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III. Utility Theory

Utility theory, over the last generation, has been split into two warring camps: (1) those who cling to the old concept of cardinal, measurable utility, and (2) those who have thrown over the cardinal concept, but have dispensed with the utility concept as well and have substituted an analysis based on indifference curves.

In its pristine form, the cardinalist approach has been abandoned by all but a rearguard. On demonstrated preference grounds, cardinality must be eliminated. Psychological magnitudes cannot be measured since there is no objectively extensive unit — a necessary requisite of measurement. Further, actual choice obviously cannot demonstrate any form of measurable utility; it can only demonstrate one alternative being preferred to another.19

Ordinal Marginal Utility and "Total Utility"

The ordinalist rebels, led by Hicks and Allen in the early 1930s, felt it necessary to overthrow the very concept of marginal utility along with measurability. In doing so, they threw out the Utility baby together with the Cardinal bathwater. They reasoned that marginal utility itself implies measurability. Why? Their notion rested on the implicit neoclassical assumption that the "marginal" in marginal utility is equivalent to the "marginal" of the differential calculus. Since, in mathematics, a total "something" is the integral of marginal "somethings," economists early on assumed that "total utility" was the mathematical integral of a series of "marginal utilities."20 Perhaps, too, they realized that this assumption was essential to a mathematical representation of utility. As a result, they assumed, for example, that the marginal utility of a good with a supply of six units is equal to the "total utility" of six units minus the "total utility" of five units. If utilities can be subjected to the arithmetical operation of subtraction, and can be differentiated and integrated, then obviously the concept of marginal utility must imply cardinally measurable utilities.21

The mathematical representation of the calculus rests on the assumption of continuity, that is, infinitely small steps. In human action, however, there can be no infinitely small steps. Human action and the facts on which it is based must be in observable and discrete steps and not infinitely small ones. Representation of utility in the manner of the calculus is therefore illegitimate.22

There is, however, no reason why marginal utility must be conceived in calculus terms. In human action, "marginal" refers not to an infinitely small unit, but to the relevant unit. Any unit relevant to a particular action is marginal. For example, if we are dealing in a specific situation with single eggs, then each egg is the unit; if we are dealing in terms of six-egg cartons, then each six-egg carton is the unit. In either case, we can speak of a marginal utility. In the former case, we deal with the "marginal utility of an egg" with various supplies of eggs; in the latter, with the "marginal utility of cartons" whatever the supply of cartons of eggs. Both utilities are marginal. In no sense is one utility a "total" of the other.

To clarify the relationship between marginal utility and what has been misnamed "total utility" but actually refers to a marginal utility of a larger- sized unit, let us hypothetically construct a typical value scale for eggs:

Ranks in Value

  • 5 eggs
  • 4 eggs
  • 3 eggs
  • 2 eggs
  • 1 egg
  • 2nd egg
  • 3rd egg
  • 4th egg
  • 5th egg

This is a man's ordinal value, or preference, scale for eggs. The higher the ranking, the higher the value. At the center is one egg, the first egg in his possession. By the Law of Diminishing Marginal Utility (ordinal), the second, third, fourth eggs, and so on, rank below the first egg on his value scale, and in that order. Now, since eggs are goods and therefore objects of desire, it follows that a man will value two eggs more than he will one, three more than he will two, and so on. Instead of calling this "total utility," we will say that the marginal utility of a unit of a good is always higher than the marginal utility of a unit of smaller size. A bundle of 5 eggs will be ranked higher than a bundle of 4 eggs, and so on. It should be clear that the only arithmetic or mathematical relationship between these marginal utilities is a simple ordinal one. On the one hand, given a certain sized unit, the marginal utility of that unit declines as the supply of units increases. This is the familiar Law of Diminishing Marginal Utility. On the other hand, the marginal utility of a larger-sized unit is greater than the marginal utility of a smaller-sized unit. This is the law just underlined. And there is no mathematical relationship between, say, the marginal utility of 4 eggs and the marginal utility of the 4th egg except that the former is greater than the latter.

