A Problem as Much of Economic Theory as of Statistical Technique? The Walsh-Edgeworth Debate on Index Numbers, 1921–25

Abstract

Correa Moylan Walsh (1862–1936) was a monetary economist of the early twentieth century, mostly remembered for his contributions to the theory of index numbers. Francis Ysidro Edgeworth (1845–1926) was a renowned mathematical economist, remembered as one of the great masters of the discipline. Walsh was an adamant defender of a deterministic, test approach to the theory of index numbers, whereas Edgeworth favored probabilistic approaches and thought index-number theorists should not close themselves to alternative methods. Holding contrasting views about the making of index numbers, they engaged in a fierce debate in the first half of the 1920s—a period marked by several quarrels on the subject. The main points in dispute between Walsh and Edgeworth were the existence of an ideal formula for index numbers and the use of probability in the measurement of price variations. The goal of this article is to explore this debate and its underpinnings. It is argued that the main points of contention between them relate to their conflicting perspectives on the nature of index numbers.

1. Introduction

In 1934, the statistical economist Arthur Lyon Bowley advised anyone interested in doing serious research on index numbers to read the following works: Irving Fisher's The Making of Index Numbers, Correa Moylan Walsh's The Problem of Estimation, and Francis Ysidro Edgeworth's papers, reports, and reviews. At the very minimum, Bowley (1934: 118) asserted, these works, all by economists, would allow an aspiring index-number theorist to have “an insight into the psychology of the writers.” Bowley was right. Many of the predilections and reservations of Fisher, Walsh, and Edgeworth indeed shine through their writings, especially the writings related to the debates in which they took part. Two of them, Walsh and Edgeworth, actually engaged in a fierce debate against one another. The goal of this article is to scrutinize the debate and its underpinnings, which relate both to economic theory and to statistical technique.

Correa Moylan Walsh (1862–1936) was an early twentieth-century American monetary economist whose work was long neglected by historians of economics. He is usually recalled—rather briefly—for his contributions to the field of index numbers.¹ Only recently has his work been scrutinized at length to show that Walsh was a full-fledged monetary economist who dwelled upon index numbers as a means to an end within his approach to monetary issues (Cruz-e-Silva and Almeida 2024). Francis Ysidro Edgeworth (1845–1926) was an Anglo-Irish economist who, unlike Walsh, has always figured among the great masters of economics. Accordingly, it is not hard to find papers in the history of economics literature focused on his work.²

The debate between them constitutes both a chapter within Walsh's approach to monetary economics worth a closer investigation and an episode in Edgeworth's scholarly work that has eluded the historical accounts of his contributions. It is true that some of their contemporaries acknowledged the debate—such as Bowley (1934: 118) and Keynes ([1930] 1971: 72)—and that Balk 2008 and Stapleford 2003 are rare cases of works in the history of economics literature that mention some of its elements.³ Nonetheless, the fundamentals of the disagreement between Walsh and Edgeworth remain uncharted.

The main argument of this article is that the roots of the debate relate, above all, to their conflicting perspectives regarding the very nature of index numbers. In this sense, it was the theoretical considerations of their debate that determined the technical aspects of their disagreement. To defend this idea, this article will present Walsh's and Edgeworth's early views on index numbers, the context of the 1920s overarching debates, and the Walsh-Edgeworth debate itself.

2. Edgeworth and Walsh: Early Views on Index Numbers

In the second half of the nineteenth century, the witnessed instability of prices took the minds of economists by assault. Both in the United States and the United Kingdom, prices fell consistently between the 1870s and the 1890s, triggering several debates within monetary economics (see Bordo 1977: 23–24; Friedman and Schwartz 1963: chaps. 3–4). One of the most fundamental issues at hand within these agitations was how to measure the changes in prices, which led to discussions about index numbers. Such an interest in index numbers would be further aroused by the systematic rise in prices that lasted from the late 1890s to the years following World War I in the United States and the United Kingdom (see Broadberry and Howlett 2005: 219; Friedman and Schwartz 1963: 196–97).

This later inflationary process, aggravated by World War I, constitutes the scenery that stimulated Fisher's search for an ideal formula for index numbers in the early 1920s, and it was in the context of Fisher's pursuit that the Walsh-Edgeworth debate unfolded. Nonetheless, the first remarks of Walsh and Edgeworth on the subject had appeared already in the last decades of the nineteenth century. Their 1920s debate followed the tracks of such early propositions.

2.1. Francis Edgeworth and Index Numbers, 1883–1901

According to John Maynard Keynes ([1933] 2013: 261), the theory of index numbers, “or the application of mathematical method to the measurement of economic value,” was the last topic to capture Edgeworth's scholarly interests, in an 1883 paper. As Stigler (1978: 297–98) recounts, Edgeworth, inspired by William Stanley Jevons to work on index numbers, broadened the scope of the field, as he was the one “who first thoroughly investigated their properties from the standpoint of statistical theory.” His contributions to the theory of index numbers and to monetary economics were gathered, under the broad label “Money,” in his Papers Relating to Political Economy, a three-volume compendium of his economic studies, organized in 1925.

The 1883 paper Keynes identifies as Edgeworth's first approach to index numbers is not included in the compendium. Already in the 1883 work, Edgeworth, whom Aldrich (2008: 269) identifies as the authority on the theory of index numbers, acknowledged the usefulness of constructing a ratio indicating the depreciation or appreciation of gold. This ratio should start from the ratios between the prices of articles in two different periods, and it should aim at the preservation of an individual's total utility regardless of the depreciation or appreciation of gold. His goal was to find a result “as such that the total utility which a person would derive from a certain income at the former of the two epochs under consideration, is equal to that which he would derive at the latter epoch from the same quantity of money multiplied by the sought number” (Edgeworth 1883: 714). Accordingly, already in 1883, Edgeworth's analysis as an index-number theorist was explicitly subject to the principle of utility.

Nonetheless, Edgeworth's most important early discussions on index numbers are to be found in the memoranda he prepared for the British Association for the Advancement of Science, in the triennium 1887–89. The memoranda open the “Money” section of his Papers Relating to Political Economy (Edgeworth 1925b: 195–343), but the ideas laid down in them appear throughout the other “Money” papers of his compendium as well.

