Mapping inequality in power law and exponential distributions

We’ve just published a new paper that advances our longer-running programme on inequality, entropy, and structure. The core move is simple but consequential: instead of collapsing inequality into a single number (Gini, Theil, entropy), we show how to locate inequality inside a distribution. Using an entropy-based decomposition, we turn inequality into a multiscale map that identifies which parts of a system are more uniform, which are more concentrated, and how those contrasts intensify as you move across the distribution.

Applied to two foundational cases—power-law and exponential distributions—we derive closed-form, analytic cutoffs that replace heuristic rules (e.g. “80/20”) with principled thresholds. For power laws, this shows precisely how inequality concentrates along heavy tails. For exponentials (including the Boltzmann distribution), the entropy-defined cutoff turns out to equal the conditional mean, giving it a clear physical interpretation as a balance point between concentration and dispersion. More broadly, the paper pushes our work from global inequality summaries toward a structural, diagnostic view: inequality not as a single score, but as something layered, ordered, and navigable across a system.