II. IRVING FISHER ON MEASURABLE UTILITY

Irving Fisher (Reference Fisher and Hollander1927, pp. 157–158) regretted that:

Fisher added, “In my first economic publication [Reference Fisher1892, pp. 11–24, 86–89], I endeavored to show that this magnitude is measurable—in theory, at least. My object here is to go one step further and to show that even the problem of statistically measuring it should not be considered impossible.” In a footnote (Reference Thurstone1927, p. 158n), he stated, “So far as I know this is the only attempt (other than Edgeworth’s therein cited [in Fisher Reference Fisher1892]) of treating ‘utility’ or ‘want’ as a definite mathematical quantity.”

Fisher there drew attention to a remarkable aspect of his and Edgeworth’s works. W. Stanley Jevons, Carl Menger, and Léon Walras, the marginalist pioneers of the 1870s, had reasoned in terms of cardinal utility. In contrast, Edgeworth (Reference Edgeworth1881) and Fisher (Reference Fisher1892) pioneered an ordinal approach to preferences and choice, drawing indifference curves (before Vilfredo Pareto and William E. Johnson) and, for more than two commodities, indifference surfaces, and showing that choice theory did not require knowing the unobservable marginal utility of any good but only the ratio between the marginal utilities of any two goods, the marginal rate of substitution represented by the slope of the indifference curve, a curve given (in principle) by observable choices of an individual. All that needed to be observed was whether an individual preferred bundle A to bundle B, not how many utils of pleasure or desire or want the individual would derive from either bundle. And yet, having shown that cardinally measurable utility was not needed to analyze choice by an individual among bundles of goods, Edgeworth (Reference Edgeworth1881, Reference Edgeworth1887) and Fisher (Reference Fisher1892, pp. 11–24, 86–89) concerned themselves with how to measure marginal utility whereas Jevons, Menger, and Walras did not.Footnote ⁷ While ordinal utility, without interpersonal comparisons of utility, sufficed for understanding individual demand for goods, Edgeworth (Reference Edgeworth1919) and Fisher (Reference Fisher and Hollander1927) later advanced interpersonal equalization of the utility of taxes paid as the criterion for just taxation. In the case of Fisher (Reference Fisher1892), concern with measuring utility, even if cardinally measurable utility was not needed to analyze individual choice, was consistent with his engineering background as a student of the physicist Josiah Willard Gibbs, which contributed to Fisher’s focus on concrete results. For his 1891 dissertation and 1892 book, Fisher did not merely rediscover the Walrasian system of simultaneous equations for general equilibrium but, acting as an engineer and under Gibbs’s guidance, constructed a hydraulic mechanism to simulate the determination of equilibrium prices and quantities, an anticipation of the much later development of computable general equilibrium that required Fisher to assume specific measures of utility represented by the shape of the vessels holding the water (Scarf Reference Scarf and Fellner1967; Dimand Reference Dimand2019, ch. 2).

Edgeworth (Reference Edgeworth1881, Reference Edgeworth1887) appealed to experimental psychology, equating a util to the smallest perceptible increment in pleasure or sense of well-being (see Colander Reference Colander2007). Fisher (Reference Fisher1892, p. 5) protested, “This foisting of Psychology upon Economics seems to be inappropriate and vicious. Others besides Prof. Edgeworth have done it. Gossen and Jevons appeared to regard the ‘calculus of Pleasure and Pain’ as part of the profundity of their theory.… The result has been that ‘mathematics’ has been blamed [by J. K. Ingram] for ‘restoring the metaphysical entities previously discarded.’” Edgeworth took no offense at the allegation of viciousness, reviewing Fisher’s thesis enthusiastically in the Economic Journal, inviting Fisher to Oxford, visiting Fisher in New Haven, and, as editor of the Economic Journal, publishing Fisher’s first four journal articles.Footnote ⁸ In contrast to Edgeworth’s psychological approach, Fisher (Reference Fisher1892) looked to the observable behavior of economic agents rather than the psychology behind that behavior. A great admirer of Fisher’s dissertation, Ragnar Frisch, described what Fisher did in his thesis as “the behavioristic approach to the quantitative definition of economic utility … the choice point of view in the analysis of utility is perhaps best known in connection with Pareto’s name. However, the theory of choice was, in fact, first introduced and developed with great consistency by Fisher” (Frisch’s lectures at Yale in 1930,Footnote ⁹ in Bjerkholt and Qin Reference Bjerkholt and Qin2010, pp. 83–84).