Edgeworth (1925b: 195, 295, 384) pointed out that an index number might serve different purposes. Personally, he championed a utility-driven consumption standard, which, in its most refined form, “compares the amounts of money required to procure the same satisfaction at different epochs” (195) and, therefore, has “the mass of final utility” at its basis (263). This would become more clear-cut in his post–World War I works (Stapleford 2003: 98–99). For him, therefore, different standards, given their specific purposes, arrive at different results and conclusions regarding the “real value” of money and its stability (Edgeworth 1925b: 196), because “even where there are no fundamental differences of theory,” Edgeworth declared, “practices may vary according to the practical end in view” (206).⁴

In relation to the specific purpose of measuring the variations in prices, the formula Edgeworth favored by then was the geometric mean—belittled by him in 1883 for the measurement of changes in the satisfaction of individuals. He declared that prices do not follow a symmetrical curve in their arrangement around a mean but “an unsymmetrical curve like that which corresponds to the geometric mean.”⁵ The reason for such asymmetry was the existence of a lower bound for prices (zero) but the nonexistence of an upper bound, insofar as prices may rise indefinitely (Edgeworth 1925b: 237–38). If, however, symmetry in the dispersion of prices was supposed, then, according to Edgeworth, the most appropriate method would be the median, because of its insensibility to biased weights and observations (248–49, 295, 331, 404–5). For Edgeworth, therefore, the choice of the “mean” to be used as index number “depended upon the distribution of prices and the economic purpose to which the index would be put” (Stigler 1978: 298).

His appreciation for the law of errors—or probability theory, for that matter—and its hallmarks also shines through his writings. The hallmarks include, for instance, the assumption of independence among observations and the uniformity of errors (Edgeworth 1925b: 298–99, 305, 359). For him, probabilities are important especially because the work with index numbers necessarily involves building averages from samples of prices, “specimens taken at random from a much larger, if not an indefinitely large series” (288), bearing consequences for the methods and the results (357). He conceived of prices as the manifestation of the relation between the volume of goods and the mass of currency, as in metallic money expanded by credit (426). One of his fundamental theoretical premises was, in fact, that commodity prices reflected a universal factor, that is, the supply of gold.

Edgeworth's approach to index numbers, therefore, came forth from an eagerness to grasp “the relationship between the supply of gold and the value of the pound [or any other currency]” (Stapleford 2003: 85). Edgeworth believed that, as the supply of gold varied, the set of moving prices behaved like “a fleet of yachts which under the influence of a common cause—it may be rising wind or tide—are variously accelerated according to ‘the build of the various yachts or seamanship of the crews’” (Edgeworth 1925b: 357). His idea was that the supply of gold influences the central tendency of the distribution of prices, but that each individual price behaves according to particular dynamics. The set of moving prices was thus taken to be formed by mutually independent magnitudes. This way, “the deviation of the observations [in a sample] is doubtless sufficiently random to justify Laplace's application of the Calculus of Probabilities” (357).

In any index number, these aggregates (of prices and quantities) are arranged to form a ratio. Since, as Edgeworth (1925b: 305) claimed, “any weight affects both the numerator and denominator [of said ratio] in the same direction, whether of excess or defect,” any bias is to a certain extent self-corrected. Such notions led Edgeworth to dismiss weighting procedures and led to the classification of his work by others as adhering to the stochastic approach to index numbers. This approach is usually identified as that whose “aim was to learn about the causes of price changes, either to isolate the cause of a price change or to measure the force of a particular cause, the monetary cause” (Aldrich 1992: 658).

It is not that Edgeworth thought weighting to be a wrong operation. He simply considered it to be an unnecessary precaution. He did so for three reasons. First, when working with large samples, different systems of weights generate nearly identical results. Second, any divergence diminishes in response to the addition of independent observations to the sample. And third, each observation should count for one (Edgeworth 1925b: 284, 287, 359). The use of the “Calculus of Probabilities,” therefore, eliminated the need for weighting in the making of index numbers. Edgeworth's message was that index numbers were “insensitive to weighting—provided there were enough observations (commodities)” (672).

Bowley (1934: 118), furthermore, would yet claim that the logical basis of Edgeworth's investigation was “further developed in later papers, especially those which deal with Mr. Correa Walsh's work.” Here, Bowley referred precisely to the debate Edgeworth waged against Walsh, which indicates the relevance of this controversy for the understanding of Edgeworth's attitude toward index numbers. In fact, Edgeworth's long review of Walsh's 1901 book was considered sufficiently important to be included in his 1925 compendium.

2.2. Correa Walsh and Index Numbers, 1901–3

Walsh touched upon monetary economics already in his first scholarly publications (Walsh 1896, 1897, 1901, 1903), as documented by Cruz-e-Silva and Almeida (2024). Enticed by the bimetallist controversy, these works represent Walsh's painstaking effort to establish a science of money equipped with an airtight definition of value, with a reference for the stability of money, and with a solid instrument for the measurement of this stability—index numbers.

The main concern of Walsh's approach to monetary economics was the establishment of which kind of value money should measure and store in a stable manner. This was the primary purpose of his archetypical book, The Fundamental Problem in Monetary Science, published in 1903, and of his 1926 booklet, The Four Kinds of Economic Value (see Cruz-e-Silva and Almeida 2024). In these works, Walsh identified four kinds of economic value: use-, esteem-, cost-, and exchange-value. “Use-value is a thing's power to serve our ends [its total utility]. Esteem-value is its power to make us desire to possess it [its final utility]. Cost-value is its power to impose upon us effort to acquire it [its cost of production]. Exchange-value is its power to procure other things in its place” (Walsh 1926: 15), or, alternatively, “the purchasing power of one thing over another thing or over things in general” (Walsh 1903: 6). Among these kinds of economic value, Walsh pointed to exchange-value as the kind that should serve as reference for monetary economics, because only the stability of money in exchange-value is compatible with higher levels of prosperity and welfare.⁶

At this point, it is important to note that Walsh's approach—as did Edgeworth's—had a remarkable Jevonian touch. Jevons, in the second edition of his Theory of Political Economy, indicated that value was an ambiguous term that should have its definition distinguished between three different meanings: “value in use,” which is the same as total utility; “esteem,” representing the final degree of utility; and “purchasing power,” which amounts to the ratio of exchange between two commodities (Jevons 1879: 87–88).⁷ The definitions of these three kinds of value are the same for Jevons and Walsh—Walsh adds cost-value to the mix—and both of them are concerned primarily with exchange-value. Walsh, however, extends the analysis from exchange-value to general exchange-value.