We must conclude then that there is no such thing as total utility; all utilities are marginal. In those cases where the supply of a good totals only one unit, then the "total utility" of that whole supply is simply the marginal utility of a unit the size of which equals the whole supply. The key concept is the variable size of the marginal unit, depending on the situation.23

A typical error on the concept of marginal utility is a recent statement by Professor Kennedy that "the word 'marginal' presupposes increments of utility" and hence measurability. But the word "marginal" presupposes not increments of utility, but the utility of increments of goods, and this need have nothing to do with measurability.24

Professor Robbins's Problem

Professor Lionel Robbins, in the course of a recent defense of ordinalism, raised a problem which he left unanswered. Accepted doctrine, he declared, states that if difference between utility rankings can be judged by the individual, as well as the rankings themselves, then the utility scale can in some way be measured. Yet, Robbins says, he can judge differences. For example, among three paintings, he can say that he prefers a Rembrandt to a Holbein far less than he prefers a Holbein to a Munnings. How, then, can ordinalism be saved?25 Is he not conceding measurability? Yet Robbins's dilemma had already been answered twenty years earlier in a famous article by Oskar Lange.26 Lange pointed out that in terms of what we would call demonstrated preference, only pure rankings are revealed by acts of choice. "Differences" in rank are not so revealed, and are therefore mere psychologizing, which, however interesting, are irrelevant to economics. To this, we need only add that differences of rank can be revealed through real choice, whenever the goods can be obtained by money. We need only realize that money units (which are characteristically highly divisible) can be lumped in the same value-scale as commodities. For example, suppose someone is willing to pay $10,000 for a Rembrandt, $8,000 for a Holbein and only $20 for a Munnings. Then, his value-scale will have the following descending order: Rembrandt, $10,000; Holbein, $9,000, $8,000, $7,000, $6,000 …, Munnings, $20. We may observe these ranks and no question of the measurability of utilities need arise.

That money and units of various goods can be ranked on one value scale is the consequence of Mises's money-regression theorem, which makes possible the application of marginal utility analysis to money.27 It is characteristic of Professor Samuelson's approach that he scoffs at the whole problem of circularity which money-regression had solved. He falls back on Léon Walras, who developed the idea of "general equilibrium in which all magnitudes are simultaneously determined by efficacious interdependent relations," which he contrasts to the "fears of literary writers" about circular reasoning.28 This is one example of the pernicious influence of the mathematical method in economics. The idea of mutual determination is appropriate in physics, which tries to explain the unmotivated motions of physical matter. But in praxeology, the cause is known: individual purpose. In economics, therefore, the proper method is to proceed from the causing action to its consequent effects.

The Fallacy of Indifference

The Hicksian Revolutionaries replaced the cardinal utility concept with the concept of indifference classes, and for the last twenty years, the economic journals have been rife with a maze of two- and three-dimensional indifference curves, tangencies, "budget lines," and so on. The consequence of an adoption of the demonstrated preference approach is that the entire indifference-class concept, along with the complicated superstructure erected upon it, must fall to the ground.

Indifference can never be demonstrated by action. Quite the contrary. Every action necessarily signifies a choice, and every choice signifies a definite preference. Action specifically implies the contrary of indifference. The indifference concept is a particularly unfortunate example of the psychologizing error. Indifference classes are assumed to exist somewhere underlying and apart from action. This assumption is particularly exhibited in those discussions that try to "map" indifference curves empirically by the use of elaborate questionnaires.

If a person is really indifferent between two alternatives, then he cannot and will not choose between them.29 Indifference is therefore never relevant for action and cannot be demonstrated in action. If a man, for example, is indifferent between the use of 5.1 ounces and 5.2 ounces of butter because of the minuteness of the unit, then there will be no occasion for him to act on these alternatives. He will use butter in larger-sized units, where varying amounts are not indifferent to him.

The concept of "indifference" may be important for psychology, but not for economics. In psychology, we are interested in finding out intensities of value, possible indifference, and so on. In economics, however, we are only interested in values revealed through choices. It is immaterial to economics whether a man chooses alternative A to alternative B because he strongly prefers A or because he tossed a coin. The fact of ranking is what matters for economics, not the reasons for the individual's arriving at that rank.

