Supporters of cardinal utility theory in the 19th century suggested that the amount of utility obtained had to have a repercussion on the market price although they did not say much about the problem of dealing with the subjectivity behind this alleged effect. Accurately measuring subjective pleasure (or pain) seemed to be an awkward goal. Therefore 19th-century's specialized literature on this subject is full with terms that tried to circumvent the -then vague - concept of utility: subjective wealth overall happiness moral worth psychic satisfaction ophlimit among others terms were used. During the second half of the 19th century many studies related to this fictional magnitude -utility- were conducted but the conclusion was always the same: unlike with distance or time one cannot simply use a ruler or stopwatch to observe the number of "utils" obtained (that was the name actually given to the units in a utility scale). Not only it proved impossible to definitively say whether a good is worth 50 75 or 125 utils to a person or to two different people but the mere dependence of utility on notions of hedonism led academic circles to be skeptical of this theory.7

Francis Edgeworth felt necessary to have utility analysis extended into the real world. He discussed the quantitative estimates that a person can make of his own pleasure or the pleasure of others borrowing methods developed in psychology to study hedonic measurement: psychophysics. This field of psychology was built on work by Ernst H. Weber but its ideas were abandoned as the Stanley Steven's Power Law became more accepted.8

In the late 19th century Carl Menger and his followers from the Austrian school of economics undertook the first successful departure from measurable utility in the clever form of a theory of ranked uses. Despite abandoning the thought of quantifiable utility (i.e. psychological satisfaction mapped into the set of real numbers) Menger managed to establish a body of hypothesis about decision-making resting solely on a few axioms of ranked preferences over the possible uses of goods and services. Around the turn of the 19th century a fraction of the marginalists known as the neoclassicals started to embrace alternative ways to deal with the measurability issue: Pareto proposed that a choice-based ordinal utility function be adopted precisely because he was concerned that one could not measure utility as a precise psychological quantity. In a letter to Benedetto Croce he writes (1900 p. 183) "I was worried about the pleasure and that pain which had to be measured because in reality nobody is capable of measuring pleasure. Who can say what pleasure is double another pleasure" Thus in his 1906 Manuale although Pareto does talk of pleasure he does so in such a way as to avoid implying that this pleasure is a well-defined numerical quantity. Shira Lewin Economics and psychology: lessons for our own day from the early twentieth century9 The works and manuals of Vilfredo Pareto Francis Edgeworth Irving Fischer and Eugene Slutsky were pivotal to the development of utility theory. According to Viner10 these economic thinkers were striving to come up with a theory that would explain the negative slopes of demand curves. Their method consisted of rejecting the measurability of utility by pursuing the construction of some abstract indifference curve map. The new theory of an ordinal utility was put together by John Hicks and Roy Allen in 193411 in a seminal paper for the theory of consumer behaviour under perfect competition and certainty. Measurability of utility and "cardinality" A utility function is considered to be measurable if the strength of preference or intensity of liking of a good or service is determined with precision by the use of some objective criteria. For example suppose that the act of eating an apple gives to a person exactly half the pleasure of that of eating an orange. This would be a measurable utility if and only if the test employed for its direct measurement is based on an objective criterion that could let any external observer to accurately reproduce the results.12 One hypothetical way to achieve this would be by the use of an hedonometer which was the instrument suggested by Edgeworth to be capable of registering the height of pleasure experienced by an individual diverging according to a law of errors.13 Before the 1930s the measurability of utility functions was erroneously labeled as cardinality by economists. A different meaning of cardinality was used by economists who followed the formulation of Hicks-Allen. Under this usage the cardinality of a utility function: ...is simply the mathematical property of uniqueness up to a linear transformation. Von Neumann-Morgenstern utility functions and time-discounted utility functions are two examples of this cardinality. When the von Neumann-Morgenstern axiomatization of expected utility was published there was some discussion about whether the authors had resurrected measurability from