III. FISHER AND FRISCH AT YALE IN 1930–31

Fisher and Frisch first met when Frisch was in the United States on a Rockefeller Fellowship in 1927–28, at which time Frisch told Fisher about Frisch (Reference Frisch1926), if Fisher had not previously heard of it. Their shared commitment to “the choice point of view in the analysis” led to Fisher (Reference Fisher and Hollander1927) and Frisch (Reference Frisch1926, Reference Frisch1932; Bjerkholt Reference Bjerkholt1995, pt. I), and, together with their shared support of the applicability of mathematics to economics (e.g., Fisher Reference Fisher1930), initiated a conversation between them that brought Frisch to Yale, where he lectured on “A Dynamic Approach to Economic Theory” from February to December 1930 (Bjerkholt and Qin Reference Bjerkholt and Qin2010), a visiting professorship that resulted in Frisch and Fisher collaborating in the creation of the Econometric Society (Bjerkholt Reference Bjerkholt and Strøm1998, Reference Bjerkholt2017). Yale’s offer to Frisch of a professorship provoked the University of Oslo to promote him to full professor of economics and statistics from July 1931. As Olav Bjerkholt and Duo Qin (Reference Bjerkholt and Qin2010, p. 11) observe, the University of Oslo did so even though, since Frisch was a largely self-taught mathematician and statistician, “his [formal] educational background was unimpressive—only a two-year economics program at the University of Oslo,” although he successfully defended a doctoral dissertation in statistics in Oslo in 1926.

Frisch (Reference Frisch1926) and Fisher (Reference Fisher and Hollander1927) both approached measurement of marginal utility of money by seeking some class of commodities whose marginal utility was independent of the consumption of other classes of commodities. In the absence of such a commodity class, Fisher and Frisch, who were both index number theorists (Fisher Reference Fisher1922; Frisch Reference Frisch1930, Reference Frisch1936), each favored an approximation based on knowing that the constant-utility index of the cost of living is bounded by the Paasche and Laspeyres indexes. Bjerkholt and Qin (Reference Bjerkholt and Qin2010, p. 21) report, “After Frisch’s arrival at Yale in 1930, [Fisher and Frisch] set out to write jointly a monograph on utility measurement. Their plan was somehow aborted; instead Frisch completed the manuscript of a monograph entitled New Methods for Measuring Marginal Utility while at Yale in 1931 and had it published in 1932.” Frisch (Reference Frisch1932) was dedicated to “Irving Fisher, the pioneer of utility measurement.” Frisch (Reference Frisch1947) recalled that he and Fisher quarreled over Fisher’s refusal to adopt the concise mathematical style and axiomatic approach of Frisch (Reference Frisch1926), even though, as Bjerkholt and Qin (Reference Bjerkholt and Qin2010, p. 21) observe, Frisch (Reference Frisch1932) “does not pursue axiomatization or dynamization of the utility concept.”

Although Frisch and Fisher never wrote their projected joint monograph, they collaborated in other ways during Frisch’s year at Yale. They joined with Charles Roos of Cornell and Joseph Schumpeter on a circular letter in June 1930 soliciting interest in creating an international society for the advancement of economic theory in connection with mathematics and statistics. Then, in late November, they followed this with a letter of invitation to the organizing meeting of the Econometric Society at the end of December at the annual meeting of the American Economic Association and the American Statistical Association. The following year Alfred Cowles 3rd wrote to Fisher (the founding president of the Econometric Society) offering to finance what became Econometrica and the Cowles Commission for Research in Economics. Fisher’s acquaintance with the Cowles family (having been a Yale undergraduate and member of the undergraduate senior society Skull and Bones at the same time as Cowles’s father and uncle),Footnote ¹⁰ the trust between Fisher and Frisch, and a visit by Frisch to Cowles in Colorado Springs combined to reassure the European members of the society’s council, who were understandably skeptical of windfall offers of funding in the depths of the Depression (Bjerkholt Reference Bjerkholt and Strøm1998, Reference Bjerkholt2017; Dimand Reference Dimand2019). Fisher and Frisch were the most active of the five members of the commission’s advisory council (appointed by the Econometric Society), as well as speaking at Cowles summer research conferences in Colorado, and Frisch edited Econometrica from its beginning in 1933 until 1960.

