Abstract
In the introduction to Activity Analysis of Production and Allocation (Cowles Monograph No. 13), Tjalling C. Koopmans recalled that he developed the model of his “Optimal Utilization of the Transportation System” (in the proceedings of 1947 International Statistical Congress, which were reissued as an Econometrica supplement, 1949) “under the stimulation of statistical work for the Combined Shipping Adjustment Board, the British-American board dealing with merchant shipping problems during the second world war.” Similarly, the contributions of George B. Dantzig and Marshall K. Wood to Cowles Monograph No. 13 (two revised journal articles and five new chapters) emerged from wartime work for the US Army Air Force and postwar work for the Department of the Air Force. This article examines the context and consequences of the wartime roots of these foundational contributions to activity analysis and linear programming, with particular attention to Koopmans's 1942 memorandum for the Combined Shipping Adjustment Board titled “Exchange Ratios between Cargoes on Various Routes” (first published in his Scientific Papers, 1970).
Introduction
As Mina Rees (1979: 29) wrote, “Linear programming owes its discovery and development in the United States to the pressures growing out of wartime experience with staff planning on the gigantic scale needed by the U. S. military establishment.” From being a useful mathematical technique invented by George Dantzig and his colleagues in the US Air Force's Project SCOOP (Scientific Computation of Optimum Programs) for military purposes, linear programming became the basis both for deeper mathematical insights into duality by Albert Tucker and his students in the Princeton game theory community and, through the work of Tjalling Koopmans and the 1949 Cowles Commission conference he organized, for activity analysis, a new perspective on the efficient organization of economic activity that also drew on Koopmans's wartime work on the transportation problem.
Although war gaming goes back to at least 1644 and perhaps even to ancient China (see Young 1952; Wilson 1969) and World War I had inspired early research (e.g., on antiaircraft OR by the physiologist and future Nobel laureate A. V. Hill, the brother-in-law of J. M. Keynes; see Hill 1960: 265–67, 306–10), practical military and naval requirements in World War II gave a tremendous stimulus to the development of the mathematical techniques that became known in the United States as operations research and in Britain as operational research (OR) (see, e.g., Air Ministry 1963; Waddington 1973; Lardner 1984; Tidman 1984; Morse 1986; McCloskey 1987a, 1987b, 1987c; Fortun and Schweber 1993; Kirby 2003; Shrader 2006; Thomas 2007; Budiansky 2013; Kennedy 2013),1 extending to such techniques as gaming to find antisubmarine strategies (Parkin 2020). Philip Morse and George Kimball's Methods of Operations Research (1951), a foundational work of OR, was originally a classified wartime document written within the US Office of Naval Research by a physicist and a chemist. The extent of the effort, and the range of backgrounds of those involved, is shown by a 350-page American Mathematical Society history titled Operational Analysis in the U. S. Army Eighth Air Force in World War II (McArthur 1990) about an OR group directed by a future Supreme Court justice, John M. Harlan (Canadian Keynesians will note a cameo appearance by the economist Lorie Tarshis as operations researcher for the US Army 15th Air Force).
The practical wartime needs of the armed forces and of a wartime economy created a vast demand for quantitative modeling and optimization in forms that later would be considered characteristic of economics but that were then undertaken by people from a variety of backgrounds with stronger mathematical training than most economists then possessed. The wartime involvement of economists in the Allied war effort extended from Lionel Robbins and James Meade directing the Economic Section of the British War Cabinet Offices to the economic expertise clustered around John Kenneth Galbraith and Nicholas Kaldor in the US Strategic Bombing Survey and its British counterpart,2 and most famously John Maynard Keynes and Harry Dexter White at Bretton Woods. But economists were largely absent from operations research, the area where the postwar evolution of economics might lead one to expect to see mathematical economists. For example, Philip Morse's Anti-Submarine Warfare Operations Research Group consisted of six mathematicians, fourteen actuaries, eighteen physicists, three chemists, two biologists, and an architect but no economists (Morse 1986; Budiansky 2013: 177; see Hickman and Heacox 1998 on the extensive role of actuaries in wartime operations research). The logistical needs of the armed forces directed operations research toward the allocation of resources for war production, which meant not just the location of ships and planes or tactics against enemy submarines or bombers; this was done primarily by mathematicians, scientists, or actuaries rather than economists. Paul Samuelson at the Radiation Laboratory MIT was the exception that proves the rule, someone whose wartime role followed from his recognized standing as a mathematical economist. W. Allen Wallis and Milton Friedman, the director and associate director, respectively, of the Statistical Research Group at Columbia University (a component of the Applied Mathematics Panel), were appointed for their expertise as statisticians, not as economists (Wallis 1980; Friedman and Friedman 1998: 131–46). Similarly, Koopmans, as someone with a doctorate in mathematical statistics who had written a monograph on freight rates and tanker construction, was hired by the Shipping Board as a statistician, not as an economist. The emergence in the late 1940s of activity analysis, bringing together strands of research with wartime roots, brought mathematical economics from the fringes of the economics profession to share with its new disciplinary neighbors in operations research in the military and governmental funding of and stimulus to such research. Mina Rees, postwar director of the Mathematical Sciences Branch of the Office of Naval Research (ONR), quoted an emphatic letter from Stanford's engineering dean on the ONR's “profound effect on the development of the mathematical sciences in the United States since the end of World War II” (Rees 1977: 106). Mathematical economics, previously peripheral to economics, was among those mathematical sciences receiving unprecedented impetus and funding from ONR, RAND, and other military sources.
Many wartime operational researchers were conscious of potential peacetime economic applications, with Budiansky (2013: 253) reporting that “Blackett, Zuckerman, Bernal, Watson-Watt, Gordon, and others on the [British] scientific left briefly entertained great expectations that their wartime triumphs had opened the door to the scientifically planned society they had long dreamed of, one in which central planning would organize industry and the economy for the benefit of all” (see Werskey 1968 on the British scientific left, including several prominent wartime operational researchers).
