Journal of Economic Literature 2015, 53(2), 326–350 http://dx.doi.org/10.1257/jel.53.2.326

Revisiting Samuelson’s Foundations of Economic Analysis†

Roger E. Backhouse*

Paul Samuelson’s Foundations of Economic Analysis played a major role in defining how economic theory was undertaken for many years after the Second World War. This paper fills out Samuelson’s account of the book’s origins and corrects some details, making clear his debt to E. B. Wilson and establishes that turning the thesis into a book was a long process. The contents of the book and its reception are then reviewed. ( JEL A22, B20, B31)

1. Preliminaries decade witnessed the takeoff in the idea that mathematical modeling was central to 1.1 Samuelson and Modern Economics1 both economic theory and empirical work, two facets of economics that were increas- he formative period in the emergence ingly separated. Whereas some of the most Tof modern economics was the decade prominent theorists of the interwar period centered on the Second World War. This (e.g. Frank Knight, Jacob Viner, Allyn Young, John Maurice Clark, or Edward Chamberlin) were uncomfortable about * University of Birmingham. This research was sup- using what would now be considered basic ported by a three-year Leverhulme Trust Major Research Fellowship to write an intellectual biography of Paul Sam- mathematics, after the Second World War, uelson. I am grateful to the Trust for its support. This economic theory became progressively more article marks the centennial of Samuelson’s birth, on May mathematical. There was a similar transition 15, 1915. He died on December 13, 2009. I am grateful to Juan Carvajalino, Wade Hands, Steven Medema, Janice in empirical work, where there was a change Murray, Robert Solow, and Roy Weintraub and to anony- from the empirical work represented by mous referees for helpful comments on this paper. Wesley Mitchell and Gardner Means—data † Go to http://dx.doi.org/10.1257/jel.53.2.326 to visit the article page and view author disclosure statement(s). intensive, but involving no formal statisti- 1 EBWP—Edwin Bidwell Wilson Papers, Harvard cal inference—to the methods of modern University, HUG4878.203. Quotations are courtesy of econometrics, though limitations of the com- Harvard University Archives. HUP—Archives of Harvard University Press, Foundations of Economic Analysis folder. puting power available to economists held I thank Harvard University Press for providing access to back the spread of formal statistical model- the editorial files from which I quote with their permission. ing and inference for many years. Symbolic PASP—Paul A. Samuelson Papers, Rubenstein Rare Book of this change in economics was the rise to and Manuscript Library, Duke University (followed by Box number and folder name). prominence of the economics department at

326 Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 327 the Massachusetts Institute of Technology As a graduate student at Harvard, he was (MIT). A department of social science that ­considered a specialist in a field called in 1940 was still focused on providing service “­mathematical economics,” in which posts teaching for engineers became, by the 1960s, were so difficult to find that his teachers one of the most important economics gradu- feared that he might never land an academic ate schools in the world. Economics at MIT position. At MIT, his mathematical skills were was part of not just an engineering school, more highly valued because every student but the engineering school that had received in the institute was required to study math the lion’s share of wartime science funding and physics, but he moved very reluctantly and, with its massive research laboratories because he wanted to remain in Harvard’s had become one of the preeminent locations much stronger department. Even this oppor- of “big science.”2 tunity presented itself only because a vacancy One of the leading figures in this transition arose at short notice when staff were lost was the winner of the AEA’s first (1947) John through preparations for war (see Backhouse Bates Clark Medal, Paul Anthony Samuelson. 2014). In contrast, by 1948, Samuelson was a As Junior Fellow at Harvard from 1937 to full professor who was being sounded out by 1940, he wrote a series of papers that changed most of the leading departments, including the basis on which theoretical debates were Chicago, which came closer than any other conducted in several fields: consumer the- university to luring him away.4 He committed ory (revealed preference), the theory of himself to MIT, where he spent the rest of international trade (the Heckscher–Ohlin– his long career. According to President James Samuelson model), Keynesian economics, Killian, the success of MIT’s economics and the business cycle (the multiplier-accel- department was due to Samuelson and the erator model). He formalized the problem distinguished economists who found satisfac- of dynamic analysis in economics, bringing tion in being his colleagues.5 these ideas together, along with a codification Clearly, other works were important in the of welfare economics, in a book, Foundations rise of mathematical modeling in econom- of Economic Analysis (1947). For decades, ics, notably John Hicks’s Value and Capital this book was a core graduate microeco- (1939), which introduced many economists nomics textbook and, for many economists, to more formal general equilibrium analy- defined the way to do economic theory. For sis, and The Theory of Games and Economic example, in 1960, having read in one of his Behavior by John von Neumann and Oskar textbooks that it was “the most important Morgenstern (1944). However, though The book in economics since the war,” Robert Theory of Games and Economic Behavior E. Lucas (1995) taught himself economic was well known among mathematical econ- theory by reading its early chapters, apply- omists, game theory did not acquire its pres- ing the techniques he learned from it to the ent status in the discipline until much later, problems that Milton Friedman posed in his and when it did, it was the noncooperative Chicago price theory course.3 theory of John Nash to which most econo- Samuelson’s own career illustrates the mists turned, rather than to von Neumann’s way economics changed during this period. and Morgenstern’s model of coalition ­formation. And though Hicks might have

2 On economics at MIT, see the articles in Weintraub (2014a), in particular Cherrier (2014). 4 See Maas (2014). Harvard did not try because they 3 Friedman’s treatment of these problems was largely knew he would prefer to stay at MIT. intuitive and graphical. 5 Killian (1985). 328 Journal of Economic Literature, Vol. LIII (June 2015) been first, it was Samuelson’s Foundations, War and partly because of the anti-theory with its focus on providing the mathematical prejudices of Harold Burbank, the Head of and conceptual tools with which comparative Harvard’s economics department, on whose statics results could be generated, that came desk it sat gathering dust before being sent to define the way most economic theorizing to Harvard University Press. was done. Samuelson’s best-selling elemen- This paper reexamines the origins tary textbook (1948), the increasing promi- of Foundations using evidence from nence of the MIT economics department, Samuelson’s papers (archived at Duke his prominent involvement in debates over University) and the archives of the public policy, and a succession of widely cited Harvard Economics Department and theoretical innovations all served to confirm Harvard University Press. Samuelson’s own his dominance of economics worldwide, and account remains indispensable, but needs with it the status of Foundations.6 In consid- ­reexamination. As people reflect on past ering the origins of Foundations, we come events, points that seem significant in view closer to seeing the origins of modern, tech- of their later interpretation of their own lives nical economics than we do by looking at any get elaborated and modified so that their other work. stories become more coherent (see Tribe 1997, pp. 6–9; Weintraub 2007). Samuelson 1.2 “How Foundations Came to Be” was no exception to this rule. He was a great Given the importance of the book, it is raconteur who was forever recalling events not surprising that when Samuelson offered from his own past. The inevitable result was “How Foundations Came to Be,” an abridged that the details he recalled most clearly were version of a paper first published in German those that fit his own perception of his life. (1997), John Macmillan immediately A good illustration is provided by the accepted it for the JEL, telling Samuelson, account of the publication process of “How one of the great economics books Foundations in “How Foundations Came to came to be written makes a wonderful JEL Be.” Late in life, Samuelson paid much more topic.”7 In this article, Samuelson reflected attention than he had earlier to the antisem- on the book’s origins as a Harvard PhD the- itism that pervaded Harvard in the 1930s, sis (1940) and the reasons for the long delay department chair Harold Burbank heading a between his first having the main ideas and “roll of dishonor” he sent to his friend Henry the book’s eventual publication. The thesis Rosovsky in 1989.8 That animosity could was not written until 1940 because he was explain why he blamed Burbank for holding precluded from working towards a higher up publication of Foundations, for trying to degree by the conditions of his junior fel- limit the print run to only 500 copies, and for lowship at Harvard; publication was then destroying the type after the first print run. delayed, partly because of the Second World He also remembered Burbank as critical of him as a young theorist, writing, “Harvard’s long-time department chairman [Harold 6 These include the consumption-loan model Burbank] was no admirer of me; long before (Samuelson 1958), the pure theory of public goods he had counseled me against working in eco- (Samuelson 1954), turnpike theory (Dorfman, Samuelson, and Solow 1958), the use of the Phillips curve (Samuelson nomic theory before I had reached (his) ripe and Solow 1960), and the theory of efficient markets old age of 50+” (Samuelson 1998, p. 1378). (Samuelson 1965a). 7 McMillan J, January 26, 1998, Letter to Samuelson, AEA Papers, Duke University 2004–0007, Box 3 8 Samuelson, September 26, 1989, Letter to Henry (Samuelson). Rosovsky, PASP Box 63 (Rosovsky). Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 329

