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Multi-Dimensional Utility and the Index Number Problem: Jeremy Bentham, J. S. Mill, and Qualitative Hedonism

  • Tom Warke (a1)

This article develops an unconventional perspective on the utilitarianism of Bentham and Mill in at least four areas. First, it is shown that both authors conceived of utility as irreducibly multi-dimensional, and that Bentham in particular was very much aware of the ambiguity that multi-dimensionality imposes upon optimal choice under the greatest happiness principle. Secondly, I argue that any attribution of intrinsic worth to any form of human behaviour violates the first principles of Bentham's and Mill's utilitarianism, and that this renders both authors immune to the claim by G. E. Moore that they committed a ‘naturalistic fallacy’. Thirdly, in light of these contentions, I find no flaw in Mill's ‘proof of utility’. Fourthly, I use the notion of intrapersonal utility weights to provide an interpretation of Mill's qualitative hedonism that is entirely consistent with his value monism.

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1 Perhaps the most familiar index number in general use is the Retail Price Index (in the USA, the Consumer Price Index), designed to measure the change over time in the cost of a ‘representative bundle’ of goods and services purchased by households for final consumption. The weights on individual prices within the KPI are meant to reflect the significance of the corresponding items in a ‘typical’ family budget. The index number problem shows itself in at least two ways. First, any actual family at any given time is highly unlikely to purchase goods in the exact same proportions as the representative bundle, and to that extent the RPI does not measure precisely changes in that particular family's cost of living. Secondly, in order to measure price change alone, the representative bundle must remain stable, but in fact the items in a ‘typical’ family budget change continuously over time, in part induced by the very changes in relative prices that the RPI is meant to average out.

2 Modern economic theory begins with an axiomatic presumption that agents can formulate a complete and transitive preference set over all conceivable combinations of ‘goods’ (and bads), thus finessing the index number problem at the individual level and degrading utility to a one-dimensional indicator of the resultant rank order. As one contemporary text puts it: ‘A world in which preferences need be neither transitive nor complete’ is ‘terrain alien to most of economics’ because ‘without transitivity or completeness, utility functions do not exist and indifference contours lose their meaning’ (Ellickson B., Competitive Equilibrium, New York, 1993, pp. 314 f.). The process by which Bentham's utility concept was stripped of its multi-dimensionality within economics is described in Warke T., ‘Mathematical Fitness in the Evolution of the Utility Concept from Bentham to Jevons to Marshall’, Journal of the History of Economic Thought, xxii (2000).

3 This particular expression for egalitarian interpersonal weights was attributed to Bentham by Mill (Utilitarianism, in Collected Works of John Stuart Mill, 33 vols., ed. Robson John M., Toronto, 1969, x. 257). I am informed by Fred Rosen of the Bentham Project that his colleague Philip Schofleld has recently located the original source, after many years of looking, in vol. 4, p. 475 of the original 1827 edition of Bentham's Rationale of Judicial Evidence (edited by Mill as a teenager). The only difference from Mill's rendition is the word ‘tell’ in place of ‘count’. I retain the more familiar version in the text to follow.

4 See Broome J., ‘Utility’, Economics and Philosophy, vii (1991).

5 The assumption that most uncertainties take the form of a probability distribution is prevalent in contemporary economic theory. It is often assumed further, both in game theory and in certain schools of macroeconomics, that agents come to know these distributions through experience. Unlike devotees of ‘rational expectations’, I make no claim for the realism of either assumption. As stated in the text, they have been adopted here solely to facilitate concentration upon a set of problems that would be associated with optimal choice even under perfect foresight.

6 Bentham J., An Introduction to the Principles of Morals and Legislation, ed. Burns J. H. and Hart H. L. A., Oxford, 1996 (hereafter IPML), pp. 39 f.

