Введение
В отличие от ординальной полезности, в экономике
In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value of one index u, occurring at any 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. 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 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 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. 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". 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.
В экономике кардинальная функция полезности или шкала представляет собой индекс полезности, сохраняющий порядок предпочтений однозначно, с точностью до положительных аффинных преобразований. Два индекса полезности связаны аффинным преобразованием, если для значения одного индекса u, соответствующего любому количеству оцениваемого набора товаров, соответствующее значение другого индекса v удовлетворяет соотношению вида
In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value of one index u, occurring at any 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. 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 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 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. 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". 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.
,
In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value of one index u, occurring at any 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. 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 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 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. 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". 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.
где a и b – фиксированные константы. Таким образом, сами функции полезности связаны между собой. Вымышленная логарифмическая функция полезности Бернулли и функция Крамера были разработаны не для теории спроса, а для решения Санкт-Петербургской задачи. Бернулли предположил, что «бедный человек обычно получает больше полезности от равного прироста, чем богатый», – подход, более глубокий, чем простое математическое ожидание денег, поскольку он включает в себя закон морального ожидания. Ранние теоретики полезности считали, что она обладает физически измеримыми атрибутами. Они полагали, что полезность ведет себя как величины расстояния или времени, где простое использование линейки или секундомера дает различимый результат. «Утили» – так фактически назывались единицы измерения в шкале полезности. В викторианскую эпоху многие аспекты жизни поддавались количественной оценке. Теория полезности вскоре стала применяться к дискуссиям в области моральной философии. Суть утилитаризма заключается в оценке решений людей на основе изменения их утилит и определении, стали ли они лучше. Основным предшественником утилитарных принципов с конца XVIII века был Джереми Бентам, который считал, что полезность можно измерить посредством сложного самоанализа и что она должна определять социальную политику и законы. Для Бентама единицей измерения интенсивности в шкале удовольствия является «степень интенсивности, присущая удовольствию, которое является наименьшей из тех, что можно различить как удовольствие»; он также утверждал, что по мере увеличения интенсивности удовольствия более высокие числа могут их представлять. Леон Вальрас, Альфред Маршалл). Однако ни один из них не представил убедительных аргументов в пользу предположения об измеримости. Джевон в более поздних изданиях своей работы добавил примечание о сложности точной оценки полезности. Маршалл занимал неоднозначную позицию в отношении измеримости гедонизма, поскольку он придерживался его психологических гедонистических свойств, но также утверждал, что это «нереалистично». Сторонники теории кардинальной полезности в XIX веке предполагали, что рыночные цены отражают полезность, хотя и не говорили много об их совместимости (то есть об объективности цен и субъективности полезности). Точное измерение субъективного удовольствия (или боли) казалось затруднительным, что, несомненно, осознавали мыслители того времени. Они переименовывали полезность в такие образные понятия, как субъективное богатство, общее счастье, моральная ценность, психическое удовлетворение или офелимите. Во второй половине XIX века было проведено множество исследований, связанных с этой вымышленной величиной – полезностью, – но вывод всегда был один и тот же: невозможно однозначно сказать, стоит ли товар 50, 75 или 125 утиль для одного человека или для двух разных людей. Более того, зависимость полезности от понятий гедонизма вызвала скептицизм в академических кругах. Фрэнсис Эджворт также осознавал необходимость обоснования теории полезности в реальном мире. Он обсуждал количественные оценки, которые человек может сделать относительно своего собственного удовольствия или удовольствия других, заимствуя методы, разработанные в психологии для изучения гедонического измерения: психофизики. Эта область психологии была основана на работах Эрнста Х. Вебера, но во время Первой мировой войны психологи разочаровались в ней. В конце XIX века Карл Менгер и его последователи из Австрийской школы экономики предприняли первый успешный отход от измеримой полезности, в форме теории ранжированного использования. Отказавшись от идеи количественной полезности (то есть психологического удовлетворения, отображенного в множество действительных чисел), Менгеру удалось сформулировать ряд гипотез о принятии решений, основанных исключительно на нескольких аксиомах ранжированных предпочтений относительно возможных вариантов использования товаров и услуг. Его численные примеры «иллюстрируют ординальные, а не кардинальные отношения». Около рубежа XIX–XX веков неоклассические экономисты начали принимать альтернативные подходы к проблеме измеримости. К 1900 году Парето сомневался в точной оценке удовольствия или боли, поскольку считал, что такая субъективная величина, основанная на самоотчетах, не имеет научной ценности. Он хотел найти альтернативный способ рассмотрения полезности, который не зависел бы от непредсказуемых восприятий чувств. Основным вкладом Парето в ординальную полезность было предположение, что более высокие кривые безразличия соответствуют большей полезности, но степень этого увеличения не нужно определять для получения результата в виде убывающей предельной нормы замещения. Работы и учебники Вильфредо Парето, Фрэнсиса Эджворта, Ирвинга Фишера и Эжена Слуцкого отошли от кардинальной полезности и стали отправной точкой для дальнейшего развития ординальности. По мнению Винера, эти экономисты разработали теорию, объясняющую отрицательный наклон кривых спроса. Их метод избегал измеримости полезности путем построения абстрактной карты кривых безразличия. В первые три десятилетия XX века экономисты из Италии и России ознакомились с паретовской идеей о том, что полезность не обязательно должна быть кардинальной. По словам Шульца, к 1931 году идея ординальной полезности еще не была принята американскими экономистами. Прорыв произошел, когда в 1934 году Джон Хикс и Рой Аллен объединили теорию ординальной полезности. Фактически, страницы 54–55 этой работы содержат первое использование термина «кардинальная полезность». Однако первое рассмотрение класса функций полезности, сохраняемых при аффинных преобразованиях, было проведено в 1934 году Оскаром Ланге. В 1944 году Фрэнк Найт убедительно отстаивал кардинальную полезность. В 1960-х годах Пардуччи изучал человеческие суждения о величинах и предложил теорию частотности диапазона. С конца XX века экономисты проявляют возобновленный интерес к проблемам измерения счастья. В этой области разрабатываются методы, опросы и индексы для измерения счастья. Некоторые свойства кардинальных функций полезности можно вывести с помощью инструментов теории меры и теории множеств.
In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility indices are related by an affine transformation if for the value of one index u, occurring at any 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. 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 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 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. 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". 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.
Межвременная полезность
Модели полезности с несколькими периодами, в которых люди дисконтируют будущую полезность, требуют использования кардинализма для обеспечения корректного поведения функций полезности. Согласно Полу Сэмюэлсону, максимизация дисконтированной суммы будущей полезности подразумевает, что человек способен оценивать различия в уровнях полезности.