Cardinal utility
Encyclopedia
In economics
, cardinal utility refers to a property of mathematical indices that preserve preference orderings uniquely up to positive linear transformations. Two utility indices are related by a linear transformation if for every value
of one index u, occurring at quantity 
of the goods bundle being evaluated, the corresponding value 
of the other index v satisfies a relationship of the form
,
for fixed constants a and b. Thus the utility functions themselves are related by
The two indices differ only with respect to scale and origin.
Cardinal utility is mostly considered to be an outdated idea. Only within specific contexts such as decision making under risk, utilitarian welfare evaluations
, and discounted utilities for intertemporal evaluations
, cardinal utility is usually accepted. Elsewhere, such as in general consumer theory, ordinal utility
is preferred.
proposed a logarithmic
utility function as a solution to the St. Petersburg paradox
. Logarithmic utility accounts for the diminishing marginal utility of wealth since people consider money gains to be less and less satisfying the more they possess. Attaching an act of human beings (consuming monetary units) to a functional form (the logarithm) was a useful analogy, but it was yet to be linked to the theory of demand.
Jeremy Bentham
believed that some complex introspective examination could somehow measure satisfaction or utility
. Driven by a desire to establish "the principle of utility" as the basis of all legislation and social policy, Bentham wrote a manuscript circa 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"; he also stated that, as these pleasures are perceived to be more and more intense, they may be represented by higher and higher numbers. In the 18th and 19th centuries the issue of the measurability of "utility" received plenty of attention from European schools of political economy, most notably through the work of William Stanley Jevons, Leon Walras
and Alfred Marshall
. Jevons defined utility as a useful object, or something which could increase pleasurable feeling or remove pain. 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. Overall, there was an enthusiasm for measurement in the Victorian era
, and there were many aspects of life succumbing to quantification.
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, and the thinkers of the time were aware of it; otherwise, they would not had 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 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.
Francis Edgeworth felt necessary 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 its ideas were abandoned as the Stanley Steven's Power Law became more accepted.
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 .
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, 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 1934, in a seminal paper for the theory of consumer behaviour under perfect competition and certainty.
, 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.
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.
The confusion between cardinality and measurability was not to be fully solved until the works of Armen Alchian, 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.
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:
During the past decade, economists are having a renewed interest in the measurement issues of happiness.
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.
Suppose you are asked to rank your preferences of
to 
and your preferences of 
to 
. If you find that you can state, for example, that your preference of 
to 
exceeds your 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, you could vary the sum of money represented by 
until you could tell us that you found your preference of 
to 
equal to your preference of 
to 
. If you find 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 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
:
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. Moreover, human senses adapt to a given level of stimulation and then register changes from that baseline.
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. 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 transferred between people; for instance, if John is enjoying certain amount of utility, some of it can be shaven off the top of him and given to Joseph, who would find it 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 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.
, 
, 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 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.
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
Assume that someone has the following preference structure under risk:
By modifiying the values of
and 
in 
, eventually there will be some appropriate values (
) for which she 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 paper 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.
There remain economists who believe that utility can be measured. These measures are not perfect but can act as a proxy for the utility. Lancaster’s characteristics approach to consumer demand illustrates this point.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, cardinal utility refers to a property of mathematical indices that preserve preference orderings uniquely up to positive linear transformations. Two utility indices are related by a linear transformation if for every value
of one index u, occurring at quantity of the goods bundle being evaluated, the corresponding value of the other index v satisfies a relationship of the form,for fixed constants a and b. Thus the utility functions themselves are related by
The two indices differ only with respect to scale and origin.
Cardinal utility is mostly considered to be an outdated idea. Only within specific contexts such as decision making under risk, utilitarian welfare evaluations
Utilitarianism
Utilitarianism is an ethical theory holding that the proper course of action is the one that maximizes the overall "happiness", by whatever means necessary. It is thus a form of consequentialism, meaning that the moral worth of an action is determined only by its resulting outcome, and that one can...
, and discounted utilities for intertemporal evaluations
Intertemporal choice
Intertemporal choice is the study of the relative value people assign to two or more payoffs at different points in time. Most choices require decision-makers to trade-off costs and benefits at different points in time. These decisions maybe about savings, work effort, education, nutrition,...