In the United States, two strands of wartime OR, one concerned with the efficient routing of cargo ships and the other with the efficient placement of aircraft, came together in a 1949 conference of the Cowles Commission for Research in Economics, described by Richard Cottle as “perhaps the most influential gathering of its kind in the history of mathematical programming” (in Dantzig 2003: 19). The conference was supported by the Cowles Commission's research contract from the RAND Corporation (which then had the US Air Force as its only client) and was published as Activity Analysis of Production and Allocation (Koopmans et al., Cowles Monograph No. 13, 1951)—a conference, then at the most technical fringe of the discipline, whose influences on subsequent economics range from the Kuhn-Tucker conditions familiar to all economics students to bringing Robert Dorfman and Paul Samuelson together to work on linear programming and economic analysis. Although the participants in the conference collaborated on their shared research agenda, their interpretation of what their field of research was reflected their different paths to the conference, Dantzig seeing his field as linear programming (and later more generally mathematical programming), part of applied mathematics, and Koopmans (though he coined the term “linear programming”)3 viewing it as activity analysis, a new approach within economics to understanding how to efficiently allocate resources. At the Cowles Commission conference in 1949, and at meetings of the Econometric Society (closely linked with the Cowles Commission), the wartime development of operations research and related techniques met a small group of mathematical economists in an institution that was then at the outer fringe of the economics profession that, partly through the stimulus and funding that came with activity analysis, was to have a deep and lasting influence on the direction taken by the discipline of economics.
The practical needs that promoted operations research in wartime, as exemplified by the work of Koopmans and Dantzig on activity analysis, continued after the war to support the advancement of formal and mathematical approaches in economics and related disciplines in the United States, through the air force's funding of RAND (which in January 1948 contracted the Cowles Commission to study resource allocation and activity analysis), through the Office of Naval Research and its Naval Research Logistics Quarterly coedited by the Princeton game theorist Oskar Morgenstern (a participant in the Cowles Commission conference on activity analysis), and through the army's Stanford Research Institute. The move of such formal analysis and quantitative techniques from the fringes of postwar economics toward the mainstream, stimulated by such new sources of research funding and the immigration of European scholars such as Koopmans, Marschak, Haavelmo, and Wald, brought into American economics and related fields (or fields that were to come to be seen as related to economics) such as operations research scholars and ideas from other disciplines such as physics or mathematical statistics and from other national traditions.
This article explores the wartime origins of the approaches that came together in the Cowles Commission's conference volume on activity analysis, in Tjalling Koopmans's work at the Combined Shipping Board, and in what George Dantzig referred to as “early models developed for Air Force use” and how these approaches recast economic analysis of allocation and production in terms of mathematical programming.4 Complementary but distinct emphases and interests were reflected in the conference volume: Dantzig concerned with programming as applied mathematics, Albert Tucker and his students with the mathematical implications of duality, and Koopmans with activity analysis as a new way of understanding economic analysis in terms of efficient allocation of resources. But beyond the specifics of the new avenues of research, the confluence of these approaches had a transformative effect on the development of mathematical economics. From being a fringe activity, conducted within the Econometric Society, Econometrica, and the Cowles Commission, institutions peripheral to the discipline of economics, mathematical economics became allied to and overlapped with what emerged from the war as the flourishing and well-funded field known as operations research, management science, or decision sciences. These alliances and the associated funding helped bring mathematical economics into the mainstream of economics.
Koopmans (1953) held that activity analysis, as an approach to modeling the allocation of resources, differed from the classical theory of production and prices in three ways. Activity analysis started from a model of technological possibilities rather than from a production function that was already the result of optimization by engineers or managers. Activity analysis abstracted from institutional arrangements such as firms. And perhaps most importantly, “it provides us with methods of computation indicating what program will best serve a given objective and how to translate given valuations of final commodities into valuations of intermediate and primary commodities” (1953: 406–7). Initially, activity analysis had developed linear models and techniques most fully and the 1949 Cowles conference was initially referred to as the Conference on Linear Programming (as by Robert Dorfman in Koopmans et al. 1951: 348; and Morgenstern [1950] 1963: preface to 1st ed., v). The change from linear programming to activity analysis in the title of the published proceedings reflected that, while Dantzig's focus was on techniques of mathematical programming, Koopmans saw activity analysis as a more general approach to resource allocation, superseding and subsuming the classical theory of production and prices and going beyond theory to computation and practical application.
Koopmans, the Shipping Board, and the Transportation Problem
Koopmans was a Dutch physicist turned econometrician (see Scarf 1995 on Koopmans's life and contributions). His first journal articles, in 1933 and 1934, were in physics (published in German, and the second reprinted in Koopmans 1970), but his 1936 Leiden dissertation in mathematical statistics and 1937 book, published in English, was on “linear regression analysis of economic time series.” He was strongly influenced by the pioneering econometricians Jan Tinbergen and Ragnar Frisch, who were to win the inaugural Nobel Prize in economics. Although Koopmans's degrees were from the universities of Utrecht and Leiden (with, according to Koopmans's Nobel autobiography, Tinbergen as an outside adviser on economic aspects of his doctoral thesis), he also studied (not for a degree) with Tinbergen at the University of Amsterdam in 1934 and spent five months at the University of Oslo, working with Frisch and lecturing on statistical inference. Koopmans took over Tinbergen's classes at the Rotterdam School of Economics when Tinbergen went to the League of Nations in Geneva in 1937 to study statistical testing of business cycle theories and then took Tinbergen's position in Geneva when Tinbergen returned to the Netherlands in 1939.