It was within this frame of mind that Walsh produced his magnum opus, The Measurement of General Exchange-Value, published in 1901, a book concerned with the development of index numbers. This book, despite being published two years before The Fundamental Problem in Monetary Science, was conceived in preparation for his 1903 work. The Measurement of General Exchange-Value offered an encyclopedic discussion about value and the most appropriate instruments for its measurement, because Walsh's goal was to discover the appropriate method for the measurement of exchange-value. In this process, since money acts as a measure of exchange-values, the measurement of the stability in the purchasing power of money became the central issue in his analysis. Accordingly, Walsh sought the most appropriate formula for such a measurement—an index number that ought to deal with technical features such as averaging, weighting, periods and base periods, and the measurement of errors (Walsh 1901).

Walsh understood index numbers strictly as price indexes—a perspective in line with the Fisherine approach later developed in the 1920s. This strict meaning of index numbers as conceived by Walsh was not relaxed until 1932, when he defined “the broad and generic meaning” of an index number as “an average of relatives or variations” applicable to a wide array of measurable phenomena (Walsh 1932: 652). Also Fisherine-to-be is the formula Walsh singled out as the universally best one available to the index-number theorist, that is, the geometric mean between the Laspeyres's and Paasche's indexes.

Accordingly, Walsh's approach to index numbers would be defined as axiomatic—as was Fisher's. “The core idea of the test [or axiomatic] approach is that the most appropriate formula for index numbers should be selected based on its ability to satisfy a number of mathematical criteria in the form of tests” (Cruz-e-Silva and Almeida 2022: 656). Walsh thus designated a handful of tests that ought to be the benchmark for index-number formulas. The best formula for index numbers would always be that which coped better with the tests, offering the most accurate result. The causes behind a certain movement in prices, hence, did not concern Walsh, because “we can be scientifically prepared for investigating the cause of variations only after measuring the variations with scientific precision,” and this was an important feature of Walsh's axiomatic approach to index numbers (Walsh 1901: 22).

This is the gist of Walsh's early approach to index numbers, and this agenda encapsulated his efforts until around the publication of The Fundamental Problem in Monetary Science. From 1904 onward, Walsh shifted the attention of his writings toward issues such as intellectual history, philosophy, ethics, and political science. He would then return to his previous monetary discussions in the 1920s, stimulated especially by the ongoing debates on index numbers.

3. A Brief Account of the Debates on Index Numbers in the First Half of the 1920s

It was within the disruptive scenario fostered by World War I that the debates on index numbers gained momentum in the early 1920s. Within that context of worldwide economic distress, attempts to confront the postwar reality with the earlier economic conditions were thwarted.⁸ Rampant price instability, for one, and the ensuing labor unrest in face of the need to adjust wages properly, posed a serious problem. On its own, this matter would already impose the need for the development of appropriate index numbers to be used in policymaking, in the formulation of business-cycle theories, and in the monetary debates (see Stapleford 2009: chap. 3). Another problem was posed by the fact that prewar consumption patterns had vanished, giving way to new consumption practices and rendering the very understanding of the new scenario nearly unworkable in terms of normalcy. In this regard, using Bowley's (1919: 346) insightful analogy, it might be said that “the problem has often been compared to that of deducing the proper motion of the sun from the apparent movements of selected stars; but what is the astronomer to do if some of his stars move off jazzing in constellations and others disappear from the firmament?”

Fueled by this scenario of economic distress, the early 1920s was the most exciting period in the history of the theory of index numbers, a period in which methodologies in the making of index numbers were questioned and the search for alternatives was aroused. Such alternatives, clashing against each other, would oppose both the theoretical keenness of index-number theorists and their methodological predilections. Furthermore, this newly discovered prestige of index numbers as a compass for policymaking may justify the rhetorical fireworks in the related debates.

Therefore, as Ragnar Frisch (1936: 1) aptly remarks, in general, the problem of how to construct an index number in the 1920s was “as much one of economic theory as of statistical technique,” because the index-number problem “arises whenever we want a quantitative expression for a complex that is made up of individual measurements for which no common physical unit exists.” Accordingly, it was with economics and statistics as battlegrounds that such debates unraveled.

The most comprehensive debate in this period revolves around Fisher's presentation at the 1920 meeting of the American Statistical Association (ASA), when he launched a search for the ideal formula for index numbers. Walsh was one of Fisher's three discussants at the ASA meeting—the others were Wesley Mitchell and Warren Persons—and played the role of Fisher's only supporter in the defense of the geometric mean between Laspeyres's and Paasche's indexes as the ideal formula for index numbers regardless of the purpose at hand (Fisher 1921, 1922; Walsh 1921a). This formula, Fisher's index (⁠$IF$⁠)—thus labeled by Walsh (1921a) in his “weighty approval” (Young 1921: 572) of Fisher's proposal—is

where $p0$ and $p1$ represent the prices in periods 0 and 1, respectively, while $q0$ and $q1$ are the quantities in the same two periods.

In fact, this formula had already been recommended by Bowley (1899) and approximated by Walsh (1901) himself, while Fisher (1902) had discredited it at first. Now, however, Fisher's exercise conformed to his equation of exchange and to the necessity of ascertaining price movements accurately in order to render the measured general purchasing power a standard for deferred payments (Banzhaf 2004: 605–6; Cruz-e-Silva and Almeida 2022: 674; Dimand 1997: 2; Persons 1921: 113). In this sense, after originally discrediting Walsh's approach, Fisher would come to change his position “to (essentially) that of Walsh” (Fisher 1924: 91).

