 31,704 
 
 9,196- 
 
 9,211 
 
 99.8 
 
 30,000- 40,000 . . . 
 
 39,644 
 
 10,711 
 
 10,872 
 
 98.5 
 
 40,000- 50,000... 
 
 52,319 
 
 12,639 
 
 13.192 
 
 95.8 
 
 50,000- 60,000 . . . 
 
 64,327 
 
 14,963 
 
 15;066 
 
 99.3 
 
 60,000- 70,000... 
 
 74,848 
 
 16,576 
 
 16,539 
 
 100.2 
 
 70,000- 80,000 . . . 
 
 90,437 
 
 18,764 
 
 18,459 
 
 101.7 
 
 80,000- 90,000... 
 
 98,379 
 
 19,273 
 
 19,351 
 
 99.6 
 
 90,000- 100,000 . . . 
 
 111,515 
 
 20,447 
 
 20,682 
 
 98.9 
 
 100,000- 150,000... 
 
 139,520 
 
 22,212 
 
 23,163 
 
 95.9 
 
 150,000- 200,000... 
 
 211,959 
 
 27,758 
 
 27,829 
 
 99.7 
 
 200,000- 250,000... 
 
 259,487 
 
 29,107 
 
 30,068 
 
 96.8 
 
 250,000- 300,000. . . 
 
 317,578 
 
 34,076 
 
 32,226 
 
 105.7 
 
 300,000- 400,000 . . . 
 
 409,756 
 
 44,393 
 
 34,786 
 
 127.6 
 
 400,000- 500,000... 
 
 514,882 
 
 38,967 
 
 36,847 
 
 105.8 
 
 500,000- 750,000... 
 
 765,905 
 
 27,582 
 
 39,765 
 
 69.4 
 
 750,000-1,000,000. .. 
 
 1,013,846 
 
 61,183 
 
 41,229 
 
 148.4 
 
 1,000,000-1,500,000... 
 
 1,426,182 
 
 89,710 
 
 42,199 
 
 212.6 
 
 1,500,000-2,000,000. . . 
 
 2,084,715 
 
 37,118 
 
 42,199 
 
 88.0 
 
 2,000,000-3,000,000. .. 
 
 3,263,673 
 
 50,178 
 
 40,729 
 
 123.2 
 
 3,000,000-4,000,000. . . 
 
 4,515,732 
 
 11,013 
 
 38,753 
 
 28.4 
 
 The data of table 30E move rather erratically in the intervals above 
 ,000 per annum income. This is natural in view of the small number 
 
412 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 of cases in these upper intervals. There were only 627 returns reporting 
 net incomes of over $300,000 per annum; this is less than one seventieth 
 of one per cent, of the total number of returns. In the 28 intervals under 
 $300,000 per annum 14 of the percentages show the data within one and 
 one half per cent, of the mathematical values. 
 
 These mathematical curves have not been introduced as being in any 
 sense the "law" of the data but merely to emphasize how smoothly the 
 data curves run and yet how unmistakable a sensation they give us of two 
 parts, one above about $6,700 total income and one below that figure. 1 
 It would, of course, be quite impossible to get any sort of approximation 
 to the lower range data by producing the parabola fitted to the upper 
 income ranges. How impossible may be seen from Table 30EE. 
 
 TABLE 30EE 
 
 WAGES AND INCOME IN THE 1918 INCOME TAX RETURNS 
 
 
 Average 
 
 total 
 
 income 
 
 Average income from wages 
 and salaries 
 
 Percentages that 
 data are of 
 
 intervals (1918) 
 
 Data 
 
 Hyper- 
 bola 
 
 Para- 
 bola 
 
 Hyper- 
 bola 
 
 Para- 
 bola 
 
 $4,000-$5,000 
 3,000- 4,000 
 2,000- 3,000 
 1,000- 2,000 
 
 $4,866 
 3,710 
 2,583 
 1,566 
 
 $2,181 
 1,897 
 1,690 
 1,169 
 
 $2,117 
 1,955 
 1,652 
 1,178 
 
 $1,574 
 
 1,152 
 
 745 
 
 391 
 
 103.0 
 97.0 
 
 102.3 
 99.2 
 
 138.6 
 164.7 
 226.8 
 299.0 
 
 The 1919 data show the same two-curve appearance as the 1918 data. 
 This may be clearly seen from chart 30F. 2 The intersection of the two 
 curves would be at about $7,100 instead of $6,700 as on the 1918 chart. 
 Is there any sign of such a change from one curve to another on the 1917 
 data? There seems to be. Chart 30D shows the 1917 data with a parabola 
 fitted to the observations above the first interval. This curve and Table 
 30D give us a strong impression that the first interval cannot be described 
 by any simple curve which describes the remainder of the data. The same 
 two-curve characteristics as the 1918 and 1919 data are strongly suggested. 
 
