# A cross-country analysis of feasible income equality using the sigmoid function and the Boltzmann distribution

## TL;DR

This study calculates optimal income distributions for 71 countries using a mathematical model, finding that better institutions reduce income inequality gaps.

## Contribution

A new method combining the sigmoid function and Boltzmann distribution to estimate feasible income equality across countries.

## Key findings

- Optimal income distributions and Gini indices across 71 countries show similar patterns.
- Better economic and political institutions correlate with smaller gaps between actual and optimal income distributions.
- A relationship between actual and optimal quintile income shares is estimated for policy use.

## Abstract

This study uses the sigmoid function in combination with the Boltzmann distribution, originally developed by Park and Kim (2021), in order to calculate the optimal income distribution that represents feasible income equality and maximizes total social welfare. Feasible income equality refers to optimal income distribution that is realistically attainable. By employing the data on quintile income shares and the Gini index of 71 countries in 2021 from the World Bank, the results indicate that the optimal income distributions representing feasible income equality, the corresponding values of the Gini index, and the respective shapes of the Lorenz curves of 71 countries are somewhat similar to each other. These results confirm Park and Kim (2021)’s conjecture in that the universal feasible equality line, as depicted by the Lorenz curve, can be identified and applied across multiple countries, potentially serving as a quantitative benchmark. In addition, this study finds that the correlations between the quality of economic and political institutions and the difference between actual and optimal income distributions are negative, suggesting that the better the quality of economic and political institutions is, the closer the gap between actual and optimal income distributions representing feasible income equality. Furthermore, this study estimates the relationship between actual quintile income shares and optimal quintile income shares representing feasible income equality of 71 countries which can be conveniently used to find any approximate level of feasible income share for a particular level of actual income share. Given that high income inequality is associated with health, social, economic, and environmental problems, the overall findings from this study could be useful for designing income redistributive policies and measures.