Later, in the midst of a published denuncia- price.9 When the type was killed, the war tion of his teachers’ antisemitism, Samuelson had ended and it was assumed that the War (Samuelson 2002, p. 51) recalled Burbank’s Production Board would soon revoke con- advice and his own response to that advice:, trols on the use of paper and it would be easy to scale up production quickly.10 Burbank’s “Samuelson, you are narrow. Keynes and ­antisemitism had affected Samuelson whilst Hawtrey are narrow. Don’t take up economic theory until after you are fifty. This is what he was a student, and there were ample our great Allyn Young used to say.” Alas, I had grounds for resentment that could have already lost my heart, and aspired to become affected his later memory, but the delay in even more narrow; and furthermore, Young the publication of Foundations can be fully had died young, just before his rendezvous explained by production difficulties.11 with greatness. . . . I was always a young man in a hurry. In addition, because we are effectively tracing the roots of modern economics back The now-puzzling ageism-in-reverse to the inter-war period, there is also a need reported here reflects the different, non- to tie down the intellectual climate out of mathematical conception of economic the- which Foundations arose. Samuelson was ory then prevailing, at least in Burbank’s extremely generous in giving credit to others, mind. Theory was not thought something notably his teacher and mentor, E. B. Wilson, that required mathematical agility, but the with whom he remained in lifelong contact, mature reflection and experience of an older but he was often vague about what people scholar. Samuelson was in a hurry because, contributed to Foundations. This may be having been diagnosed as having high blood the result of a certain defensiveness in “How pressure, he expected to die young like his Foundations Came to Be,” for one of his father. goals in that article was to contest the history In contrast, the archival record shows that, of dynamics provided in Roy Weintraub’s while the book did sit on Burbank’s desk for Stabilizing Dynamics (1991), about which he a few months (presumably, as Department had been upset for several years.12 Though Chair, he had a responsibility to review it this chapter sought to place Foundations before sending it to the publishers), it is in the intellectual climate in which it was clear that the timing of the book’s publica- ­written, Samuelson took him to be suggesting tion was determined by the availability of that there was “some mystery, and maybe even typesetting capacity (not all printers could some cover up” (Samuelson 1998, p. 1381). handle the complex mathematics involved), Thus, Samuelson (1998, p. 1383) stressed the and ultimately by the availability of paper extent to which he was self-taught through (a rationed commodity in wartime). It also establishes that negotiations with Harvard University Press were handled by Abbott 9 Wilson, C.W., April 18, 1947, Letter to Alfred V. Jules, Usher, not Burbank, that the initial order was HUP; Jules, A.V., June 18, 1947, Letter to C. W. Wilson, HUP. for 1,200 copies (it is a mystery why only 887 10 Wilson, C.W., October 2, 1947, Letter to Alfred V. were eventually shipped, the likely expla- Jules, HUP. Email from Michael Aronson to the author, nation being the availability of paper), and April 9, 2012. 11 The issue of antisemitism is discussed in more detail that the type was “killed” at the instigation in Weintraub (2014b) and Backhouse (2014). of the printers, who pleaded with Harvard 12 Weintraub says that this was reported to him by University Press that the department had no Robert Summers (Samuelson’s brother) and Lawrence Klein (Samuelson’s first PhD student). The original ver- understanding of the shortage of type metal, sion of the JEL article, published in German, discussed then selling for almost double its normal Weintraub at length. 330 Journal of Economic Literature, Vol. LIII (June 2015) hours spent in the Widener Library, asserting explained that calculus was a prerequisite that he found nothing of interest in Lawrence for a proper understanding of indifference Henderson’s sociology, and recalling that curves and production possibility frontiers, when he tried to discuss Willard Gibbs with though admitting that he did not understand him, Henderson preferred to enumerate the it himself (Samuelson 1972b, p. 911). Having failings of Franklin Roosevelt. There is no taken Elementary Mathematical Analysis at reason to doubt this, but his remark hardly the end of his junior year, Samuelson appears addresses the question of whether three years to have taken as much mathematics as he of weekly dinners at the Society of Fellows, could in his senior year, with courses in sta- dominated by Henderson, may, directly or tistics, algebra, plane analytic geometry, and indirectly, have left more of a mark on his calculus. After graduating, he took a summer thinking than Samuelson remembered. course in differential equations. For these and other reasons, then, it is However, though it was at Chicago that he instructive to revisit the process whereby had decided to become an economist and that Foundations came to be. he had realized the importance of mathemat- ics, the most important part of Samuelson’s education took place at Harvard, where he 2. Graduate Study at Harvard took courses from 1935–37, before becom- ing a junior fellow, turning out paper after 2.1 Chicago and Harvard paper, from 1937–1940. By the time his the- Samuelson, who was born a century ago sis was submitted, he had published twenty as of May 15, 2015, began his account of articles covering consumer theory, capital the origins of Foundations by telling sto- theory, international trade, unemployment, ries about his childhood, speculating on the and business cycle theory and was widely link between being precocious and having considered to be the leading young economic “ridiculous” levels of self-confidence, and his theorist. He was using mathematical tech- undergraduate education at Chicago. Aaron niques that left most of his teachers behind. Director introduced him to economics with a Though he acknowledged the importance very traditional economic principles course, of Joseph Schumpeter, Wassily Leontief, but one in which he drew attention to the Gottfried Haberler, and Alvin Hansen in equations for general equilibrium in Gustav teaching him modern economic analysis (see Cassel’s Theory of Social Economy (1923). In also Samuelson 1976, p. 27), he made it clear his final year, Samuelson took Jacob Viner’s that the main influence on Foundations was graduate price theory course in which, to Edwin Bidwell Wilson. He took Wilson’s the delight of the graduate students—many courses in mathematical economics and of whom lived in terror of Viner’s socratic mathematical statistics, and spent hours method combined with a three-mistakes- talking to him after lectures (Samuelson and-you-are-out rule—Samuelson used to 1972a). In “How Foundations Came to Be,” point out mistakes on the blackboard. It was in he confined himself to saying that Wilson Chicago that Samuelson decided that learn- had vaccinated him to understand that eco- ing mathematics would help his economics, nomics could use the same mathematics as something that was far less obvious then physics without resting on the same empir- than it is today. A graduate student, George ical foundations and certainties (Samuelson Stigler, had exposed Samuelson to the joys of 1998, p. 1376). Wilson had, he claimed, large determinants, showing him notes on a