In recent years, the indifference concept has been subjected to severe criticism. Professor Armstrong pointed out that under Hicks's curious formulation of "indifference," it is possible for an individual to be "indifferent" between two alternatives and yet choose one over the other.30 Little has some good criticisms of the indifference concept, but his analysis is vitiated by his eagerness to use faulty theorems in order to arrive at welfare conclusions, and by his radically behaviorist methodology.31 A very interesting attack on the indifference concept from the point of view of psychology has been leveled by Professor Macfie.32

The indifference theorists have two basic defenses of the role of indifference in real action. One is to cite the famous fable of Buridan's Ass. This is the "perfectly rational" ass who demonstrates indifference by standing, hungry, equidistant from two equally attractive bales of hay.33 Since the two bales are equally attractive in every way, the ass can choose neither one and starves therefore. This example is supposed to indicate how indifference can be revealed in action. It is, of course, difficult to conceive of an ass, or a person, who could be less rational. Actually, he is not confronted with two choices but with three, the third being to starve where he is. Even on the indifference theorists' own grounds, this third choice will be ranked lower than the other two on the individual's value-scale. He will not choose starvation.

If both bundles of hay are equally attractive, then the ass or man, who must choose one or the other, will allow pure chance, such as the flip of a coin, to decide on either one. But then indifference is still not revealed by this choice, for the flip of a coin has enabled him to establish a preference!34

The other attempt to demonstrate indifference classes rests on the consistency-constancy fallacy, which we have analyzed above. Thus, Kennedy and Walsh claim that a man can reveal indifference if, when asked to repeat his choices between A and B over time, he chooses each alternative 50 percent of the time.35

If the concept of the individual indifference curve is completely fallacious, it is quite obvious that Baumol's concept of the "community indifference curve," which he purports to build up from individual curves, deserves the shortest possible shrift.36

The Neo-Cardinalists: the von Neumann-Morgenstern Approach

In recent years, the world of economics has been taken by storm by a neo-cardinalist, quasi-measurement theory of utility. This approach, which has the psychological advantage of being garbed in a mathematical form more advanced than economics had yet known, was founded by von Neumann and Morgenstern in their celebrated work.37 Their theory had the further advantage of being grounded on the most recent and fashionable (though incorrect) developments in the philosophy of measurement and the philosophy of probability. The Neumann-Morgenstern thesis was adopted by the leading mathematical economists and has gone almost unchallenged to this day. The chief consolation of the ordinalists has been the assurance by the neo-cardinalists that their doctrine applies only to utility under conditions of uncertainty, and therefore does not shake the ordinalist doctrine too drastically.38 But this consolation is really quite limited, considering that some uncertainty enters into every action.

The Neumann-Morgenstern theory is briefly as follows: an individual can compare not only certain events, but also combinations of events with definite numerical probabilities for each event. Then, according to the authors, if an individual prefers alternative A to B, and B to C, he is able to decide whether he prefers B or a 50:50 probability combination of C and A. If he prefers B, then his preference of B over C is deduced as being greater than his preference of A over B. In a similar fashion, various combinations of probabilities are selected. A quasi-measurable numerical utility is assigned to his utility scale in accordance with the indifference of utilities of B as compared with various probability combinations of A or C. The result is a numerical scale given when arbitrary numbers are assigned to the utilities of two of the events.

The errors of this theory are numerous and grave:

  1. None of the axioms can be validated on demonstrated preference grounds, since admittedly all of the axioms can be violated by the individual actors.
  2. The theory leans heavily on a constancy assumption so that utilities can be revealed by action over time.
  3. The theory relies heavily on the invalid concept of indifference of utilities in establishing the numerical scale.
  4. The theory rests fundamentally on the fallacious application of a theory of numerical probability to an area where it cannot apply. Richard von Mises has shown conclusively that numerical probability can be assigned only to situations where there is a class of entities, such that nothing is known about the members except they are members of this class, and where successive trials reveal an asymptotic tendency toward a stable proportion, or frequency of occurrence, of a certain event in that class. There can be no numerical probability applied to specific individual events.39

Yet, in human action, precisely the opposite is true. Here, there are no classes of homogeneous members. Each event is a unique event and is different from other unique events. These unique events are not repeatable. Therefore, there is no sense in applying numerical probability theory to such events.40 It is no coincidence that, invariably, the application of the neo-cardinalists has always been to lotteries and gambling. It is precisely and only in lotteries that probability theory can be applied. The theorists beg the entire question of its applicability to general human action by confining their discussion to lottery cases. For the purchaser of a lottery ticket knows only that the individual lottery ticket is a member of a certain-sized class of tickets. The entrepreneur, in making his decisions, is on the contrary confronted with unique cases about which he has some knowledge and which have only limited parallelism to other cases.