Fisher had suggested forming a group or society to promote quantitative and mathematical economics, under the aegis of the American Association for the Advancement of Science, as early as 1912 (see Dimand Reference Dimand2019, pp. 204–205), but no concrete action was taken until Frisch’s visiting position at Yale in 1930. Such a society would no doubt have eventually been formed, but Frisch’s visiting appointment as Fisher’s temporary colleague determined the timing, which resulted in the Econometric Society, upon Fisher’s reassurance and Frisch’s visit to Colorado to investigate, being in a position to respond when Alfred Cowles became disillusioned with stock market forecasters. In the wake of the 1929 Wall Street crash, Cowles, an investment counselor in Colorado Springs and heir to a publishing fortune (his grandfather was a cofounder of the Chicago Tribune), lost confidence both in the stock market letters to which he subscribed and in the one that he produced. Seeking to demonstrate statistically that stock market forecasters did no better than random predictions (see Cowles Reference Cowles1933, Reference Cowles1944; Dimand and Veloce Reference Dimand and Veloce2010),Footnote ¹¹ Cowles asked the advice of Harold Thayer Davis, an Indiana University mathematics professor who summered in Colorado Springs and who told him of the newly established Econometric Society. Given Cowles’s devastating critique of stock forecasts, he needed, and demonstrated, great tact in approaching Fisher, sensitive and defensive about his notorious gaffe about stock prices reaching a permanently high plateau in October 1929.Footnote ¹² The Econometric Society was a small-scale organization when Cowles guaranteed a minimum of $12,000 a year for the society, a journal, and a research laboratory: the society’s total assets were $24.13 when Cowles began more than two decades as the society’s treasurer in April 1932. The shared interest of Fisher and Frisch in measuring marginal utility, which brought Frisch to spend an academic year with Fisher and thus determined the timing of the creation of the Econometric Society, had institutional consequences for the advancement of formal mathematical and statistical methods in economics.

IV. FRISCH ON MEASUREMENT OF UTILITY

“Intermediate between mathematics, statistics and economics, we find a new discipline which, for lack of a better name, may be called econometrics,” announced Ragnar Frisch (Reference Frisch1926, p. 386).

He thereby discarded as an evasion of the problem the assumption by Alfred Marshall (and, at least implicitly, by Jules Dupuit half a century before Marshall) of a constant marginal utility of money for sufficiently small changes in consumption bundles. Although Frisch published his essay in a Norwegian series of mathematical publications, he did so in French and stated, “The study was made during a stay in Paris, 1923,” when he was studying mathematics there. Frisch was already familiar with at least some of Fisher’s work when writing his 1926 study, mentioning Fisher’s preference for Charles Gide’s term “desirability” rather than “utility” (Frisch Reference Frisch1926, p. 395, perhaps with Fisher Reference Fisher1918 in mind) and discussing the basis on which “Mr. Fisher and Mr. Pareto after him have studied the static equilibrium of exchange” (Frisch Reference Frisch1926, p. 391, clearly with reference to Fisher Reference Fisher1892 since Fisher was stated to precede Pareto). Bjerkholt and Ariane Dupont-Kiefer (Reference Bjerkholt and Dupont-Kieffer2009) report that Frisch had already studied Fisher (Reference Fisher1892) while in Paris.

The Paris connection was important to Frisch’s involvement with Fisher, the formation of the Econometric Society, and measurement of marginal utility. Upon publication, Frisch sent a copy of Frisch (Reference Frisch1926) to François Divisia (still famous for the Divisia index), initiating an exchange of fourteen letters leading to the creation at the end of 1930 of the Econometric Society, of which Frisch was the moving spirit (and from 1933 founding editor of Econometrica), Divisia the first vice-president and second president, and Fisher the first president (Bjerkholt Reference Bjerkholt and Strøm1998, Reference Bjerkholt2017). In 1933 Frisch gave a series of lectures on econometrics in Paris, in the prestigious Poincaré series noted, for example, for Bruno De Finetti’s lectures on subjective probability (De Finetti [1937] Reference De Finetti1964; see also De Finetti Reference De Finetti1935 on utility measurement). While New Methods of Measuring Marginal Utility (Frisch Reference Frisch1932), drafted at Yale, was forthcoming, Frisch’s French summary of the book, presented to the first European meeting of the Econometric Society in Lausanne in September 1931, was published in Revue d’Économie Politique with an expository introduction by Jacques Moret that was as lengthy as Frisch’s paper (Moret and Frisch Reference Moret and Frisch1932). Frisch had first encountered Fisher’s work through Moret’s translation of Fisher’s dissertation (Fisher Reference Fisher and Moret1917). Moret, a follower of the Lausanne school of general equilibrium (Moret Reference Moret1915), was, along with Divisia, one of the four French economists among twenty-six recipients of a letter from Fisher, Frisch, Charles Roos, and Joseph Schumpeter about organizing an Econometric Society (letter of invitation, 17 June 1930, and list of invitees on the Econometric Society website). Moret shared both Frisch’s admiration for Fisher’s dissertation as the origin of the choice-theoretic approach to utility and Fisher’s and Frisch’s interest in trying to measure marginal utility, and published a short review of Frisch (Reference Frisch1932) in the next issue of Revue d’Économie Politique after his and Frisch’s articles (Moret Reference Moret1932).

