Comparing the welfare of one individual with that of another. The welfare level of an individual is measured by a utility function. Utility can be ordinal so that it is no more than a numbering of indifference curves. An ordinal utility function can be subjected to any monotonic increasing transformation, f, without changing its meaning: the initial utility function U and the transformed utility
U* = f(U)
are equivalent. Utility is cardinal when the initial utility function U is equivalent to the transformed function
U* = a + bU
only under affine transformation. An example of cardinal utility is an expected utility function. Non-comparability means that different transformations can be applied to different consumers' utilities. Let U1 be the utility function of consumer 1 and U2 the utility function of consumer 2. These utilities are non-comparable if the transformation f1 can be applied to U1 and a different transformation f2 to U2, with no relationship between f1 and f2. With non-comparability a suitable choice of f1 and f2 can change the ranking of initial and transformed utilities (i.e., U1 > U2 becomes f1(U1) < f2(U2), so the utility information does not provide a welfare ranking. Utility is comparable when the transformations that can be applied to the utility functions are restricted. The only form of comparability with ordinal utility is ordinal level comparability: the same transformation must be applied to the utility functions of all consumers. Denoting the transformation by f, then if U1 ≥ U2 it must be the case that f(U1) ≥ f(U2): the transformation preserves the ranking of utilities between different consumers. If the underlying utility functions are cardinal, there are two important forms of comparability. For cardinal unit comparability the constant multiplying utility in the transformation must be the same for all consumers, but the constant that is added can differ. For two consumers the transformed utilities are
U1* = a1 + bU1
U2* = a2 + bU2.
This transformation allows gains in utility for one consumer to be measured against losses to another. Cardinal full comparability further restricts the constant a in the transformation to be the same for both consumers. For all consumers the transformed utility becomes
Uh* = a + bUh
and it is possible for both changes in utility and levels of utility to be compared. See also Arrow's impossibility theorem.