contempt for social scientists who aped the course taught by Henry Schultz, and Viner Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 331 natural sciences “in a parrot-like way.”13 To important because Samuelson’s theory of find more detailed statements about what he revealed preference (1938b; 1939c) grew learned from Wilson, it is necessary to look out of Haberler’s theory of index numbers. elsewhere in his writings (e.g. 1947, p. 81; My own work in this direction grew out of a Samuelson 1972c, p. 254; Samuelson 1965b, remark made to me by Professor Haberler p. 438; Samuelson 1967 [CSP], pp. 655, 673) in his 1936 international trade seminar at Schumpeter acted as Samuelson’s mentor Harvard. “How do you know indifference for many years. He taught the main graduate curves are concave?” My quick retort was economic theory course, which Samuelson “Well, if they’re not, your whole theory of index numbers is worthless”. Later I got to later praised for the range of topics and thinking about the implications of this answer authors it covered (Samuelson 1951, p. 102). (disregarding the fact that it is not worded They became close and Schumpeter con- quite accurately). Being then full of Professor tinued to support him, commenting on his Leontief’s analysis of indifference curves, I papers and giving feedback on his first pre- suddenly realised that we could dispense with almost all notions of utility: starting from a few sentation to an AEA meeting (in a session logical axioms of demand consistency, I could Schumpeter chaired). Schumpeter was an derive the whole of the valid utility analysis as enthusiast for mathematical economics but corollaries (Samuelson 1950, pp. 369–70). his command of mathematics was weak. He inspired many students and it is pos- Samuelson’s theory of revealed preference sible to see Samuelson as having achieved resulted from seeing that Haberler’s own the- the breakthrough in mathematical econom- ory of index numbers could be used to show ics that Schumpeter sought but could not that Leontief, who used indifference curves achieve himself. It was Leontief, better in his trade theory, was right to assume that trained in mathematics and, by 1935, teach- indifference curves were concave. ing the course that Schumpeter had set up 2.2 Edwin Bidwell Wilson to improve the teaching of mathematical economics, who provided Samuelson with Edwin Bidwell Wilson is not well known a more rigorous training in formal eco- among economists, but his importance to nomic theory than Schumpeter was able to Samuelson and hence to Foundations can- provide. not be overstated. Wilson was one of the two Though Samuelson disparaged Edward protégés of the great American physicist, Chamberlin (placed just below Burbank Willard Gibbs, leading Samuelson repeat- in his roll of antisemitic dishonor), claim- edly to express his admiration for Wilson ing that at the age of thirty-five, he was by claiming Gibbs as his intellectual grand- already at the zenith of his scholarly career father (this analogy would, of course, imply (Samuelson 2004, p. 4), his publications that Irving Fisher, Gibbs’s other protégé, show that he thoroughly absorbed the con- was his uncle) (e.g. Samuelson 1969, p. 10). tent of Chamberlin’s Theory of Monopolistic Wilson had been trained in mathematics at Competition (1933), covered extensively in Yale, where he wrote a book on vector anal- a course Samuelson took from him in 1935 ysis based on Gibbs’s lectures, establishing and also covered in Schumpeter’s economic what has since become standard notation, theory course. Haberler, whom Samuelson and the Ecole normale supérieure in Paris. remembered much more positively, was “The so-called foundations of geometry” (Wilson 1903) took on the leading pure math- 13 This countered the claims of Mirowski (1989), whose ematician of his day, David Hilbert, criticizing work he considered completely mistaken. his use of set theory and logic. After a brief 332 Journal of Economic Literature, Vol. LIII (June 2015) spell at Yale, Wilson joined MIT as Professor years, and even chance remarks which of Mathematical Physics in 1907, where he you have let fall concerning Gibb’s [sic] wrote textbooks on calculus and aeronautics. ­thermodynamical systems have profoundly altered my views in corresponding fields of In 1922, he moved to Harvard as Professor of economics.16 Vital Statistics in Harvard’s School of Public Health, where his publications included arti- cles on confidence intervals in statistics. His For Wilson, the main things that economists contribution to the Economics Department’s had to learn from physics were how to build teaching was courses in mathematical eco- up an exact argument and how, when faced nomics and mathematical statistics that he with difficult factual problems, to apply taught in alternate years. In the course on existing mathematical tools and develop mathematical statistics taken by Samuelson new ones. After attending his lectures, in the spring of 1936, Wilson spent several Schumpeter wrote to Wilson, explaining why weeks on the first twelve pages of Arthur he found the lectures valuable. Bowley’s Mathematical Groundwork of I was strongly impressed with the immense Economics (1924), his reason for proceeding value to the economist of such lectures as you so slowly being that “some of our high-pow- gave in the first part of the course. I perfectly ered mathematical economists did not know agree with those who object to the practice of their fundamental definitions and would read some economists, simply to copy out what they right over a pair of statements that were con- believe is an economic argument from text- 14 books of pure mathematics or of theoretical tradictory and assume that both were right.” mechanics or physics, and I hope you will not The previous year, he had published an article interpret what I am about to say in the sense in which he pointed out that, though Bowley of that practice, which sometimes comes near referred to the marginal utility of money in being ridiculous. discussing consumer theory, it was only its But it is one thing to copy and another thing rate of change that entered into his proof. to learn how to apply existing, and to derive However, whilst it seems certain that the stimulus for constructing new, mathe- matical tools in the face of difficult factual Wilson will have raised questions about the patterns. We must not copy actual arguments significance of the notion of utility to con- but we can learn from physics how to build sumer theory, on which he had himself just up an exact argument. Moreover we can learn published (Wilson 1935), Samuelson’s main to understand the relation of mathematics to debt to him lay elsewhere. It was Wilson the reality to which it is applied. Most import- ant of all is the consideration that there are who stimulated Samuelson’s lifelong inter- obviously a set of concepts and procedures est in thermodynamics and the realization which, although belonging not to the field of that economists could learn from physics.15 pure mathematics but to the field of more or Though politeness no doubt lay behind less applied mathematics,­ are of so general a the remarks, there is no reason to doubt character as to be applicable to an indefinite number of different fields. The concepts of Samuelson’s observation, made to Wilson Potential or Friction or Inertia are of this kind, when he left Harvard, and the problems and difficulties to which they give rise are formally much the same, whatever I have benefitted from your suggestions, per- the subject matter.17 haps more than from anyone else in recent