 The equation of the 1917 parabola on a double log scale is y + 1.8417 — 
 1.8346 x + .124 x 2 = 0. The poorness of the fit to the first interval and 
 the comparative goodness of the fit to the remainder of the data as high as 
 $250,000 per annum may be seen from Table 30D. If the data were 
 numerous enough to permit us fitting two curves they would probably 
 intersect at about $4,500. 
 
 1 An alteration in the size of the intervals in which the data are quoted by the Income Tax 
 Bureau would of course change the data curve to some extent. However, taking the intervals 
 as they come and fitting the curves to them we get the unmistakable impression of great regu- 
 larity. It seemed scarcely worth while to fit the curves to areas rather than points. 
 
 s The story told by Chart 30F is so plain it seemed hardly necessary to fit another set of 
 curves. 
 
AMERICAN INCOME TAX RETURNS 
 
 413 
 
 TABLE 30D 
 
 WAGES AND INCOME IN THE 1917 INCOME TAX RETURNS 
 
 
 Average 
 total income 
 
 Average income from 
 wages and salaries 
 
 Percentages 
 that data are of 
 
 intervals (1917) 
 
 Data 
 
 Mathematical 
 curve 
 
 mathematical 
 curve 
 
 $ 2,000-$ 4,000... 
 
 4,000- 5,000... 
 
 5,000- 10,000... 
 
 10,000- 20,000... 
 
 20,000- 40,000... 
 
 40,000- 60,000... 
 
 60.000- 80,000... 
 
 80,000- 100,000... 
 
 100,000- 150,000... 
 
 150,000- 200,000. .. 
 
 200.000- 250,000. .. 
 
 250,000- 300,000 . . . 
 
 300,000- 500,000... 
 
 500,000-1,000,000... 
 
 1,000,000-1,500,000... 
 
 1,500,000-2,000,000... 
 
 2,000,000 and over 
 
 $ 3,059 
 
 4,818 
 
 7,210 
 
 14,623 
 
 29,236 
 
 51,940 
 
 72,811 
 
 93,742 
 
 126,979 
 
 181,156 
 
 233,880 
 
 293,905 
 
 398,517 
 
 740,769 
 
 1,294,619 
 
 1,812,388 
 
 4,551,718 
 
 $1,280 
 
 1,619 
 
 2,442 
 
 4,517 
 
 7,368 
 
 11,024 
 
 13,516 
 
 16,510 
 
 19,108 
 
 21,758 
 
 27,501 
 
 25,587 
 
 38,204 
 
 43,558 
 
 33,973 
 
 64,201 
 
 99,132 
 
 $1,101 
 
 1,688 
 
 2,422 
 
 4,374 
 
 7,411 
 
 11,038 
 
 13,699 
 
 15,992 
 
 19,081 
 
 23,147 
 
 26,388 
 
 29,478 
 
 33,877 
 
 43,632 
 
 52,845 
 
 58,358 
 
 71,945 
 
 116.3 
 
 95.9 
 
 100.8 
 
 103.3 
 
 99.4 
 
 99.9 
 
 98.7 
 
 103.2 
 
 100.1 
 
 94.0 
 
 104.2 
 
 86.8 
 
 112.8 
 
 99.8 
 
 64.3 
 
 110.0 
 
 137.8 
 
 Both the regularity of the data curves and the positions of the inter- 
 sections of the mathematical curves ! might suggest that heterogeneity 
 of the wages and salaries data was the primary cause of the irregularity 
 in the total income curve. The position of the points of intersection of the 
 mathematical curves might seem inconsistent with a sudden change in 
 accuracy of reporting at exactly $5,000. 
 
