6. So far, then, as tested by the principle of transfers, the standard deviation, whether absolute or relative, and the mean difference, whether absolute or relative, are good measures; Professor Bowley's quartile measure is a very indifferent measure; the mean deviation, whether absolute or relative, is a bad measure; and Professor Pareto's measure evades judgment. But the scope of the principle of transfers, as a test of measures of inequality, is narrowly limited. It can only be applied to some cases -and by no means to all-in which both the total income and the number of income-receivers are constant, and distribution varies. It cannot be applied when either the total income or the number of income-receivers varies, or when both vary simultaneously. For these more general cases further tests are required, and three general principles suggest themselves as serviceable for this purpose.

7. We have, first, what may be called the principle of proportionate additions to incomes. It is sometimes. suggested that proportionate additions to, or subtractions from, all incomes will leave inequality unaffected. But, if the definition of inequality given above be accepted, this is not so. It appears, rather, that proportionate additions to all incomes diminish inequality, and that proportionate subtractions increase it. This is the

Professor Pigou (Wealth and Welfare, p. 25 n.) uses the followig argument to prove that, in these circumstances, a reduction in the standard deviation will probably increase aggregate satisfaction. "If A be the mean income and a1, a2, deviations from the mean, aggregate satisfaction, on our assumption,

I

I

=nf(A)+(a1+a2...) ƒ' + 1, (ay2+ag2+.....) ƒ”+'-,(a13+aq3+.....) ƒ''+...

2!

3!

But we know that a1+ag+...=0. We know nothing to suggest whether the sum of the terms beyond the third is positive or negative. If, therefore, the third and following terms are small relatively to the second term, it is certain, and, in general, it is probable that aggregate satisfaction is larger, the smaller is (a+a+...). This latter sum, of course, varies in the same sense as the ... standard deviation." This argument would be strong, if all deviations were small, i.e., if inequality were already very small. But when, as is the case in all important modern communities, a number of the deviations is very large, it is quite likely that successive terms in the expansion will go on increasing (numerically) for some time, and this is specially likely as regards the series of alternate terms, which involve deviations raised to even powers. This likelihood will vary according to the form of the function f, but it seems clear that the third and following terms cannot, in general, be neglected. It follows that, in general, there is no certainty and only a somewhat low and problematical degree of probability, that a reduction in the standard deviation will increase satisfaction. There is no reason to suppose tha it is not at least equally probable that a reduction in certain other measures of dispersion will have the same effect. One good test of the relative appropriateness of various measures of the inequality of incomes would be the relative probability that a reduction in such measures would increase economic welfare (or satisfaction), on the assumption that both the total income and the number of income-receivers were constants. But the evaluation of such relative probabilities presents difficulties.

2 See, e.g., Taussig, Principles of Economics, II, p. 485.

principle of proportionate additions to incomes just referred to. A general proof of this principle presents difficulties, and is not attempted here, but the proof in two important special cases is easy. For, first, assume, using the same notation as in Section 3 above, that the relation of income to economic welfare is w=log x. Then, if d be the inequality of any given distribution, we have

=

log xa log Xg

Let all incomes be multiplied by and let ♂ be the inequality of the new distribution.

Then '=

=

x

log 0+log * and, since xx, we have log 0+log xg

S>', if log >0, that is to say, if (>1.

That is to say, proportionate additions to all incomes diminish inequality and proportionate subtractions increase it.1 This is true, if x is the total income of any individual. A fortiori, it is true, if x is surplus income in excess of " bare subsistence." For equal proportionate additions to surplus income involve larger proportionate additions to total income, when the latter is large, than when it is small. A series of transfers from richer to poorer will, therefore, transform proportionate additions. to surplus incomes into proportionate additions to total incomes.

Next assume that the relation of income to economic

Let all incomes be multiplied by and let ♪ be the inequality of the new distribution.

If we write =x/xg, instead of 8=log/log, proportionate additions or subtractions will leave inequality unaffected. It follows that */* is not a mere simplification of the measure log */log * arrived at in section 3 above, but is a distinct, and interior, measure.

and we have, &>d', if (x-xh) (0 − 1)>0.

.. d>S', if 0>I.

S<d', if <1.

That is to say, proportionate additions to all incomes diminish inequality, and proportionate subtractions increase it.

