An inquiry into the Ramsey-Hotelling connection

Abstract

Ramsey’s 1928 paper on saving and Hotelling’s 1931 article on exhaustible resources are considered to be two seminal contributions in economic dynamics. They have been associated because of their temporal proximity, use of the calculus of variations, and because of Hotelling’s citation of Ramsey. This connection however needs to be precisely investigated and characterized. On the basis of archival material, this paper shows that, on the interpersonal and theoretical ground, the connection is quite thin, but that significant parallels are found in Ramsey’s and Hotelling’s expectations with mathematical economics for the progress of science and for informing public decision.

Keywords:

• Ramsey
• Hotelling
• mathematical economics
• dynamics
• intertemporal distribution
• saving
• exhaustible resources
• history of economic thought

JEL codes:

• B2
• C02
• O4
• Q3

Acknowledgements

This paper benefited from the support of the European Society for the History of Economic Thought (ESHET), through the project ‘Bifurcations in Natural Resources Economics (1920s–1930s)’. We thank R. Ferreira da Cunha, M. P. V. Franco, T. M. Mueller and F. Nadaud for stimulating discussions in the working-group. Thanks also to the participants in the 2016 ESHET Conference in Paris and in the 2018 Environmental Economics Conference in Orléans, and to the readers of previous versions of this paper. We are grateful to the staff of the Rare Book and Manuscript Library of Columbia University for their help during our archival visit in Hotelling papers, and to the staff of the Special Collections Library of the University of Adelaide. We finally thank the reviewers for their suggestions to improve the paper.

Archives

FPRP: Frank P. Ramsey Papers, Archives and Special Collections, University of Pittsburgh, USA. Catalogue: https://www.library.pitt.edu/archives-special-collections

HHP: Harold Hotelling Papers, Rare Book and Manuscript Library, Columbia University, USA. Catalogue: http://findingaids.cul.columbia.edu/ead/nnc-rb/ldpd_4078401

JMKP: John Maynard Keynes Papers, King’s College Archive Centre, Cambridge University, UK. Catalogue: https://janus.lib.cam.ac.uk/db/node.xsp?id=EAD/GBR/0272/PP/JMK

RAFP: Ronald A. Fisher Papers, Special Collections Library, University of Adelaide, Australia. Catalogue: https://digital.library.adelaide.edu.au/dspace/handle/2440/3860

Notes

1 The calculus of variations is a technique of integrals optimization, coming from pure mathematics and physics. After some developments from the seventeenth to the nineteenth centuries, mathematicians particularly investigated the calculus of variations in the beginning of the twentieth century (Bolza 1904; Hadamard 1910; Hancock 1904; Hilbert 1900; Kneser 1900; Mayer 1905). See also Goldstine 1980.

2 See also Darnell 1988, 60; Duarte 2009b, 163f.

3 Letter from Hotelling to Fisher, July 23rd, 1930 (RAFP, digitized version online).

4 Ramsey’s biography is well documented (Duarte 2009a; 2016; Paul 2012; Sahlin 1990; Taylor 2006).

5 Margaret Paul, Ramsey’s sister, relates (2012, 47) that Ramsey assisted to his father’s 1918 lectures on dynamics, and that he read “Besant and Ramsey, dynamics” in January 1920 (probably Arthur’s 1913 Treatise on Hydromechanics, part II). This textbook adopts a pure theoretical point of view and presents Euler’s and Laplace’s methods to deal with problems of fluid dynamics in various conditions.

6 Ramsey listened to Russell’s lectures in Cambridge and London in 1920–1921, and had occasions to discuss some elements of the Principia Mathematica with him (Paul 2012, 81, 111, 155).

7 Mellor (1990, xiii) considers “The Foundations of Mathematics” (Ramsey, 1925a) as “the culmination of the logicist programme”. On Ramsey and logicism, see also Sahlin 1990.

  8 Logical atomism, as developed by Russell (1914) and Wittgenstein (1922), focused on the conditions of knowledge. It intended to warrant that knowledge was based on logical compositions of so-called atomic (or elementary) propositions. Atomic propositions were supposed to express atomic facts, described as ultimate and immediate elements of knowledge, independent from each other. Atomic propositions were true or false, by virtue of the truth of the fact they expressed.

  9 The manuscript of this on-going work may be consulted in Ramsey papers (digitized version online).

10 Ramsey states (1919, quoted by Paul 2012, 67): “The destruction of the wage fund theory is comparable to Galileo’s discovery of acceleration. Other parts are like pure mathematics; the question of whether utility or cost of production governs value though extraordinary interesting, seems no more useful than the theory of infinite series. But no doubt the invention of Cartesian Coordinates seemed equally useless at first”.

11 According to the index cards of the John Maynard Keynes’s Papers (JMKP), no editorial correspondence survived on Ramsey’s paper on taxation (1927b). In contrast, a set of letters, dated July 1928 (JMKP, Box EJ, Folder 1–3), testifies that Keynes carefully read Ramsey’s paper on optimal saving. He asked for clarification, offered an intuitive formulation of the optimal saving rule in the context of zero discounting, and discussed Ramsey’s assumption of a constant discount rate. Ramsey changed the manuscript, following the requirements, and mentioning Keynes’ intuitive result in main text. He however considered that introducing a variable discount rate would bring only “manipulative interest” (Ramsey in Keynes 1983, 788). Both sides of the correspondence may be found in Keynes (1983, 784–789).

12 Duarte (2009a) extensively discusses the possible impact of Ramsey’s results on Pigou’s work.

13 When studying the optimal national saving rule, Ramsey defines discounting as an “ethically indefensible” practice, arising “from the poverty of our imagination” (Ramsey 1928, 543).

14 This section has been published by Duarte (see Ramsey 2009).

15 Hotelling’s biography can be found in Darnell (1988, 1990), with information mostly coming from autobiographical material (Hotelling 1948, 1963). See also Levene 1974; Smith 1978.

16 In his autobiography, he even indicates that he studied mathematics to improve his abilities in economics (1948, 17, 20). See also Darnell (1990, 4).

17 Crabbé (1986, 2) and Darnell (1990, 4f) suggest that Hotelling could have heard about the calculus of variations during a summer school in Chicago in 1920. Crabbé (1986, 3) also reports a letter to Coffey (dated Sept. 11th, 1964) in which Hotelling indicates that he learnt “advanced portions of the subject […] in J. Hadamard, Calculus of Variations”.

18 Some generalized results in topology were published a few months later (Hotelling 1926a).

19 Letter from Alsberg to Hotelling, March 3rd, 1924 (HHP, Box 6, Folder ‘Wallis-Fry’).

20 Monthly report from Hotelling to the directors of the Food Research Institute, April 3rd, 1925 (HHP, Box 41, Folder ‘Agriculture III’).

21 In the 1920s, Evans, who contributed to functional analysis (Morrey 1983), had the project to improve the theory of production, through the introduction of price variations in demand functions (Evans 1922, 1924, 1925). He was an advocator of the calculus of variations, whose relation with economics was considered as “not accidental” (1925, 94). On the mathematical tradition of economic analysis coming after Evans and focusing on the calculus of variations, see Pomini 2018.

22 Letter from Hotelling to Dresden, Dec. 11th, 1924 (HHP, Box 10, Folder ‘AMS Reports and Correspondence (3)’). Hotelling’s notes for the talk have also been preserved in the same folder.

23 Monthly report from Hotelling to the directors of the Food Research Institute, Dec. 1st, 1924 (HHP, Box 41, Folder ‘Agriculture III’). Hotelling talks about the calculus of variations as an “abstruse branch of mathematics” probably because his report was intended for supervisors who were neither mathematicians nor physicists.

24 Evans mostly used the calculus of variations by considering demand functions as functions of current prices and current inflation. In the 1931 article, Hotelling introduces such demand representations at the end of the paper (1931a, 162 and following).