 However this argument does not appear so conclusive when we examine 
 the actual amount of wages in each income interval. The constitution of 
 the reported income each year may be seen rather plainly in Charts 28T, 
 28U, 28V, 28W, 28X, 28Y, 28Z, and 28AA. 2 These charts show the number 
 of dollars per dollar income interval reported in each income interval by 
 sources for the years 1916 to 1919. 3 They not only illustrate the fact that 
 the constitution of the income curve changes radically as we move from 
 small to large incomes but also picture the salient characteristics of these 
 changes; each source curve, being charted on a double log scale, may be 
 
 1 Particularly the 1919 intersection which is above the $5,000 to $6,000 net income interval. 
 
 2 See pages 385 to 392. 
 
 3 The five lines representing wages, business, rents, interest, and dividends were found to 
 interweave to such an extent when drawn on one chart that two charts were drawn for each 
 year, one representing wages and business and the other incomes from property. 
 
 Wages includes "salaries, wages and commissions" and in 1916 and 1917 "professions and 
 vocations." 
 
 Business includes "business," "partnerships, personal service corporations, estates, and 
 trusts," and "profits from sales of real estate, stocks, bonds, etc.," and in 1918 and 1919 
 "professions." 
 
 Rents includes royalties. 
 
 Interest includes unclassified investment income. 
 
414 PERSONAL DISTRIBUTION OF INflgME IN U. S. 
 
 seen at a glance in its entirety. We see from Charts 28X and 28Z that, 
 though the ratio of the income from wages and salaries to total income 
 may, when charted, show an angle above $5,000, the entire "bulge" on 
 the wages and salaries curve itself occurs in the under-$5,000 intervals 
 both in 1918 and 1919. Moreover, while "wages and salaries" is the larg- 
 est item in these lowest income intervals, and hence is the controlling factor 
 in determining the peculiar shape of the total curve in this region, it is not 
 the only item showing irregularities and "bulges." Some of these move- 
 ments are extremely difficult to explain. Why should a "bulge" appear 
 on the lower income ranges of the "rent" curve in 1918 and by 1919 be- 
 come pronounced? l The appearance of a bulge on the wage curves in 
 1918 and 1919 seems quite explicable on the basis of heterogeneity within 
 the wage and salary data themselves but one feels a shade less confidence 
 in any explanation of why that curve moved in this peculiar manner if the 
 explanation does not seem also clearly applicable to the rents curve which 
 moved in an apparently similar manner. 
 
 1 A mere increase in rents will not, of course, account for this unevenness in their distribu- 
 tion. 
 
CHAPTER 31 
 
 INCOME DISTRIBUTIONS FROM OTHER SOURCES THAN 
 INCOME TAX RETURNS 
 
 Concerning the frequency distribution of incomes over $3,000 or $4,000 
 per annum we have almost no information aside from the income tax 
 returns. Existing wage distributions and non-tax income distributions 
 almost never reach higher than $2,500 or $3,000 per annum. 
 
 Even in the lower income ranges (under say $2,500 or $3,000) most of 
 the existing non-tax income distributions are of little use in our problem. 
 In the first place there are less than half a dozen distributions of this sort 
 which are not such small samples as to prevent us feeling much confidence 
 in their representative nature. 1 An even more serious defect of every such 
 distribution known to us, with one exception 2 is that the purpose for which 
 the data have been collected almost inevitably makes them extremely 
 ill-adapted to our use. For example, one of the largest recent samples is 
 prefaced by almost a page of introduction explaining what types of re- 
 cipients were purposely excluded. 3 This is rather typical. To base upon 
 such distributions any wide generalizations with respect to the income 
 curve for the country as a whole or even for the localities from which such 
 data were collected would be unwarranted. 
 
 Furthermore, almost without exception these studies in income distri- 
 bution are on a family basis. While it is sometimes possible to make a 
 
 1 For example, Chapin's well-known investigation into the distribution of incomes includes 
 only 391 workingmen's families, and the best distribution of farmers' incomes includes only 
 401 farmers from a single state. 
 