Koopmans's second monograph was titled Tanker Freight Rates and Tankship Building (Netherlands Economic Institute, 1939), which, after the German invasion prevented his return to the Netherlands, led to work as a statistician for the Combined Shipping Adjustment Board, an Anglo-American joint body in Washington, from 1942 until he joined the Cowles Commission for Research in Economics at the University of Chicago as a research associate in 1944 (and from 1946 also an associate professor of economics). Already from 1940–41, when he was a special lecturer in statistics at New York University, Koopmans attended an NBER econometrics seminar run on weekends by Jacob Marschak, then teaching in the University in Exile at the New School for Social Research. Marschak, who became research director of the Cowles Commission from the first day of 1943, recruited Koopmans to the commission. Although employed by the Shipping Board as a statistician and publishing in the Annals of Mathematical Statistics in 1942 a paper titled “Serial Correlation and Quadratic Forms in Normal Variables” (Koopmans 1970: 29–49), Koopmans identified as an economist in the choice of topics and journals for his publications before joining the Cowles Commission, starting with two articles in Econometrica, edited by Frisch, but then turning to more general economics journals. From January 1940 through December 1943, Koopmans published on the degree of damping in business cycles, distributed lags in dynamic economics, the logic of econometric business cycle research, the dynamics of inflation, and durable consumer goods and the prevention of postwar inflation in Econometrica, the Journal of Political Economy, the Review of Economic Statistics, and the American Economic Review (Koopmans 1970: 11–28, 50–76, 595).
At the shipping board in Washington, where he spent time at the British Shipping Mission, Koopmans wrote a twelve-page memorandum titled “Exchange Ratios between Cargoes on Various Routes (Non-refrigerated Dry Cargoes)” (1942, first published in his Scientific Papers, 1970),5 which he regarded as providing the stimulus for the more developed model in his “Optimal Utilization of the Transportation System” (in the proceedings of the International Statistical Congress, 1947, which were reissued as a supplement to Econometrica, 1949) and his chapter with Stanley Reiter titled “A Model of Transportation” (in Koopmans et al. 1951: 222–59).6 The opening footnote of the Koopmans and Reiter chapter in the Cowles conference volume stated that “the theory presented in this chapter was originally developed by the former author partly during, but mostly after, his association as statistician with the (British-American) Combined Shipping Adjustment Board and with the British Merchant Shipping Mission in Washington during World War II” (Koopmans et al. 1951: 222n1). Koopmans and Reiter referred readers to Koopmans's paper at the 1947 Statistical Congress (Koopmans 1948, 1949b) “for a nonmathematical exposition of this model . . . where another illustrative example is also given.” As Arrow stated (in Arrow et al. 1991: 11), “Koopmans, drawing upon analogies from physics, produced a perfectly constructive solution for the transportation problem,” but it was Dantzig's slightly later work for the air force that “provided the effective solution for linear programming in general.”
Koopmans did not retain many documents from the early part of his career. However, a second important draft memorandum by Koopmans for the Combined Shipping Adjustment Board titled “Methods for Comparing Availability and Requirements, United Nations Tonnage (Dry Cargo)” is preserved in the Koopmans Papers (MS 1439, box 9, folder 153). Pages 4 and 5 summarize the “method employed in C. S. A. B. (W) (42) 150 to estimate the ratios governing compensating changes in the program corresponding to the vessel employment situation in the summer of 1942.” “The planning of a most economical pattern of routing can now proceed as follows,” wrote Koopmans (IIA-6).
To find the most economical manner of executing a given program is a mathematical problem that can be analyzed in a systematic manner. Or, following the practical man's preferences, it can be worked out on the map by a common sense method of trial and error. We shall follow the latter method first. By way of illustration of the argument, the map given on an inset opposite page shows the pattern of ballast trade that prevailed in the second half of 1942 [IIA-7]. . . . In a simple case like that of figure 2 it is relatively easy to find the most economic pattern of routing. When a greater number of points or areas has to be considered, which are moreover not neatly arranged in a simple geometrical figure, a long procedure of trial and error must be followed in order to arrive at the correct solution. (IIA-9)
“With respect to more complicated situations, two questions will naturally arise in the mathematical mind. Firstly, when this process comes to an end . . . is the solution arrived at unique?” (IIA-10). Koopmans concluded that the process would reach a solution and that, while final patterns of ballast traffic might not be unique, all “final patterns must necessarily involve the same amount of tonnage employed” (IIA-10-11). Tantalizingly, Koopmans states that “in order to satisfy [replacing “for the benefit of”] the mathematical mind the answers here to be given to these questions are derived mathematical proofs in Appendix I” (IIA-10), but there is no such appendix in the folder with the draft memorandum. The draft ends with the sentence, “In the next [blank space] this table will be used to calculate the shipping cost or saving connected with a change in loading requirements for the given route,” followed by the reminder to “mention somewhere 1. tonnage in repairs 2. size of area, port time, intra-area time, inter area time” (IIA-15). There is no indication in the Koopmans Papers that the draft memorandum or its mathematical appendix was completed and circulated.
Koopmans might reasonably have expected his economic argument about how to plan convoy routing and the trial and error method, but not the mathematical appendix, to be understood and sympathetically received at the top of the Allied shipping organization, because the British Shipping Mission in Washington was headed by the man who had delivered the inaugural series of Alfred Marshall Lectures, Sir Arthur Salter (later Lord Salter), the independent member of Parliament for Oxford University and, until his appointment in Washington, parliamentary secretary to the Ministry of Shipping. Salter had been director of ship requisitioning at the Admiralty in World War I, worked with Jean Monnet to establish the Allied Maritime Transport Council, and written the volume on Allied shipping control in the Carnegie Endowment's history of the war before heading the economic and financial section of the League of Nations secretariat (see Salter 1967). After leaving the League of Nations, first for journalism and then for All Souls, Oxford, Salter had published the Salter Report of the Conference on Road and Rail Transport and then a thirty-one-page pamphlet, Toward a Planned Economy (Salter 1934). But although Salter was sympathetic to economic planning in a capitalist society and was regarded as an economist,7 he was decidedly at the nonmathematical end of political economy, his Oxford chair being that of Gladstone Professor of Political Theory and Institutions. Koopmans's two wartime memoranda read as if crafted with Salter as their intended audience: someone at the frontier between economic policymaking, politics, and academia, long involved with directing shipping in two world wars, closely associated with leading economists, notably Keynes. But there is no evidence that Koopmans succeeded in attracting Salter's attention. Although Robert Skidelsky (2003: 984) characterized Salter as “a good example of the middle way in search of a theory,” Salter did not react to the theory for planning shipping offered in Koopmans's memoranda even without a mathematical appendix.
