ity to its possessor depend upon the quantity of that commodity alone. 2 Further work, partly induced by hostile criticism, transformed this ‘psychological’ or ‘subjective’ or ‘modern’ theory of value before long. In order to convey the essentials of a story that cannot be told satisfactorily in the space at our command, we shall confine ourselves to a minimum of names and reduce to a sequence of logical steps what actually was a sequence of controversies, which were sometimes as acrimonious as they were pointless. [3. THE CONNECTION WITH UTILITARIANISM] The first task that confronted the sponsors of the ‘new’ theory of value was to defend it against all the misunderstandings—some of them quite puerile—to which it had given rise. 1 Ever fuller restatements resulted—nourished by applications to particular cases, which were not valueless though they were sneered at as futile casuistry—that did something to clear the ground for further advance. For instance the Austrians, who faced German opponents of strongly anti-utilitarian tastes, pretty quickly realized the necessity of clearing their skirts of hedonism. The historical alliance of utility theory with utilitarian philosophy was obvious. We cannot blame men who were no theorists for suspecting that there was also a logical one. Moreover, some of the most prominent exponents of marginal utility were in fact convinced utilitarians: Gossen was, and Jevons, and Edgeworth. They, and others too, had used language that was apt to create the impression that marginal utility theory depended upon utilitarian or hedonist premisses— Bentham certainly thought so—and could be attacked successfully by attacking these. Jevons was the chief culprit: he even went so far as to call economic theory a ‘calculus of pleasure and pain’—Verri had done so before—and elicited from Marshall the rebuke that he was mixing up economics with ‘hedonics.’ It was one of the many merits of Marshall’s treatment of utility that he deplored and renounced the alliance with utilitarianism (see especially his footnote, pp. 77–8 of his Principles, Book I, ch. 5). But in one respect he followed 2 But, unlike Gossen, they did not postulate linearity of the marginal utility function. That this is not a harmless and insignificant detail can be shown by asking ourselves such questions as how a moderate inflation affects the marginal utility of money income for those whose money income remains constant in the process. The answers differ according to the form of the function. And since the straight-line form is certainly unrealistic (except for infinitesimal intervals), the answer derived from it is practically sure to be wrong. See R.Frisch, New Methods of Measuring Marginal Utility (1932). 1 The protagonist of the Austrian group who did most of this work was Böhm-Bawerk. I shall mention only his controversy with Dietzel in the Jahrbücher für Nationalökonomie (1890–92), and both the text of and the appendices to the third edition of his great treatise on capital and interest (Kapital und Kapitalzins). A brilliant and compact survey of arguments and counterarguments has been presented by P.N. Rosenstein-Rodan in the article ‘Grenznutzen’ in the German encyclopedia (Handwörterbuch der Staatswissenschaften, 4th ed., vol. IV, 1927). History of economic analysis 1022 Jevons in teaching a doctrine that comes more naturally from a utilitarian although, again, the relation is one of association rather than logic. From the standpoint of a calculus of pleasure and pain, ‘disutilities’—the term is Jevons’—should be in fact introduced on the same level as utilities. This is what Jevons did. Walras did not do so and the Austrians, Böhm-Bawerk in particular, were strongly opposed to doing so. But Marshall and Pigou kept to the Jevonian standpoint: Marshall developed it into his doctrine of real cost (efforts and sacrifices), which, in a way, was the olive branch presented to his ‘classical’ predecessors. J.B.Clark and, in Vienna, Auspitz and Lieben also accepted it. Notice that this standpoint, however independently arrived at, stands in line with old tradition (compare, e.g., what has been said above on Galiani’s theory of value); and that, outside of the utility theory tradition, it had the support of A.Smith (and of many philosophers of natural law). In England, Cairnes sponsored it, but it was renounced by Wicksteed and, more effectively, by Keynes. The analytic importance of the question lies in its bearing upon the concept of supply of labor and, if we adopt an abstinence theory of interest, of capital. In all other respects it makes little difference whether we take the available amount of labor as given or insert into our system another equation (marginal utility of real wages=marginal disutility of labor) in order to determine it. Actually it is not difficult to show that the utility theory of value is entirely independent of any hedonist postulates or philosophies. For it does not state or imply anything about the nature of the wants or desires from which it starts. 