, cardinal utility is usually accepted. Elsewhere, such as in general consumer theory, ordinal utility
Ordinal utility
Ordinal utility theory states that while the utility of a particular good or service cannot be measured using a numerical scale bearing economic meaning in and of itself, pairs of alternative bundles of goods can be ordered such that one is considered by an individual to be worse than, equal to,...
is preferred.
History
Modern work in cardinal utility theory began in the 18th century, when Daniel BernoulliDaniel Bernoulli
Daniel Bernoulli was a Dutch-Swiss mathematician and was one of the many prominent mathematicians in the Bernoulli family. He is particularly remembered for his applications of mathematics to mechanics, especially fluid mechanics, and for his pioneering work in probability and statistics...
proposed a logarithmic
Logarithmic scale
A logarithmic scale is a scale of measurement using the logarithm of a physical quantity instead of the quantity itself.A simple example is a chart whose vertical axis increments are labeled 1, 10, 100, 1000, instead of 1, 2, 3, 4...
utility function as a solution to the St. Petersburg paradox
St. Petersburg paradox
In economics, the St. Petersburg paradox is a paradox related to probability theory and decision theory. It is based on a particular lottery game that leads to a random variable with infinite expected value, i.e., infinite expected payoff, but would nevertheless be considered to be worth only a...
. Logarithmic utility accounts for the diminishing marginal utility of wealth since people consider money gains to be less and less satisfying the more they possess. Attaching an act of human beings (consuming monetary units) to a functional form (the logarithm) was a useful analogy, but it was yet to be linked to the theory of demand.
Jeremy Bentham
Jeremy Bentham
Jeremy Bentham was an English jurist, philosopher, and legal and social reformer. He became a leading theorist in Anglo-American philosophy of law, and a political radical whose ideas influenced the development of welfarism...
believed that some complex introspective examination could somehow measure satisfaction or utility
Utility
In economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
. Driven by a desire to establish "the principle of utility" as the basis of all legislation and social policy, Bentham wrote a manuscript circa 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"; he also stated that, as these pleasures are perceived to be more and more intense, they may be represented by higher and higher numbers. In the 18th and 19th centuries the issue of the measurability of "utility" received plenty of attention from European schools of political economy, most notably through the work of William Stanley Jevons, Leon Walras
Léon Walras
Marie-Esprit-Léon Walras was a French mathematical economist associated with the creation of the general equilibrium theory.-Life and career:...
and Alfred Marshall
Alfred Marshall
Alfred Marshall was an Englishman and one of the most influential economists of his time. His book, Principles of Economics , was the dominant economic textbook in England for many years...
. Jevons defined utility as a useful object, or something which could increase pleasurable feeling or remove pain. 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. Overall, there was an enthusiasm for measurement in the Victorian era
Victorian era
The Victorian era of British history was the period of Queen Victoria's reign from 20 June 1837 until her death on 22 January 1901. It was a long period of peace, prosperity, refined sensibilities and national self-confidence...
, and there were many aspects of life succumbing to quantification.
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
Pleasure
Pleasure describes the broad class of mental states that humans and other animals experience as positive, enjoyable, or worth seeking. It includes more specific mental states such as happiness, entertainment, enjoyment, ecstasy, and euphoria...
(or pain
Pain
Pain is an unpleasant sensation often caused by intense or damaging stimuli such as stubbing a toe, burning a finger, putting iodine on a cut, and bumping the "funny bone."...
) seemed awkward, and the thinkers of the time were aware of it; otherwise, they would not had 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 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
Hedonism
Hedonism is a school of thought which argues that pleasure is the only intrinsic good. In very simple terms, a hedonist strives to maximize net pleasure .-Etymology:The name derives from the Greek word for "delight" ....
, led academic circles to be skeptical of this theory.
Francis Edgeworth felt necessary 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
Psychophysics
Psychophysics quantitatively investigates the relationship between physical stimuli and the sensations and perceptions they effect. Psychophysics has been described as "the scientific study of the relation between stimulus and sensation" or, more completely, as "the analysis of perceptual...
. 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.
In the late 19th century, Carl Menger
Carl Menger
Carl Menger was the founder of the Austrian School of economics, famous for contributing to the development of the theory of marginal utility, which contested the cost-of-production theories of value, developed by the classical economists such as Adam Smith and David Ricardo.- Biography :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
Vilfredo Pareto
Vilfredo Federico Damaso Pareto , born Wilfried Fritz Pareto, was an Italian engineer, sociologist, economist, political scientist and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices....
