Cardinal utility

In economics, a cardinal utility function or scale is a utility index that preserves preference orderings uniquely up to positive affine transformations. 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

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.

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