# Cardinal utility

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A simple example of two cardinal utility functions of y=2x+3

In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations.[1][2] Two utility indices are related by an affine transformation if for every value ${\displaystyle u(x_{1})}$ of one index u, occurring at quantity ${\displaystyle x_{1}}$ of the goods bundle being evaluated, the corresponding value ${\displaystyle v(x_{1})}$ of the other index v satisfies a relationship of the form

${\displaystyle v(x_{1})=au(x_{1})+b\!}$,

for fixed constants a and b. Thus the utility functions themselves are related by

${\displaystyle v(x)=au(x)+b.}$

The two indices differ only with respect to scale and origin.[1]

The idea of Cardinal utility is considered outdated except for specific contexts such as decision making under risk, utilitarian welfare evaluations, and discounted utilities for intertemporal evaluations where it is still applied.[3] Elsewhere, such as in general consumer theory, ordinal utility is preferred.

## History

Modern work in cardinal utility theory began in the 18th century, when mathematician Daniel Bernoulli noted how the diminishing marginal utility of wealth would satisfactorily explain why a riddling game like the St. Petersburg paradox should not lead to an infinite expected value. Bernoulli thought that a logarithmic utility function accounted well for the diminishing marginal utility of wealth since people consider money gains to be less and less satisfying the more they possess of it. This way, by attaching an act of human beings (consuming monetary units) to a functional form (the logarithm), the rational ground for a theory of utility was laid down, though it was yet to be linked to the economic analysis of demand.

Early theorists of utility considered that it had physically quantifiable attributes. For example, one could tell if doing something would make people better off by looking at their change in utility. In this vein, Jeremy Bentham reflected on how satisfaction or utility could be measured by some complex introspective examination. He stated that utility should guide the design of social policies and laws. Bentham wrote a manuscript c. 1782 where he defines the unit of intensity as "the degree of intensity possessed by that pleasure which is the faintest of any that can be distinguished to be pleasure";[4] he also stated that, as these pleasures increase in intensity higher and higher numbers could represent them.[4] 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 William Stanley Jevons, Léon Walras and Alfred Marshall. Jevons defined utility as a useful object, or something which could increase pleasurable feeling or remove pain.[5] Nonetheless, he was characteristically frank and confused when referring to the measurability of utility, since he argued -in later editions of his works- that it was difficult to imagine how estimations of utility and summations can be made with any approach to accuracy.[4] Overall, there was an enthusiasm for measurement in the Victorian era, and there were many aspects of life succumbing to quantification.[6]

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 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 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: 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 did 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 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.[8][9]

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. 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.[10]

The works and manuals of Vilfredo Pareto, Francis Edgeworth, Irving Fischer, and Eugene Slutsky, were pivotal to the development of utility theory. According to Viner,[11] 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 idea of Pareto and Slutsky that utility did not need to be cardinal. According to Schultz, by 1931 the idea of ordinal utility was not yet embraced by American economists. A theory of ordinal utility was put together by John Hicks and Roy Allen in 1934,[12] in a seminal paper for the theory of consumer behavior under perfect competition and certainty. You can watch someone making an actual choice between a pastrami sandwich or, for the same money, a beef sandwich. By this choice, he reveals that he gets more pleasure from the pastrami.

The term 'cardinal utility' was first used by Hicks and Allen in pages 54–55 of their 1934 paper in Economica.[13] The first treatment of a class of utility functions preserved by affine transformations, though, was made in 1934 by Oskar Lange in his Review of Economic Studies' paper.

In the decade of 1960 Parducci studied human judgements of magnitudes and suggested a range-frequency theory.[14] Since the late 20th century economists are having a renewed interest in the measurement issues of happiness.[15][16] 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.[17] 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 people, diverging according to a law of errors.[8]

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. Around the end of the 1940s, some economists even rushed to argue that von Neumann-Morgenstern axiomatization of expected utility had resurrected measurability.[10]

The confusion between cardinality and measurability was not to be solved until the works of Armen Alchian,[18] William Baumol,[19] and John Chipman.[20] 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.[21] 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.

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 measurement[18]

### Order of preference

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.[22]

Suppose you are asked to rank your preferences of A to B and your preferences of B to C. If you find that you can state, for example, that your preference of A to B exceeds your preference of B to C, we could summarize this information by any triplet of numbers satisfying the two inequalities: UA > UB > UC and UA - UB > UB - UC

If A and B were sums of money, you could vary the sum of money represented by B until you could tell us that you found your preference of A to B equal to your preference of B to C. If you find such a B, then the results of this last operation would be expressed by any triplet of numbers satisfying the relationships: (a) UA > UB' > UC , and (b) 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. In this case, "cardinality" means nothing more being able to give consistent answers to these particular questions. 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 uncertainty[23]

According to this view, those situations where a person just can not 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. Moreover, human senses adapt to a given level of stimulation and then register changes from that baseline.[24]

## Applications

### 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.[25]

Under this framework, actions (including production of goods and provision of services) are judged by their contributions to the subjective wealth of people. 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.

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 people. 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 everyone. 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 people.[26]

### Marginalism

• 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.

### Expected utility theory

Template:Rellink This type of indices involves choices under risk. In this case, A, B, and C, are lotteries associated with outcomes. Unlike Cardinal utility theory under certainty, in which 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 behavior of people 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.[27]

Suppose that certain outcomes are associated with three states of nature, so that x3 > x2 > x1; This set of outcomes, X, 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 L1 > and L2 be two lotteries such that

${\displaystyle L_{1}=(0.6,0,0.4)\!}$
${\displaystyle L_{2}=(0,1,0)\!}$

Assume that someone has the following preference structure under risk:

${\displaystyle L_{1}\succ L_{2}\!}$

By modifying the values of p1 and p3 in L1, eventually there will be some appropriate values (L1') for which she is found to be indifferent between it and L2.

${\displaystyle L_{1}'=(0.5,0,0.5)\!}$

Expected utility theory tells us that:

${\displaystyle UE(L_{1}')=UE(L_{2})\!}$
${\displaystyle p_{1}*u(x_{1})+p_{2}*u(x_{2})+p_{3}*u(x_{3})=p_{1}*u(x_{1})+p_{2}*u(x_{2})+p_{3}*u(x_{3})\!}$
${\displaystyle (0.5)*u(x_{1})+(0.5)*u(x_{3})=p_{2}*u(x_{2})\!}$

In this example from Majumdar[28] it is shown how by fixing the arbitrary origin at x1 (the utility index being =0 at x1, and by choosing the scale so that the utility index at x2=1, then:

${\displaystyle (0.5)*u(x_{1})=1\!}$
${\displaystyle u(x_{1})=2\!}$

Thus the utility function is cardinal (it preserved the order of preference over xi 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.

### Intertemporal utility

Template:Rellink Models of utility with several periods, in which people discount future values of utility, need to employ cardinalism in order to have well-behaved utility functions. According to Paul Samuelson the maximization of the discounted sum of future utilities implies that a person can rank utility differences.[29]

## 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 utility[20]

There remain economists who believe that utility can, if not be measured, proximated somewhat to provide some form of measurement, similar to how prices, which has no uniform unit to provide actual price level, could still be indexed to provide "inflation rate" (which is actually a level of change in the prices of weighted indexed products). These measures are not perfect but can act as a proxy for the utility. Lancaster's[30] characteristics approach to consumer demand illustrates this point.

## References

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