On the shoulders of giants: from Lange (1934) to Samuelson (1938) on the “unique” measure of utility

Abstract

This contribution discusses Samuelson’s reply to Lange’s paper (1934) on the unique measure of utility. It proposes an interpretation of the debate drawing on the theory of scales later introduced by Stevens in 1946. This shows that, contrary to an intuitive perception, their divergence on the possibility of a cardinal measure of utility was rooted less in mathematical than in cognitive arguments related to the way transitions between allocations are considered. Consequently, although Samuelson succeeded in giving appropriate conditions for cardinality, he based his own mistrust towards its plausibility on arguments later used in the framework of reference-dependent approaches.

Keywords:

• Paul Samuelson
• Oskar Lange
• cardinal utility
• mathematical theory of scales
• reference-dependency

JEL CLASSIFICATION:

• B21
• B31
• D01
• D90

Acknowledgements

This paper owes a lot to the many discussions with André Lapidus, whom I would like to thank in particular. In a previous version, it was presented at the 23rd ESHET Summer School in History of Economic Thought, Economic Philosophy and Economic History (Paris, France, 2021) and at the Conference of the European Society for the History of Economic Thought (Sofia, Bulgaria, 2021). I benefited on these occasions from comments and suggestions by Richard Arena, Herrade Igersheim, Jean-Sébastien Lenfant and James Morrison, to whom I am most grateful. I would also like to thank the two anonymous referees of this Journal for their insightful remarks. Nonetheless, the usual caveat holds.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 See infra, n. 4, p. 3.

2 Moscati (2013) provided a review of this literature for a broader period from 1909 to the end of the war.

3 Although Jevons and Marshall did not explicitly use the term uniqueness, the content of the debate was already present. We can find comparisons of marginal utilities in both Jevons and Marshall. For instance, when they support the idea that it is decreasing. We also find ratios of marginal utilities. Now, since marginal utility is the limit of a finite difference in utility, this perspective appears to be the origin of the idea of uniqueness found in the following generation. It is clear that, in the 1930s, uniqueness referred to what will be identified as an “interval scale” (see Stevens (1946) and Coombs (1950)), which permits the conservation of utility difference ratios and therefore reflects a measurement scale in which functions are related to each other by positive affine transformations.

4 The “uniqueness” of each measurement scale amounts to the possibility of identifying a specific class of transformations of functions. The more general class, at least for economic matters, is given by the conservation of an order (rigorously, a preorder): in this case, which refers to an “ordinal scale,” uniqueness is guaranteed for any positive monotonous transformation of the utility function. By contrast, the more specific class identifies uniqueness with a case where transformations of functions preserve ratios between differences and the point of origin: this is the “ratio scale,” which would allow not only a cardinal expression of utility, but also a distinction between utility strictly speaking and disutility. Intermediary situations correspond to two other types of scales, the “ordered metric scale” and the “interval scale”. The first preserves the order of differences, and the second, the ratios of differences. Textual evidence shows that, in the 1930s, the term unique denoted an interval scale, that is, a scale for which not only order, but also ratios of differences matter. In the following, uniqueness refers specifically to the interval scale, for which functions are related to each other by positive affine transformations.

5 Note that Samuelson, at the end of his paper (Samuelson, 1938, p. 70), pointed out the arbitrariness of the additive structure (the one that characterizes positive affine transformations) defining cardinality and suggested that alternative conventions could be used. Put differently, there is no particular reason to give supremacy to a definition of cardinality as resulting from an additive structure. This remark by Samuelson can be related to that of Frisch (1926) when the latter criticized taking an additive structure as the only structure for transformation functions leading to a unique measure.

6 The term cognition here refers to cognitivism as it will be found later in the works of cognitive psychologists (see, for example, Abraham S. Luchins (1942), Arthur S. Reber (1996), Walter Schneider and Richard M. Shiffrin (1977), Richard M. Shiffrin and Walter Schneider (1977) and John A. Bargh and Tanya L. Chartrand (1999)). Indeed, although Samuelson did not refer explicitly to psychology (neither to social psychology nor to cognitive psychology) in 1937, he already had in mind intuitions that can be linked to it, especially when he rejected, at the end of his article, the cardinality he had just obtained by means of a decision procedure in time. What seemed to be Samuelson’s vision of cognition (even if he did not use the term) is less restrictive than that based on economic calculation: he questioned the existence of a specific mental capacity of economic agents, that of ordering utility transitions. By pointing out a difficulty linked to the mental representations of individuals, Samuelson thus anticipated the questions that will be addressed by psychologists from the 1960s onwards.

7 In the remainder of this article, for sake of clarity, we will use Lange’s analytical framework, i.e. the three assumptions, to discuss both the procedure used by Pareto to reach a unique measure of utility and the unique measure itself.

8 See, for instance, Christian E. Weber (2001), Luigino Bruni and Francesco Guala (2001) or Giocoli (2003, pp. 69-71), whose discussion of Pareto’s position is related to the role of psychology.

9 Rorty (1984, p. 55) distinguished between two concepts: the “meaning” and the “significance” of a contribution to the history of ideas. The meaning depends on what an author did for him and his contemporaries; whereas the significance depends on what can be understood of what he did, independently of his intention. For a transposition to the methodology of the history of economics, see André Lapidus (2019).

10 Something similar can be found in Fishburn (1976). He distinguished four types of measurement: ordinal measurement, quasi-cardinal measurement, cardinal measurement, and measurement on a ratio-scale. The last three measurements are the homologous, respectively, of ordered metric scale, interval scale and ratio-scale. Note that Fishburn understood cardinality restrictively, reserving it to the interval scale.

11 See Lange (1934, pp. 220-221) and Basu (1982, Theorem 2, p. 310). Basu relied on an explicit axiomatic basis given by [4] and [5]. Rigorously speaking, however, note that [4] and [5] cannot be regarded as independent axioms, since assuming $u_1=u_3$ in [5] yields [4] as a particular case.

12 Here is an example of a transformation function, belonging to ${\Omega }_{5^{*}},$ i.e. preserving the order in the differences of utility and which cannot be reduced to a positive affine function. Assume $X$ contains only four elements: $x_1=1;x_2=2;x_3=3;x_4=4.$ Assume also that the utility function $u(x)$ = $x^2.$ The order of differences is preserved by any transformation function $F= a{u}^r+b,$ with $a>0,$ as long as $r\ge 0.5.$ Note that the positive affine transformation is obtained when $r=0.5.$

13 Note that in a way which announces further axiomatizations initiated by Kenneth Arrow in his pioneering book on social choice (Arrow, 1951), in which the primitive has moved from utility to a binary relation of preference, Samuelson introduced a condition of transitivity of preferences:

“(1) $A$ equally preferred to [indifferent to] $B,$ and $B$ equally preferred to $C,$ must imply $A$ equally preferred to $C.$

(2) $A$ either preferred to $B$ or equally preferred to [indifferent to] $B,$ and $B$ preferred to $C,$ must imply $A$ preferred to $C.$” (Samuelson, 1938, p. 67)

Since Gerard Debreu (1954), we know that provided additional conditions are fulfilled (continuity of the preferences for Debreu), transitivity (and, of course, completeness) allows preferences to be represented by a real-valued utility function.

14 Samuelson explained this from the relationships between the norms of vectors $x_1,$ $x_2$ and $x_3.$ The case of relation [8] corresponds to the case where points $x_1,$ $x_2$ and $x_3$ are aligned. This means that the ratios between distances can be interpreted as ratios of differences in utility intensities (see Samuelson, 1938, pp. 68-69).