 2 Arthur T. Emery's distribution of income among 1960 Chicago households. 
 
 s "In studying the sources of income and the importance of each source with relation to 
 the total income of a family the following limitations to the type of family schedules should 
 be kept in mind. No families were scheduled in which there were children who lived as 
 boarders, that is, paid a certain sum per week or per month for board and spent the remainder 
 of their earnings or salary as they saw fit. No families were scheduled which kept any board- 
 ers. The number of lodgers to be kept by a family was limited to three at any one time. No 
 families were scheduled in which the total earnings of the family did not equal 75 per cent, or 
 more of the total income. It will be seen that these limitations excluded a large number of 
 families and this materially affects the percentage of families having earnings from children 
 and income from lodgers, and also results in showing a larger percentage of the total income 
 as coming from the earnings of the husband than would be the case if the type of families 
 named had not been excluded from the study. It also reduces the actual amount per family 
 earned by children and received from boarders or lodgers that would be shown in case a cross 
 section of a community including all the types mentioned were used. The object in making 
 the exclusions named was to secure families dependent for support, as largely as possible, 
 upon the earnings of the husband. Of course, it was impracticable to secure a sufficient 
 number of families in which the only source of income was the earnings of the husband, but 
 in following the course named the percentage of families having an income from other sources 
 has been very largely reduced." "Cost of Living in the United States — Family Incomes," 
 Monthly Labor Review, Dec, 1919, p. 30. 
 
 415 
 
416 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 rough estimate of the individual incomes from the family data, such es- 
 timates are not what are needed for our purposes. They can show nothing 
 but the distribution of income among the individuals constituting these 
 families and these families are almost inevitably so chosen as to make the 
 individuals composing them not representative of income recipients at 
 large. Analysis of the distribution of earnings among the individual mem- 
 bers of such families discloses an heterogeneity so extreme as to result in a 
 pronouncedly duomodal distribution curve. The fathers' incomes have one 
 mode while the children's incomes have another. Chart 31A showing a 
 natural scale frequency distribution of earnings among 2811 individuals 
 in 2170 families in 1918 * exhibits this duomodal appearance in a striking 
 manner. The " families" had been so chosen as to exclude both young 
 
 frequency distribution 
 
 OF 
 ANNUAL EARNIN6S OF Z8II INDIVIDUALS 
 IN 
 2170 FAMILIES IN THE U.S. IN 1618. 
 
 Source. Burehu or I anon Sthtisticz 
 Scmxs Mrrt/K/f*. 
 
 ANNUAL EARNINGS 
 
 1,000 1,200 1,400 1,600 
 
 I I I I 
 
 married couples having no children and unmarried but independent wage 
 earners. Investigations planned to bring out the economic character- 
 istics of such "typical families," while they may be extremely valuable 
 for the purposes for which they were undertaken, are necessarily of but 
 little use in the construction of a frequency distribution of all individual 
 incomes in the community. Moreover, even if we were attempting to 
 construct a family and not an individual distribution these data would not 
 generally be particularly helpful for, in addition to the exclusions just 
 mentioned, further narrow and rigid restrictions are usually, and for the 
 purposes in view quite properly, imposed upon the definition of the " typical 
 family." 
 
 1 This is a sample from the 12,096 white families referred to in note 3, page 415 The 
 detailed figures of this sample were tabulated for us by the Bureau of Labor Statistics. 
 They cover 15 cities chosen as representative of the whole list. Each one of the 15 cities 
 shows the duomodal appearance referred to in the text. 
 
DATA FROM OTHER SOURCES THAN TAX RETURNS 417 
 
 As incidentally remarked above, there is one non-tax income frequency 
 distribution to which many of the above criticisms do not apply. It is 
 the distribution of income among 1960 Chicago "households" in 1918 from 
 an investigation made by Mr. Arthur T. Emery for the Chicago Daily 
 News. 1 Instead of attempting to describe a "typical family" Mr. Emery 
 attempted to discover the "household" income of each person whose name 
 came at the top of a page in the Chicago city directory. Mr. Emery en- 
 countered many difficulties in attempting to follow out this scheme and 
 has himself pointed out sources of error. 2 Notwithstanding the inevitable 
 difficulties, Mr. Emery seems to have made a real effort to obtain a scien- 
 tific sample. While his distribution shows unmistakable irregularities, 
 it is in many respects for our purposes the most interesting and suggestive 
 recent non-tax income distribution available. 
 
 Finally, it seems impossible to obtain from these distributions any but 
 extremely general conclusions concerning the relation between income 
 from effort and income from property. The data have almost always 3 
 been so chosen as to eliminate any families obtaining an appreciable frac- 
 tion of their income from property. While they may give us some clues 
 as to the shape of the upper range tail of the wage-earners' income distri- 
 bution curve 4 they can tell us little about even the upper tail of the general 
 income curve and almost nothing about the lower income tail of either the 
 wage-earners' or the general income curve. 
 
 1 While the Bureau is not at liberty to publish this material we were permitted to make 
 what use we could of it in constructing our income curve for the country. 
 