Fisher's approach was challenged by some of the most important statistical economists of his time, mainly because of the arbitrariness and absoluteness in his selection of a formula fit for all purposes.⁹ This reaction came from the United States and from across the Atlantic, with a somewhat organized British reaction to Fisher (and Walsh). In fact, according to Stapleford (2003: 116), especially inflamed by Walsh's 1921 dismissal of statistics and utility as irrelevant in the field of index numbers, “the British statistical establishment rose up en masse” to challenge the general premises of Fisher's proposal and to defend “constant-utility” index numbers. The idea of a general exchange-value was hence chided among British index-number theorists as an “indefinite proposition”—even though “they had nothing concrete to offer as a substitute” that matched their own criteria for constant-utility index numbers in an environment marked by flickering consumption patterns and evolving consumer preferences (119). Fisher's faith in his approach, nonetheless, remained unshaken (see Fisher 1923, 1924). The same is true of Walsh (1921a, 1921b, 1924).

Accordingly, much influenced by Fisher's work, a series of debates on the matter of index numbers would mark the first half of the 1920s.¹⁰ One of these was the one starring Walsh and Edgeworth.

4. The Walsh-Edgeworth Debate on Index Numbers

In addition to the transcript of his 1920 participation in the meeting of the ASA, 1921 also brought to light Walsh's The Problem of Estimation, another book of his on index numbers. This work complemented the arguments Walsh had presented in The Measurement of General Exchange-Value and reinforced Walsh's defense of Fisher's index. It also ignited his debate with Edgeworth.

In the preface to The Problem of Estimation, Walsh (1921b) defined the issue at hand as a mathematical one and saw little room for dispute regarding its solution. Walsh built his argument in the book oriented toward the appreciation of “the theoretically true method” (34). For him, “if index numbers are ever to be put to any serious use, the closest possible approximation to the truth will be necessary” (84). Strenuous as this may be, accuracy must be the benchmark, and the precision demanded by theory can make “no compromise with sloth” (85). Walsh, as Fisher, focused on the identification of the changes in prices as dictated by the equation of exchange.

Through testing, the best method ought to be singled out, and this method has to be accounted as appropriate for measuring price variations at any time or place (Walsh 1921b: 118). He concluded that the method capable of offering the most accurate results is the geometric mean between two evenly weighted price-variations (94). No other alternative survived the negative test with two objects, as required by the axiomatic test, which meant “that the same mean or average be applicable to any possible number of items” (74–76).¹¹ In this process, weighting occupies a high place in Walsh's reasoning, because it is necessary to average the deviations from a common point of reference, and, for this, commodities should be considered according to their relevance (81–82).

The best geometric mean available, according to Walsh, was Fisher's index—thus named by Walsh (1921a). He regarded Fisher's ideal formula as the method that came closest to satisfying the necessary tests, including the circular test (Walsh 1921b: 102)—which became a point of contention in the Walsh-Edgeworth debate.

4.1. The Ignition

On top of that, The Problem of Estimation also ignited the quarrel between Walsh and Edgeworth. The book brought to light Walsh's response to the objections Edgeworth had raised twenty years earlier against Walsh's approach to index numbers. The debate would then unfold following two main lines. The first regarded the existence of an ideal formula for index numbers and the formula's purpose. The second revolved around the use of probability theory in the measurement of price variations, which Edgeworth (1925b: 369) regarded as “the main ground” of his differences with Walsh. Nonetheless, other secondary issues were also present in the debate, such as weighting and the circular test and how to cope with its violation, because both Walsh and Edgeworth attached more importance to the circular test than did Fisher.¹² These points of contention, it is here argued, may all be traced back to their conflicting perspectives on the nature of index numbers.

Edgeworth had drawn first blood in the dispute already in 1901, as he found himself “at variance with Mr. Walsh on certain fundamental issues” (Edgeworth 1901: 408). The first target of Edgeworth's criticism is the exaggerated ambition of Walsh, because it was futile “to seek an exact measure of the change in the value of money” taking as reference only the variations of two prices. At most, this would allow for the measurement of general exchange-value at each period separately (408)—even though Edgeworth struggled to comply with any definite perception whatsoever about the exchange-value of money. Accordingly, here he introduced their first great point of disagreement: “I can form no idea of such a general exchange-value, except the somewhat indefinite notion of the relation between an amount of money and the quantity of utility which it will procure” (408).

Edgeworth reinforced, therefore, that his attitude toward index numbers privileged the measurement of variations in utility rather than in the purchasing power of money, highlighting along the way his exasperation with approaches to index numbers that aimed at absoluteness. Besides, Edgeworth already hinted at the notion that consumption standards differed across different social groups, which made it impossible to aim at an absolute index number (further elaborated in Bowley 1919: 365). As such, Edgeworth thought that index-number theorists would never reach an “exactly true method” along Walsh's lines, at least not until final utility—“or what he [Walsh] calls ‘esteem value’”—was appropriately handled (Edgeworth 1901: 409). Maybe not even then. In fact, following Marshall, Edgeworth deemed “an absolutely perfect standard ‘unthinkable’” (410), and this leads us to their second great point of disagreement: the use of probabilities.

Walsh's neglect of positive science and disregard for probabilities in the making of index numbers was a serious omission in Edgeworth's eyes. Even if it was granted that the primary problem at stake was the measurement of the value of money, Walsh had, by rejecting the calculus of probabilities, “thrown away an instrument necessary for the performance of that measurement” (Edgeworth 1901: 409). Great uncertainty is involved in such a measurement, in the sense that this initiative depends upon as many observations as possible. This necessarily involves samples of prices, so that observed prices included in the analyses of the index-number theorist may be tarnished by errors. Any definite measurement of the quantities thus involved in this exercise is difficult. Edgeworth also restated his belief that weighting “is of less importance than at first sight appears” to the making of index numbers and, accordingly, represents an “unnecessary precaution” to which Walsh misleadingly pays much more attention than he should (410).

This was Edgeworth's inaugural report on Walsh's approach to index numbers. He regretted that Walsh's great learning had not saved him from “an inability to perceive the many-sidedness of the problem” (Edgeworth 1901: 416). Furthermore, he also highlighted that this was nothing but his own opinion on the subject, and he suggested that several approaches to index numbers could and should coexist: “There are those who conceive the problem in a sense more favorable to Mr. Walsh. To me he seems unfortunate in his subject; to others perhaps, only in his critic” (416).