2 [4. PSYCHOLOGY AND THE UTILITY THEORY] Once we recognize the purely formal character of the theorist’s utility concept, we are naturally led to question the relations between the utility theory of value and psychology. Some of the early Austrians seem to have believed that their theory was rooted in psychology and even that they were developing what in essence was a branch of ‘applied psychology.’ This belief was encouraged by some Austrian psychologists such as von Meinong and von Ehrenfels, who held that Menger had made a valuable contribution to psy- 2 We have also seen above that the theory does not imply any hypothesis concerning the role of egotism in human behavior and that it is not particularly ‘individualistic.’ It is, however, interesting to notice, first, how difficult it is for people to realize all this whose whole thinking runs in ‘philosophic’ terms and who are primarily interested in possible philosophic implications; and that this difficulty is greatly increased by the presence of cases where sponsorship of the theory is actually combined with hedonist or individualist philosophies or politics or where, even in the absence of such philosophical or political preferences, the language of an author invites interpretation in a hedonist or individualist sense. In the latter case it may be next to impossible to get rid of undesired associations evoked by the words used. This explains the many attempts that have been made to replace the word utility, which seems to convey more than the fact that a thing is actually being desired, by other terms such as desiredness (Fisher) or ophelimity (Pareto). Equilibrium analysis 1023 chology which was capable of more general application. Certain applications, for instance to the psychology of religion, were in fact made which it is impossible to report without a smile though they were far from being nonsense. Thus, von Ehrenfels actually spoke of marginal piety and of the marginally pious individual. But many non-Austrian economists, who sympathized with the Austrian theory, also thought (and even think) a great deal of the importance of its psychological aspects. On this compare: Maurice Roche-Agussol, La Psychologie économique chez les Anglo-Américains (1918) and Étude bibliographique des sources de la psychologie économique (1919); also the same author’s ‘Psychologische Ökonomie in Frankreich,’ Zeitschrift für Nationalökonomie, May 1929 and January 1930. Let us note in passing a side issue that has never received the attention it deserves. If psychology is to render effective assistance to economics at all, economists must not, of course, neglect experimental psychology and especially the work that turns upon the measurement of sensations. It is, to say the least, a curious fact that one of the early exploits in this field, the one that was undertaken by E.H.Weber, has led to a result amplified by G.T.Fechner (see above ch. 3, sec. 3) into the ‘fundamental law of psycho- physics,’ which is formally identical with the Bernoulli-Laplace hypothesis about the marginal utility of income: it postulates that, if y be the intensity of sensation, x the physically measurable external stimulus, and k an individual constant, then dy=k dx/x. This was in fact noticed by some economists. But it was brushed aside by the leading Austrians, Wieser, for example, declaring (Theorie der gesellschaftlichen Wirtschaft, 1) that this law had nothing whatever to do with Gossen’s law of satiable wants. But however that may be, the efforts of psychologists to measure psychical quantities is not a matter of indifference to any economist who is not entirely lacking in scientific imagination. For examples of recent progress in the measurement of sensation, see especially Professor S.S.Stevens’ ‘A Scale for the Measurement of a Psychological Magnitude: Loudness,’ Psychological Review, September 1936, and his and J. Volkmann’s, ‘The Relation of Pitch to Frequency,’ American Journal of Psychology, July 1940. But both the Austrians and others soon came to realize that their ‘psychology’ was a mistake: the utility theory of value has much better claim to being called a logic than a psychology of values. Opponents, however, at first did not see this, any more than did adherents. In consequence, the sponsors of the ‘psychological theory of value’ had to face two additional indictments: first, that they were exploring psychological aspects of value- in-use that were irrelevant to the objective facts of the economic process; second, that their psychology was bad. The first indictment has no other basis than a failure to understand the import of the theory. 