  1. The neo-cardinalists admit that their theory is not even applicable to gambling if the individual has either a like or a dislike for gambling itself. Since the fact that a man gambles demonstrates that he likes to gamble, it is clear that the Neumann-Morgenstern utility doctrine fails even in this tailor-made case.41
  2. A curious new conception of measurement. The new philosophy of measurement discards concepts of "cardinal" and "ordinal" in favor of such labored constructions as "measurable up to a multiplicative constant" (cardinal); "measurable up to a monotomic transform" (ordinal); "measurable up to a linear transform" (the new quasi-measurement, of which the Neumann-Morgenstern proposed utility index is an example). This terminology, apart from its undue complexity (under the influence of mathematics), implies that everything, including ordinality, is somehow "measurable." The man who proposes a new definition for an important word must prove his case; the new definition of measurement has hardly done so. Measurement, on any sensible definition, implies the possibility of a unique assignment of numbers which can be meaningfully subjected to all the operations of arithmetic. To accomplish this, it is necessary to define a fixed unit. In order to define such a unit, the property to be measured must be extensive in space, so that the unit can be objectively agreed upon by all. Therefore, subjective states, being intensive rather than objectively extensive, cannot be measured and subjected to arithmetical operations. And utility refers to intensive states. Measurement becomes even more implausible when we realize that utility is a praxeological, rather than a directly psychological, concept.

A favorite rebuttal is that subjective states have been measured; thus, the old, unscientific subjective feeling of heat has given way to the objective science of thermometry.42 But this rebuttal is erroneous; thermometry does not measure the intensive subjective feelings themselves. It assumes an approximate correlation between the intensive property and an objective extensive event — such as the physical expansion of gas or mercury. And thermometry can certainly lay no claim to precise measurement of subjective states: we all know that some people, for various reasons, feel warmer or colder at different times even if the external temperature remains the same.43 Certainly no correlation whatever can be found for demonstrated preference scales in relation to physical lengths. For preferences have no direct physical basis, as do feelings of heat.

No arithmetical operations whatever can be performed on ordinal numbers; therefore, to use the term "measurable" in any way for ordinal numbers is hopelessly to confuse the meaning of the term. Perhaps the best remedy for possible confusion is to avoid using any numbers for ordinal rank; the rank concept can just as well be expressed in letters (A, B, C …), using a convention that A, for example, expresses higher rank.

As to the new type of quasi-measurability, no one has yet proved it capable of existence. The burden of proof rests on the proponents. If an object is extensive, then it is at least theoretically capable of being measured, for an objective fixed unit can, in principle, be defined. If it is intensive, then no such fixed unit can apply, and any assignment of number would have to be ordinal. There is no room for an intermediate case. The favorite example of quasi-measurability that is always offered is, again, temperature. In thermometry, centigrade and Fahrenheit scales are supposed to be convertible into each other not at a multiplicative constant (cardinality) but by multiplying and then adding a constant (a "linear transform"). More careful analysis, however, reveals that both scales are simply derivations from one scale based on an absolute zero point. All we need to demonstrate the cardinality of temperature is to transform both centigrade and Fahrenheit scales into scales where "absolute zero" is zero, and then each will be convertible into the other by a multiplicative constant. Furthermore, the actual measurement in temperature is a measurement of length (say, of the mercury column) so that temperature is really a derived measure based on the cardinally measurable magnitude of length.44

Jacob Marschak, one of the leading members of the Neumann-Morgenstern school, has conceded that the temperature case is inappropriate for the establishment of quasi-measurability, because it is derived from the fundamental, cardinal measurement of distance. Yet, astonishingly, he offers altitude in its place. But if "temperature readings are nothing but distance," what else is altitude, which is solely and purely distance and length?45