For unknown reasons, it took Walsh twenty years to strike back. His reaction may have been triggered by Bowley's paper “The Measurement of Changes in the Cost of Living,” which was published, in 1919, alongside comments by Edgeworth, Edward Gonner, and Leo Chiozza Money. In his comment, Edgeworth revisited his accusation that Walsh dispensed “with the aid of Probabilities” in his “stout volume on Exchange Value, a great part of which was devoted to the investigation of a formula proper to the case, in which there were only two prices!” (Bowley 1919: 366). Walsh (1921b: 107) opened his reaction to Edgeworth with a citation precisely to the comments laid down by Edgeworth in 1919, and he spent the last thirty-two pages of The Problem of Estimation engaged in his “violent and tedious polemic against Edgeworth,” to use Yule's (1921: 626) words. The dispute then moved from the back burner to the forefront of the debates.

Walsh (1921b) criticized Edgeworth particularly for his fixation on the use of probabilities, which, albeit interesting and useful for some purposes, are not, unlike weighting, relevant to the study of price variations insofar as the measurement of changes in the exchange-value of money is concerned (81, 137). Referring to Edgeworth's arguments in the 1888 memorandum prepared for the British Association for the Advancement of Science, Walsh regretted Edgeworth's indifference toward weighting, because weighting mattered not only in theory but also in practice (see Edgeworth 1925b: 316–17, 320). Walsh complained that, even if Edgeworth were right and other problems were given precedence over weighting, this would not be a good reason “for not giving all the care that we can to weights” (Walsh 1921b: 136; also 128–32).

As he strived for precision, Walsh could not wrap his head around the discrepancies Edgeworth tolerated between weighted and unweighted index numbers. “If a government, whose business it is to regulate the currency, were to use index-numbers liable to such errors, which might accumulate from year to year,” Walsh (1921b: 137) declared, “that government would do better to leave the currency alone.” Also, for Walsh, Edgeworth relied on the theory of probabilities because he did not know how to identify the best method, so he had to assume different methods “are likely to err equally on opposite sides” (107).

Walsh (1921b: 107–8) believed that it was exactly because it disregarded probabilities and strove for precision that The Measurement of General Exchange-Value had any merit: “That work aimed at precision, at the attainment of absolutely correct results as far as within our power.” In this search, Walsh claimed, “the only way to compare methods is to try them on cases where the true result is known, whereby we can measure their comparative erroneousness and their comparative accuracy.” To the charge that he had relied only on two commodities and two prices, Walsh replied that the simplest case can certainly disprove a method—but it cannot necessarily prove it, because such a proof would come only after the generalization of the simplest case (108). In this regard, a special reason for Edgeworth's rejection of the two-price system, according to Walsh, was that it does not admit the use of the calculus of probabilities (110). Walsh was then categorical: “Where plain algebra is sufficient for dealing with a problem, it is simple pedantry to lug in the so-called higher mathematics” (136), meaning that Edgeworth's resort to the calculus of probabilities was an unnecessary extravagance.

Walsh conceded that Edgeworth was right in recognizing the existence of different methods for measuring price variations. He claimed, however, that Edgeworth failed to acknowledge that one of them had to be the true or best method, the one that came closest to fulfilling the sole purpose of averaging price variations, that is, measuring the “true variation of the exchange-value of money” (Walsh 1921b: 117). Walsh, therefore, tossed aside Edgeworth's concern with utility altogether; nor was it necessary to design other purpose-oriented formulas. “There is no other object or purpose. . . . The single object or purpose has been stated. It is specific. No further varietal subdivision of it is needed, nor is it possible, so far as the method of averaging the price-variations is concerned” (117). For him, the data used may change from case to case, but the method of averaging must be the same in all cases (117–18). For Walsh, index numbers were necessarily price indexes, and nothing else.

Walsh hence went back to Edgeworth's first works on index numbers and condemned Edgeworth's recommendation for averaging the results obtained by different index-number formulas, on the basis that this would do nothing but vitiate the results achieved by the best method (Walsh 1921b: 106–7). For Walsh, he himself had succeeded in providing a precise approach to index numbers, whereas Edgeworth had not even tried to do so. Either Edgeworth had to disprove Walsh's propositions or have his claims utterly ignored (111–12).

For all of that, he denounced the high place of authority reached by Edgeworth in the matter of index numbers, an authority that had actually undermined the progress of the field (Walsh 1921b: 138). “So enamoured is he of his own probabilistic way of looking at the subject, that he refuses to be instructed, and shuts his eyes to the truth, just because it has come from another mode of treating the subject more analytical (dialectical he called it) and systematic than his own” (136).

4.2. The Boiling Point

Following Walsh's overdue and rather acrimonious reply, the debate reached its boiling point, as both authors tried to point out the irreconcilable flaws in each other's approach. With this avowed purpose in mind, Edgeworth wrote, in 1923, two papers meant to deal with Walsh's “hostile criticism” (Edgeworth 1925b: 198). He highlighted that it was important to break away from Walsh's prejudice regarding more complex mathematical methods. For him, index numbers necessarily relied on sampling, which was governed by errors and, therefore, could not forgo the use of probabilities even in the simpler cases (Edgeworth 1923a: 348–49; 1923b: 570). Accordingly, only through differential calculus and the theory of probabilities could the theory of index numbers be transformed from a mere “comparison of subjective quantities” and assume “the character of objective science” (1923a: 347).

Edgeworth (1923b: 573–77) further condemned Walsh's reliance on weighting, because “even where weights are available it may be convenient to dispense with the trouble of using them when the result sought is rather curious than important” (582). What is a “rather curious than important” result he did not say; he used this argument, nonetheless, to ascertain the superiority of the median in relation to the arithmetic mean in the absence of weights.

For Edgeworth (1923a: 345), Walsh made a big mistake when he considered that an entity called exchange-value could have its quantity periodically measured with the same precision with which physical objects are measured. He agreed with Walsh that the circular test “ought to be fulfilled,” but this was not sufficient for him to subscribe to Fisher's ideal formula, because not even that formula passed the circular test (345–46).

Indeed, as has long been known, Fisher's index fails to satisfy the circular test (see Dimand 1997: 4; Frisch 1936: 6; Walsh 1901: 396). The test establishes that the “price index should be independent of the choice of another time point (decomposing it into the product of two similar price indexes)” (Dimand 2019: 142). According to Young's (1923: 346) notation, and given the successive periods a, b, and c, the circular test thus demands that $Ia;c=Ia;b×Ib;c$⁠.

Furthermore, Edgeworth reiterated their fundamental disagreement regarding purpose: he made it clear that, unlike Walsh, he did not restrict the use of index numbers to the measurement of variations in the exchange-value of money alone but envisioned other applications to that tool as well, which called for alternative methodological designs (Edgeworth 1923a: 350; 1923b: 572). In his own approach to the subject, Edgeworth (1923a: 344–45) reinforced a predilection for the measurement of variations in utility.

Walsh responded to Edgeworth's latest attack in 1924. This was, as intended, his last word on the controversy. Hence, Walsh (1924: 500) was clear-cut in his opening statement: “Professor Edgeworth's and my views on index-numbers are diametrically antagonistic. Either his or mine must be wrong; and if the wrong ones are those of an economist, mathematician, and statistician so authoritative as Professor Edgeworth, it would be well that others should be warned.”

The quintessential difference between the two approaches, for Walsh, was that he was striving for absolute accuracy, while Edgeworth contended himself with indefiniteness and vagueness, aiming at the stability of a “certain vague word” because he could not see a definite measurement for the exchange-value of money (Walsh 1924: 500–501). By “certain vague word,” Walsh made reference to Edgeworth's allegedly unclear theoretical notion of “value,” insofar as he, for Walsh, mixed up the concepts of exchange-value and esteem-value.¹³ He correctly indicated that Edgeworth apparently favored stability of money in esteem-value rather than in exchange-value, but that such a predilection was not clear enough in Edgeworth's work. “If Professor Edgeworth has reached a firm conviction as to which of these standards is the proper one, let him tell us so; he should then advocate the one he prefers, and try to get its best measurement” (501–2). For Walsh, exchange-value is the one in which money must be kept stable, and no efforts should be spared in finding the best possible method for its measurement (502). In this regard, for Walsh, the adoption of mathematical tests eliminated the need for the use of probabilities (515), even though Edgeworth insisted on introducing probabilities in places where they did not belong.

Somewhat paradoxically, nonetheless, Walsh argued that the collection of data relates to sampling, which brings about errors, and this involves the theory of probabilities. For him, however—and this is the crux of the matter—sampling was “not a peculiarity of our subject,” that is, index numbers (Walsh 1924: 517). His concern as an index-number theorist was the search for the proper method of weighting and for the proper method of averaging price variations—the proper formula. These were the problems to be solved in the making of index numbers. In this regard, hence, Walsh reserved no place for the use of probabilities in the theory of index numbers.

For Walsh, although not perfect—because he knew it failed the circular test, for instance—the best existing formula was Fisher's index. Therefore, index-number theorists should rely on Fisher's index for the time being but keep looking for the perfect formula (Walsh 1924: 502–4, 508–10). Walsh regretted that Edgeworth found this result to be unattainable, and he wished he could persuade Edgeworth to put his great mathematical talents to use in searching for the perfect formula (505). Nonetheless, since one could not expect from Edgeworth a search for the perfect formula, Walsh issued the challenge: “Can he demonstrate that a perfectly correct average is impossible? If he can, I wish he would do so. He has not done so yet” (510).¹⁴

At last, Edgeworth published two interdependent papers in 1925, which were not explicit responses to his antagonist but revisited much of the criticism previously directed at Walsh. Based on his previous arguments, Edgeworth (1925a: 559–61) reinforced the case both for probabilities in the theory of index numbers and for utility as a measure of value. Here, he denounced Walsh for his denial “that utility has anything to do with the measurement of price variation,” and he seized the opportunity to charge him for giving away “the case against the use of Probabilities when he admits that sampling plays a part in the determination of index-numbers” (559). The fundamental difference we may identify at this point is that, unlike Edgeworth, Walsh does not understand sampling as a part of the theory of index numbers, in the sense that, for Edgeworth, it was through sampling that probabilities played a distinctive role in the theory of index numbers (573).

Furthermore, Edgeworth (1925c: 379) now offered a clear statement of his definition of an index number: “A number adapted by its variations to indicate the increase or decrease of a magnitude not susceptible of accurate measurement.” This definition is much wider in scope than Walsh's, which was restricted to the measurement of the variation of the exchange-value of money. Consequently, Edgeworth (1925c) once again defended the position that index numbers should rely on as many formulas and methods as there are purposes, including, for example, the measurement of variations in the cost of living of different social groups, in welfare, and in labor income. In this regard, which closed the controversy, Edgeworth did not waste the opportunity to isolate Walsh's approach from those of the other students of index numbers, lending Walsh an aura of unrivaled eccentricity in the field: “The uncertainty which the nature of things has attached to estimates of utility and probability is intolerable to Mr. Correa Walsh. . . . Not content to walk with the rest of the world in the twilight of Probability, he cherishes a light visible only to himself” (Edgeworth 1925a: 575).

Walsh did not respond.

In 1926, however, and therefore after Edgeworth's death, in The Four Kinds of Economic Value, Walsh partially gave in on the issue of probabilities. He retained, nonetheless, the idea that exchange-value should be the benchmark for the stability of money, that weighting was a necessary feature of the making of index numbers, and that one best formula had to be suited to all purposes. In The Problem of Estimation, he had already declared that “the calculus of probabilities should not be invoked—until it is needed” (Walsh 1921b: 110). In The Four Kinds of Economic Value, in a brief and unelaborated remark, Walsh went one step further and admitted the usefulness of probabilities but only after checkmarking all the criteria he had previously defended—that is, provided that data be accurately gathered in a sufficient quantity and that the mathematical construction of the index number be sound. “When this has been done, the mathematical theory of probabilities may be invoked to calculate what is the probability of error in the result; for it is well known that no observation, no measurement, made by man, ever is absolutely correct” (Walsh 1926: 61).

5. Reactions, Underpinnings, and the Aftermath of the Debate

Keynes's perspective on the making and application of index numbers did not coincide with Walsh's or Edgeworth's. On the one hand, in A Treatise on Money, he criticized Fisher's—and as a consequence Walsh's—approach to index numbers on the grounds that their method takes the crossing of formulas further than it is appropriate or advisable. For Keynes, their ideal formula ignores “the nature and degree of error which is involved” in the measurement of price variations and allows for “a numerical comparison between any two price levels whatever—regardless of whether tastes have changed and of everything else”—a great fault, because when tastes and environment are not constant, the indexes of Paasche and Laspeyres cannot stand as appropriate limiting values (Keynes [1930] 1971: 100–102).

