Cardinal utility

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 ${\displaystyle u(x_{i})}$ of one index u, occurring at any quantity ${\displaystyle x_{i}}$ of the goods bundle being evaluated, the corresponding value ${\displaystyle v(x_{i})}$ of the other index v satisfies a relationship of the form

${\displaystyle v(x_{i})=au(x_{i})+b\!}$,

for fixed constants a and b. Thus the utility functions themselves are related by

${\displaystyle v(x)=au(x)+b.}$

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.