its prior disrepute. A consensus was then reached that cardinality and measurability (in the above sense) were different concepts and the former in no way implied the latter. Shira Lewin Economics and psychology: lessons for our own day from the early twentieth century9 The confusion between cardinality and measurability was not to be fully solved until the works of Armen Alchian14 William Baumol15 and John Chipman.16 Baumol's paper in particular was titled "The cardinal utility which is ordinal" reflecting the semantic mess existing at the time. It is helpful to consider the same problem as it appears in the construction of scales of measurement in the natural sciences.17 In the case of temperature there are two degrees of freedom for its measurement - the choice of unit and the zero. Different temperature scales map its intensity in different ways. In the celsius scale the zero is chosen to be the point where water freezes and likewise in cardinal utility theory one would be tempted to think that the choice of zero would correspond to a good or service that brings exactly 0 utils. However this is not necessarily true. The mathematical index remains cardinal even if the zero gets moved arbitrarily to another point or if the choice of scale is changed or both. Every measurable entity maps into a cardinal function but not every cardinal function is the result of the mapping of a measurable entity. The point of this example was used to prove that (as with temperature) it is still possible to predict something about the combination of two values of some utility function even if the utils get transformed into entirely different numbers as long as it remains a linear transformation. The following quote from Alchian served to clarify once and for all the real nature of utility functions emphasizing that they no longer need to be measurable: Can we assign a set of numbers (measures) to the various entities and predict that the entity with the largest assigned number (measure) will be chosen If so we could christen this measure "utility" and then assert that choices are made so as to maximize utility. It is an easy step to the statement that "you are maximizing your utility" which says no more than that your choice is predictable according to the size of some assigned numbers. For analytical convenience it is customary to postulate that an individual seeks to maximize something subject to some constraints. The thing -or numerical measure of the "thing"- which he seeks to maximize is called "utility". Whether or not utility is of some kind glow or warmth or happiness is here irrelevant; all that counts is that we can assign numbers to entities or conditions which a person can strive to realize. Then we say the individual seeks to maximize some function of those numbers. Unfortunately the term "utility" has by now acquired so many connotations that it is difficult to realize that for present purposes utility has no more meaning than this. Armen Alchian The meaning of utility measurement14 During the past decade economists are having a renewed interest in the measurement issues of happiness.1819 Preferences in cardinal utility theory For more details on this topic see Preference (economics). In 1955 Patrick Suppes and Muriel Winet solved the issue of the representability of preferences by a cardinal utility function and derived the set of axioms and primitive characteristics required for this utility index to work.20 Suppose an individual is asked to rank his preferences of to and his preferences of to . If he finds that he can state for example that his preference of to exceeds his preference of to we could summarize this information by any triplet of numbers satisfying the two inequalities: (a) and (b) . If and were sums of money we could ask the man to vary the sum of money represented by until he could tell us that he found his preference of to equal to his preference of to . If he finds such a then the results of this last operation would be expressed by any triplet of numbers satisfying the relationships: (a) and (b) . Any two triplets obeying these relationships must be related by a linear transformation; they represent utility indices differing only by scale and origin. In this case "cardinality" means nothing more than the subject was able to give consistent answers to these particular questions. In other words there is a rank of goods on the one hand and the quantification of the utility given by those goods on the other. If the function is cardinal then the order of the former must be consistent with the order of the latter. Note that this experiment does not require measurability of utility. Itzhak Gilboa gives a sound explanation of why measurability can never be attained solely by introspection: It might have happened to you that you were carrying a pile of papers or clothes and didn't notice that you dropped a few. The decrease in the total weight you were carrying was probably not large enough for you to notice. Two objects