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 .
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, 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
Ordinal utility
Ordinal utility theory states that while the utility of a particular good or service cannot be measured using a numerical scale bearing economic meaning in and of itself, pairs of alternative bundles of goods can be ordered such that one is considered by an individual to be worse than, equal to,...
was put together by John Hicks
John Hicks
Sir John Richard Hicks was a British economist and one of the most important and influential economists of the twentieth century. The most familiar of his many contributions in the field of economics were his statement of consumer demand theory in microeconomics, and the IS/LM model , which...
and Roy Allen
Roy Allen
Roy Allen was an American, born in the north Philadelphia neighborhood of Olney. He was a bomber pilot during World War II shot down over France and sent to Buchenwald concentration camp.-France and captivity:...
in 1934, 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 repeat the results accurately. One hypothetical way to achieve this would be by the use of an hedonometerHedonometer
A hedonometer or hedonimeter is a device used to gauge happiness or pleasure. Conceived of at least as early as 1880, the term was used in 1881 by the economist Francis Ysidro Edgeworth to describe "an ideally perfect instrument, a psychophysical machine, continually registering the height of...
, 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.
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.
The confusion between cardinality and measurability was not to be fully solved until the works of Armen Alchian, 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
Level of measurement
The "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement"...
in the natural sciences. In the case of temperature
Temperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
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
Armen AlchianArmen Albert Alchian is an American economist and an emeritus professor of economics at the University of California, Los Angeles....
, The meaning of utility measurement
During the past decade, economists are having a renewed interest in the measurement issues of happiness.
Preferences in cardinal utility theory
In 1955, Patrick SuppesPatrick Suppes
Patrick Colonel Suppes is an American philosopher who has made significant contributions to philosophy of science, the theory of measurement, the foundations of quantum mechanics, decision theory, psychology, and educational technology...
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.
Suppose you are asked to rank your preferences of
to and your preferences of to . If you find that you can state, for example, that your preference of to exceeds your 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, you could vary the sum of money represented by until you could tell us that you found your preference of to equal to your preference of to . If you find 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 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 introspectionIntrospection
Introspection is the self-observation and reporting of conscious inner thoughts, desires and sensations. It is a conscious and purposive process relying on thinking, reasoning, and examining one's own thoughts, feelings, and, in more spiritual cases, one's soul...
:
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
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. Moreover, human senses adapt to a given level of stimulation and then register changes from that baseline.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.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. 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 transferred between people; for instance, if John is enjoying certain amount of utility, some of it can be shaven off the top of him and given to Joseph, who would find it 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 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.
Applications to 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 derivativeDerivativeIn calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
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
This type of indices involves choices under risk. In this case,, , and , are lotteriesLottery (probability)
In expected utility theory, a lottery is a discrete distribution of probability on a set of states of nature. The elements of a lottery correspond to the probability that a certain outcome arises from a given state of nature...
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 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.
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 thatAssume that someone has the following preference structure under risk:
By modifiying the values of
and in , eventually there will be some appropriate values () for which she 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 paper 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
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.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
There remain economists who believe that utility can be measured. These measures are not perfect but can act as a proxy for the utility. Lancaster’s characteristics approach to consumer demand illustrates this point.
See also
- UtilityUtilityIn economics, utility is a measure of customer satisfaction, referring to the total satisfaction received by a consumer from consuming a good or service....
- Ordinal utilityOrdinal utilityOrdinal utility theory states that while the utility of a particular good or service cannot be measured using a numerical scale bearing economic meaning in and of itself, pairs of alternative bundles of goods can be ordered such that one is considered by an individual to be worse than, equal to,...
- Marginal utilityMarginal utilityIn economics, the marginal utility of a good or service is the utility gained from an increase in the consumption of that good or service...
- Expected utility theory
- Level of measurementLevel of measurementThe "levels of measurement", or scales of measure are expressions that typically refer to the theory of scale types developed by the psychologist Stanley Smith Stevens. Stevens proposed his theory in a 1946 Science article titled "On the theory of scales of measurement"...
External links
- Ordinal utility vs. Cardinal utility
- Murray N. Rothbard, "Towards a Reconstruction of Utility and Welfare Economics"