Kinetic Exchange Income Distribution Models with Saving Propensities: Inequality Indices and Self-Organised Poverty Level

Paul, Sanjukta; Mukherjee, Sudip; Joseph, Bijin; Ghosh, Asim; Chakrabarti, Bikas K.

doi:10.48550/arxiv.2108.12343

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“…Using this expression for L(f ) and using an estimate for the pre-factor B of f 3 in eqn. (8), we get the analytical expressions ( 9) and (10) for g (λ) and k(λ). We compare the MC results for g and k at large values of λ with the numerical solutions of the Gamma distribution and those from analytical expressions ( 9) and (10) in Fig.…”

Section: Summary and Discussionmentioning

confidence: 99%

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

Joseph,

Chakrabarti

2021

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We study analytically the change in the wealth (x) distribution P (x) against saving propensity λ 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 λ = 0 we derive L(f ), and from there the index values g and k. We then proceed with an approximate Gamma distribution form, Pn(x), of P (x) for non-zero values of λ (n(λ) > 1 for λ > 0 and n(λ) − → ∞ for λ − → 1). Then we derive the results for g and k at λ = 0.25 and as λ − → 1. We note that for λ − → 1 the wealth distribution, P (x) becomes a Dirac δ-function. Using this and assuming that form for larger values of λ (= 1 − , 0 < << 1) we proceed for an approximate estimate for P (x) centered around the most probable wealth (a function of λ), and utilize that for evaluating the L(f ) to estimate in the end g(λ) and k(λ) for large λ values. These analytical results for g and k at different λ are compared with numerical results from the study of the Chakraborti-Chakrabarti model using the Kinetic exchange theory. 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.

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Section: Summary and Discussionmentioning

confidence: 99%

“…In ref. [8], a numerical analysis of the variations of g and k for different λ is reported, and it shows that as λ increases, g and k decrease (for λ = 0 the index value for g = 0.5 and k ≈ 0.681, and for λ = 1 the index value for g = 0 and k = 0.5).…”

Section: Introductionmentioning

confidence: 99%