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In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations.〔Strotz, Robert. (1953). "Cardinal utility". ''American economic review'' Vol. 43, No. 2, pp. 384–397〕 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.〔 Thus if one is concave, so is the other, in which case there is said to be diminishing marginal utility. Thus the use of cardinal utility imposes the assumption that levels of absolute satisfaction exist, so that the magnitudes of increments to satisfaction can be compared across different situations. This contrasts with ordinal utility, in which concavity or convexity of the utility function has no economic relevance. The idea of cardinal utility is considered outdated except for specific contexts such as decision making under risk, utilitarian welfare evaluations, and discounted utilities for intertemporal evaluations where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.〔Köbberling, Veronika. (2006). "Strength of preference and cardinal utility". ''Economic theory'', No. 27, p. 375〕 Elsewhere, such as in general consumer theory, ordinal utility with its weaker assumptions Is preferred because results that are just as strong can be derived. == History == The first one to theorize about the marginal value of money was Daniel Bernoulli in 1738. He assumed that the value of an additional amount is inversely proportional to the pecuniary possessions which a person already owns. Since Bernoulli tacitly assumed that an interpersonal measure for the utility reaction of different persons can be discovered, he was then inadvertedly using an early conception of cardinality. 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 that 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.〔Bernstein, Peter. (1996). ''Against the gods. The remarkable story of risk''. New York: John Wiley and Sons, p. 191〕 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";〔Stigler, George. (1950). ("The development of utility theory. I" ). Journal of political economy Vol. 58, No. 4, pp. 307–327〕 he also stated that, as these pleasures increase in intensity higher and higher numbers could represent them.〔 In the 18th and 19th centuries utility's measurability received plenty of attention from European schools of political economy, most notably through the work of marginalists (e.g. William Stanley Jevons, Léon Walras, Alfred Marshall). However, neither of them offered solid arguments to backup up 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.〔 Walras, too, struggled for many years before he could even attempt to formalize the assumption of measurability. 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 market prices reflected utility, although they did not say much about them being incompatible (i.e. prices are objective measures but 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.〔Stigler, George. (1950). "The development of utility theory. II". Journal of political economy Vol. 58, No. 5, pp. 373–396〕 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.〔Colander, David. (2007). "Retrospectives: Edgeworth's hedonimeter and the quest to measure utility". ''Journal of economic perspectives'', Vol. 21, No. 2, pp. 215–226.〕 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.〔Lewin, Shira. (1996). ( "Economics and psychology: lessons for our own day from the early twentieth century" ). ''Journal of economic literature'' 34 (3), 1293–1323.〕 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,〔Viner, Jacob. (1925a). "The utility concept in value theory and its critics". ''Journal of political economy'' Vol. 33, No. 4, pp. 369–387〕 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.〔Hicks, John and Roy Allen. (1934). "A reconsideration of the theory of value". ''Economica'' Vol. 1, No. 1, pp. 52–76〕 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.〔Kornienko, T. (2004). A cognitive bases for cardinal utility. Retrieved November 4, 2012 from the website of the University of Stirling: http://staff.stir.ac.uk/tatiana.kornienko/tape.pdf, p. 3〕 Since the late 20th century economists are having a renewed interest in the measurement issues of happiness.〔Kahneman, Daniel., Peter Wakker, and Rakesh Sarin. (1997). "Back to Bentham? Explorations of experienced utility?". ''Quarterly journal of economics'' Vol. 112, No. 2, pp. 375–405.〕〔Kahneman, Daniel., Ed Diener and Norbert Schwarz. (1999). Well-being: the foundations of hedonic psychology. New York: Rusell Sage Foundation〕 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. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「cardinal utility」の詳細全文を読む スポンサード リンク
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