Affine Virtual Values and the Generalized Pareto distribution

This paper establishes a precise link between the tail behavior of a distribution and the structure of its Virtual Value. We show that the only continuously differentiable distributions whose Virtual Value is linear on an interval coincide, on that interval, with a Generalized Pareto distribution. This equivalence yields explicit expressions for optimal cutoffs and implies that, within this class, comparative statics of allocation and pricing rules are governed by a single primitive, the tail index. Thin tails lead to the familiar monotone adjustment of prices, whereas heavy tails reverse the sign of the response and produce countercyclical pricing. The analysis also demonstrates that widely used textbook distributions acquire an approximately affine Virtual Value in their upper tails whenever their expectations are finite. Under standard regularity conditions from Extreme Value Theory, the tail Virtual Value converges to an affine form, with slope equal to the underlying tail index. These results provide a unified framework for studying allocation, mechanism design and monopoly pricing in environments where high valuations play a central role.