überproportionales

Überproportionales is a term used in German-language discussions of proportional relationships to describe a situation in which a dependent variable grows more than in direct proportion to an independent variable. In other words, when the independent variable increases by a given relative amount, the dependent variable increases by a larger relative amount. The concept is often summarized by the idea of a superproportional or non-linear strengthening effect.

Formally, if y depends on x, y is überproportional to x when the ratio y/x increases as x increases. In mathematical terms this is sometimes captured by a power-law relation y = k x^α with α > 1, or more generally by a convex, nonlinear relationship where the marginal effect of x on y rises with x. This distinguishes it from subproportional (sublinear) relationships, where y grows more slowly than x, and from proportional (linear) relationships, where y/x remains constant.

Applications appear across disciplines. In economics, überproportionales can describe increasing returns to scale, where doubling inputs yields more than double output. In market and network analysis, network effects can generate überproportional growth in value or revenue as the user base expands. In taxation or risk assessment, certain burdens or losses may rise more than proportionally with income or exposure, respectively. The term is useful for describing and analyzing nonlinearities without committing to a single formal model.

See also: proportionality, elasticity, returns to scale, over/proportionality.