  • 19. Mises's priority in establishing this in establishing this conclusion is acknowledged by Professor Robbins; cf. Lionel Robbins, "Robertson on Utility and Scope," Economica (May 1953): 99–111; Mises, Theory of Money and Credit, pp. 38–47 and passim. Mises's role in forging an ordinal marginal utility theory has suffered almost total neglect.
  • 20. The error began perhaps with Jevons. Cf. W. Stanley Jevons, Theory of Political Economy (London: Macmillan, 1888), pp. 49ff.
  • 21. That this reasoning lay at the base of the ordinalists' rejection of marginal utility may be seen in John R. Hicks, Value and Capital, 2nd ed. (Oxford: Oxford University Press, 1946), p. 19. That many ordinalists regret the loss of marginal utility may be seen in the statement by Arrow that: "The older discussion of diminishing marginal utility as aiming for the satisfaction of more intense wants first makes more sense" than the current "indifference-curve" analysis, but that, unfortunately it is "bound up with the untenable notion of measurable utility." Quoted in D.H. Robertson, "Utility and All What?"
  • 22. Hicks concedes the falsity of the continuity assumption but blindly pins his faith on the hope that all will be well when individual actions are aggregated. Hicks, Value and Capital, p. 11.
  • 23. The analysis of total utility was first put forward by Mises, in Theory of Money and Credit, pp. 38–47. It was continued by Harro F. Bernardelli, especially in his "The End of the Marginal Utility Theory?" Economica (May 1938): 206. Bernardelli's treatment, however, is marred by laborious attempts to find some form of legitimate mathematical representation. On the failure of the mathematical economists to understand this treatment of marginal and total, see the criticism of Bernardelli by Paul A. Samuelson, "The End of Marginal Utility: A Note on Dr. Bernardelli's Article," Economica (February 1939): 86–87; Kelvin Lancaster, "A Refutation of Mr. Bernadelli," Economica (August 1953): 259–62. For rebuttals see Bernadelli, "A Reply to Mr. Samuelson's Note," Economica (February 1939): 88–89; and "Comment on Mr. Lancaster's Refutation," Economica (August 1954): 240–42.
  • 24. See Charles Kennedy, "Concerning Utility," Economica (February 1954): 13. Kennedy's article, incidentally, is an attempt to rehabilitate a type of cardinalism by making distinctions between "quantity" and "magnitude," and uasing the Bertrand Russell concept of "relational addition." Surely, this sort of approach falls with one slash of Occam's Razor — the great scientific principle that entities not be multiplied unnecessarily. For a criticism, cf. D.H. Robertson, "Utility and All What?" pp. 668–69.
  • 25. Robbins, "Robertson on Utility and Scope," p. 104.
  • 26. Oskar Lange, "The Determinateness of the Utility Function," Review of Economic Studies (June 1934): 224ff. Unfortunately, Lange balked at the implications of his own analysis and adopted an assumption of cardinality, solely because of his anxious desire to reach certain cherished "welfare" conclusions.
  • 27. See Mises, Theory of Money and Credit, pp. 97–123. Mises replied to critics in Human Action, pp. 405ff. The only further criticism has been that of Gilbert, who asserts that the theorem does not explain how a paper money can be introduced after the monetary system has broken down. Presumably he refers to such cases as the German Rentenmark. The answer, of course, is that such paper was not introduced de novo; gold and foreign exchange existed previously existing moneys. Cf. J.C. Gilbert, "The Demand for Money: the Development of an Economic Concept," Journal of Political Economy(April 1953): 149.
  • 28. Samuelson, Foundations of Economic Analysis, pp. 117–18. For similar attacks on earlier Austrian economists, cf. Frank H. Knight, "Introduction" in Carl Menger, Principles of Economics (Glencoe, Ill.: The Free Press, 1950), p. 23; George J. Stigler, Production and Distribution Theories (New York: Macmillan, 1946), p. 181. Stigler criticizes Bv?hm-Bawerk for spurning "mutual determination" for "the older concept of cause and effect" and explains this by saying that Bv?hm-Bawerk was untrained in mathematics. For Menger's attack on the mutual determination concept, cf. Terence W. Hutchison, A Review of Economic Doctrines, 1870–1929 (Oxford: Clarendon Press, 1953), p. 147.
  • 29. The "indifference theorists" also err in assuming infinitely small steps, essential for their geometric representation but erroneous for an analysis of human action.
  • 30. Wallace E. Armstrong, "The Determinateness of Utility Function," Economic Journal (1939): 453–67. Armstrong's point that indifference is not a transitive relation (as Hicks assumed), only applies to different-sized units of one commodity. Also cf. Armstrong, "A Note on the Theory of Consumers' Behavior."