On the other hand, nonetheless, in his direct assessment of the Walsh-Edgeworth debate, also in A Treatise on Money, Keynes issued a verdict favoring Walsh. In Keynes's ([1930] 1971: 72) view, Edgeworth, Jevons, and Bowley chased a “mythical creature” named the “intrinsic value of money” or “value of money as such”—the utility level affordable by a given amount of money—whereas they should be, like “the Americans,” concerned with the purchasing power of money. The main flaw in the British reasoning, for Keynes, resided in the fact that prices did not fluctuate randomly, as assumed by Edgeworth, and thus were not mutually independent (76). The aim should not be, therefore, the isolation of a general cause for price variations from random movements in the prices of individual commodities. Keynes then praised Walsh as the one who actually condemned the hunters of this “mythical creature” and declared that “in the long controversy waged between Mr. Walsh and Edgeworth . . . on the appropriateness of the application to price index numbers of certain ideas drawn from the calculus of probabilities, I am, in main substance, on the side of the former” (72).

Frisch also wrote about the approaches to index numbers from the 1920s—although not about the Walsh-Edgeworth debate per se. He classified Edgeworth's approach to index numbers as stochastic, whereas Walsh's conceptions may be understood as belonging to the test/axiomatic approach to index numbers (Frisch 1936: 3–4). In Frisch's conception, Edgeworth's stochastic approach referred to the identification and isolation of general changes in the price level—the inverse of the value of money—from the individual prices of commodities, whereas the axiomatic approach, concerned as it was with a fixed-goods consumption basket, consisted in the formulation of formal tests to indicate the most appropriate formula. “Economists calculating a fixed-goods index sought to measure the changing of a collection of goods; the stochastic approach sought to separate a universal cause of price changes (the changing supply of gold) from individual prices” (Stapleford 2003: 90). Edgeworth's stochastic approach, therefore, looked for the causes of changes in prices (Aldrich 1992: 658); Walsh's axiomatic approach, on the other hand, had no regard whatsoever for “the causes of their [prices'] constancy or of their variations” (Walsh 1901: 22).

Nonetheless, in Frisch's (1936: 7) view, neither Edgeworth nor Walsh would “lead to one particular formula that may be taken as the definition of the price level”¹⁵—Edgeworth, because his method, stochastic as it is, prevents “‘exact’ statements like those we make about other magnitudes in an economic theoretical scheme” and because it—erroneously—assumes that monetary movements would manifest themselves proportionally in price changes (4–5); Walsh, because no formula is capable of conforming to all of his tests simultaneously (7).

From these acknowledgments, we may identify the roots of the Walsh-Edgeworth debate in the divergent perspectives held by them on the very nature of index numbers. This is what anchors the debate. For Walsh, a perfect method was attainable, and index-number theorists should strive for determinacy. For Edgeworth, determinacy was not possible, and such impossibility demanded the use of probabilities and a plurality of approaches.

In fact, Edgeworth held a pluralistic view of index numbers and economics in general. In 1925, he wrote, “It would be too much to ask economists, what Cromwell asked theologians, to think it possible that they might be mistaken. Each maker of index numbers is free to retain his conviction that his own plan is the very best. I only ask him to think it possible that others may not be entirely mistaken” (Edgeworth 1925c: 388). This notion was also present in his 1887 memorandum: “The path which we have to investigate has many bifurcations. To decide at each turn which is the right direction is either impossible or at least presumptuous” (Edgeworth 1925b: 206).

Within this pluralistic frame of mind, nonetheless, Edgeworth had preferences that oriented his own approach. No perfect method was conceivable, for him. He also favored monetary stability in terms of utility and reserved a special role for probabilities, because the “theory of Probabilities lends to Economics, as to other sciences, certain premises which are evidenced, neither by pure intuition, nor by formal induction, but by general impressions, and what may be called mathematical common sense” (Edgeworth 1925c: 388). Like Bowley and Yule, Edgeworth was determined to show the usefulness of probability theory in generalizing from statistical data. In fact, the failure to acknowledge the undisputable role for advanced mathematical methods in economic analysis was among Edgeworth's standing grievances against his fellow economists. For instance, in their own debate regarding tax incidence on monopolies, Edgeworth had offered a similar complaint about Edwin Seligman (see Moss 2003).

For Bowley (1934: 117–18), therefore, Edgeworth showed a refined notion on both sides of the controversy. On the technical, statistical side, he examined the merits of various types of averages, the little relevance of weights, and the law of errors. On the theoretical side, he offered a careful and systematic treatment of the purposes involved in the making of index numbers, of the quantities appropriate for each purpose, of utility as a benchmark for the stability of money, and of the relation between these issues and currency problems.

In this sense, Edgeworth's exercise relates to what he identified as hedonimetry, which aimed at measuring utility (Edgeworth 1891: 98–102). Edgeworth recognized the difficulties related to this enterprise—such as the comparison of interpersonal units of pleasure—but deemed its development a necessary step to the accomplishment of a mathematical reasoning in economics. Ideally, he conceived of a “perfect instrument, a psychophysical machine, continually registering the height of pleasure experienced by an individual, exactly according to the verdict of consciousness, or rather diverging therefrom to a law of errors,” in a way that a “delicate index now flickering with the flutter of the passions” might be formed (101). Naturally, such an instrument is—as he knew it to be—unattainable.¹⁶ A “less perfect instrument” was then needed, an instrument that, following the theory of probabilities, relied on averaging and on a large number of observations (102). Index numbers come to mind—even though index numbers are altogether absent from this specific discussion of his. Edgeworth's (1894) agreement with Marshall that utility (or a proxy for it) could be measured in terms of money was, henceforth, a natural development—one that brought about Edgeworth's shift from the defense of cardinal utilities to ordinal utilities (see also Colander 2007; Mueller 2020).

