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The Law of Income Distribution and Various Statistical Complements

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Vilfredo Pareto: An Intellectual Biography Volume II

Part of the book series: Palgrave Studies in the History of Economic Thought ((PHET))

Abstract

Pareto’s investigations in statistics, during the period being examined in this volume, were not limited to the discovery of what is now the well-known law of income distribution (Sects. 10.2, 10.3) but also extended to an organic and analytically detailed conceptual approach to topics such as the computation of probabilities (Sect. 10.1) and methods of interpolation (Sect. 10.4).

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Notes

  1. 1.

    Pareto to Pantaleoni, 22nd July 1893, see Pareto (1984, p. 387).

  2. 2.

    Pareto (1893–1894, p. 27).

  3. 3.

    Ibid., p. 29. Pareto to Pantaleoni, 3rd October 1891, see Pareto (1984, p. 70), claimed that the Bertrand (1889) “is not a good book” due to its erroneous premises and that his author is “a good mathematician but a very bad philosopher”. Among various other alternative works on the computation of probability Pareto described as “excellent”, ibid., Liagre (1879). His author, the Belgian general and mathematician Joseph Liagre, sets out to study, for didactic purposes, a priori probabilities (where causes which are treated as known are combined in order to reconstruct consequences) and a posteriori probabilities (where consequences which are treated as known are used to reconstruct causes) and enlarges on their applications, particularly in the field of geodesics, pp. 17, 19. Furthermore, Pareto (1892, p. 393) did not feel able to “accept Mill’s statements on the use of computations of probability”, possible alluding to the objectivist claim that “before applying the doctrine of chances to any scientific purpose, the foundation must be laid for an evaluation of the chances, by possessing ourselves of the utmost attainable amount of positive knowledge. The knowledge required is that of the comparative frequency with which the different events in fact occur”, Mill (1974, p. 539).

  4. 4.

    Pareto (1893, p. 25).

  5. 5.

    By cause, Pareto, ibid., p. 26, note 1, meant “simply a conjuncture which, when it occurs, generates a certain probability of a consequent conjuncture”.

  6. 6.

    Ibid.

  7. 7.

    Pareto (1893–1894, p. 28).

  8. 8.

    Ibid.

  9. 9.

    Ibid., pp. 29–32.

  10. 10.

    Ibid., p. 33.

  11. 11.

    Pareto (1893, pp. 7–8).

  12. 12.

    Ammon (1900).

  13. 13.

    See above, §3.3.

  14. 14.

    Pareto (1895a).

  15. 15.

    Ibid., p. 60.

  16. 16.

    Ibid., p. 67.

  17. 17.

    Pareto (1902–1903, p. 9, note 1).

  18. 18.

    Pareto (1896–1897, §961, note 4).

  19. 19.

    Pareto also informed Ammon of his income law in a letter dated 30th November 1900, Banca Popolare di Sondrio Vilfredo Pareto letter archive, saying that it showed that “the distribution of income did not depend only on chance, but implied a certain law of heterogeneity among individuals”.

  20. 20.

    Ammon (1900, p. 180).

  21. 21.

    Ibid., p. 178.

  22. 22.

    Ibid. No account is taken of the final range, from 9600 marks up, because the normalised percentage value would be even smaller than for the two preceding ones, which in turn “are not representable in graphic form due to their minute dimensions”.

  23. 23.

    Ibid., p. 179.

  24. 24.

    Ibid., pp. 178, 330. Ibid., Ammon underlined that for an in-depth investigation of income distribution “one must … be capable of grasping the variations of a mathematical function resulting from the variations of specific scale dimensions”.

  25. 25.

    Ibid., pp. 179–180.

  26. 26.

    Ibid., pp. 180–181.

  27. 27.

    Ibid., pp. 183.

  28. 28.

    Ibid.

  29. 29.

    Ibid., p. 184.

  30. 30.

    Ibid., p. 330.

  31. 31.

    Ibid., p. 332.

  32. 32.

    Ibid.

  33. 33.

    Pareto (1895a, p. 60).

  34. 34.

    Ibid.

  35. 35.

    Pareto (1896a).

  36. 36.

    Benini (1894) presented the table shown in the first two columns of Table 10.3, ibid. p. 866:

    Pareto (1896a, p. 4), reformulated Benini’s table with the addition of the third and fourth columns, without specifying that the reformulation was his work and ignoring small incomes, because “they are easy to disguise”, Pareto (1896–1897, §959).

  37. 37.

    Pareto (1896a, p. 2).

  38. 38.

    Ibid and Pareto (1896–1897, §958, note 1).

  39. 39.

    Pareto (1896b, p. 83).

  40. 40.

    Pareto (1895a, p. 60).

  41. 41.