7 Given his meticulous use of language, Bentham's adoption of the term ‘species’ for kinds of pleasure (or pain) is significant. As will be shown in section II below, the genus ‘pleasure’ included, so far as Bentham was concerned, anything perceived by an individual as ‘good’. In chapter 5 of IPML he classifies all such perceptions into mutually exclusive categories, defined by one or more characteristics unique to the class. For example, pleasures of piety are those which accompany a person's belief of being in the favour of the Supreme Being. In similar fashion, a botanist classifies an apple species by the characteristics that distinguish it from other members of its genus. Distinct biological species lack a natural common denominator in a reproductive sense (one plus one equals zero) and in the sense of unambiguous aggregation (hence the admonition ‘That's like adding apples and oranges!’).

8 Bentham J., Deontology, together with A Table of the Springs of Action and Article on Utilitarianism, ed. Goldworth A., Oxford, 1983 (CW).

9 IPML, p. 42.

10 Ibid., pp. 49 f.

11 Bentham J., ‘Legislator of the World’: Writings on Codification, Law, and Education, CW, ed. Schofleld P. and Harris J., Oxford, 1998, 251.

12 Ibid., p. 254.

13 Bentham's JeremyEconomic Writings, ed. Stark W., 3 vols, London, 19521954, i, pp. 117 f.

14 Stark, i, p. 113.

15 UC xcvi. 117.

16 For purposes of illustration, let us assume that a victim's total wealth is sufficiently great that its marginal utility is constant over a £40 range. It follows that the privation pain of losing £40 to a non-threatening thief is twice that of losing £20, and we may denote the latter pain as x. Whether 2x (theft of £40) is greater or less than x + y (being robbed of £20) then depends entirely on whether x (privation of £20) is greater or less than y (being threatened with a gun) – ‘a matter of conjecture and opinion’.

17 As Lord Robbins phrased it: ‘There is much talk in the Benthamite literature of a felicific calculus, and the term naturally suggests a most pretentious apparatus of measurement and computation. But, in fact, this is all shop window.… [Benthamite] use of the felicific calculus lay in quite another direction – in rough judgements of the expediency of particular items of the penal law, in general estimates of the suitability of existing institutions or the desirability of other institutions to take their place’ (The Theory of Economic Policy, 2nd edn, London, 1978, p. 181). In his recently published book, Roger Crisp presents as ‘Bentham's account’ a precise quantitative utility calculation (Mill on Utilitarianism, London, 1997, pp. 22 f.), having asserted: ‘Bentham assumes that any particular pleasure or pain has a determinate value…’ (ibid., p. 21). As I have shown, Bentham specifically acknowledged that particular pleasures or pains do not have determinate values, and so far as I am aware, there is no example of what Crisp terms ‘Bentham's account’ in any of Bentham's actual output. Nonetheless, Crisp's account demonstrates correctly that detailed utilitarian decision-making implies precise relative values on pleasures and pains. The point I am making is that Bentham avoided pronouncements on fine-grained choices in the public domain, largely, I would argue, because he realized that impartial utilitarians can legitimately disagree in their fine-grained estimates of the representative pleasure/pain values that such decisions require.

18 Many commentators have pointed out the unrealistic burden imposed on decision-makers by a rigorous application of the utilitarian standard, and the consequent necessity for utilitarians to retreat to derived rules or secondary principles on most occasions. These routine choices rely upon implicit rather than explicit weights on pleasures and pains, and it is only when differing implicit weights generate conflicting designations of right and wrong action that a return to first principles is required (see Hare R. M., ‘Ethical Theory and Utilitarianism’, Utilitarianism and Beyond, ed. Sen A. and Williams B., Cambridge, 1982). In this paper, I confine my main interest to the utilitarian first principles.