 2 In a letter to the Bureau he writes, "There was, however, one important source of error 
 in this method — the poorer and middle class residents were willing to talk, and with the care- 
 fully trained approach of the investigator, the upper class was also won over, but we found 
 in the wealthy districts that the butler and 'not at home' caused a large amount of travel 
 on the part of the investigator," and often a final failure to obtain any report. 
 
 3 These remarks do not apply to the distribution of income among the 401 farmers or Mr. 
 Emery's distribution. However, the Bureau has no figures, in the case of Mr. Emery's dis- 
 tribution, for income from property. 
 
 * Compare pages 378, 379, 380. 
 
CHAPTER 32 
 WAGE DISTRIBUTIONS 
 
 There is in all an immense amount of American wage data. On the other 
 hand, as an investigator gets into his subject, he begins to realize that the 
 material is more remarkable for its fragmentary nature than for its amount 
 — great as that may be. For no recent year can he obtain wage distribu- 
 tions for more than about 8 per cent, of those gainfully employed. Of 
 course, if these 8 per cent, were scattered over the different types of em- 
 ployment and localities in any truly random fashion, and if their wages 
 were uniformly reported, much might be done with the material. As 
 things are, however, whole occupations as important as agricultural labor 
 and trade are almost unrepresented. Moreover, as we are interested in 
 the amount of wages actually received during the year, it is rather dis- 
 couraging to find that this is the one type of distribution which practically 
 never occurs. Distributions of amounts actually earned in a month are 
 almost as rare. There are a few distributions of amounts actually earned 
 in a week or fortnight, but the great majority of wage distributions are 
 distributions of wage rates — figures by the hour being the commonest — or 
 of hypothetical earnings, generally known as full-time earnings per week. 
 
 Now it is in general impossible to construct a wage distribution for earn- 
 ings from a distribution of rates. Earnings depend, of course, not only on 
 rates but also on hours worked. However, we seldom know anything about 
 the distribution of hours worked and almost never do we know anything 
 about the relation between rates and hours worked. Chart 32 A illustrates 
 how violent may be the difference in shape of the earnings and rates curves 
 for the same individuals. 1 The earnings distribution in this particular 
 case shows not only a much greater scatter than the rates distribution but 
 is of an entirely different shape, as may be seen from Chart 32B where the 
 data are drawn on a double log scale. Chart 32C shows the distribution 
 of hours worked in a week for the same individuals. Now, though the 
 slaughtering and meat packing industry may be an extreme example, 
 what evidence we have suggests that distributions of rates and of earnings 
 are rarely in close agreement. Moreover the relation of the one distribu- 
 tion to the other changes as we pass from industry to industry. 2 
 
 1 43,063 Male Employees in the Slaughtering and Meat Packing Industry in 1917. Bureau 
 of Labor Statistics, Bulletin 262. For purposes of comparison the two distributions are so 
 placed that the frequency curves show the same arithmetic means and areas. 
 
 1 Resulting largely, of course, from the varying types of distributions of hours-worked-in- 
 
 418 
 
WAGE DISTRIBUTIONS 
 
 419 
 
 n 
 
 u — i 
 
 % 
 
 CHART 32 A 
 
 FREQUENCY DISTRIBUTIONS OF RATES OF 
 WAGES PER HOUR AND EARNINGS PER WEEK 
 FOR 43,063 MALE EMPLOYEES IN THE 
 SLAUGHTERING AND MEAT PACKING INDUSTRY 
 IN THE U.S. IN 1917. 
 
 S0UKC£: BuKCRU OF IpBOR STHTItTICS. BOLLMTW ZSZ 
 
 Sc/rjusa ///rre/x/rt. 
 
 \. 1 
 
 
 \ 
 
 . 
 
 -\~2 
 
 CHART 32B 
 
 FREQUENCY DISTRIBUTIONS OF RATES OF WAGES PER HOUR 
 
 AND EARNINGS PER WEEK FOR 43,063 MALE EMPLOYEES 
 
 IN THE SLAUGHTERING AND MEAT PACKING INDUSTRY IN 
 
 THE U.S. IN 1317. 
 
 Scukcg: Burehu of Ojsor Sr/fTtsTics Bulletin zsz 
 
 Scfltfs LoonmrHM/c. 
 
420 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 FREQUENCY DISTRIBUTION 
 
 OF 
 
 HOURS WORKED IN A WEEK 
 
 FOR 
 43,063 MALE EMPLOYEES 
 5,000 IM THE 
 
 SLAUGHTERING AND MEAT PACKING INDUSTRY 
 IN THE 
 U.S. IN I SIT 
 SouRC&Bwtcfio of lhbo# Statistics. 
 
 B(Ji.L£TW,2SZ 
 
 4,000 g SCH/.CS HfnVRHU 
 
 3,000 9 
 
 s 
 
 -2,000 z 
 
 HOURS WORKED IN THE WEEK 
 50 60 
 
 The same difficulty as we find in any attempt to estimate the distribu- 
 tion of earnings per week from the distribution of rates per hour seems 
 inherent in any attempt to estimate the distribution of earnings in a year 
 from the distribution of earnings in a week. The unknown distribution of 
 weeks worked in the year must seriously affect our results. 1 
 
 Estimating the frequency distribution of wages earned in a year for an 
 industry from the frequency distribution of wages earned in another year 
 in the same industry, if we had such data, would involve us in a similar 
 difficulty. Even though we knew the total number of individuals gainfully 
 employed and their total wage bill each year and also the frequency dis- 
 tribution of earnings for one of the years, estimating the frequency dis- 
 tribution for the other year would be hazardous. While some rates dis- 
 tributions for the same industry in the same locality show symptoms of 
 not changing in shape very radically from year to year, 2 this does not seem 
 
 the-week (month or year) in different industries. Illustrations of lack of uniformity in the 
 relation between rates and earnings of the same persons for the same period but in different 
 industries were worked up from Professor Davis R. Dewey's Special Report on Employees 
 and Wages for the 12th Census. 
 
 1 We have no distributions of amounts earned in a week and in a year for the same industry, 
 with which to illustrate this point directly. 
 
 2 For example, the distribution curve for wages per week among Massachusetts factory 
 workers shows a moderate degree of similarity of shape from year to year. 
 
 Professor H. L. Moore (Political Science Quarterly, vol. XXII, pp. 61-73) discussed the 
 fluctuation from 1890 to 1900 in the variability of wage rates in a total made up of thirty 
 
WAGE DISTRIBUTIONS 
 
 421 
 
 a sufficient reason for assuming the same of earnings distributions. The 
 shape of the distribution representing hours or days worked in the year 
 may be expected to change greatly from year to year with alternations of 
 prosperity and depression. 1 
 
 What little evidence we possess suggests that wage distributions 2 for 
 individuals of the same sex in the same industry at the same date, but in 
 different localities, though generally more dissimilar in shape than distri- 
 butions for the same industry in the same place but at different dates, 
 are less unlike one another than distributions for different industries though 
 in the same place and at the same time. The variation in shape of such 
 distributions for different industries is often extreme. 3 
 
 selected manufacturing industries. These distributions (for 1890 and 1900) illustrate both 
 the similarity and the difference in rates distributions between the two years. 
 
 1 For example, what little information we have points to the "scatter" of the days-worked- 
 in-a-year distribution being much greater in a year of depression than in a year of prosperity. 
 
 The extreme variations in shape of the income distributions for the same 1240 individuals 
 in the years 1914 to 1919 as seen in the Statistics of Income, 1919, page 30, are interesting in 
 this connection. 
 
 2 Whether earnings or rates. 
 
 ' Examples of this are numerous. Charts 32D and 32E show the distribution of wages 
 per week among Massachusetts males working in (a) the boot and shoe industry and (b) the 
 paper and wood pulp industry. For purposes of comparison the two distributions are so 
 placed on the natural scale chart that the frequency curves show the same arithmetic means 
 and areas. The double log chart is based directly upon the natural scale chart. It was 
 necessary to break up the "over $35" interval before calculating the arithmetic means. 
 
422 
 
 PERSONAL DISTRIBUTION OF INCOME IN U. S. 
 
 CHART 32 E 
 FREQUENCY DISTRIBUTION OF RATES OF WAGES PER WEEK 
 FOR MALES IN THE BOOT AND SHOE INDUSTRY AND FOR MALES 
 IN THE PAPER AND WOOD PULP INDUSTRY IN MASSACHUSETTS 
 
 IN I9IS. 
 Source.. Mff&s/rcHosrrrs Sr/tvsr/cs or MnNorfiCTo/ZES J9/8. 
 
 5C»1£S J.OOR/?/THM/C 
 
 •j i — . 
 
 In conclusion, the order of importance of the variables as affecting the 
 shape of the distribution curve seems to be — industry, place, time. 
 