Where Fisher (Reference Fisher and Hollander1927) began with a policy-relevant problem, that of justifying a progressive income tax by Edgeworth’s principle of equal sacrifice (equalizing marginal utility of income across persons, Edgeworth Reference Edgeworth1919), the first numbered section (after the unnumbered introduction) of Frisch (Reference Frisch1926) presented two sets of three axioms (axioms of choice, transitivity, and addition), first for an individual at a given position choosing between two displacements and then for an individual who knows that on two different occasions he will be in two different positions, choosing among displacements from those positions. Independently of the merits of his method to measure marginal utility, Frisch (Reference Frisch1926) is thus important as a pioneering axiomatization of utility theory, an approach he did not take further in Frisch (Reference Frisch1932) but did discuss in Yale lectures (Bjerkholt and Qin Reference Bjerkholt and Qin2010, pp. 93–94) and which was carried further, with reference to Frisch (Reference Frisch1932) and in a presentation in Oslo, by Franz Alt ([1936] Reference Alt1971, Reference Alt1937). The axiomatization of utility theory, continued by John von Neumann and Oskar Morgenstern in the 1940s for maximization of expected utility and culminating in the work of Gérard Debreu (Reference Debreu1959),Footnote ¹⁴ paralleled the axiomatization of probability theory, notably by Andrei N. Kolmogorov in the 1930s. Frisch’s axiomatic presentation was a major difference between his and Fisher’s essays on measuring marginal utility, leading to what Frisch (Reference Frisch1947) termed “quite an argument” between them.

Based on his axioms, Frisch (Reference Frisch1926, pp. 401–402) ruled out the famous function of “moral expectation” or marginal utility proposed by Daniel Bernouilli ([1738] Reference Bernoulli and Sommer1954),Footnote ¹⁵ which in Frisch’s notation expressed the function g(r) as a constant divided by (r – a), where r stands for wealth and a for some constant, as well as the utility function proposed by Charles Jordan (Reference Jordan1924) equating g(r) to a constant divided by the square of (r – a). According to Frisch, the simplest function for the marginal utility of money consistent with his axioms would be a constant divided by (log r – log a).

Where Fisher (Reference Fisher and Hollander1927) proposed only a method for measuring marginal utility, if it was possible to define a class of commodities such that its marginal utility was independent of the consumption of other classes of commodities, Frisch (Reference Frisch1926) attempted to actually perform such a calculation, using for illustration monthly data from June 1920 to December 1922 for French real income, sales of sugar at the Union des Coopérateurs, and the price of sugar divided by the cost-of-living index. Frisch used more extensive data in his 1932 monograph. As Fisher remarked in his 1942 book on income taxation (reprinted in Fisher Reference Fisher, Barber, Dimand, Foster and Tobin1997, vol. 12, p. 264), Frisch’s Reference Frisch1932 statistical results, following from his novel procedures of measurement, “were, however, startlingly different from what many people would have expected. According to his results (very tentative, of course), if a man’s income is halved, his marginal want for one more dollar is not more than doubled, as commonly supposed—not even doubled—but less than doubled. Or conversely if his income is doubled his desire for one more dollar, although reduced, is reduced by less than half” so that Edgeworth’s principle of equal sacrifice would imply “not progressive, but regressive rates” (Fisher’s italics).

