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Front Page Titles (by Subject) Chapter II: PARETO'S LAW - The Economics of Welfare

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## Chapter II: PARETO'S LAW - Arthur Cecil Pigou, The Economics of Welfare [1920]

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The Economics of Welfare (4th ed.) (London: Macmillan, 1932).

Liberty Fund, Inc. is a private, educational foundation established to encourage the study of the ideal of a society of free and responsible individuals.

### PARETO'S LAW

§ 1. THE mere statement of this problem brings us into contact with an interesting thesis, which, if valid, would immediately dispose of it. This thesis is that no cause operating in opposite senses upon the aggregate amount of the dividend and upon the absolute share of the poor can possibly exist. It is backed by an inductive proof. The data for the induction are derived from some remarkable investigations conducted by Pareto and published by him in his Cours d' économie politique. Statistics of income in a number of countries, principally during the nineteenth century, are brought together. It is shown that, if x signify a given income and N the number of persons with incomes exceeding x, and if a curve be drawn, of which the ordinates are logarithms of x and the abscissae logarithms of N, this curve, for all the countries examined, is approximately a straight line, and is, furthermore, inclined to the vertical axis at an angle, which, in no country, differs by more than three or four degrees from 56°. This means (since tan56° = 1.5) that, if the number of incomes greater than x is equal to N, the number greater than mx is equal to (1/m1.5)·N, whatever the value of m may be. Thus the scheme of income distribution is everywhere the same. "We are confronted, as it were, with a great number of crystals of the same chemical composition. There are large crystals, middle-sized crystals and small crystals, but they are all of the same form."2 These are the facts as found by Pareto. The inference which he appears to draw from them in the Cours d' économie politique contains two parts. He defines diminished inequality among incomes thus: "Incomes can tend towards equality in two quite different ways; that is, either because the larger incomes diminish, or because the smaller incomes increase. Let us give this latter significance to the diminution of inequality among incomes, so that this will take place when the number of the individuals having an income less than an income x diminishes compared with the number of persons having an income greater than x."3 On this basis he finds: First, "we may say generally that the increase of wealth relatively to population will produce either an increase in the minimum income, or a diminution in the inequality of incomes, or both these effects in combination."4 Secondly, "to raise the level of the minimum income or to diminish the inequality of income, it is necessary that wealth should grow more rapidly than population. Hence we see that the problem of improving the condition of the poor is, before everything else, a problem of the production of wealth."5 Now, on Pareto's definition, "to increase the minimum income, or to diminish the inequality of income, or both," is substantially equivalent to "to increase the absolute share of the national dividend accruing to the poor." Hence, what this thesis amounts to in effect is that, on the one hand, anything that increases the national dividend must, in general, increase also the absolute share of the poor, and, on the other hand—and this is the side of it that is relevant here—that it is impossible for the absolute share of the poor to be increased by any cause which does not at the same time increase the national dividend as a whole. Hence disharmony of the type referred to in the preceding chapter is impossible: we cannot be confronted with any proposal the adoption of which would both make the dividend larger and the absolute share of the poor smaller, or vice versa.

§ 2. Now it is quite evident that a sweeping proposition of this kind, based upon an inductive argument, requires very careful consideration. It is, therefore, necessary at the outset to call attention to certain defects in its statistical basis. The sum of what has to be said is that, though the various distributions that are brought under review are similar in form, the likeness among them is by no means complete. In all of them, it is true, the logarithmic income curve—at least for incomes of moderate size—is approximately a straight line; but the inclination of this line, though it does not differ widely, still does differ distinctly, for the different groups of statistics that have been observed. Pareto's lowest figure from adequate data for the tangent of the angle made with the vertical axis is, for instance, 1.24 (Ba&circ;le, 1887), and his highest 1.89 (Prussia, 1852). Nor is this all. As Dr. Bowley has pointed out, in the most important set of figures observed over a long period (those for Prussia) the slope of the curve has been decreasing with the lapse of time. The figures which Dr. Bowley gives differ slightly from those of Pareto, but the general effect is the same in both sets. According to Pareto, however, a smaller slope of the curve means a greater equality in his sense—a sense the appropriateness of which, it will be remembered, is matter for debate—in the distribution of income.6 Dr. Bowley, therefore, naturally offers as an explanation of the Prussian figures: "The incomes are becoming more uniformly distributed in Prussia, and the result is, from these figures, that the Prussian income is getting to the more uniform distribution of the English."7 Hence, interesting as Pareto's comparisons are, to build upon them any precise quantitative law of distribution is plainly unjustifiable.