1 The second would be quite true, if any psy- 1 It was formulated by many Marxists, e.g. by Karl Kautsky in his Preface to Marx’s Theorien über den Mehrwert: the psychological theory describes how individuals feel about the process of valuation which, determined by hyperindividual social forces, runs its course irrespective of these feelings exactly as a railroad accident happens irrespective of what the passengers feel about it. The reader should carefully distinguish between the error in this—which consists in overlooking the measure of success with which the theory explains precisely those very objective facts that this argument holds are beyond its reach—and the perfectly sound principle that the facts of a social process must never be confused with the images of them in the individual psyches. But many non- History of economic analysis 1024 chology were involved in the utility theory of value considered as a theory of economic equilibrium. If we ask how consumers come to behave as they do in all those wider problems of human behavior for which particular psychological propositions become relevant, we must in fact appeal to all that modern professional psychology—of all varieties, from Freudianism to behaviorism—might have to give us. As a rule, however, the necessity of such an appeal does not arise in technical economics—it is different, of course, in economic sociology. Most of us would indeed find it difficult or at least highly inconvenient to avoid entirely all reference to motives, expectations, comparative estimates of present and future satisfactions, and the like, however fervently we might hope for an economic theory that would use nothing but statistically observable facts. But such use of psychical observations must not be confused with the use of methods or results borrowed from professional psychology. Like all other research workers, whatever their field, we take our facts where we find them, irrespective of whether or not they are also dealt with by other sciences. We do not become dilettantes in physics when we use the physical facts that are implied in the classical law of decreasing returns in agriculture. No more do we become dilettantes in psychology—or borrow from professional psychology—when we speak of motives or, for that matter, of wants or satisfactions. But though this practice does not present any problems concerning the relation between economics and psychology, it does present another. Early utility theorists talked about psychical facts with the utmost confidence. They included them in the stockpot of common experience—that source of knowledge of the course of everyday life, no element of which a reasonable man could possibly call into question. But so far as these psychical facts are known to us only from observation of what goes on in our own individual psyches—from introspection—their standing evidently leaves something to desire, even though most of them, such as the satisfaction incident to quenching one’s thirst, are so simple and so little problematical that he who quibbles about them might easily compromise himself in the eyes of men of less delicate methodological conscience. In any case, nobody will deny that it is preferable to derive a given set of propositions from externally or ‘objectively’ observable facts, if it can be done, than to derive the same set of propositions from premisses established by introspection. And, as we shall presently see, this can actually be done in the case of the utility theory of value, at least so long as we do not ask it to do more for us than to furnish the assumptions or ‘restrictions’ that we need within the equilibrium theory of values and prices. This is the Leitmotiv of subsequent developments. 2 Marxists also held that, by its probings into the ‘psychology’ of value-in-use, the utility theory contributed nothing to our understanding of economic processes. For an example see the article ‘Grenznutzen’ by W.Lexis, in the second edition of the Handwörterbuch der Staatswissenschaften. 2 Before proceeding I wish to advert to a type of pseudo-psychology which is nothing but an abuse. Keynes’s well-known psychological law about the propensity to consume is an outstanding example. It avers that both individuals and societies will, if Equilibrium analysis 1025 5. CARDINAL UTILITY Let me repeat once more: in the beginning, utility, both total and marginal, was considered a psychic reality, a feeling that was evident from introspection, independent of any external observation—hence, to repeat this also, not to be inferred from those externally observable facts of behavior in the market which were to be explained by it— and a directly measurable 1 quantity. I believe that this was the opinion of Menger and Böhm-Bawerk. Marshall, though he spoke boldly of utility as a measurable quantity, refined upon this, in the remarkably careful argument of Sections 2–9, Chapter 5, Book I of his Principles, by adopting the weaker assumption that, though we cannot measure utility or ‘motive’ or pleasantness and unpleasantness of sensations directly, we can measure them indirectly by their observable effects, a pleasure for instance by the sum of money a man is prepared to give up in order to obtain it rather than go without it. 2 This was no doubt a step in advance. But we shall henceforth merge both these theories of utility measurement into one conception which we shall call (the theory of) Cardinal Utility. Both present difficulties and are open to objection. But neither is simply nonsense. However, even on this level and apart from mere defense and elaboration, there was plenty to do. In order to illustrate, I shall mention three contributions of major importance. First, none of the founding fathers, not even Walras, had bestowed adequate care upon fundamentals. 