V. FURTHER CONTRIBUTIONS TO THE FISHER–FRISCH INVESTIGATION OF MEASURABLE UTILITY

A few authors followed up Frisch’s studies. Frederick Waugh (Reference Waugh1935, p. 376) credited Fisher (Reference Fisher and Hollander1927) and Frisch (Reference Frisch1926, Reference Frisch1932) with having “shown that under certain conditions it is possible to measure quantitatively the variation in the marginal utility of money from one period of time to another or from one group of persons to another,” although he cautioned that the results of Fisher and Frisch applied to the marginal utility of money of selected groups of consumers, not necessarily representative of “the whole populations of France and the United States.” Like Fisher, Waugh cited the principle of “equal sacrifice” in taxation as a motivation for his study. Waugh, at the US Bureau of Agricultural Economics when his article appeared, studied with Frisch in 1932–33, on a one-year fellowship from the Social Science Research Council, and applied Frisch’s methods to US data on income and on the prices and consumption of sugar, coffee, meat, and butter: “With the kind assistance of Professor Frisch and his assistants at the Universitets Økonomiske Institutt in Oslo, these data were studied both by graphic and by mathematical-statistical methods” (Waugh Reference Waugh1935, p. 381). Waugh (Reference Waugh1935, p. 385) reached “the conclusion that in the period 1916 to 1921 and again in the period 1922 to 1932 the per capita consumption of foods was almost constant from year to year … if the average diet in a given period was constant, we can assume that the marginal utility of that diet was very nearly constant and thus we have a convenient and reliable base with which to compare other utilities” (italics in original). Later writers did not, however, find such a convenient simplification offered by the data.

In contrast to Waugh’s enthusiasm for Frisch’s approach, Roy G. D. Allen (Reference Allen1933, p. 187) rejected

(See Dupont-Kieffer 2013 on the debate between Frisch and Allen.)

But, contrary to Allen’s insistence on the full set of variables that theory states should be arguments of a utility function, Fisher and Frisch, apart from the exact representation possibility if the marginal utility of one commodity class was independent of the consumption of other commodity classes, sought to approximate empirical reality by making use of the true constant-utility index of the cost of living being bounded by the Paasche and Laspeyres indexes. Fisher’s candidate for such an index would be the Fisher ideal index, the geometric mean of the Paasche and Laspeyres indexes (Fisher Reference Fisher1922), which satisfied more of Fisher’s seven “statistical tests” or criteria for a desirable index than any other index formula (Frisch Reference Frisch1930, Reference Frisch1936, were to show that no index could satisfy more than six of the seven criteria; see Dimand Reference Dimand2019 and my introduction and afterword to Fisher Reference Fisher, Barber, Dimand, Foster and Tobin1997, vol. 7). Like Allen, Harold Hotelling (Reference Hotelling1932) and Henry Schultz (Reference Schultz1933, Reference Schultz1938), while sympathetic to Frisch’s attempt to find an indirect method of measuring utility, were skeptical of the auxiliary assumptions (such as a consumer’s utility from a commodity depending only on the quantity of that commodity) that Frisch needed to get concrete results (Moscati Reference Moscati2019, p. 122).

Schultz held “that the feelings of the human heart are not directly measurable by any known procedure, and I doubt whether such a procedure will ever be invented. But it is these feelings which continually prompt us to buy and sell, borrow and lend, labor and rest, produce and consume; and it is from the quantitative effects of these feelings that we can estimate their comparative intensity” (1933, p. 110; Schultz’s italics). Schultz noted that data limitations obliged Frisch to try to assume the existence of an average person, rather than analyzing individuals. “Although the lack of the necessary data compels Frisch in his numerical illustrations to make this assumption and even less plausible assumptions,” argued Schultz (Reference Schultz1933, p. 111), “there can be doubt but that, had the relevant data for an individual been available, Frisch’s procedure would give his money flexibility, i.e., the relative change in his marginal utility of money corresponding to a relative change in his income when changes are infinitesimal.” Schultz (Reference Schultz1933, pp. 114–115) noted that Frisch’s assumption that the marginal utility of money is a function only of income and the cost of living, while less general than the assumption of Fisher (Reference Fisher and Hollander1927) that the marginal utility of money depends on income and the prices of commodities, “makes it possible to use observations relating to different price situations in order to determine points on the same utility curve” (Schultz’s italics).

Frisch’s Reference Frisch1926 axiomatic approach to measurement of marginal utility (which contrasted to Fisher’s more discursive, non-axiomatic presentation)Footnote ¹⁶ was taken up by the mathematician and physicist Franz Alt, who, in addition to publishing in German in the Austrian journal Zeitschrift für Nationalökonomie (Alt [1936] Reference Alt1971), came to Oslo to present his results at the International Congress of Mathematicians in 1936 (summarized in English as Alt Reference Alt1937), placing his research in a discourse with Frisch. Alt ([1936] Reference Alt1971, p. 430) mentioned Frisch (Reference Frisch1932) “and a few shorter articles by the same author” (not further identified) but as writings in which “methods for the practical measurement of utility of commodities were proposed—provided that certain conditions were satisfied—but whereas there the conditions were usually not specified precisely or completely, we have been concerned mainly with the formulation of these conditions, that is, with the logical side of the problem,” which overlooks the axiomatic approach of Frisch (Reference Frisch1926). Instead, Alt focused on Oskar Lange (Reference Lange1934), which had sketched a proof of the existence of a utility function without offering the proof itself or giving a complete list of the necessary conditions.Footnote ¹⁷ John Chipman, Leonid Hurwicz, Marcel Richter, and Hugo Sonnenschein (Reference Chipman, Hurwicz, Richter and Sonnenschein1971, pp. 327–329) noted that Alt’s first three axioms, regarding transitivity and connexity of binary preference relations, corresponded to the axioms stated by Frisch, but Alt went further to established necessary and sufficient conditions for solvability, making use of the topological property of connectedness used two decades later by Gérard Debreu: “To a large extent many of the most essential features of contemporary models of measurable utility were anticipated in this pioneering study by Franz Alt” (p. 329).

In addition to Alt’s study, another important contribution to economic theory was inspired by the attempts of Frisch and Fisher to measure utility. William Vickrey’s “Measuring Marginal Utility by Reactions to Risk” (Reference Vickrey1945) transformed the discussion of measuring marginal utility by appealing to the treatment of expected utility by von Neumann and Morgenstern (Reference Von Neumann and Morgenstern1944), recognizing the importance of that analysis even before the axiomatization of rationality as expected utility maximization in an appendix to the 1947 second edition. Fisher (Reference Fisher and Hollander1927), cited in Vickrey’s first and last footnotes, and Frisch (Reference Frisch1932), cited in Vickrey’s second footnote, found a utility function defined up to a positive linear transformation by assuming that commodity classes could be defined so that the marginal utility of one commodity class is independent of the consumption of another class. Vickrey showed that comparing risky alternatives allowed deriving such a utility function without the Fisher–Frisch assumption: the probability weights attached to two risky alternatives to make an individual indifferent between the gamble and a sure thing served as an index of utility. Because those probability weights are bounded between zero and one, that index of utility is measurable. Vickrey, a Nobel laureate who was president of the American Economic Association seventy-five years after Fisher, was Fisher’s last and most distinguished student. Vickrey graduated in engineering and mathematics from Yale and went on to doctoral study in economics at Columbia in 1935, the year Fisher retired from Yale. Fisher, whose PhD was jointly in mathematics and political economy and who began his teaching career as an assistant professor of mathematics, was the only Yale economist of that era likely to attract a Yale mathematics major to become an economist. Fisher thanked Vickrey in a 1939 American Economic Review on double taxation of savings for “kindness and patience in reading and criticizing sundry preliminary drafts of this article,” and in the preface of a 1942 book on income taxation for reading most of the manuscript and making important suggestions (Fisher Reference Fisher, Barber, Dimand, Foster and Tobin1997, vol. 12, pp. 118, 161), while Vickrey’s dissertation and first book devoted a chapter to Fisher’s income concept and expenditure tax proposal (Dimand and Koehn Reference Dimand and Koehn2002).

While Vickrey (Reference Vickrey1945) is the outstanding example of the influence of the Fisher–Frisch connection within economics, William P. Fisher Jr. (Reference Fisher2021) draws attention to another beyond economics. Following from Frisch’s Reference Frisch1930 article on the possibility of an index number that would satisfy Fisher’s tests, Frisch prompted the Danish mathematician Georg Rasch “to formalize a separability theorem that continues today to serve as the basis of a wide range of theoretical and applied developments in psychological and social measurement” (Fisher Reference Fisher2021, p. 21). Rasch’s separability theorem has been sufficiently influential that forums at which William Fisher presented his paper include the Rasch Measurement Special Interest Group of the American Educational Research Association and the Copenhagen conference marking the fiftieth anniversary of Rasch’s book (Rasch Reference Rasch1960).