§ 3. But, if the position is to be fully understood, it is well that this point should be waived. Let us suppose that the statistical basis of Pareto's reasoning is not defective in the way that has been indicated. Even so, much material for criticism remains. For let us consider what exactly this scheme, or form, of distribution is, for the existence of which a mysterious necessity seems to have been discovered. If we were to plot it out, not as Pareto does, but in the simpler form of a curve so drawn that the abscissae represent amounts of income, and the ordinates the number of people in receipt of these amounts, the curve would rise very quickly to its highest point and, thereafter, fall much less quickly. This would express in a picture the well-known fact that there are a very large number of people with incomes much below the average income, and, comparatively, a very small number with incomes above the average income. In short, the essential characteristic of current income distributions is that the great bulk of incomes are massed together near the lower end of the income scale. This fact is significant for the following reason. There is clear evidence that the physical characters of human beings—and considerable evidence that their mental characters—are distributed on an altogether different plan. When, for instance, a curve is plotted out for the heights of any large group of men, the resulting picture will not, as with incomes, have a humped and lop-sided appearance, but it will be a symmetrical curve shaped like a cocked-hat. It will, in short—to use a technical term—be the characteristic Gaussian curve, or curve of error, symmetrical about the mean in such wise that there is no massing near either end, but an equal number of heights above the mean and below it and a lessening number of people at every height as the distance from the mean in either direction is increased. Now, on the face of things, we should expect that, if, as there is reason to think, people's capacities are distributed on a plan of this kind, their incomes will be distributed in the same way. Why is not this expectation realised? A partial answer to this question may, perhaps, be found in a closer analysis of the word "capacities." For our purpose this must mean income-earning capacities. But people earn incomes by means of several different sorts of capacity, among which the principal division is between manual capacity and mental capacity. From the point of view of income-getting, therefore, it cannot properly be assumed that we are dealing with a single homogeneous group. If we examined together the members of a University and the members of a junior school, the table of heights obtained for these two groups jointly would not accord with the normal curve. If the number of persons in the University was much smaller than the number in the junior school group, heights very much above the average of the two groups combined would be unduly numerous. It may be that brain-workers constitute a homogeneous group and handworkers a homogeneous group, but that, for the purpose of income earning, they do not jointly constitute a homogeneous group; that the normal law rules in each separately, but, as with the University and the school, not in both together. On these lines the peculiar form of the income-distribution curve might be partly explained. There is, however, a more important and more certain explanation. Income depends, not on capacity alone, whether manual or mental, but on a combination of capacity and inherited property. Inherited property is not distributed in proportion to capacity, but is concentrated upon a small number of persons. Even apart from the fact, to be referred to in a moment, that the possession of a large property enables the property-owner to improve his capacity by training, this circumstance necessarily deflects the curve of income distribution from the "normal" form. The significance of this, from the standpoint of our present problem, is obvious. If the form of the income-distribution curve is partly determined by the facts of bequest and inheritance, the particular form which is found to be dominant in current conditions cannot possibly be necessary, except upon the assumption that the broad scheme of inheritance now generally in vogue is maintained. An alleged law, then, that should speak of any form as necessary in an absolute sense, runs counter to this apparently irrefutable reasoning.8

§ 4. The statistics adduced by Pareto do not provide a basis for any counter-argument. For, as a matter of logic, it is plain that, if all the different groups to which his statistics refer possess any common characteristic in addition to the fact that they are all in receipt of income, no general inference about income distribution that is based upon them can be extended to groups not possessing these characteristics. But, in fact, all these groups are communities enjoying the general type of inheritance laws common to modern Europe.9 It follows at once that no inference can be drawn as to how the form of the income distribution would be affected if these laws were abolished or fundamentally changed. In his Manuale di economia politica, published some years after the Cours, Pareto himself explicitly recognised this. He wrote: "We cannot assert that the form of the curve would not change if the social constitution were to change radically; if, for example, collectivism were to take the place of the system of private property."10