3 The theory badly needed rigorous restatement. This was accomplished, in a manner that anticipated they experience an increase in income, normally increase their expenditure or consumption but by less than the increase in income. Whether this is so or not, it is a statement of statistically observable fact which Keynes raised to the rank of an assumption. Nothing is gained, except a spurious dignity, by calling it a psychological law. Our experience with such ‘laws of human nature,’ from the seventeenth century on, is certainly not encouraging. But even Jevons would not do without them (Theory of Political Economy, p. 59). 1 The meaning of direct measurability is best instanced by the measurement of length. It may be defined as the association, with every utility sensation, of a real number, unique except for the choice of a unit which is to be interpreted as a unit sensation. Nobody held that this could be done as easily as it can in the case of length. But some authors did hold that there was no difficulty of principle involved. The presence of a practical difficulty—that would reduce utility measurements to rough ‘esti-mates’—was recognized by Böhm-Bawerk (Kapital und Kapitalzins, 3rd ed., Appendix). 2 He guarded this carefully against circularity. The exact definition of measurability in this sense would run as follows: it is possible to associate, with every utility sensation, a real number, unique except for the choice of a unit, which is to be interpreted as a unit quantity of an externally observable incentive producing an externally observable reaction. An illustrative analogy, which is however not quite satisfactory, is provided by the method of measuring heat by means of a thermometer. 3 This may surprise readers who remember the prolix commentaries of the Austrians. But then Wieser and Böhm-Bawerk were fatally handicapped by their lack of the necessary mathematics. History of economic analysis 1026 many a later performance, by Antonelli. 4 Second, Edgeworth did away with the assumption that the utility of every commodity is a function of the quantity of this commodity alone, and made the utility enjoyed by an individual a function of all the commodities that enter his budget. Marshall welcomed this step coldly (to say the least), perhaps because he thought of the mathematical complications involved in making the equations of utility theory partial instead of ordinary differential equations. As a third example we choose Marshall’s attempt to make the measurement of utility operational by means of the concept of Consumer’s Rent. The term Consumers’ Surplus or Rent is Marshall’s, but the essential idea—not every detail—is Dupuit’s. The reader should, if necessary, refresh his memory from Principles, Book III, Chapter 6, so that this space may be reserved for comments. There, Marshall does not mention Dupuit’s name, and only inadequate amends are made for this by means of a statement occurring in another and far distant place (Book V, ch. 12, concluding footnote), namely, that ‘the graphic method has been applied in a manner somewhat similar to that adopted in the present chapter by Dupuit in 1844 and, independently, by Fleeming Jenkin in 1871.’ The idea of ‘measuring’ the total utility accruing to an individual from the consumption of a given quantity of a given commodity by the sum of money represented by the definite integral, taken from zero to the given quantity, of his individual demand function (the consumers’ surplus is then the difference between this integral and the price actually paid times the quantity bought) is at first sight open to a number of objections, which were in fact raised but most of which rest upon misunderstandings of Marshall’s meaning. Appreciation of the value of the tool will be best conveyed by a frank recognition of the limitations to which, at least in the original Marshallian formulation, it must be understood to be subject. First, it was meant to be essentially a tool of partial analysis; the price of one commodity only is made to vary, all other prices being kept constant. Second, even within this range, the concept of consumers’ rent embodies a method of approximation (though it may be exact in certain cases). For it assumes that the marginal utility of income does not change if the individual, having acquired a first unit of the commodity in question for, say, $100, a second for, say, $99, a third one for, say, $90, goes on spending more and more money on additional units as they are offered to him at decreasing prices. Strictly, this is inadmissible. But if this expenditure is but a small part of his total expenditure—so that his other expenditures are not perceptibly affected by this one—we may neglect, as of second order of magnitude, the variations in the marginal utility of income that actually occur. Of course, this limits the method severely: it cannot be applied to such things as food in general or house room or it can be applied only to small ranges of variations in the prices of these, and Marshall knew why he used tea as an example by which to display it. But within these limits the method is neither incorrect nor valueless. Even the sum of all consumers’ rents enjoyed by an individual—a concept that looked absurd to some critics—and the sum of all consumers’ rents enjoyed by all individuals buying an 4 G.B.Antonelli, Sulla teoria matematica della economia