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.
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.
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.
[5] In the 18th and 19th centuries utility's measurability received plenty of attention from European schools of political economy, most notably through the work of marginalists (e.g., William Stanley Jevons,[6] Léon Walras, Alfred Marshall).
They renamed utility in imaginative ways such as subjective wealth, overall happiness, moral worth, psychic satisfaction, or ophélimité.
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.
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.
[10][11] 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.
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.
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"[15] 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.
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.
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.
[25] 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.
[30] The title of Baumol's paper, "The cardinal utility which is ordinal", expressed well the semantic mess of the literature at the time.
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.
If A and B were sums of money, the agent could vary the sum of money represented by B until he could tell us that he found his degree of preference of A over the revised amount B′ equal to his degree of preference of B′ over C. If he finds such a B′, then the results of this last operation would be expressed by any triplet of numbers satisfying the relationships UA > UB′ > UC and UA − UB′ = UB′ − UC.
Any two triplets obeying these relationships must be related by a linear transformation; they represent utility indices differing only by scale and origin.
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 same would apply to your perception of sound (volume, pitch), light, temperature, and so forth...According to this view, those situations where a person just cannot tell the difference between A and B will lead to indifference not because of a consistency of preferences, but because of a misperception of the senses.
[35] Under this framework, actions (including production of goods and provision of services) are judged by their contributions to the subjective wealth of people.
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.
[39] Some authors have commented on the misleading nature 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.There remain economists who believe that utility, if it cannot be measured, at least can be approximated somewhat to provide some form of measurement, similar to how prices, which have no uniform unit to provide an actual price level, could still be indexed to provide an "inflation rate" (which is actually a level of change in the prices of weighted indexed products).