may be too close in terms of weight for us to notice the difference between them. This problem is common to perception in all our senses. If I ask whether two rods are of the same length or not there are differences that will be too small for you to notice. The same would apply to your perception of sound (volume pitch) light temperature and so forth... Itzhak Gilboa Theory of decision under uncertainty21 According to this view those situations where a person just can't tell the difference between and will lead to indifference not because of a consistency of preferences but because of a misperception of the senses. Applications to welfare economics Among welfare economists of the utilitarist school it has been the general tendency to take satisfaction (in some cases pleasure) as the unit of welfare. If the function of welfare economics is to contribute data which will serve the social philosopher or the statesman in the making of welfare judgements this tendency leads perhaps to a hedonistic ethics.22 Under this framework actions (including production of goods and provision of services) are judged by their contributions to the subjective wealth of individuals. In other words it provides a way of judging the "greatest good to the greatest number of persons". An act that reduces one person's utility by 75 utils while increasing two others' by 50 utils each has increased overall utility by 25 utils and is thus a positive contribution; one that costs the first person 125 utils while giving the same 50 each to two other people has resulted in a net loss of 25 utils. Some utilitarians thought that wealth could actually be redistributed to those with lower levels of utility. They also believed that since utility has diminishing marginal returns it could be shaven off the top of an individual and given to another individual who would find the utility more useful. If a class of utility functions is cardinal intrapersonal comparisons of utility differences are allowed. If in addition some comparisons of utility are meaningful interpersonally the linear transformations used to produce the class of utility functions must be restricted across individuals. An example is cardinal unit comparability. In that information environment admissible transformations are increasing affine functions and in addition the scaling factor must be the same for all individuals. This information assumption allows for interpersonal comparisons of utility differences but utility levels cannot be compared interpersonally because the intercept of the affine transformations may differ across individuals.23 Applications to marginal utility For more details on this topic see Marginal utility. Under cardinal utility theory the sign of the marginal utility of a good is the same for all the numerical representations of a particular preference structure. The magnitude of the marginal utility is not the same for all cardinal utility indices representing the same specific preference structure. The sign of the second derivative of a differentiable utility function that is cardinal is the same for all the numerical representations of a particular preference structure. Given that this is usually a negative sign there is room for a law of diminishing marginal utility in cardinal utility theory. The magnitude of the second derivative of a differentiable utility function is not the same for all cardinal utility indices representing the same specific preference structure. Applications to expected utility theory For more details on this topic see Expected utility theory. This type of indices involves choices under risk. In this case and are lotteries associated with outcomes. Unlike Cardinal utility theory under certainty where the possibility of moving from preferences to quantified utility was almost trivial here it is paramount to be able to map preferences into the set of real numbers so that the operation of mathematical expectation can be executed. Once the mapping is done the introduction of additional assumptions would result in a consistent behaviour of the individual regarding fair bets. But fair bets are by definition the result of comparing a gamble with an expected value of zero to some other gamble. Although it is impossible to model attitudes toward risk if one doesn't quantify utility the theory should not be interpreted as measuring strength of preference under certainty.24 Suppose that certain outcomes are associated with three states of nature so that ; This set of outcomes can be assumed to be a calculable money-prize in a controlled game of chance unique up to one positive proportionality factor depending on the currency unit. Let and be two lotteries such that Suppose that the individual has the following preference structure under risk: By modifiying the values of and in eventually there will be some appropriate values () for which the individual is found to be indifferent between it and . Expected utility theory tells us that: Using the information of this