25 Before publication in the Economic Journal in 1929, a version was presented to the AMS in 1928.

26 During the publishing process, Hotelling got in touch with Keynes, who also alerted him on Sraffa’s on-going works (Darnell 1990, 14). Keynes’ letter to Hotelling (dated Aug. 7th, 1928) is available both in Keynes papers (Cambridge University) and in Hotelling papers (HHP, Box 42, Folder ‘Calculations Relating to: “Differential…”’).

27 Letter from Hotelling to Van Sickle, April 22nd, 1929 (HHP, Box 1, Folder ‘Van Sickle, John V.’).

28 Hotelling also tried to invite Fisher in Stanford in 1927–28, but faced administrative obstacles (Stigler 1999, 264).

29 Fisher joined Rothamsted in 1919. See Mahalanobis 1964; Fisher Box 1978.

30 Letter from Hotelling to Wildman, April 22nd, 1929 (HHP, Box 51, Folder ‘Finances and Personal’).

31 Letter from Hotelling to Fisher, April 28th, 1929 (HHP, Box 14, Folder “IMS Draft for National Roster”).

32 Letter from Hotelling to Van Sickle, April 5th, 1929 (HHP, Box 1, Folder ‘Van Sickle, John V.’).

33 From the 1920s onwards, Hotelling kept bibliographical notes on statistics and probability. None of Ramsey’s works are mentioned in these notes. By contrast, Ramsey’s 1928 paper on saving was reviewed by Hotelling for “Social science abstracts” (HHP, Box 25, Folder ‘Causes of Birth Rate Fluctuations’).

34 Letter from Hotelling to Fisher, May 20th, 1929 (HHP, Box 14, Folder ‘IMS Draft for National Roster’).

35 Two letters from Hotelling to Fisher and Chamberlin (dated July 23rd, 1930 and August 8th, 1930) indicate that he was working on the final version of the paper in the summer of 1930 (RAFP, digitized version online; Guicherd 2017, 179). A letter from Mills (dated Jan. 17th, 1931) shows that the final version already circulated at the very end of 1930 (HHP, Box 1, Folder ‘Mills, Frederick C.’).

36 The expression “Hotelling rule” was popularized by Solow in 1974 in his article “The Economics of Resources or the Resources of Economics”.

37 In fact, Hotelling’s rule was not such a decisive innovation in the history of environmental economics. Lewis C. Gray (1913, 1914) and Gustav Cassel (1918) had already emphasized the same intuitive principle some years before (see Missemer 2017, chap. 4; Robinson 1989).

38 Hotelling’s interest in Ramsey’s article was however partial, even on these points, since “total utility”, “better called the social value of the resources” (1931a, 144) was not perceived, like in Ramsey, as the representation of the collective (or national) preferences. This total utility rather consisted of “concrete quantities, not symbols for pleasure” (145). This means that Hotelling’s representation of social utility referred to consumers’ surplus (through aggregate demand curves) rather than to a collective intertemporal utility function.

39 For instance on taxation, his study of Edgeworth’s paradox started in 1926 (HHP, Box 39, Folder ‘Misc. (5)’).

40 For instance quantum theory challenged the habit of explaining the world with continuous and differentiable functions, and the theory of relativity called for rethinking the notions of space and time.

41 Weintraub (2002, 9–10) retains three causes for this crisis: the difficult integration of non-Euclidian geometry; the failures of the set theory; and the paradoxes in the foundations of arithmetic and logic, associated with Frege and Peano. Russell’s logicism was a response to these paradoxes. It was opposed to the formalist (Hilbert) and the intuitionist (Brouwer) re-foundation programs. On these competing programs, see also Grattan-Guinness 2000.