politica (1886). Equilibrium analysis 1027 individual commodity may be made to carry meaning by means of further assumptions that are not worse than others we habitually make. However, consumers’ rent had a bad reception from the first, and Professor Pigou, who developed Marshall’s teachings so faithfully in other respects, did not throw the weight of his authority into its scale. But of late, Professor Hicks, impressed by its usefulness in welfare economics (see below, sec. 8), recalled it—or something like it—from the limbo of dead issues to what looks like another lease on life. See his note to Chapter 2 of Value and Capital and his articles ‘The Rehabilitation of Consumers’ Surplus’ (Review of Economic Studies, February 1941); ‘Consumers’ Surplus and Index Numbers’ (ibid. Summer 1942); and ‘The Four Consumer’s Surpluses’ (ibid. Winter 1943). [Cf. also R.L.Bishop, ‘Consumer’s Surplus and Cardinal Utility,’ Quarterly Journal of Economies, May 1943.] 6. ORDINAL UTILITY Of course, if measurability were the only stumbling block in the way to acceptance of the marginal utility theory, critics could be satisfied by a reformulation that retains the concept of utility or satisfaction but makes it a non-measurable quantity. 1 For there is in fact no compelling necessity of insisting upon measurability so long as we are interested only in a maximum problem: there are means of telling whether or not we are on the top of a hill without measuring the elevation of the place where we stand. And since the objection to measurability was the most serious of the objections that were raised from the first by nonmathematical opponents of the nonmathematical exponents of the marginal utility theory, some of these, Wieser especially, soon discovered that they could afford to yield the point, 2 at least with respect to total, as distinguished from incremental, utility. Pareto, who, after having at first accepted the marginal utility theory in the Walrasian form, turned against it around 1900, 3 also raised primarily this objection which then was anything but new, to wit: ‘show me a utility or satisfaction that is, say, three times as great as another!’ But nobody questioned people’s ability to compare satisfactions expected from the possession of different sets of goods without measuring them, that is to say, people’s ability to array such sets in a unique ‘scale of preference.’ This is what we mean by Ordinal Utility. Only the briefest reference can be made to a point on which economists have not been able to reach agreement to this day. We can, as has just been stated, array hypo- 1 A quantity or magnitude (the Greek ) is defined as anything that is capable of being greater or smaller than some other thing. This property implies only transitivity, asymmetry, and aliorelativity (the last term meaning that no thing can be greater or smaller than itself). It also covers the relation of equality, which is however symmetrical and reflexive (the latter term meaning the opposite of aliorelative). Now, quantity in this very general sense does not imply measurability, which requires fulfillment of two more conditions: (1) that it be possible to define a unit; (2) that it be possible to define addition operationally, i.e. so that it can be actually carried out. 2 This is, I suppose, what Wieser meant when he said that utility had no ‘extension’ but only ‘intensity.’ If my interpretation be correct, this turn of phrase was no doubt highly infelicitous. 3 Pareto’s publications during the nineties, the Cours in particular, are substantially pristine utility theory (or ophelimity theory, as he called it). I think that his change of heart was first revealed in History of economic analysis 1028 the lectures he gave in 1900 at the École des Hautes Études in Paris. The first publication on the new line that I know is his ‘Sunto di alcuni capitoli di un nuovo trattato di economia pura,’ in the March and June numbers of the Giornale degli Economisti, 1900. thetical sets of goods ordinally. Suppose that an individual tells us that he prefers a set of goods (B) to a set of goods (A) and a set of goods (C) to the set of goods (B); therefore he prefers (C) to (A) (transitivity). But can we go further and assume that the increase in satisfaction which, on the showing of the experiment, he must experience when, having been promised (A), he is then promised (B), is capable of being greater or less than, or equal to, the increase of satisfaction he would experience if, having been promised (B), he is then promised (C)? This question is by no means otiose because it has been asserted by some and denied by others that admissibility of this assumption opens a way back to measurability (even though, by itself, it is not sufficient to insure it). We cannot go into this question and must content ourselves with a reference to the three most important papers about it. They are: O.Lange, ‘The Determinateness of the Utility Function’ (Review of Economic Studies, June 1934); P.A.Samuelson, ‘The Numerical Representation of Ordered Classifications and the Concept of Utility’ (ibid. October 1938); and especially F.Alt, ‘Über die Messbarkeit des Nutzens’ (Zeitschrift für Nationalökonomie, June 1936). For readers who can muster sufficient interest in questions of this kind, I shall however add this: the merit of having seen the importance of this assumption is Lange’s. But he failed to see that it was only necessary, but not sufficient, in order to prove the possibility of measurement. Samuelson’s argument points this out correctly. Alt’s argument, however (which was not known to Samuelson), is logically adequate and reduces the problem satisfactorily to one of empirical verification of the seven assumptions involved (which it is true has not been attempted so far). Pareto proceeded to develop the idea of ordinal utility and eventually worked out what must in fairness be considered the fundament of the modern theory of value. 