Koopmans demurred from a claim by Mina Rees (1979: 39) that he “had the task of planning convoy assignments so as to accomplish prescribed deliveries of cargo with the smallest amount of travel in ballast” as one of “many that exaggerate the extent of my responsibility and the practical effect of my study in the work of the Board” (in a letter to Mina Rees, July 11, 1979, Koopmans Papers, MS 1439, box 10, folder 183). But while Koopmans certainly was not in charge of planning Allied convoys and while his study may not have influenced Salter or Koopmans's other superiors at the board, his wartime memoranda led directly to his subsequent work, and he expressed no reservation about Rees's statement that Koopmans's study was “a special case of the Transportation problem, the easiest case of linear programming and virtually the only case that can be solved without the use of an electronic computer. But the logic of the solution is identical with the logic of the most general case, and Koopmans discovered that logic.”
Dantzig and the Needs of the Air Force
The other strand of activity analysis originated in the wartime US Army Air Force and postwar Department of the Air Force, leading to two 1949 Econometrica articles on “programming of interdependent activities,” the first by Marshall K. Wood and George B. Dantzig and the second, more mathematical one by Dantzig alone, revised as the first two chapters of the Cowles monograph. The Cowles conference volume also included three new chapters by Dantzig (one of which proved that two-person, zero sum games could be written as linear programming problems and thus solved by the simplex algorithm) and two by Wood. Dantzig had gone to work for the Army Air Force as a statistician in 1939, becoming mathematical adviser to the Defense Department before moving to the Mathematics Department of RAND in 1952.8
While Koopmans's disciplinary identity had evolved from mathematical statistics to mathematical economics and econometrics, Dantzig's interests and identity remained in mathematics, including mathematical statistics and operations research as applied mathematics. After taking his MA in mathematics at the University of Michigan in 1937, Dantzig became a junior statistician at the Bureau of Labor Statistics. Inspired by reading a paper by the statistician Jerzy Neyman as part of an assignment at the bureau, Dantzig followed Neyman to become a doctoral student in mathematics at the University of California at Berkeley in 1939. There, arriving late for one of Neyman's lectures, Dantzig saw two problems on the blackboard and, thinking they were homework assignments (although harder ones than usual), submitted solutions—not realizing that they were unsolved problems (see Dantzig, interviewed in Albers with Reid 1986; Albers, Alexanderson, and Reid 1990). Dantzig thus had his doctoral dissertation and two Annals of Mathematical Statistics articles, the first in 1940 (he also published another article in the same journal that year) and the second, “On the Fundamental Lemma of Neyman and Pearson,” with Abraham Wald in 1950 (reprinted as the first two chapters of Dantzig 2003). Dantzig, occupied with linear programming, might never have published his solution to the second problem had not Wald, with an accepted article already in galley proof, discovered that his contribution had already been made in Dantzig's dissertation. Because the war had brought him to US Army Air Force Headquarters in Washington as a statistician, Dantzig did not submit his dissertation until 1946, more than five years after his 1940 article; but long before his PhD was conferred, he was already singled out as a prodigy among mathematical statisticians.
While Koopmans continued to publish during his time at the Shipping Board, Dantzig, working for the Army Air Force during the war and the Department of the Air Force after the separation of the air force from the army, did not publish from 1940 until 1949. Within the Department of the Air Force, Dantzig was part of a research group directed by Marshall K. Wood. In addition to their joint paper on the programming of interdependent activities that appeared in Econometrica and as the first chapter of the Cowles conference volume (Wood and Dantzig 1949), preceding in each case Dantzig's single-authored, more mathematical paper, Wood contributed chapters to the Cowles conference volume on “representation in a linear model of nonlinear growth curves in the aircraft industry” and, with Murray A. Geisler,9 also of the Department of the Air Force, “development of dynamic models for program planning” (Koopmans et al. 1951: chaps. 13 and 12, respectively). Wood and Geisler emphasized that “the work presented here is the work of the entire staff of the Planning Research Division, Comptroller, U. S. Air Force. . . . We are attempting to solve these problems by the construction of a mathematical model of Air Force operations which can be manipulated with a large scale digital electronic computer or, as an interim measure, with the punched card electrical accounting equipment now available” using “a simplified model of the Berlin airlift” for illustration (Koopmans et al. 1951: 189, 194, 195). Other participants in the Cowles conference listed their affiliation as the Department of the Navy (Walter H. Keen, Fred D. Rigby) or the US Naval Proving Ground (Francis W. Dresch, a Berkeley mathematician who had spoken on general equilibrium at two prewar Cowles conferences), the Federal Reserve Board (Donald Fort), the Department of Commerce (Walter Jacobs), or the Bureau of the Budget (David Rosenblatt, also affiliated with the Carnegie Institute of Technology). Dantzig's development of the simplex algorithm for solving linear programming problems was thus not an idiosyncratic interest of a lone mathematician but had the support of an active research group in the Department of the Air Force and the interest of other areas of the federal government. Such governmental interest and participation had been absent from all the prewar Cowles summer research conferences on economics and statistics (see Dimand 2021).
