In economics, a cardinal utility expresses not only which of two outcomes is preferred, but also the intensity of preferences, i.e. how much better or worse one outcome is compared to another.
In consumer choice theory, economists originally attempted to replace cardinal utility with the apparently weaker concept of ordinal utility. Cardinal utility appears to impose the assumption that levels of absolute satisfaction exist, so magnitudes of increments to satisfaction can be compared across different situations. However, economists in the 1940s proved that under mild conditions, ordinal utilities imply cardinal utilities. This result is now known as the von Neumann–Morgenstern utility theorem; many similar utility representation theorems exist in other contexts.
History
In 1738, Daniel Bernoulli was the first to theorize about the marginal value of money. He assumed that the value of an additional amount is inversely proportional to the pecuniary possessions which a person already owns. Since Bernoulli tacitly assumed that an interpersonal measure for the utility reaction of different persons can be discovered, he was then inadvertently using an early conception of cardinality.
Bernoulli's imaginary logarithmic utility function and Gabriel Cramer's function were conceived at the time not for a theory of demand but to solve the St. Petersburg's game. Bernoulli assumed that "a poor man generally obtains more utility than a rich man from an equal gain" an approach that is more profound than the simple mathematical expectation of money as it involves a law of moral expectation.
Early theorists of utility considered that it had physically quantifiable attributes. They thought that utility behaved like the magnitudes of distance or time, in which the simple use of a ruler or stopwatch resulted in a distinguishable measure. "Utils" was the name actually given to the units in a utility scale.
In the Victorian era many aspects of life were succumbing to quantification. The theory of utility soon began to be applied to moral-philosophy discussions. The essential idea in utilitarianism is to judge people's decisions by looking at their change in utils and measure whether they are better off. The main forerunner of the utilitarian principles since the end of the 18th century was Jeremy Bentham, who believed that utility could be measured by some complex introspective examination and that it should guide the design of social policies and laws. For Bentham a scale of pleasure has as a unit of intensity "the degree of intensity possessed by that pleasure which is the faintest of any that can be distinguished to be pleasure"; he also stated that as these pleasures increase in intensity, higher and higher numbers could represent them. Léon Walras, Alfred Marshall). However, neither of them offered solid arguments to support the assumption of measurability. In Jevon's case he added to the later editions of his work a note on the difficulty of estimating utility with accuracy. Marshall was ambiguous about the measurability of hedonism because he adhered to its psychological-hedonistic properties but he also argued that it was "unrealistical" to do so.
Supporters of cardinal utility theory in the 19th century suggested that market prices reflect utility, although they did not say much about their compatibility (i.e., prices being objective while utility is subjective). Accurately measuring subjective pleasure (or pain) seemed awkward, as the thinkers of the time were surely aware. They renamed utility in imaginative ways such as subjective wealth, overall happiness, moral worth, psychic satisfaction, or . 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: it proved impossible to definitively say whether a good is worth 50, 75, or 125 utils to a person, or to two different people. Moreover, the mere dependence of utility on notions of hedonism led academic circles to be skeptical of this theory.
Francis Edgeworth was also aware of the need to ground the theory of utility 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 around the time of World War I, psychologists grew discouraged of it.
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. His numerical examples are "illustrative of ordinal, not cardinal, relationships".
However, there are other interpretations of Carl Menger's work. Ivan Moscati and J. Huston McCulloch argue that Menger was a classical cardinalist, as his numerical examples are not merely illustrative but represent explicit arithmetic proportions of value between economic goods. Arithmetic proportions, sums, and multiplications are inherently cardinal and do not exist in an ordinal paradigm. Menger also explicitly states the following: "Only the satisfaction of our needs has direct and immediate significance to us. In each concrete instance, this significance is measured by the importance of the various satisfactions for our lives and well-being. We next attribute the exact quantitative magnitude of this importance to the specific goods on which we are conscious of being directly dependent for the satisfactions in question"
Around the turn of the 19th century neoclassical economists started to embrace alternative ways to deal with the measurability issue. By 1900, Pareto was hesitant about accurately measuring pleasure or pain because he thought that such a self-reported subjective magnitude lacked scientific validity. He wanted to find an alternative way to treat utility that did not rely on erratic perceptions of the senses. Pareto's main contribution to ordinal utility was to assume that higher indifference curves have greater utility, but how much greater does not need to be specified to obtain the result of increasing marginal rates of substitution.
The works and manuals of Vilfredo Pareto, Francis Edgeworth, Irving Fischer, and Eugene Slutsky departed from cardinal utility and served as pivots for others to continue the trend on ordinality. According to Viner, these economic thinkers came up with a theory that explained the negative slopes of demand curves. Their method avoided the measurability of utility by constructing some abstract indifference curve map.
During the first three decades of the 20th century, economists from Italy and Russia became familiar with the Paretian idea that utility does not need to be cardinal. According to Schultz, by 1931 the idea of ordinal utility was not yet embraced by American economists. The breakthrough occurred when a theory of ordinal utility was put together by John Hicks and Roy Allen in 1934. In fact pages 54–55 from this paper contain the first use ever of the term "cardinal utility". The first treatment of a class of utility functions preserved by affine transformations, though, was made in 1934 by Oskar Lange.
In 1944 Frank Knight argued extensively for cardinal utility. In the decade of 1960 Parducci studied human judgements of magnitudes and suggested a range-frequency theory. Since the late 20th century economists are having a renewed interest in the measurement issues of happiness. This field has been developing methods, surveys and indices to measure happiness.
Several properties of cardinal utility functions can be derived using tools from measure theory and set theory.
Measurability
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 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 repeat the results accurately. One hypothetical way to achieve this would be by the use of a hedonometer, which was the instrument suggested by Edgeworth to be capable of registering the height of pleasure experienced by people, diverging according to a law of errors. Around the end of the 1940s, some economists even rushed to argue that von Neumann–Morgenstern axiomatization of expected utility had resurrected measurability. William Baumol, and John Chipman. The title of Baumol's paper, "The cardinal utility which is ordinal", expressed well the semantic mess of the literature 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. 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 if both the scale and the zero are changed. 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.
Von Neumann and Morgenstern stated that the question of measurability of physical quantities was dynamic. For instance, temperature was originally a number only up to any monotone transformation, but the development of the ideal-gas-thermometry led to transformations in which the absolute zero and absolute unit were missing. Subsequent developments of thermodynamics even fixed the absolute zero so that the transformation system in thermodynamics consists only of the multiplication by constants. According to Von Neumann and Morgenstern (1944, p. 23), "For utility the situation seems to be of a similar nature [to temperature]".
The following quote from Alchian served to clarify once and for all the real nature of utility functions: