Editing Cardinal utility (section)

=== 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.<ref>{{cite journal |last1=Bernadelli |first1=H. |date=May 1938 |title=The End of the Marginal Utility Theory? |journal=Economica |volume=5 |issue=18 |page=196 |doi=10.2307/2549021 |jstor=2549021}}</ref> 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.<ref name="Collander, David 2007"/>

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, where two cardinal utility functions are considered the same if they preserve [[Preference (economics)|preference]] orderings uniquely up to positive [[affine transformation]]s.<ref name="Ellsberg, Daniel 1954">{{cite journal |last=Ellsberg |first=Daniel |year=1954 |title=Classic and Current Notions of 'Measurable Utility' |journal=Economic Journal |volume=64 |issue=255 |pages=528–556 |doi=10.2307/2227744 |jstor=2227744}}</ref><ref>{{cite journal |last1=Strotz |first1=Robert |year=1953 |title=Cardinal Utility |journal=American Economic Review |volume=43 |issue=2 |pages=384–397}}</ref> Around the end of the 1940s, some economists even rushed to argue that von Neumann–Morgenstern axiomatization of expected utility had resurrected measurability.<ref name="teaching.ust.hk"/> 

The confusion between cardinality and measurability was not to be solved until the works of [[Armen Alchian]],<ref name="www2.uah.es">{{cite journal |last1=Alchian |first1=Armen A. |date=March 1953 |title=The Meaning of Utility Measurement |url=http://www2.uah.es/econ/MicroDoct/Alchian-Utility%20Measurement_1953.pdf |journal=American Economic Review |volume=43 |issue=1 |pages=26–50 |jstor=1810289 |access-date=2010-03-21 |archive-date=2012-03-21 |archive-url=https://web.archive.org/web/20120321224736/http://www2.uah.es/econ/MicroDoct/Alchian-Utility%20Measurement_1953.pdf |url-status=dead }}</ref> William Baumol,<ref>{{cite journal |last1=Baumol |first1=William |date=December 1958 |title=The Cardinal Utility Which is Ordinal |journal=Economic Journal |volume=68 |issue=272 |pages=665–672 |doi=10.2307/2227278 |jstor=2227278}}</ref> and John Chipman.<ref name="Chipman, John 1960">{{cite journal |last1=Chipman |first1=John |date=April 1960 |title=The Foundations of Utility |journal=Econometrica |volume=28 |issue=2 |pages=215–216 |doi=10.2307/1907717 |jstor=1907717}}</ref> 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 [[level of measurement|scales of measurement]] in the natural sciences.<ref>{{cite journal |last1=Allen |first1=Roy |date=February 1935 |title=A Note on the Determinateness of the Utility Function |journal=Review of Economic Studies |volume=2 |issue=2 |pages=155–158 |doi=10.2307/2967563 |jstor=2967563}}</ref> In the case of [[temperature]] there are two ''degrees of freedom'' for its measurement{{snd}} 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.&nbsp;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{{citation needed|date=August 2017}} the real nature of utility functions:
{{quote
|text=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 {{snd}} or numerical measure of the "thing"{{snd}} 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.
|author=[[Armen Alchian]]
|source=The meaning of utility measurement<ref name="www2.uah.es"/>}}