4 He was not quite consistent about it and slid back again and again into the habits of thought he had acquired in his formative years. Further advance was made, however, by Johnson and Slutsky although it was not until 1934 that the job was completely done by Allen and Hicks. 5 Additional prob- 4 See the Appendix to his Manuel in its entirety. But the later article in the French edition of the encyclopaedia of mathematical sciences (Encyclopédie des sciences mathématiques pures et appliquées, 1911), contains several improvements (the earlier article in the German ed. is of no importance). 5 W.E.Johnson, ‘The Pure Theory of Utility Curves,’ Economic Journal, December 1913. This important paper contains several results that should secure for its author a place in any history of our science. But, having apparently been written in ignorance of Pareto’s work, it aroused not unnatural resentment on the part of Italian economists because of its failure to acknowledge Pareto’s priority in most essentials. The Russian economist and statistician Eugen Slutsky, Professor in the University of Kharkov, published in the Giornale degli Economisti, July 1915, an article entitled ‘Sulla teoria del bilancio del consumatore,’ the complete neglect of which outside of Italy may perhaps be excused on account of the conditions prevailing in that year. It keeps to the idea that utility is a quantity, though an unmeasurable one; posits certain assumptions about its properties; and then develops the theory of consumers’ behavior with which little fault can be found so long as that view of utility is accepted. Ample amends for that neglect were made by Equilibrium analysis 1029 Henry Schultz (‘Interrelations of Demand, Price, and Income,’ Journal of Political Economy, August 1935); by R.G.D.Allen (‘Pro fessor Slutsky’s Theory of Consumers’ Choice,’ Review of Economic Studies, February 1936); and by J.R.Hicks, who in Value and Capital gave Slutsky’s name to the lems cropped up in the process, some of them in several different forms, but the familiar outcome may be briefly stated as follows. 6 Cardinal utility had been conceived as a uniquely determined 7 real function of the quantities of commodities (per stated period of time) at the disposal of the individual or household. Ordinal utility cannot be so conceived. But it is still possible to describe its behavior by means of any real function of the same quantities that increases whenever we proceed from any given set of commodities to another which the individual prefers, decreases whenever we proceed from any given set of commodities to another which is less acceptable to the individual, and assumes constant values (does not change) whenever we proceed from any given set of commodities to one which is equally acceptable to the individual—just as the two bundles of hay were to Buridan’s ass. Such a function will represent the individual’s ‘scale of preference’ mentioned above but, unlike the function that represents cardinal utility, it will not do so in a uniquely determined way, because all it is devised to tell us is whether there is increase, decrease, or equality of utility. Everything else about it, any further algebraic or numerical features it may display, is entirely arbitrary and has in fact no economic meaning. Hence, if φ be any such function, 8 any monotonically increasing function of φ, call it f(φ), will do just as well. Pareto called such a function an Index Function (funzione-indice). They were to play the same role in the value theory that works with ordinal utility as had been played by the utility function in the value theory that worked with cardinal utility—in fact, we might call them utility functions that obviate the objection against measurability. As a matter of fact, however, it was not the index function as such, but another construct that became characteristic of this stage of value theory, namely, the indifference surfaces or, in the case of two commodities, the indifference curves (curves of equal choice, curve di scelti uguali). It is very interesting to notice that historically these were independently ‘discovered,’ for purposes that fundamental equation of the modern theory of value. Perusal of Professor Allen’s article, a shining example for what in this book is considered correct behavior in the case of unexpected discovery of predecessors, will tell readers unfamiliar with Italian all there is to know about Slutsky’s performance. No comment is, I trust, necessary on the famous ‘Reconsideration of the Theory of Value’ by Allen and Hicks (Economica, February and May 1934), which marks substantial advance beyond Slutsky. 