Interactions and Conferences Leading to the Cowles Commission Monograph
The June 20–24, 1949, Cowles Commission conference on activity analysis in Chicago followed from the Econometric Society's September 7–10, 1948, meeting at the University of Wisconsin–Madison, held jointly with the American Mathematical Society, the Mathematical Association of America, and the Institute of Mathematical Statistics.10 The Cowles Commission and the Econometric Society were intimately connected, even if from 1939 the commission's advisory committee was appointed by the University of Chicago rather than by the Econometric Society. Alfred (Bob) Cowles III, the president and founder of the Cowles Commission, was treasurer of the Econometric Society for a quarter of a century from 1932, the society's secretary from 1937, and business manager of Econometrica, with Cowles Commission administrator Dickson Leavens, and after his retirement in 1948 his successor William Simpson, as managing editor of Econometrica. The organizations, which shared offices, both depended on an annual gift from Cowles, whose grandfather had been one of the founders of the Chicago Tribune. Jacob Marschak, research director of the Cowles Commission from 1943 to 1948 and a leading figure at Cowles until 1960, was president of the Econometric Society in 1946, just after John Maynard Keynes and before Jan Tinbergen. Tjalling Koopmans, research director of the Cowles Commission from 1948 to 1954 and of its successor the Cowles Foundation at Yale from 1961 to 1967, was president of the Econometric Society in 1950.
At the Econometric Society meeting in Madison, the morning of September 8 was devoted to the two pioneers of game theory, with the mathematician, physicist, and computer scientist John von Neumann offering “A Survey of the Theory of Games” complemented by Princeton economist Oskar Morgenstern on “economics and the theory of games.” The afternoon session was a panel discussion chaired by von Neumann, with talks by, among others, Dantzig, Morgenstern, Samuel Karlin, J. C. C. McKinsey of RAND, and future Nobel laureate Lloyd Shapley, then a Princeton graduate student in mathematics, with Harold Hotelling as commentator. The two-paragraph abstract of Dantzig's “Programming in a Linear Structure” (1949a) began by stating that
W. Leontief, [Karl] Schlesinger, [Abraham] Wald, von Neumann, and T. C. Koopmans have studied economic models of the type considered here. This paper differs essentially from those of the above authors in that it is concerned with the basic problem of programming in a rapidly changing “economy.” . . . It is proposed that computational techniques such as those developed by J. von Neumann and by the author be used in connection with large scale digital computers to implement the solution of programming problems. (Dantzig 1949a; 2003: 23)
Two days later, on September 10, Koopmans presented an early version of “Analysis of Production as an Efficient Combination of Activities” (Koopmans et al. 1951: chap. 3). His abstract (Koopmans 1949a) appeared immediately following Dantzig's abstract in the report of the meeting in Econometrica.
But, although the brief abstract in the January 1949 report of the September meeting was the first published mention of his work, Dantzig had been discussing linear programming with Koopmans, and with other Cowles researchers notably future Nobel laureate Leonid Hurwicz, for more than a year before his presentation in Madison. In the opening footnote of his chapter titled “Maximization of a Linear Function of Variables Subject to Linear Inequalities” (Koopmans et al. 1951: 339n1), Dantzig recorded that “his work on this subject stemmed from discussions in the spring of 1947 with Marshall K. Wood, in connection with Air Force programming methods” and that “the general nature of the ‘simplex’ approach (as the method discussed here is known) was stimulated by discussions with Leonid Hurwicz.” But Dantzig went on to state that “the author is indebted to T. C. Koopmans, whose constructive observations regarding properties of the simplex led directly to a proof of the method in the early fall of 1947.” Following these discussions, Dantzig announced the discovery of the simplex method at a meeting of the American Statistical Association on December 29, 1947, and a week later circulated a paper in which he recounted an October 1947 meeting with John von Neumann in which von Neumann emphasized the importance of duality and conjectured that linear programming problems were equivalent to two-person, zero-sum games (Dantzig 1966, 1982; Rees 1979: 31; Dantzig, unpublished 1976 memoir quoted in Dorfman 1984: 291–92 and in Düppe and Weintraub 2014a: 96). Similarly, the acknowledgment footnote at the start of Koopmans's paper, as revised in the subsequent Cowles conference volume, stated that “in conversation, George B. Dantzig introduced me to the wider applicability of models involving constant production coefficients to the discussion of allocation problems” (Koopmans et al. 1951: 33n1).
Dantzig recalled that he first met Koopmans
in Chicago and not earlier than late June 1947. Probably before August 1947 because Leo Hurwicz visited me sometime during the summer of 1947 at Koopmans’ suggestion. I do not have an exact fix either of the date when I first saw Von Neumann's paper on a model of an expanding economy. It was after my first meeting with him in October 1947. I refer to it in my February 1948 paper on Programming in a Linear Structure [presented in September 1948 and abstracted in Dantzig 1949a] but not in my January 1948 paper about my meeting with Von Neumann. (Dantzig to Robert Dorfman, December 6, 1972, copy in Koopmans Papers, Yale University Library 1986, MS 1439, box 10, folder 183; see Dorfman 1984)
Koopmans recalled that “I met Dantzig first in my office in the Cowles Commission in Chicago, where I think he was brought in by Jacob Marschak. Up to that point, I did not know that he was visiting and who he was” (Koopmans to Joseph F. McCloskey, October 5, 1976, Koopmans Papers, MS 1439, box 10, folder 183).
In addition to revised versions of their Econometrica articles, Koopmans and Dantzig wrote additional chapters for the Cowles Commission's conference and proceedings. Koopmans, writing on the “analysis of production as an efficient combination of activities” (chap. 3), turned linear programming into the basis for an activity analysis view of the economy, and, with George W. Brown, offered some concluding “computational suggestions for maximizing a linear function subject to linear inequalities” (chap. 25). Dantzig discussed the “maximization of a linear function of variables subject to linear inequalities” (chap. 21) and, building on Koopmans's model of the transportation problem, offered an “application of the simplex method to a transportation problem” (chap. 23). Much of part 1 of the conference volume focused on linking activity analysis to leading developments in quantitative economics, asserting the place of linear programming and activity analysis in the mainstream, if not of the discipline of economics as a whole, then at least of mathematical economic theory and quantitative economics. Although Wassily Leontief of Harvard was ill and so, although invited, did not attend the conference or contribute a paper, substitutability and other properties of his input-output modeling (Leontief 1941, 1949; cf. Bollard 2020: chap. 6) were considered by five chapters in part 1 of the conference volume, by Harlan Smith, Paul Samuelson,11 Tjalling Koopmans, Kenneth Arrow, and Nicholas Georgescu-Roegen, while, in chapter 4, Georgescu-Roegen related the aggregate linear production function to von Neumann's expanding economy model (von Neumann [1937] 1945).