 We have but little basis for estimating total income from earnings. In 
 the preceding chapter on Income Distributions from other Sources than 
 Income Tax Returns attention was drawn to the difficulty of arriving at 
 any reliable statement of relationship between earnings and income from 
 such distributions because of the way in which the data were selected. It 
 is even less possible to discover the nature of any such relationship from the 
 income-tax material. Though there is no such apparent "selection" in 
 the income-tax data as in the case of non-tax income distributions, the 
 material is not arranged to answer our particular question. 
 
 The non-statistical reader examining Charts 30D, 30E and 30F, on which 
 are plotted average total income and average income from wages in each 
 income interval, might think that it would be quite simple to estimate the 
 probable average total income of persons having any specified wage. How- 
 ever there is a profound statistical fallacy involved in the use of this ma- 
 terial for any such purpose. As given in the official tables, income is the 
 independent variable, wages the dependent. This condition cannot be 
 reversed without retabulation of the original returns. The statistical 
 student recognizes the problem as one involving the impossibility of de- 
 riving one regression line from the other when neither the nature of the 
 
WAGE DISTRIBUTIONS 423 
 
 equation representing the regression line ! nor the degree of relationship 
 (correlation in the broad, non-linear sense) is known. Even if we knew 
 that the average net income of those persons reporting in 1918 in the $5,000 
 to $6,000 net income class was $5,474 and the average wage obtained by 
 these persons was $2,192, we would be quite unwarranted in concluding 
 that the average income of persons receiving $2,192 per annum wages was 
 $5,474. If no wage earner received income from any other source than 
 wages we still would have a condition where the average income in the 
 income class would be greater than the average wage. Total wages would 
 be necessarily less than total income, because in the income class are in- 
 cluded not only wage earners but capitalists and entrepreneurs. But both 
 total wages and total income are divided by the same number to get an 
 average — namely total number of persons in that income class. 
 
 This suggests a technical criticism of the material contained in the 
 Statistics of Income. All data concerning the relation between two vari- 
 ables are always there published in such a manner as to give information 
 concerning only one of the regression lines and no information whatever 
 concerning the " scatter." If such data were published in the form of " cor- 
 relation tables" the increase in usefulness for statistical analysis would 
 be very great. Such " correlation tables" keep closer to the original data 
 than the usual type of statistical tables. Freer use of them is much to be 
 desired, particularly in cases where it is difficult to anticipate all the prob- 
 lems for whose solutions investigators will go to the tabulated materials. 
 
 1 The difficulty of the problem is, if possible, increased in this particular case because of 
 the fact that the regression is radically non-linear. 
 
CHAPTER 33 
 
 THE CONSTRUCTION OF A FREQUENCY CURVE FOR ALL 
 INCOME RECIPIENTS 
 
 The direct and only adequate method of discovering what is the fre- 
 quency distribution of income in the United States would be to define 
 very carefully the terms income and income recipient and then have a care- 
 fully planned census taken by expert enumerators upon the basis of these 
 definitions. The returns brought in by the enumerators should moreover 
 be sworn to by the persons making them and heavy penalties attached to 
 the making of false or inaccurate returns. A less satisfactory method but 
 one which would probably give excellent results would be to have a large 
 number of truly random samples taken by such a census. The results of 
 either procedure could then be adjusted in the light of other statistical 
 information concerning the National Income and also in the light of theo- 
 retical conclusions derived from the data themselves. 
 
 Constructing an income frequency distribution for all income recipients 
 in the United States from the existing data, a few of whose peculiarities 
 have been noted in the preceding chapters, necessarily involves an ex- 
 tremely large amount of pure guessing. It is only because of the practical 
 value of even the roughest kind of an estimate that any statistician would 
 think of attacking the problem. The method followed in the actual con- 
 struction of the income frequency distribution has been outlined in vol- 
 ume I. 1 This method contains one assumption after another that is open 
 to question. Moreover we feel in many cases quite unable to estimate 
 the probable errors involved in these assumptions. Their only excuse is 
 their necessity. What is the amount of under-reporting for income tax 
 and how is it distributed? What is the effect upon the returns of "legal 
 evasion? " To what extent is the "bulge" on the income-tax returns in the 
 region under about $5,000 in 1918 the result of the "intensive drive?" 
 What is the relation between wages and total income by wage intervals? 
 What is the relation between wage rates and earnings in any particular 
 industry? Etc., etc. These are all questions which must be answered over 
 and over again and yet they are questions the answers to which must be, 
 in many instances, almost pure guesses. And, to repeat, the margin of 
 possible error is often large. 
 