  • 31. Little, "Reformulation" and "Theory." It is another defect of Samuelson's revealed preference approach that he attempts to "reveal" indifference-curves as well.
  • 32. Alec L. Macfie, "Choice in Psychology and as Economic Assumption," Economic Journal (June 1953): 352–67.
  • 33. Thus, cf. Joseph A. Schumpeter, History of Economic Analysis (New York: Oxford University Press, 1954), pp. 94 n. 1064.
  • 34. Also see Croce's warning about using animal illustrations in analyses of human action. Croce, "Economic Principle I," p. 175.
  • 35. Kennedy, "The Common Sense of Indifference Curves" and "On Descriptions of Consumer's Behavior."
  • 36. William J. Baumol, Welfare Economics and the Theory of the State (1952; Cambridge, Mass.: Harvard University Press, 1965), pp. 47ff.
  • 37. John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, 2nd ed. (Princeton, N.J.: Princeton University Press, 1947), pp. 8, 15–32, 617–32.
  • 38. Thus see the excellent expository article by Armen A. Alchian, "The Meaning of Utility Measurement," American Economic Review (May 1953): 384–397. The leading adherents of the Neumann-Morgenstern approach are Marschak, Friedman, Savage, and Samuelson. Claims of the theory, even at its best, to measure utility in any way have been nicely exploded by Ellsberg, who also demolishes Marschak's attempt to make the theory normative. Ellsberg's critique suffers considerably, however, from being based on the "operational meaning" concept. D. Ellsberg, "Classic and Current Notions of Measurable Utility," Economic Journal (September 1954): 528–56.
  • 39. Richard von Mises, Probability, Statistics, and Truth (New York: Macmillan, 1957). Also Ludwig von Mises, Human Action, pp. 106–17. The currently fashionable probability theories of Rudolf Carnap and Hans Reichenbach have failed to shake the validity of Richard von Mises's approach. Mises refutes them in the third German Edition of his work, unfortunately unavailable in English. See Richard von Mises, Wahrscheinlichkeit, Statistik, und Wahrheit, 3rd ed. (Vienna: J. Springer, 1951). The only plausible critique of Richard von Mises has been that of W. Kneale, who pointed out that the numerical assignment of probability depends on an infinite sequence, whereas in no human action can there be an infinite sequence. This, however, weakens the application of numerical probability even to cases such as lotteries, rather than enabling it to expand into other areas. See also Little, "A Reformulation of the Theory of Consumers' Behavior."
  • 40. Compare Frank Knight's basic distinction between the narrow cases of actuarial "risk" and the more widespread nonactuarial "uncertainty." Frank H. Knight, Risk, Uncertainty, and Profit (2nd ed.; London, 1940). G.L.S. Schackle has also leveled excellent criticism at the probability approach to economics, especially that of Marschak. His own "surprise" theory, however, is open to similar objections; cf. C.F. Carter, "Expectations in Economics," Economic Journal (March 1950): 92–105; G.L.S. Schackle, Expectations in Economics (Cambridge: Cambridge University Press, 1949), pp. 109–23.
  • 41. It is curious how economists have been tempted to discuss gambling by first assuming that the participant doesn't like to gamble. It is on this assumption that Alfred Marshall based his famous "proof" that gambling (because of each individual's diminishing utility of money) is "irrational."
  • 42. Thus, cf. von Neumann and Morgenstern, Theory of Games and Economic Behavior, pp. 16–17.
  • 43. Cf. Morris R. Cohen, A Preface to Logic (New York: Henry Holt, 1944), p. 151.
  • 44. On measurement, see Norman Campbell, What is Science? (New York: Dover, 1952), pp. 109–34; and Campbell An Account of the Principles of Measurement and Calculation (London: Longmans, Green, 1928). Although the above view of measurement is not currently fashionable, it is backed by the weighty authority of Mr. Campbell. A description of the controversy between Campbell and S. Stevens on the issue of measurement of intensive magnitudes was included in the unpublished draft of Carl G. Hempel's Concept Formation, but was unfortunately omitted from Hempel's published Fundamentals of Concept Formation in Empirical Science (Chicago: University of Chicago, 1952). Campbell's critique can be found in A. Ferguson, et al. Interim Report (British Association for the Advancement of Science Final Report, 1940), pp. 331–49.
  • 45. Jacob Marschak, "Rational Behavior, Uncertain Prospects, and Measureability," Econometrica (April 1950): 131.
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