19 IPML, p. 67.

20 Ibid, p. 62.

21 The issue of individual sacrifice for the sake of maximum aggregate happiness is a familiar topic in utilitarian studies, discussed recently and cogently by Rosen and his commentators (Rosen F., ‘Individual Sacrifice and the Greatest Happiness: Bentham on Utility and Rights’, Utilitas, X (1998)). Although I share Rosen's view that Bentham rejected moral rights as a check on sacrifice because he regarded their delineation as necessarily subjective, hence falling under the principle of sympathy and antipathy, I disagree with Rosen that Bentham's ‘principle of utility meant the provision of an equal quantity of happiness to all concerned’ (ibid., 140, emphasis added). I do not doubt that evidence can be adduced in support of Rosen's assertion that ‘Bentham's commitment to maximization was mainly a commitment to a distributive rather than an aggregative principle’ (ibid.). The problem, as stated in the text, is that these two principles in their pure forms are mutually exclusive. Thus I argue that the most accurate consistent representation of Bentham's lifelong position is a commitment to aggregate maximization constrained by an egalitarian distributive rule. In any case, IPML lacks altogether any distributive rule and, given chapter IVs explicit instructions on aggregation, its omission by Bentham was a betrayal of his own egalitarian instincts. If egalitarian interpersonal weights are imposed as a constraint on aggregate maximization, this constrained principle of utility provides a considerable (though not absolute) check on any state-imposed individual sacrifices. In fact, if a utilitarian legislature were persuaded to allocate sufficiently large weights to particular species of pleasure, such as being confident about personal security and imagining the happiness of future generations, it might well adopt a remarkably strong (if less-than-absolutist) Bill of Rights.

22 For example, J. C. Harsanyi refers to ‘the equiprobability model of moral value judgements,’ according to which ‘a rational individual… will be a utilitarian, who defines social utility as the mean of individual utilities’ (‘Morality and the Theory of Rational Behaviour’, Sen and Williams, pp. 44 f.). As noted in the text, a mean value of individual utilities is equivalent to equal interpersonal weights (it does not matter for this point that, as in all economic models, Harsanyi's individual utilities are one-dimensional). R. M. Hare takes the same position: ‘I try to base myself… entirely on the formal properties of the moral concepts as revealed by the logical study of moral language; and in particular on the features of prescriptivity and universalisability which I think moral judgements… all have.… In this position, I am prescribing universally for all situations just like the one I am considering, and thus for all such situations, whatever role, among those in the situations, I might myself occupy. I shall therefore give equal weight to the equal interest of the occupants of all the roles in the situation; and, since any of these occupants might be myself, this weight will be positive’ (‘Ethical Theory and Utilitarianism’, Sen and Williams, p. 29).

23 In Halévy E., La Jeunesse de Bentham, Paris, 1901, p. 427, trans. Mack M. P., Jeremy Bentham: An Odyssey of Ideas, London, 1962, p. 449.

24 IPML, p. 100n.

25 Ibid., pp. 25 f.

26 Ibid., p. 17.

27 Ibid., p. 100.

28 Moore G. E., Principia Ethica, Cambridge, 1903, p. 2.

29 Ibid., pp. 16 f.

30 Ibid., p. ix.

31 Ibid., p. 41.

32 Ibid., p. 10.

33 There is no indication within Principia Ethica that Moore had ever read a word of Bentham directly. His account of Bentham's principle of utility is taken entirely from Sidgwick. The only citation by Moore of Bentham is the pushpin-poetry maxim, and he almost certainly got that from reading Mill.

34 Moore, pp. 17–19.

35 This point is emphasized in Goldworth A., ‘Bentham's Concept of Pleasure: Its Relation to Fictitious Terms’, Ethics, lxxxii (1972). Goldworth interprets Bentham's ‘good’ as supervenient on ‘pleasure’: ‘Although supervenience constitutes a dependence between the things named by fictitious terms and those named by real terms, it does not entail the reducibility of fictitious terms to real terms’ (342).