42 Ramsey borrows the expression from Charles Sanders Peirce.

43 Paul states (2012, 11): “F. P. Ramsey said that applied mathematics had afforded no stimulus to advance since the middle of eighteenth century. All the recent advances in mathematical physics had been made by pure mathematicians. With regard to applied mathematics as a subject of study, he was of the opinion that it would be better to do away with it in the university, as it was merely a collection of standardized puzzles.” The Tripos in mathematics was the core of final exams in Cambridge. The first part of the exam, common to all, basically consisted in Euclidian Geometry, arithmetic and classical tools of applied mechanics (restricted to static problems). Students’ skills were tested through endless problems combining these methods. Weintraub (2002, chap. 1) shows how the Tripos in mathematics shaped the minds and the teaching of Cambridge’s elites until the 1920s, making Cambridge science quite impervious to other kinds of mathematics.

44 Reminding about Ramsey’s lectures, Elphinson (1990, quoted by Paul 2012, 187) indicates: “In the 1920s, few schoolboys were taught this subject properly. Of course, we knew how to differentiate and understood the use of this operation of differentiation for discovering the properties and shapes of plane curves […]. But in a very few schools was the subject dealt with in any sort of rigorous style. Ramsey introduced us to that magic number e and to rigorous logical methods in analysis […]. The differential geometry course showed me for the first time that we could really use them constructively to clarify a subject, rather than obscure it by treating it, in effect, as if in a foreign language”.

45 Ramsey and Hotelling seem to have occupied a peculiar position in the period between the end of the nineteenth century, when economists looked for a “mechanical analogy” (Ingrao and Israel 1990) and “physical-model-based analysis” (Weintraub 2002), and the middle of the twentieth century, when they opted for a “mathematical analogy” (Ingrao and Israel 1990) and “mathematical-model-based analysis” (Weintraub 2002). In a sense they were two pioneers of the “mathematical-model-based analysis”, two decades before its development. The fact that Hotelling trained several major post-war economists (Arrow being a typical case) is a signal of this pioneering position. For extensive works on the history of mathematical economics, see also Israel 1996; Mirowski 1991.

46 Hardy’s essay was much discussed in the 1950s and 1960s, and occupies an important place in the history of mathematics in the middle of the twentieth century (see Mordell 1970).

47 For instance, Ramsey’s inquiry shows that the only way to reach “Bliss” and to maximize intertemporal national welfare is to apply the rule with no discount for future utilities and a planning framework. The introduction of a positive discount rate modifies the optimal path of saving, with a “modified Bliss” path (554), where national utility is lower than previously. Finally, in a decentralized economy, the existence of different discount rates leads to divergent familial trajectories inside the economy, condemning in particular families with a too high discount rate to eternal poverty (559). The existence of different discount rates thus generates inequalities of income and welfare in each generation.

48 Hotelling establishes that a private monopoly would be the appropriate structure for the conservation of resources, but that competition would bring a higher intertemporal “surplus”. A public monopoly – or “a benevolent and all-wise state” (Hotelling 1936, 166) – targeting the “total utility” or “social value of the resource” (1931, 143) would adopt the same schedule of exploitation as producers under (perfect) competition, shortening the exhaustion time. An unexpected tax on the private monopoly rent, or a severance tax on private monopoly could both slow exploitation and lower prices.

49 Paul (2012) indicates that “[Ramsey] was interested in using economics as a tool for reform rather as a way of explaining why things are as they are” (260).

50 The fact that the calculus of variations was not the most important feature of the Ramsey-Hotelling connection is reinforced by the existence of another tradition after Evans, with Charles Roos and some Italian scholars (Amoroso, La Volpe), which precisely focused on the calculus of variations in the 1920s, 1930s and 1940s (see Pomini 2018).

51 Duarte (2009b) explains that Ramsey’s 1928 article was known in the 1930s, even if it failed to find a large readership until economists were ready to associate “mathematical formalism, utility maximizing agents and aggregate level control”. Hotelling was sometimes cited in the 1930s (Hart 1937; Smithies 1939), even if major developments came later (e.g. Gaffney 1967).