2

8. If the principle of proportionate additions to incomes thus enunciated be provisionally accepted as true generally, and not merely for the particular hypotheses just examined, a second principle follows as a corollary.1 This may be called the principle of equal additions to incomes, and is to the effect that equal additions to all incomes diminish inequality and equal subtractions increase it. Here, again, a direct general proof presents difficulties, though several writers have regarded the principle as so obvious that no proof is required. But as a corollary of the preceding principle the proof is easy. For, let the total additional income involved in proportionate additions to all incomes be redistributed among income-receivers in such a way as to make equal, instead of proportionate, additions to all incomes. Then the addition to maximum economic welfare attainable is the same in both cases. But the addition to economic welfare actually attained is obviously greater when additions to incomes are equal than when they are proportionate. Therefore, inequality is smaller after equal additions have been made than after proportionate additions have been made, the total additional income being the same in both cases. But proportionate additions reduce inequality. Therefore, a fortiori, equal

additions reduce inequality."

The additions must, of course, be genuine. Inequality in this country would not be diminished by reckoning everyone's income in shillings, instead of in pounds. Units of money income in any two cases to be compared must have approximately equal purchasing power. "An equal addition to everyone's income obviously makes incomes more equal than they would otherwise be." Cannan, Elementary Political Economy, p. 137. See also Loria, La Sintesi Economica, p. 369.

Or alternatively, the total additional income being given, a distribution involving equal additions to all incomes may be evolved from a distribution involving proportionate additions to all incomes by means of a series of transfers from richer to poorer.

9. The third principle may be called the principle of proportionate additions to persons, and is to the effect that inequality is unaffected if proportionate additions are made to the number of persons receiving incomes of any given amount. This, again, is easily proved. For the maximum economic welfare attainable and the economic welfare actually attained will both have been increased in the same proportion, and hence their ratio I will be unaltered.

10. We may now test, by means of these three principles, the measures of inequality which have already been tested by means of the principle of transfers. Simple mathematical operations yield the following results :

Here the three absolute measures of dispersion give one set of identical results, and the four relative measures another. None of the seven measures pass the test of proportionate additions to incomes, but the relative measures come nearer to doing so than the absolute measures.1 The relative measures pass the test of equal additions to incomes, but the absolute measures do not. All seven measures pass the test of proportionate additions to persons. We may therefore eliminate the three absolute measures from further consideration. As between the four relative measures, the order of merit established by reference to the principle of transfers may stand, so far, unchanged, viz. :

I. and 2. Relative standard deviation and relative mean difference (bracketed equal)

3. Bowley's quantity measure.

4. Relative mean deviation.

1 It should be noticed that, if we are comparing the inequality of two distributions by means of a measure which is unchanged by proportionate additions to incomes, it is not

II. Can Professor Pareto's measure be brought into this order of merit? This is a relative measure, which is only applicable when distribution is approximately of the

form A=2, where x is any income, y the number of

2

incomes greater than x, and A and a constants for any given distribution, but variables for different distributions.1 Assuming this formula for distribution, which, as Professor Bowley has shown, is the same thing as assuming that the average of all incomes greater than x is proportional to x, Professor Pareto treats a as the measure of inequality, in the sense that, the greater a, the greater inequality. It follows mathematically that "neither an increase in the minimum income nor a diminution in the inequality of incomes can come about, except when the total income increases more rapidly than the population." In other words, increased production per head is both a necessary condition and a sufficient guarantee of a diminution of inequality.

Professor Pareto's law, about which much has been written both by way of criticism and of qualified appreciation, implies a uniformity in distribution, which makes it impossible to apply either the principle of transfers or the principle of equal additions to incomes. Like the four other measures just considered, it is unchanged both by proportionate additions to incomes and by proportionate additions to persons. It has been suggested that this measure, where it is applicable, will be in general accord with other plausible measures of dispersion. But, in view of the investigations of Italian economists,"

necessary that the unit of money income in the two distributions should have approximately the same purchasing power.

1 Compare Pareto, Cours d'Economie Politique, II, pp. 305 ff., and Manuel d'Economie Politique, pp. 391 ff.

Measurement of Social Phenomena, p. 106.

Cours, II, pp. 320-1.

4

See, e.g., Pigou, Wealth and Welfare, pp. 25 and 72.

5 See Bresciani, Giornale degli Economisti, August, 1905, pp. 117-8, and January, 1907, pp. 27-8, Ricci, L'Indice di Variabilità, pp. 43-6, Gini, Variabilita, p. 65 and pp. 70-1. Compare also Persons, Quarterly Journal of Economics, 1908-9, pp. 420-1, and Benini, Principii di Statistica Metologica, p. 187. Professor Benini inverts Professor Pareto's measure, but apparently without realising that he has done so.