A simple example of two cardinal utility functions u (first column) and v (second column) whose values in all circumstances are related by v=2u+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 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.[1] Thus if one is concave, so is the other, in which case there is often said to be diminishing marginal utility.
In consumer choice theory, economists originally attempted to replace cardinal utility with the apparently-weaker concept of ordinal utility. Cardinal utility appears to impose the assumption that levels of absolute satisfaction exist, so magnitudes of increments to satisfaction can be compared across different situations. However, economists in the 1950s proved that under mild conditions, ordinal utilities imply cardinal utilities. This result is now known as the von Neumann-Morgenstern utility theorem; many similar utility representation theorems can be proven under different assumptions.
^ abEllsberg, Daniel (1954). "Classic and Current Notions of 'Measurable Utility'". Economic Journal. 64 (255): 528–556. doi:10.2307/2227744. JSTOR 2227744.
^Strotz, Robert (1953). "Cardinal Utility". American Economic Review. 43 (2): 384–397.
cardinalutility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. Two utility...
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ordinal preferences. The expected utility hypothesis imposes limitations on the utility function and makes utilitycardinal (though still not comparable across...
specification. Further, efficiency dispenses with cardinal measures of utility, replacing it with ordinal utility, which merely ranks commodity bundles (with...
economic efficiency despite dispensing with interpersonally-comparable cardinalutility, the hypothesization of which may merely conceal value judgments, and...
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In economics, additive utility is a cardinalutility function with the sigma additivity property.: 287–288 Additivity (also called linearity or modularity)...
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concepts of ordinal and cardinalutility. Cardinalutility allows the relative magnitude of utilities to be discussed, while ordinal utility only implies that...
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Marginal Utility by Reactions to Risk". Econometrica. 13 (4): 319–333. doi:10.2307/1906925. JSTOR 1906925. Harsanyi, J. C. (1953). "CardinalUtility in Welfare...
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