14 Wilson, July 14, 1936, Letter to John D. Black, 16 Samuelson, October 9, 1940, Letter to Wilson, EBWP Box 27 (B 1935-6). EBWP Box 35 (S). 15 See Samuelson’s answers to questions posed by 17 Schumpeter, J.A., May 19, 1937, Letter to Wilson, Donald A. Walker in 1992, PASP Box 152 (Walker), p. 3. EBWP, Box 29 (S 1937). Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 333

The examples Schumpeter cited (potential, the general case where functions were not friction, inertia) were presumably ones that necessarily smooth and differentiable. Their Wilson had discussed. correspondence makes it clear that Wilson This view is precisely the one that pushed Samuelson to analyze finite changes informed Samuelson’s use of mathematics. and, as Gibbs had done, to base conclusions He learned from Wilson that there were on inequalities linked to generalized notions similarities between the structures of eco- of convexity. Directing Samuelson to the nomic problems and certain problems in types of mathematics on which he was to rely physics and that, even though the evidential in much of his work, Wilson recommended basis for the two disciplines was different, he The Calculus of Observations (Whittaker could make use of these similarities to solve and Robinson 1926), which discussed many economic problems without implying that of the techniques that Samuelson was later there was any deeper relation between the to use in Foundations, and which marked his physical and the economic concepts. The book out from previous work on the subject. main example of this was the Le Chatelier It focused on methods for obtaining numer- principle, the idea for which he attributed ical solutions, beginning with interpolation, specifically to Wilson’s lectures: difference equations, determinants, lin- ear equations, and statistical theory (linear In particular, I was struck by his statement that regression and correlation analysis). the fact that an increase in pressure is accom- panied by a decrease in volume is not so much Wilson was also important in stimulating a theorem about a thermodynamic equilibrium Samuelson’s views on dynamics and com- system as it is a mathematical theorem about parative statics. Crucial evidence comes in surfaces that are concave from below or about a letter dated December 30, 1938, in which negative definite quadratic forms. Armed with Wilson wrote detailed comments on a paper this clue, I set out to make sense of the Le 19 Chatelier principle. (Samuelson 1972c, p. 254) Samuelson had recently shown him. Even if Samuelson’s paper were an early attempt at tackling the relations between dynamics Though the status of the Le Chatelier princi- and comparative statics, the letter showed ple in physics and chemistry has been much that he had not yet sorted out the material to debated, for Samuelson it was a result that Wilson’s satisfaction.20 Wilson thought that held in any equilibrium system, whether Samuelson’s analysis might not be “so gen- physics, chemistry, or economics. This was eral in some respects as Willard Gibbs would an important idea in his later work. It should have desired.” Samuelson had written of cer- be noted that though accounts of it are often tain “secondary conditions” and of “a certain couched in terms of differential calculus, the form being positive definite.” This caused principle is much more general and applies, Wilson to tell Samuelson that Gibbs used to as Samuelson showed, equally to systems in talk of “non-negative quadratic forms” and which discrete choices mean that there is no that he used to lay great stress on the fact smooth substitution between variables.18 that it was important “to remain within the Samuelson learned from Wilson the impor- limits of stability.” Wilson appears to have tance of basing economic theory on pre- cisely specified postulates and of ­analyzing 19 Wilson, December 30, 1938, Letter to Samuelson, EBWP Box 31 (S 1938). 18 Milgrom (2006) discusses subsequent attempts to 20 It not clear which paper this letter referred to, though generalize the conditions under which the principle holds, it seems most likely that it was either this one or a draft of including ones by Samuelson. the corresponding chapter of his thesis. 334 Journal of Economic Literature, Vol. LIII (June 2015) been telling Samuelson not only that he had 2.3 Marion Crawford not stated the second-order conditions for optimization correctly (positive definiteness At Harvard he audited courses in was sufficient but not necessary), but also thermodynamics and Fourier analysis. that second-order optimization conditions However, given the mathematics he used in and stability were related, an idea that was to Foundations, a particularly important course become central to Samuelson’s work. would seem to be one he took in the summer of 1936, at summer school at the University Now if one wishes to postulate the derivatives of Wisconsin–Madison. This was the theory including the second derivatives in an abso- lutely definite quadratic form one doesn’t need of equations, taught by one of the few women to talk about the limits of stability because the in American mathematics, Margarete Wolf. definiteness of the quadratic form means that She had obtained her PhD the year before, one has stability. (ibid.) and two years later was to publish, jointly with her sister, a paper on the problems of In an optimizing system, second-order con- deriving necessary and sufficient conditions ditions implied stability and vice versa. for the existence of solutions to linear matrix It is also significant that in the same letter, equations and determining the number of Wilson reminded Samuelson of the need to solutions (Wolf 1936). Though this material consider the more general case where func- may not have been covered in the course he tions were not continuously differentiable. took, it was very close to the topic of “systems of linear equations and determinants with He [Gibbs] doesn’t use derivatives but intro- applications” described in the course catalog. duces a condition which is equivalent to say- Given his exposure to Leontief’s input-out- ing that his function has to be on one side or in a tangent plane to it. He doesn’t even put analysis the term before, and his use of assume that there is a definite tangent plane the textbook by Whittaker and Robinson, but merely that at each point of his surface it is Samuelson could hardly have failed to see possible to draw some plane such that the sur- the significance of this material. face lies except for that point and some other One reason for his going to Wisconsin that points entirely to one side of the plane. (ibid.) summer was that it was not far from the home of Marion Crawford, an economics major at The mathematics of convex sets and separat- Radcliffe College, whom he married in 1938. ing hyperplanes, which is what Wilson was While he was taking courses in the theory of implicitly describing, was not well developed, equations and functions of complex variables, and he went on to say “Just how general a she was studying money and banking (they theorem one can get I don’t know because took German together). After graduation, I have never worked it out as carefully as I she worked as an assistant to Schumpeter, ought to have done.” However, it remained who rated her extremely highly. For her the case that Gibbs had pointed out that if undergraduate dissertation (Crawford one exceeded the limits of stability, the con- 1937), she undertook a mathematical inves- ditions Samuelson assumed might not hold. tigation of a concept that had only recently Wilson’s final advice was that he needed to appeared in the literature, the elasticity of explain himself better: “a little more text and substitution, applying to it techniques in not so many formulae in proportion to the linear algebra that would have been beyond text might make the whole decidedly easier most ­economists of the time (and which reading”—advice that Samuelson did not she must have discussed with Samuelson). always take. Her only academic ­publication was on ­tariff Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 335

­protection in Australia (Samuelson 1939), as statements about the world that were in and Samuelson later suggested that he might principle refutable (ibid., pp. 2, 3). This idea have learned more from this than he had had been completely absent from his pub- acknowledged when he wrote the paper lications in 1937, the first hint of it coming with Stolper on protection and real wages in February 1938, in which he referred to that was the first to use the term Heckscher– his consumer theory, based on index num- Ohlin trade theory (Stolper and Samuelson ber theory, as being “more directly based on 1941; Samuelson 1994, p. 346). They wrote those elements which must be taken as data a joint paper on population dynamics, which by economic science” and “more meaning- remained unpublished. It is clear that her ful” in its formulation (Samuelson 1938b, role in Foundations extended beyond the p. 71). But this is no more than a hint. His typing, for he wrote that her contributions to first clear commitment to operational- the book were “too many” and that “the result ism came in the paper he presented to the has been a vast mathematical, economic, and Econometric Society in December 1937 stylistic improvement” (Samuelson 1940, (Samuelson 1938a). It is tempting to link this preface; Samuelson 1947, p. vii). Given her to discussions he had in the preceding three qualifications it seems likely that Samuelson months in the Society of Fellows, where the was not exaggerating, though it is impossi- weekly discussions over dinner were domi- ble to say where, on the spectrum running nated by physiologist Lawrence Henderson. from copy editing to coauthorship, her input Samuelson later played down the signifi- should be placed.21 cance of his contact with Henderson, though he had cited him when discussing opera- tionalism in Foundations (1947, p. 5, n. 2). 3. The Thesis Henderson had used Gibbs’s mathematics In the middle of 1940, as his Junior to apply the idea of general equilibrium to Fellowship was coming to an end, he took physiology, and when he discovered Pareto’s the decision, with Marion, that it would be sociology he saw that it could also be applied ­prudent to obtain a PhD.22 He composed and to social systems. Henderson organized the rearranged material “at fever pace,” dictating “Pareto Circle,” an interdisciplinary group some of the material to Marion, who typed exploring Pareto’s ideas, though Samuelson the entire first draft of the thesis (Samuelson claims he found it of no interest and attended 1998, p. 1377). It bore the subtitle, The only once.24 Observational Significance of Economic The term “operational” inevitably con- Theory.23 His use of the term observational jures up the operationalism of the Harvard in the title is consistent with his definition physicist, Percy Bridgman, which was widely of “operationally meaningful theorems” known in the 1930s.25 Even if Samuelson had not attended Bridgman’s lectures on ther- modynamics,26 he could hardly have avoided 21 Samuelson always praised his fellow-students, argu- ing that it was they, as much as their teachers, who were responsible for what he called Harvard’s golden age. However, aside from specific instances, such as his collabo- 24 Weintraub (1991, chapter 3) presents the Pareto rations with Abram Bergson and Gustav Stolper, their pre- circle as part of the context out of which Foundations cise role in the evolution of his thinking is unclear. emerged. 22 Backhouse (2014) discusses this period of his life in 25 For example, Harvard’s Stanley Stevens argued for detail. operationalism in psychology. 23 In Foundations (1947, p. 7) he incorrectly claimed 26 He has stated that he took a course in thermodynam- that the subtitle of the thesis was “The operational signifi- ics, and Bridgman taught the only course listed under that cance of economic theory.” title. 336 Journal of Economic Literature, Vol. LIII (June 2015) him, given Bridgman’s closeness to those priori truth but a hypothesis. The second running the Society of Fellows. However, was the assumption that an equilibrium must the way Samuelson described operationally be stable, for otherwise it made no sense to meaningful theorems, emphasizing “data” analyze comparative statics results. This was and “observation” (a term that is given necessary for the analysis of group behav- prominence by its use in the thesis title) has ior because at this level nothing was being much stronger echoes of the Vienna Circle maximized.28 ideas being developed by Rudolf Carnap, whom Samuelson’s friend Willard Quine However, when we leave single economic units, the determination of unknowns is found had helped bring to the United States in to be unrelated to an extremum position. In 1936, and whom Samuelson might also have even the simplest business cycle theories there encountered through Senior Fellow Alfred is lacking symmetry in the conditions of equi- North Whitehead.27 librium, without which there is no possibility of Operationally meaningful theorems could reducing the problem to that of a maximum or a minimum. Here the hypothesis is made that be derived using mathematics. Samuelson the system is in stable equilibrium or motion opened the thesis with a quotation, repro- in terms of an assumed dynamical system duced exactly in Foundations, from the (Samuelson 1940, p. 4-5). mathematician E. H. Moore: “the existence of analogies between central features of The justification for assuming stability was various theories implies the existence of a that “positions of unstable equilibrium, general theory which underlies the partic- if they exist, are transient, non-persistent ular theories and unifies them with respect states” that will be observed less often than to these central features” (Samuelson 1940, stable ones. p. 1). He sought to apply this idea to eco- After these methodological and mathe- nomics, presenting the theory underlying matical preliminaries, Samuelson turned to the theories of production, the consumer, “cost and production,” stressing that produc- international trade, the business cycle, and tion, value, and distribution were all aspects income determination. It had long been of a single problem. He was proposing a recognized that these all involved systems general theory and was trying to derive “all of interdependent equilibrium conditions, possible operationally meaningful theorems” but the novelty was seeing that there existed (Samuelson 1940, p. 68). Given the tradi- “formally identical meaningful theorems” tional emphasis on derivatives and marginal in all of these fields. Not only would he conditions, he emphasized the need to con- derive operationally meaningful theorems, sider finite changes. but these theorems would be derived from a common set of underlying principles. The It is curious to see the logical confusion into which many economists have fallen. The pri- first source of results was that equilibria were mary end of economic analysis is to explain often the solutions to optimization problems, a position of minimum (or maximum) where for propositions about individuals could be it does not pay to make a finite movement in derived from the hypothesis that “conditions any direction. Now in the case that all func- of equilibrium are equivalent to the maximi- tions are continuous, it is possible as a means towards this end to state certain equalities on zation (minimization) of some magnitude” differential coefficients which will (together (Samuelson 1940, p. 4). This was not an a with appropriate secondary conditions) insure

