The Geometry of Heterogeneous Extremes: Optimal Transport and Entropic Design

Extreme economic outcomes are not shaped by tails alone. They are also shaped by unequal access to opportunities. This paper develops a theory of heterogeneous extremes by taking the distribution of opportunity access as the object of study. In a mixed Poisson search setting, normalized maxima admit a Laplace mixture representation that yields order comparisons and a clean benchmark against the homogeneous economy. The main contribution is geometric: a canonical coupling turns differences in heterogeneity into optimal transport bounds for the whole induced law of extremes, the full schedule of top quantiles, and structured counterfactual paths between economies. The paper also derives a second order expansion that separates classical extreme value approximation error from heterogeneity effects. As a complementary normative exercise, it studies an entropy regularized design problem for reallocating opportunities under a mean constraint. A stylized labor market network application interprets heterogeneity as unequal access to job opportunities and shows how the framework can be used for tail counterfactuals and robustness analysis of top wage distributions.