Four chapters in part 3, “Mathematical Properties of Convex Sets,” linked activity analysis to the game theory established as a promising field of study by von Neumann and Morgenstern (1944) and followed a short chapter by Morgenstern abstracting his On the Accuracy of Economic Observations ([1950] 1963).12 Chapters by David Gale titled “Convex Polyhedral Cones and Linear Inequalities” and by Gale, Harold Kuhn, and Albert Tucker titled “Linear Programming and the Theory of Games” linked the activity analysis conference to the game theory community of Princeton's Mathematics Department and the development of the Kuhn-Tucker conditions for optimization. But the most important chapter in part 3 was perhaps Dantzig's “Proof of the Equivalence of the Programming Problem and the Game Problem.” Von Neumann had asserted the equivalence of the linear programming problem and the solution to a two-person, zero-sum game in his October 1947 conversation with Dantzig, but now Dantzig and Gale, Kuhn, and Tucker had proved that equivalence. Von Neumann had proved, using advanced techniques from topology, the existence of a minimax solution (not necessarily unique) to any two-person, zero-sum game, and von Neumann and Morgenstern (1944) had adopted the nontopological 1938 proof by Jean Ville (translated as an appendix to Ben-El-Mechaiekh and Dimand 2010; see also Ben-El-Mechaiekh and Dimand 2011 for a simpler proof based on Ville). But proving that a solution must exist or else there was a contradiction did not show how to find the solution in general. The initial burst of enthusiasm for game theory expressed in review articles of von Neumann and Morgenstern's book (surveyed by Dimand and Dimand 1995) subsided in the absence of a general method of finding solutions. By proving that any two-person, zero-sum game could be rewritten as a linear programming problem, for which his simplex algorithm would provide a solution, Dantzig provided a general method of solving such games.
Impact and Significance
Tjalling Koopmans (1951) expounded the activity analysis view of “Efficient Allocation Resources” to a joint session of the Econometric Society, the American Economic Association, and the American Statistical Association at the end of December 1949 and in Econometrica, speaking to a mathematically literate audience but without use of mathematical notation. Drawing on his introduction to the Cowles Commission conference volume, Koopmans (1953) summarized activity analysis for a more general audience of economists in the American Economic Review and then again at greater length in his Three Essays on the State of Economic Science (1957), in contrast to Dantzig, who preferred to direct his work to mathematicians through mathematical journals and mathematics conferences. When Koopmans received his Nobel Prize, Lars Werin (1976: 84) endorsed the importance of activity analysis for the wider audience of economists: “The activity set-up has not only penetrated—in fact, made possible—much of the most basic and abstract theory of allocation. It is also becoming the framework of very down-to-earth empirical studies of production and resource allocation phenomena.” In his 1951 article and introduction to the Cowles monograph (but not in the 1953 AER piece), Koopmans related activity analysis to the discussions of existence of general equilibrium in Karl Menger's prewar Vienna mathematics colloquium and to the arguments by Oskar Lange (reprinted in Taylor and Lange 1938) and Abba Lerner, who defended the possibility of rational calculation in a socialist economy against the critiques of Ludwig von Mises (translated in Hayek 1935).
As Düppe and Weintraub (2014a: 82; 2014b) observe, the high level of abstraction of the papers in the Cowles monographs made the use of planning tools and references to Lerner and Lange less conspicuous and controversial. Mathematical techniques of planning became associated with the wartime and Cold War needs of the armed services rather than with socialism.13 But such references did nothing to assuage the concerns about the Cowles Commission's interventionist inclinations held by Koopmans's Chicago colleague Milton Friedman, who had written biting review articles, reprinted in Friedman 1953, of Lerner's Economics of Control (1944) and Lange's Cowles monograph (1945). Friedman restated in his memoirs (Friedman and Friedman 1998) his lasting belief that the mathematicians such as the mathematical economists of the Cowles Commission tended to be planners, interventionists, and socialists because mathematicians became used to thinking themselves capable of find the solution to any problem. His suspicions found support in the examples of Oskar Lange, who took leave from the Cowles Commission to represent the new Communist government of Poland, or of British operations researchers such as Cecil Gordon, who attempted, through the Board of Trade's Special Research Unit, to put operations research to work guiding the economy under Britain's postwar Labour government (Rosenhead 1989, 1991; Budiansky 2013: 255). Friedman was far from alone in his concern about activity analysis and related techniques as the basis for a planned economy. When Charles E. Wilson, president of General Motors, became secretary of defense in the new Republican administration in 1953, he abolished Project SCOOP, abandoned its goal of an interindustry input-output model of the US economy to plan for wartime mobilization, and renamed the air force's Planning Research Division as the Computation Division (Erickson et al. 2013; Klein 2015: 17).
Although, as Düppe and Weintraub (2014a: 101) point out, the participants in the Cowles Commission activity analysis conference, with the sole exception of Paul Samuelson (who was, in 1947, the inaugural winner of the AEA's John Bates Clark Medal), “were marginalized in the larger economics community in the late 1940s,” their position was quite different from that of mathematical economists attending Cowles Commission or Econometric Society conferences in the 1930s (see Dimand 2021) or from the situation of economists with regard to operations research during the war. The mathematical economists in the 1930s were marginalized within economics without being part of any other influential community. Wartime operations research drew on mathematicians, physicists, statisticians, and actuaries but not on economists (except for those regarded as statisticians rather than as economists). Such marginalization was reflected in the financial straits of the Econometric Society, which had assets of twenty-four dollars and thirteen cents when Alfred Cowles became treasurer, and the Cowles Commission, which in 1938 had to abandon its first attempt to recruit Jacob Marschak when its application for a Rockefeller Foundation grant was denied. Linear programming and activity analysis gave mathematical economists access to the funding and support flowing from RAND and ONR to operations research and related fields. It had not occurred to wartime operations researchers, either American or British, that mathematical economists could be relevant to their work, but that changed with activity analysis. The participants in the Cowles activity analysis conference, Samuelson excepted, were marginal to economics in the late 1940s but even counting the number of future Nobel laureates among them (five, plus a sixth invited but unable to attend because of illness) shows that they did not remain marginalized. The funding that activity analysis brought to the mathematical economists, together with identification with the armed forces rather than with socialist planning, helped bring the mathematical economists in from the margin.
