## Quick Reference

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 *U*^{1} be the utility function of consumer 1 and *U*^{2} the utility function of consumer 2. These utilities are non-comparable if the transformation *f*^{1} can be applied to *U*^{1} and a different transformation *f*^{2} to *U*^{2}, with no relationship between *f*^{1} and *f*^{2}. With non-comparability a suitable choice of *f*^{1} and *f*^{2} can change the ranking of initial and transformed utilities (i.e., *U*^{1} > *U*^{2} becomes *f*^{1}(*U*^{1}) < *f*^{2}(*U*^{2}), 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 *U*^{1} ≥ *U*^{2} it must be the case that *f*(*U*^{1}) ≥ *f*(*U*^{2}): 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

*U*^{1}* = *a*_{1} + *bU*^{1}

and

*U*^{2}* = *a*_{2} + *bU*^{2}.

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

*U*^{h}* = *a* + *bU*^{h}

and it is possible for both changes in utility and levels of utility to be compared. See also Arrow's impossibility theorem.

*Subjects:*
Economics.