27 One of Samuelson’s friends, Shigeto Tsuru, worked 28 Note that phrase he coined to describe this, the “cor- closely with Whitehead and took an interest in psychology. respondence principle,” was not used at this point. Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 337

that certain inequalities will hold for finite needed explicit dynamic models that ­movements. It is no exaggeration to say that explained how, starting from arbitrary initial infinitessimal analysis was developed with just conditions, all variables would change over such finite applications in view (Samuelson 1940, p. 92). time. This took the mathematical analysis to an even higher level, for it might be modeled using “differential, difference, mixed differ- Though influenced by Wilson on this point, ential-difference, integral, integro-differen- it meant his arguments applied not just to tial and still more general” sets of equations traditional theory, but also to Leontief’s (Samuelson 1940, p. 196). Even if some of input-output models. the ideas would have been familiar to econ- This theme, that the existing literature omists, he was using mathematical language was confused, continued in the next three in a way that only a tiny minority of math- chapters, in which he sought to show pre- ematical economists would have previously cisely what was contained in traditional encountered. He then proceeded to discuss utility theory. After a few pages on the evo- different types of stability, illustrating this lution of the concept of utility, he turned to with several economic examples, including a mathematical treatment of the problem, the multimarket equilibrium system used drawing on his early articles. Revealed pref- by Hicks (1939) and a Keynesian system, erence and notions of indifference were such as James Meade, Hicks, and Oskar used to show how meaningful propositions Lange were using. Though these were not could be derived. He clarified the meaning the same type of supply and demand sys- of concepts such as cardinal utility, inde- tem, he treated the Keynesian system as pendent utilities, and complementarity, and something analogous to the other market devoted a whole chapter to the problem systems he had been discussing in his book. of the constancy of the marginal utility of Given that it was not based on optimiza- income, important because of the legacy of tion, this was the example that showed most Marshall’s ideas and because of attempts clearly the importance of dynamics to com- that had been made by Chicago’s Henry parative statics. Schultz to measure demand functions. Samuelson achieved a number of things When he turned to dynamics, in the last with these examples. The first was to show two chapters of the thesis (covering mate- that dynamic processes were implicit in rial that also appeared in 1941 and 1943a), familiar economic examples, implying that he carried on in the same vein, criticizing economists could not argue that dynam- ­previous generations of economists for fail- ics was unnecessary. They might not talk ing to investigate the character of the laws in terms of explicit dynamic systems, but they claimed existed. If nothing is known they were nonetheless using them. The other than that supply and demand deter- second was to illustrate some of the dif- mine prices, without knowing the shape of ferent types of mathematics that could be the functions, he claimed, “the economist used: ­differential, difference, and integral would be truly vulnerable to the gibe that equations. One of his examples also illus- he is only a parrot taught to say ‘supply trated stability of the second kind and the and demand’” (Samuelson 1940, p. 192). notion that systems might be subject to ran- Economists needed to derive comparative dom shocks—that they might “take a ran- statics results. The first of his chapters on dom walk” (Samuelson 1940, p. 207). He dynamics linked this to the stability of equi- referred to Brownian motion of large mole- librium. He then argued that ­economists cules undergoing random collisions, though 338 Journal of Economic Literature, Vol. LIII (June 2015) argued that before applying such ideas, brought the recipient $500 (equivalent to with the possibility that movements might two months’ salary when he was appointed become more variable the longer a system Assistant Professor at MIT in October 1940) was observed, in economics, “some statisti- and also covered the costs of publication of cal evidence of its possible validity­ should the thesis by Harvard University Press. He be adduced” (Samuelson 1940, p. 208). remembered being awarded the Wells Prize Another example showed, through famil- in 1941, saying that he then revised his ­thesis iar examples, that stability analysis was not whilst working on fire control (designing an esoteric matter that economists could radar guidance systems for naval artillery) at ignore, but that it was important for deriv- MIT’s Radiation Laboratory, handing it in in ing comparative statics results. As in his 1944 (Samuelson 1998, p. 1378). However, previous chapters, the tone was of showing the Harvard Corporation ruled his thesis, economists how to do properly the things examined in December, would not be con- that they had previously been trying to do sidered that year. It was not until February without success. 10, 1942, that the department unanimously Dynamic analysis also made it possible endorsed the recommendation, made by a for Samuelson to introduce randomness committee comprised of Seymour Harris, (also in one of his earlier examples) and Haberler, Leontief, and Hansen, to award to relate economic theory to econometrics Samuelson the prize. (as the term eventually came to be under- By this time, Samuelson was in his second stood). He provided a justification for rep- year at MIT, teaching courses in economic resenting economic equilibrium “as simply theory, mathematical economics (which a statistically fitted trend,” implying that the had a course in differential equations as a approach to data analysis represented by prerequisite), and business cycles. He had Henry Ludwell Moore and Henry Schultz, also become involved, through Hansen, whose work he cited, might have rigor- as a consultant for the Full-Employment ous theoretical justification. Citing Lotka’s Stabilization Unit at the National Resources Elements of Physical Biology (1925), he Planning Board (NRPB), traveling to noted that a dynamic equilibrium of supply Washington every other week. His role and demand was “essentially identical with there was to help assemble and supervise the moving equilibrium of a biological or a team that was forecasting consumer chemical system undergoing slow change” demand under varying assumptions (Samuelson 1940, p. 236). Here, and in about the distribution of income across his discussions of concepts of stability, he households. He was also getting involved in was moving away from economics into the debates over the multiplier and the theory realm of mathematics and physical systems of income determination (e.g. Samuelson in general. He was sketching directions in 1942b, 1943b). In June 1943, the NRPB was which economic analysis might develop. disbanded, and his teaching commitments at MIT rose.29 In addition to his economics 4. From Thesis to Book teaching, he also taught basic mathematics to U.S. Navy officers.30 4.1 Chronology

Samuelson’s thesis was an obvious can- 29 This is discussed in Maas (2014). didate for the Wells Prize for the best 30 The exact date and content of this teaching has not ­publishable thesis in Economics, which been established. Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 339

