# Trade-Off Between Entropy and Gini Index in Income Distribution

## TL;DR

The paper explores how income inequality metrics like entropy and the Gini index interact, showing that extreme equality can harm societal stability.

## Contribution

The paper introduces a new framework combining entropy and K-spread metrics to better assess income inequality and societal stability.

## Key findings

- Moderate Gini reductions have little effect on entropy, but aggressive equalization harms stability.
- Entropy declines are linked to political instability in countries like Argentina and South Africa.
- Geopolitical powers currently operate in a high stability area according to the proposed metrics.

## Abstract

We investigate the fundamental trade-off between entropy and the Gini index within income distributions, employing a stochastic framework to expose deficiencies in conventional inequality metrics. Anchored in the principle of maximum entropy (ME), we position entropy as a key marker of societal robustness, while the Gini index, identical to the (second-order) K-spread coefficient, captures spread but neglects dynamics in distribution tails. We recommend supplanting Lorenz profiles with simpler graphs such as the odds and probability density functions, and a core set of numerical indicators (K-spread K2/μ, standardized entropy Φμ, and upper and lower tail indices, ξ, ζ) for deeper diagnostics. This approach fuses ME into disparity evaluation, highlighting a path to harmonize fairness with structural endurance. Drawing from percentile records in the World Income Inequality Database from 1947 to 2023, we fit flexible models (Pareto–Burr–Feller, Dagum) and extract K-moments and tail indices. The results unveil a concave frontier: moderate Gini reductions have little effect on entropy, but aggressive equalization incurs steep stability costs. Country-level analyses (Argentina, Brazil, South Africa, Bulgaria) link entropy declines to political ruptures, positioning low entropy as a precursor to instability. On the other hand, analyses based on the core set of indicators for present-day geopolitical powers show that they are positioned in a high stability area.