    Pareto (1896a, p. 5).

  42. 42.

    Pareto (1896b, pp. 84–87).

  43. 43.

    Pareto (1896–1897, §§958–959).

  44. 44.

    Pareto (1896b, p. 83).

  45. 45.

    See, for example, Benini (1906, pp. 185–186) and Niceforo (1935, pp. 475–478).

  46. 46.

    Pareto (1896–1897, §959).

  47. 47.

    Pareto (1895a, pp. 61–62), Pareto (1896–1897, §§959–960). Pareto (1895a, p. 64) admitted in passing that α (in his essay substituted with h) “depends on all the quantities characterising the economic context” and that only as a first approximation could it be considered as a constant, although he notoriously neglected this caveat. This constitutes a quite interesting example of a conflict between the General Equilibrium framework and the search for constants in Pareto’s work during this period.

  48. 48.

    Pareto (1986–1897, §957). Pareto to Gustave de Molinari, 18th November 1896, see Pareto (1975, p. 307) justified the debate over the “chance” hypothesis by reference to the need to present a complete coverage of the topic for didactic purposes.

  49. 49.

    Pareto (1896–1897, §962, note 1).

  50. 50.

    Ibid. §962.

  51. 51.

    Ibid. §1026. By “prime subjects” Pareto was referring to “individuals capable of more intense vital activity”, ibid., §1030. Pareto, ibid., tome 2, pp. 416–419 showed, through a complicated exercise in the calculation of probability, that the law of income distribution could be conceived as a consequence of the hypothesis postulating the varying capabilities of differing social categories to generate income.

  52. 52.

    Pareto to Tullio Martello, 15th December 1895, see Pareto (1975, p. 275).

  53. 53.

    Pareto (1895a, p. 64). Pareto (1896a, pp. 8–15) determines the proportional rate of taxation which would yield the same revenues as any given progressive tax.

  54. 54.

    Leroy-Beaulieu (1881), whose importance appears essentially to be more as a target for attack than as a source of inspiration for Pareto’s law of income distribution.

  55. 55.

    Ibid., p. 515.

  56. 56.

    Pareto (1895b, p. 33).

  57. 57.

    Pareto (1896–1897, §965, note 1).

  58. 58.

    Pareto, ibid., §964 was able to establish the unambiguous nature only of the concept of income equality, as well as the fact that it was attainable in two different ways, that is, the impoverishment of the rich or the betterment of the poor. It was Leroy-Beaulieu’s proposition that the incomes of the poor increased more rapidly than those of the rich which prompted Pareto to adopt the second definition.

  59. 59.

    Where a represents a constant which is very small for total income, is positive for capital and negative for wages, ibid., §959, note 3.

  60. 60.

    Since the logarithms of ux = (h + a/x + a)α, will yield logux = αlog(h + a) − αlog(x+a), an expression whose differentiation relative to a and to α lead to the formula appearing in the text.

  61. 61.

    Since h < x, log(h + a/x + a) is negative whereas [(1/h + a)−(1/x + a)] is positive.

  62. 62.

    Since, on the basis of the formula N = A/(x + a)α, the number of income earners between x and x + dx is given by –dN/dx = (αA)/(x + a)α + 1.

  63. 63.

    Given that the primitive of the integral is –[A(a + x)–α(a + αx)]/(α–1)where –α is an apex.

  64. 64.

    Pareto (1896–1897, §965). Customs protection (and in general all forms of despoliation), by depriving victims of more than is received by beneficiaries, have the consequence of increasing income inequality and reducing the lowest incomes, ibid., §1044. Pareto, ibid., §1063 added that at the end of the 1800s the French national income was increasing more rapidly than the population thanks to measures limiting population growth, as also in England, thanks to the substantial increase in national income stemming from economic liberty.

  65. 65.

    Ibid., §1036.

  66. 66.

    Ibid., §965.

  67. 67.

    Ibid., §965.

  68. 68.

    Ibid., §1008.

  69. 69.

    Ibid., §1009.

  70. 70.

    Ibid., §1012.

  71. 71.

    Pareto (1897, p. 367).

  72. 72.

    Ibid., p. 369.

  73. 73.

    Ibid., p. 376.

  74. 74.

    Pareto (1899, pp. 89–92).

  75. 75.

    Ibid., p. 116.

  76. 76.

    Ibid., p. 117.

  77. 77.

    Ibid.

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    Fiorenzo Mornati

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Mornati, F. (2018). The Law of Income Distribution and Various Statistical Complements. In: Vilfredo Pareto: An Intellectual Biography Volume II. Palgrave Studies in the History of Economic Thought. Palgrave Macmillan, Cham. https://doi.org/10.1007/978-3-030-04540-1_10

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