superproportional

Superproportional describes a relationship in which an output grows faster than proportionally with respect to an input. If x is the input and f(x) is the output, the relationship is superproportional when the ratio f(x)/x increases with x, so higher inputs yield disproportionately larger outputs. In discrete terms, doubling the input produces more than doubling the output: f(2x) > 2 f(x) for relevant x.

Mathematically, many superproportional relationships are modeled by functions with exponent p>1, such as f(x) = a x^p with a>0. Then f(x)/x = a x^{p-1} increases with x, and f(2x) = 2^p a x^p > 2 a x^p = 2 f(x).

Applications of the concept appear in various fields. In economics, a tax or burden may be described as superproportional to income if the marginal impact rises with income, causing the overall burden to grow faster than proportional to earnings. In risk assessment or pharmacology, responses that accelerate with exposure or dose can be characterized as superproportional. In reliability analysis, failure rates that increase more than proportionally with load or stress may also be described this way. The term is typically descriptive and comparative rather than a single formal modeling standard, and it is related to but distinct from concepts like proportionality, convexity, and superlinear growth.

See also: proportional, superlinear, convex function, allometric scaling.