Notwithstanding the economic emphasis of the Cowles conference, the research on activity analysis continued to be appreciated in the military context that had done so much to promote operations research. Mina Rees (1977: 111) recalled that
when, in the late 1940’s, the staff of our office [in the Mathematics Division of the Office of Naval Research] became aware of some mathematical results obtained by George Dantzig, who was then working for the Air Force, could be used by the Navy to reduce the burdensome costs of their logistics operations, the possibilities were pointed out to the Deputy Chief of Naval Operations for Logistics. His enthusiasm for the possibilities presented by these results was so great that he called together all those senior officers who had anything to do with logistics, as well as their civilian counterparts, to hear what we always referred to as a “presentation.” The outcome of this meeting was the establishment in the Office of Naval Research of a separate Logistics Branch with a separate research program. This proved to be a most successful activity in the Mathematics Division of ONR, both in its usefulness to the Navy, and in its impact on industry and the universities. Two recent Nobel Laureates in economics, Kenneth Arrow and Tjalling Koopmans, have contributed to the effort.
The military and naval support for operations research extended to activity analysis and mathematical economics as well as to other areas of pure and applied mathematics. War and revolution reinvigorated mathematical economics and econometrics in the United States with European scholars such as Koopmans, Marschak, Haavelmo, and Wald, just as the influx of European scholars refreshed other American academic disciplines and arts. In these ways World War II transformed the hitherto marginal role of mathematical economics in American economics. The part played by Koopmans, Dantzig, activity analysis, and the Cowles Commission conference and monograph, growing out of the wartime involvement of Koopmans at the Shipping Board and the wartime and postwar role of Dantzig with the air force, in that transformation was of greater lasting significance than any of the specific technical advances made in that research.
I am grateful for helpful comments from participants in the HOPE conference “Economists at War,” including Michele Alacevich, François Allisson, Marcel Boumans, Pedro Garcia Duarte, Ariane Dupont-Kieffer, Steven Medema, and E. Roy Weintraub, and from two referees.
Notes
Apart from Kennedy 2013, these references focus solely on Allied military operations research in World War II and its influence on postwar developments in OR, rather than on the full range of wartime applications of mathematics (let alone of science) in general. Several are by participants: the Canadian physicist Lardner and later the British biologist Waddington directed operational research for RAF Coastal Command, Morse antisubmarine operations research for the US Navy. Shrader 2006 is the official history of US Army operations research. Joseph McCloskey’s three 1987 articles survey prewar, wartime British, and wartime American developments in OR. For wartime applications of mathematics generally, and especially the Applied Mathematics Panel of the US Office of Scientific Research and Development, see accounts by leading participants Warren Weaver (1970) and Mina Rees (1977, 1979), and, on Rees’s remarkable role, two works by Amy Shell-Gellasch (2001, 2002). Sapolsky (1990) discusses the Office of Naval Research. The scale of the literature reflects the scale of the wartime effort. For the history of Project SCOOP from its establishment in 1947 to its dissolution in 1953, see the chapter “The Bounded Rationality of Cold War Operations Research” in Erickson et al. 2013.
The trained intuition of economists noticed, as air force officers might not, that destroying places of civilian employment increased the supply of labor to war production. Crossing disciplinary boundaries in the opposite direction, George Dantzig reported in a briefing to the Air Staff in August 1948 that “one ranking mathematical economist at a recent conference at Rand confessed to me that it had remained for Air Force technicians working on the Air Force programming problems to solve one of the most fundamental problems of economics” (quoted by Judy Klein [2015: 11n8]).
Dantzig wrote that “as to the origin of the ‘linear programming,’ Koopmans suggested it and I promoted it” (Dantzig to Robert Dorfman, December 6, 1972, copy in Koopmans Papers, Yale University Library 186, MS 1439, box 10, folder 183). For recollections by pioneers of the development of linear and nonlinear programming, see Lenstra, Kan, and Shrijver 1991.
There were also a few other, isolated publications on similar problems. Koopmans (introduction to Koopmans et al. 1951: 4) emphasized that his “static model of transportation [was] developed, in ignorance of an earlier study by Hitchcock (1941), under the stimulus of statistical work for the Combined Shipping Adjustment Board . . . during the second world war.” Dantzig (in Koopmans et al. 1951: 32) noted that “A. Cahan (1948) has proposed a warehouse problem that can be solved by linear programming techniques” and that “the minimum-cost adequate diet problem was formulated by Jerome Cornfield in 1941 [in an unpublished memorandum] and by G. J. Stigler (1945).” In a footnote added in proof, Koopmans (in Koopmans et al. 1951: 33n) mentioned “a fascinating article by Remak (1929), which contains in intuitive form some of the ideas concerning productive efficiency more fully elaborated in the present chapter and some other chapters of this volume.” Leonid V. Kantorovich ([1939] 1960), in the Soviet Union, was in an altogether different category from these, because, as Koopmans wrote to Joseph McCloskey (October 5, 1976, Koopmans Papers, MS 1439, box 10, folder 183), “He was not a ‘precursor’ but a first inventor. He had the mathematical insight in the crucial role of convexity and of Minkowsky’s separation theorem for convex sets. He also had an amazing foresight of the range of applications.” Hildreth (1986: 76) observed that Kantorovich’s method of “revolving multipliers” was not as general as Dantzig’s simplex method “but did essentially use a Lagrangian formulation and properties of the dual problem that have great theoretical interest.” Kantorovich’s work did not become known to Dantzig and Koopmans until the mid-1950s. On Kantorovich, see Bollard 2020: chap. 5 and references given there.