6 I cannot do more than indicate the most important milestones on the main road. Many other things must go by the board. For instance, part of the development I am trying to describe in the text was paralleled by the thought of the later Austrians, though, owing to the inefficiency of their nonmathematical method, they did not get very far. On these Viennese developments, see A.R.Sweezy, ‘The Interpretation of Subjective Value Theory in the Writings of the Austrian Economists,’ Review of Economic Studies, June 1934. 7 This must of course be qualified in two directions: we are always free to choose a unit and we are always free to choose our zero point. In these two respects cardinal utility is also arbitrary—but not more so than is any other method of measurement. History of economic analysis 1030 8 For technical reasons, we do, however, require certain other properties, such as continuity and differentiability. had nothing to do with ordinal utility, by Edgeworth, 9 who fully accepted the doctrine of cardinally measurable utility. Let us for a moment return to this doctrine. Confining ourselves to the two commodity case, we can then lay off the quantities of these commodities on two of the co-ordinates of a three-dimensional diagram and represent by the third co-ordinate the varying amounts of total utility enjoyed that correspond to all the possible combinations of the two commodities. The result is a utility surface that rises from the origin as the quantities of the two commodities increase, and possibly flattens out later on, presenting a shape not unlike that of a loaf of bread (Pareto called it la colline du plaisir). A succession of horizontal planes—that is, of planes parallel to the plane of the two commodity co-ordinates—will cut out from this loaf curves along which total utility is constant, the quantities of the commodities varying in such a way that the increase of one just compensates the individual for the corresponding decrease of the other. These curves, the whole meaning of which seems to rest upon the assumption that utility is measurable, are what Edgeworth called indifference curves. If we project them on the commodity plane, we get the familiar ‘indifference map.’ Edgeworth used it very elegantly in his theory of barter, particularly in order to delimit the range of possible barter terms or exchange ratios. 10 But so soon as we project the indifference lines on the commodity plane, the utility dimension vanishes from the picture so that their meaning is no longer dependent on any hypothesis of measurability. They then tell us no more than (1) that the individual considers certain combinations of the two commodities as equally eligible and (2) that he prefers combinations represented by any ‘higher’ indifference curve to combinations represented by any ‘lower’ one. The first man to see the implications of this was Irving Fisher. 11 He had no objection to measurability. On the contrary, he tried to make it operational (see below editor’s note between sections 7 and 8). But in doing so, he encountered certain difficulties when, in the second Part of his work, he discarded the untenable assumption that the utility of each good depends on its own quantity only (‘independent goods’). 12 At this point doubts were bound to arise not only about the measurability but also about its very existence. Accordingly, Fisher presented an analysis completely free from utility assumptions that worked only with indifference maps in the modern sense. With him—as later 9 They put in appearance in his Mathematical Psychics (1881) and therefore antedate ordinal-utility analysis of the Pareto type by about twenty years. 10 Marshall was sufficiently impressed with this brilliant piece of work to reproduce the gist of it in a note in the appendix to his Principles. But this is all he had to do with indifference curves. It is incorrect to say that he anticipated the idea by the apparatus of curves which he used in his Pure Theory of Foreign Trade (1879). 11 Mathematical Investigations (see above, ch. 5, sec. 7b). It is not sufficiently recognized that, partly explicitly, partly by implication, this book anticipated the better part of modern value theory. 12 The nature of these difficulties will be indicated in the next footnote. It is my guess—not, I think, a very hazardous one—that it was these difficulties which motivated Marshall’s adherence to the conception of independent goods. Equilibrium analysis 1031 . with the images of them in the individual psyches. But many non- History of economic analysis 1024 chology were involved in the utility theory of value considered as a theory of economic equilibrium ed., vol. IV, 1927). History of economic analysis 1022 Jevons in teaching a doctrine that comes more naturally from a utilitarian although, again, the relation is one of association rather. mathematics. History of economic analysis 1026 many a later performance, by Antonelli. 4 Second, Edgeworth did away with the assumption that the utility of every commodity is a function of the quantity