To focus resources on the war effort, MIT had decided to include new material (the closed down much of its economics teach- changes he made are discussed below). ing for academic year 1944–45, and it was at Given what he told Stolper, it seems unlikely this point, in March 1944, that Samuelson that it could have been submitted before the joined the Radiation Laboratory as a mathe- summer of 1944. There was then the delay, matician in the theory group under physicist discussed above, before the manuscript Ivan Getting. The pressure he was under, was submitted to the Press, during which working 8.30 a.m. until 6 p.m., six days a he saw it gathering dust as he checked its week, able to work on his manuscript only progress once a month. (Samuelson 1998, in the evenings and on Sundays, no doubt p. 1378). Given that the book was sent to explains why this stage of the process stuck Harvard University Press in February 1945, in his memory. However, he had started this delay seems unlikely to have been revising the book much earlier, for on May more than a few months. If, as is possible, 29, 1942, he had told Harvard University it had taken him a long time to overcome Press that he was undertaking “fairly exten- the difficulties he described to Stolper, in sive revisions” and that, due to his work at the limited time allowed by his work in the the NRPB, he would not be able to com- Radiation Laboratory, it could have been plete it in that year.31 The description he less than that. provided of the book suggested that he was 4.2 New Material planning to extend its coverage of business cycle theory, a topic that he had covered The modifications he made to the thesis only briefly in his thesis. were very substantial and are summarized Progress continued to be slow, despite in table 1. The most prominent change was Samuelson having reported to Abbott the introduction of the “correspondence Usher, Chair of the Harvard Economics principle.” In the introduction to the thesis, Department’s publications committee, who Samuelson had explained the link between was responsible for liaison with Samuelson comparative static and dynamic theory by and the Press, that he hoped to be able to writing, “Here the hypothesis is made that finish it by September 1943.32 In August the system is in stable equilibrium or motion 1943, he told Walter Salant, a friend from in terms of an assumed dynamical system. Harvard working in Washington with whom This implies no ideological or normative he was discussing the multiplier, that he was significance. . .” (Samuelson 1940, p. 4). In “on the home stretch.”33 However, in April Foundations, that became: 1944, he told Wolfgang Stolper that it was only with great difficulty that he could make Instead, the dynamical properties of the sys- tem are specified, and the hypothesis is made any ­progress on “the remaining chapters 34 that the system is in “stable” equilibrium or of my own manuscript.” It would appear motion. By means of what I have called the that at some point between these dates he Correspondence Principle between compara- tive statics and dynamics, definite operation- ally meaningful theorems can be derived from 31 Samuelson, June 28, 1942, Letter to David T. so simple a hypothesis. One interested only in Pottinger, PASP Box 34 (H, 1940-57). fruitful statics must study dynamics. 32 Samuelson, July 8, 1943, Letter to A. P. Usher, PASP The empirical validity or fruitfulness of the Box 85 (Foundations). 33 Samuelson, August 11, 1943, Letter to Walter Salant, theorems, of course, cannot surpass that of PASP Box 67 (Salant). the original hypothesis. Moreover, the stability 34 Samuelson, April 6, 1944, Letter to Abram Bergson, hypothesis has no teleological or normative sig- PASP Box 16 (Bergson). nificance … (Samuelson 1947, p. 5). 340 Journal of Economic Literature, Vol. LIII (June 2015)

Table 1 Foundations in relation to Samuelson’s PhD Thesis

Chapters in Foundations Pages Material added to/removed from thesis I. Introduction 5–6 Discussion of correspondence principle II. Equilibrium Systems and Comparative Statics None III. The Theory of Maximizing Behavior 23–29 New section: A Calculus of Qualitative Relations­ IV. A Comprehensive Restatement of the Theory 78–80 New section: Indeterminacy in Purest of Cost and Production Competition?­ V. The Pure Theory of Consumer’s Behavior 117–124 New section: A Note on the Demand for Money VI. Transformations, Composite Commodities, 125–171 New chapter and Rationing VII. Some Special Aspects of the Theory of 179 Paragraph on independence added Consumer’s­ Behavior 184–189 Discussion of Value and Capital and complementarity added 189–202 Three sections, Constancy of Marginal Utility of Income and two on consumer’s surplus, are a completely rewritten version of a thesis chapter, reflecting the content of Samuelson 1942a VIII. Welfare economics 203–253 New chapter IX. The Stability of Equilibrium: Comparative 258 Sentence on correspondence principle added Statics and Dynamics since Samuelson 1941 266 Figure illustrating offer curves added in Samuelson 1941 272 Two sentences on imperfect stability omitted since Samuelson 1941 274–275 Discussion of Lange 1942 and Lange 1944 added since Samuelson 1941 X. The Stability of Equilibrium: Linear and 284–310 New chapter (Samuelson 1942c) Non-linear­ Systems XI. Some Fundamentals of Dynamic Theory 311–317 New Section: Statics and Dynamics. All but one page is included in Samuelson 1943a. 327 Final paragraph on stability of equilibrium changed 330–332 Discussion of “what processes shall be described as equilibrium processes?” added 334 Phrase “physical systems” corrected to “conservative physical systems” 335–349 Six sections on business cycles added XII. Conclusions 350–355 New chapter Appendix A: Maxima and Quadratic Forms 379 Two clarificatory sentences added Appendix B: The Theory of Difference and Other 380–439 New chapter Functional Equations Note: Pages are those on which significant changes have been identified. Only those articles published after the thesis are mentioned. Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 341

The content of his argument had not he made was between endogenous models changed, but Samuelson had introduced a (which explained the cycle as ­self-generating, name for the idea and had elevated it to the determined by factors within the system being status of a “principle.”35 Samuelson clearly analyzed) and exogenous theories (which thought that his work was in the forefront explained fluctuations in terms of factors out- of a radical change that was taking place in side the model). The problem of endogenous economics. theories was that they required that there was no damping—that the parameters of the An understanding of this principle [the Correspondence Principle] is all the more economic system were such as to generate a important at a time when pure economic the- system where fluctuations neither faded away ory has undergone a revolution of thought— nor exploded. In physics there were constants from statical to dynamical modes. While many that might generate such systems, but there earlier foreshadowings can be found in the lit- was no reason to assume such constants in erature, we may date this upheaval from the publication of Ragnar Frisch’s Cassel Volume economics. Thus he was critical of Michał essay of only a decade ago. The resulting Kalecki (1935) for imposing the condition change in outlook can be compared to that that cycles not be damped. This was a milder of the transition from classical to quantum version of an even more critical appraisal of mechanics. And just as in the field of physics it Kalecki that he had made privately in a letter was well that the relationship between the old and the new theories could be in part clarified, to Hurwicz: so in our field a similar investigation seems in order (Samuelson 1942c, p. 1; Samuelson By the way, have you read Kalecki’s most 1947, p. 284, emphasis added). recent “Studies in Economic Dynamics”? He has a chapter on “pure” business cycle which, in my humble opinion, hits the low as far as It is hard not to speculate that he had in method is concerned. In order to obtain his mind Keynes’s implied comparison of the favorite mixed difference-differential equa- revolution to be wrought by his General tions he approximates differences by deriva- Theory with that brought about by Einstein tives, but not all the way thru—then he would 36 have had a simple differential equation. Also, in physics. he makes it non-linear to maintain stability The new material on business cycle theory­ regardless of coefficients, but he does not inte- (Samuelson 1947, pp. 335–49) was less a grate the system explicitly, nor even write out 37 survey of business cycle than a survey of the non-linear term. the different mathematical methods that could be used to model the cycle. This was Samuelson then explained that the problem a topic on which many economists in several with linear endogenous models was that they government agencies were actively working, could not explain the amplitude of the cycle: Samuelson being one of many economists in as with a pendulum, the amplitude could be government service and academia trying to of any magnitude, depending on where the forecast whether the end of the war would be system started. One way round the problem followed by depression, as had happened after was to drop the assumption of a purely endog- the First World War. The basic distinction enous cycle and to assume that external fac- tors kept the system going (though he did not 35 The index entry for “Correspondence principle” gives cite it at this point, this was Frisch’s “rocking three additional pages. On one, the phrase is not used, and horse” model of the cycle, according to which the remaining two are in a new chapter. 36 He would be familiar with Pigou’s (1936, p. 115) remark that Einstein actually did for physics what Keynes 37 Hurwicz, L., August 29, 1944, Letter to Samuelson, claimed to have done for economics. PASP Box 39 (Hurwicz). 342 Journal of Economic Literature, Vol. LIII (June 2015) a rocking horse hit periodically by outside consumer demand. This provided him with shocks would exhibit a continuing cycle). The the opportunity to tackle a range of issues other was to go for a nonlinear model, citing in consumer theory, notably generalizing several references to the literature on the use a theorem from Value and Capital (Hicks of such models in physical systems. An exam- 1939) on aggregation. However, the most ple of such systems was “billiard table” theo- substantial addition to part I was a fifty-page ries, such as Harrod’s (1936) in which output chapter on welfare economics that became moved between a full-employment ceiling and the canonical statement of what came to be a floor. A problem here was that Hansen had known as Bergson–Samuelson social welfare shown that there was “no (relevant) natural functions. He included this, he explained in bottom to the economic system” (Samuelson a letter to Viner, “because the condition of 1947, p. 340). He favored mixed endogenous– the current literature seemed to me to be in exogenous systems, citing his own work on a rather scandalous state of confusion and the multiplier-accelerator model (Samuelson ignorance.”38 Though there was nothing on 1939b; Samuelson 1939c). His point here this topic in his thesis, the origins of these was that if the multiplier-accelerator model ideas go back to 1936–37 when Samuelson’s were augmented with a periodic movement fellow student Abram Bergson (formerly such as a sine wave (though not necessarily a Burk) kept asking him, “What can Pareto sine wave), the system would, subject to cer- mean by this 1898 use of the French singu- tain conditions, settle down to a periodicity lar when he speaks of ‘the social optimum?’” determined by the exogenous factor. He lik- (Samuelson 1981, p. 224). The problem was ened this to the physical phenomenon of res- that Pareto referred to a social optimum as if onance. In such a system, shocks would break it were a single point, yet the conditions he into this, producing shorter cycles. proposed did not seem to define any unique Finally, he turned to “Mixed systems of a point. They solved it by making a clear sepa- linear stochastic type”—linear models sub- ration between ethical value judgements and ject to random shocks. Such models had been empirical propositions. Bergson (Burk 1938) analyzed by the Russian economist, Eugen used this distinction to sort out confusing Slutsky, and Frisch (1933), whose paper claims that had been made about welfare in he described as brilliant, but Samuelson the 1930s. Though Samuelson (1939a) had linked it to Fourier analysis and his MIT col- discussed welfare in the context of inter- league, Norbert Wiener. What is significant national trade, his first comprehensive dis- about this is that he linked this approach cussion of welfare came in this chapter in to the cycle to the problem of estimation, Foundations. citing Trygve Haavelmo’s “The Probability Samuelson’s starting point was a func- Approach in Econometrics” (1944) and work tion that was even more general, and by Abraham Wald. Though it was technically with less content, than Bergson’s, for much more difficult, he also outlined the it began with a social welfare function, problem of modeling the cycle as a nonlinear W = W(​​z​ 1​, z​ 2​, …) where the z’s are any stochastic system. variables that are thought relevant to social New material in part I included a chapter, welfare. Any statement about what the “Transformations, Composite Commodities z’s were involved making ethical judge- and Rationing” that tackled problems that ments. Unlike Hicks, who tried to eliminate were of great practical importance in war- time (rationing and index numbers) and 38 Samuelson, April 9, 1948, Letter to Jacob Viner, which he had confronted in his work on PASP Box 74 (Viner). Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 343