 In view of the sparsity and inadequacy of the data, our first approach 
 to the problem was an attempt to discover, if possible, some general mathe- 
 matical law for the distribution of income. Were we to get any very defin- 
 
 1 Income in the United States, Vol. I, pp. 122-139. 
 
 424 
 
FREQUENCY CURVE FOR INCOME RECIPIENTS 425 
 
 ite and reliable clues as to the mathematical nature of the frequency dis- 
 tribution of income from small sample income distributions and from wages 
 distributions, etc., such clues might of course be invaluable in checking 
 the results obtained from piecing together existing wage distributions, 
 income distributions, and other scattered information. We would be in 
 the position of the astronomer who is able to "adjust" the results of his 
 observations in the light of some known mathematical law. It soon be- 
 came clear, however, that it is quite impossible to discover any essential 
 peculiarities of the income frequency distribution. The available material 
 is not only insufficient for purposes of such generalizations, but moreover 
 the distribution from year to year is so dissimilar, that any generalization 
 of this nature is too vague to be of any practical value. 
 
 The method finally used for the construction of the income curve has 
 therefore, we are sorry to say, practically all the weaknesses of the data 
 from which it has been constructed. The occupations of the country 
 were tabulated and to each occupation was assigned those wage and income 
 distributions which seemed applicable with the least strain. We had then 
 a series of income and wage distributions which nominally covered nearly 
 all the income recipients in the United States, though for some occupations 
 the inadequacy of the wage and income samples was little short of absurd. 
 The wage distributions were converted into income distributions on the 
 assumption that the smaller the wage the larger is its percentage of total 
 income. Beyond this simple assumption the particular functional relation- 
 ships used for many industries were almost pure guess work. Moreover, 
 not only was there the danger of error in moving from wage distribution 
 to income distribution and the danger of error resulting from estimating 
 a wage distribution for a particular industry in a particular locality from 
 a similar though not identical industry in a different locality, but also there 
 was the danger of error resulting from estimating a wage distribution for 
 one year from a wage distribution for another. 
 
 The final results are probably not quite so bad as they might have been 
 had we not had a number of collateral estimates with which roughly to 
 check up and otherwise adjust the first results of our estimates. For ex- 
 ample, such independent information as Mr. King's estimate of the total 
 income of the country and Mr. Knauth's estimate of the total amount of 
 income from dividends were pieces of information with which the results 
 of the frequency curve calculations were made to agree. 
 
 Some hypothetical reasoning is inevitable in such a statistical study as 
 the present one. The investigator must not lose heart. Sir Thomas 
 Browne in his rolling periods sagely remarks that "what song the Syrens 
 sang, or what name Achilles assumed when he hid himself among women, 
 though puzzling questions, are not beyond all conjecture!" 
 
VITA 
 
 Frederick Robertson Macaulay was born in Montreal, Canada, August 12, 1882. He 
 attended McGill University, 1899-1902; Colorado College, 1906-1907; the University 
 of Arizona, 1907-1908; the University of Colorado, 1908-1911. From the University 
 of Colorado he received three degrees, B. A. 1909, M. A. 1910, LL. B. 1911. 
 
 He attended Columbia University for three years, 1912-1915. During that time he 
 studied under Professors Edwin R. A. Seligman, Benjamin M. Anderson, Jr., Robert E. 
 Chaddock, John B. Clark, William A. Dunning, Frank A. Fetter, Franklin H. Giddings, 
 Wesley C. Mitchell, Henry L. Moore, Henry R. Mussey, Karl F. Th. Rathgen, James H. 
 Robinson, Joseph Schum peter, Henry R. Seager, James T. Shotwell, Vladimir G. 
 Simkhovitch. He attended the seminars of Professors Seligman, Seager, and Schum- 
 peter. 
 
 He taught miscellaneous economic subjects for one year (1915-1916) in the University 
 cf Washington, Seattle, Washington. He then taught Economic Theory and Statistics 
 for three years (1916-1917, 1917-1918, and 1919-1920) in the University of California, 
 Berkeley, California. During the year 1918-1919 he was California District Statistician 
 for the Emergency Fleet Corporation. Since May, 1920, he has been on the research 
 staff of the National Bureau of Economic Research, New York City. 
 
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