How much income inequality is fair? Nash bargaining solution and its connection to entropy

TL;DR

This paper explores the concept of fair income inequality using the Nash Bargaining Solution, revealing its connection to entropy and showing that ideal income distribution aligns with a lognormal distribution.

Contribution

It introduces a new approach using NBS to derive the same lognormal income distribution, clarifying the economic meaning of NBS as fundamentally about fairness.

Findings

NBS leads to a lognormal income distribution at equilibrium.

NBS's fairness property is central to understanding income inequality.

A connection between Nash product and entropy is established for rational agents.

Abstract

The question about fair income inequality has been an important open question in economics and in political philosophy for over two centuries with only qualitative answers such as the ones suggested by Rawls, Nozick, and Dworkin. We provided a quantitative answer recently, for an ideal free-market society, by developing a game-theoretic framework that proved that the ideal inequality is a lognormal distribution of income at equilibrium. In this paper, we develop another approach, using the Nash Bargaining Solution (NBS) framework, which also leads to the same conclusion. Even though the conclusion is the same, the new approach, however, reveals the true nature of NBS, which has been of considerable interest for several decades. Economists have wondered about the economic meaning or purpose of the NBS. While some have alluded to its fairness property, we show more conclusively that it is all about fairness. Since the essence of entropy is also fairness, we see an interesting connection between the Nash product and entropy for a large population of rational economic agents.