­ethical judgements­ from welfare ­economics, Reviewers greeted it with great enthusiasm Samuelson sought to establish what ethical as a very important contribution to eco- judgements needed to be made in order to nomic theory (for example, Boulding 1948; derive conclusions about welfare. He began Tintner 1948; Reder 1948; Allen 1949; with widely accepted judgements, such as Stolper 1949). This may be because most of that individuals’ preferences “count” (not them were mostly young, as few of the older true, he claimed, for Nazi and Communist generation would have been able to follow “totalitarian” states) before moving on to the book.39 The mathematical appendices more controversial ones, such as those were welcomed as providing much-needed involved in assuming that welfare was the teaching materials, though the book stim- sum of cardinal utilities. Assuming no more ulated discussion of the role of mathemat- than that individuals’ preferences counted ics in economics, still a controversial issue (implying a social welfare function of the form (e.g., Boulding 1948, Savage 1948). The W = W(​​U​ 1​(·), U​ 2​(·), …), and that utilities teaching of economic theory was still dom- depended on goods consumed and produc- inated by Alfred Marshall’s Principles of tive services supplied, he showed that, to Economics (1920), which reflected the view use terminology introduced by Ian Little that mathematical reasoning should be kept (1950) shortly afterwards, there was an strictly subordinate to verbal explanations. infinity of Pareto optima. All that the pro- In his introduction, Samuelson (1947, p. 6) duction and exchange conditions could do argued that Marshall’s position should be was to establish that there was an equation “exactly reversed” on the grounds that “the relating the well-being of different people laborious literary working over of essen- in the system: a “utility possibility function” tially simple mathematical concepts such that determined the maximum utility any as is characteristic of much of modern eco- person could achieve given the utilities of nomic theory is not only unrewarding from everyone else. the standpoint of advancing the science, but Though it rested on different founda- involves as well mental gymnastics of a par- tions, Samuelson claimed that this analysis ticularly depraved type.” Where Marshall of welfare economics completed his static had relegated mathematics to footnotes and analysis of maximization. Because it was less appendices, Foundations sought to demon- clear what should be maximized, the nature strate what could be achieved through using of the discussion had to be very different, mathematics in economics. Not surprisingly, focusing on conceptual, more philosoph- some reviewers took up the comparison ical problems. Scientific economic analy- with Marshall. For example, Reder (1948, sis might not be able to arbitrate between p. 516) characterized Marshall as having competing value judgements, but it could sought to clarify concepts, using mathemat- analyze the implications of different sets of ics when it served that purpose, whereas values. Samuelson “begins with systems of equa- tions and attempts to deduce their empirical