§ 5. Nor is it necessary to imagine so large a change as the destruction of inheritance laws, in order that the form of the income-curve may be largely affected. There is ground for believing that a like result would come about in consequence of anything that affected, in a marked way, the proportion between "earned" income and income derived from investments. The reason for this opinion is twofold. First, it is found by experience that incomes from property are distributed much more unevenly than incomes from either head-work or hand-work. Mr. Watkins, in his Growth of Large Fortunes, after printing an interesting table, comments on it as follows; "In making the comparisons made possible by this table, the criterion must be relative, not absolute. Convenient relative numbers are the ratio of the upper decile, or the upper centile, to the median. It will be observed that, in the statistics of wages, the upper decile is always somewhat less than twice the median, and, in one occupation of the nine, it is little more than one-fourth greater. In the distribution of salaries the upper decile is approximately twice the median, the inequality thus being not greatly different from that prevailing among wage-incomes. But there is a great gap between this and the prevailing distribution of income from property. In the Massachusetts probate statistics the upper decile is eight or nine times the median, and the error is doubtless in the direction of under-statement, since the figures are not net, so that large deductions for debts should be made from the smaller estates, and also since many very small properties do not pass through the courts. Among French estates the upper decile is thirteen times the median."11 For this country also the available statistics show a much more marked concentration of wealth than of income. This is well illustrated by Professor Clay's comparison between his own estimate for the distribution of capital in the United Kingdom in 1912 and Dr. Bowley's estimate for the distribution of income in 1910. He wrote: "94.5 per cent of persons have 56 per cent of the national income, while 96.2 per cen tof persons have only 17.22 per cent of the national capital; 98.9 per cent of persons have 71 per cent of income, while the same percentage of persons have only 33 per cent of the capital."12 Secondly, the distribution of earned income itself is likely to be more uneven, the greater is the importance of the unevenly distributed income from investments. This result comes about because differences in income from investments make possible different degrees of educational training and afford different opportunities for entering lucrative professions. The correlation between the two sorts of income is illustrated by Benini in a table, in which he divides the figures for certain Italian incomes into two parts: "The one represents the income that people derive from property, supposed to be invested for all the different categories at a uniform rate of, say, 5 per cent; the other represents the strictly personal income, due to work, enjoyed by the same people. For example, a total income of 2000 lire, accompanied by a property of 9016 lire, may be regarded as composed of 451 lire, the fruit of investment, and of 1549 lire, the fruit of professional activity. Calculating in this manner, we obtain the following table:

ECONOMICS OF WELFARE
Income derived
Total IncomeIncome derivedfrom Personal
(lire)from Property.Activity.
1,000 =143 +857
2,000 =451 +1549
4,000 =1,458 +2542
8,000 =4,285 +3715
16,000 =11,665 +4335
20,000 =15,885 +4115
32,000 =28,640 +3360
40,000 =37,500 +2500

It will be noticed, of course, that, so soon as total incomes begin to exceed 16,000 lire, the part derived from personal activity diminishes; but this does not mean that the remuneration of the profession followed diminishes; it only means that many will now live wholly on the income derived from their property without following any gainful profession, and that this conduct of theirs reduces the average of the income due to work for the class to which they belong."13 Moreover, there is yet another way in which the form of the income-curve might be modified. A change in the distribution of training and so forth, that is, of investment of capital in people, may take place apart from variations in income from investments. When this happens, the change must tend directly to alter the distribution of earned income, even though original capacities are distributed in accordance with some (the same) law of error. It is perhaps some change of this kind that accounts for the conclusion, which Professor Moore derives from his study of American wage statistics, that the variability of wages (as between different people at the same time) was less in 1900 than it had been in 1890.

§ 6. When these points are conceded, the general defence of "Pareto's Law" as a law of even limited necessity rapidly crumbles. His statistics warrant no inference as to the effect on distribution of the introduction of any cause that is not already present in approximately equivalent form in at least one of the communities—and they are very limited in range—from which these statistics are drawn. This consideration is really fatal; and Pareto is driven, in effect, to abandon the whole claim which, in the earlier exposition of his formula, he seemed to make. In the Manuale di economia politica he insists that that formula is purely empirical. "Some persons would deduce from it a general law as to the only way in which the inequality of incomes can be diminished. 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."14 This means that, even if the statistical basis of the "law" were much securer than it is, the law would but rarely enable us to assert that any contemplated change must leave the form of income distribution unaltered. As things are, in view of the weakness of its statistical basis, it can never enable us to do this. Disharmony between movements of the national dividend as a whole and of the absolute share accruing to the poor cannot be proved by statistical evidence to be impossible, and a detailed study of the matter must, therefore, be made.

[2.][2] Cours d'économie politique, ii. pp. 306-7.

[3.][3] Manuale di economia politica, p. 371.

[4.][4] Cours d' économie politique, ii. p. 324.

[5.][5] Ibid. p. 408

[6.][6] Cf. ante, p. 96.

[7.][7] Select Committee on the Income Tax, 1906, Evidence, p. 81.

[8.][8] Cf. Benini, Principii di statistica metodologica, p. 810.

[9.][9] Of course it is not suggested that the inheritance laws of all modern European countries are exactly identical. They differ considerably in detail. The French laws, for example, force a more even division of estates among children than the English laws and deny special privileges to the eldest son. It is interesting to connect this fact with the observation of Benini (Principii di statistica metodologica, p. 191), that the distribution of wealth is more even in France than it is here. (Cf. also Ely, Property and Contract, vol. i. p. 89.)

[10.][10] Loc. cit. pp. 370-71.

[11.][11] The Growth of Large Fortunes, p. 18.

[12.][12] Proceedings of the Manchester Statistical Society, 1924-26, pp. 64-5. For a useful summary of the available statistics concerning income and capital distribution, cf. Carr-Saunders and Jones, Social Structure in England and Wales (1927), chapters ix. and x.

[13.][13] Principii di statistica metodologica, pp. 836-7.

[14.][14] Manuale di economia politica, pp. 371-2.