Variation of Gini and Kolkata Indices with Saving Propensity in the Kinetic Exchange Model of Wealth Distribution: An Analytical Study

TL;DR

This paper analytically investigates how Gini and Kolkata indices vary with saving propensity in a kinetic exchange model of wealth distribution, deriving explicit formulas and comparing them with numerical simulations.

Contribution

It provides the first analytical derivation of Gini and Kolkata indices as functions of saving propensity in a kinetic wealth exchange model.

Findings

Gini and Kolkata indices increase with saving propensity.

Analytical expressions match well with Monte Carlo simulations.

Established a thermodynamic analogy linking indices to entropy and inverse temperature.

Abstract

We study analytically the change in the wealth ($x$) distribution $P(x)$ against saving propensity $\lambda$ in a closed economy, using the Kinetic theory. We estimate the Gini ($g$) and Kolkata ($k)$ indices by deriving (using $P(x)$) the Lorenz function $L(f)$, giving the cumulative fraction $L$ of wealth possessed by fraction $f$ of the people ordered in ascending order of wealth. First, using the exact result for $P(x)$ when $\lambda = 0$ we derive $L(f)$, and from there the index values $g$ and $k$. We then proceed with an approximate gamma distribution form of $P(x)$ for non-zero values of $\lambda$. Then we derive the results for $g$ and $k$ at $\lambda = 0.25$ and as $\lambda \rightarrow 1$. We note that for $\lambda \rightarrow 1$ the wealth distribution $P(x)$ becomes a Dirac $\delta$-function. Using this and assuming that form for larger values of $\lambda$ we proceed for an approximate estimate for $P(x)$ centered around the most probable wealth (a function of $\lambda$). We utilize this approximate form to evaluate $L(f)$, and using this along with the known analytical expression for $g$, we derive an analytical expression for $k(\lambda)$. These analytical results for $g$ and $k$ at different $\lambda$ are compared with numerical (Monte Carlo) results from the study of the Chakraborti-Chakrabarti model. Next we derive analytically a relation between $g$ and $k$. From the analytical expressions of $g$ and $k$, we proceed for a thermodynamic mapping to show that the former corresponds to entropy and the latter corresponds to the inverse temperature.