5. The Reception of the Book 39 Reviewers included (ages at the time in brackets) Roy Allen (41), Gerhard Tintner (40), Kenneth Boulding Despite his youth, Samuelson was (37), George Stigler (36), Wolfgang Stolper (35), Lloyd already well known by 1947, with around Metzler (34), Kenneth May (32), Leonard Savage (30), Charles Carter (28), Melvin Reder (28), William Baumol fifty academic papers to his credit, and the (25). Samuelson’s fellow students in this list were Stigler (at ­publication of his thesis was eagerly awaited. Chicago), Metzler, and Stolper (at Harvard). 344 Journal of Economic Literature, Vol. LIII (June 2015) or ­operational implications,” concluding that could be no confidence in the results; they if one follows Samuelson, mathematics is far focused on the mathematics, with very little more likely to be useful. In a similar vein, economic content; and no attention was paid Allen (1949, p. 111) compared Marshall’s use to expectations. Gerhard Tintner (1948, p. of mathematics to the use of steel to provide 499), at the Cowles Commission, concluded scaffolding, whereas Samuelson used it as that because Samuelson’s results were very part of the structure. limited, his methods needed to be combined Foundations was also compared with with econometric methods so as to quan- Hicks’s Value and Capital (1939), Reder tify the theory. Samuelson, though he had and Allen providing extensive comparisons. undertaken empirical work on consumption Samuelson (1998, p. 1382) later described and published papers on statistical theory, Hicks’s book as “an expository tour de force and though he worked with students (e.g. of great originality, which built up a reader- Lawrence Klein) and coauthors (e.g. Robert ship for the problems Foundations grappled Solow) who did econometric work, never with and for the explosion of mathematical went down that route himself.40 economics that soon came.” However, few Several reviewers recognized that reviewers saw the relationship in this way. Samuelson did not see economics as synon- According to Roy Allen (1949, p. 112), who ymous with constrained maximization using had collaborated with Hicks on consumer calculus-based techniques. Metzler (1948, theory in the early 1930s, Hicks had tried to pp. 905, 906) emphasized Samuelson’s work out “if not a complete economic theory, departures from traditional theory, pointing at least a full development of one particular out that he believed that the most important line of approach”; in contrast, Samuelson problems, including those dealing with the tried to do no more than show the common economy as a whole, could not be reduced to mathematical basis underlying different maximization problems and that he was cyn- fields of economics. More critically, William ical about consumer theory. Kenneth May Baumol (1949, p. 159) noted that the book (1948, p. 94), a mathematician, was left feel- lacked a theoretical unity, some chapters ing that the book contained “an implicit basic amounting to “collections of his miscella- critique of economic theory” in that “much neous thoughts and brilliant analytical sor- economic theory turns out to be banal or ties lumped together on the basis of some meaningless when stripped of its vague liter- tenuously established common characteris- ary formulation.” Allen (1949, p. 113) noted tic.” Tintner (1948) considered Samuelson’s that Samuelson stressed finite changes and treatment of uncertainty inferior to Hicks’s. discontinuity and Leonard Savage (1948, p. There was disagreement over which was 202), a statistician, noted that the economic the most original part of the book. For Lloyd theorist needed little math beyond “a few Metzler (1948, p. 906), who had used simi- ideas about continuity and convexity.” lar methods, the part dealing with dynamics Though Samuelson clearly attached was most important. Samuelson had gone importance to the Le Chatelier principle, beyond Hicks in constructing formal dynamic this was not something that impressed his ­models. Yet many reviewers were disap- reviewers, whether economists or mathe- pointed by this part of the book (e.g., Savage maticians. The only reviewer to mention it 1948; Tintner 1948; Stigler 1948; Allen 1949; (Baumol 1949, p. 160) did so in a footnote Baumol 1949). Three major problems were identified. Discussions of dynamics rested on 40 This is a substantial point that is beyond the scope of very special assumptions meaning that there this paper and will be discussed in detail elsewhere. Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 345 pointing out that Samuelson had gotten a on such methods, Linear Programming and minus sign wrong. The reason was presum- Economic Analysis (Dorfman, Samuelson, ably that, though the Le Chatelier principle and Solow 1958). Foundations came to be had been an important step for Samuelson, associated with what became the standard it was not a necessary step: it was sufficient approach to theoretical modeling, relying to begin with results on optimality conditions heavily on the use of Lagrange multipliers and the correspondence principle. Though and differential calculus, for Samuelson did Samuelson had stated (1947, pp. 38, 81) that not become involved in literature on the the Le Chatelier principle played a role in existence of general competitive equilib- natural science, his reference to the “meta- rium, and he remained a skeptic about game physical vagueness” with which it was stated theory.41 As such methods came to be more will not have encouraged economists to fol- widespread, and as more elegant techniques low this up. involving cost and expenditure functions The range of responses to Foundations came to be more widely used, the influence showed that, unlike Value and Capital, of Foundations declined.42 In part, of course, Foundations could be interpreted in many the decline was because methods that were ways. Though Samuelson remained skeptical once innovative came to be routine and unre- about much of the “neoclassical” orthodoxy markable. This explains the pattern of cita- that emerged in the 1960s and 1970s—and tions shown in Figure 1: citations reached a his friend Metzler (1948, p. 905) had gone so peak in the 1970s, the decade by which the far as to claim that Samuelson believed that use of mathematical methods became firmly “most of the important economic problems” established (see Backhouse 1998), but after could not “be reduced to simple problems that, Foundations was increasingly displaced of maximization”—Foundations provided a by books covering newer and different toolbox to which those who developed that techniques.43 orthodoxy could turn (exemplified by the case of Lucas, cited earlier). The techniques 6. Conclusions Lucas and others learned from Foundations increasingly dominated economic theoriz- Samuelson’s Foundations of Economic ing. For example, Don Patinkin’s reformula- Analysis did much to define the way eco- tion of monetary economics, Money, Interest nomic theorizing was undertaken after the and Prices (1956), central to discussions of Second World War. It was important both macroeconomics in the 1960s and 1970s (see because of its contents—it provided an Backhouse and Boianovsky 2013), made use of Samuelson’s theory of revealed prefer- 41 This is another story that will be told elsewhere. ence, the correspondence principle, and his 42 Later, when trying to generalize the theorem to apply discussion of dynamics. to Leontief and other models (Samuelson 1960), he admit- Samuelson had seen the importance of ted that he had failed to derive a result that applied more than locally. However, it seems unlikely that recognition of methods that went beyond differential and this played a role in the decline in citations of Foundations. integral calculus, and from the late 1940s 43 Sales figures for the period for the book’s early years onwards played a major role in the devel- tell an interesting story. In 1948–49 it sold over 3,000 cop- ies. After the initial demand was met, average sales from opment of linear modeling techniques, 1950–1954 were under 300 per year, which rose steadily to clarifying the relation of Leontief and activ- 873 by 1962. The background to this steady rise in the late ity-analysis models to traditional theory 1950s is no doubt both the increased use of mathematics and the increased number of economists. To put this in (1949), and working with Robert Dorfman perspective, AEA membership rose from 5,329 in 1947 to and Robert Solow to produce a major textbook­ 11,285 in 1962, with similar expansions in other countries. 346 Journal of Economic Literature, Vol. LIII (June 2015)

45

40

35

30

25

20

15

10

5

0

1948 1951 1954 1957 1960 1963 1966 1969 1972 1975 1978 1981 1984 1987 1990 1993 1996 1999 2002 2005 2008

Figure 1. Citations of Foundations in JSTOR articles, 1948–2010

Source: http://dfr.jstor.org. Search for “Samuelson” and “Foundations of Economic Analysis,” restricted to academic articles. up-to-date toolbox for doing economics— ­long-awaited revision of his PhD dissertation, and because Samuelson, despite being only a significance it would not otherwise have thirty-two when the book was published, had. As economics became progressively was already a well-known figure. He was more mathematical, with graduate students widely considered the star of the generation increasingly expected to construct formal of economists that had come of age during models of maximizing consumers and firms, the Second World War, acknowledged by Foundations was widely seen as the canoni- the award of the AEA’s first Clark Medal. cal exposition of such methods. In addition, He was already making his mark at MIT, it was an important resource for anyone tack- where his presence was a major factor in ling dynamics or welfare economics. the transformation­ of a service department This paper fills out Samuelson’s account of focused on teaching engineers into one of the the book’s origins, as well as correcting some leading American economics departments. details. It makes clear the extent of his debt His articles on fields as diverse as consumer to E. B. Wilson, who not only drew his atten- theory, international trade, business cycle tion to thermodynamics, but also impressed theory, and the Keynesian multiplier demon- upon him the importance of looking carefully strated the way mathematics could be used at all the assumptions being made, the impor- to resolve confusions in literary accounts of tance of analyzing finite changes and mod- economic theory. This gave Foundations, the els in which functions were not ­necessarily Backhouse: Revisiting Samuelson’s Foundations of Economic Analysis 347

­continuous and differentiable, and the rela- mark on Foundations than did Samuelson’s tionship between comparative statics and other Harvard teachers, it is tempting to stability. It also establishes that turning the conjecture that Samuelson’s belief that thesis into a book was a long process, extend- there were strict limits to what could be ing over two to three years, during which it achieved through the analysis of optimizing almost doubled in length. The most notable behavior reflected the influence of Alvin additions included much of the material on Hansen, who was emphatically not a mathe- dynamics and a completely new chapter on matical economist. For much of the period welfare economics, as well as the term “the when Samuelson was revising Foundations, correspondence principle.” he was working with Hansen on macroeco- Central to Samuelson’s book was the nomic policy problems, notably estimating idea that there were common mathemat- consumers’ expenditure and trying to fore- ical structures underlying different prob- cast postwar national income, for which lems, both within economics and across the mathematical apparatus constructed disciplines. Operational theorems could in Foundations was of little use. Though be obtained by analyzing the properties the main influence of such work was on his of the appropriate equilibrium systems. introductory textbook (Samuelson 1948), For problems involving consumers and which expresses a very different view of the firm, this involved maximization and economics—more institutional and data- hence, the use of second order conditions. driven—it is hard not to speculate that the For problems involving aggregates where view on which Metzler picked up reflected optimization was not involved, comparative this work. His wartime teaching at MIT and statics results could be derived by assuming interaction with policymakers had no doubt that the equilibrium was stable. The paper given Samuelson the opportunity to develop contends that, contrary to what some com- his expository skills. This was evident in the mentators have claimed, what mattered to material he added, notably on index num- Samuelson was not the analogies with ther- bers and welfare economics, though in gen- modynamics per se, but the mathematical eral he failed to respond to Wilson’s advice structures on which certain physical, chem- to expand greatly the explanations of what ical, and biological theorems were based. he was doing with the mathematics, and Results that might appear to be physical most of the material taken from the thesis in origin could actually be mathematical was unchanged. The book might be flawed, implications of a system being in equilib- but neither the minor mistakes nor the fail- rium. This was an argument for the use of ure to provide an integration of economic mathematics in economics at a time when theory prevented it from being a resource most economic theory was conducted using on which many economists drew heavily verbal reasoning. during a period that was to witness a major Contrary to popular belief, as his close transformation of the subject along lines friend Metzler noted, Foundations reflected that Samuelson helped lay down. the view that there was much more to economics than optimizing behavior. References Macroeconomics required different foun- Allen, R. G. D. 1949. “The Mathematical Foundations dations, for aggregate behavior could not be of Economic Theory.” Quarterly Journal of Econom- explained as the outcome of optimizations: ics 63 (1): 111–27. Backhouse, Roger E. 1998. “The Transformation of hence the need for the correspondence U.S. Economics, 1920–1960, Viewed through a Sur- principle. Though Wilson left a stronger vey of Journal Articles.” History of Political Economy 348 Journal of Economic Literature, Vol. LIII (June 2015)

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