Walsh, on the other hand, sought a deterministic method capable of accurately enacting the single purpose of index numbers, that is, the measurement of variations in the exchange-value or purchasing power of money. In this, Walsh followed the deterministic worldview that was still dominant in economics in the first decades of the twentieth century (Biddle and Boumans 2021: 13). Accordingly, if the systematic treatment of purposes was not one of Walsh's concerns on the theoretical side of the controversy, this quarrel reinforces Walsh's quest for precision, which is manifest in his intransigent defense of the most appropriate formula for index numbers.

Walsh was in principle an axiomatic index-number theorist, in the sense that he privileged tests in the choice of the best formula. He resigned himself, however, to what Boumans (2005) defined as an instrumental approach, because the “best instrument sometimes has to be a compromise between incompatible requirements” (149) while the perfect formula could not be singled out through tests designed for this purpose. Fisher, as did Walsh, allowing for the impossibility of a formula to checkmark all of his tests, also recognized this need for a compromise. Therefore, in the theoretical corner of the debates on index numbers, both of them aimed at dealing with prices as defined in Fisher's equation of exchange.

Walsh would rather mostly reject the rising field of probabilities, which worked with levels of confidence instead of with mathematical determinisms, than recognize that his approach generated—or could generate—anything other than the best method, that is, the axiomatically proven correct method.

In consequence, Walsh identified his proposed exercise as axiometry, which was “the most important field in which index numbers can be used” (Walsh 1932: 652) and aimed at “the measurement of general exchange-value, especially of money” (Walsh 1921b: 69–81). It concerns “the deviations of prices at a later period from what they were at an earlier period, when they had a level which it is desirable to maintain—and it is desirable to maintain any price-level that is more or less settled” (69). Here, Walsh is talking about the general price level, whereas individual prices must be allowed to fluctuate according to supply and demand. Price levels, however, are not single objects, and one price level “exists only in comparison with another level of prices at another period” (70)—an ordinal perspective. For that reason, Walsh clearly stated that, in axiometry, “what we really do is to average the variations of prices,” or, put differently, the variations of the purchasing power of money (70).

Be that as it may, this lively debate marks the last record of Edgeworth's and Walsh's engagements in scholarly disputes. Walsh is not wrong in defining their standpoints as diametrically opposed. In fact, Edgeworth's hedonimetry and Walsh's axiometry are in opposition to each other in almost every aspect of the theory of index numbers. It is, therefore, curious to realize that, despite the feud between them, the approaches of Edgeworth and Walsh stemmed from a shared Jevonian background (see Stapleford 2003: 106–7). Walsh, on the one hand, kept the Jevonian flame burning with respect to the description of the “‘value’ of gold in terms of exchange ratios” and in using the geometric mean. Edgeworth, on the other, carried on Jevons's analysis regarding “how changes in the supply of gold affected prices.”

At last, in 1926, Walsh stated that mathematical economists—such as himself—had already worked the “method of measuring the variation or constancy of the general exchange-value of money on the supposition that prices and quantities are given.” The statistician, for him, should rather be concerned exclusively with collecting relevant data (Walsh 1926: 60). Yet again, Edgeworth was anything but a simple data collector: he was a philosopher of statistics rather than a mere practitioner (Bowley 1934: 116). For Edgeworth, “the task of the statistician is to study the techniques that permit a systematic comparison of means and to evaluate whether differences in figures are also differences in facts” (Edgeworth, quoted in Baccini 2007: 92–93).

In any case, after their success in the 1920s, the axiomatic and the stochastic approaches ended up overshadowed by the so-called economic approach to index numbers from the 1930s onward (Cruz-e-Silva and Almeida 2022: 681). Curiously, Fisher's ideal index number for prices and quantities would be revived in the 1970s as “the best of the superlative indexes to use in empirical applications,” even in cases where utility functions are taken as aggregator functions (Diewert 1978: 891; see also Diewert 1976; for an opposing view, see Samuelson and Swamy 1974). Once again, however, no consensus around this proposition emerged.¹⁷

6. Concluding Remarks

The matter of index numbers was brought to life in the 1920s in the intersection between economics and statistics. Recalling once again Frisch's (1936: 1) definition, it was therefore a problem “as much of economic theory as of statistical technique.” Generally speaking, Frisch was right. However, in the case of the Walsh-Edgeworth debate, it might be said that the debate was determined more by economic theory than by statistical technique. In their discussions, Walsh and Edgeworth dwelled upon technical issues, such as the use of probabilities and weighting, and theoretical issues, such as the very definition of an index number and the purpose of its use. And it was this latter group of considerations that determined the former.

Accordingly, the Walsh-Edgeworth debate may be boiled down to their dissonant perspectives on the very nature of index numbers. For Edgeworth, whose approach to index numbers is more holistic, sampling is an inherent part of the making of index numbers, so probabilities are important; for Walsh, making index numbers is one thing, which relates to the mathematical search for the most accurate method, whereas sampling data is something else entirely, so probabilities should not be on the agenda of the index-number theorist. Thus, both Walsh and Edgeworth recognized a role for probabilities in sampling, but only Edgeworth understood sampling as a part of the theory of index numbers.

For Edgeworth, the making of index numbers relates to multiple objectives, so different formulas are necessary; for Walsh, index numbers refer exclusively to the measurement of variations in the exchange-value of money, to the extent that one formula simply has to be the best one. For Edgeworth, prices cannot be intertemporally compared unless utility is factored in; for Walsh, exchange-value should be the benchmark for the stability of money.

Finally, for Edgeworth, the statistician, such as himself, ought to be the main character in thinking through the design of index numbers; for Walsh, au contraire, statisticians should concern themselves simply with the less elevated yet necessary task of collecting data following the dictum of mathematical economists like Walsh himself.

The author would like to thank Felipe Almeida, Rogério Arthmar, and Bert Balk for their comments on previous versions of this article. This appreciation is extended to the audiences at the Fiftieth Brazilian Economics Meeting, hosted by ANPEC, and the Ninth Latin American Conference on the History of Economic Thought, hosted by ALAHPE, who gave valuable feedback on this research. The author is also grateful for insightful remarks by Kevin Hoover, Thomas Stapleford, and an anonymous referee during the editorial process, which significantly improved the arguments presented in the article. Finally, thanks are due to Paul Dudenhefer for his careful reading of the paper and for helping to convey the intended message.

Notes

References