36 Moore, p. 180.

37 Although there are contrary interpretations of the relationship between Bentham's psychological hedonism and his principle of utility, I take the same line as H. L. A. Hart: ‘[Bentham's] view is, I think, that reasonable men can accept the principle of utility as having the social function of controlling behaviour even if there will be some occasions when it will be impossible for them to comply with it voluntarily because it is not in their interest on such occasions to do so’ (‘Bentham's Principle of Utility and Theory of Penal Law’, IPML, p. xcii). And further: ‘On this account… Bentham's statement that this principle is the standard of right and wrong “in the field of morality in general” and of government in particular must be taken to mean that it is a standard of morality not for the guidance of individual conduct but only for the critical evaluation of it’ (ibid., p. xciii).

38 IPML, p. 11.

39 Mill J. S., Utilitarianism, CW, x. 234.

40 Ibid., p. 234.

41 Moore, p. 67.

42 Scarre G., Utilitarianism, London, 1996, p. 96.

43 Ibid., p. 97.

44 Ibid., p. 97.

45 Mill J. S., Utilitarianism, CW, x. 235.

46 Ibid., p. 237.

47 Ibid., p. 239.

48 Ibid., p. 211.

49 Moore, p. 80.

50 Mill J. S., Utilitarianism, CW, x. 235.

51 This interpretation of qualitative hedonism is very close to Mill's own answer to the self-directed question, ‘If I am asked, what I mean by difference of quality in pleasures, or what makes one pleasure more valuable than another,…’ (ibid., p. 211). Two differences are: (i) Mill seeks a consensus among ‘those who are competently acquainted with both’ as to which of (only) two pleasures is preferred, rather than applying his strength-of-preference criterion to a single individual; and (ii) there is a very large, perhaps infinite discrepancy in degree of preference between the two pleasures in Mill's illustration. Jonathan Riley interprets Mill as finding a qualitative difference only if the discrepancy is infinite, so that ‘pleasure is a heterogeneous phenomenon consisting of irreducibly plural kinds or dimensions arranged in a hierarchy’ (‘On Quantities and Qualities of Pleasure’, Utilitas, v (1993), 292). Riley's hierarchy is a special case for ordering a multi-dimensional entity, known as lexicographical ordering. As Geoffrey Scarre points out, lexicographical ordering ‘is false to the psychological reality and misrepresents the way in which we actually estimate, compare and rank our pleasures’ (‘Qualitative Hedonism’, Utilitas, ix (1997), 356). I doubt very much if it is what Mill had in mind as a general rule.

52 Mill J. S., Utilitarianism, CW, x. 211.

53 Ibid., p. 211.

54 Ibid., p. 213.

55 For an explanation and critique of this approach, see Brennan T. J., ‘A Methodo-logical Assessment of Multiple Utility Frameworks’, Economics and Philosophy, v (1989).

56 Aside from the implications for egalitarian interpersonal weights, the persuasive force of Mill's two-stage argument, in which the supposed superiority of a pleasure is ‘empirically’ derived from the judgements of a set limited to those who have acquired and maintained a taste for it, was satirized superbly by W. S. Jevons: ‘Mill entirely begs the question when he assumes that every witness against him is an incapacitated witness, because he must have lost his capacity for the nobler feelings before he could have decided in favour of the lower. The verdict which Mill takes in favour of his high-quality pleasures is entirely that of a packed jury. It is on a par with the verdict which would be given by vegetarians in favour of a vegetable diet’ (‘John Stuart Mill's Philosophy Tested. IV. – Utilitarianism’, Contemporary Review, xxxvi (1879), 532).

57 Mill J. S., Utilitarianism, CW, x. 213.

58 Ibid., p. 213.

59 Ibid., p. 212.

60 I wish to acknowledge the encouragement given by Philip Schofield and Fred Rosen whilst pursuing my interest, as an economist, in the philosophical side of classical utilitarianism. I have also benefited from the very insightful comments of Geoffrey Scarre, Roger Crisp and Ross Harrison, none of whom is responsible for the positions I have taken, but all of whom are responsible for vastly improving my expression of them.

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