example Tapas Majumdar proved in a 1958 paper25 that by fixing the arbitrary origin at (the utility index being 0 at ) and by choosing the scale so that the utility index at then: Thus the utility function is cardinal (it preserved the order of preference over except for an arbitrary choice of origin and scale). This is the same as representing these preferences by a class of utility functions related only by linear transformations. Applications to intertemporal utility For more details on this topic see Intertemporal choice. Models of utility with several periods in which individuals are assumed to discount future values of utility need to employ cardinalism in order to have well-behaved utility functions. Controversies Some authors have commented on the ambiguity of the terms "cardinal utility" and "ordinal utility" as used in economic jargon: These terms which seem to have been introduced by Hicks and Allen (1934) bear scant if any relation to the mathematicians' concept of ordinal and cardinal numbers; rather they are euphemisms for the concepts of order-homomorphism to the real numbers and group-homomorphism to the real numbers John Chipman The foundations of utility16 There remain economists who believe that utility can be measured. These measures are not perfect but can act as a proxy for the utility. Lancasters 26 characteristics approach to consumer demand illustrates this point. See also Utility Ordinal utility Marginal utility Expected utility theory Level of measurement References a b Ellsberg Daniel. (1954). "Classic and current notions of 'Measurable utility'". Economic Journal 64(255): 528-556 Strotz Robert. (1953). "Cardinal utility". American economic review Vol. 43 No. 2 pp. 384-397 Kbberling Veronika. (2006). "Strength of preference and cardinal utility". Economic theory No. 27 p. 375 a b Stigler George. (1950). "The development of utility theory. I". Journal of political economy Vol. 58 No. 4 pp. 307-327 Jevons William Stanley. (1866). "Brief account of a general mathematical theory of political economy". Bernstein Peter. (1996). Against the gods. The remarkable story of risk. New York: John Wiley and Sons p. 191 Stigler George. (1950). "The development of utility theory. II". Journal of political economy Vol. 58 No. 5 pp. 373-396 Colander David. (2007). "Retrospectives: Edgeworth's hedonimeter and the quest to measure utility". Journal of economic perspectives Vol. 21 No. 2 pp. 215-226. a b Lewin Shira. (1996). "Economics and psychology: lessons for our own day from the early twentieth century". Journal of economic literature Vol. 34 No. 3 pp. 1293-1323. Viner Jacob. (1925a). "The utility concept in value theory and its critics". Journal of political economy Vol. 33 No. 4 pp. 369-387 Hicks John and Roy Allen. (1934). "A reconsideration of the theory of value". Economica Vol. 1 No. 1 pp. 52-76 Bernadelli H. (1938). "The end of the marginal utility theory". Economica Vol. 5 No. 18 p. 196 Colander David. (2007). "Retrospectives: Edgeworth's hedonimeter and the quest to measure utility". Journal of economic perspectives Vol. 21 No. 2 pp. 215-226 a b Alchian Armen. (1953). "The meaning of utility measurement". American economic review Vol. 43 No. 1 pp. 26-50. Baumol William. (1958). "The cardinal utility which is ordinal". Economic journal Vol. 68 No. 272 pp. 665-672 a b Chipman John. (1960). "The foundations of utility". Econometrica Vol. 28 No. 2 pp. 215-216 Allen Roy. (1935). "A note on the determinateness of the utility function". Review of economic studies Vol. 2 No. 2 pp. 155-158 Kahneman Daniel. Peter Wakker and Rakesh Sarin. (1997). "Back to Bentham Explorations of experienced utility". Quarterly journal of economics Vol. 112 No. 2 pp. 375-405. Kahneman Daniel. Ed Diener and Norbert Schwarz. (1999). Well-being: the foundations of hedonic psychology. New York: Rusell Sage Foundation Suppes Patrick and Muriel Winet. (1955). "An axiomatization of utility based on the notion of utility differences". Management scienceNo. 1 pp. 259-270 Gilboa Itzhak. (2008). Theory of Decision under uncertainty. Cambridge University Press. Viner Jacob. (1925b). "The utility concept in value theory and its critics. II. The utility concept in welfare economics". Journal of political economy Vol. 33 No. 6 pp. 638-659. Blackorby Charles. Walter Bossert and David Donaldson. (2002). "Utilitarianism and the theory of justice" in: Kenneth Arrow Amartya Sen and Kotaru Suzumura (eds.) Handbook of social choice and welfare. Vol. 1 p. 552 Shoemaker Paul. (1982). "The expected utility model: its variants purposes evidence and limitations". Journal of economic literature Vol. 20 No. 2 pp. 529-563 Majumdar Tapas. (1958). "Behaviourist cardinalism in utility theory". Economica Vol. 25 No. 97 pp. 26-33 Lancaster Kelvin. (1966). "A new approach to consumer theory". Journal of political economy Vol. 74 No. 2 pp. 132-157 External Links Ordinal utility vs. Cardinal utility Murray N. Rothbard "Towards a Reconstruction of Utility and Welfare Economics"