The manuscript in the Koopmans Papers has an additional thirty-five pages of appendixes, maps, and calculations (Yale University Library MS 1439, box 9, folder 172).
Koopmans remained interested in applications of economic theory to transportation, contributing an introduction and a chapter (Koopmans 1956) to Martin Beckmann et al., Studies in the Economics of Transportation (1956), a Cowles Foundation Special Publication described in the Cowles Foundation’s annual research report as directed by Koopmans (see also Koopmans and Beckmann 1957). However, as pointed out by Ariane Dupont-Kieffer at the HOPE conference, Koopmans’s more general works such as his Three Essays (1957) were prone to neglect transport costs, even when written at the same time as specific studies of highway and rail transport.
According to R. A. Butler, Winston Churchill called Salter “the greatest economist since Jesus Christ” when appointing Salter (by then a Conservative MP, after the abolition of university seats) as minister of state for economic affairs in 1951 (an opinion not shared by Butler, then chancellor of the exchequer). Salter had been a classics scholar, graduating in Literae Humaniores (“Greats”), so his knowledge of economics was acquired on the job as a civil servant, and he had no formal training in mathematics. At the Paris Peace Conference in 1919, Salter was the secretary of the Supreme Economic Council, on which Keynes sat as deputy for the chancellor of the exchequer, and he commented on the proofs of Keynes’s Economic Consequences of the Peace. Salter served with Keynes on the Committee on Economic Information from 1931 to 1937, visited Keynes at Tilton in 1938 to discuss Keynes’s paper on buffer stocks, and from September 1939 took part in weekly meetings at Keynes’s home of the “Old Dogs,” who had been prominent in economic policy in World War I, a series of meetings leading to Keynes’s How to Pay for the War (Skidelsky 2003: 236, 489, 493, 566, 584). See Salter 1967.
Dantzig did not share the 1975 Nobel Memorial Prize with Koopmans and Leonid Kantorovich, presumably being regarded as a mathematician rather than a mathematical economist; but after Koopmans died, it became known that he had only accepted the share of the prize money that he would have received if there had been three winners, anonymously donating the remainder to a research institute closely associated with Dantzig (and with Koopmans), the International Institute for Applied Systems Analysis (Düppe and Weintraub 2014a: 110). An indication of the close friendship between Dantzig and Koopmans is that while other professional peers on first-name terms with Tjalling Charles Koopmans, such as Robert Dorfman, addressed letters “Dear Tjalling,” Dantzig always wrote “Dear Charlie.” Dantzig continued to visit Koopmans and the Cowles Foundation after Cowles moved from Chicago to Yale, for example, giving Cowles Foundation seminars on linear programming in April 1958 and on complementary theory in mathematical programming in April 1966. Marshall Wood, by then with the National Planning Association, took part in a Cowles conference in April 1961 and spoke at Cowles in May 1963 about PARM, a detailed model of the US economy.
Geisler, later head of RAND’s Logistics Systems Laboratory, also presented a single-authored paper titled “Nonlinear Aspects of Air Force Programming” at the conference, but it was not included or referred to in the conference volume. Koopmans remarked in his introduction that “the term ‘linear’ still applies to all the models discussed here,” even though the change of the volume’s title “is intended to convey that the work has in part already outgrown the designation and may be expected to outgrow it further” (Koopmans et al. 1951: 5).
The Institute of Mathematical Statistics and Annals of Mathematical Statistics played a role in the development of mathematical statistics within statistics in the 1930s comparable to the role of the Cowles Commission, the Econometric Society, and Econometrica in the development of econometrics and mathematical economics within economics. Defending “the claim that mathematical statistics began in 1933,” Stephen Stigler ([1996] 1999: 157, 158) remarked in his IMS presidential address that “mathematical statistics as a field is not identical with the Institute of Mathematical Statistics (IMS), but they are coterminous and highly correlated.” As with Cowles in economics, the financial support of a single individual was crucial: “The hero of the day was editor Harry C. Carver, who in January 1934 took over the Annals at his own expense and maintained it without institutional base or support” (160).
In addition to his “Abstract of a Theorem Concerning Substitutability in Open Leontief Models” (Koopmans et al. 1951: 142–46), Samuelson wrote a much longer paper on “market mechanisms and maximization,” which was presented (with discussion by Abba Lerner, one of the few listed formal discussants of a paper at the conference) and was circulated in 1949 as two RAND memoranda, but not published in the volume, presumably because of its length and because, rather than making an original contribution, it drew on memoranda by Koopmans and Dantzig “to provide an exposition of linear programming that students of economic theory can understand” (Samuelson 1966: 425). It was finally published in 1966 in Samuelson’s Collected Scientific Papers (1:425–92). Together with a short monograph by Robert Dorfman (1951), Samuelson’s 1949 RAND memoranda led to Linear Programming and Economic Analysis (Dorfman, Samuelson, and Solow 1958). The conference brought Samuelson into contact with Dorfman (then of the University of California, Berkeley), who contributed a paper titled “Application of the Simplex Method to a Game Theory Problem” (Koopmans et al. 1951: 348–58).
Not only was Morgenstern’s monograph too long for publication as a chapter (101 pages in the 1950 first edition, more than three times that in the second edition), but its message about the limitations of available data cannot have been welcome to an audience with high hopes for quantitative understanding of the economy.
Similarly, the high level of abstraction in the writings of Soviet mathematical economists such as Kantorovich and the labeling of their work as “planometrics” allowed them to engage in disguised reasoning about markets, prices, and incentives to an extent not possible for less-mathematical economists whose work would be intelligible to official ideologues.
